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Library · docs/37-methods.md

37. Methods: the reusable identities, lemmas, controls and techniques, by theme

51,278 words · 5,342 lines · source

This is the cross-hunt index of method. hunts/README.md logs each hunt by outcome; nothing there says which trick a later hunt could pick up. This file does. It was built by a retroactive sweep of every hunt directory (99 hunts, about 775k words of record), and it is meant to be kept current: a hunt that produces something on the list below adds an entry here in the same change.

What earns an entry

An entry is a technique with a stated scope and a recorded provenance: an exact identity, a proved lemma, a construction, a control that discriminated a real fault, a computational technique that a second hunt reached for, or an obstruction that closes a route so nobody walks it again. The bar is one of:

A larger computation is not a method. A number is not a method. A heuristic that was measured once and never carried is recorded under "Seen and not admitted" so the next sweep does not re-judge it; it is not an entry.

How to read an entry

Every entry carries kind, originating hunt, and the grade as the hunt's own record states it. This index upgrades nothing: an ordinary derivation stays an ordinary derivation, a measured number stays measured, and a hunt entry is still a hunt entry (nothing under hunts/ is a result). The certainty ladder in AGENTS.md governs the words: measured, hardened or enclosure-carrying, kernel-checked. Prior art is reported as the hunt reported it: cited, searched-and-found, searched-and-absent, or unsearched. "Unsearched" is a statement about the record, not a novelty claim.

Statements are condensed from the hunt's RESULTS and theorem files by a reader, not copied; the evidence line points at the section that carries the actual proof or table. Read that before using the method.

Adding an entry

Same fields, same order: name; kind, hunt, grade; statement; Evidence; Prior art; Reused in; Why it travels. Put it under the theme it belongs to, or open a theme if none fits. Grade it with the hunt's words. The reserved word for zeta/rigor.py and the Lean arm stays out of this file. No em dashes.

Themes

Totals: 167 entries from 60 hunts. Kinds: identity 21, lemma 41, bound 12, construction 15, calibration 7, computational 15, control 35, obstruction 21.

Prime pairs and the circle method

Shifted prime correlations at a modulus, the centered circle-method budget, character-twisted pair sums, exceptional-zero corrections and the controls that kept the fits honest.

Modulus-q four-piece decomposition of the prime-pair count

identity | hunts/prime_pair_error/ | grade: exact identity, numerically pinned (I3 at every k<=5000 for q=1,2,3,5,30,210; closed form vs general code to 2e-6); the corrections built on it are measured

Fix q, phi=phi(q), w=q/phi, M_q(n)=w on (n,q)=1 else 0, and split Lambda on reduced classes as M_q+delta'. For 1<=k<=N, psi_2(N,k)=A_q+L_q+X_q+D_q exactly, where A_q=sum M(n)M(n+k)=S_q(k)(N-k)+rho_q(N,k) with S_q(k)=q nu_q(k)/phi^2=prod_{p|q} sigma_p(k) (sigma_p=p/(p-1) if p|k, else p(p-2)/(p-1)^2) and |rho_q|<=w^2 nu_q(k); L_q is linear in psi(x;q,b) at x=N, k, N-k only (I4); X_q is the O(log^2 N) sum over pairs containing a power of a prime dividing q; D_q is the centered-correlation remainder. The singular series factors S(k)=S_q(k)S^(q)(k) at every k including odd k (I3). At q=1 the endpoint term is R(N)+R(N-k)-R(k), R=psi-x. Under the one heuristic (H_q) the predicted signed part is c_q=S^(q)(k)[rho_q+L_q]+X_q with every coefficient 1.

Evidence: RESULTS.md Section 8 'The decomposition at a modulus q' (lines 232-288, items I1-I6); Section 9 lines 290-332; residue.py FROZEN table; tests/test_prime_pair_residue.py Prior art: searched-and-found: Bogomolny-Keating (arXiv:1307.6010) finite products over primes dividing the modulus are (I3); BHMS (Mathematika 2019) is the sum-side analogue; Korevaar-te Riele (Math. Comp. 2010) is the q=1 small-k average. Hunt states 'not claimed new'. Reused in: residue.py (q=1,3,30 corrections, q=210 diagnostic); RESULTS.md Section 12 statement (T); hunts/README.md summary Why it travels: Any shifted correlation of an arithmetic function can be split into periodic baseline, endpoint prime-count terms, exceptional pairs and a centered remainder at any modulus, with the singular series factoring cleanly.

Mod-3 character antisymmetry and the Wronskian reduction of the character-weighted pair sum

identity | hunts/prime_pair_error/ | grade: exact identities checked on Gaussian random weights at N=7,50,301,1000 to 5e-12 and on Lambda; reviewed written proof (REFEREE.md), not kernel-checked

For chi the non-principal character mod 3 and 3 dividing neither n nor m, chi(m-n)=(chi(n)-chi(m))/2 (E0); this is special to modulus 3. Hence for any weight f supported off multiples of 3, with A,B the prefix sums on the classes 1,2 mod 3, P=A-B, r=A+B-n: sum_{n<m<=N} f(n)f(m)chi(m-n) = sum_{j=0}^{N-1}P(j) - ((N-1)/2)P(N) + W(N), W(N)=(1/2)sum_n[P(n-1)dr(n)-r(n-1)dP(n)] (E1). Pairs with a power of 3 give X_chi(N)=(log 3)sum_{3^j<=N}[P(N)-2P(3^j)] (E2). With T(N)=sum_k chi(k)[psi_2(N,k)-S(k)(N-k)], I(N)=sum_{j<N}P(j)-(N/2)P(N), H_chi=sum chi(k)S(k)(N-k): T-I = W + P(N)/2 + X_chi - H_chi (E3). Partial summation gives W = R(N)P(N)/2 - J(N) + Q(N), J(N)=sum_{m<=N}Lambda(m)chi(m)(psi(m)-m), |Q|=O(N log N) (E4).

Evidence: RESULTS.md Section 14 lines 532-583 (E0-E3), Section 16 lines 629-658 (E4-E5); REFEREE.md Section 1 lines 34-60 (verified, corrects the symmetric identity's missing P(N)); wronskian.py; tests section 5 Prior art: searched-and-absent: web search 2026-09-06 for a character-weighted difference-side pair sum returned nothing; BHMS is the sum side; hunt claims no novelty Reused in: RESULTS.md Sections 17-18 (Theorems A and B); THEOREM_B_SEQUENCE.md; FAREY_BASELINE_REPAIR.md Appendix B Why it travels: Turns any antisymmetric character-weighted pair correlation into a Wronskian of two prime-counting remainders plus explicit one-point terms; the random-weight check is a generic test for such identities.

q=1 mixed moment is a completed square of prime-counting remainders

identity | hunts/prime_pair_error/ | grade: ordinary derivation, independent model review 'no substantive defect'; identity pinned on arbitrary weights

With K_N the Dirichlet kernel, F_N the von Mangoldt exponential sum and Delta(t)=psi(t)-t at integers: T_N := int_T |K_N|^2|F_N-K_N|^2 = sum_{t=1}^N Delta(t)^2

sum_{t<N}[Delta(t)-Delta(N)/2]^2 (29), because the coefficient of e(m beta) in K_N(F_N-K_N) is Delta(m-1) for 2<=m<=N+1 and Delta(N)-Delta(m-N-1) for N+2<=m<=2N. The q=1 arc integral U_1 differs from T_N by at most (N^2/4Q^2) sum(Lambda(n)-1)^2 << N^2 L (30). Consequence: the sufficient bound (31) sum Delta(t)^2 << N^{2+eps} implies Delta(N)=O(N^{1/2+eps}), i.e. the q=1 component of any centered circle-method budget for E(N) contains the RH-strength remainder; Siegel-Walfisz at q=1 yields only T_N << N^3 L^{-2H}.

Evidence: UPPER_BOUND.md Section 7 lines 476-540 (eqs 29-31); Section 8 review note (convolution and completed square checked on 640 exact signed-integer-weight examples, N<=64); tests/test_prime_pair_upper_bound.py Prior art: unsearched Reused in: RESULTS.md 'The doors' rank 1; RANK3_SCOPE.md, RANK3_Z_COMPONENT.md (Z_(1) >= 3U_1^2/(2N^3+N)); LOCALIZED_MIXED_ENERGY.md; FRONTIER_2026_09_12.md Why it travels: Exhibits, for any centered exponential-sum energy, exactly where the one-point remainder hides; the Parseval-plus-complete-the-square step works for any coefficient sequence a_n-1.

Exact p=2 Mobius pairing partition inside a hyperbola block

identity | hunts/prime_pair_error/ | grade: ordinary reviewed derivation plus Class A exact rational checks at 20 cutoffs to N=10000; no asymptotic gain; ATTEMPT_UNRESOLVED

For N>=4, K=floor(sqrt N), M=floor(N/2), U=floor(sqrt M), b in [2,U], V=floor(M/b), F=floor(V/2) and any kernel w: M_b=sum_{U<a<=V}mu(a)w(ab) equals P_b+T_b+H_b+Z_b exactly, with P_b=sum_{m odd, U<m<=F}mu(m)[w(mb)-w(2mb)] (paired differences via the bijection m -> 2m and mu(2m)=-mu(m) for odd squarefree m), T_b=sum_{m odd, max(U,F)<m<=V}mu(m)w(mb), H_b=sum_{t<=U, U<2t<=V}mu(2t)w(2tb), and Z_b (m=2t, t even) identically zero since 4|m. Under the gate ab>Y=N/K (proved for every Sigma_2 pair, N>=12), w(mb)=1-floor(N/(mb)) and the paired difference Delta(m,b)=floor(N/(2mb))-floor(N/(mb)) <= 0. Under absolute-value majorants B(N)=(1/48)N log^3 N+O(N log^2 N) and Ptot=(1/192)N log^3 N+O(N log^2 N): the majorant is pinned Theta(N log^3 N) from both sides, so this explicit majorant cannot improve, while signed pairing, more primes, or a Sig1 bound remain open.

Evidence: MOBIUS_PAIRING.md Sections 1-2 lines 10-70 (theorem and proof), Section 5; MOBIUS_PAIRING_REVIEW.md Sections 3-4; RESULTS.md Fourth pass lines 926-972 Prior art: cited as standard: mu(2m)=-mu(m); no prior pairing work found in the hunt (narrow grep only) Reused in: FINAL_ACCEPTANCE.md; RESULTS.md Fourth pass 'Doors of this mechanism' Why it travels: A general bookkeeping identity for pairing a Mobius-weighted sum with its 2-multiples, with the head/tail boundary written out exactly; applies to any kernel w.

Theorem A: double-zero kernel bound for the Wronskian via Perron with a moving weight

lemma | hunts/prime_pair_error/ | grade: ordinary derivation, independently reviewed written proof (REFEREE.md); not kernel-checked

Let Theta, Theta_chi be the suprema of real parts of zeros of zeta and of L(s,chi_3). Then W(N) << N^{Theta+Theta_chi}(log N)^4 unconditionally, so T(N)=I(N)+O(N^{Theta+Theta_chi}log^4 N), and O(N log^4 N) under RH for both. Mechanism: insert the truncated explicit formula (T=N) for R(m) into J(N); for each zeta zero rho apply Perron to -L'/L(w-rho,chi) with c=1+beta+1/log N and a height V_rho in [2N,2N+1] chosen at distance >= a/log(2N) from every L-zero ordinate (measure argument via O(log) zeros per unit interval); shift to Re w=-delta with delta in {1/8,3/16} avoiding the trivial zero; the residues give A_rho(N) = -sum_{rho' in D_rho} N^{rho+rho'}/(rho+rho')

K(rho,rho')=(rho'-rho)/(2 rho rho'(rho+rho')) and sum |K| over the same sets is O(log^4 N), the central bin |gamma+gamma'|<1 handled by |rho+rho'|>=beta>>1/log N from the classical zero-free region. No simplicity, distinct-ordinate or attained-supremum assumption.

Evidence: RESULTS.md Section 17 lines 674-801 (Theorem A, Steps 0-3, F1-F5); REFEREE.md Sections 2-3 lines 101-256 (R1-R6, verdict 'proof valid after specified repair') Prior art: searched-and-found for comparison only: BHMS Theorem 2 (Mathematika 65, 2019) sum-side with Gamma-factor kernels; Fujii 1991; Languasco-Zaccagnini 2012/2015; Goldston-Yang arXiv:1601.06902. No difference-side statement found. Reused in: RESULTS.md Section 18 (Theorem B uses A for the transfer); REFEREE.md Section 7 Why it travels: The good-height selection, slanted cutoff sets and the near-diagonal handling via the zero-free region apply to any bilinear sum over zeros of two L-functions with 1/(rho(rho+rho')) type weights.

Theorem B: character-weighted Cauchy-Schwarz plus Landau pole gives an unconditional Omega bound

lemma | hunts/prime_pair_error/ | grade: ordinary derivation, independently reviewed written proof; not kernel-checked

E(N) >= 2T(N)^2/N by Cauchy-Schwarz since sum chi(k)^2 <= N. For Re s>1, int_1^inf I(x)x^{-s-2}dx = -(L'/L)(s,chi)(1-s)/(2s(s+1)) with I(x)=sum_{m<=x}Lambda(m)chi(m)(x/2-m); a zero rho' of multiplicity h gives residue h(rho'-1)/(2rho'(rho'+1)) != 0, so absolute convergence implies I(x) is not O(x^{1+Theta_chi-eps'}) (only 'absolute convergence implies analyticity', no sign condition). With eta=min(eps,1-Theta)/2 and Theorem A, along an unbounded sequence |T(N)|>=N^{1+Theta_chi-eta}/2 and E(N)>=N^{1+2Theta_chi-eps}/2. Case Theta_chi<=Theta is CHHL Theorem 2. Conclusion is along a sequence, not eventual; sequence may depend on eps.

Evidence: RESULTS.md Section 18 lines 803-852; REFEREE.md Section 4 lines 258-321 ('Proof verified'); THEOREM_B_SEQUENCE.md Section 2 (why it is nonconstructive: obstructions A and B) Prior art: cited: CHHL Theorem 2 (same mechanism with weight 1); Ingham ch. V / Montgomery-Vaughan 15.1 for the Landau direction used Reused in: REFEREE.md Section 7 strongest surviving statement; FRONTIER_2026_09_12.md table ('settled nonconstructive Theorem B is unaffected') Why it travels: Replaces the k-mean weight 1 by any character; RESULTS Section 18 notes the general-modulus version needs only a fresh main term since the bilinear antisymmetric form is handled pairwise by Theorem A.

Signed exceptional-correction bridge from a corrected pair energy to the prime-counting remainder

lemma | hunts/prime_pair_error/ | grade: handwritten deduction; the mean criterion and parameter substitution independently re-derived in FRONTIER_INDEPENDENT_REVIEW.md; not kernel-checked

With r_N(h)=psi_2(N,h)-(N-h)S(h), an exact correction C_N(h) (zero when no exceptional zero is assigned), and E_corr(N)=2 sum_{h<=N}|r_N(h)-C_N(h)|^2: from CHHL's identity psi(N)^2 = d_N + 2 sum_h psi_2(N,h) and the singular-series first moment 2 sum (N-h)S(h) = N^2+O(N log N), Cauchy-Schwarz on the corrected vector only gives |psi(N)-N| <= sqrt(2E_corr(N)/N) + (2/N)|sum_h C_N(h)| + O(log N) (18), using psi(N)+N>=N. The signed first moment satisfies |sum_h C_N(h)| << N(log N)^{2/5} (2) by a divisor expansion of S_*(h) with the parity factor retained and cancellation of the nonprincipal character over complete periods. Hence E_corr << N^{2+eps} for every eps implies RH (one direction only; the two-way scalar criterion is RH iff M_N=2|sum_h(r_N-C_N)|^2/N << N^{2+eps}).

Evidence: CORRECTED_RH_BRIDGE.md Sections 1-5 lines 40-330 (eqs 2, 3, 8-9, 17-20); FRONTIER_2026_09_12.md lines 28-52 (two-way scalar criterion); FRONTIER_INDEPENDENT_REVIEW.md ('Mean criterion epsilon quantifiers: Passed'; signed-mean equivalence reviewed) Prior art: cited: CHHL Section 3 identity and first moment; Tao-Teraevaeinen Definition 2.1 for the exceptional data Reused in: SIGNED_MEAN_RENEWAL.md Section 2; FRONTIER_2026_09_12.md; ENDPOINT_BOUND_REVIEW.md dependency 4; S8_CONTROL.md Why it travels: Shows how to keep an explicit exceptional-zero correction inside a Cauchy-Schwarz bridge without bounding its second moment; the same shape applies to any corrected correlation energy.

Bounded-order Bonferroni divisor approximant to the sieve indicator with explicit error

lemma | hunts/prime_pair_error/ | grade: ordinary derivation, independently reviewed (Codex review 2026-09-09, PASS; finite illustration corrected 2026-09-11)

For P=prod_{p<Z}p, even m, B_m(n)=sum_{d|P, d|n, omega(d)<=m} mu(d): B_m(n)=1 if no prime below Z divides n, and B_m(n)=binom(s-1,m) otherwise (s the number of such primes), so 0 <= B_m(n)-1_{s=0} <= binom(s,m+1) (4). Hence sum_{n<=N}|B_m(n)-1_{(n,P)=1}| <= sum_{omega(d)=m+1} floor(N/d) <= N H_Z^{m+1}/(m+1)!, H_Z=sum_{p<Z}1/p (5), with no accumulated O(D_0) remainder, every coefficient in {0,+-1}, every divisor below D_0=Z^m. With m=2ceil(sqrt(log N)) and Z=exp((log N)^{1/10}) this is N exp(-c sqrt(log N) log log N). Retuning at the endpoint kappa=1/2 requires m+1=ceil((2c/kappa)l^kappa/log l) and Mertens for H_Z rather than H_Z<=1+log Z; ENDPOINT_SHARP_REVIEW found the document's finite retuned-m table too small by exactly 2 at every row while confirming the asymptotic (A).

Evidence: ENDPOINT_BOUND.md Section 2 lines 138-190 (eqs 4-6); ENDPOINT_BOUND_REVIEW.md (PASS); ENDPOINT_SHARP.md Section 2.1; ENDPOINT_SHARP_REVIEW.md Item 1 (asymptotic confirmed, finite table defective) Prior art: unsearched (Bonferroni/Brun truncation is textbook; the explicit prefix-uniform error form is the lab's) Reused in: ENDPOINT_SHARP.md, ENDPOINT_HALF.md, ARC_SPLIT_BUDGET.md (later dropped from the chain as unnecessary) Why it travels: A drop-in replacement for the sieve indicator whenever a sum over divisors of bounded length is needed with an error uniform over all prefixes and no boundary remainder.

Centered circle-method budget for E(N): geometric leakage and large-denominator mixed-moment bounds

bound | hunts/prime_pair_error/ | grade: ordinary derivation, independent model review 2026-09-06 found no substantive defect; finite identity tests only

Arcs I_{q,a}={|alpha-a/q|<=Q/(qN)}, q<=Q, 2Q^2<N (disjoint, measure <=2Q^2/N). With A_Q=sum|P_{q,a}|^2 1_{I_{q,a}}, P_{q,a}=(mu(q)/phi(q))K_N(alpha-a/q), H_Q=V_Q-A_Q>=0 and w(Q)=max_{q<=Q}q/phi(q): ||H_Q||_2^2 << N^3 Q^{-2} log(2Q) w(Q)^2 (13), proved by int H_Q << N (14) times ||H_Q||_inf << w(Q)^2((N^2/Q^2)log(2Q)+N) from dyadic center-counting sum ||alpha-x||^{-2} << delta^{-2}+R^2/delta. Total: E(N) <= 4M_Q+4I_Q+4||H_Q||_2^2+2D_tail(N,Q) (12), and at Q=floor(sqrt N/3) E(N) <= 4M_Q+4I_Q+O(N^2L^3) (16). Splitting M_Q into U_Q=sum int|P|^2|R|^2 and Z_Q=sum int|R|^4 gives E(N)<=32U_Q+8Z_Q+4I_Q+O(N^2L^3) (23); the large sieve at unshifted centers in dyadic blocks R<q<=2R gives U_{q>R_0} << w(Q)^2 N^3 L/R_0^2 (26), hence O(N^2L^5) at R_0=Q/L. The tail identity |sqrt E - sqrt J_ms(N,z)| <= sqrt D_tail(N,z) with D_tail << N^3/z^2 from GHN Theorem 2 (doubled, exact tail sum T(z)) is (4)-(6).

Evidence: UPPER_BOUND.md Sections 2-6 lines 101-474 (eqs 4-6, 12-13, 16, 23-27); Section 8 table; ARC_SPLIT_BUDGET_REVIEW.md Item 1 (identities confirmed) Prior art: cited: CHHL Sections 3-4 (centered setup), Goldston-Hunts-Ngotiaoco Theorem 2 (tail), Montgomery-Vaughan MNT II Theorems 17.1 and 19.4 (Vaughan, large sieve) Reused in: MINOR_LEVEL_SETS.md, CENTERED_DISPERSION.md, ARITHMETIC_FOURTH.md, RANK3_* series, ARC_SPLIT_BUDGET.md, LOCALIZED_MIXED_ENERGY.md, RESULTS.md 'The doors' Why it travels: A fully itemised difference-side budget with every component's exponent, reusable as the starting ledger for any sharp-cutoff pair-correlation mean square.

Farey-block residual fourth-moment baseline by bounded overlap

bound | hunts/prime_pair_error/ | grade: ordinary derivation, independently reviewed; not kernel-checked

For integer N>=16, 1<=Q<=N^{2/5}, reduced a/q with Q<=q<2Q and q<=sqrt N, and R_{q,a}(theta)=F_N(a/q+theta)-(mu(q)/phi(q))K_N(theta): sum_q sum_a^* int_{|theta|<=1/(q sqrt N)} |R_{q,a}|^2 << N log N (E) and sum sum int |R_{q,a}|^4 << N^3 (log N)^9/Q (A). Proof of (E): arcs in one block have radius <=1/(Q sqrt N) and centers 1/(4Q^2)-separated, so any point is covered at most 9 times; Parseval gives the F part << NL, and sum_{q in block} mu(q)^2/phi(q) << 1 (dyadic reciprocal-totient bound (C)) gives the model part << N. (A) follows from (E) times the supremum bound |R|^2 << N^2 L^8/Q from Vaughan with all three terms O(N/sqrt Q) for Q<=N^{2/5}. Supersedes a faulty AP-variance/character-orthogonality argument; explicitly not an optimality or closure statement.

Evidence: FAREY_BASELINE_REPAIR.md Sections 1-3 lines 19-133 (eqs A-E); Section 4 (exact corrections to the AP-variance argument); FRONTIER_INDEPENDENT_REVIEW.md Section 1 and obligation matrix (Farey overlap, residual second moment, fourth-moment range: Passed) Prior art: cited: Vaughan (MNT II 17.1), Gallagher's lemma, primitive large sieve for the optional Gauss-first route (Appendix A) Reused in: FRONTIER_2026_09_12.md route table; SHARP_EXPONENT.md Section 1.2 correction Why it travels: A clean template for block-wise residual moments on Farey arcs with an explicit overlap constant and an explicit Q-range where Vaughan's N^{4/5} term stays harmless.

Explicit exceptional-zero correction to the pair count with exact residue-average identities

construction | hunts/prime_pair_error/ | grade: handwritten deduction with bounded algebra checks (artifacts/siegel_uniformity/check.py); consumed as a checked dependency by ENDPOINT_BOUND_REVIEW (PASS) and ENDPOINT_SHARP_REVIEW

Model a(n)=nu(n)(1-n^{beta-1}chi(n)) with nu(n)=b_Z 1_{(n,P(Z))=1}, Z=exp((log N)^{1/10}) (TT Definition 2.1). Uniformly for 1<=h<=N: sum_{n<=N-h}a(n)a(n+h)-(N-h)S(h) = C_{q,beta,Z}(h)+O(N e^{-c L^{1/10}}) (18), where C = L_h[-u_q(h)J_1(h)-v_q(h)J_2(h)+(c_q(h)/q)J_{12}(h)], L_h=(q/phi(q))^2 prod_{p<Z,p not|q}alpha_p(h), J_1=int_0^{N-h}max(1,t)^{-delta}, J_2=int_0^{N-h}(t+h)^{-delta}, J_12 their product integral, delta=1-beta. Exact identities (17): u_q(h)=mu(q)chi(-h)/q, v_q(h)=mu(q)chi(h)/q for odd primitive real conductor q, u=v=0 for 4|q, and sum_r chi(r)chi(r+h)=c_q(h) (Ramanujan sum). Supporting tools: fundamental-lemma sieve count per residue class with sharp endpoint N-h (10); |sigma_Z(h)-S(h)| << L^2/Z (12); transfer ||C(|F|^2-|H|^2)||_2 <= ||W||_4(||F||_4+||H||_4) (6) with exact U^2 normalization A_N=(2N^3+N)/3 (4); moment bound sum_h|C|^2 << 1_{q odd}N^{2beta+1}q^2/phi(q)^4 + N^{4beta-1}q^2/phi(q)^3 (23) and matching lower bound A_exc >= (1/192)N^{4beta-1}q^2/phi(q)^3 (EXCEPTIONAL_ENERGY eq 2).

Evidence: SIEGEL_UNIFORMITY.md Sections 1-5 lines 26-350 (eqs 3-6, 10-12, 17-19, 23); EXCEPTIONAL_ENERGY.md lines 6-30 (eqs 1-3); ENDPOINT_BOUND_REVIEW.md 'Checked dependencies' item 3 Prior art: cited: Tao-Teraevaeinen arXiv:2107.02158v4 (Definition 2.1, Theorem 2.7, Lemma 5.1); CHHL Section 7 for the Ramanujan approximant Reused in: CORRECTED_RH_BRIDGE.md, EXCEPTIONAL_ENERGY.md, LOCALIZED_MIXED_ENERGY.md, ENDPOINT_BOUND.md, ENDPOINT_SHARP.md, ARC_SPLIT_BUDGET.md, MAJOR_ARC_EXPLICIT.md, PAGE_ZERO_ASYMPTOTIC.md, S8_CONTROL.md Why it travels: Gives the exact shape a Siegel zero imprints on any shifted correlation with sharp endpoint, with the residue averages in closed form for every primitive real conductor. Index note: Consumed by seven downstream documents in the same hunt as a checked input.

Lifted balanced-seed factorial certificate for psi(N)

construction | hunts/prime_pair_error/ | grade: exact-rational feasibility checks plus elementary general argument; pilot independently reviewed (all 87 seeds reproduced); later refinement rules 'no separate proof review'

Choose L>=M>=2 and rationals a_j indexed by j|L with sum_j a_j/j=0 (balance), g(t)=sum a_j floor(t/j) >= 0 on r=0..L-1 and >=1 on r=1..M-1; balance makes g periodic with period L and integer breakpoints, so one period certifies g>=0 for all real t>=0 and g>=1 on [1,M). For N put K=floor(log_M N) and W_N(t)=sum_{k<=K} g(t/M^k); then W_N(t)>=1 for 1<=t<=N (the term k=floor(log_M t) is >=1, the rest >=0), and B_N=sum_k sum_j a_j log floor(N/(jM^k))! = sum_{d<=N}Lambda(d)W_N(N/d) >= psi(N) with no RH input. Cost: B_N = C N + O(A (log N)^2) with kappa=-sum a_j log(j)/j, C=kappa/(1-1/M), using |log floor(y)! - (y log y - y)| <= 1+log^+ y. Period 30 recovers Chebyshev; best period-2310 seed C~1.0699; structural enlargement to 30030 gives 1.0558; the refinement rule with carry b_n(t)=floor(t/n)-floor(t/(n+1))-floor(t/(n(n+1))) (nonnegative, 0 before n, 1 at n) and h_p=sum_{d|210}mu(d)b_p(t/d) reaches C~1.0500, and the combined-weight variant (W_g>=1 below R, g>=0 above R suffices) reaches C~1.04866. Every fixed seed has C>1; the omitted-prime lemma forces W(p)>=2 for primes absent from the denominator set.

Evidence: frontier/2026-09-06/factorial_certificate_pilot/PILOT.md Sections 1-3 (lines 7-75), REVIEW.md; checkpoint/CHECKPOINT.md Sections 4.8, 4.10-4.12 (lines 95-143); certificate_refinement_rule/REFINEMENT.md; certificate_route_test/ROUTE_ASSESSMENT.md Prior art: cited: Chebyshev's factorial construction (the period-30 seed); Nyman-Beurling noted as a different known criterion, not executed Reused in: hunts/paid_shortfall/RESULTS.md (inherits the pilot main term and remainder); certificate_lp_frontier (the family whose LP floor is computed); hunts/quotient_certificate/ Why it travels: A self-certifying majorant template: one finite integer-period check plus a radix lift gives an unconditional psi(N) ceiling with an explicit leading constant for any seed.

Asymptotic of the character-twisted singular-series sum H_chi(N) and its constant c_H

calibration | hunts/prime_pair_error/ | grade: reviewed written proof; constant is a truncated-product value, not an enclosure

For chi mod 3, H_chi(N)=sum_{k<=N}chi(k)S(k)(N-k) = c_H N + O(N^{3/4}) with c_H = D(0) = -(2C_2/3) prod_{p>3}(1+chi(p)/(p-2)) < 0, from the Dirichlet series D(s) = -2^{1-s}C_2 L(s,chi)F(s), F(s)=L(1+s,chi)K(s), K an Euler product absolutely convergent for Re s>-1/2, contour shifted to Re s=-1/4 using L(s,chi)<<(1+|t|)^{3/4} and L(1+s,chi)<<(1+|t|)^{1/8+nu}. Numerically c_H is near -0.31305 (JSON value -0.31306242 is a product truncated at 2e6, not the limit); measured H_chi(N)/N = -0.3130 to -0.3131 at five cutoffs. Also M(x)=sum_{k<=x}chi(k)S(k) << log x by Mertens. Hence P(N)/2+X_chi-H_chi = -c_H N + o(N), so the auxiliary term in (E3) is Theta(N), not O(N log^2 N) as first proposed.

Evidence: RESULTS.md Section 15 lines 585-627; REFEREE.md Section 5 lines 323-416 ('proof valid after specified repair'; finite products must be labelled approximations); wronskian.py H_chi_by_divisor_expansion (independent evaluation agreeing to 1e-10) Prior art: cited: Friedlander-Goldston / Korevaar-te Riele for the untwisted sum sum S(k) = h - (1/2)log h + ...; twisted version derived here Reused in: RESULTS.md Section 16 (E5); REFEREE.md Section 7 Why it travels: Template for any character-twisted or otherwise weighted sum of the singular series: write the Dirichlet series as a product of L-factors and an absolutely convergent Euler product, shift the Riesz-mean contour, read the constant at s=0.

Replication convention for CHHL Table 1: truncate, do not round

calibration | hunts/prime_pair_error/ | grade: measured

Chou-Haag-Huryn-Ledoan report E(N)/(N^2 log^2 N) truncated to five decimals, not rounded: at N=1e4 the computed value 0.1232782 rounds to 0.12328 and truncates to the paper's 0.12327; all eleven rows match after truncation. Supporting pins: C_2 from the Euler product to 4e6 is 0.6601618261 against the pinned 0.6601618158 inside the tail bound; the identity sum_{|k|<=N}psi_2(N,k)=psi(N)^2-sum Lambda(n)^2 holds at every N to float precision and forces the k-mean of e to psi(N)-N+c(N), c in [1.21,1.32]; odd separations carry E-share 2e-6 at 1e7 and only involve powers of 2.

Evidence: RESULTS.md Section 1 lines 10-41; MISSION.md required_oracles and kill_conditions Prior art: cited: CHHL arXiv:2308.14888 Table 1 and Section 3 Reused in: CHALLENGE.md Attack 1 (uses the 1e5 row 0.16857 as an external sanity target); S8_CONTROL.md Why it travels: Anyone re-deriving or extending Table 1 must truncate to match; the 1e4 row is the discriminating check.

Freeze-then-test protocol with derived coefficients and post-hoc slope diagnostic

control | hunts/prime_pair_error/ | grade: measured

Every coefficient of a candidate correction is fixed by derivation (here all equal to 1), recorded in a FROZEN table that the test suite refuses to let move, and the correction is evaluated only at cutoffs the discovery pass never used (2e5, 5e5, 2e6, 5e6, 7e6 after discovery on 1e3..1e7), with nothing inspected in between. Two read-outs: reduction 1-sum(e-c)^2/sum e^2, and the post-hoc least-squares slope of e on c whose derived value is 1 (measured within 0.2 percent of 1 for q=30 at all five cutoffs). The first pass used the same pattern with held-out N=3e6 and 1e7 and a kill condition 'coefficient moves by more than its scale between discovery and held-out N'.

Evidence: RESULTS.md Section 9 lines 305-311 ('Every coefficient is fixed at 1 by the derivation... the test refuses any other value'); Section 10 lines 336-364; Section 4 lines 131-133 (held-out rows); MISSION.md kill_conditions Prior art: unsearched (standard held-out practice; the specific convention of derived-not-fitted coefficients plus slope-should-be-1 diagnostic is the lab's) Reused in: residue.py and tests/test_prime_pair_residue.py; S8_CONTROL.md and CHALLENGE.md reuse the independent-reimplementation cross-check pattern Why it travels: Separates 'the model explains X percent' from 'the fit absorbed X percent' for any predicted profile on any family of cutoffs.

Davenport-Heilbronn battery for pair energies via the periodic-sequence singular series

control | hunts/prime_pair_error/ | grade: measured (float) plus written argument

To test whether an exact identity or domination for E_corr carries arithmetic content, feed the identical construction a real, period-5 sequence a(n) built from the DH coefficients (1,kappa,-kappa,-1,0), kappa=0.284079..., whose L-function satisfies F(s)=F(1-s) and has a located zero at Re s ~ 0.8085. The exact analogue of the singular series for a periodic sequence is its period autocorrelation rho(h)=(1/5)sum_j a(j)a(j+h), which is the exact main term of sum a(n)a(n+h) with no error term. If the identity holds for the DH sequence to the same precision (here 8.2e-16), it distinguishes nothing about zeta. Attack 3 pairs this with a line-by-line audit: does any proof step use Lambda, S, or C_N beyond being fixed real numbers? For the dyadic Haar/martingale energy the answer was no (P_aP_b=P_max(a,b), orthogonal projections, Pythagoras).

Evidence: CHALLENGE.md Attacks 2 and 3 (lines 73-172) and Overall verdict; CANDIDATE_ENERGY.md Section 3 lines 98-165 (the identity being tested); s8_challenge.py Prior art: cited: the repository's standing counterexample-battery rule (zeta/epstein.py docstring, NULLCONTROLS.md, REDTEAM.md attack A3); the periodic-sequence main term is the hunt's adaptation Reused in: POSITIVITY_ENERGY.md (runs the battery against the per-scale arithmetic facts of a second energy); CHECKPOINT.md Why it travels: Any 'exact energy decomposition' or 'domination' claimed for the primes can be falsified as arithmetic evidence in one run by substituting a periodic DH-type sequence with its exact period autocorrelation as prediction.

Counterfeit greedy-deficit sequence: falsifier for coarse-property arguments

control | hunts/prime_pair_error/ | grade: elementary written argument with computed table to N=2e6; unreviewed draft per CHECKPOINT.md 4.7

Fix beta=3/4, t=3, c=1/10, A(x)=x+c x^beta cos(t log x)+b. Keep true Lambda through n0=10000; above n0 allow weight only at (n,30)=1, adding log n exactly when A(n)-Psi_a(n-1)>=log n, else 0. Since |(c x^beta cos(t log x))'|<1/2 and gaps between permitted n are at most 6, the deficit stays in [0, log n+9), so Psi_a(N)=N+cN^{3/4}cos(3 log N)+b+O(log N) (C1), sum a_n^2 = N log N - N + O(N^{3/4}log N) (C2), and the support count is ~N/log N. The sequence is nonnegative, has the right density and second moment, avoids multiples of 2,3,5, yet violates any N^{1/2+eps} remainder and fails the divisor identity sum_{d|n}Lambda(d)=log n first at n=10007 (defect about -9.211). Any argument that uses only positivity, leading density, prime-sized second moment and small-prime avoidance therefore cannot prove (U).

Evidence: frontier/2026-09-06/DIRECT_ATTACK.md Section 5 lines 139-194, Section 6; check_attack.py, checks.json (run to 2e6) Prior art: unsearched Reused in: frontier/2026-09-06/checkpoint/CHECKPOINT.md 4.7 ('counterfeit control'); FRONTIER_2026_09_12.md route table ('existing DH control') Why it travels: A constructive negative control for any proposed prime-counting upper bound: list the coarse properties the proof consumes, build a sequence with those properties and a planted oscillation, check the proof does not exclude it.

Exact-rational planted-lesion discrimination with three evidence classes

control | hunts/prime_pair_error/ | grade: Class A exact finite checks for the stated cutoffs only; general identities rest on written derivations

Finite identities in a proof package are checked in exact Fraction arithmetic with zero tolerance (Class A), transcendental constants at high precision without enclosure (Class B, mpmath dps=80, 'not exact'), and floats only for envelopes and ratios (Class C, 'diagnostic only'); every claim is labelled with its class. Each identity check is paired with planted lesions that must produce strictly nonzero exact defects: for the p=2 Mobius pairing M_b=P_b+T_b+H_b+Z_b, lesions missing_head/missing_tail/wrong_sign give defects 11/5/2 at N=100 and 39/17/20 at N=400; for the hyperbola factorization, four lesions (E-sign flip, smooth/fractional minus sign, empty-cell guard omission, partition boundary omission) all fail with positive defect. The independent-review script must not import the author's checker (mobius_pairing_independent_check.py vs mobius_pairing_check.py; 15794 pairs over 20 cutoffs). A first review draft citing an ephemeral scratch script was retracted and replaced by a durable artifact.

Evidence: FINAL_ACCEPTANCE.md Section 3 lines 86-98 (evidence classes), Section 1 item 2; MOBIUS_PAIRING.md Section 5 lines 140-170 and Section 6 run record; MOBIUS_PAIRING_REVIEW.md lines 244-260; ARITHMETIC_CANCELLATION_REVIEW.md lines 347-363; RESULTS.md Fourth pass lines 947-959 Prior art: unsearched Reused in: MOBIUS_PAIRING.md and its review; FACTORIZATION_NEXT_STEP.md; RESULTS.md Fourth pass Why it travels: A checking convention that separates 'identity holds exactly at these cutoffs' from 'float agreement' and proves each guard is load-bearing by removing it.

Exact scale relation from the factorial identity, and the zeta-multiplier obstruction to contraction

obstruction | hunts/prime_pair_error/ | grade: ordinary derivation, independently reviewed (FRONTIER_INDEPENDENT_REVIEW.md, conditional on named classical inputs); not kernel-checked

From sum_{d|m}Lambda(d)=log m: sum_{k<=N}psi(N/k)=log N!, so sum_{k<=N}R(N/k)=G(N)=log N! - N H_N = -(1+gamma)N + (1/2)log N + O(1). With I=int_1^inf R(u)u^{-2}du = -(1+gamma) (proved via -zeta'/(s zeta)-1/(s-1) as s->1+ and absolute integrability from the unconditional PNT rate), for every integer 1<=K<=N: sum_{k<=K}R(N/k) - N int_{N/K}^inf R(u)u^{-2}du = G(N)+(1+gamma)N - Q_{N,K}, |Q_{N,K}| <= Var_{[1,N/K]}R << N/K, hence = O(N/K+log N) (16); at K=sqrt N the forcing is O(sqrt N). Obstruction: for f_rho(u)=u^rho, 0<Re rho<1, the same operator gives (L_K f_rho)(N) = N^rho(sum_{k<=K}k^{-rho} - K^{1-rho}/(1-rho)) = zeta(rho)N^rho + O_rho((N/K)^beta) (22), by Euler-Maclaurin; at a zeta zero the N^beta mode passes every scale within the O(N/K) allowance. Equivalently the Mellin transform of R has residue -m/rho at a zero of multiplicity m, which no exact forcing cancels. Hence no absolute-value or positivity-preserving contraction on (16) can exclude off-line zeros; the missing estimate is |D_N| << N^{1/2+eps} with D_N = N int_{N/K}^inf R u^{-2} - sum_{k=2}^K R(N/k).

Evidence: SIGNED_MEAN_RENEWAL.md Sections 3-6 lines 132-340 (eqs 9-16, 20-23), Section 8 (positive monotone model); frontier/2026-09-06/DIRECT_ATTACK.md Sections 2-4 (F1-F5, M); FRONTIER_INDEPENDENT_REVIEW.md Section 2 obligation matrix (integral constant, all-K estimate, multiple-zero claim: Passed); FRONTIER_2026_09_12.md lines 54-102 Prior art: cited: NIST DLMF 25.11.E5 (Euler-Maclaurin for zeta); text says the zeta(s) multiplier is 'the familiar Mellin/Dirichlet-convolution multiplier, not a newly discovered phenomenon' Reused in: FRONTIER_2026_09_12.md; FINAL_ACCEPTANCE.md and FACTORIZATION_NEXT_STEP.md (D_N target); MOBIUS_PAIRING.md Why it travels: Any renewal or feedback argument for a summatory function through Dirichlet convolution must check the multiplier at the zeros of the convolving series; if it vanishes there the recursion is blind to those modes.

Rudin-Shapiro-type polynomial obstruction to minor-arc fourth-moment saving from global norms

obstruction | hunts/prime_pair_error/ | grade: written derivation with exact finite algebra check; 'bounded independent agent check', not external verification

For r>=2 put d=2^{2r}, m=2^{3r}, N=dm=2^{5r}. Define A_0=B_0=1, A_{s+1}=A_s+z^{2^s}B_s, B_{s+1}=A_s-z^{2^s}B_s; then |A_s|^2+|B_s|^2=2^{s+1} on |z|=1 with coefficients in {-1,1}. Let H=e(alpha)A_{2r}(e(alpha))D_m(d alpha) (or B). Then H has frequencies 1..N with unimodular coefficients, ||H||_2^2=N, ||H||_inf<=sqrt2 N^{4/5}, and one of the two has int|H|^4 >= (2/3)N^{13/5} since int|D_m(d alpha)|^4=(2m^3+m)/3. Fubini over translates places the mass in the actual minor set: some g=H(.-theta) has int_{m_Q}|g|^4

= (14/27)N^{13/5}, while int|g|^p <= 2^{(p-2)/2}N^{p-1} for every p>2. A

nonnegative-coefficient variant f_omega=sum[1+Re(omega c_n)]e(n alpha) keeps the conclusion. So the minor-arc supremum (Vaughan), Parseval, and every Green-Tao restriction-type bound int|F|^p<<N^{p-1} together cannot imply I_Q << N^{13/5-delta}; additional prime arithmetic must enter.

Evidence: MINOR_LEVEL_SETS.md Section 4 lines 157-266 (eqs 11-20), Section 5 lines 293-300; artifacts/minor_level_sets/check.py (exact integer convolutions at N=32,1024) Prior art: unsearched in the text (the recursion is the classical Rudin-Shapiro construction; the hunt does not name it and claims no novelty) Reused in: RESULTS.md 'The doors' rank 2; RANK3 documents; ARITHMETIC_FOURTH.md Why it travels: A ready-made extremal example showing which scalar norm inputs are insufficient for any minor-arc L^4 target; reusable whenever a circle-method budget is stuck at sup times L^2.

LP floor for floor-sum prime certificates, its dual, and the rough-spike lemma

obstruction | hunts/prime_pair_error/ | grade: LP values measured (HiGHS float, certificates re-verified cell by cell); Lemmas 1-3 and Proposition 4 proved elementarily and pinned in tests; barrier law is conjecture

Every certificate of the form W(t)=sum_{j<=y}c_j floor(t/j)>=1 on cells 1..N gives psi(N) <= B(N)=sum_j c_j log floor(N/j)!, and the excess is exactly B(N)-psi(N)=sum_n (W(n)-1) m_n with m_n=psi(N/n)-psi(N/(n+1)) (E). The best certificate at (y,N) is the LP V*(y,N)=min c.L s.t. W>=1; its dual maximises sum nu_n over nonnegative nu with sum_n nu_n floor(n/j)=log floor(N/j)! for j<=y, i.e. a nonincreasing measure passing the first y Chebyshev tests, the primes being feasible with value psi(N). Lemma 1 (dual): any delta with sum delta_n floor(n/j)=0 (j<=y) and delta_n>=-m_n gives E(c)>=sum delta_n. Lemma 2: all admissible U(k)=sum_{n>=k}delta_n arise as U(k)=sum_m mu(m)T(km) for T on (y,N], gain sum_{m>y}mu(m)T(m), constraint U(k)-U(k+1)>=-m_k. Lemma 3 (rough spikes): E(c) >= sum_{y<q<=N, q y-rough} m_q, since the jump of W at a y-rough q equals c_1=W(1)>=1. For one cutoff N only, W>=1 is needed only on the attainable quotients Q_N={floor(N/d)}, about 2 sqrt N cells (T*, prime-blind). Measured floor at y=sqrt N is about 0.32 N^{3/4} over three decades; the barrier V*-psi >= cN/sqrt y is a conjecture, and the Section 4.2 dimension-count argument is refuted by N=27,y=9.

Evidence: frontier/2026-09-06/certificate_lp_frontier/RESULTS.md Sections 1-3, 5-6 (lines 22-94, 344-398) and correction header; BARRIER.md Sections 1-2 lines 16-123 (Lemmas 1-3, 5, Proposition 4); barrier_lemmas.py; tests/test_certificate_lp_barrier.py Prior art: cited: Chebyshev's construction (recovered at period 30); Selberg's identity as a test dictionary; Huxley's divisor bound for the Voronoi alternative Reused in: hunts/quotient_certificate/ (T* relaxation, N=27 counterexample, PR #203); hunts/paid_shortfall_scaling/RESULTS.md (cites BARRIER.md); hunts/outband_certificate/, hunts/paid_shortfall/ Why it travels: Converts any 'hand-built majorant' family into a computable floor plus a dual witness showing where the freedom sits; the excess identity (E) and rough-spike lemma hold for any floor-sum certificate.

Nonnegative squared-term obstruction to an integrated Barban-Davenport-Halberstam theorem at N^{1+eps}

obstruction | hunts/prime_pair_error/ | grade: measured (exact enumeration, small Q only) plus heuristic argument; the text calls it 'strong evidence, not a proof' and does not extrapolate to Q polynomial in N

Sigma(N,Q)=sum_{q<=Q}sum_b^* sum_{t<=N}Delta(t;q,b)^2 with Delta(t;q,b)=psi(t;q,b)-t/phi(q) is a sum of nonnegative terms, so no cancellation across t is possible; if Delta(t;q,b)^2 is of order t/phi(q) (up to logs) for a positive proportion of t in [N/2,N], which is what square-root cancellation and the proved sharp BDH asymptotic assert, then Sigma(N,Q) >> QN^2, not QN^{1+eps}. Measured exactly (true Lambda, exact cumulative class sums, no sampling) for Q in {5,10,20,30} and N from 1e4 to 3.2e5: Sigma/(QN^2 log N) stable within a factor 2 and log-log exponent 1.97-2.00 in N. Hence the t-integrated mean-value theorem RANK3_ROUTE_D.md's (D11) would need does not exist because the quantity does not have that size; the obstruction is the object, not the technique.

Evidence: RANK3_INTEGRATED_BDH.md Section 3 lines 126-216 and Section 4; rank3_integrated_bdh_probe.py, results_rank3_integrated_bdh_probe.json Prior art: cited: Montgomery 1970, Hooley (sharp BDH); Barban-Vehov and Motohashi examined by name and found inapplicable Reused in: RANK3_ROUTE_D.md, RANK3_MEAN_VALUE_TOOLS.md Section 5, RANK3_BDH_VERIFY.md Section 6 Why it travels: The general principle: before searching the literature for a mean-value theorem of a given strength, check whether the target is a sum of squares whose typical term already has the forbidden size.

Factorial and LP certificates for psi(N)

Chebyshev-type factorial certificates for psi(N) as linear programs: the quotient relaxation, its floor, exact duals, repair budgets and the rank and transport obstructions.

Exact paid-cost excess identity for factorial certificates

identity | hunts/paid_shortfall | grade: ordinary derivation, independently checked algebra; small deterministic checks in construction.py and tests/test_paid_shortfall.py

For N >= 2, Q_N = {floor(N/d) : 2 <= d <= N}, W_c(q) = sum_j c_j floor(q/j), B_N(c) = sum_j c_j log(floor(N/j)!), m_q = sum_{floor(N/d)=q} Lambda(d), and any cap U_q >= m_q, define C_N^U(c) = B_N(c) + sum_q U_q (1 - W_c(q))+. Then by the divisor identity log n = sum{d|n} Lambda(d) one has B_N(c) = sum_q m_q W_c(q), and exactly: C_N^U(c) - psi(N) = sum_q m_q (W_c(q)-1)_+ + sum_q (U_q - m_q)(1 - W_c(q))_+ >= 0. The first term charges surplus coverage; the second charges slack in the cap precisely where coverage is missing; improving a cap helps only where the candidate has a deficit.

Evidence: RESULTS.md section 1 'Definitions and the complete paid cost', equation (1), lines 24-60 Prior art: paid functional and capped dual preserved from a September 7 conversation; main term inherited from ../prime_pair_error/frontier/2026-09-06/factorial_certificate_pilot; novelty not assessed Reused in: hunts/paid_shortfall_saturation (full total = factorial + P_Lambda = psi(N) + S, the surplus S = sum Lambda(d)(W_d - 1)_+) Why it travels: Splits the excess of any Chebyshev-type factorial upper bound into two nonnegative, separately attributable terms, so every coefficient or cap change can be priced exactly.

Perfect-power cap mass: exact Möbius-type identity and asymptotic bound

identity | hunts/paid_shortfall_scaling/ | grade: elementary written derivations, not Lean, independently reviewed once; numerics carry two-backend rational enclosures

For d ≥ 2 let r(d) be its largest perfect-power exponent, h(d) = log d·(1 − 1/r(d)), T(X) = Σ_{2 ≤ d ≤ X} h(d), L(X) = log(⌊X⌋!). With b_k = −Π_{p | k}(1 − p) over distinct primes (b_2..b_12 = 1,2,1,4,−2,6,1,2,−4,10,−2), for X ≥ 4 and K = ⌊log₂X⌋: T(X) = Σ_{k=2}^{K} b_k L(X^{1/k}), using exact integer roots (root^k ≤ n < (root+1)^k, no floating-point root rounding). Proof: with b_1 = −1, Σ_{k | r} (−b_k)/k = Π_{p^a ∥ r}(1

expansion counts d = a^r exactly at k | r. Asymptotic: L(√N) ≤ T(N) ≤ L(√N) + 8N^{1/3}log N, hence T(N) = ½√N log N − √N + O(N^{1/3}log N) with the coefficient of N^{1/3}log N at most 2/3 + 12/(e log 2) < 8 (using max_{x ≥ 1} (log x)x^{−1/12} = 12/e). For any finitely supported coefficient vector the cap saving is exactly S_N = Σ_{d=2}^{N} h(d)δ_{⌊N/d⌋} with 0 ≤ S_N ≤ D_N T(N), so if D_N = o(√N/log N) the perfect-power cap cannot change a leading term CN + o(N).

Evidence: RESULTS.md §1 (eqs. 1–3, lines 25–83), §2 (eq. 6, lines 105–131); computations/check-001 (257 cap identities, two interval log implementations at 35 and 70 digits) Prior art: 'Novelty has not been assessed' Reused in: §3 eq. 10 (saving bound D_N T(N/y) for the balanced-prefix family); §1 bound (5) for the old truncated lifts, S_N = O(N^{1/4}log N) Why it travels: An O(log X) evaluation of a perfect-power-weighted log sum, and a clean criterion for when a capacity refinement cannot move a leading constant.

Attainable-cell (quotient) relaxation of the factorial LP certificate

identity | hunts/quotient_certificate/ | grade: identity exact (integer arithmetic on all 889 retained cells); optimum values and logarithms measured (float LP, 50/80-digit replays)

Let W_c(t) = sum_{j<=y} c_j floor(t/j), B_c(N) = sum_{j<=y} c_j log(floor(N/j)!), and Q_N = {floor(N/d) : 2<=d<=N}. Expanding log(m!) = sum_{d<=m} Lambda(d) floor(m/d) and interchanging finite sums gives exactly B_c(N) - psi(N) = sum_{d=2}^N Lambda(d) [W_c(floor(N/d)) - 1] (floors commute under successive integer division). Hence W_c(n)

= 1 for n in Q_N alone suffices for B_c(N) >= psi(N); no constraint at other integers,

no sieve, no prime locations. For r = floor(sqrt N), Q_N = ({1..r} union {floor(N/d): d<=r}) minus {N}, so |Q_N| <= 2r-1. The relaxed optimum T* satisfies psi(N) <= P* <= T* <= V* (P* = prime-cell, V* = all-cell relaxations). Measured savings of 30-42% of the excess at N = 10^3, 10^4, 10^5 with rational coefficients (denominator 10^12) checked feasible in Python integers.

Evidence: hunts/quotient_certificate/RESULTS.md, Section 1 'A relaxation that uses no prime locations' (lines 14-50) and Section 2 table (lines 52-79) Prior art: Chebyshev identity is classical and used as such; Section 5 corrects cited literature (Brent-Platt-Trudgian, CHHL, Burnol, Bettin-Conrey-Farmer, Baez-Duarte, Diamond) but the relaxation itself is unsearched; no novelty claimed Reused in: Reused as the base object in FINITE_TRANSFER.md, HEIGHT_KERNEL.md, COMPENSATED_REPAIR.md, TRANSPORT_CAPACITY.md, FOLD_UPPER.md, CREDITED_BUNDLE.md (same hunt) and consumed by the prime_pair_error/frontier certificate_lp_frontier lane (PR #208 DUAL_WITNESS.md) Why it travels: Any floor-dictionary LP bounding a Lambda-weighted sum only needs constraints at the distinct values floor(N/d); cuts constraint count from N to about 2 sqrt N with no loss of validity.

Exact full-period sawtooth covariance and Jordan-totient variance identity, with primorial refutation of the diagonal comparison

identity | hunts/quotient_certificate/ | grade: exact (finite enumeration and exact algebra); ordinary derivation, self-reviewed

For f_j(n) = {n/j} - (j-1)/(2j) and n uniform on a common integer period, Cov(f_i, f_j) = (gcd(i,j)^2 - 1)/(12 i j) exactly, and Var(sum_j c_j f_j) = (1/12) sum_{d=2}^y J_2(d) [sum_{j<=y, d|j} c_j/j]^2 with J_2(d) = d^2 prod_{p|d}(1 - 1/p^2), via gcd(i,j)^2 - 1 = sum_{d | gcd, d>=2} J_2(d). Corollary (obstruction): no uniform kappa>0 gives 12 Var >= kappa sum c_j^2 for balanced coefficients: for a squarefree primorial P with delta = phi(P)/P, tau = 2^omega(P), take c_d = mu(d) on d|P but c_1 = 1 - delta; then sum c_d/d = 0, 12 Var = tau delta - delta^2 and sum c_d^2 = tau - 2 delta + delta^2, ratio -> 0. Also W_c(n) = n A_c - sum_j c_j {n/j} with A_c = sum c_j/j, and the LP does not impose A_c = 0 (measured A_c = 0.0081388771875 at (1000,31)).

Evidence: hunts/quotient_certificate/RESULTS.md, Section 4 'The random-sawtooth argument discards structure' (lines 119-158); covariance checked by exact full-period enumeration for all 400 pairs i,j<=20; primorial formulas checked at P=6,30,210,2310 Prior art: unsearched (full-period sawtooth correlations are classical Franel-Landau territory; hunt does not cite) Reused in: FINITE_TRANSFER.md Section 3 uses the period matrix K^per_{ij} and its lower bound ||c - A_c e_1||^2/(24 H_y^2) (issue #204) as the comparison target Why it travels: Closed-form second-moment structure for any linear combination of sawtooths; the primorial family is a ready-made witness against unrestricted diagonal (l2) comparisons.

True-rate prefix transport identities and the zero-mass drain obstruction

identity | hunts/quotient_certificate/ | grade: hardened (integer moment identities, independent triangular solve and factorization, interval enclosures); not formalized or externally reviewed

With {1..y} in Q_N, define for q > y: R_q(s) = sum_{k<=y/s} mu(k) floor(q/(sk)), r_s(q) = R_q(s) - R_q(s+1) (R_q(y+1) = 0), z_q = e_q - sum_{s<=y} r_s(q) e_s. Summation by parts and sum_{k|n} mu(k) = 1_{n=1} give sum_{s<=y} r_s(q) floor(s/j) = floor(q/j), so A^T z_q = 0 and sum z_q = 1 - W_mu(q). Endpoint correction: r_s need not be 0 or 1 (r_51 = 2, r_100 = 50 at q=5000,y=100). Exact suffix identity: for y/2 < u <= y, sum_{s=u}^y r_s(q) = floor(q/u); hence aggregate drains satisfy sum_{s=u}^y C_s = sum_{q>y} eta_q floor(q/u) against sum_{s=u}^y m_s = psi(floor(N/u)) - psi(floor(N/(y+1))). Obstruction: the proposed shortcut 'source capacity plus top-half prefix conditions imply a positive-fraction feasible exchange' is false at a=103, b=101 (N=10000): the exchange withdraws -theta at cell 33 whose integer cell {295..303} contains no prime power, so m_33 = 0 and every theta > 0 fails; the exact feasible amount for that direction is theta_max = min(nu_3, nu_5, nu_17, nu_33, nu_50, nu_103).

Evidence: hunts/quotient_certificate/TRANSPORT_CAPACITY.md, Section 2 'True integer rates and an exact suffix identity' (eqs. 3-4), Section 3 'A precise capacity shortcut and its exact counterexample' (eqs. 5-6), Section 4 (eqs. 7-9) Prior art: identities consumed from PR #208 and corrected here; unsearched Reused in: Feeds the capacity bookkeeping (7)-(9) that COMPENSATED_REPAIR.md Section 4 restates as the uniform-family obligation Why it travels: Exact signed-rate calculus for moving dual mass through the prefix of a floor dictionary, and a concrete demonstration that zero-mass cells (not source mass) are the binding constraint.

Contractive repair lemma: explicit budget for compensating empty-cell deficits

lemma | hunts/quotient_certificate/ | grade: ordinary derivation with independent mathematical review and rational algebra control (four-row example with C_12 = 1/4, C_21 = 1/2, S = 4); hardened evidence, not formalized

Fix N, y, the exact measure m, empty cells Z = {q : m_q = 0}. Let U (seed) and R_1..R_r (repairs) be zero-moment vectors on Q_N, and Z_1..Z_r disjoint nonempty blocks of empty cells; outside the blocks require U_q >= 0 and (R_i)_q >= 0. Hypotheses: (1) R_i supplies >= 1 at every cell of Z_i and >= -C_{ki} on Z_k (C >= 0, C_ii = 0), d_k = max_{Z_k}(-U_q)_+; (2) prices p_i > 0 and 0 <= rho < 1 with sum_k p_k C_{ki} <= rho p_i; (3) envelopes (-U_q)_+ <= b_0(q), (-(R_i)q)+ <= p_i b(q) on positive-mass cells; (4) seed gain >= G_0, repair gains >= -ell p_i. Put S = sum p_i d_i/(1-rho), t = (I - C)^{-1} d = sum C^n d >= 0, D = U + sum t_i R_i. Then D >= 0 on all empty cells, p^T t <= S, and if b_0(q) + S b(q) <= K m_q on P then m + D/K >= 0 and E_c >= (G_0 - ell S)/K. Proof: p^T C^n d <= rho^n p^T d gives convergence; on Z_k, D_q >= -d_k + t_k - sum_i C_{ki} t_i = 0. Positive repair gain may be retained for a sharper bound (CREDITED_BUNDLE: (sum V + t_1 sum R)/K = 615 log229/146). Limitation identified: the lemma forces aggregate repair >= 0 on every empty cell, so it cannot spend seed credits; regroup the seed to fit.

Evidence: hunts/quotient_certificate/COMPENSATED_REPAIR.md, Section 2 'Conditional lemma: repair budgets from a contraction' (eqs. 5-9) and Section 3; CREDITED_BUNDLE.md Sections 3-4 (why the 2U seed fails, regrouped decomposition V = F_103, R = D - V) Prior art: unsearched; 'No novelty claim is made' Reused in: CREDITED_BUNDLE.md applies it to the coordinator's N=10000 bundle with Z_1 = {33}, K = 146/log 229 Why it travels: A general sufficient rule for bounding the total amount of a cascade of compensating perturbations under a weighted-l1 contraction; applicable to any dual-measure repair where secondary deficits are created.

Halving folds are an integer basis of ker(A^T); nonnegative-fold gain has a finite rational upper certificate

lemma | hunts/quotient_certificate/ | grade: hardened (exact integer column inequalities, fresh interval arithmetic at 90/105 digits, independent algebra review, computation manifest); not formalized or externally reviewed

Assume {1..y} is contained in Q_N and let H = Q_N minus the prefix. The fold at attainable a > y is F_a = -e_a + 2 e_{floor(a/2)} + sum_{i<=y} (T_i(a) - T_{i+1}(a)) e_i with T_i(a) = sum_{k<=y/i} mu(k) (floor(a/(ik)) mod 2), T_{y+1} = 0; the parity identity floor(a/j) - 2 floor(floor(a/2)/j) = floor(a/j) mod 2 plus prefix Mobius inversion gives A^T F_a = 0 and gain g(a) = 1 + sum_{j<=y} mu(j)(floor(a/j) mod 2). At rows q > y, (Fx)_q = -x_q + 2x_{2q} + 2x_{2q+1}, so any zero-moment delta supported on Q_N has the unique signed fold coordinates x_q = 2x_{2q} + 2x_{2q+1} - delta_q evaluated in descending q, and the prefix residual h = delta - Fx vanishes because the prefix matrix (floor(i/j))_{i,j<=y} is unit lower triangular. The high-row matrix is unimodular, so folds form an integer basis of ker(A^T) on the lattice. Separately, at N=10000, y=100, a 52-entry nonnegative integer vector B with beta = B/1387 satisfies -1387 g_a - sum_q B_q F_{qa} >= 0 for all 98 columns, hence x >= 0, m + Fx >= 0 implies g^T x <= beta^T m in [66.3384089619581189, ...190] (exact cost 16456 log2 + 3052 log7 + ... over 1387).

Evidence: hunts/quotient_certificate/FOLD_UPPER.md, Section 1 (fold convention, eq. 3), Section 2 'The rational certificate and the inequality signs' (eqs. 4-6), Section 4 'Signed recurrence, including prefix reconstruction' (eqs. 7-8); COMPENSATED_REPAIR.md Section 1 eq. 3 Prior art: fold construction consumed from PR #208 (other lane); basis lemma and upper certificate 'No novelty claim is made'; unsearched Reused in: Used to prove the 132.729535 prefix-exchange witness must have a negative fold coordinate (FOLD_UPPER Section 4 'Scope consequence') Why it travels: Gives an explicit unimodular coordinate system for all zero-moment dual perturbations of a floor-dictionary LP, and shows how a finite rational dual certificate caps an entire cone of constructions at once.

Strictly positive dual as an exact uniqueness certificate for the floor-LP optimizer

lemma | hunts/quotient_certificate/ | grade: exact integer/rational identities plus fresh interval enclosures; independent agent review; not formalized or externally reviewed

Let S be a basis of 100 attainable cells with A_S = (floor(s/j)) invertible (det -1328), primal c* with A_S c* = 1 and W_{c*} >= 1 on all 198 cells. Reconstruct the integer matrix E_{jp} = sum_k floor(floor(N/j)/p^k) over the 1229 primes p <= N, L_j = sum_p E_{jp} log p, V = (A_S^T)^{-1} E, so nu_s = sum_p V_{sp} log p satisfies A_S^T nu = L with no rationalization of logarithms; 80-digit intervals prove every nu_s > 0 (min in [0.000612063301151538696782037139, ...140]). Then for every feasible c, B_c - T* = sum_{s in S} nu_s (W_c(s) - 1) with T* = sum_s nu_s, so B_c = T* forces c = c*: all nonzero cost-preserving feasible displacements are excluded. Every feasible vector is c*

(control: z = e_1 gives W(20) = -13/83, so they cannot be dropped).

Evidence: hunts/quotient_certificate/HEIGHT_KERNEL.md, Section 1 'Exactly which ingredients of #208 were consumed' and Section 2 'Keep the full slack parameterization and budget' Prior art: complementary slackness, standard; the exact prime-log reconstruction is unsearched Reused in: Consumed within the same hunt (HEIGHT_KERNEL Section 4 uses T* in bound (3)) Why it travels: Pattern for certifying LP optimality and uniqueness exactly when the objective involves logarithms: keep coefficients as integer matrices over log p, enclose only at the end.

Early-truncation lemma: repair bill of a square-root-support balanced seed lift

bound | hunts/paid_shortfall | grade: ordinary derivation, self-reviewed with independently checked algebra; finite checks only falsify the implementation

Fix M >= 2 and finitely supported a_j (support in j <= J) with sum a_j/j = 0, g(t) = sum a_j floor(t/j), full lift W_inf(t) = sum_{k>=0} g(t/M^k) >= 1 for t >= 1, and g <= H. With A = sum |a_j|, kappa = -sum a_j log j / j, C = kappa/(1 - 1/M), L(x) = log(floor(x)!), truncation K, R = M^{K+1}, c_n = sum_{j M^k = n, k <= K} a_j: psi(N) <= C_N(c) <= C(1 - 1/R) N + E_N + H sum_{l>=0} L(N/(R M^l)), E_N = A sum_{k<=K} (1 + log^+(N/M^k)), and the repair term is at most (H/(1-1/M)) x log^+ x with x = N/R. For J M^K <= sqrt N the bill is at most (H J/(1-1/M)) sqrt N log(J sqrt N), so C_N(c) - N = (C-1)N + O_{a,M}(sqrt N log N) for a fixed seed. Explicit base-6 seed g(t) = floor t - floor(t/2) - floor(t/3) - floor(t/6) (values 0,1,1,1,1,2 on residues) has (1 - W_K(q))+ = 1{R | q} exactly, coefficient mass 2r+2, C_6 = (4 log 2 + 3 log 3)/5 > 1.

Evidence: RESULTS.md section 2 'Early-truncation lemma' with proof, equations (2)-(7), lines 62-179; section 3 base-6 construction, equation (8), lines 181-234 Prior art: balanced-seed main term inherited from the factorial_certificate_pilot; not searched Reused in: hunts/paid_shortfall_saturation imports the cutoff selection from paid_shortfall_scaling Why it travels: A fully charged truncation for any balanced floor-sum seed at any base M; the exact base-6 indicator identity (8) is a clean test object.

Finite prime-weighted variance decomposition (R), atom-loss conversion (M), and kernel-distance bound (F)

bound | hunts/quotient_certificate/ | grade: algebraic proofs and finite exact checks; 'not formalized or externally reviewed' (hunt's words); moment values measured at 50/80 digits

Let p_q = m_q/Psi on the positive-mass attainable cells P (m_q = sum of Lambda over the exact integer cell, Psi = psi(N)), S_c = sum_j c_j {q/j}, V_q = Var_p(q), beta = Cov_p(q,S_c)/V_q. Then (R): Var_p(W_c) = Var_p(S_c - beta q) + (A_c - beta)^2 V_q exactly, so Var_p(W_c) >= c^T R_f c with R_f = C_f - h h^T/V_q, ker R_f = {c : Uc in span(1, t)}. (M): for X = W_c - 1 >= 0, mu = E_p X, v = Var_p X, alpha = min p_q: E X^2 <= mu^2/alpha, hence E := B_c - psi(N) = Psi mu >= Psi sqrt(alpha v/(1-alpha)); for N>=4, alpha = log2/Psi exactly (cell floor(N/2) holds only d=2), and (M) is sharp at N=4, y=2, c=(1,t-1). (F): with D_W the anchored integer difference matrix of rank r and L = ||D_W||F^2, v >= p{q0} alpha L^{-(r-1)} dist(c, K_W)^2, because the product of positive eigenvalues of D_W^T D_W is a sum of squared integer minors >= 1. Zero-mass rows must be excluded from D_W.

Evidence: hunts/quotient_certificate/FINITE_TRANSFER.md, Section 1 'Exact finite measure', Section 4 'Drift is a separate obstruction, with an exact repair' (eq. R), Section 5 'Second moments to nonnegative linear excess' (eqs. M, F); N=9 drift control with masses (log210, log2, log3, log2) Prior art: unsearched; no novelty claimed Reused in: HEIGHT_KERNEL.md Section 5 applies (M) with the Cauchy-Schwarz bound v >= M^2/D; TRANSPORT_CAPACITY.md Section 1 uses (M) to cap the constant-variance/smallest-atom procedure at 17.507 Why it travels: Generic bookkeeping for converting a variance lower bound on a nonnegative slack vector under a finite positive measure into a linear-excess bound, with the smallest-atom loss made explicit and shown sharp.

Perfect-power cap and an exact finite LP-dual optimum removing an artificial-mass obstruction

construction | hunts/paid_shortfall | grade: exact finite example (ordinary derivation); no growth estimate claimed

For d >= 2 let r(d) be the largest r with d = b^r, u(d) = log d / r(d), U_q = sum_{floor(N/d)=q} u(d); then u = Lambda on prime powers and m_q <= U_q <= w_q, computable with integer power tests only. At N = 14 with support j <= 3 and cells (1,2,3,4,7), c = (1,-1,-3/2) gives C^w = psi(14) + (1/2) log 2 and C^U = psi(14). The raw-cap excess is unavoidable for every c' with that support: with epsilon = (1/2) log 2, v = (-1,1,2,0,-1), x = m + epsilon v is feasible (0 <= x <= w) with zero moments sum_q v_q floor(q/j) for j = 1,2,3 and sum v_q = 1; since x(W-1) + w(1-W)_+ >= 0 for 0 <= x <= w, summing gives C^w(c') >= sum x_q = psi(14) + (1/2) log 2. The saving from the cap is exactly sum_d log d (1 - 1/r(d)) (1 - W_c(floor(N/d)))_+.

Evidence: RESULTS.md section 4 'A finite obstruction removed by arithmetic capacity', equation (9) and the dual vector argument, lines 236-301 Prior art: unsearched Reused in: hunts/paid_shortfall_saturation (perfect-power capacity retained on prime powers, saturated by trial division through sqrt(X)) Why it travels: The dual-vector certificate of a finite optimum (feasible x with matching zero moments) is a reusable way to prove a paid-cost floor exactly, and the perfect-power cap is a reusable arithmetic refinement.

Balanced Möbius-prefix coefficient family with bounded mass

construction | hunts/paid_shortfall_scaling/ | grade: elementary written derivations, not Lean; finite identities checked exactly at N = 144…36864

For integer y ≥ 2 and S(m) = Σ_{j ≤ m} μ(j)/j, set c_j^{(y)} = μ(j) for j < y and c_y^{(y)} = −yS(y−1). Then Σ_j c_j^{(y)}/j = 0 (zero harmonic drift), W_y(q) = 1 for 1 ≤ q < y (coverage, from Σ_{j ≤ q} μ(j)⌊q/j⌋ = 1), and mass A_y = Σ|c_j| ≤ 2y − 1 because mS(m) = 1 + Σ_{j ≤ m} μ(j){m/j} with the j = 1 fractional part zero gives |S(m)| ≤ 1. Consequences without positivity: |W_y(q)| ≤ A_y, D_N ≤ 1 + A_y ≤ 2y; with κ_y = Σ_{j < y} (μ(j)/j)log(y/j), |B_N(c^{(y)}) − κ_y N| ≤ A_y(1 + log N) for every N ≥ 2; deficits occur only for d ≤ N/y, so 0 ≤ P_N^U ≤ D_N L(N/y) and 0 ≤ C_N^w − C_N^U ≤ D_N T(N/y). At y = ⌊√N⌋ the factorial error has square-root scale but the crude penalty bound does not; the missing control is on the combined (κ_y − 1)N + P_N^U with its actual signs.

Evidence: RESULTS.md §3 (eqs. 7–10, lines 133–182); scaling.py; MISSION.md 'First construction' Prior art: prior lab work: finite Möbius-prefix and harmonic-drift identities in prime_pair_error/frontier/…/BARRIER.md; novelty not assessed Reused in: §4–§5 finite discriminators; cutoff selection rule Why it travels: A scale-dependent coefficient family with explicit coverage, drift and mass bounds, usable as a baseline in any paid-bound or factorial-route experiment.

Zero-surplus rational balanced vector at N = 144 and the exact cost identity C = psi(N) + S

construction | hunts/paid_surplus_obstruction/ | grade: 'ordinary finite argument with exact rational executable checks and two independent logarithm enclosure implementations' by the same producer; review status 'pending independent challenge'; no Lean, no novelty claim, no uniform estimate

Class: rational c_1..c_12 with harmonic balance sum c_j/j = 0, coverage W(q) = sum c_j floor(q/j) = 1 for 1 <= q <= 5 (forcing c_1 = 1), and mass sum |c_j| <= 23. Construction: 20(c_1..c_12) = (20,-20,-20,0,-20,-13,7,6,0,0,0,13) has balance (840 sum b_j/j = 16800-8400-5600-3360-1820+840+630+910 = 0), coverage (q - floor(q/2) - floor(q/3) - floor(q/5) = 1 for q <= 5), mass 119/20, and W_d := W(floor(144/d)) <= 1 at every prime power 2 <= d <= 144: for d >= 25 the quotient lies in [1,5] so coverage gives W_d = 1 automatically (34 of 47 prime powers), and the 13 prime powers <= 24 are checked by integer substitution (values 3/10, 3/10, 0, 1, 0, 0, 13/20, 1, 1, 1, 1, 7/10, -13/20). Identity: with B = sum_j c_j log(floor(144/j)!) = sum_{d<=144} Lambda(d) W_d (finite rearrangement of the Legendre count, no limit interchange) and P_Lambda = sum Lambda(d)(1 - W_d)+, applying w + (1-w)+ = 1 + (w-1)_+ row by row gives C := B + P_Lambda = psi(144) + S, S = sum Lambda(d)(W_d - 1)_+; the table proves S = 0 exactly so C = psi(144), the elementary lower bound, attained. The previous endpoint (1,-1,-1,0,-1,1,-1,0,0,1,-1,2/385) has S_old = (2/55) log 2 + (8/385) log 3 + (387/385) log 7 + (387/385) log 11 > 0 exactly. The vector lies outside the convex hull of all balanced Möbius prefixes (c_7 = 7/20 > 0 where every prefix has c_7 <= 0). Controls: LP used only as discovery aid with exact rational reconstruction; old endpoint rejected on five surplus rows; a class-preserving perturbation (+3/5 on c_6, -6/5 on c_12) rejected on four rows; zero vector fails coverage; domain/support/balance/coverage/mass lesions rejected.

Evidence: RESULTS.md 'Exact class and source boundary' (lines 22-50), 'Finite proof' (lines 52-90), 'Full cost and what was improved' (lines 92-145), 'Controls, reproducibility and remaining uncertainty' (lines 147-185); construction.py; computations/check-001/ Prior art: unsearched ('An unrun literature search establishes nothing about priority') Reused in: relaxes the balanced-prefix construction of hunts/paid_shortfall_scaling/RESULTS.md §3; none else stated Why it travels: The row-wise identity turning a signed Chebyshev-type sum plus repair cost into psi(N) plus a surplus term, and the coverage trick that discharges all large-d constraints for free, apply to any finite Möbius-style coefficient design at other cutoffs or supports.

Sampling-null feasible witness from a residue-matrix kernel

construction | hunts/quotient_certificate/ | grade: exact rational witness with independent integer checks (all 198 constraints, every sawtooth at every positive cell); general construction proved algebraically, not externally reviewed

Let R be the integer matrix with rows (q mod j) - (q_0 mod j) over positive-mass cells q in P minus {q_0} and columns j = 2..y. Whenever rank R < y-1, pick nonzero rational z in ker R, set v_j = j z_j (denominators cleared), v_1 = 0, D = sum_{j>=2}|v_j|, c = e_1 + v/D. Then S_c is constant on P (finite sawtooth variance zero), the full-period variance is strictly positive (the largest index k contributes J_2(k) c_k^2/(12 k^2) > 0), sum_{j}|c_j| = 2, and W_c(q) >= q - sum_{j>=2}|c_j| floor(q/j) >= q - floor(q/2) >= 1 for every integer q >= 1, so c is feasible on all of Q_N without repair. This refutes any comparison Var_p(S_c) >= kappa Var_period(S_c) over the full feasible class (witness at N=10000, y=100: 98x99 residue matrix of rank 95, 42 nonzero tail coefficients). Boundary: its cost is B_c = log(N!), so it says nothing about low-cost certificates, and its direction is blocked at the saved optimizer by two zero-mass cells (q=60, 333).

Evidence: hunts/quotient_certificate/FINITE_TRANSFER.md, Section 3 'One candidate comparison, and its feasible counterexample' and subsection 'What the same direction does to the saved low-excess certificate'; witness in finite_transfer_counterexample.json Prior art: unsearched Reused in: HEIGHT_KERNEL.md Section 1 extends the two-cell blocking to all cost-preserving displacements via the strictly positive dual Why it travels: Cheap, prime-blind-in-coverage feasibility construction (sum|c| = 2 guarantees W_c >= 1 everywhere) for producing counterexamples to sampling/transfer inequalities in floor-dictionary LPs; also shows why zero-mass constraints must be retained.

Finite duality lower bound with exact integer product capacity checks

computational technique | hunts/quotient_certificate/ | grade: hardened (integer identities, independent factorization and triangular reconstruction, interval enclosures); not formalized or externally reviewed

For the exact measure m_q on Q_N and floor matrix A_{qj} = floor(q/j): if H is supported on Q_N with A^T H = 0 (zero moments) and m + H >= 0, then every feasible c (W_c = Ac >=

  1. has E_c = B_c - psi(N) >= sum_q H_q. To find the maximal scale lambda with m + lambda

D >= 0 without rounding logarithms, let M_q be the product of prime bases p once per prime power in cell q, so m_q = log M_q exactly; then m_q + lambda D_q >= 0 with lambda = log(P)/K is the integer inequality M_q^K >= P^{-D_q}. Instances: N=1000,y=31, D = F_76

only 49 = 7^2), so E_c >= (10/3) log 7 in [6.48636716351771101701, ...02]; N=10000: E_c

= (615/146) log 229 and the two-cell control delta = log(97)(e_1 + e_102 - e_103) gives

E_c >= log 97.

Evidence: hunts/quotient_certificate/COMPENSATED_REPAIR.md, Section 1 'Fixed definitions and the independently checked bundle' (eqs. 2, 4); TRANSPORT_CAPACITY.md Section 1 'The two coordinator deductions' (eq. 1); CREDITED_BUNDLE.md Section 2 (eq. 3) Prior art: ordinary finite LP duality, used as such; the exact-product capacity check is unsearched; no novelty claimed Reused in: Used in COMPENSATED_REPAIR.md, CREDITED_BUNDLE.md, TRANSPORT_CAPACITY.md, FOLD_UPPER.md Section 2 (eq. 6, grouped log p evaluation) Why it travels: Turns 'is this dual perturbation feasible against the prime measure' into exact integer comparisons; no N/q^2 approximation or interval-length proxy for m_q, and the binding cell is identified exactly.

Rank obstruction: more columns than prime cells does not force zero excess (N=27 exact counterexample)

obstruction | hunts/quotient_certificate/ | grade: elementary finite proof with exact-arithmetic checks of row relation, rank, prime weights and attaining vector (hunt: 'not a kernel proof or an asymptotic result')

The claim 'P*(y,N) = psi(N) whenever y exceeds the number of prime-looking cells' is false. At N=27, y=9 the prime-cell set S = {1,2,3,4,5,6,9,13} has |S| = 8 < 9, yet the rows R_n = (floor(n/j))_{j<=9} satisfy the integer relation -R_1 + R_3 - R_6 - R_9 + R_13 = 0 whose coefficients sum to -1, so the matrix (rank 7) cannot interpolate the constant vector on S. For feasible W with e_n = W(n)-1 >= 0 the relation forces e_3 + e_13 >= 1, and the excess identity weights at cells 3 and 13 (log 42, log 2) give B - psi(27) >= log 2, attained by c = (1,-1,-1,0,-1,1,0,0,-1). Hence P*(9,27) - psi(27) = log 2 exactly. General lesson: the zero-excess question is a feasibility/rank question about integer row relations, not a variable count.

Evidence: hunts/quotient_certificate/RESULTS.md, Section 3 'More columns than prime cells does not force zero excess' (lines 81-117) Prior art: unsearched Reused in: Generalised in HEIGHT_KERNEL.md Section 3 (21-cell integer relation at N=10000 excludes the height kernel) and TRANSPORT_CAPACITY.md Section 1 (normalized signed measures with A^T w = 0, sum w = 1) Why it travels: Template for refuting dimension-count optimality claims in floor-matrix LPs: exhibit an integer row relation with nonzero coefficient sum, read off a forced excess from the positive weights.

Weil positivity, Gram forms and kernels

Weil-form truncations, window optima, incidence laws, cell-table and interval certificates for kernel inequalities, and the ceilings on what out-of-band positivity can buy.

Window Rayleigh ceiling and the sqrt(2) harmonic orthogonality

identity | hunts/amtopa_ceiling/ | grade: VERIFIED in the hunt's own label (recomputed from the primary source; float confirmation max |M[0,1:]| = 1.7e-16); not an enclosure

For a window family v = sum_j c_j w_j in the AF2026/AMTOPA construction, the window constant is H(v) = 2 - 1/c1 with c1 = (u.c)^2 / (c^T M c), a Rayleigh quotient in the coefficients, so its supremum over the whole coefficient space is the closed form H_max = 2 - 1/(u^T M^{-1} u), attained at c proportional to M^{-1} u. For the frequency set {w_0} union {2 j pi}: u_j = sinc(w_j/2) = 0 for every harmonic, and the off-diagonal entry M[0,j] times 4 j^2 pi^2 D / S (S = (-1)^j sin(w_0/2), D = w_0^2/4 - j^2 pi^2) reduces to 2 j^2 pi^2 (w_0^2 - 2)/w_0, which vanishes iff w_0 = sqrt(2). Hence at the fundamental sqrt(2), and only there, the harmonics are M-orthogonal and each strictly lowers H (about -0.59 c_j^2), and H_max = 0.67250070367941172655 (Theorem D, HD(1)) for 1, 2, 3, 7, 13, 17 and 25 terms alike.

Evidence: RESULTS.md section '4.1 The window: 16 free coefficients, and an exact ceiling of zero gain' (lines 356-404); probe_window.py; section 8 'Knownness' Prior art: unsearched; the hunt explicitly claims no novelty (Conrey-Ghosh-Gonek shape; Rayleigh-quotient maximiser is standard); whether the w_0^2 = 2 decoupling is in the literature was not checked Reused in: hunts/outband_certificate/RESULTS.md Why it travels: Gives an exact ceiling for any linear window family in a Rayleigh-quotient functional without search, and the orthogonality criterion tells you when adding basis functions cannot raise the constant.

Finite Poisson lattice formula for the infinite-chain energy

identity | hunts/family_wall/ | grade: DERIVED, VERIFIED against brute force; minimisation over P ≤ 6 is MEASURED (an upper bound on the true minimum)

With f(t) = cos(√2 t)·1_{[−1/2,1/2]}(t), K = f̂ and w = FT(G)/K(0)², G = f∗f supported on [−1,1] with G(u) = (1−|u|)cos(√2u)/2 + sin(√2(1−|u|))/(2√2) for |u| ≤ 1. For a period-T configuration with P points x_j per period, Poisson summation terminates: W_inf = (1/(PTK(0)²)) Σ_{|ν| ≤ T} G(ν/T)|Σ_j e^{−2πiνx_j/T}|² − 1, since G(ν/T) = 0 for |ν| > T. Verified against a brute-force sum over 800,001 images to 7.9e-9 (the brute force's own truncation error). The resulting W_inf(mean gap) curve is violently non-monotone (commensurability with kernel zeros), which is why the finite-n ladder's convex envelope, not the curve, governs the peak.

Evidence: FAMILY-LIMIT.md §2.6 (lines 499–529); periodic_energy.py; artifacts/periodic-energy-curve.json Prior art: none (standard Poisson summation; the finite termination is specific to the compact-support kernel) Reused in: FAMILY-LIMIT.md §1.4 chord reading Why it travels: Exact, cheap periodic energies for any kernel whose autocorrelation has compact support; replaces truncated lattice sums.

LAW D: alias-free grid incidence identity and the rigid bilinear diagonal

identity | hunts/frontier_math/ | grade: kernel-checked (Lean 4 + Mathlib, sorry-free, standard axioms) for real arguments; the complex-argument form is hardened (truncation ladder 1.7e-13 → 5.4e-19 at K = 150/300/600, lesion 65/64 grid stretch flat at 0.3038, decoy refuted, DH rival defect 2.7e-14)

For a window φ that is even, bounded, measurable and supported in [−1/2, 1/2] (no continuity, no decay), with φ̂(x) = ∫φ(u)e^{ixu}du, the family n ↦ φ̂(x−n)φ̂(y−n) is summable over ℤ and Σ_n φ̂(x−n)φ̂(y−n) = 2π·FT(φ^2)(x−y) for real x, y (kernel-checked; evenness is necessary, counterexample the indicator of (0,1/2] with grid sum 0 against π). Proof route: polarised Parseval on ℝ/2πℤ rather than Poisson summation, which is why bounded-measurable suffices. In the paper's units (window supported in [−L/2, L/2], critical grid τ_k at spacing h = 2π/L, normalisation aL^2) and extended to complex arguments the identity reads Σ_k φ̂(z−τ_k)φ̂(z'−τ_k) = L·Φ2(z−z') with Φ2 = FT(φ^2), and taking z' = z gives Σ_k φ̂(z−τ_k)^2 = aL^2 for every z ∈ ℂ: the bilinear self-incidence of a zero is exactly +m_ρ per unit multiplicity at every depth and position, so a conjugate pair contributes bilinear trace +2m_ρ, while the Hermitian mass Σ|φ̂|^2 = L·Φ2(2iy) is what grows like X^{|2β−1|}.

Evidence: hunts/frontier_math/law_d_incidence.lean (tsum_phihat_mul_phihat_even, tsum_phihat_windowA/B, tsum_phihat_of_continuous, grid_incidence_needs_even); TRANSPLANT-LEMMA.md §Seam (i), core: LAW D exactness is kernel-checked; SIGNED-INCIDENCE-LAW.md §The three laws (LAW D) and §Controls ledger Prior art: cited: the source paper's Lemma 2.2 (real arguments) and its Appendix B numerical check at complex arguments; new here: the complex statement used structurally, the bilinear/Hermitian split, the no-continuity hypothesis, and the Parseval proof found by the Aristotle prover Reused in: LEVEL2-GAP-CONSISTENCY.md (LAW G); LEVEL3-THETA-RECOVERY.md (LAW K); LEVEL5-ENCLOSURE-AND-PAIRS.md (LAW L); LEVEL7-VCELL.md (LAW M); PREPRINT.md §What is kernel-checked item 2; docs/27-state-of-the-transplant.md §2 Why it travels: Any Gabor/critical-grid Gram computation with a compactly supported window gets exact incidence values with no aliasing term and no smoothness assumption, including windows that jump at the box edge.

k-pair identity and the three-term Gram form of the multi-pair slack (two-species reduction)

identity | hunts/frontier_math/ | grade: measured (identities checked numerically to 1e-17 and 1e-25; the hunt's own label 'identity, checked, sufficient'); B ≥ 0 rests on the kernel-checked energy_F_ge

With F(w) = Σ_a e^{i x_a w}, P(w) = Σ_p 2cosh(y_p w)e^{i t_p w} and D(y,s) = Qim(y,s)^2 − Qre(y,s)^2: margin_k = Eng(F+P) − (199/200)Eng(F) − n/200 − 4k = (4/A^2)·slack_k with slack_k = Σ_p Shq(y_p)/2 − Σ_{p,a} D(y_p, x_a − t_p) + (1/400)Σ_{a<b} φ_r(x_a − x_b)^2 − (1/2)Σ_{p≠q}[D(y_p+y_q, τ_pq) + D(y_p−y_q, τ_pq)] (checked against quadrature on Eng for k = 1..4, mixed depths, worst residual 4.21e-17). Rewritten against the measure c_2 dw: slack_k = B(T,y) + Cross + R(X)/400 with B(T,y) = ∫c_2(w)[cosh^2(yw)|T̂(w)|^2 − k]dw ≥ 0 (free: energy_F_ge on the pair centres, cosh^2 ≥ 1, c_2 ≥ 0), R = Σ_{a<b}φ_r(x_a−x_b)^2 ≥ 0 (sum of squares), Cross = −Σ_{a,p}D(y, x_a − t_p) the only sign-indefinite term; −D(y,·) = ∫c_2(w)cosh(yw)cos(sw)dw is positive definite by Bochner since c_2 cosh(y·) is a positive measure on [−1,1]. The additive budget k·Shq/2 was never the budget (coincident centres give B 175× larger than spread ones). D(0,τ) = −Kpair(τ) exactly (Qim vanishes at depth 0), so the pair centres are atoms of a second species with the same repulsion kernel at 400× the rate and the same window damage at doubled depth; general depths split per pair as K_{y_p+y_q} + K_{|y_p−y_q|}. Two-species counting: m atoms and j centres in one window (width ≤ 0.9860008) close by a square completion iff D_1^2 ≤ 4(γ^2/800)κ(w_max), which holds with margin 327.9×, and maximising m j D_1 − m(m−1)γ^2/800 − j(j−1)κ(w_max) over integers gives j = 1 optimal (recovering the k = 1 optimum m = 3): a second pair in the same window is never profitable.

Evidence: hunts/frontier_math/PROOF-LEDGER.md §The k-pair identity, and coordinator defect #20; §ROAD B, step 2: the slack is three terms and two of them are free; §ROAD B, step 3: the counting lemma; K2-TWO-SPECIES.md §0–§1; kpair_identity.py, gram_form.py, two_species.py, counting_lemma.py Prior art: unsearched Reused in: K2-TWO-SPECIES.md (k = 2 equal-depth closure over the tau-table); LATTICE-EXTREMALITY-ROUTE.md (centre-gas cost); hunts/r_a97060 (interval pass); CROSS-ARM-REPLY.md Why it travels: Shows how to rewrite a multi-source quadratic energy against its positive spectral measure so that every term but one is nonnegative for free, before charging anything per source; the failure of additive per-source budgets is diagnosed by the same rewrite.

Closed forms for the windowed sine-Gram moments m2, m3 as functionals of the window, with exact rationals for polynomial windows

identity | hunts/rogue_frontier/window_opt | grade: hardened (continuum derivation cross-checked by the exact finite-N CUE lattice count at two windows; exact rationals; two-backend enclosures for the strict improvement)

For the windowed Gram kernel K_v(x) = int_{-1/2}^{1/2} v(xi) e^{2 pi i x xi} dxi over the unit-density sine process, with l1 = int v, W = 1_B * v, tri = 1_B * 1_B, B = [-1/2,1/2]: E tr H^2/N = l1^2 + [int v^2 - int W^2]; E tr H^3/N = l1^3 + 3 l1 [int v^2 - int W^2] + [int v^3 - 3 int (tri*v) v^2 + 2 int W^3]. With D2 = (int v^2 - int W^2)/l1^2 and D3 = (int v^3 - 3 int (tri*v) v^2 + 2 int W^3)/l1^3: m2 = 1 + D2, m3 = 1 + 3 D2 + D3, F = 2 m2 - m3 = 1 - D2 - D3. Everything reduces to V0(x) = int_0^x v and P0(x) = int_0^x t v via W(eta) = V0(1/2) + V0(1/2 - eta) and (tri*v)(s) = l1 - [2 s V0(s) - 2 P0(s) + 2 P0(1/2)], so polynomial v gives exact rationals. Witness: v*(s) = 1 - (1467/1000)s^2 + (1159/1000)s^4 gives F = 2245228120295149280/3276332462159207451, beating cos(8s/5) by 4.3148e-5 (both ball backends).

Evidence: RESULTS.md section 1 'Setup and derivation', lines 46-83; section 2 validation (four routes, exact finite-N CUE cross-check), lines 85-120; section 5, lines 174-193 Prior art: cited: the 10 Aug 2026 'more than two thirds' preprint section 7.5(g) (m2 closed form sketched by the coordinator; m3 formula new here) Reused in: RF-C003 promoted claim, re-run by hunt R-F00E48 (hunts/r_f00e48/probe.py pins the rational) Why it travels: Any certificate whose input is a spectral moment of a windowed Gram matrix can now be optimised over windows exactly; the reduction to V0, P0 is the reusable trick.

Bridge identity between the BBLS basis Gram and the Vasyunin Gram, and the enclosure pipeline for Baez-Duarte d_N^2

identity | hunts/rogue_frontier/nyman_beurling | grade: measured with enclosure-checked arithmetic (arb balls conditional on Vasyunin's published formula, validated five independent ways)

With e_k(t) = {1/(kt)} on L^2(0, inf) and the repo's BBLS basis A_k on L^2(0,1): <A_j, A_k>_{L^2(0,1)} = G_jk - G_j1/k - G_1k/j + G_11/(jk), where G is Vasyunin's Gram matrix; derived from e_k(t) = 1/(kt) for t > 1 and validated across all 1225 entries of the repo's digamma-based cache to 4.9e-61. Pipeline: the Vasyunin sum V(p/q) is an exact integer matrix (reduced weights 2(mp mod q) - q) times a ball vector of cot(pi m/q), using the pairing m <-> q-m and reflection V((q-p)/q) = -V(p/q); d_N^2 = 1 - b^T G^{-1} b via arb_mat.solve at 192 bits (radii ~1e-56 to N = 1536). Calibration: cond(G) grows almost exactly quadratically in N (not the folklore severe ill-conditioning); each extra solve bit buys one enclosure bit; the obstruction to pushing the criterion is the 1/log N decay, not conditioning. Exact anchor d_1^2 = 1 - (1-gamma)^2/(log 2pi - gamma).

Evidence: RESULTS.md section 1 'What was computed', lines 18-37; section 2 validation battery and bridge identity, lines 39-64; section 6 'Conditioning, measured', lines 123-138 Prior art: cited: Baez-Duarte 2003, Vasyunin's formula, BCF arXiv:1211.5191, Landreau-Richard 2002 (float values to n = 20000 as anchors) Reused in: ties to zeta.criteria._bd_gram_block and zeta.criteria.baez_duarte_table (N = 50 cache) Why it travels: The bridge identity lets any BBLS-basis result be checked against a Vasyunin-basis computation and vice versa; the exact-integer-times-ball-cot decomposition is the way to make cotangent sums enclosure-carrying at large q.

n-point certificate-to-proportion bridge with block cap

lemma | hunts/ainta_seven_point/ | grade: kernel-checked (Lean 4, sorry-free, axioms [propext, Classical.choice, Quot.sound]); conditional on hCert, which at n=7,8 is an interval-verifier acceptance, at n=3,4 discharged in Lean

For n ≥ 2, c > 0, integers m ≥ n, p > 0: if the finite inequality hCert: ∀ g ∈ ℝ^{n−1}_{≥0}, c ≤ F_n,p(g) holds (F_n,p(g) = (1/p)Σg_i + Σ_{i<j} (2/(n−(j−i))) w(y_j−y_i), w = (K/K(0))², K(x)=∫_{−1/2}^{1/2} cos(√2 t)cos(2πxt)dt) and the side condition A₀ = c(m−(n−1)) ≤ 1 holds, then liminf N₀^s(T,2T)/N(T,2T) ≥ Φ_n(c,m,p) = (H − (n−1)(m−1)/(pm)) / (1 − c(m−(n−1))/m), H = 3/2 − (1/√2)cot(1/√2). The paper's numerals 6 and m−6 are the per-gap charge n−1 and window count m−(n−1); m is capped at (n−1)+⌊1/c⌋ (the min{1,·} in the block-defect lemma), so the bound is sawtoothed, not monotone, in c. One genuine change was needed for general n: the pre_solve tolerance η = min(εm/(m+3+(n−1)(m−1)), A₀) instead of the paper's min(ε/10, A₀).

Evidence: BRIDGE.md §1 'The n-point theorem, and the eight-point instance' (lines 85–147); TRUST-MAP.md §1.1–1.3 (lines 61–168); Lean: lean/bridge/Zeta23Ext/Bridge/Main.lean n_point_bound Prior art: cited: Ainta paper/riemann.tex (n=7 only, 1/500 hardcoded); generalisation to n and the derivation of the cap are the lab's Reused in: hunts/family_wall (uses Φ_n and the cap as its starting point); THREE-POINT.md and FOUR-POINT.md (n=3,4 instances, unconditional bounds Φ₃, Φ₄) Why it travels: Any finite-window pressure certificate for simple zeros plugs into one parametric theorem; only (n, c, m, p) change.

Sharp stability rank–trace inequality with no hypothesis on V

lemma | hunts/ainta_seven_point/ | grade: kernel-checked (eleven declarations, [propext, Classical.choice, Quot.sound], zero sorry)

For V : Matrix n r 𝕜 and Hermitian Q with positive index ≤ b: 2·tr(VVᴴ) + 4·tr Q − r − 4b + tr Ψ(VᴴV) ≤ ‖VVᴴ + Q‖_F², with Ψ(t) = (t−1)² for t ≤ 2 and 2t−3 otherwise (stable_rank_trace_sharp). Ainta's form 4·tr(VVᴴ+Q) − 3r − 4b + tr Ψ(VᴴV) ≤ ‖VVᴴ+Q‖_F² (which assumes column norms ≤ 1) follows. Identification: Ψ = gc 2 + 1 of anthropics/zeta-23-lean, so the paper's 'new' lemma S2 is an instance of the upstream kernel-checked rank_trace_mult at c = 2 evaluated in the eigenbasis of P = VVᴴ; the column-norm hypothesis hV is decoration.

Evidence: ARISTOTLE-PROBE.md 'Verdict' and §1 'What was proved' (lines 7–75), §6 'What hV is actually for'; Lean: Zeta23Ext/StableRankTrace.lean Prior art: searched-and-found: Zeta23.ZeroSide.RankTraceMult.rank_trace_mult (zeta-23-lean) is the same theorem at c=2; TRUST-MAP had filed it as 'a different one' Reused in: BRIDGE.md skeleton (D(M) = rtrace (specMap hM Psi) with Psi = gc 2 + 1) Why it travels: A defect-carrying rank–trace inequality valid for every Hermitian Q and every V, usable wherever a Gram/projector energy is compared to a trace.

Pinching inequality via row-stochastic spectral mixture (no Peierls, no unitary invariance)

lemma | hunts/ainta_seven_point/ | grade: kernel-checked (part of the sorry-free bridge)

For Hermitian M = U diag(λ) Uᴴ and an injective index map g selecting a principal submatrix with spectrum μ: the matrix Y = Vᴴ(U restricted to rows g) has orthonormal rows and diag(μ) = Y diag(λ) Yᴴ, so each μ_j is a convex combination of the λ_i with weights summing to exactly 1 per eigen-direction (eigenvalues_submatrix_eq_mix). Scalar Jensen then gives Σ_B tr Ψ(G_B) ≤ tr Ψ(M) for concave Ψ with NO positivity of Ψ needed; positivity of Ψ enters only in the separate step D(M) ≥ D(M°). This replaces the paper's one-sentence appeal to 'pinching is an average of unitary conjugations and X ↦ tr Ψ(X) is convex and unitarily invariant', both halves of which are absent from Mathlib and heavier than needed.

Evidence: BRIDGE.md §6 'Pinching (S14)' (lines 294–311); bridge/ARISTOTLE-pinching.md Prior art: searched-and-found partial: upstream sum_gc_diag_le_sum_gc_eigenvalues (RankTraceMult.lean:119) is the 1×1-blocks fibre at f = gc c; the general block form is the lab's Reused in: Bridge step S14 for every n Why it travels: Block-diagonal pinching of trace-concave functionals appears in every operator-counting argument in this programme; this is the cheapest formal route.

Interior optimum in the pressure denominator and the affine-infimum concavity argument

lemma | hunts/ainta_seven_point/ | grade: VERIFIED by inspection (the four structural facts); the location of the optimum is INFERRED/float

Split F_n^{(p)}(g) = (1/p)S(g) + W(g), S = Σg_i, W the w-sum, and c(p) = inf_{g≥0} F^{(p)}. As a function of u = 1/p, c is an infimum of affine functions of u, hence concave and nondecreasing in 1/p; c(p) → W(0) = Σ_s (2/(n−s))(n−s) as p → 0⁺ (=12 at n=7) and c(p) → 0 as p → ∞. Feeding into Φ: at large p the gain c(m−(n−1))/m → 0 and Φ → H; at small p m_max = n−1 kills the gain while the penalty diverges, so Φ has an interior maximum in p and the paper's p = 3000 is a tuning constant, nearly optimal (peak at 3400 for n=7). Approximation Φ*(p) ≈ (H − (n−1)/p)/(1 − c(p)), accurate to ~1e-5.

Evidence: TRUST-MAP.md §1.5 'The third leg' (lines 228–314); RESULTS.md §3 'The peak is at p = 3400' (lines 154–165) Prior art: none; nothing in Ainta derives 3000 Reused in: hunts/family_wall §1.2–1.3 (closed-form monotonicity within a minimiser family and the crossover formula) Why it travels: Any bound with a linear-pressure certificate has the same trade-off; the concavity argument fixes the shape without a search.

Witness-leg barrier for the n-point pressure family, with the case split that makes it a proof

lemma | hunts/family_wall/ | grade: DERIVED (audit's argument re-derived line by line), VERIFIED numerically; nothing machine-checked

Let k = n−1 ≥ 2, c any uniform floor for F_k on nonnegative gap vectors, m at the cap q0

H, then Φ_n ≤ H(1+W(g)) for any g since W ≥ 0. If Φ_n > H, then Φ_n − H = [Hcd − q0(m−1)/p]/D > 0 forces pcH > 1 + c(q0−1), which is exactly the sign condition for Φ to increase in m, so moving to m_max is legitimate, N > 0 there so replacing D by m−1 raises the quotient (step A), and m_max − 1 ≥ 1/c (step B). Then Φ_n ≤ Hm/(m−1) − k/p ≤ H + Hc/(1+(k−2)c) − k/p ≤ H + Hc − k/p ≤ H + HW(g) + (HS(g) − k)/p for every g ≥ 0 (since c ≤ F(g,p)). Witness leg: any g with S(g) = Σg ≤ (n−1)/H makes the pressure term nonpositive, so Φ_n ≤ H(1+W(g)) for every p; reaching the ceiling would need W ≥ 0.0138706 for every admissible witness. Trivial leg: Φ_n ≤ H(n−1)/(n−2). Sharp form with finitely many witnesses: Φ_n ≤ H + max_p min_j [Hc_j(p)/(1+(k−2)c_j(p)) − k/p], c_j(p) = W_j + S_j/p. The one-line chain as first written was invalid at two steps (denominator replacement with negative numerator; evaluating at the cap without monotonicity), exhibited by admissible counterexamples at n=3, c=0.01.

Evidence: FAMILY-LIMIT.md §2.1–2.2 (lines 189–314), §2.1a (AUDIT, lines 216–273); chain_repair_check.py (6,475 increment-sign checks, 5,729 admissible triples, zero violations) Prior art: unsearched; novelty not declared Reused in: RESULTS.md verdict sup_n Φ_n ≤ 0.675142509660254; the lab's reading of the ainta ceiling (RESULTS.md 'Where the family runs out') Why it travels: Converts a pressure-certificate bound into a pure energy statement at fixed density; the case-split pattern applies whenever a Möbius-in-m bound is evaluated at a cap.

Integer trade with window decomposition: the k=1 retention inequality with no separation hypothesis

lemma | hunts/frontier_math/ | grade: hardened (four independent instruments agree plus an exact-rational certificate and a from-definitions reproduction); obligations O1, O2, O5–O8 kernel-checked, O3, O4, O9, O10, O11 not; O9's 699-cell table decides at kernel grade but its soundness seam lemmas are unwritten, so the composite is not kernel-checked

Setting (bespoke, in Zeta23Ext/EForm3/Defs.lean): g(u) = cos(√2 u) on |u| ≤ 1/2, A = ∫g, c_2 = g⋆g, Eng G = (1/A^2)∫_{−1}^{1} c_2 |G|^2, Qre/Qim the cosh/sinh transforms, Shq y = Qre(2y,0)^2 − A^2, Kpair(u) = Qre(0,u)^2, Dam(y,s) = max(0, Qim(y,s)^2 − Qre(y,s)^2). Claim (★★), for every n, every x, every t, every y ∈ [0,1/2]: 4Σ_j Dam(y, x_j−t) ≤ (1/200)Σ_{j≠k} Kpair(x_j−x_k) + 2 Shq(y); it implies (199/200)Eng(F) + n/200 + 4 ≤ Eng(F+P). The damage-only route (retention_of_damage) is false from n = 8 (8 × 0.00439642 > Shq(1/2)/2 = 0.03375420), so the repulsion term must be kept. Three facts make (★★) true: Kpair ≥ 0 pointwise (cross-group pairs may be dropped with no sign bookkeeping); Kpair ≥ 39/50 on |u| ≤ 1 and ≥ 1/125 on |u| ≤ 6; Dam(y,s) = 0 for |s| ≤ 28/5 and otherwise supported in nine windows I_k of length < 1 with rational endpoints, left endpoints spaced > 6.16, caps c_k = 21/20 × sup(Dam/y^2) (table in §4), far field by the proved majorant Wt with Wt(w) ≤ (637/1000)/w for w ≥ 1368. Lemma 5.1 (integer trade): for q > 0, c ≥ 0, m ∈ ℕ, 4cm − q m(m−1) ≤ P(c,q) := max over m_0 ∈ {⌊1/2 + 2c/q⌋, ⌈1/2 + 2c/q⌉} of (4c m_0 − q m_0(m_0−1)). Summing window by window (q = 39/10000, q_far = 1/25000) the deficit at y = 1/2 is 8.5555138e-2 against budget (51944/100000)/4 = 0.12986, surplus 4.4304862e-2, margin 1.5179×; n never appears. Closing argument against the obvious alternative: relaxing the n unit atoms to a measure gives inf gap = 0.114022 − n/200 exactly linear from n = 6 on and unbounded below (mass escapes as dust), so no convex relaxation of the atom constraint can close this; integrality is spent only in Lemma 5.1.

Evidence: hunts/frontier_math/RETENTION-PROBLEM.md §1–§5, §7, §8, §9; PROOF-LEDGER.md §The SINGLE-PAIR retention closed at hardened grade (four independent routes, salvage_audit 7/7); NOVELTY-CHECK-RETENTION.md; exact_gap_attack.py, repulsion_trade.py, near_coincident.py Prior art: searched-and-absent, in the weakest sense: NOVELTY-CHECK-RETENTION.md finds no occurrence of retention/Eng/Kpair/Dam in the upstream 316-file formalisation and grades the inequality novel only because nobody had asked it Reused in: K2-TWO-SPECIES.md (the three pillars transplanted to the centre species; k = 2 equal-depth closed over 6601 tau-cells); PROOF-LEDGER.md ROAD B step 3 (two-species square completion, 'at most one pair per window'); hunts/r_a97060 (interval pass); O9-SCOPING.md, O9-LEAF-REPAIR.md, LATTICE-EXTREMALITY-ROUTE.md Why it travels: The pattern 'positive pair kernel with a short-range floor + damage confined to sparse narrow windows + integer maximisation per window' bounds any atom-configuration energy uniformly in the number of atoms, where convex relaxations provably cannot.

Homogeneity closure of the shallow end (an interval with no smallest point)

lemma | hunts/frontier_math/ | grade: hardened (double precision; 18 cells tiling (0,1/2] exactly, closes at 0.995 and 0.996, fails at 0.997 on the deepest cell)

When the adversary's damage scales as the square of the depth parameter (f^+(g,y) ≤ y^2 F̂(g); in the retention setting every deficit piece is P(c_k v, q) with v = y^2), each cap piece is a maximum of finitely many affine functions of v vanishing at v = 0, hence convex through the origin with P(λv) ≤ λP(v) for λ ∈ [0,1], and the budget is linear in v (2Shq(y) ≥ (51944/100000)v; at the paper window slack(y)/y^2 ≥ 8L_2/A = 0.6199944, its y→0^+ limit via sinh t ≥ t). Then a single finite check at the deepest point of a cell covers every shallower depth, and the check at y = 1/2 gives the inequality for all y ∈ [0,1/2]. A geometric ladder of depth cells can never reach 0 and would leave (0,ε) open, the same missing quantifier one decade lower. Measured margin runs from 1.72× at y = 1/2 to 2.51× as y → 0, so the shallow end is the easy case. Extension route for two depths: every piece convex in (v_1,v_2) so the vertex argument would finish it, but the honest convex majorant at the (1/4,0) vertex is too fat by ~0.015 (named obligation).

Evidence: hunts/frontier_math/RETENTION-PROBLEM.md §6 Uniformity in y, for free; PROOF-LEDGER.md §Blocker 1 closed (single-pair layer): depth-uniform retention, row 'The shallow end (no smallest point)'; PREPRINT.md §Statement item 1; K2-TWO-SPECIES.md §3 and §4; depth_uniform.py Prior art: unsearched; elementary convexity Reused in: K2-TWO-SPECIES.md §3 (k = 2 depth uniformity for free) and §4 (unequal-depth route); k2_closure.py; PREPRINT.md Why it travels: Any depth- or amplitude-parametrised inequality whose bad term is homogeneous of degree 2 and whose budget is linear needs one check at the extreme, not a ladder toward zero.

Lattice extremality via the structure-factor identity and Newton's identities

lemma | hunts/frontier_math/ | grade: measured (double precision, no enclosure); §1–§4 a proof sketch, not formalised; hypothesis H1 (periodic, ρ ≥ 1/(2π)) explicit; lattice extremality as a whole not established

For a P-periodic centre configuration with offsets a_1..a_m and density ρ = m/P, the per-centre cost J_h(T) = (2/m)Σ'_{p,q,n} h(a_p − a_q + nP) satisfies J_h(T) = 2ρĥ(0) − 2h(0) + (2/(mP))Σ_{j≠0} ĥ(2πj/P)|A_j|^2 with A_j = Σ_p exp(2πi j a_p/P) (Poisson summation; residual < 4e-9 against direct summation on five configurations). If ĥ ≤ 0 is supported in [−1,1] and vanishes at ±1 (here h = −κ, κ̂(ξ) = 2π c_2(|ξ|)(1 + cosh|ξ|), and c_2 = g⋆g > 0 on (−1,1) because g > 0 on its support, c_2(±1) = 0 because the supports meet in a point), every j ≠ 0 term is ≤ 0, so J ≤ LP(ρ) = 2ρĥ(0) − 2h(0), strictly decreasing in ρ. For ρ ≥ 1/(2π): J(T) ≤ −4c_2(0) + 2κ(0) = 0.11433003938654052…, with equality iff ρ = 1/(2π) and A_j = 0 for j = 1..m−1, which by Newton's identities (power sums vanish ⇒ elementary symmetric functions vanish) forces the polynomial z^m − c, i.e. T is a translate of the 2π lattice; the threshold is sharp because the constrained modes are exactly j = 1..m−1. Rectification gap closed by the explicit majorant v(x) = c·(sin(x/2)/(x/2))^2 with c = K_1(0) = 0.9115647 (needs c ≤ cos^2(√2/2)(1 + cosh 1) = 1.4698290, margin 0.558; ∫v = 2πv(0) keeps tightness). Sparse side ρ < 1/(2π): the bound is vacuous; the only route (a density-independent bound, forced by linearity) is infeasible at bandwidth 1 by Boas–Kac/Shannon rigidity, short by 29% (3.5076780 vs 4.9441838). A sampled-point LP for u ≥ 0 is unsound (returns 4.908534, dips negative between samples); the right tool is an SDP through the Fejér–Riesz representation u = |h|^2.

Evidence: hunts/frontier_math/LATTICE-EXTREMALITY-ROUTE.md §1 The identity, §2 Three facts about kappa_hat, §3 The bound, §4 Uniqueness, §5, §5a Gap B closed, §6a Gap A; K2-TWO-SPECIES.md §5 T1 (amended 2026-08-20); lattice_extremality.py Prior art: unsearched (Poisson summation, Newton's identities, Boas–Kac, Shannon used as standard tools; the assembly for this kernel is the lab's) Reused in: K2-TWO-SPECIES.md T1 row (closes the dense side of the centre-gas obligation) Why it travels: Template for 'is the lattice extremal' for any periodic pair energy whose kernel has a nonnegative compactly supported Fourier transform vanishing at the band edge, including the uniqueness step and the diagnosis of why sampled LPs cannot certify positivity.

Edge lemma for autocorrelations of real even factors

lemma | hunts/outband_certificate | grade: proved (prose grade, not kernel-checked); named as the natural next Lean target

Let u be real, even, in L^2, supported in [-s,s] with s the true support edge, and Khat = u*u (autocorrelation). Then Khat is not nonpositive on any interval (2s-eta, 2s). Hence a kernel K=|k|^2 with real even spectral factor either has supp Khat inside [-1,1] or fails any 'nonpositive on the out-of-band strip' condition. Proof: near the edge only the two ends of u meet, Khat(2s-c) = (W*W)(c) with W the edge profile; Titchmarsh's convolution theorem puts 0 in supp(W*W); Laplace transforms at large real lambda give (LW)^2 = L(W*W), a square equal to a negative quantity, contradiction.

Evidence: RESULTS.md section 5 'Its death, with proof', lines 154-173 Prior art: cited: Titchmarsh convolution theorem (used as published); the lemma itself unsearched Reused in: none recorded Why it travels: Kills any signed-spectral-profile / difference-of-squares certificate with real even factor before any price is paid; applies to every inertia-type argument on band-limited kernels.

Second-order small-depth bound from evenness of ĝ, closing the u → 0 corner

lemma | hunts/r_a97060/ | grade: enclosure-carrying (python-flint balls at 96 bits, cross-checked with mpmath.iv)

For D(u,τ) = −Re ĝ(u/2 + iτ)² with ĝ even with real Taylor coefficients, ĝ'(iτ) is purely imaginary, so ∂u D(u,τ)|{u=0} = −2G(τ)Re ĝ'(iτ) = 0 identically (G(τ) = ĝ(iτ)): D has no depth-linear term. Hence for every u ≤ b, D(u,τ) ≤ −G(τ)² + (u²/2)sup_{[0,b]}|D''|, and the ratio bound 4max(0,D)/u² ≤ 4max(0, sup|D''|/2 − G²/b²). Taking the smaller of this and the direct box bound closes the sup over the open depth interval (0, 1/2] at the resonance cell τ ∈ [6.62,6.64] (a root of G) where a plain branch-and-bound in u stalls at 0.1142 against a true 0.0673, bringing C1 ≤ 0.0703 and the table to 0 nonpositive cells. No sampling grid starting at y = 0.05 can check y < 0.05, which is where the 0/0 lives; the enclosure pass had to prove it.

Evidence: RESULTS.md §2 'The one thing that genuinely resisted, and what fixed it' (lines 63–102); probe.py lines 255–287 (second_order); Loose thread 4 Prior art: unsearched; hunt notes evenness of ĝ was used here for the first time in the tree Reused in: suggested for the unequal-depth majorant of K2-TWO-SPECIES.md §4 (not done) Why it travels: Any ratio sup D(u)/u² with an even generating function has a vanishing linear term; the second-order bound converts an unboundable open-interval corner into one enclosed inequality.

Kernel-checked count of pairings (2m-1)!! with lesion tests and a proper-subset oracle

lemma | hunts/rogue_frontier/matchings | grade: kernel-checked per the coordinator's recorded recompilation (EXIT=0, standard axioms); the landing did not recompile, so a fresh lake build is owed before the grade is cited outside the hunt

In Lean 4 / Mathlib (v4.33.0-rc2, rev 51e6992e), for pairings of a Finset s represented as fixed-point-free involutions extended by the identity outside s (PairsUp), card(pairings s) = (|s|-1)!! for even |s| and 0 for odd |s|; also card_pairings_two_mul and 2^m m! card = (2m)!. Proof by Nat.twoStepInduction with a card_nbij' bijection dropPair/addPair; 430 lines, 0 sorry, axioms [propext, Classical.choice, Quot.sound], no native_decide. Controls: the oracle (three routes: brute force, partner recursion, (2m)!/(2^m m!)) run before writing Lean; a kernel decide check on a proper subset {0,1,2,3} of Fin 5 to exercise the extension-by-identity clause that the univ case leaves vacuous; two lesions (wrong double factorial; deleted extension clause) each compiled to EXIT=1. Not bridged to SimpleGraph.Subgraph.IsPerfectMatching (estimated 80-150 lines).

Evidence: NOTE.md; LOG.md entries 7-10 (lines 169-273); Matchings.lean; LANDING.md verification paragraph Prior art: searched-and-absent in Mathlib at the pin (compiled run_cmd over getEnv sweeps: no Isserlis/Wick, no perfect-matching counting API) Reused in: closes the blocker named at step A2c of hunts/r_8c3b94 (Erdos-Kac in Lean); same count enumerated by sine_gram's moment engine Why it travels: Library content usable by any Wick/Isserlis or Erdos-Kac formalisation; the lesion-plus-proper-subset control pattern is the template for testing that a Lean statement is load-bearing.

Global window optimum for xi' via the coercive operator A = I + T_{F_1}

lemma | hunts/wide_search/ | grade: ordinary derivation, self-reviewed (the theorem); the constant is measured (five routes, three agreeing to 14 digits; 'converged high-precision numerical evaluation, not an interval enclosure')

Let I = [-1/2, 1/2], H = L^2_even(I), F_1(x) = |x| - 4x^2 + sum_{k>=1} ((k-1)!/(2k)!) (2|x|)^{2k+1} for |x| <= 1 (Farmer-Gonek-Lee form factor minus its spike), (T v)(s) = int_I F_1(s-t) v(t) dt, A = I + T. Then: (1) ||T||_{2->2} <= 4/9; (2) <Av, v> >= (5/9)||v||^2, so A^{-1} exists with norm <= 9/5; (3) w = A^{-1} 1 has a continuous even representative with w >= 1/5 on I; (4) sup over 0 != v >= 0 of (int v)^2 / (int v^2 + iint F_1(s-t) v v) = <1, A^{-1} 1> = c*, with equality exactly on the rays v = c w; (5) the source functional's optimum c_lambda* is nondecreasing in lambda on (0, 1], so the bandwidth-one optimum sits at lambda = 1. Numerically c* = 0.8838931253605797508..., H* = 2 - 1/c* = 0.8686415005297670641..., Hd* = 0.9343207502648835320...; the source paper's quartic is 1.5005e-6 below sharp and no admissible window reaches Wu's unconditional 0.86957 (short by 9.285e-4).

Evidence: RESULTS-xiprime-global-optimality.md sections 'Theorem' (lines 12-84), 'Proof', 'Numerical status'; RESULTS-xiprime.md sections 'The answer', 'The functional', 'How it was checked'; xiprime.py Prior art: searched-and-absent for the unconditional xi' window optimisation (confidence 0.85 stated); Farmer-Gonek-Lee Thm 1.1 and Chirre-Goncalves-de Laat (RH-conditional SDP) cited; Wu 2015 read at primary source Reused in: hunts/frontier_map/RESULTS-frontier-map.md; hunts/frontier_math/RESULTS-frontier-math.md; hunts/outband_certificate/RESULTS.md; hunts/wide_search/HANDOFF.md Why it travels: The operator-coercivity route (norm bound on the kernel operator, coercive A, optimum = <1, A^{-1} 1>, positivity of A^{-1} 1, monotonicity in the scale) solves the window problem for any pair-correlation kernel with a bounded convolution operator, e.g. higher derivatives once F_k is available.

Out-of-band envelope: frequencies at or above 2L are free in the window reduction

lemma | hunts/oob_envelope | grade: ordinary derivation, independent referee PASS (different model family); not kernel-checked

For supp f in [-L, L], |F|^2 is the Fourier transform of an autocorrelation supported in [-2L, 2L], so any bounded almost-periodic H whose frequencies all satisfy |lambda| >= 2L (boundary included) has integral of |F|^2 H equal to 0, for every complex f. The Weil symbol may therefore be replaced by Psi_L + H without changing Q on the window, and Zhu's one-stroke reduction (arXiv:2608.24827, Theorem 1.1) runs with S = sup(P_L - H) in place of A_L = sup P_L. The best such S is exactly lambda_max of the windowed comb operator (strong duality), reached at rate N^{-2} by explicit trigonometric H, and it grows like e^L against Zhu's 4 e^L: the threshold stays doubly exponential, but the matrix size needed at a fixed window drops by one to two orders.

Evidence: hunts/oob_envelope/theory/RESULTS.md sections 1 to 3 (Lemma 1, Theorem 1', Theorem 2, Theorem 3); referee/REVIEW.md sections 2 to 4 Prior art: searched-and-found: Burnol 2000 (math/0101068, Theoreme 3.7) uses one boundary cosine for the same purpose; Liu 2026 Theorem B is the operator form. The systematic prime-comb use, the duality and the e^L constant were not found in the searched scope (theory/RESULTS.md section 4.2). Reused in: hunts/oob_envelope/numerics (support 1.6 and 2.38 window bounds) Why it travels: Any positivity argument on a band-limited class may spend the symbol outside the band for free; it turns a pointwise envelope question into an operator one.

LAW E/F/K: depth envelope for signed on/off incidence and the exact pair spectrum

bound | hunts/frontier_math/ | grade: hardened (random-subgrid envelope floors never violated and nearly attained, worst margin +3.3e-4; LAW K eigenvalues to 8 digits at three depths, x·y ~ 1e-16); derivations ordinary, self-reviewed

Define σ^2(y) by Σ_k (Im φ̂(t−iy−τ_k))^2 = L∫φ^2(u)sinh^2(yu)du = aL^2 σ^2(y), independent of t (the imaginary mass is band-limited to [−L/2, L/2] with density φ(u)sinh(yu), so it is alias-free). Since every omitted term of LAW D's real part is (Re φ̂)^2 ≥ 0, for every subgrid S (any truncation, any placement) and real x: Re Σ_{k∈S} φ̂(z−τ_k)^2 ∈ [−aL^2σ^2(y), aL^2(1+σ^2(y))] and |Im Σ_{k∈S} φ̂(x−τ_k)φ̂(z−τ_k)| ≤ aL^2σ(y) (Cauchy–Schwarz against the two alias-free masses), hence every normalised on/off cross cell obeys 2Re(B̂(x,z)^2) ≥ −2σ^2(y), with σ(0) = 0: negative incidence mass requires depth. LAW F: by convexity of sinh, σ^2(y) ≤ sinh^2(yL/2), and since 0 < β < 1 unconditionally, σ(y) < sinh(L/4) for every zero at bandwidth L; a pair whose cross column reaches |Im B̂| = m needs depth y ≥ y_min(m) := inf{y : σ^2(y) ≥ m^2} and is impossible once m ≥ σ(1/2^−) (at L = 8, σ(1/2^−) = 1.315, so every m ≥ 2 dies). LAW I sharpening: W(g,y) ≥ −σ^2(y)(1−ω(2g)) ≥ −(1+m_0)σ^2(y) with m_0 = −min ω = 0.2137172540 for this window. LAW K: for a pair u = x+iy at depth y, LAW D's u·u = 1 (real) forces x ⊥ y with |x|^2 = 1+σ^2, |y|^2 = σ^2, so spec(2(xx^T − yy^T)) = {2(1+σ^2), −2σ^2} exactly, and against the flat charge 4 the pair retains slack exactly 8σ^2 + 8σ^4; consequence (three-zero lemma): a pair with at most three on-line zeros in its negative cells has net ≥ (8 − 6(1+m_0))σ^2 = 0.7177σ^2 > 0, placement- and depth-free.

Evidence: hunts/frontier_math/SIGNED-INCIDENCE-LAW.md §The three laws (LAW E, LAW F), §The exclusion, wall by wall, §Controls ledger; LEVEL2-TWO-GAP-MARKED.md §LAW I: the one-cell tightening; LEVEL3-THETA-RECOVERY.md §LAW K: the exact pair spectrum, §The three-zero lemma Prior art: cited: the paper knew the (1,1) signature of the pair block and Remark 5.10's Hermitian mass; the envelope, the depth floor and cap, the eigenvalues pinned by depth and the three-zero lemma are the lab's; no literature search recorded Reused in: LEVEL2-GAP-CONSISTENCY.md; LEVEL3-THETA-RECOVERY.md; LEVEL4-COUNTING-DUAL.md (cell sups use σ, E); LEVEL5-ENCLOSURE-AND-PAIRS.md (depth split σ^2(y±y')); K2-TWO-SPECIES.md Why it travels: Gives explicit, depth-graded lower bounds for the sign-indefinite part of any windowed Gram matrix with complex (off-line) evaluation points, using only alias-free masses and Cauchy–Schwarz.

LAW G/H: a correlation is a gap, and n-independent anti-duplication caps

bound | hunts/frontier_math/ | grade: hardened (float mirror vs mpmath closed form 4.5e-7; majorant, cap, decoy planted 4x too small fires 11/12, collapse lesion, n-independence and duplication-cheat controls); derivation ordinary, self-reviewed

On the full grid the normalised on-line correlation of two zeros at gap g is exactly ω(g) = Φ2(g)/Φ2(0), with ω(0) = 1 and |ω(g)| < 1 for g ≠ 0 (φ^2 ≥ 0 continuous with interval support), so a declared correlation ≥ c caps the gap (at L = 8: c = 0.99 buys g ≤ 0.0719, c = 0.5 buys g ≤ 0.5557) and a marked two-gap word's outer correlation must be ω(g_1+g_2), not a free value (the independence guess ω(g_1)ω(g_2) is refuted with residual 0.13228305 at (0.7, 1.3)). With the paper's majorant |Φ2(r)| ≤ ψ(r) = min(L, 2/|r|, c_ρ/(w r^2)) (c_ρ = 4‖ρ'‖_∞ + 4‖ρ''‖_1 = 105/4 exactly for the septic ramp) and local density ν (ν ≤ A_0 log T by Titchmarsh Thm 9.2), the incidence mass of one point against any configuration is bounded independently of the configuration's size: Σ_{j≠i} ω(x_i−x_j)^2 ≤ κ(ν) = 2ν[1 + Σ_{k≥1}(ψ(k)/aL)^2] and Σ_j |B̂(x_j,z)|^2 ≤ κ_cross(ν,y) = 2ν[E(y)^2 + Σ_{k≥1}(ψ_y(k)/aL)^2] with E(y) = Φ2(iy)/Φ2(0) and ψ_y(r) = min(aL E(y), (c_ρ/w)cosh(yL/2)/r^2). Hence Σ_j 2Re(B̂(x_j,z)^2) ≥ −2κ_cross(ν,y), an n-independent floor where level 1 had −2nσ^2(y) (crossover n > κ_cross/σ^2 = 72 at L = 8, y = 0.3), and R(P) ≤ nκ(ν) for any real configuration, so n − 1 ≤ κ(ν) = O(log T): the obstruction family's realisable size at height T is explicitly finite.

Evidence: hunts/frontier_math/LEVEL2-GAP-CONSISTENCY.md §Pinned inputs, §The two laws, §The kill control, run, §The robust exclusion, §Controls ledger; gap_consistency.py; test_gap_consistency.py Prior art: cited: the paper's (2.17) majorant and Titchmarsh Theorem 9.2 as inputs; the laws and the assembly are the lab's Reused in: LEVEL3-THETA-RECOVERY.md (dense packing self-defeat); LEVEL4-COUNTING-DUAL.md; LEVEL7-VCELL.md; SIGNED-INCIDENCE-LAW.md addendum (removes n-extensivity) Why it travels: Converts per-cell floors that scale with the number of points into a per-point cap using only kernel decay plus a local density bound; applies to any decaying-kernel Gram sum over a configuration of bounded local density.

Outer bound sup F <= 1 - (inf D2)^2 via the central-moment identity and a Neumann-series rational bound on inf D2

bound | hunts/rogue_frontier/window_opt | grade: derived (exact rational chain end to end on the hardened closed forms); slice sup derived in exact arithmetic; landscape statements measured

F = 2 m2 - m3 = 1 - int lam (lam-1)^2 dmu_v(lam) where mu_v is the limiting spectral measure of A = H/l1, a positive measure on [0, inf) since H is Gram. Hence F <= 1 for every admissible window, F <= 1 - D2^2 pointwise (Cauchy-Schwarz m2^2 <= m1 m3), and sup F <= 1 - (inf D2)^2. D2 = <v,(I-T)v>/l1^2 with T the tri-kernel operator (Fourier multiplier sinc^2, tr T^2 = 1/2, ||T|| <= 1/sqrt2), and for int v = 1, <v,(I-T)v> >= 2c

with tail t_44/(1 - 707107/1000000) give U >= <1,(I-T)^{-1}1>, inf D2 >= 1/U = 0.327499295198, sup F <= 0.892744211644411 (exact rational). Sharp for its inputs: the two-point measure (d/(1+d)) delta_0 + (1/(1+d)) delta_{1+d} attains 1 - d^2. Companion: the quartic slice sup is settled exactly by compactification (interior critical points via Groebner eliminant, endpoint-zero and double-root boundary families, arc at infinity), giving 0.685287032176998 <= sup F <= 0.892744211644412.

Evidence: RESULTS.md section 9.1 'The outer bound', lines 335-399; section 9.3 'Exact slice suprema', lines 452-512; global_bound.py, global_slice.py Prior art: unsearched; hunt notes closing the gap needs a time-band-limiting-style exclusion of near-two-point spectral measures Reused in: upgrades RF-C003's caveat from 'local search outcome' to 'unique critical point, globally capped by a derived bound' Why it travels: The moment-relaxation ceiling (positive spectral measure, Cauchy-Schwarz, exact Neumann tail) applies to any 'optimise a polynomial in spectral moments over windows' problem.

Lean cell-table architecture for certifying a kernel functional inequality

construction | hunts/ainta_seven_point/ | grade: kernel-checked (three_point_cert, three_point_bound sorry-free, standard axioms; GitHub Actions run 32689888754 green); four-point likewise per FOUR-POINT.md §5 axiom audit

Prove ∀ g ≥ 0, c ≤ F_n,p(g) in Lean without an interval tactic: (1) Kfun_eq_sinc: K(x) = (sinc((√2−2πx)/2) + sinc((√2+2πx)/2))/2, total, no singularity case split; (2) cos_sin_taylor12: for |θ| ≤ 1, twelve-term Taylor enclosures of cos and sin with error |θ|^{12}·13/5748019200 (from Complex.exp_bound at n=12), wrapped as four monotone one-sided bounds; (3) kfun_closed: k(x) = (cos b − 2γ b sin b)/(1 − 2b²), b = πx, γ = (1/√2)cot(1/√2) = 3/2 − H, so one transcendental constant is enclosed once (gam_bounds width 1.11e-8); (4) one general lemma wfun_ge (x nlo dhi …): (nlo/dhi)² ≤ w(x), instantiated by generated cell lemmas wc_k with rational literals, angles reduced exactly to half-integer anchors so θ ≤ π/4; (5) window [0,1/2] done from the sinc form (sinc_taylor valid at 0), w ≥ 19/100 across the removable singularity; (6) pressure cutoff Σg ≥ c·p closes everything outside a simplex by w ≥ 0 and linarith; (7) 1-D cover lemma exporting only the near-zero intervals of the kernel, then bisection trees over products of those intervals with leaf = sum of cell constants + linarith; (8) all rationals rounded outward to fixed denominators, generator in fractions.Fraction, never float. At n=3: 368 cell lemmas, 487 leaves, ~65 min cold CI.

Evidence: THREE-POINT.md §3 (lines 176–296); CERTIFICATE-ROUTE.md §4.1–4.4 (lines 223–306); FOUR-POINT.md §3 (lines 229–350); three_point_gen.py, four_point_gen.py Prior art: searched-and-absent in pinned Mathlib: no numerical sin/cos evaluator, no interval tactic (Real.cos_bound error 5/96·|x|⁴ is far too weak); cos_sin_taylor12 and sinc_taylor are the lab's Reused in: FOUR-POINT (imports the cell machinery from three_point_gen.py); lean/bridge ThreePoint and FourPoint libraries; Palomar surface Why it travels: Any inequality over a bounded region of a trigonometric/sinc kernel functional can be kernel-checked in Lean by the same generator; the primitives (Taylor-12 enclosure, anchor reduction, sinc window) are problem-independent.

Period-37 Sturmian witness word with closed-form length margin and uniform tail estimate

construction | hunts/family_wall/ | grade: AUDIT (mpmath.iv 100 digits, unsafe direction) and VERIFIED here (famlib float and 60-digit integer-window sum agree to 1.3e-17); not machine-checked

g_i = 1 + ⌊18i/37⌋ − ⌊18(i−1)/37⌋ (nineteen 1s and eighteen 2s per period). Its prefix of length k has S_k = k + ⌊18k/37⌋ ≤ (55/37)k, and 55/37 = 1.48648… < 1/H = 1.48698…, so S ≤ k/H holds for every k in closed form with no per-n check. Tail estimate for n ≥ 12: at least 5/12 of prefix gaps are 2s (min over 11 ≤ k ≤ 20000 of ⌊18k/37⌋/k is exactly 5/12 at k=12); every scale-s window has integer length ≥ s and w decreases at integers, w(s) ≤ w(1)/s⁴ since 2π²s² − 1 ≥ s²(2π²−1); hence W ≤ 2[(7/12)w(1) + (5/12)w(2)] + 2w(1)(π⁴/90 − 1) < 0.003928331920529310 and H(1+W) < 0.675142509660253902. At integer j the kernel collapses exactly to k(j) = (−1)^{j+1}/(2π²j² − 1), so W is a finite sum of exactly representable terms. Small n (3–7, 11) closed with separately polished interval-checked witnesses. Replaced a 367-value scan plus bolted-on trivial tail with one word.

Evidence: FAMILY-LIMIT.md §2.3a (lines 357–437), §2.4 table; RESULTS.md §2; period37_check.py; audit/periodic_certificate.py (directed intervals) Prior art: unsearched Reused in: RESULTS.md headline bound Why it travels: A balanced word whose density beats 1/H by a rational margin gives an all-n witness for any density-constrained energy bound; the integer-argument kernel collapse makes W exactly computable.

Chain counting dual: configuration-free cap by cell partition, exact integer DP, and one-sided Lipschitz cell sups

construction | hunts/frontier_math/ | grade: hardened (level-5 ball-arithmetic pass at 128 bits, radii 1e-16 to 1e-20, secures θ* = 0.1 at the hunt window; the MT band dual hardened in balls with no Lipschitz margin at θ* = 0.9988); the counting theorem itself is elementary, ordinary derivation

Partition the offset line into cells of width δ. Same-cell pairs are within δ so each pays internal mass at least K_δ = min_{[0,δ]} ω^2; adjacent-cell pairs pay at least K_{2δ}; non-adjacent payments are dropped (one-sided in the adversary's favour). With F_j ≥ sup over cell j of the damage's positive part and c = 1−θ, the adversary's value is at most max over n_j ∈ ℕ of Σ_j [n_j F_j − c K_δ n_j(n_j−1) − c K_{2δ} n_j n_{j+1}], a tridiagonal chain program solved exactly by dynamic programming (pentadiagonal refinement adds K_3 = min_{[δ,3δ]} ω^2 with 3δ ≤ 0.9; with K_3 = 0 it reduces to the chain within 1e-6). The chain term is load-bearing: without it the free density is 1/δ and the scan fails by an order of magnitude (19.2 vs the plain bound at y = 0.45); at θ = 1 every charge vanishes and the cap is infinite, matching the measured dense-regime failure. Cell sups come from closed-form centre values of C = Re Φ2(g+iy), S = −Im Φ2(g+iy) inflated by explicit constants |dS/dg| ≤ (L/2)aLσ(y), |dC/dg| ≤ (L/2)aL E(y), |dS/dy| ≤ (L/2)aL E(y), |dC/dy| ≤ (L/2)aLσ(y), so f^+ ≤ 4(S̄^2 − C_und^2)^+/(aL)^2 on the whole cell; the far tail uses the depth-scaled majorant |S| ≤ ψ_S(y)/g^2 with ψ_S = (c_ρ/w)sinh(yL/2) + 4y cosh(yL/2) + y^2 aL sinh(yL/2), which vanishes linearly in y so shallow cells (where slack also vanishes) are not silently failed. The sup grid must be decoupled from the capacity partition δ; the Lipschitz inflation scales with the sup-grid step, so coarsening the grid changes the theorem's constants (a 2x coarser hardened run failed by −3.0 while the original resolution holds with +0.27 to +2.03 margins). Projection control: the bound must dominate every lower-level measured adversary, which caught a factor-2 error in the damage. Band-partition variant (MT window): partition by the damage field's own negative bands rather than a ruler, with the off-band allowance exactly zero (an unresolved band would need an interior dip excluded whenever q > (1/8)|q''|step^2), cap(θ) = 2Σ_k max_m[mF_k − (1−θ)m(m−1)K_k] + closed-form 1/g^2 tail.

Evidence: hunts/frontier_math/LEVEL4-COUNTING-DUAL.md §The theorem, §One-sided cells, §The soundness incident, §Measured record; LEVEL5-ENCLOSURE-AND-PAIRS.md §Gate 1: the hardened scan, and what almost went wrong; LEVEL6A-THETA-FULL.md §Lever 1: the pentadiagonal counting dual; TRANSPLANT-LEMMA.md §The dual; counting_bound.py, enclosure_pass.py Prior art: unsearched; elementary counting on top of the lab's LAW E/G/H Reused in: LEVEL5, LEVEL6A, LEVEL6B, LEVEL7-VCELL.md (direct route: the same DP on the joint damage field); TRANSPLANT-LEMMA.md band dual (θ* = 0.995 at the paper window); PREPRINT.md; K2-TWO-SPECIES.md and k2_closure.py (tau-table); hunts/r_a97060 (interval pass over the k=2 table) Why it travels: Any problem of the form 'an adversary places atoms to maximise damage minus a quadratic internal charge' reduces to a small integer DP once the kernel has a positive short-range floor, and the one-sided cell-sup recipe makes the result configuration-free.

Odd-factor kernel: pointwise nonnegative, compactly supported transform nonpositive outside the band

construction | hunts/outband_certificate | grade: proved and verified on a grid of step 1e-3 (min K = 0, max Khat on 1<|alpha|<3.05 = 1.5e-6 FFT noise, min -0.5)

k(x) = sqrt2 sinc(x) sin(2 pi x), real and odd, gives K = |k|^2 = sinc^2(x)(1 - cos 4 pi x) >= 0 with Khat = tri(alpha) - tri(alpha-2)/2 - tri(alpha+2)/2, supported in [-3,3] and nonpositive on |alpha| > 1. With 1 - c cos in place of 1 - cos the same holds with K(0) = 1-c > 0. Together with the edge lemma this is a dichotomy: even factor is Gram-able but strip-blind; odd/complex factor is strip-capable but never the kernel of a Gram matrix (antisymmetric under difference of ordinates).

Evidence: RESULTS.md section 5 'And the lemma is false for the class one needs' and 'The dichotomy', lines 175-193 Prior art: cited: Krein factorisation (as published); the explicit odd witness unsearched Reused in: none recorded Why it travels: An explicit witness for the class of kernels that can spend out-of-band positivity; usable as a test object for any future certificate class claim.

Davenport-Heilbronn port of the truncated Weil form (structure-matched rival control) with attribution by dictionary decomposition

construction | hunts/rogue_frontier/weil_trunc | grade: hardened for every sign statement and the transition curve (ball LDL^T, Rayleigh, acb_mat.eig agreeing); measured for localisation, dictionary decomposition, and the mechanism reading

The CvS/CCM truncated Weil form Q = W02 - WR - Wp on the Fourier basis of L^2([0, L]), L = log c, ports to the DH function with three changes dictated by its explicit formula: (i) coefficient measure Lambda_f(n) = a_n log n - sum_{d | n, 1<d<n} Lambda_f(d) a_{n/d} on all n >= 2 (no Euler product), band-limited to n <= c; (ii) no pole block (f entire); (iii) archimedean kernel from Re psi_Gamma(3/4 + ir/2) - log(pi/5), x-space rho_1(x) = e^{-3x/2}/(1 - e^{-2x}). The ported form is positive at every cell with N <= 32, c <= 47 (enclosure-checked), so 'positive with ground state locating zeros' distinguishes nothing; the first negative cell on the integer lattice is (c*, N*) = (31, 60), even sector, lam_min = -1.874e-31 (three rigorous routes), while zeta at the same cell is +4.8e-100. Attribution: the ported G2 Thm 2.5 dictionary sum has exactly one negative entry, the off-line quadruple 4 Re g_v(gamma_off - i delta) = -6.7e-29 (359x the eigenvalue); crossing happens once the band edge 2 pi N / L reaches the off-line ordinate 85.7 with amplification e^{delta L} = c^{0.3085}.

Evidence: RESULTS.md section 4 'The DH control', lines 137-192; section 8.1-8.3, lines 255-366; SOURCE.md section 4 'Davenport-Heilbronn portability' Prior art: cited: Connes-van Suijlekom arXiv:2511.23257 Prop 4.1, CCM arXiv:2511.22755, Groskin arXiv:2605.20224 / 2607.02828 (chi_3 port corroborates the shift), Connes arXiv:2602.04022; the DH port and failure height searched-and-absent in those sources Reused in: zeta.epstein.battery discipline (docs/09 gate #3); thread raised for an issue on the N -> infinity limit at fixed c <= 30 Why it travels: A recipe for running any explicit-formula compression on the RH-violating rival, and a way to attribute a negative eigenvalue to a specific off-line zero rather than to 'the form went negative'.

Source-admissible closure certificate: exact rational strict-concavity plus C^3 endpoint tapers

construction | hunts/wide_search/ | grade: exact rational bounds (fractions.Fraction and SymPy), self-reviewed, pinned by tests and two lesions; not kernel-checked

To show an L^2 optimiser w = A^{-1} 1 is the supremum over the source paper's physically admissible (nonnegative, radially monotone) window class: (i) take an exact rational even polynomial trial u (degree 10, coefficients c_0..c_5 = 427163/446844, -205089/684401, -2976898/824779, -13369/15690, -104561/672519, -32375/630751), a truncated kernel A_0 (M = 20 terms) with tail bound rho = ||A - A_0|| <= 45088768/2828846926917599723269509375 < 1.6e-20, residual r_0 = 1 - A_0 u with ||r_0||_2 < 7.875e-10 and ||r_0||_inf < 2.171e-5, and derive a uniform second-derivative bound showing w is strictly concave hence strictly decreasing in |s| (concavity margin +0.59326318); (ii) build source-admissible approximants v_L = w times a taper using eta(x) = 35x^4 - 84x^5 + 70x^6 - 20x^7 on (0,1) (C^3, eta' = 140 x^3 (1-x)^3 >= 0, ||eta''||_1 = 35/8, int eta'^2 = 700/429, ||(eta^2)''||_1 <= 20615/1716), converging to w in L^2 at rate 2/L; (iii) boundedness of A closes the quotient. Lesions: setting the delta_0 coefficient 2 to zero is rejected; multiplying residual bounds by 101 flips the concavity margin to -0.0128 and the verdict to false.

Evidence: RESULTS-xiprime-admissible-closure.md sections 'Theorem', 'Exact strict-concavity bound' (lines 90-224), 'Explicit source-admissible sequence' (225-312), 'Closing the quotient', 'Executable evidence and lesions'; admissible_closure.py; tests/test_pub1_admissible_closure.py Prior art: unsearched Reused in: none recorded Why it travels: A reusable recipe for certifying that an abstract Hilbert-space optimum lies in the closure of a constrained physical class: rational trial + tail bound + derivative bound for shape, then an explicit C^3 taper with exact norms for approximation.

Cover level c(n−1)/2 for the near-zero cover at n ≥ 4

calibration | hunts/ainta_seven_point/ | grade: kernel-checked as part of the FourPoint build; the level rule itself is an ordinary derivation

At n=3 the adjacent-pair coefficient in F is 2/(n−1) = 1, so a gap x with c ≤ w(x) closes the certificate alone and the 1-D cover can run at level c. For n ≥ 4 the coefficient is 2/(n−1) < 1, so the cover must run at level c(n−1)/2 (3c/2 at n=4) so that (2/(n−1))·level = c exactly; the near-zero intervals widen by ≈1.22× in half-width (w quadratic at a simple zero). A sizing pass at level c gave leaf counts ~30% too optimistic and would have produced a tree that looks right and proves nothing. Also: beyond x ≈ √(0.06938/level) the local maxima of w fall below the level (envelope γ²/(π²x²)), so the last basin merges into a tail interval running to the cutoff.

Evidence: FOUR-POINT.md §3.1 'The cover level: the correction that matters' (lines 237–277); four_point_preflight.py §2 checks the level explicitly Prior art: none; lab-internal correction Reused in: four_point_preflight.py (explicit level check so the mistake cannot recur) Why it travels: Any n-point extension of the cell-table route must set the cover level from the smallest pair coefficient, not from c.

Exhaustive kernel-zero seeding for the F_n minimiser

computational technique | hunts/ainta_seven_point/ | grade: INFERRED (float; every c_p is an upper bound on the true floor)

Uniform multistart Nelder–Mead, differential evolution and sample-then-polish all miss the global minimiser of F_6 at p=3000 (they return 0.003868–0.004140 against 0.0038262312), because the true argmin (1.046,1.989,1.986,1.042,1.977,1.045) is non-palindromic and every symmetric basin is higher. What works: polish all 4^{n−1} combinations of the first four positive zeros of k (1.057278, 2.030068, 3.020243, 4.015236) as seeds, then refine the best ~30 basins. Recovers the argmin to 1.2e-13 at p=3000, used as the gate before applying the same method at every p and n.

Evidence: TRUST-MAP.md §1.5 lines 272–283; RESULTS.md §2–3 (float floors at n=7,8,9 with 1-2-2-1 structure) Prior art: unsearched Reused in: modal_npoint_sweep.py / artifacts/npoint-sweep.json; hunts/family_wall (minimiser families as balanced 1-2 words) Why it travels: Any energy over gap vectors whose kernel has zeros has minimisers on the lattice of those zeros; the seeding rule generalises immediately.

Interval-table certification discipline: inflate caps off the attained supremum and round-trip the generator against the kernel's arithmetic

computational technique | hunts/frontier_math/ | grade: measured (cell counts, inflation wall); the 699-cell table is kernel-checked for consistency only (all 18 chunks decide, no sorry/native_decide/axiom), soundness open

(i) An inequality that is an equality at an interior point of a box cannot be proved by interval arithmetic at any table size: any enclosure of a box containing the argmax has an upper bound strictly above the supremum, so the caps must carry documented slack. Here c_k = 21/20 × sup(Dam/y^2) over I_k × [0,1/2] (sup attained at y = 1/2, interior); the budget absorbs inflation up to 1.3945× before the surplus reaches zero, and a 1.02× table closes with 0 undecided in 196 cells, so 1.05× sits inside both; buying the margin costs 6.104e-3 of surplus. (ii) A generator that builds its transcendental leaves by one route (Arb at 300 bits, outward rounding onto the 2^-64 grid, 4-ulp pad) while the kernel builds them by another (truncated Taylor via hornerI, widen by one ulp, reduction modulo a 2^-64 enclosure of 2π with Lean's truncating integer division) can only confirm itself: decide +kernel refuted the 339-cell table on 7 of 9 chunks. The control is a kernel-faithful model (same coefficient lists, SQ2 and PI2iv integers copied not recomputed) plus #eval round-trips requiring the fixed-point integers to be equal, not close (14/14 leaves, then 10/10 compositions one level up before the table was believed). The faithful model doubled the cell count (339 → 699 at 1.20×; 325 → 618; 309 → 568), with the width loss entirely in the trig leaves through argument reduction (cosX2 39.9× wider, hyperbolics 1.0×, shc 0.7×). (iii) A green decide +kernel build then certifies self-consistency of the table, not the enclosed statement, until the _mem seam lemmas (rIv_mem, qreIv_mem, sqrScaled_mem, the shcSmall truncation lemma, the y = 0 case split) connect enclosure to quantity.

Evidence: hunts/frontier_math/RETENTION-PROBLEM.md §4 'The caps carry deliberate slack, and this is load-bearing'; O9-LEAF-REPAIR.md §1–§6; O9-SCOPING.md §2 The two knobs; tests/test_o9_leaves_kernel.py; Zeta23Ext/EForm3/O9RoundTrip.lean, O9CompEval.lean Prior art: unsearched Reused in: hunts/README.md entry on the o9_leaf.py repair (476-cell kernel-model reproduction); K2-TWO-SPECIES.md §4 ('exactly the O9-table technology, in the tau dimension'); hunts/r_a97060 Why it travels: Applies to every numerically generated Lean or interval certificate: budget the inflation explicitly against the attained supremum, and prove the generator mirrors the checker's arithmetic bit for bit before trusting any cell count.

Control-ladder calibrated extrapolation for LP ladders

computational technique | hunts/outband_intake/ | grade: measured

When extrapolating a discretised LP ladder v(X) to its limit, run a matched control ladder whose true limit is known (here the in-band LP, whose limit is the Montgomery-Taylor dual 0.6725007) on the same grids (X, J) = (40,200), (80,320), (120,480), (160,640), (240,960), fit the same three-parameter law a + b X^{-p} to both, and report the control's miss as the method error. Here the difference ladder d(X) = v_out - v_in extrapolates to +0.0065 while the control's known-zero excess extrapolates to +0.0018, so the gain exceeds the method error by about 3.5x and the honest statement is a range ([0.679, 0.682] for the class value, [0.005, 0.009] for the gain) rather than a third digit. A ratio test (2.808, 2.714, 2.780, 2.869 looked flat) was tried and withdrawn: over this range both ladders decay at nearly the same rate so the ratio is uninformative under either hypothesis.

Evidence: RESULTS.md section '1. The measurement' (table, 'The difference is the signal, and the method is calibrated before it is believed', 'A ratio test was tried first and is withdrawn'); refit.py; RUNS.md Prior art: unsearched Reused in: docs/35-the-unspent-fact.md; hunts/outband_certificate/RESULTS.md Why it travels: Any relaxation ladder converging from above can borrow the same calibration: a sibling ladder with known limit on identical grids turns a fit into a measurement with an error bar, and exposes ratio-type tests that carry no information.

Ball LDL^T inertia ladder with an exact-dyadic Rayleigh witness

computational technique | hunts/r_ac9ca3/ | grade: hardened / enclosure-carrying (ball LDL^T inertia, exact-dyadic Rayleigh upper endpoint and Rump eigenvalue enclosure agree, plus an independent mpmath float scout)

For a Galerkin/Gram matrix family indexed by band limit N in which the (N+1)-band matrix is the leading principal submatrix of the N_max-band one, a single ball (Arb, python-flint) LDL^T factorisation at N_max (precision 600–700 bits) returns the pivot signs, and by Sylvester's law the number of negative pivots among the first N+1 equals the number of negative eigenvalues at band N, so one factorisation per cutoff c yields the entire N-ladder (first negative N, even and odd sectors separately). The factorisation is declared conclusive only if every pivot ball excludes zero; a pivot straddling zero returns inconclusive rather than a sign. To certify a specific negative eigenvalue rigorously: take a high-precision float eigenvector (mpmath eigsy at 60 dps), convert its entries to exact dyadics (mantissa × 2^exp), and evaluate v^T M v / v^T v in balls; the upper endpoint is a rigorous upper bound on λ_min (−1.873935689e-31 < 0 at (c,N) = (31,60)), cross-checked by acb_mat.eig Rump enclosure (radius ~3e-192) and by the pivot ladder. Result of the sweep: c ≤ 30 positive to N = 128 (256 at c = 29, 30); c = 31 first negative at N = 60 (even sector), N = 59 positive (+8.365e-31); c = 32..60 first negative N in 48..54 with crossing band edge 2πN/log c averaging 83.64 ± 2.44 against γ_off = 85.6993.

Evidence: hunts/r_ac9ca3/RESULTS.md §1 Positivity Horizon and the Crossing Curve; §2 Marginal Cell (31, 60) Multi-Route Verification; probe.py pivot_signs, first_neg_from_signs, eig_enclosure_min, and the exact-dyadic Rayleigh block (~lines 432–460); origin hunts/rogue_frontier/weil_trunc/RESULTS.md §2 and §8.2 ('one factorization per c gives the whole N-ladder') Prior art: unsearched for the ball-arithmetic variant; LDL^T/Sylvester inertia and Rayleigh quotients are textbook; the CvS/CCM truncated Weil form is used as published Reused in: hunts/r_f00e48 (salvage of weil_trunc; agrees digit for digit on the DH eigenvalue −1.87393568857018838648… and the zeta control); hunts/rogue_frontier/weil_trunc (origin; r_ac9ca3 is the stable reference per hunts/README.md) Why it travels: Any nested-truncation positivity question (Weil forms, Li-type Gram matrices, moment matrices) gets a rigorous first-failure ladder at one factorisation per parameter, and any single negative eigenvalue gets a certificate from a float eigenvector made exact.

One ball LDL^T factorisation yields the inertia of every leading principal submatrix (whole N-ladder per c)

computational technique | hunts/rogue_frontier/weil_trunc | grade: hardened (conclusive-pivot ball factorisations)

The (N+1)-band even-sector matrix is the leading principal submatrix of the Nmax-band one, so the signs of the LDL^T pivots of one conclusive ball factorisation at Nmax give the number of negative eigenvalues at every band N <= Nmax at once (negative pivots among the first N+1). One factorisation per integer c (Nmax = 128, prec 600, about 1 s each) replaces a per-(c,N) eigenproblem sweep; used with the Weyl-shift logic (a uniform diagonal error eps shifts every eigenvalue by eps) to validate the DH diagonal constant to 1e-30 via Gate H (DH diagonal = zeta diagonal + log 5 + a difference-kernel integral decaying like r^-4).

Evidence: RESULTS.md section 8.2 'The transition curve (one factorization per c gives the whole N-ladder)', lines 280-308; Gate H rationale, lines 44-50 Prior art: standard linear algebra (Sylvester inertia via LDL^T), unsearched as applied; the cost-class change is the lab's Reused in: dhneg_scan.py ladders for c in 6..60 Why it travels: Changes the cost class of any nested-band positivity sweep from O(#cells eigenproblems) to O(#c factorisations).

Residual-enclosed shifted Cholesky: a rigorous lambda_min lower bound where interval LDL^T is undecided

computational technique | hunts/oob_envelope | grade: hardened / enclosure-carrying (two independent implementations, GL-96 and Clenshaw-Curtis-192, both Arb)

At condition numbers near 1e48 a ball LDL^T of A - lambda0 I leaves hundreds of pivots straddling zero. Instead: take the exact dyadic midpoint matrix M of the assembled balls, compute a plain Cholesky factor C of M - lambda0 I, freeze every entry of C as an exact dyadic, and enclose Rres = M - lambda0 I - C C^T in Arb with every operand exact. With r the max absolute row sum of Rres and e the max row sum of (entry radius + quadrature radius), symmetry and Weyl give lambda_min(A) >= lambda0 - r - e. The factor need not be accurate: its only job is to make r small, and r is enclosed. lambda0 is a proposal (0.99 of a measured Ritz value, or an exact rational fixed in advance); a Cholesky at 1.01 of the Ritz value must break, which is the built-in lesion.

Evidence: hunts/oob_envelope/numerics/stage_b_modal.py (positivity) and RUNS.md stage B; referee/REVIEW.md "L = 1.19" positivity audit. At L = 1.19, N = 500: r ~ 1e-113, e ~ 2e-63 against lambda0 ~ 5.7e-48. Prior art: unsearched for this exact pipeline; residual-based eigenvalue inclusion is standard verified numerics (Rump-style) Reused in: none recorded beyond the two oob_envelope lanes Why it travels: Any ill-conditioned positive Gram or Galerkin block where interval LDL^T gives up but the gap to zero is many orders above the arithmetic noise.

Exact finite-N CUE engine for band-Gram trace moments, with the fit-and-check polynomial identification protocol and the Wick regime boundary

computational technique | hunts/rogue_frontier/sine_gram | grade: integers exact; polynomial and limit identifications measured (heavily overdetermined exact agreement on the computed grid, Monte Carlo consistent); lambda forms conjectured on the sampled grids

For T_{m,m'} = Tr U^{m-m'} over a band of d consecutive integers, U ~ CUE(N), E tr T^k is an integer computed exactly: E prod_j Tr U^{h_j} reduces through determinantal correlations and the Dirichlet kernel to lattice-point counts max(0, N - spread) per signed permutation cycle over set partitions; validated by E|Tr U^h|^2 = min(|h|, N) including h > N, Fubini term counts, and E tr T^3 = 2N^4 - N^2. Identification: fit a degree-(k+1) polynomial in N through the k+2 smallest grid points, demand exact integer agreement at every remaining computed value and at even N off the grid; m_k(lambda) is the leading coefficient over q^k p on families N = qt, d = pt. Results: m_5(1) = 101/18, m_6(1) = 640/63; m_k(lambda) piecewise with Wick polynomial below lambda = 1/floor(k/2) and defects -(j lambda - 1)^{2j+1} g_{k,j}(lambda)/lambda above each 1/j. Calibration: the Gaussian (Diaconis-Shahshahani) regime for joint trace moments needs total positive frequency <= N, which is why a Wick-pairing engine gave the wrong m_4(1) = 49/15 against the true 13/4; exact arithmetic was load-bearing because the defects vanish to order 2j+1 at breakpoints.

Evidence: RESULTS.md 'What happened so far' items 1-4, lines 19-43; moments_report.md sections 1-3 (engine validation, lambda = 1 table, lambda structure and its fit-and-check bookkeeping) Prior art: cited: the source paper's m_k(1) = 1, 4/3, 2, 13/4 (7.5(f)); literature surveyed stops at m_4 (searched-and-absent for m_5, m_6) Reused in: window_opt/crosscheck_finiteN.py (read-only import of exact_finite_N.py) as the independent validation of the m3 functional Why it travels: An exact integer instrument for any CUE band-Gram moment, plus a reusable identification protocol with explicit spare-point counts; the Wick-boundary calibration warns every Gaussian-approximation moment computation at lambda near 1.

Target-derived pressure-cutoff soundness rule for the box verifier

control | hunts/ainta_seven_point/ | grade: measured (verifier re-run on Modal, artifacts/modal-rerun-sound-cutoff.json)

In the Arb branch-and-bound verifier for F_n ≥ c, the compactification prune 'discard any box whose gap-sum lower corner ≥ CUTOFF' is sound iff CUTOFF/p ≥ c, i.e. CUTOFF_cells ≥ ceil(GRID·p·c). The published constant 45600 at GRID 4000, p 3000 encodes exactly c = 19/5000; every run that raised only the target (Gohms 191/50000, and the lab's own probes) pruned 3,087 boxes on unproved grounds. Repair: derive the cutoff from the target (46,400 cells sound for c ≤ 0.003867) and re-run; node counts moved by tens, all acceptances stood. Node-for-node reproduction of a published run is agreement, not soundness.

Evidence: TRUST-MAP.md §5.1 'The Gohms variant's compactification prune is unsound at its own target' (lines 590–634); RESULTS.md §3 'A defect in every raised-target run' (lines 100–122) and §7 (lines 289–298) Prior art: searched-and-absent: defect present in the published verifier and in the Gohms issue; reported upstream Reused in: verify_n.py / modal_verify_n.py (cutoff derived from target rather than hardcoded); the eight-point certificate Why it travels: Any verifier that prunes by a linear term must have its prune constant tied to the target; hardcoded prune constants silently encode one target.

Fault-injection harness for an interval certificate verifier

control | hunts/ainta_seven_point/ | grade: MEASURED, both runs

To test that a branch-and-bound interval verifier can fail (not merely that it reproduces): (i) a known-false target just above the float floor must be REFUSED at a terminal cell; (ii) four planted unsoundnesses (prune ignoring the target, kernel lower table inflated by 2e-4, coefficients scaled by 1.01, the minimiser's boxes dropped) must each flip that refusal to ACCEPTED. Pitfall recorded: when the functional has a symmetry (F is reversal-symmetric), all symmetric images of the minimiser's box must be dropped, else the verifier refuses at the mirror and the fault looks undetected. Passed 4/4 at n=3 and n=7.

Evidence: RESULTS.md §7 (lines 289–298); RUNS.md runmanifest 'ainta_seven_point-2026-08-31-verify-n-rescue-and-fault-injection' (lines 395–408); verify_n_faults.py Prior art: unsearched Reused in: none recorded Why it travels: Applies to any accept/refuse interval verifier: it is the only way to distinguish a verifier that is right from one that cannot say no. Index note: Fault-injection of a verifier is the lab standard; this is the interval-certificate instance.

Two-sided bracket on an infimum: interval certificate below, Arb point evaluation above

control | hunts/ainta_seven_point/ | grade: hardened (interval-enclosed lower end, Arb enclosure upper end)

To make an 'apparent floor' rigorous on both sides: the lower end is the largest rational target the interval verifier ACCEPTS (a proof that inf F ≥ c), the upper end is an Arb ball evaluation of F at the float argmin (any point value is an upper bound on the infimum). Gives 0.003826 ≤ inf F_6 ≤ 0.0038262312115073 (width 2.3e-7) and 0.0041763 ≤ inf F_7 ≤ 0.0041773221. The upper end is a value at a rounded point, not the infimum; family_wall later found two independent minimisers 2.0e-13 below it, which is the correct behaviour against an upper bound.

Evidence: RESULTS.md §3 'And the bracket is rigorous on both sides' (lines 89–98) and eight-point bracket (lines 181–182); family_wall/FAMILY-LIMIT.md §3 (lines 533–556) on how to read the upper end Prior art: unsearched Reused in: hunts/family_wall (control on n=7,p=3000; corrected its own reading of the number) Why it travels: Standard shape for any 'how low does this functional go' question where a refusal-capable verifier exists.

Preflight arithmetic filter that reads the generated Lean, plus a fault-injected, must-fail axiom audit

control | hunts/ainta_seven_point/ | grade: MEASURED / VERIFIED (preflight exit 0, audit fixtures)

Before spending a CI build: (a) a Python preflight re-reads the emitted Lean text (not the generator's in-memory tree, so emission bugs are caught) and checks against the true w from the sinc form that every cell constant is a lower bound (401-point sweep per cell), the cover is contiguous and hits the cutoff exactly, and at every leaf the invoked cells cover the ranges the branch conditions force and the linarith combination is true; fault-injected four ways, 4/4 caught. (b) The #print axioms audit is scoped to the advertised declaration names, accepts choice/Classical.choice as one axiom, REQUIRES all advertised names to be present (an empty log cannot pass), and is tested against three fixtures: the real log passes, a planted sorryAx fails, an empty log fails. (c) Set autoImplicit := false in a package whose point is an advertised theorem; with it on, unresolved names (HD, Ncount, N0simple) silently became implicit variables and the statements were briefly about nothing.

Evidence: THREE-POINT.md §4 (lines 299–347), §5 'The axiom audit, VERIFIED' (lines 515–538), §7 autoImplicit bullet (lines 600–608); FOUR-POINT.md §4 (lines 353–358); RUNS.md line 380 Prior art: unsearched Reused in: four_point_preflight.py (extends three_point_preflight.py); .github/workflows/three-point.yml audit step Why it travels: Generated-proof pipelines fail at emission and at audits that cannot fail; both checks are generic to any Lean table generator.

Shared-invariance test before promoting a repeated null to a constraint

control | hunts/director_run/ | grade: argument with an in-tree counterexample; disposition of the universal: REJECTED (director's ledger)

Before a repeated failure across N instruments is promoted to a class-level constraint ('coefficient-side statistics cannot see the critical line'), check whether the instruments share an invariance. The three instruments (factorization defect D(f), Fourier quasicrystal separation, local-positivity c_p) are each invariant or nearly so under the twist a_n -> n^delta a_n, which is what produced the common blind spot; the universal is refuted in-tree by Titchmarsh 14.25(B)/(C) (zeta/criteria.py face 1), an RH-equivalence in the coefficients of 1/zeta alone. Rule: a repetition across instruments is evidence about the instruments before it is evidence about the subject; a 'counterexample' offered against a universal must be nontrivial in the sense the claim intends (sigma_a, d_p died to this; theta survived only for zeta).

Evidence: GRAVEYARD.md entries G1 and G2 (lines 10-60); CLAIMS.md entries C-SHIFT-01, C-SHIFT-02, C-SHIFT-03 Prior art: searched-and-found for the shift computation itself (C-SHIFT-02 is the Selberg-class theta < 1/2 axiom, Conrey-Ghosh 1992; Jacquet-Shalika); the control rule is the run's own Reused in: docs/25-the-director-run.md Why it travels: Applies to any 'N independent probes all failed, therefore the class is closed' argument: first exhibit the symmetry the probes share, or the closure is about the probes.

Lesion the reference, not the instrument: mis-set a constant in the closed form to prove the agreeing comparison can disagree

control | hunts/frontier_map/ | grade: measured (float; 'a map, not a result')

When a numerical optimiser is cross-checked against a published closed form (here the lambda-landscape optimum c*_lambda = sqrt2 tan(theta)/(1 + theta tan theta), theta = lambda/sqrt2, paper eq. 7.4, not used in building the optimiser; max deviation 9.5e-15 over lambda in [0.1, 1]), also run the same comparison against the closed form with one constant deliberately mis-set (sqrt2 -> 1.5): the minimum deviation must jump by orders of magnitude (here 1.5e-2, twelve orders), which shows the comparison is capable of disagreeing and the 9.5e-15 is a real agreement. Paired with a basis/quadrature convergence ladder (stability < 6e-15 per rung against pinned constants) and a monotonicity check on the refinement grid (0 decreases), reported in a controls ledger with the rival control explicitly marked 'not run here, quoted'.

Evidence: RESULTS-frontier-map.md §1 'Cross-check (control 1)' and lesion (lines 40-48), 'Controls ledger' (lines 109-117); probe.py (crosscheck_zeta_curve, lesion, convergence_response, monotonicity) Prior art: cited: the 10 August 2026 paper eq. 7.4 for the closed form; the optimiser is shared with hunts/wide_search Reused in: inherits caveats from hunts/wide_search RESULTS files; none else stated Why it travels: A cheap way to make any 'numeric matches closed form' cross-check falsifiable: lesion the oracle side, not the instrument, and report the minimum deviation the lesion produces alongside the agreement.

Clean-kill exact witness against the first algebraic lemma (transpose vs conjugate-transpose)

control | hunts/frontier_math/ | grade: kernel-checked (Lean 4, axioms propext/Classical.choice/Quot.sound only) for the integer obstruction; exact Gaussian-integer checker for the witness

Before any numerical search on a proposed inequality built over a pinned upstream object, rebuild the upstream summand literally and attack the first lemma the chain needs with the smallest exact-arithmetic witness. Here the upstream zero-side summand is m·u_z·u_z^T (transpose, not conjugate transpose), so an off-line conjugate pair with u_z = a+ib contributes 2m(aa^T − bb^T), a hyperbolic block, and the class interactions are on/on m_x m_y B(x,y)^2, on/off 2 m_x m_z Re(B(x,z)^2), off/off 2 m_z m_w Re(B(z,w)^2 + B(z,conj w)^2), of which only on/on is automatically nonnegative. The old instrument had built u u* (always a squared modulus) and could not see the sign. Witness in one dimension: u_x = 1, u_z = i, u_{conj z} = −i, all multiplicity one gives P1 = [1], Q' = [−2], tr(P1 Q') = −2, refuting the asserted tr(P1 Q') ≥ 0; with five unit on-line labels the proposed additive inequality demands 9 ≥ 13. The kill happens before taper, truncation, census, bootstrap or LP questions can matter, and all downstream values (0.6725124, 0.672529, 0.6725318) lose their zeta implication at once.

Evidence: hunts/frontier_math/CLEAN-KILL-REPORT.md §First false statement, §Smallest exact obstruction, §Permanent controls; hunts/frontier_math/clean_kill.py; lean/ZetaLean/FrontierMathObstruction.lean; RESULTS-frontier-math.md §0 Prior art: cited: the transpose summand is the upstream paper's own definition (Zeta23/Defs.lean:298-305, ZeroSide.lean:314-379 at the pinned commit); the obstruction and the witness are the lab's Reused in: RESULTS-frontier-math.md §0 and §3; SIGNED-INCIDENCE-LAW.md (pins the transpose reading as input); INTERACTION-CONTROL-REPORT.md; tests/test_frontier_math_clean_kill.py; hunts/README.md frontier_math entry Why it travels: Any transplant onto a formalised upstream must reproduce the upstream bilinear form literally and be attacked by a minimal exact witness before a single float scan is run.

Refinement-direction ladder with a structurally adjacent control configuration

control | hunts/frontier_math/ | grade: measured

A discretised lower bound that is real rises toward its tight relaxation under refinement; a bound that falls under refinement is manufacturing floor. The gap-distribution LP's first implementation assigned bins to cells by midpoint, crediting straddling bins wholesale, and produced a false conditional 0.6728294 that would have beaten Cheer–Goldston; the bin-width ladder caught it (the floor fell), and snapping cell edges onto the bin grid restored the monotone ladder 0.69 → 1.02 → 1.44 → 1.47 ×1e-5 for h = 0.02 … 0.0025. Conversely a numerically found counterexample must survive a full ladder, not one refinement: sparse-24 read −0.004396 at step 0.005 / G = 60 and closed at every finer setting; every-5 was +0.000075 coarse, −0.002288 at the first refinement (step 0.0025 / G = 120), and +0.000069 to +0.000073 on the rest of the ladder, identified as an artifact of one setting because the structurally adjacent every-4 control closed at every step. Also: a per-cell allowance granted uniformly (a Lipschitz margin of ~1.7e-3 on ~800 cells) manufactured ~1.3 units of damage and a false obstruction; any per-cell allowance must be local and scale with the quantity it protects (three occurrences: level-5 coarse-grid scare, level-7 mid-zone blanket, transplant first session). And the evaluator must be read inside its calibrated domain: the reported y = 0.979 'counterexample' sat outside the strip (y < 1/2 always) and vanished when the search guard was clamped.

Evidence: hunts/frontier_math/RESULTS-frontier-math.md §5 ('A control earned its keep here, twice'); BLOCKER2-INDOMAIN-FLOOR.md §1, §2, §3; LEVEL5-ENCLOSURE-AND-PAIRS.md §Gate 1: the hardened scan, and what almost went wrong; TRANSPLANT-LEMMA.md §The first session's obstruction was not one Prior art: unsearched Reused in: LEVEL4-COUNTING-DUAL.md (sup grid decoupled from the capacity partition); TRANSPLANT-LEMMA.md band dual (off-band allowance exactly zero); enclosure_pass.py defaults pinned at original resolution with the coarse failure kept as a negative control Why it travels: A generic acceptance test for any discretised floor (must rise under refinement) and for any numerically found counterexample (must survive a ladder and differ from an adjacent control).

Hardening control set for converting a sampled table to an enclosure table

control | hunts/r_a97060/ | grade: enclosure-carrying for the τ-table step; the composite k=2 claim takes the grade of its weakest step (the convexity transfer, argued not enclosed)

Replacing scans by bounds in the k=2 τ-table used six paired controls: (H1) enclosure vs unpadded scan on the binding cells must be within a small ratio (1.0000–1.0004; the deleted flat 1.05 pad was 100–400× the actual enclosure error); (H2) any heuristic prune in the adversary's search is on the unsound side, so re-run exhaustively on binding cells and report the delta (0.0); (H3) the enclosure's lower witness never exceeds the scan value, so the old grid's blindness is qualitative (y < 0.05), not numerical; (H4) planted cap fault: inflate the table's own caps by 1.01/1.02/1.05/1.10 and record where the resonance cell dies (1.10×; the measured pass fired at 1.02×), showing the detector still has power; (H5) cross-backend: mpmath.iv rectangles must contain the Arb balls (they did, 1.8–3.5× wider); (H6) clamp check: verify the geometric assumption behind any clamp (Kpair(min(dmax,6)) is a valid lower bound only if Kpair is monotone on [0,dmax]; widest component 1.9894 so it never bound). Replacement for the clamp: a running minimum of the Kpair envelope over [0,dmax], which needs no monotonicity argument.

Evidence: RESULTS.md §1 table (lines 41–58), §3 'Controls' (lines 104–113), §4 (lines 115–134); ball_field.py Prior art: unsearched Reused in: named as the template for k ≥ 3 and the depth-1 far constant (loose threads 1, 5); K2-TWO-SPECIES.md §6 updated Why it travels: A checklist for any 'measured → hardened' upgrade: it catches pads that were standing in for enclosures, prunes on the unsound side, and clamps whose soundness rests on unstated geometry.

Zeta control at the identical cell (rival discrimination in both directions)

control | hunts/r_ac9ca3/ | grade: hardened (ball LDL^T at precision 2400 plus Rump eigenvalue enclosure; inertia conclusive)

A positivity failure on a structure-matched RH-violating rival (Davenport–Heilbronn, off-line zero ρ ≈ 0.8085 + 85.6993i, δ = 0.3085) is evidence only if the same truncation at the same (c, N) is strictly positive for ζ: at (31, 60) the zeta form has even inertia (61, 0), odd (60, 0), λ_min(ζ) = +4.82160175e-100 (ball LDL^T at precision 2400 plus eigenvalue enclosure) against λ_min(DH) = −1.8739e-31, a discrimination of 100 orders of magnitude at one cell. The zeta control failing to remain positive is a stated kill condition. The complementary direction is used by frontier_math for structural lemmas: LAW D, LAW E and LAW H must also hold on the DH off-line zero (LAW D defect 2.7e-14 at depth 0.30851718; level-2 cap 31.856 obeyed), because a lemma failing on the rival would be a bug and passing distinguishes nothing.

Evidence: hunts/r_ac9ca3/RESULTS.md §2 table row 'Riemann Zeta Control (31, 60)' and the paragraph following; MISSION.md kill_conditions; hunts/frontier_math/SIGNED-INCIDENCE-LAW.md §What is not achieved (third bullet) and §Controls ledger row 'rival (Davenport–Heilbronn off-line zero)'; LEVEL2-GAP-CONSISTENCY.md controls ledger row 'rival' Prior art: unsearched; the Davenport–Heilbronn function is used as published Reused in: hunts/r_f00e48; hunts/rogue_frontier/weil_trunc; hunts/frontier_math levels 1–2 (rival checks on DH) Why it travels: Every 'the instrument detects the off-line zero' claim needs the same instrument at the same parameters on the object without one, and every structural lemma about zeros must pass on the rival too.

Dictionary attribution: isolating the off-line quadruple as the sole negative term of the explicit-formula decomposition

control | hunts/r_ac9ca3/ | grade: measured (float dictionary sums with a 7% bookkeeping residual; the eigenvalue itself is enclosed)

For the minimising eigenvector v of a truncated Weil form, expand the form value through the zero-side identity λ = ⟨v, Qv⟩ = Σ_{γ>0} 2 g_v(r_γ). Every on-line zero contributes a nonnegative term; the off-line pair contributes the quadruple 4Re g_v(γ_off − iδ). At (31, 60): λ_min = −1.8739e-31, quadruple = −6.734989e-29, on-line partial sum (T ≤ 120, 64 zeros, all ≥ 0) = +5.9537e-29, tail model (T > 120 via mean density) +2.956e-30, second off-line pair +7.57e-32, bookkeeping residual 4.67e-30 (~7% of the quadruple); λ − quadruple = +6.716250e-29 > 0. Since all on-line terms are nonnegative the quadruple is the sole negative contributor and removing it flips the sign, which attributes the failure to that zero; corroborated by beam profiling of the eigenvector (95.63% of coefficient mass within ±6.0 of γ_off at the deep cell (47, 64), peak mode k = 52 vs target 52.51) and by the crossing band edge tracking γ_off across c ∈ [32, 60]. A second negative eigenvalue appearing at c ≥ 44 matches the second off-line zero (δ_2 = 0.15083, γ_2 = 114.1633), later because c^{δ_2} amplification is weaker. Kill condition: removal of the quadruple not flipping the sign.

Evidence: hunts/r_ac9ca3/RESULTS.md §3 Mechanism and Localization (3.1 beam profile, 3.2 dictionary decomposition); §4 Second Off-Line Pair Detection; §1 band-edge tracking; MISSION.md kill_conditions; probe.py dictionary_attribution_31_60 Prior art: unsearched; the explicit formula is standard Reused in: hunts/rogue_frontier/weil_trunc/RESULTS.md (same decomposition at (31, 60) and (47, 64)) Why it travels: Converts 'the form went negative' into 'this zero made it negative' for any quadratic form with an explicit-formula expansion over zeros, and gives a falsifiable kill condition for the attribution.

Scalar and moment-matched obstruction families: no universal recovery coefficient from rank, trace and positive-index inputs

obstruction | hunts/frontier_math/ | grade: exact integer/rational arithmetic throughout (no optimizer, sampled grid, or floating tolerance); ordinary derivation, self-reviewed

Given only the data the pinned rank-trace lemma consumes (P psd, rank P ≤ r, tr P ≤ s, Q Hermitian, n_+(Q) ≤ b, A = P+Q), the sharp available inequality is ‖A‖F^2 ≥ ((2 tr A − tr P)+)^2/(r+4b), which at c = 2 reduces to the paper's census form ‖A‖F^2 ≥ 4 tr A − 3s − 4b and contains no on-line cross mass R(P) = Σ{i≠j}|⟨u_i,u_j⟩|^2. Exact scalar family: for integer m ≥ 1 take n = 2m^2+2 on-line labels with scalar evaluation vector 1 and one off-line pair with vectors im, −im; then P = n, Q = −(n−2), A = 2, R(P) = n(n−1), census slack 3n, so any universal strengthening ‖A‖_F^2 ≥ 4 tr A − 3s − 4b + θ R(P) forces θ ≤ 3/(2m^2+1) → 0. A direct-sum dilution (M = kn orthogonal positive off-line blocks of eigenvalue 2+1/k plus L unit on-line blocks with L the nearest integer to (‖A_0‖_F^2 − C N_0)/(C−1)) simultaneously matches tr A = N and drives ‖A‖_F^2/N to the paper's printed Frobenius ratio C = 1327499296/10^9 while R(P)/N exceeds twice the proposed gap floor (checked exactly at m = 10, k = 100000). Conclusion: separate on-line gap statistics plus aggregate off-line counts cannot yield the strengthening; the missing datum is a signed joint on/off incidence law, and the report specifies the level hierarchy (state retained / exact object / kill control) needed to obtain one.

Evidence: hunts/frontier_math/INTERACTION-CONTROL-REPORT.md §Sharp inequality available at this interface, §Exact scalar family, §Matching the paper's prime-side moments, §Missing invariant, §Next configuration hierarchy; interaction_obstruction.py; test_interaction_obstruction.py Prior art: cited: the upstream rank-trace lemma (Zeta23/LinAlg/RankTrace.lean:157-195) and the paper's §7.5(a); the obstruction families are the lab's Reused in: SIGNED-INCIDENCE-LAW.md (walls W1–W4 exclude the family); LEVEL2-GAP-CONSISTENCY.md (epsilon-robust exclusion via LAW H); LEVEL3-THETA-RECOVERY.md Phase 5 battery; PROOF-LEDGER.md post-kill audit Why it travels: Template for pricing any proposed strengthening of an inequality: build in exact arithmetic the extremal family satisfying every retained hypothesis, show it drives the coefficient to zero, then dilute it to match the printed moments so the unrealistic-moments objection is closed too.

Sieve wall: constant-factor prime-pair upper bounds cannot open the λ > 1 band

obstruction | hunts/frontier_math/ | grade: ordinary derivation, self-reviewed

The paper's §4 machinery is support-agnostic; only the prime-side second moment caps the support parameter λ at 1 (its §7.5(a)). The tempting unconditional route, bounding the off-diagonal prime sums by a Selberg-sieve upper bound Σ_{n≤N}Λ(n)Λ(n+h) ≤ C·𝔖(h)·N with classical C, fails structurally: for X = T^λ with λ > 1 the off-diagonal and expected-value terms are each of scale (x/T)·N = T^{λ−1}·N and cancel to O(N) only under Hardy–Littlewood with error; a sieve constant C multiplies the x-scale term, so the loss is (C−1)·T^{λ−1}·N/log T, unbounded relative to N for any fixed C > 1. Only C = 1 + o(1), Hardy–Littlewood itself, closes it. This makes the paper's Remark 1.1 wall ('0.70 needs support ≈ 1.04') mechanism-explicit: no constant-factor upper bound on prime pair correlations, however sharp, opens the band.

Evidence: hunts/frontier_math/RESULTS-frontier-math.md §2 The λ > 1 sieve wall, quantified; cited by docs/35-the-unspent-fact.md §4 Prior art: cited: the paper's Remark 1.1 and §7.5(a); the mechanism-explicit accounting is the lab's Reused in: docs/35-the-unspent-fact.md §4 (uses it to distinguish a wall from Hunt #110's gap); hunts/README.md frontier_math entry; hunts/README.md line ~436 (contrast with a construction) Why it travels: Tells any future attempt at widening support that sieve-grade constants are structurally insufficient and the required input is Hardy–Littlewood grade; the T^{λ−1} scale argument transfers to any prime-side second moment beyond the diagonal.

Measure-level LP collapse: multiplicity types reduce out to the Montgomery–Taylor dual

obstruction | hunts/frontier_math/ | grade: measured (LP ladder, last rung ~78 min) plus an exact reduction (ordinary derivation, self-reviewed)

Minimising the density of simple on-line points over multiplicity types p_m, off-line pair density q and off-diagonal pair measure ρ ≥ 0, subject to the bandwidth-one data R̂_2(α) = δ(α) + |α| on [−1,1], the type structure reduces out exactly: eliminating (p_m, q) against the density constraint gives p_1 = 2 − D + Σ_{m≥3}(m^2 − 2m)p_m, where the m ≥ 3 and off-line types enter with nonnegative coefficients m^2 − 2m and 0, so the LP value is the dual of the Montgomery–Taylor extremal problem and integrality devices beyond (m−1)(m−2) buy nothing at the measure level. Measured: the (X, J, ε) ladder descends monotonically 0.6794 → 0.6776 → 0.6765 → 0.6756 → 0.6750823 (X = 40…640, J = 5X, ε = 0.4/X) with D climbing 1.3206 → 1.324918 toward the MT constant 1.3274993, consistent with convergence to 0.6725007 from above and nothing in between; τ = −sinc^2 satisfies the data rows (GUE anchor), and the value moves in the predicted directions as ε and X change. Consequence: the ceiling gap (0.6725007, 0.68185) is not about the pair measure; it measures what configuration realizability (ordered real sequences) adds beyond measure positivity. With BGSTB's unconditional out-of-band positivity added as data the class value at (X = 80, J = 320) is 0.6863 and still descending.

Evidence: hunts/frontier_math/RESULTS-frontier-math.md §1 THREAD 1 answered: the measure-level LP collapses, §4, §Controls ledger; configuration_lp.py Prior art: cited: the paper's §1.2 scoping of Theorem D's optimality to F on [−1,1]; Cheer–Goldston 1993 closing remark; BGSTB 2023 (arXiv:2306.04799) Theorem 1 for the out-of-band positivity, which the lab re-derived before finding Reused in: hunts/README.md frontier_math entry; the ordered-gap LP of RESULTS §5 and the whole level hierarchy build on 'the gap is configuration realizability'; docs/35-the-unspent-fact.md Why it travels: Closes the class 'add multiplicity or type variables to the pair-correlation LP' for any kernel with the same bandwidth-one data, and identifies the realizability constraint as the only remaining lever.

Place-local Selberg-bound gate and the non-assembly of local norms

obstruction | hunts/local_positivity/ | grade: measured (float; the file states it carries no enclosure at any step)

For a degree-d object with Satake parameters alpha_j at p, the place-local kernel K_p^{(d)}(theta) = d + 2 sum_m lambda_m p^{-m/2} cos(m theta) has closed form sum_j (1 - |alpha_j|^2/p)/|1 - alpha_j p^{-1/2} e^{i theta}|^2, so the coefficient-computable test c_p <= d is exactly the local bound |alpha_j| <= sqrt p (zeta's threshold 2/(sqrt p + 1) to 12 digits). Calibration: decoy swap moves the verdict by 15 orders; 300 random period-5 nulls fail 100% with DH at the 6th percentile; lesion blindness threshold eps* = 0.184 on the zeta -> DH interpolation; degree-2 family keeps margin >= 0.343 over 60 Satake angles; a legitimate degree-2 product with alpha = 2.3, 1/alpha is rejected at p = 5 (c_p = 65.24), so the gate tests the local Selberg/Ramanujan bound, not 'has an Euler product'. Obstruction: the prime side of the explicit formula decomposes place by place as -sum_p log p (Q_p(f) - ||f||^2) with Q_p = (1 - 1/p) ||Phi_p f||^2 a genuine norm at every place (reconstruction to 22 digits), but Q_p - ||f||^2 is not of definite sign (52 of 60 places positive, 8 negative), so local positivity does not assemble into a global one and is compatible with either sign of W. Director's adjudication: c_p(zeta(. - delta)) = 2x/(1 + x), x = p^{delta - 1/2}, c_p <= d iff delta <= 1/2, is KNOWN (Selberg-class theta < 1/2).

Evidence: CORRECTIONS.md section '2. New hunt: hunts/local_positivity/' (paragraphs 'What it establishes', 'Reference table', 'Controls', 'Where it dies', 'The honest boundary'); localpos.py; director_run/CLAIMS.md C-SHIFT-01, C-SHIFT-02 Prior art: searched-and-found by the director run for the shift computation (Conrey-Ghosh 1992 theta < 1/2; Jacquet-Shalika, stated sharp by Sarnak); the closed form is standard Satake arithmetic; the assembly obstruction is the hunt's own Reused in: docs/24-the-local-positivity-attempt.md; hunts/director_run/CLAIMS.md Why it travels: The place-local decomposition with the measured sign-indefiniteness closes the 'prove Weil positivity place by place' route with a mechanism, and the gate with its calibrated blindness threshold is a ready coefficient-side instrument for any rival battery.

Ceiling: positive-definite (Gram) kernels cannot spend out-of-band form-factor positivity

obstruction | hunts/outband_certificate | grade: proved for the edge lemma and dichotomy; 'read' for the compression structure (checked against the paper's full text); prose grade

Any certificate whose positivity input is Weil's Hermitian form compresses to G = V V^H

positive-definite, so Khat >= 0 everywhere; the out-of-band term int_{|alpha|>1} Khat F is then nonnegative and unbounded above because F has no unconditional upper bound outside the band, forcing Khat = 0 outside [-1,1]. A kernel with signed transform has odd/complex factor (dichotomy above), equivalently an indefinite inner product, which hunt #110 refuted by a 2x2 witness. So the strip is worth exactly zero to unconditional Gram-form certificates; the LP value [0.679,0.682] is the RH-conditional pointwise-positivity class.

Evidence: RESULTS.md section 8 'The ceiling, and why the LP priced the wrong class', lines 240-322; kill condition 2 in MISSION.md Prior art: cited: Chirre-Goncalves-de Laat 0.6792 (conditional), BGSTB arXiv:2306.04799, Guth-Maynard arXiv:2405.20552, hunt #110 outband_intake witness Reused in: placed hunts amtopa_ceiling, four_point_pressure, cycle_moments, wide_search, prime_pair_error (RESULTS.md section 9) Why it travels: Closes the whole class 'inertia/isolation argument on a kernel signed only on a strip' and names the two inputs that would reopen it; any future out-of-band proposal must first escape this argument.

Autocorrelation-kernel obstruction: inertia arguments cannot spend out-of-band positivity

obstruction | hunts/outband_intake/ | grade: measured; the 2x2 counterexample is exact arithmetic

Any diagonal-isolation / inertia argument of the CGdL type requires the evaluation form to be positive semidefinite, which forces the window spectral density v = phi^2 >= 0 and hence the pair weight Khat = v * v >= 0 everywhere; the framework never has a free signed ghat. The requirement cannot be relaxed to a weighted inner product tr(X^H S Y S) with indefinite S: the rank half of the lemma survives but the inertia half needs S^{1/2}; minimal counterexample Q = t [[0,1],[1,0]], S = diag(1,-1), c = 2 gives slack +2.0 at t = 1 and -0.5 at t = 1.5, while 4000 random PSD pairs never violate (worst slack +0.001321). First-order law: adding out-of-band mass -eps where F >= Lbar moves the bound by dJ/deps = (J - Lbar)/g(0), adverse for Lbar below about 1.3275, so with BGSTB's Lbar = 0 the direct channel pays nothing and the entire measured gain is indirect (it licenses an in-band profile that is not an autocorrelation). Calibration of where the information lives: at (X, J) = (80, 320) enforcing nonnegativity on (1, A_out] gives 60.2% of the full gain by A_out = 1.25 and 91.4% by 1.5, so a certificate only needs to be signed on a narrow strip just past the band.

Evidence: RESULTS.md sections '1b. The information is local: 91% of it sits in (1, 1.5]', '2. Why no certificate can spend it', '3. So the result is a gap, not a wall'; price_the_band.py Prior art: cited: BGSTB arXiv:2306.04799 Thm 1 for the positivity; Chirre-Goncalves-de Laat section 4 for the argument that does not transfer; frontier_math/paper_pin.py for the int |Khat| discharge Reused in: docs/35-the-unspent-fact.md; hunts/outband_certificate/RESULTS.md Why it travels: Names the exact structural property (kernel must be a square) that any attempt to use signed or out-of-band constraints in a Gram/inertia certificate must escape, with a two-by-two witness that kills the obvious weakening.

Universal certificate pinning and band-limited exclusion for the centre-gas gap

obstruction | hunts/r_b9552d/ | grade: measured (double precision, no enclosures); section 3's factorisation and sampling steps are quoted standard results, not proved or probed

Let f(s) = Dam(1,s) - Kpair(s) = -kappa(s) + K_1(s)^+, J(T) the per-centre row, L = 2 kappa(0) - 4 c2(0) = 0.11433003938654052 its value on the uniform 2 pi lattice, and let a certificate be g >= f pointwise with ghat <= 0, giving J <= B_g(rho) = 2 rho ghat(0) - 2 g(0). (P) If one rho-independent g proves J <= L at every density then necessarily ghat(0) = 0, g(0) = -L/2, sup|g| <= L/2, ghat(j) = 0 for every integer j and g(2 pi n) = f(2 pi n) for n != 0; the cheap necessary test sup_{s != 0} f <= L/2 passes with margin 3.05 (sup f = 0.0187431348 at s = 6.3974 vs L/2 = 0.05716501969). For the gap-B family g = -kappa + c s, s = (sin(x/2)/(x/2))^2, ghat(0) = 0 iff c = 2 c2(0) = 1.6984559986366083, but admissibility caps c at cos^2(sqrt2/2)(1 + cosh 1) = 1.4698290136, a shortfall factor 1.15554665 (witness: ghat > 0, max +0.0839 at xi = 0.87493). By Fejer-Riesz factorisation plus critical sampling on 2 pi Z, any band-limited (support in [-1,1]) nonnegative v = g + kappa vanishing on 2 pi Z minus 0 is c times the Fejer kernel, so the whole band-limited family is excluded: any certificate closing gap A must carry Fourier mass outside [-1,1]. Consolation bound at the cap: J(T) <= 2 kappa(0) - 2 cos^2(sqrt2/2)(1 + cosh 1) = 0.5715840115651507 for every periodic configuration at every density.

Evidence: RESULTS.md sections '1. What a certificate that closes gap A must be, exactly', '2. The family that closed gap B cannot close gap A, and by an exact amount', '3. Why this is a statement about a family and not about one ansatz', '4. What the family does give' (lines 40-170); section 5 controls (inflation-ladder planted fault caught) Prior art: unsearched for the pinning argument; Fejer-Riesz / Krein and Paley-Wiener sampling cited as standard Reused in: hunts/r_c7f779/MISSION.md; hunts/frontier_math/K2-TWO-SPECIES.md; hunts/dps_cap/README.md; hunts/support_5418c63e/MISSION.md Why it travels: Template for any Cohn-Elkies-style LP with a known extremal lattice: equality at the lattice plus affine dependence on density pins the certificate's zeroth moment, value at zero, sup norm and lattice interpolation, and gives a one-line necessary test before any LP is run.

Scalar-moment joint-window LP collapses to the best single window

obstruction | hunts/wide_search/ | grade: argument, self-reviewed (the hunt labels the file 'measured'); frontier_math later confirmed the collapse at the pair-measure level

If, for each admissible window v, the only data retained from the Weil-form matrix is its trace and Frobenius norm, the resulting constraint is exactly s1/N >= H(v) and the joint feasible set over all windows is the intersection of half-lines, i.e. s1/N >= sup_v H(v). Therefore no LP or SDP built from per-window scalar moments can move the single-window constant 0.6725007037...; any non-collapsing formulation must retain cross-window information or act on the whole bandwidth-one form-factor measure before reduction to one Rayleigh quotient. The full bandwidth-one certificate (c0 + sum_j s_j r(j/N) <= p configuration by configuration) is an infinite LP dual and does not reduce to single-window bounds.

Evidence: RESULTS-pair-ceiling.md sections '1. Joint scalar trace constraints collapse' and '2. Full bandwidth-one certificates do not reduce to one window' (lines 31-66); HANDOFF.md 'What is now closed, and what is left' Prior art: unsearched Reused in: hunts/frontier_math/RESULTS-frontier-math.md; hunts/wide_search/HANDOFF.md Why it travels: Closes a whole class of 'combine many windows' proposals in one line and tells any successor exactly what information a non-collapsing relaxation must keep.

Higher xi derivatives and moments

Exact resolvent identities for xi derivatives, resummed Lambda-convolutions, mean-value inputs and the moment obstructions to counting simple zeros.

Exact distinct-cycle Möbius corrections with overlap statistics

identity | hunts/cycle_moments/ | grade: D_3 identity and the frame example Lean-checked (AXLE, standard axioms); D_4 identity ordinary proof with exact-fraction enumeration checks

For complex symmetric N×N K with K_ii = 1, M_j = tr K^j, S = Σ_i(Σ_j K_ij²)², Q = Σ_ij K_ij⁴, and D_j the ordered j-cycle sum over pairwise distinct indices: D_3 = M_3 − 3M_2

Π(−1)^{|B|−1}(|B|−1)!, with the opposite-pair identification giving S and the double-opposite giving Q, not power traces). With V_3 = Σ_{i,j,k distinct} K_ij²K_ik², S = 2M_2 + Q − 2N + V_3 and Δ_4 = −D_4 − 2V_3 + M_2 − Q. Replacing S and Q by power traces is an incorrect simplification: the frame v_1=(1,0), v_2=(0,1), v_3=(c,s), v_4=(−s,c) has K² = 2K (all M_j = 2^{j+1}) but D_4 = −8c²s².

Evidence: MOBIUS-CORRECTIONS.md §1–§3 (eqs. 1–5); DistinctCycles.lean (D_3 over any commutative ring), IsospectralCycles.lean (the frame example); distinct_cycles.py Prior art: cited: Rudnick–Sarnak §4 (partition-lattice inversion, eq. 4.4–4.9) for the general mechanism; the explicit overlap-statistic form is the lab's Reused in: COUNTING-OVERLAP.md, OVERLAP-BOUND.md (S and Q as counting statistics) Why it travels: Anyone converting all-index cycle sums to distinct-index correlation sums (or back) needs exactly these coefficients; the overlap terms are where naive simplifications go wrong.

Exact resolvent identity for xi'''/xi'' and its frozen rational generating function Q(z), with strict sign alternation

identity | hunts/higher_xi/ | grade: hardened (three independent exact q_j generators and two scalable mean-square routes agree exactly through index 40; symbolic identity checked by resummed_bridge.py); third exact route in hunts/r_2ac05f

With U = xi'/xi = L + D, D = zeta'/zeta, direct differentiation gives xi'''/xi'' = U + (2UU' + U'')/(U^2 + U'), an identity with no coefficient-order cutoff; on Re s >= 1 + epsilon the denominator inverts in the weighted Dirichlet algebra via b(1,s) = A_0^{-1}, b(n,s) = -A_0^{-1} sum_{d|n, d>1} a_+(d,s) b(n/d,s). Freezing L' = L'' = 0, z = 1/L, and using the convolution atoms A = Lambda, B = Lambda log, C = Lambda log^2, the arithmetic part becomes exactly Q(z) = -A + (2Bz - (2A*B + C)z^2)/((1 - Az)^2 + Bz^2), whose coefficients q_n have the closed binomial form q_n = 2 sum_j (-1)^j C(n-1,2j) A^{*(n-1-2j)} * B^{(j+1)} - sum_j (-1)^j C(n-1,2j+1) A^{(n-2-2j)} * B^{*j} * C. Every word beta in q_n has entries in {0,1,2}, len(beta) + |beta| = n + 1, sign (-1)^{n - len}; nonzero pairings need equal length with powers summing to i - 1, so sgn C_{2,i} = (-1)^{i-1} exactly with no internal cancellation, and 2r + |beta| + |delta| - 1 = i reduces the mean-square denominator to i!. Corrected sequence 1, -8, 24, -32, 64/3, -64/3, 1216/45, ... (Bian's printed -4 dropped the multiplicity weights M(v_l)M(w_k) on thesis p. 71).

Evidence: hunts/higher_xi/RESUMMED-BRIDGE.md Sections 2-3 'Exact representation without geometric order', 'The corrected Q object is the frozen exact resolvent'; CORRECTED-F2.md 'Exact generating mechanism' and 'Exact coefficient fixture'; C2_PROVENANCE.md 'The one-line obstruction' and 'Independent exact routes' Prior art: cited: Bian 2008 thesis (SHA-256 pinned), Farmer-Gonek arXiv:0803.0425 (level-1 control reproduced exactly); the rational generating function is the lab's Reused in: hunts/r_2ac05f (fourth exact route, general-kappa form), URMS1-CLOSURE.md Section 6, RAMS2-CLUSTER.md (the cluster expression is algebraically identical to Q(z)), window_certificate.py, LEAN-FRONTIER.md Why it travels: Replaces a finite geometric-order expansion by an exact resolvent whose arithmetic part is a rational function of three convolution atoms; the same freezing procedure applies to any higher logarithmic derivative of xi or of another completed L-function.

Squarefree depth identity alpha_k(n) = log(n) Lambda_k(n)/k and the support-density origin of the lost logarithm

identity | hunts/higher_xi/ | grade: exact identities checked as formal polynomials in independent symbols log p for every squarefree integer through 70 and every depth (hardened by exactness); diagnosis ordinary derivation, self-reviewed

For the level-one resummed coefficient family a_T(n) = -Lambda(n) + sum_{k>=1} z_T^k alpha_k(n), alpha_k = Lambda_{k-1} * (Lambda log), marking one of the k identical convolution positions gives alpha_k(n) = log(n) Lambda_k(n)/k. On squarefree n = p_1...p_r, alpha_k(n) = (r-1)! log n prod_{p|n} log p if k = r and 0 otherwise, so distinct convolution depths never cross on squarefree support; cross-depth terms are collision terms requiring a repeated prime. Pure prime powers: alpha_k(p^a) = C(a,k)(log p)^{k+1}, hence a_z(p^a) = log p ((1 + z log p)^a - 2), with total square contribution O_r(x^{1/2}(log x)^2). Diagnosis: the elementary majorant |alpha_k(n)| <= (log n)^{k+1} summed over all integers gives sum (log n)^2 ~ x (log x)^2 already at depth zero, whereas the prime-power support gives sum Lambda(n)^2 = x log x + O(x); the extra logarithm is support density, created by replacing exact coefficients by a pointwise envelope (planted dense-support control retains x(log x)^2).

Evidence: hunts/higher_xi/RAMS1-ATTACK.md, Section 2 'Exact squarefree identity' (lines 76-140) and Section 3 'The missing logarithm is support density' (lines 142-183, classification table); URMS1-CLOSURE.md Section 2.1 Prior art: Farmer-Gonek-Lee cited for fixed-depth prime asymptotics; identity unsearched Reused in: URMS1-CLOSURE.md Section 2 (RAMS1 theorem), RAMS2-CLUSTER.md, URMS2-ATTACK.md Section 5 Why it travels: Any mean square of iterated Lambda-convolutions should be split by squarefree support before majorising; the identity gives the exact squarefree main term and localises all cross-depth interaction to repeated-prime strata.

Concave spectral score and quartic perturbation for simple-real counting

lemma | hunts/cycle_moments/ | grade: Lean-checked (AXLE, Lean 4.33.0, standard axioms) for the spectral Jensen step, the quartic's concavity/bounds and the composition under explicit diagonal-block hypotheses; adapted-basis construction and kernel-cycle identification are ordinary proofs, self-reviewed, external review pending

Let A = Σ v_i⊗v_i + Σ m_j w_j⊗w_j + 2Σ n_l(g_l⊗g_l − h_l⊗h_l) on a finite real inner-product space with unit v_i (s of them, 'simple'), unit w_j with m_j ≥ 2, and pairs with ‖g_l‖² − ‖h_l‖² = 1 (no positivity of A). For every globally concave f with f(0)=f(2)=0 and f ≤ 1: tr f(A) ≤ s. Proof via adapted subspaces U ⊂ V ⊂ W, the diagonal-sum estimate Σ_{j≤d_U} α_j ≥ 2d_U, and the direction-sensitive comparison f(α_j) ≥ Σ_i Q_ji² f(λ_i). Instance: q_t(x) = x(2−x)[1+t(x−1)+t²(x−1)²] is concave for |t| ≤ √(5/8) and ≤ D_t = 1 + t²/(4(1−t²)), giving s ≥ [2N − M_2 − tΔ_3 + t²Δ_4]/D_t with M_j = tr A^j, Δ_3 = M_3 − 3M_2 + 2N, Δ_4 = 2N − 5M_2 + 4M_3 − M_4; t=0 recovers s ≥ 2N − M_2. Asymptotically, if M_2/N → c_2, M_3/N → c_3 with δ_3 = c_3 − 3c_2 + 2 ≠ 0 and limsup M_4/N ≤ B < ∞, an explicit t = −sign(δ_3)·min(1/2, |δ_3|/(2(C+1))) gives a strict gain. For the Fourier realisation with even p and conjugation-invariant multiset Z, M_j = C_j(Z,p) the ordered cycle sums, nonreal pairs retained. At the Montgomery–Taylor profile δ_3 = (6k−5)(6k²+6k−1)/24 < 0 with k = cot(1/√2)/√2, sign proved by elementary inequalities 3/4 < k < 5/6.

Evidence: FINITE-THEOREM.md §2–§4, §7, §8, §10; Lean: SpectralJensen.lean, QuarticScore.lean, CycleMomentAssembly.lean (FORMAL-CHECKS.json) Prior art: cited: Lamzouri, arXiv 2609.02882 Prop. 2.1 (adapted subspaces, quadratic score); the general concave score, quartic family and asymptotic condition are the lab's; novelty not established Reused in: MIXED-MOMENTS.md, THIRD-MOMENT-OBSTRUCTION.md, COUNTING-OVERLAP.md (all build on §1 hypotheses) Why it travels: Turns any bounded family of matched power-trace moments into a simple-count lower bound with an explicit, tunable concave score. Index note: Kernel-checked core; the adapted-basis construction around it is ordinary proof.

Powerful-squarefree factorisation of resummed Lambda-convolution coefficients (RAMS1 mechanism)

lemma | hunts/higher_xi/ | grade: ordinary derivation, self-reviewed (exact algebra and finite checks in test_higher_xi.py; prime-measure asymptotic and dominated convergence are classical analytic inputs); 'remaining promotion gate is external mathematical review'

Every integer factors uniquely as n = qm with q powerful, m squarefree, (q,m) = 1; put j = omega(m) and B_{q,j}(z) = sum_{h>=0} C(j+h-1,h) z^h Lambda_h(q), the coefficient at q of (1 - zA)^{-j}. For j >= 1 the positive part of the level-one family factors exactly: sum_{k>=1} z^k alpha_k(qm) = (j-1)! z^j log(qm) prod_{p|m} log p B_{q,j}(z) (checked on every eligible integer through 55). For q <= y and r_y = log y/lambda_T <= r_0 < 1, 0 <= B_{q,j} <= (1 - r_0)^{-j}, and B_{q,j}(z_T) -> 0 for each fixed powerful q > 1; since every powerful q = a^2 b^3 with b squarefree, sum_{q powerful} 1/q <= zeta(2) zeta(3), so dominated convergence with the squarefree prime-measure majorant D^j y (log y)^{2j+1}/(j!(2j-1)!) gives sum_{qm<=y, q>1, j>=1} |a_T(qm)|^2 = o(y log y) uniformly on the compact band, and the j = 0 powerful integers contribute O(sqrt y (log y)^2). Result (RAMS1, assuming RH): sum_{n<=y}|a_T(n)|^2 = y log y Phi_1(r_y) + o(y log y), Phi_1(r) = 1 - 2r + 2 sum_{j>=1} (j-1)!/(2j)! r^{2j}, uniformly for r_min <= r_y <= r_0 < 1.

Evidence: hunts/higher_xi/URMS1-CLOSURE.md, Section 2 'RAMS1 theorem' with 2.2 'Unique support split', 2.3, 2.4 'Mixed repeated-prime strata vanish' (lines 58-201); hostile controls Section 7 Prior art: cited Farmer-Gonek-Lee for fixed-depth prime asymptotics, Chebyshev estimate; the split is unsearched Reused in: RAMS2-CLUSTER.md Section 7 'Repeated primes' reuses the split after the Gaussian mixture; BANDWIDTH-FORENSICS.md finite-prime peeling Why it travels: Turns a uniform-in-depth second-moment problem for multiplicative-like coefficient families into a squarefree main term plus a finite-harmonic-mass powerful correction; applicable to any Borel/Laplace-resummed Euler-product coefficient.

Spacing-sensitive mean value with log-frequency spacing: off-diagonal O(sum n |c_n|^2) independent of polynomial length

lemma | hunts/higher_xi/ | grade: Montgomery-Vaughan application: ordinary derivation, independently agent-audited (URMS2-051-AUDIT PASS on six gates); elementary substitute and rational witness margins: kernel-checked (Lean 4, no sorry per LEAN-FRONTIER.md); not externally reviewed

For distinct real frequencies lambda_n, the weighted Montgomery-Vaughan estimate is int_U^{2U} |sum_n c_n e^{-it lambda_n}|^2 dt = U sum |c_n|^2 + O(sum |c_n|^2/delta_n), delta_n the nearest-neighbour spacing. With lambda_n = log n, log(n+1) - log n >= 1/(n+1) gives delta_n^{-1} <= 2n for n >= 2, hence int_U^{2U} |sum_{n<=W} c_n n^{-it}|^2 dt = U sum_{n<=W}|c_n|^2 + O(sum_{n<=W} n |c_n|^2) with no condition W < U. Replacing delta_n^{-1} by the worst spacing W had forced gamma < delta < 1 against the tail's 2alpha < gamma, i.e. a deficit of exactly 51/50 - 1 = 1/50 at alpha = 0.51; retaining the spacings removes it. With alpha = 51/100, delta = 3/4, gamma = 21/20, epsilon = 1/100 the margins are 6/25 (mean value), 9/1000 (infinite tail), 1/4, 7399/10000, far-cutoff ratio 14/5. An elementary application-specific substitute is kernel-checked in Lean: for real magnitudes, sum_{m<n<=W} 2|c_m||c_n|/(log n - log m) <= 3 H_W sum_{n<=W} n c_n^2 (both orientations cost <= 6 H_W sum n c_n^2), with H_W <= 1 + log W connected to Mathlib; the extra log is absorbed by the strict power margin.

Evidence: hunts/higher_xi/URMS2-051.md, Section 2 'Exact failure of the old proof at 0.51' and Section 3 'The minimal replacement lemma' (lines 108-171), Section 6 'Rational witness at 0.51'; URMS2-051-AUDIT.md Section 1; LEAN-FRONTIER.md items 1, 2, 8, 9 Prior art: cited: Montgomery and Vaughan, Hilbert's Inequality, J. London Math. Soc. (2) 8 (1974) 73-82 (weighted form); the log-spacing specialisation and Lean substitute are the lab's Reused in: URMS2-051.md Section 7 (URMS2 through |alpha| <= 51/100), Section 8 (simplicity proportion >= 0.0147728663285376 under RH and the rebuilt bridge); LEAN-FRONTIER.md modules LogMeanValue, ComplexLogMeanValue, TwoRangeWeights Why it travels: Whenever a Dirichlet-polynomial mean square is truncated at length W >> U, using the actual 1/n spacing of log n instead of the worst spacing removes an artificial length restriction; the kernel-checked substitute is available for any real-coefficient instance.

Finite-prime peeling of the powerful Euler product to reach every compact coefficient band

lemma | hunts/higher_xi/ | grade: ordinary derivation, self-reviewed, with exact rational parameter witnesses

Fix any finite rho_0 and choose 0 < R < 1 so small that the Laplace rate E_R(rho_0) < 1/2. Put P = R^{-2} and split the powerful support into primes <= P and primes > P. The tail product prod_{p>P}(1 + sum_{e>=2} R^{-2e}/p^e) converges (each local series converges and its first term is O_R(p^{-2})), with Laplace rate at most E_R(rho_0) < 1/2. The finitely many primes <= P each get their own radius R_p > p^{-1/2}; their exponent sums converge and the total logarithm of the finite set is constant, so its contribution to the Laplace rate is o(1) as z -> 0 and is absorbed by the remaining half of the gap. Dominated convergence on the powerful strata then runs as before, establishing RAMS2 on every compact fixed rho band (previously only rho <= 1/10). With RAMS2 at target, intermediate and far cutoffs, choosing delta = (2alpha+1)/2, gamma = (2alpha+delta)/2, beta = (alpha+gamma)/2 gives alpha < beta < gamma < delta < 1 and 2alpha < gamma, so RC2 and URMS2 hold on every compact band 0 < |alpha| < 1/2 (exact witness at alpha = 0.499 and split_powerful_witness(21/10) in exact arithmetic).

Evidence: hunts/higher_xi/BANDWIDTH-FORENSICS.md, Section 7 subsection 'Finite-prime peeling removes the fixed-rho endpoint' (lines 441-491); Lesion 4 in URMS2-051.md Section 9 Prior art: unsearched Reused in: URMS2-051.md Section 6 (RAMS2 at far-cutoff ratio 14/5), RESULTS-higher-xi.md 'URMS2 attack' Why it travels: Standard-shaped but explicitly quantified: when a uniform Euler-product estimate fails only because of finitely many small primes, give those primes individual radii and absorb their bounded logarithmic mass; converts a fixed-band result into every compact band.

Majorant-with-equality-recurrence bypass for tight aggregate inequalities (hprime gate)

lemma | hunts/higher_xi/ | grade: R (equality recurrence) and mass_le_of_dominated_majorant kernel-checked (ZetaLean/MajorantBypass.lean); L1, L2, L10 kernel-checked; the remaining chain is ordinary derivation with every link checked numerically at X = 250 and X = 5000 (margins 3.9x to 3582x); obstruction arguments ordinary, self-reviewed with measured crossing

The literal gate (2j)(2j+1) D_j(X) <= B(X) A_j(X) puts the true distinct-prime-support mass A_j on the right and needs an order-uniform lower bound on it; upper Chebyshev cannot supply that, and two-sided elementary Chebyshev (theta_ge with log 2 against log

  1. loses 2^j across j up to about log X/log log X, so the gate cannot close by a

Chebyshev route (obstruction, two independent arguments, measured crossing at X about 2e4 where 12 A_2^small/A_1(sqrt X) exceeds B(X)). Retarget: exhibit an explicit majorant M_j(X) = C_0^j X L^{2j+1}/(j!(2j-1)!) with C_0 = log 16, L = log X, prove A_j <= M_j (Route A chain L1-L10: theta(y) <= (log 4) y, N(y) = sum_{p<=y}(log p)^2 <= C_0 (e^u(u-1)+1), Abel comparison against int t e^t g(t) dt, ordered-tuple recursion W_{j+1}(V) = sum (log p)^2 W_j(V - log p), convolution identity int_0^V t e^t E_j(V-t) dt = E_{j+1}(V), E_j(V) <= e^V V^{2j-1}/(2j-1)!, V_j <= W_j/j!, A_j <= L^2 V_j), and note that M satisfies the recurrence with equality: (j+1)(2j)(2j+1) M_{j+1} = C_0 L^2 M_j because the factorial denominator was built so that (j+1)(2j)(2j+1) is the step ratio. Domination then transfers the downstream display A_r(X) <= (log 16)^r X (log X)^{2r+1}/(r!(2r-1)!). Route B (one-dimensional Abel, constant 60 log 4) is kept for its formalisation advantage.

Evidence: hunts/higher_xi/HPRIME-ROUTES.md, Section 0 'What both chains agree on', Section 1 'Route A' (table L1-L10, M, R), Section 3 'Why the literal gate does not close'; CROSS-ARM-TRANSFER.md Section 3 (the three-step pattern) Prior art: Mathlib Chebyshev.theta_le_log4_mul_x cited; pattern unsearched Reused in: RAMS2-CLUSTER.md (consumes the display near line 427); proposed to hunts/frontier_math in CROSS-ARM-TRANSFER.md, where the frontier arm's reply records that the transfer does not survive there Why it travels: When a target inequality is tight and every local charging argument dies, dominate the true object by a majorant that satisfies the required recurrence with equality by construction; the tightness becomes the majorant's identity. The obstruction half tells you when to stop trying Chebyshev-only routes.

Factorial-permanent tail majorant with rational atom weights and a two-step geometric ratio

bound | hunts/higher_xi/ | grade: exact rational majorant computed deterministically (hardened by exactness); the ratio-monotonicity argument is ordinary derivation, self-reviewed

For the pairing K(beta,delta) = sum_{pi in S_r} prod_j (b_j + d_{pi(j)} + 1)!/(2r + |beta| + |delta| - 1)! with entries in {0,1,2}, set w_0 = 1, w_1 = 5/2, w_2 = 11; all nine inequalities (a+b+1)! <= w_a w_b hold, so the permanent for two length-r words is at most r! w(beta) w(delta). Retaining exact word length through index 101 and, for n = i - 1 >= 25, R = floor(n/2), L = ceil(n/4), M = R - L + 1, a = 5/2, K = 47/4, using C(n,j) <= 2^n and monotonicity of r! a^{-2r}: |C_{2,i}| <= B_i = (n-1) M 4^n K^2 a^{n-2-2R} R!/(n+1)!. The two-step ratio B_{i+2}/B_i splits by n mod 4 into four rational expressions with decreasing differences (numerators -32, -2(16m^4+...), -8(16m^3+...), -32(8m^3+...)), largest start rho = 5202/64375 < 0.081 for n >= 101, so both parity tails are geometric and the series is entire. Exact tail bound E_40(alpha) = sum_{i=41}^{101} U_i alpha^i + (B_102 alpha^102 + B_103 alpha^103)/(1 - rho alpha^2); at bandwidth one sum_{i>40}|C_{2,i}| |alpha|^i < 3.279e-9, and 4.76344632220668 < sum_{i>=1} C_{2,i} < 4.76344632876331. For any integrable window v the omitted autocorrelation tail is at most E_40(1) ||v||_1^2.

Evidence: hunts/higher_xi/CORRECTED-F2.md, 'Tail majorant' (lines 116-185) and 'Direct weighted consequence' (lines 187-194); TAIL-BARRIER.md 'Split result'; checker corrected_form_factor.py Prior art: unsearched Reused in: window_certificate.py (2 - D > 0.9234015), URMS2-051.md Section 8 (0.51 window), BANDWIDTH-FORENSICS.md Section 9 (positivity of F_2 on [0,1/2]) Why it travels: A general recipe for bounding an infinite series of factorial-permanent pairings: dominate each factorial by a product of per-letter weights, keep exact lengths for a finite prefix, then bound the remainder by a proven-decreasing two-step ratio.

Isospectral multisets with different simple counts under one fixed kernel

construction | hunts/cycle_moments/ | grade: kernel-checked for the two explicit matrices (AXLE, standard axioms); Fourier realisation ordinary proof

Density p(u) = 1 + cos(2πu) + cos(4πu)/4 on |u| ≤ 1/2 (= 1/4 + (1+cos2πu)²/2 ≥ 1/4, even, integral one) has K(0)=1, K(±1)=1/2, K(±2)=1/8, K(n)=0 for |n| > 2. Multisets Z_A = (0,0,0,3,6) (multiplicities 3,1,1, s=2) and Z_B = (0,0,1,1,4) (multiplicities 2,2,1, s=1) both have weighted location Gram spectrum (3,1,1), hence M_j = 3^j + 2 for every j ≥ 1, yet s differs; S = 29 vs 26, Q = 11 vs 19/2, D_4 = 0 vs 9/2. Translating h independent blocks by 10j (cross-block kernel entries vanish) keeps all normalized power traces fixed while s/N ranges over [1/5, 2/5]. Proves the full power-trace sequence does not determine the simple count even with the kernel fixed; the rescaling q(u) = Lp(Lu) makes the frequency support arbitrarily narrow.

Evidence: COUNTING-OVERLAP.md §2–§4 (eqs. 3–7), §6; CountingOverlap.lean (equal power traces for every natural exponent, simple counts 2 and 1); counting_overlap.py (G³ = 4G² − 3G, 60-digit Fourier quadrature) Prior art: none cited; 'original finite constructions' Reused in: OVERLAP-BOUND.md §4 equality cases; README.md Why it travels: A ready-made counterexample kit for any claim that a spectral (power-trace) summary controls a simple-point count; also a test fixture for overlap-aware bounds.

Exact rational Legendre window with an exact pointwise floor

construction | hunts/higher_xi/ | grade: exact rational calculation (hardened by exactness); conditional on the unproved analytic bridge for any statement about zeros of xi''

v(s) = P_0(2s) - (185616/10^6) P_2(2s) - (111471/10^6) P_4(2s) - (20783/10^6) P_6(2s) on |s| <= 1/2. Since |P_j(x)| <= 1, v has the exact positive floor 68213/100000, integral one and L1 norm one, so it is an admissible spectral factor and the tail allowance E_40(1) ||v||_1^2 applies with ||v||_1 = 1. Exact polynomial integration of the first 40 corrected coefficients gives D_40 = 1.0765984703331668..., and with the full tail allowance 2 - D > 0.923401526388517... > 0.9234015; the floating optimizer reaches 0.923401531890862, only 5.51e-9 higher. Caveat the hunt states: v(+-1/2) = 68213/100000

0, so the autocorrelation has only a simple zero at the band edge, which is why the

window cannot repair the source's endpoint estimate.

Evidence: hunts/higher_xi/CORRECTED-F2.md, 'Direct weighted consequence' (lines 197-227); BRIDGE-CLOSURE.md Section 6 'Why the target weighting does not repair the source bound'; window_certificate.py Prior art: unsearched Reused in: URMS2-051.md Section 8 uses a different (indicator) window; RESULTS-higher-xi.md 'Weighted window result' Why it travels: Low-degree rational Legendre combinations give windows with exact floors and norms, so simplicity-proportion functionals can be evaluated in exact arithmetic with a uniform tail allowance instead of a float optimizer.

Invariance claims need a control that moves the invariant's parameter (fault-injection power measurement)

control | hunts/r_2ac05f/ | grade: hardened for the identity (two independent exact derivations plus external kappa = 1 anchor); the control-power measurement is exact rational arithmetic

When an inherited lemma asserts a quantity is invariant in a parameter (Bian Lemma 12: C_{kappa,2} = -4 for all kappa), a control that only checks the anchored instance (the Farmer-Gonek kappa = 1 row) has zero power against a defect in the invariance. Measured by planting: forcing the x^1 coefficient of Qhat_kappa to g instead of kappa g (the defect under audit) leaves the kappa = 1 control passing and reproduces the published wrong C_{2,2} = -4; corrupting the pairing denominator by one factorial step fails the control, so the control is not vacuous. Stated rule: compute the quantity independently for kappa = 1, 2, 3 and assert the values are not equal. The underlying identity: the x^1 coefficient of Qhat_kappa is kappa g, so C_{kappa,2} = 2(<q_0,q_1> + <q_1,q_0>) = -4 kappa (derived -4, -8, -12, -16, -20 for kappa = 1..5).

Evidence: hunts/r_2ac05f/RESULTS.md, Section 3 'Why the other table is wrong, in one line' (lines 85-104) and Section 4 'The control that would have caught it' (table lines 136-139, rule lines 147-164); fault_check.py Prior art: Bian 2008 thesis Lemma 12 (the corrected claim); Farmer-Gonek arXiv:0803.0425 (anchor); the general-kappa correction 'recorded nowhere but here' Reused in: Matches the failure shape recorded from run 726a6b3f on finite_height_spacing_experiment; confirms hunts/higher_xi C2_PROVENANCE.md Why it travels: A one-line design rule for validating any audit that inherits a universality/invariance lemma as an axiom, with a demonstrated fault-injection procedure for measuring a control's power.

Third-moment obstruction block diag(b+2, b+2, −2b)

obstruction | hunts/cycle_moments/ | grade: scalar power sums, discrepancy, slack and their signs Lean-checked; vector construction and limit ordinary proof

Under the signed-vector hypotheses, matched second and third moment limits alone cannot improve the quadratic counting proportion, even with strictly negative third discrepancy. Block: g_1 = √(1+b/2)e_1, h_1 = √(b/2)e_3, g_2 = √(1+b/2)e_2, h_2 = √(b/2)e_3 gives A_b = diag(b+2,b+2,−2b), N_b = 4, s_b = 0, M_2 = 6b²+8b+8, M_3 = −6b³+12b²+24b+16, M_4 = 18b⁴+16b³+48b²+64b+32, so Δ_3 = −6b²(b+1) < 0 while the slack E_b = M_2 − 2N_b + s_b = 6b²+8b, with −Δ_3 − (b/2)E_b = b²(3b+2) ≥ 0. Adding it with b_L = (dL/6)^{1/3} to a baseline of s_L simple and r_L doubled orthonormal vectors gives, for any 1 < c_2 < 2, d > 0: M_2/N → c_2, Δ_3/N → −d, s/N → 2 − c_2, while M_4/N → ∞. Hence a finite normalized fourth-moment bound is a substantive hypothesis, and any conclusion liminf s/N ≥ 2 − c_2 + ε from two moment limits is impossible in this class.

Evidence: THIRD-MOMENT-OBSTRUCTION.md §1–§3; ThirdMomentObstruction.lean (AXLE request 660ac38d…, standard axioms) Prior art: none cited; 'original finite construction; novelty has not been established' Reused in: README.md summary; closes the second+third-moment shortcut for the quartic route Why it travels: A concrete isospectral-type gadget showing which moment hypotheses are load-bearing; the shared negative direction trick generalises. Index note: Kernel-checked scalar part.

Heat flow and de Bruijn-Newman constants

Backward heat flow of xi-type functions: landing times as lower bounds, contour and winding instruments that refuse rather than round, and the calibrations that fix the frame.

Shave integral for the landing time of a crowded pair, with isolated, leading-order and dense-sea limits

identity | hunts/lambda_dh_exact/ | grade: measured (one float route; derivations ordinary, self-reviewed)

Isolated pair p_0 = (z-x)^2 + y_0^2: p_t = p_0 - t p_0'' = (z-x)^2 + y_0^2 - 2t, so t*(isolated) = y_0^2/2 exactly (verified to about 1e-13 by exact polynomial flow; backward cross-route via zeta.heatflow.polynomial_heat_flow gives -a^2/2). Frozen neighbours: with Q = -y^2 and S(y) = sum_a 1/((x-a)^2 + y^2), dy/dt = -(1 + 2 y^2 S(y))/y, hence t* = int_0^{y_0} y dy/(1 + 2 y^2 S(y)) (); S > 0 for real neighbours proves t < y_0^2/2 (sign theorem). Leading term: relative shave = 1 - t*/(y_0^2/2) = S_0 y_0^2 + O(y_0^4), S_0 = sum_a 1/(x-a)^2; two neighbours at distance d give 2(y_0/d)^2. Dense sea of density rho = 1/h: S -> pi rho/y, dy/dt = -1/y - 2 pi rho, t* = [V - log(1+V)]/(2 pi rho)^2 with V = 2 pi rho y_0; for V >> 1, t* -> y_0/L, L = 2 pi/h. Frozen-neighbour bias measured: () under-predicts t by 1% (light crowding) to about 17% (heavy), because neighbours repel. Four-root control: landing time t_+ = [(Y^2 - a^2) + sqrt((Y^2-a^2)^2 + 12 a^2 Y^2)]/12 for (z^2-a^2)(z^2+Y^2).

Evidence: hunts/lambda_dh_exact/MISSION.md, Section 3 'The isolated-pair law, derived and verified' (lines 188-222), Section 4.2 'The shave, in one integral', 4.3, 4.4 'The deep regime', 4.7 'The bias of the frozen-neighbour approximation, measured' (table lines 370-378); Section 2 quartic control (lines 173-178) Prior art: unsearched; N-body law attributed to zeta/heatflow.py four-way sign check Reused in: Explains hunts/flow_repair's empirical 'shave tracks y_0^2 times local density' (P1/P2); calibrated against its nine landings (rms 0.54%) and a census holdout (0.79%) Why it travels: Closed-form landing-time model for any conjugate pair under heat flow given the surrounding real zero configuration; the sign theorem alone converts every measured landing into a strict inequality against y_0^2/2.

Threshold bracket via upward-closed, closed set S for Hermitian (not necessarily even) heat kernels

lemma | hunts/dh_minus_heat | grade: ordinary derivation, self- and model-reviewed; numerical inputs enclosure-carrying

Let S be the set of real t at which H_t has all zeros real. If the whole-line Fourier kernel is Hermitian (K(u) = conj K(-u), e.g. K = -2i g with g real odd), then de Bruijn's strip-contraction theorem (Dobner's extended-Selberg-class form) makes S upward closed; locally uniform dependence on t plus Rouche on a small disk with zero-free boundary makes S closed; so S = [Lambda, inf). One enclosed non-real zero at t_0 gives Lambda > t_0; one zero-free strip |Im z| < b gives Lambda <= b^2/2. No trajectory tracking and no even-kernel restriction needed. Applied: 217/200 < Lambda_minus <= 567009/320000 (narrow frame); Lambda_plus <= 1/2.

Evidence: RESULTS.md section 5 'Threshold and strict lower bound', lines 232-251; section 1 for the odd-kernel construction, lines 77-101 Prior art: cited: de Bruijn strip contraction, Dobner arXiv:2005.05142v2; bounded search for second-DH heat bounds found no bracket (searched-and-absent, not claimed as novelty) Reused in: none recorded Why it travels: Gives a two-sided de Bruijn-Newman-type bracket for any L-function-like object with odd or complex theta kernel from one local disk and one zero-free strip.

Lemma M2: uniform bound on |H_t''| over a half-strip by a shifted u-contour

lemma | hunts/lambda_dh_bounds/ | grade: hardened (prose proof plus decided Arb arithmetic at 300 bits; every constant a reported ball, every hypothesis a decided predicate); not kernel-checked; read by no human; weakest input is the cited evenness Phi_DH(-u) = Phi_DH(u)

Let Phi_DH(u) = 4 e^{3u/2} sum_n n a_n exp(-pi n^2 e^{2u}/5) with |a_n| <= 1 and Phi_DH even on the strip |Im u| < pi/4, G(u) = e^{t u^2} Phi_DH(u), H_t(z) = int_0^inf G(u) cos(zu) du, t >= 0, and R(x_lo, y_hi) = {Re z >= x_lo > 0, |Im z| <= y_hi}. If (H1) 0 < v < pi/4, (H2) S >= 1 and q_S = exp(-(pi/5) e^{2S} cos 2v) <= 29/100, (H3) c = (pi/5) cos 2v - (t S^2 + beta S) e^{-2S} > 0 with beta = 7/2 + y_hi, then H_t is entire and for all z in R, |H_t''(z)| <= M2 := e^{-x_lo v - t v^2} (J + T), where J = int_0^S (s^2+v^2) e^{t s^2} 4 e^{3s/2} Omega(e^{2s} cos 2v) cosh(y_hi s) ds is bounded by an 800-panel interval sum using two Omega majorants (unimodal comparison 1/(2a) + 2e^{-1/2}/sqrt(2a) and geometric q/(1-q)^2 <= 2q for q <= 29/100), and T = 4 e^{2v} e^{-cV}/(cV), V = e^{2S}. Proof: differentiation under the integral with explicit dominant, Cauchy on the rectangle 0, R, R+iv, iv with the far side bounded by 8e^{2v}e^{-c e^{2R}}, vertical legs cancel by evenness of G (Fact E, the one load-bearing citation), then majorise the two rays. Corollary: affine interpolation error on any axis-parallel segment of half-length h is <= M2 h^2/2 (Green's-function form, valid for complex g). Decided at v = pi/4 - 1/256, S = 5: M2 <= 1.1887e-78 (t = 23/400) and 1.1371e-78 (t = 36/625); decided cushion over the true sup at most 55.65 and 53.79. Key point: pointwise ball enclosure of H_t'' cannot give a uniform bound (a z-ball of radius 1e-24 already contains 0), because at Re z ~ 240 the integrand is O(1) and the integral ~1e-80; the contour shift puts the factor e^{-x_lo v} outside the integral.

Evidence: M2-LEMMA.md §2 (lines 74-125, statement and decided instantiation), §3 Steps 0-4 (lines 131-416, proof), §4 (corollary, lines 452-475), §7 (cushion table, lines 527-600), §8 (why ball evaluation cannot give a uniform bound, lines 604-631); m2_lemma.py Prior art: cited for inputs (Cauchy, Morera, dominated convergence, Hecke theta transformation for the functional equation); the lemma itself unsearched Reused in: hunts/lambda_dh_exact/RESULTS.md §5 item 6 (inherits the M2 blind-spot lesson); docs/29-de-bruijn-newman-davenport-heilbronn.md; hunts/dh_minus_heat/RESULTS.md uses the same Taylor-remainder M2 r^2/2 pattern at a small disk Why it travels: Any argument-principle count of a heat-deformed Fourier integral H_t at large Re z needs a uniform second-derivative bound that ball arithmetic cannot supply; the recipe (shift by v just inside the kernel's analyticity strip, cancel legs by evenness, panel-plus-closed-form tail) transfers to any even kernel holomorphic on a strip.

Phase obstruction for a two-L-function combination: Theta(sigma) < tau gives a zero-free half-plane

lemma | hunts/lambda_dh_bounds/ | grade: decided constants plus exact elementary analysis (hunt's own wording); the composite Lambda_DH <= Delta^2/2 is 'cited plus decided, weakest step cited' (de Bruijn 1950 Theorem 13)

Let chi be the odd primitive character mod 5 with chi(2) = i, A = (1 - i kappa)/2, so the Davenport-Heilbronn coefficients satisfy a_n = A chi(n) + conj(A) conj(chi)(n) and f(s) = A L(s,chi) + conj(A) L(s,conj chi) for Re s > 1. Since both Euler products converge absolutely and do not vanish there, f(s) = 0 iff R(s) := L(s,chi)/L(s,conj chi) = -conj(A)/A = exp(i(pi + 2 arctan kappa)), whose smallest absolute argument is tau = pi

(1+u)/(1-u) with |u| = p^{-sigma}. Moebius-disc lemma: for |u| <= r < 1, |arg (1+u)/(1-u)| <= 2 arctan r, with equality at u = +-ir, and the argument-maximising point has modulus exactly 1 (centre C = (1+r^2)/(1-r^2), radius rho = 2r/(1-r^2), C^2 - rho^2 = 1), so the |R| = 1 constraint cannot sharpen the bound. Hence if Theta(sigma) := sum_{p = 2,3 mod 5} 2 arctan(p^{-sigma}) < tau then f has no zero on Re s = sigma, and since Theta is decreasing one decided sigma* closes the half-plane Re s >= sigma*; the gamma factor and F(s) = F(1-s) carry it to the completed F and give the strip |Im z| < Delta = sigma* - 1/2. Theta is decided with no prime counting: head summed exactly from a sieve to P, tail closed by the Euler-product identity T1 - Tchi = 2Q + (E_chi - E_1) where all even-k terms cancel because chi5(p)^2 = 1, leaving eps3 = (2/3) P^{1-3 sigma}/((3 sigma - 1)(1 - P^{-2 sigma})) (a factor 5.7e5 better than bounding the two tails separately). Decided on both backends (flint 192 bits, mpmath.iv dps 40, P = 10^5): sigma* = 1.12036249819 exactly, Delta^2/2 = 0.19242481458026887663805 narrow, improving the coefficient-domination abscissa 1.39513615823511 by factor 2.082.

Evidence: STRIP2.md §3.1-3.6 (lines 146-268), §4.1-4.3 (tail identities and backends, lines 272-348), §5 (decided numbers, lines 352-412), §6 (eight abort-on-fail controls plus the tau_- control reproducing Bombieri-Ghosh's 2.3822861089 to ten digits, lines 416-464); RESULTS.md §3.6 (lines 598-755) Prior art: searched-and-found: Bombieri and Ghosh 2011 Theorem 7 is the same equation Theta(sigma) = tau term for term; the hunt rederives only the necessary half (upper bound) without Bohr/Kronecker theory and states 'what is new here is the grade and not the number' Reused in: hunts/dh_minus_heat/RESULTS.md (lines 210-218: same Euler-factor argument for the second DH function with a simpler tail 2 P^{1-sigma}/(sigma-1)); docs/29-de-bruijn-newman-davenport-heilbronn.md Why it travels: Any Dirichlet series that is a fixed linear combination of two L-functions with Euler products gets a zero-free half-plane from a phase budget rather than from L1 coefficient domination; the k=2-cancelling tail identity is a general trick for sums over a residue class of primes.

Landing times are unconditional lower bounds for the DH de Bruijn-Newman constant; equality needs no-creation

lemma | hunts/lambda_dh_exact/ | grade: Section 1: ordinary derivation, self-reviewed, resting on cited theorems; Section 2 (NC): mechanism only, sampled not proved (hunt's own words)

Phi_DH is real and even (evenness measured to 4.2e-51), so nonreal zeros of H_t come in conjugate pairs. By Dobner (arXiv:2005.05142, Thm 1) the set {t : H_t has only real zeros} is a closed half-line [Lambda_DH, inf). For a pair P present at t = 0 define t*(P) = inf{t >= 0 : P is no longer nonreal}; it is finite by de Bruijn/Newman-Wu, y_max(t) <= sqrt(max(Delta^2 - 2t, 0)). For t < t*(P), H_t has a nonreal zero so t < Lambda_DH; hence Lambda_DH >= t*(P) for every pair and Lambda_DH >= sup_P t*(P), unconditionally. Equality Lambda_DH = sup_P t*(P) holds under (NC): forward flow never drives two real zeros off the axis. (NC) is the time-reversed Sturm/Angenent lap-number statement (since u(x,t) = H_t(x) solves du/dt = -d^2u/dx^2 on the real line), which the hunt states is a mechanism, not a proof for entire functions; sampled 400/400 gated random polynomial configurations with no decrease in real-zero count.

Evidence: hunts/lambda_dh_exact/MISSION.md, Section 1 'Why every landing time is a lower bound for Lambda_DH' (lines 89-113) and Section 2 'Does Lambda_DH equal that sup? The no-creation step' (lines 117-184) Prior art: cited: Dobner arXiv:2005.05142; de Bruijn 1950 Thm 13 / Newman-Wu Thm 7; Sturm 1836, Matano 1982, Angenent 1988; Polya-Wiman theory named as the standard route Reused in: Stated as the reason hunts/flow_repair's nine landings are floors and hunts/lambda_dh_bounds could decide one Why it travels: Fixes exactly which half of a landing-time program is unconditional for any real even entire function of the de Bruijn class, so floors can be published without the no-creation step.

Euler-phase zero-free half-plane for a two-character combination, with integer tail

bound | hunts/dh_minus_heat | grade: enclosure-carrying (Arb and mpmath.iv, arctangent Taylor series with explicit remainder); the Euler-factor argument itself is inherited from ../lambda_dh_bounds/STRIP2.md, the simpler tail is new here

For D(s) = A L(s,chi) + conj(A) L(s,conj chi) with chi the primitive quartic character mod 5, a zero with Re s = sigma > 1 needs the Euler-product ratio to reach phase 2 atan(kappa), kappa = tau_plus. Primes p = 1,4 mod 5 and p = 5 contribute phase 0; each p = 2,3 mod 5 contributes at most 2 atan(p^{-sigma}). Zero excluded if Theta(sigma) = 2 sum_{p=2,3 mod 5} atan(p^{-sigma}) < 2 atan(kappa); tail above P bounded by 2 P^{1-sigma}/(sigma-1). Decided at sigma = 953/400 with sieve through P = 10000 on two interval backends; monotone in sigma; reflected by the functional equation to give all zeros in |Im z| < 753/400.

Evidence: RESULTS.md section 4 'Upper strip from prime phases', lines 195-230 Prior art: cited as inherited from hunts/lambda_dh_bounds/STRIP2.md; literature unsearched Reused in: inherited from lambda_dh_bounds, sharpened here with the integer tail Why it travels: Any linear combination of two L-functions with conjugate coefficients gets an explicit zero-free half-plane from a finite prime sieve plus a one-line tail.

Contour-moment pair tracker in the collision-safe discriminant variable

construction | hunts/flow_repair/ | grade: measured (mpmath floats with cross-route defects; 'no enclosure claims are made anywhere in this hunt')

For a conjugate pair of zeros of an entire function H_t inside a circular contour, compute the power sums q_1 = z_1 + z_2, q_2 = z_1^2 + z_2^2 by contour integration of z^k H_t'/H_t, and track the discriminant Delta = 2 q_2 - q_1^2 = (z_1 - z_2)^2, Q = Delta/4 (Q = -y^2 off the axis, Q > 0 on it). Q is analytic through the landing Q = 0, so the landing time t* under the backward heat flow H_t = exp(-t d^2/dz^2) H_0 is read as the root of Delta(t) by bracketing, with the contour's winding number required to be exactly 2. Companion N-body law: with S(y) = sum_a 1/((x-a)^2 + y^2) over neighbours, dQ/dt = 2 - 4 Q sum_a 1/((x-a)^2 - Q), from dz_k/dt = 2 sum_{j != k} 1/(z_k - z_j). Refusals are loud: a clipped contour reports N = 1 and refuses; a grazing contour returns non-integer winding (about 1.8e9) and refuses. Precision response: t* for pair 1 identical to the last digit across dps 44/54/70 and 96/192 nodes.

Evidence: hunts/flow_repair/NOTES.md, Section 1 'The headline: the repair times, measured', Section 3 'The null control', Section 4 'Lesions', Section 5 'Precision response'; MISSION.md lines 57-79 Prior art: searched-and-found for the N-body law (Calogero-Moser: Cuenca-McSwiggen arXiv:2606.06859, Hall-Ho arXiv:2308.11685); no tabulated DH de Bruijn-Newman constant found; contour-moment discriminant tracking unsearched Reused in: hunts/lambda_dh_exact (MISSION.md Section 4.1 re-derives dQ/dt from the N-body law; uses the nine landings as calibration data); hunts/lambda_dh_bounds (decided landing referenced in lambda_dh_exact RESULTS.md Section 4) Why it travels: Works for any entire function under the heat flow (DH, Epstein, zeta): the pair discriminant is the only quantity analytic through a real-axis collision, and the winding gate makes half-pair and grazing failures impossible to feed silently.

Frame scaling law for de Bruijn-Newman constants: Lambda/Delta^2 is the frame-free quantity

calibration | hunts/lambda_dh_bounds/ | grade: derived (the hunt's word), with every row measured numerically; the earlier claim that the two deformations 'share Lambda exactly' was wrong by a factor 4 and is preserved in a correction box

For any admissible kernel Phi, a > 0, c != 0, define Phitilde(u) = (c/a) Phi(u/a) and Htilde_t(z) = int_0^inf e^{t u^2} Phitilde(u) cos(zu) du. Then Htilde_t(z) = c H_{a^2 t}(a z), so the zero set scales by 1/a, Delta -> Delta/a and Lambda -> Lambda/a^2; hence Lambda/Delta^2 is invariant and de Bruijn's threshold t >= Delta^2/2 can be quoted frame-free while Lambda and Delta separately cannot. Applied with a = 1/2, c = 1/2: Phi_F(u) = Phi_DH(2u), xi_t^F((1+iz)/2) = H_{t/4}(z/2), so Lambda(wide: s = (1+iz)/2, Newman/Rodgers-Tao/Polymath 15/Dobner/zeta.heatflow) = 4 Lambda(narrow: s = 1/2 + iz, Stopple/Newman-Wu kernel), Delta(wide) = 2 Delta(narrow). Verified numerically row by row at a = 1/2, 2, 13/10 (relative defects <= 2.2e-29) and with Dobner's Phi_F computed only from his own definition by Fourier inversion. House rule: always print the frame with the number, print both values or the factor 4 with its direction, never compare a zeta record across frames unconverted.

Evidence: FRAME.md §2 (lines 179-210, derivation), §3 (lines 214-236, explicit conversion), §4 (conversion table, lines 240-254), §5 (numerical verification), §8 (lines 455-464, what to write every time); RESULTS.md §0 (lines 33-61); THEOREM13.md §6 'Dobner's frame is not this frame: the factor is 4' Prior art: cited: Stopple arXiv:1301.3158, Dobner arXiv:2005.05142, Rodgers-Tao arXiv:1801.05914, Polymath 15 arXiv:1904.12438, Newman-Wu 2020; the scaling law and factor-4 dictionary derived in-tree Reused in: docs/29-de-bruijn-newman-davenport-heilbronn.md §1 'the frame trap'; hunts/dh_minus_heat/RESULTS.md (reports narrow and wide by the same factor 4); SEPARATION.md (frame-invariant cross-multiplication); zeta.heatflow.lambda_facts() Why it travels: Every future Lambda-type constant for any function in the extended Selberg class must be tagged with its frame; the invariant Lambda/Delta^2 is the safe way to compare bounds across papers.

Delta^2/2 calibration of de Bruijn Theorem 13 by polynomial heat flows (derived, never recalled)

calibration | hunts/lambda_dh_bounds/ | grade: measured (one float route each; the hunt's own words)

To pin the constant in de Bruijn's Theorem 13 (all zeros of H_0 in |Im z| <= Delta implies all zeros of H_t real for t >= Delta^2/2 under the multiplier e^{t u^2}) without trusting memory: first check that the e^{tu^2} multiplier under the integral and the finite polynomial series sum_k (-t)^k/k! p^{(2k)} both satisfy the backward heat equation dG/dt = -d^2G/dz^2 (residual 0 at dps 30 by quadrature; 4.2e-6 by float64 central differences consistent with h^2), which licenses calibrating the integral multiplier with polynomials. Route A: p(z) = z^2 + Delta^2 lands exactly at t* = Delta^2/2 (ratio 2t*/Delta^2 = 1.0 at Delta = 0.6, 0.895136, 1.2), refuting Delta^2/8 by a factor 4 and showing 2 Delta^2 slack by 4. Route B: cos z + c flowed to e^t cos z + c gives ratios 0.8366, 0.9839, 0.99967 at c = 2.0, 1.05, 1.001 climbing to 1, so the constant 1/2 is sharp. Route C: (z^2+1)(z^2-A^2) gives 0.9329, 0.99506, 0.99980 at A = 5, 20, 100: spectator real zeros only accelerate landing. Independently reproduced by an adversary at D = 0.3, 0.6, 0.895136, 1.0, 2.0. Reading: the threshold is t = lambda^2/2 for de Bruijn's e^{(1/2) lambda^2 u^2}, t = lambda/2 for Newman-Wu's e^{lambda u^2/2}, frame-free in either.

Evidence: THEOREM13.md §6 'Calibration of the factor Delta^2/2 (derived, never recalled)' (lines 502-549); RESULTS.md §3.3-3.4 (lines 526-586); calibrate_theorem13.py, calibration.json Prior art: cited: de Bruijn 1950 Duke Math. J. 17 Theorem 13 (transcribed from the image scan), Dobner Theorem 3, Newman-Wu 2020 Theorem 7 as typeset corroborations Reused in: RESULTS.md §3.5 (the upper bound), STRIP2.md §3.5, docs/29; the same dictionary is what dh_minus_heat's wide-frame values ride on Why it travels: A three-family numerical calibration that fixes the constant and the multiplier convention of any Polya-de Bruijn style strip-to-time theorem before it is applied to a new function; the heat-equation check is the licence for using polynomials as the oracle.

Taylor/Rouche disk certificate for a simple non-real zero of a heat-flowed entire function, with all tails charged

computational technique | hunts/dh_minus_heat | grade: enclosure-carrying (Arb at 96/128/160 bits, exact rational recheck; mpmath 55-digit quadrature as float cross-check); ordinary argument, model-reviewed only

For H_t(z) = 4 int_0^inf e^{t u^2} g(u) sin(zu) du (or cos), a disk |z-c| <= r avoiding the real axis, enclose |H_t(c)|, |H_t'(c)|, and a disk-wide majorant M2 >= |H_t''| (replace every coefficient by its bound M and the wave factor by u^k e^{yu}, y = |Im c|

endpoints, Rouche gives exactly one simple non-real zero in the disk. Tails: theta tail after n=N bounded by M(N+1)rho^{N+1}/(1-rho)^2 times 4U^{k+1}exp(tU^2+(3/2+y)U); integral tail after U by 4 M U^k exp(-CV)/(CV) with V=e^{2U}, C = pi/5 - (tU^2+(3/2+y)U)/V > 0. An insufficient cutoff returns inconclusive, never silently drops a tail.

Evidence: RESULTS.md sections 2-3 'Two local disks' and 'Error bounds actually consumed', lines 108-193; verify.py, rouche.json Prior art: cited: Rouche, Taylor remainder (standard); packaging unsearched Reused in: none recorded Why it travels: Drop-in local zero-existence certificate for any heat deformation of a theta-type kernel; converts 'a root-finder residual' into a statement with all errors retained. Index note: Enclosure-carrying zero certificate; the chord-tube winding count in lambda_dh_bounds is the sibling instrument.

Chord-tube segment winding count that refuses rather than rounds

computational technique | hunts/lambda_dh_bounds/ | grade: decided (enclosure-carrying integer count, python-flint Arb 420 bits; second float witness by an independent DHFlow argument-principle route at dps 130 sharing 0 declared layers)

To decide the number N of zeros of H_t inside an exact-rational rectangle: split the boundary into axis-parallel subsegments with exact dyadic endpoints; for each subsegment [z_a, z_b] of half-length h evaluate Arb balls A = H(z_a), B = H(z_b); the image lies in chord(A,B) + disc(M2 h^2/2) with M2 the uniform |H_t''| bound (Lemma M2). Decide (i) dist(0, chord) > M2 h^2/2 via a lower-ball _chord_clearance (cases on the foot of the perpendicular), which puts the image in an open half-plane so |Delta arg| < pi, and (ii) Re(B/A) > 0, so |Arg q| < pi/2 and Delta = Arg q exactly. Undecided subsegments are halved (evaluations cached) down to a budget; on exhaustion the routine returns status 'undecided' naming the failing segment and never an integer. The sum of per-segment Arg q balls divided by 2pi must be a ball deciding a single integer. Box validators reject boxes touching the real axis (Im lo = 0 raises before any evaluation). The countermeasure winding.measured_h2_guard requires M2 to dominate a directly sampled sup|H_t''| on a rule sharing no code with the derivation, and main() refuses the floor if it fails. Decided N = 1 at t = 23/400 and t = 36/625 on boxes with Im z >= 3/1024, prec 420 bits, winding-ball width < 1e-39.

Evidence: winding.py header lines 20-60 (the two per-segment decisions and the refusal rule), lines 593-730 (_chord_clearance, winding_rectangle, tube = M2*hh*hh/2); RESULTS.md §2.1 (lines 298-316), §5 controls table (lines 801-811: displaced box N=0, on-axis box raises, edge-through-zero returns undecided) Prior art: cited in-tree: adapts the zeta.rigor._segment_delta pattern with the no-zero step rescaled for |H_t| ~ 1e-83; argument principle standard Reused in: hunts/lambda_dh_exact/RESULTS.md §7 step 1 recommends this instrument to decide the height-10^6 zero; hunts/dh_minus_heat uses a Rouché variant of the same chord/tube inequality Why it travels: A general rigorous zero-count for any entire function with a uniform second-derivative bound on the box, with an explicit refusal path instead of rounding; the on-axis and edge-through-zero controls specify what the detector must do when the box is bad.

Arithmetic-free N-body null control: the repair clock reads geometry

control | hunts/flow_repair/ | grade: measured

To test whether a flow-time quantity carries arithmetic information, census the t=0 zero configuration in a window (line zeros by phase-refined sign scan, total strip count by the argument principle, accounting required to close exactly: line + 2 x quadruples = strip, e.g. 49 + 4 = 53), then integrate the bare N-body ODE dz_k/dt = 2 sum 1/(z_k - z_j) from those positions only (no Dirichlet series, character or conductor) in the variable Q, and compare its landing time with the PDE flow. For five DH quadruples the ODE and PDE landing times agree to within 0.04% (differences -0.036%, -0.010%, -0.002%, -0.006%, +0.024%), sign scattering with truncation knobs, so the repair time contains no information beyond the initial zero layout. Companion 'rival-as-validation' leg: the generic-Phi evaluator must reproduce the sibling module (zeta's H_t to 5.4e-42) before it is trusted on the rival.

Evidence: hunts/flow_repair/NOTES.md, Section 3 'The null control: the repair clock reads geometry, not arithmetic' (table lines 104-110), Section 0 'The instrument is telling the truth', 'Standing-checklist accounting' Prior art: unsearched for the control design; N-body universality literature cited (Hall-Ho, Cuenca-McSwiggen) Reused in: hunts/lambda_dh_exact MISSION.md Section 4.6 uses the N-body null control landing of the census pair at gamma = 531.28 as a holdout Why it travels: A matched decoy that explains the effect rather than merely failing to reproduce it; applicable whenever a claimed structural quantity might be a function of configuration geometry alone.

Deflation lesion of an analytic constant, with onset factor, blindness radius and health-metric direction

control | hunts/lambda_dh_bounds/ | grade: measured (float guard; hunt's stated verdict 'SENSITIVITY MEASURED (not a pass)'); the cushion is decided as of 2026-08-18

For a detector that consumes a derived analytic constant (here M2, the uniform |H_t''| bound feeding the chord-tube radius): hold geometry, precision and subdivision fixed and deflate only the constant by factors 1, 10, 72, 75, 100, 1000; record for each whether status is 'decided', the integer returned, whether it is correct, and the detector's own health metrics. Findings that define the control: (a) wrong_answer_onset_factor = 75 with n_wrong_and_silent = 3 (wrong integer with status 'decided', the one output the routine promises never to produce); (b) the health metric min_chord_margin_digits reads 0.02 on the correct run and 0.11, 1.11 on the wrong runs, i.e. the detector's own metric improves as the answer becomes wrong, so it may not be used as a guard on the constant; (c) countermeasure: a measured guard (sup|H''| sampled on a rule sharing no code with the derivation) that trips at deflation 55.7, before the first wrong integer at 75, with the ordering later shown structural (cushion 33-204 across a 40-unit span) but not proved on arbitrary boxes; (d) the same discipline applied to the two tail bounds gives a blindness factor 8.02 (smallest domination ratio bound/true over 18 stress rows), so a bound deflated by less than 8 is invisible. Verdict vocabulary: 'SENSITIVITY MEASURED (not a pass)', with all_pass = false reported as the headline rather than controls_1_to_4_pass.

Evidence: RESULTS.md §5 table (lines 801-811), §5.1 'Control 5, in full' (lines 825-888, deflation table at 838-847, 'the perverse metric' 855-863, countermeasure 865-871, what is still blind 873-881), §5.1a (lines 890-940); INDEPENDENCE.md §5(ii) (tail-bound domination ratios, lines 234-305, blindness factor 8.02); controls.py, controls_results.json Prior art: cited in-tree: docs/25's rule that a lesion threshold without a blindness radius is half a measurement Reused in: hunts/lambda_dh_exact/RESULTS.md §5 item 6 (lesion_degree52 records the same health-metric blind spot); GATE.md known assumption 6 Why it travels: Any enclosure-based detector with a hand-derived constant should be lesioned this way; the two numbers to report are the wrong-answer onset and the blindness radius, and the direction the health metric moves under the lesion is itself a finding.

t=0 admissibility gate for exact polynomial heat flow (float root-finding lesion)

control | hunts/lambda_dh_exact/ | grade: measured

When zeros of a heat-evolved polynomial are found by float64 numpy.roots on coefficients evolved in closed form (p_t = sum_k (-t)^k/k! p^{(2k)}), reject any configuration whose starting roots at t=0 are not recovered. Lesion that fired silently: at degree 52 with spacing 0.5, numpy.roots reported 14 nonreal roots at t=0 where there are 2 and max|Im| = 0.967 where it is 0.600; the resulting landing time was wrong by a factor 3.3 and looked ordinary. All tables in the hunt are gated; the detector's own health metric does not flag this fault (inherited blind spot recorded in hunts/lambda_dh_bounds/M2-LEMMA.md).

Evidence: hunts/lambda_dh_exact/MISSION.md, Section 4.7 'Lesion, recorded because it fired silently' (lines 387-392); RESULTS.md Section 5 item 6 'Lesion evidence is published, not resolved' Prior art: unsearched Reused in: Rule stated for every later phase using exact polynomial flow; lesion cross-referenced to hunts/lambda_dh_bounds/M2-LEMMA.md Why it travels: Cheap, mandatory sanity check for any root-tracking experiment on high-degree polynomials: the answer at t=0 is known and must be reproduced before any evolved root is believed.

Epstein zeta, precision and zero counting

Precision floors for the completed Epstein zeta, what argument-principle box counts can and cannot discriminate, and the rightmost-zero wall for the prime zeta.

Tail-subset wall lower bound via short-interval prime count

bound | hunts/prime_zeta_rightmost/ | grade: proved in THEOREM.md (ordinary derivation, self-reviewed); the numeric instance D3 is decided on two independent backends (python-flint arb 350 bits, mpmath.iv dps 40)

For the k-th prime p_k >= 23 let S = {p >= p_k} and let sigma_c(p_k) be the unique root in (1, inf) of the balance h(T) = sum_{p > p_k} p^{-T} - p_k^{-T} = 0 (the supremum of real parts of zeros of the tail prime zeta function). Then sigma_c(p_k) >= B_k := log2(3 p_k / (5 log p_k)), which tends to infinity with k. Proof template: Rosser-Schoenfeld (3.8) gives more than 3x/(5 log x) primes in (x, 2x] for x >= 20.5; each contributes at least (2 p_k)^{-T}, so h(T) > 0 whenever 2^{-T} 3 p_k/(5 log p_k) > 1, i.e. T < B_k, and monotonicity of h forces the root above B_k. Decided instance: B_9 = log2(69/(5 log 23)) in [2.1379035036560028560606113813, +1e-30], > 17/8 on both backends (D3). Corollary C3: no constant bounds the real parts of zeros over all subsets of the primes.

Evidence: THEOREM.md, section 'Theorem C2 (tail subsets have unbounded walls)' and 'Corollary C3' (lines 576-644); RESULTS.md section 7.1 item (b) and section 1 (D3) Prior art: searched-and-absent (four independent searches; the hunt itself notes this is a statement about the searches). Framework (wall = balance root, triangle-inequality proof) is prior art: Belovas-Cepaityte-Sabaliauskas 2025 Thm 1; Sepulcre-Vidal 2022 Thm 4.3. Rosser-Schoenfeld cited. Reused in: docs/30-prime-zeta-rightmost-zeros.md Why it travels: Turns any explicit short-interval prime-count inequality into an explicit lower bound on the balance root of a tail Dirichlet series over primes; the same three lines apply to any subseries where a leading term competes with a tail.

Precision-adequacy guard (evaluate at dps D and D+15) and the Epstein digit-loss rule dps = 20 + ceil(0.6822 t_max)

calibration | hunts/gate5_p6_b/ | grade: measured (two-precision replication guard, mpmath)

Before trusting an argument-principle zero count of a completed L-function at height, evaluate the integrand at the box corners at working precision D and again at D + 15 and require relative disagreement below 1e-6 (worst observed 3.5e-22 across ten decided cells). Calibration for zeta.epstein.epstein_completed: Lambda_Q is exponentially small in t while its Mellin-split terms are O(10^-2), so about pi t/(2 ln 10) = 0.6822 t digits are lost; at s = 0.8 + 85.7i, (2,1,3): 1.2e-34 at dps 20 versus 1.617e-58 at dps 60 and 90 against the analytic magnitude 2.64e-58. Preregistered rule dps = 20 + ceil(0.6822 t_max) (28 for t <= 11.5, 79 for t <= 86.2). Defect found: zeta.epstein.battery's rival interfaces cap dps at min(dps, 20), so above t about 25 they return winding numbers of round-off that still pass the integrality check.

Evidence: hunts/gate5_p6_b/RESULTS.md, 'The table' (adequacy paragraph lines 33-36), 'What was chosen, and why: The precision' (lines 85-99), 'A defect in the battery, found on the way' (lines 165-185), loose thread 'A precision-adequacy guard as a reusable instrument' Prior art: unsearched Reused in: none recorded Why it travels: Every hunt evaluating a completed L-function at height needs the digit-loss rule and the two-precision guard; the hunt states nothing in zeta/ offers it. Index note: Fourth hunt to hit the Epstein digit-loss defect; belongs with the gate5_p6_c, dps_cap and r_f7cd45 entries.

Height-dependent precision rule for the completed Epstein zeta and its digit-loss law

calibration | hunts/gate5_p6_c/ | grade: measured (three independent hunts, one route each)

epstein_completed returns d^{s/2}(first + second/√d + 1/(√d(s−1)) − 1/s), a sum of O(1/t)-to-O(1) terms, while |Λ_Q(s)| ~ |Γ(s)| ~ exp(−πt/2); so about πt/(2 ln 10) = 0.6822·t decimal digits cancel. Rule: evaluate Λ_Q at dps = 20 + ceil(0.6822·t_max) for the box, a function of the box alone written before any winding number was computed. Confirmed independently: dps_cap measured the noise floor at ≈1e-(D+13), crossover at t ≈ 50 for dps 20 (relative error passes 1 between t=45 and t=50), and r_f7cd45 measured a convergence floor of dps ≈ 60 at t ≈ 85.5 (0.6822×85.5 ≈ 58). ξ and Davenport–Heilbronn are unaffected at dps 20. The battery's hardcoded min(dps, 20) at zeta/epstein.py:1091, :1141 therefore sits below the floor for every t ≳ 30.

Evidence: gate5_p6_c/RESULTS.md 'The defect that cost this hunt its high-t Epstein arm' (lines 90–125); probe.py GUARD_PER_UNIT_HEIGHT = 0.6822 (lines 61–77); dps_cap/README.md 'Where up the strip the cap actually fails' (lines 143–168); r_f7cd45/RESULTS.md 'The precision floor, measured' (lines 101–140) Prior art: none; a defect in the lab's own module Reused in: hunts/dps_cap (crossover table), hunts/r_f7cd45 (convergence-floor ladder), gate5_p6_c B2/B4 counts Why it travels: Any completed L-function evaluated via a sum of O(1) terms loses ~πt/(2 ln 10) digits at height t; the rule sets working precision from the box before spending budget.

Parse-sensitivity and archimedean-envelope checks for detecting precision noise

control | hunts/dps_cap/ | grade: measured, one point (0.8 + 85.7i), form (2,1,3)

(a) Parse sensitivity: write the same evaluation point at several parse precisions (mp.mpc('0.8','85.7') at dps 15, 20, 60, and as Python floats); a converged evaluation must return the same magnitude for all, since the bit patterns denote the same number far past any digit that matters. At dps 20 the returned |Λ_Q(0.8+85.7i)| spread by a factor 64 across parses; at dps 60 no spread to twelve digits. This also reconciled two earlier 'conflicting' numbers (1.2e-34 vs 3.1e-33) as two samples of the same floor. (b) Archimedean envelope: divide the returned magnitude by |(√d/π)^s Γ(s)| (computable from mp.gamma alone, touching nothing under test) to get the implied |ζ_Q(s)|; a Dirichlet series continued into the strip grows polynomially, so an implied 1e25 at t=85.7 is impossible while 0.61 is ordinary. (c) Refinement stability: the value at D=60 and D=80 agreeing to 1.5e-16 is the convergence control; D=30 returning exactly 0.0+0.0i is the failure mode that looks like a root.

Evidence: README.md readings 1–5 (lines 57–141); FINDINGS.md 'Reading' (lines 18–46); probe-f12f9441.py parse_sensitivity (lines 90–105), implied_abs_zeta_Q; sanity_stirling.py Prior art: unsearched Reused in: gate5_p6_c and r_f7cd45 precision findings (same defect, independently); HANDBACK.json thread on count_zeros_box Why it travels: Both checks are routine-agnostic ways to tell a rounded answer from noise; the envelope check works for any completed L-function with a known gamma factor.

Decided-root control battery: known-answer calibration plus template-swap lesion

control | hunts/prime_zeta_rightmost/ | grade: decided (two backends), as the hunt states

A bisection root solver on a strictly decreasing function (decide.bisect_decreasing, endpoint signs decided by ball/interval enclosures with exact-Fraction bracket bookkeeping) is validated by five controls run through the identical code path: (1) calibration: feed the zeta partial-sum balance 1 = sum_{n>=2} n^{-sigma} and recover the OEIS value x* = 1.72864723899818361813... (31 digits on flint, 13 on iv); (2) input lesion: drop one term (p = 3) and require a decided shift (0.35022851787919218338...); (3) template lesion, the 'mis-port made mechanical': keep the zeta series but balance the prime-zeta template's leading term 2^{-sigma} against the tail after it, which must land at a value (2.4241112509134051...) decidedly equal to neither x* nor sigma_c, proving both the series and the balance template are read; (4) precision response: widths must shrink strictly (60/120/200 bits: 1.4e-15 / 6.2e-34 / 1.0e-57) with nested intervals; (5) subprocess rerun reproduces decided.json line for line.

Evidence: RESULTS.md section '4. Calibration control and lesions (WP5)' (lines 282-317); controls.py, controls_results.json Prior art: unsearched (standard lesion discipline of the lab; the template-swap lesion is the hunt's own) Reused in: docs/30-prime-zeta-rightmost-zeros.md Why it travels: Any 'decided constant' claim from a root solver can be pinned by the same five-control sequence; the template-swap lesion specifically catches the failure mode where a solver is ported from one balance problem to another with the wrong leading term.

Convergence-floor ladder before spending winding-number budget

control | hunts/r_f7cd45/ | grade: measured

Before running an argument-principle count on a box, evaluate each completed function at one interior point (0.75 + 85.5i) up a dps ladder and record the lowest dps at which the value stops moving; cells whose preregistered precision is below that floor are marked INADMISSIBLE and skipped rather than computed. Rationale: count_zeros_box's integrality check cannot detect noise, since noise winds to an integer as readily as signal does, so a below-floor count returns a plausible small integer that is not a zero count. Floors found: ξ ≤ 15, Davenport–Heilbronn ≤ 15, Epstein (2,1,3) and (1,1,6) 60. A preregistered dps 15 would have published a wrong integer.

Evidence: RESULTS.md 'The number this run adds: the precision floor, measured' (lines 101–134), Loose threads bullet 1 Prior art: unsearched Reused in: confirms dps_cap and gate5_p6_c; recommendation to replace min(dps,20) with a box-height floor pinned by a dps 60 vs 100 stability test Why it travels: A pre-check that turns a silent wrong integer into an explicit 'inadmissible' for any contour-count oracle whose only self-check is integrality.

Local zero-free box properties cannot discriminate RH from its rivals

obstruction | hunts/gate5_p6_c/ | grade: measured (argument-principle counts, count_zeros_box, preregistered boxes committed at c29c876 and 53c8cd1)

'In a box strictly off the critical line the completed function has no zeros' has no truth value as a property: with σ ∈ [0.7,0.9] fixed, t ∈ [85.5,85.9] gives DISTINGUISHES (Davenport–Heilbronn has its off-line zero at 0.80852+85.6993i, ζ has none) while t ∈ [10,14] and [40,44] give VACUOUS (all four functions have zero count 0). Mechanism: an RH-violating function has a positive proportion of zeros on the line and sparse off-line zeros, so any box missing them looks RH-satisfying; a local statement cannot carry a global distinction. Every repair that both has a truth value and distinguishes collapses to RH-with-a-margin (uncheckable in a box) or to 'no zeros with σ > 1' (the Euler product, which is gate property 5 already). B4 in σ ∈ [1.05,1.55], t ∈ [10,14] returned VACUOUS even though DH has zeros in σ > 1 by the 1936 theorem, because a width-4 window is silent about the band. Reproduced independently by r_f7cd45 on preregistered boxes B1 [80,81] VACUOUS, B2 [85,86] with DH count exactly 1.

Evidence: RESULTS.md 'Answer', 'The table', 'Reading the flip', 'What a well posed version would be' (lines 18–195); r_f7cd45/RESULTS.md 'The sixth property' (lines 67–99) Prior art: cited: Davenport–Heilbronn 1936 (zeros in σ > 1); the well-posedness argument is the lab's Reused in: hunts/r_f7cd45 (independent reproduction); recommendation to retire property 6 from the battery Why it travels: Closes the class of box-local zero-freeness properties as gate discriminators, with an argument (sparse off-line zeros) rather than a failed run; the blind/declared-non-blind box design is reusable for any local property.

Erdos #126 and S-unit equations

Equivalent forms of the problem, residue-class and deletion lemmas, reductions to S-unit equations, and the barriers that show which relaxations are exact.

Lattice rational identity for the depth-1 damage kernel: sign on 2 pi Z for all d without a horizon

identity | hunts/support_5418c63e | grade: derived (exact identity, checked to 8.4e-16 relative against direct evaluation; the cubic coefficients are double precision, not enclosure-carrying, with wide sign margin)

With ghat(z) = int_{-1/2}^{1/2} cos(sqrt2 t) e^{zt} dt, the exact one-term form is ghat(z) = [alpha z sinh(z/2) + beta cosh(z/2)]/(z^2+2), alpha = 2 cos(1/sqrt2), beta = 2 sqrt2 sin(1/sqrt2). For D(1,s) = -Re ghat(1+is)^2 at s = 2 pi d, cos s = 1 and sin s = 0 make cosh z, sinh z real, giving D(1, 2 pi d) = P(u)/(u^2-2u+9)^2 with u = (2 pi d)^2 and P(u) = p u^3 - (3p-3q+r)u^2 - (5p+2q-10r)u - 9(p+q+r), p = (1+cos sqrt2)(cosh1 - 1), q = 2 sqrt2 sin(sqrt2) sinh1, r = 2(1-cos sqrt2)(cosh1+1). All coefficients of P but the constant are positive, so P has one nonnegative root u* = 1.7707 < 4 pi^2, hence D(1, 2 pi d) > 0 for every integer d >= 1 (and D(1,0) < 0, harmless since P sums over p != q). This replaces the measured 'P = 0 on the critical lattice out to d = 4000' ingredient of the ceiling rho* <= 0.153216295 with a derivation for all d.

Evidence: RESULTS.md section 1 (one-term form, lines 42-59) and section 5 'The lattice is exactly solvable', lines 164-194; section 6 for the consequence Prior art: unsearched Reused in: hunts/r_c7f779 (run 872d7dce) ceiling rho* <= 0.153216295, the P = 0 ingredient; RESULTS-37fb06a9.md section 2 Why it travels: The move 'evaluate the kernel only where it becomes rational (the lattice) and decide the sign by a polynomial root count' turns a horizon-limited scan into an all-d statement; also the edge law centre - 2 pi d = K/s with odd next-order term translating rather than widening the window.

Four-subset to nondegenerate three-term S-unit equation reduction

identity | hunts/support_8ea74995 | grade: proved (elementary); the counting consequence is conditional on the unbounded multiplicity M

If a,b,c,d are distinct positive integers whose six pairwise sums are S-smooth, then (a+b)+(c+d) = (a+c)+(b+d) yields x+y+z = 1 (equivalently X+Y-Z = 1) in positive S-units of Q with no vanishing subsum (a subsum vanishing forces a=d or b=c or a zero sum). So an n-element witness supplies C(n,4) instances; if the map to solutions has multiplicity <= M then g(k) <= (4! M N_3(S))^{1/4}, and N_3(S) = exp(o(k)) would prove #126. Known lower bounds on S-unit solution counts (exp(c sqrt s / log s)) do not obstruct this. Gap named: the multiplicity/distinctness step.

Evidence: RESULTS.md section 6 'S-unit form', lines 258-294; independently in hunts/support_eccd5f5e/RESULTS.md section 7 Proposition, lines 238-271 Prior art: cited: Evertse 1984, Evertse-Schlickewei-Schmidt, Erdos-Stewart-Tijdeman 1988; the hunt says 'not a new idea in the field' but the explicit bridge was absent from the prior hunt Reused in: support_eccd5f5e and support_517b887f name it as the only door that reads more than residues Why it travels: Converts any sum-smoothness extremal problem into a unit-equation counting problem with an explicit, checkable degeneracy analysis.

Normalization lemma for S-admissible sets (dilates of primitive sets)

lemma | hunts/support_60982bf6/ | grade: proved (ordinary, three-line), exhaustively checked on the sweep's witnesses

S a finite set of primes; A ⊂ ℤ_{>0} finite is S-admissible if a + b is S-smooth for all a ≠ b in A. If |A| ≥ 2 and d = gcd(A), then (1) d is S-smooth (d | a + b, divisors of smooth numbers are smooth); (2) A/d is S-admissible and primitive; (3) for S-smooth m ≥ 1, A is admissible iff mA is. Hence the S-admissible sets are exactly mA_0 with m S-smooth and A_0 primitive admissible, and g(S) = max|A| is attained on a primitive set. Consequence: widening the search box only produces dilates of small optima (e.g. {120,2280,3720,10680,19320} = 40·{3,57,93,267,483}), which explains why box width 'changed not one row' in the predecessor hunt; the right box parameter is the height of the smallest primitive optimum, not N. Checked on all 381 sweep witnesses.

Evidence: RESULTS.md §2 'Lemma 1 (normalization). Proved.' (lines 92–124); raw2.json normalization_checks Prior art: cited context: Erdős #126, Erdős–Turán finiteness; the lemma itself unsearched Reused in: §5 Conjecture 1 (height h(S) defined over primitive sets); audit table of r_186989 Why it travels: Any smoothness-closed additive problem has the same dilation structure; it converts a box-size question into a height question.

Deletion lemma and primitive descent g*(k) <= 2^k

lemma | hunts/support_7ddfee4b | grade: proved (ordinary derivation, self-reviewed)

Let (S,A) be admissible (all distinct pairwise sums S-smooth) and p in S odd. Write A_p = {a : p | a}, B_1 = union of classes c in [1,(p-1)/2], B_2 = union of classes -c. Then A = A_p sqcup B_1 sqcup B_2 and each B_i is admissible for S minus {p}. For primitive A (no element divisible by a prime of S) A_p is empty, so g*(k) <= 2 g*(k-1), and with g*(1) = 2 (power-of-2 base case: a+b=2^x, a+c=2^y, b+c=2^z with a<b<c forces 2a<0) one gets g*(k) <= 2^k, tight at k=1,2 ({1,5,7,11}, S={2,3}). The gap to g(k): A_p/p is admissible for the same S, so the split is circular for non-primitive sets.

Evidence: RESULTS.md section 2 Theorems 2-4, lines 90-129; section 6 on the primitivity hole, lines 221-236 Prior art: cited: Erdos-Turan 1934 (3*2^{k-1}); Erdos-Suranyi form 2^k noted in support_517b887f section 1 Reused in: support_517b887f reconstructs the same architecture and finds the sketch defects Why it travels: A clean self-contained induction for smooth-sum problems; the primitivity caveat is the exact place where 'divide by gcd' is not enough.

Galois connection and six equivalent forms of Erdos #126

lemma | hunts/support_8ea74995 | grade: proved (ordinary derivation, self-reviewed; audited by a second arm)

With f(n) = min_{|A|=n} |P(A)| and g(k) = max_{|S|=k} max{|A| : P(A) subset S} (distinct-pair convention): f is non-decreasing; g(k) is a genuine maximum (Erdos-Turan bound uniform in S); f(n) <= k iff n <= g(k). Consequently the following are equivalent: f(n)/log n -> inf; log g(k) = o(k); g(k)^{1/k} -> 1; f(2^m)/m -> inf; g(k) <= e^{ck} eventually for every c; the Cesaro mean of log(g(j+1)/g(j)) -> 0. Correction recorded: finiteness of g is equivalent to f -> inf, strictly weaker than f >> log n.

Evidence: RESULTS.md sections 1-2, Theorem 1.4 and Theorem 2.1, lines 42-112; independently re-proved in hunts/support_eccd5f5e/RESULTS.md section 1 Prior art: cited: Erdos-Turan 1934 used as published Reused in: support_eccd5f5e (audit), support_7ddfee4b, support_517b887f all use the g/f inverse forms Why it travels: Template for any min/max inverse-staircase pair; the Cesaro form (6) is the one that makes direction results obvious.

Residue-class lemma for an omitted prime (correct form of the pigeonhole), with parity as the unique capping case

lemma | hunts/support_8ea74995 | grade: proved, with counterexamples verified by trial division in results.json (three independent arms)

If P(A) subset S and the odd prime p is not in S, then at most one element of A lies in class 0 mod p, the set R of occupied nonzero classes satisfies R cap (-R) = empty, so A occupies at most (p+1)/2 residue classes mod p, and no bound on |A| follows (A = {1, 1+p, ..., 1+(m-1)p} has all pairwise sums = 2 mod p). p = 2 is the unique prime where every class is self-paired, giving |A| <= 2 when 2 is not in S. The proposed claim |A| <= p-1 is false for every prime (also at p=2 by {1,2}, S={3}), and would have settled #126 in three lines.

Evidence: RESULTS.md section 4, lines 152-194; hunts/support_eccd5f5e/RESULTS.md section 5, lines 166-198; hunts/support_7ddfee4b/RESULTS.md Theorem 1 and Corollaries 1a, 1b, lines 55-88 Prior art: unsearched Reused in: support_eccd5f5e, support_7ddfee4b (three arms converge on the same statement) Why it travels: Also a control: a claim strong enough to settle a 92-year-old problem that agrees with seven data points is a warning, not evidence; the sieve reformulation (density 1/2 at every prime above max S) is the salvage.

Lemma A: residue-class cap for S-summable sets at a prime outside S

lemma | hunts/support_baf4cde6/ | grade: ordinary derivation, self-reviewed (two-line proof), with computational checks (probe.py, stdlib)

Let A be a finite set of distinct positive integers with a + b free of primes outside S for all distinct a, b in A, and let p be a prime not in S. Then (1) A contains at most one element divisible by p; (2) A never meets both classes r and -r mod p for r not 0; (3) hence A meets at most (p+1)/2 residue classes mod p; (4) for p = 2, |A| <= 2. Proof: a = -b mod p with a != b gives p | a + b, so p in S. The cap is on classes, not elements: any number of elements may share a class r with 2r not 0. Refutes the parent thread 'prove |A| <= p - 1 when p not in S': A = {1,3,7,13}, S = {2,5,7}, 3 not in S, |A| = 4 > 2; sweeps reach |A| = 5 (S = {2,5,11,17}, A = {1,9,31,79,241}). Verified on all seven parent witnesses at every prime <= 37 outside S; {1,3,7,13} is sharp mod 3.

Evidence: hunts/support_baf4cde6/RESULTS.md, Section 1 table (loose thread refutation, lines 58-67) and Section 2 'Lemma A: the exact local obstruction' (lines 69-95) Prior art: Erdos-Turan 1934 bound cited; Lemma A itself unsearched Reused in: Used in Proposition C (same hunt); proposed as a Lean target in the loose threads Why it travels: Exact local constraint for any 'pairwise sums avoid prime p' set system; the class-versus-element distinction is the correction that other arms had missed. Index note: Same statement as the support_8ea74995 residue-class lemma, reached independently.

Four-subset to non-degenerate three-variable S-unit equation reduction

lemma | hunts/support_baf4cde6/ | grade: ordinary derivation, self-reviewed; the reduction is proved, the counting bound is conditional on m

For distinct a, b, c, d in an S-summable set A, (a+b) + (c+d) = (a+c) + (b+d); dividing by a + b gives x + y + z = 1 with x = (a+c)/(a+b), y = (b+d)/(a+b), z = -(c+d)/(a+b) in the group of rationals supported on S (rank k plus sign). The solution is non-degenerate: x + y = 0 is impossible for positive elements, x + z = 0 iff a = d, y + z = 0 iff b = c. Hence C(|A|,4) <= m E_3(k), so |A| <= (24 m E_3(k))^{1/4}, where E_3(k) bounds non-degenerate solutions and m bounds the fibers of the map 4-subsets -> solutions. This is an upper-bound route; with Evertse-Schlickewei-Schmidt's exp((6n)^{3n}(r+1)) the exponent constant is about 5e10 against Erdos-Turan's log 2, and the fiber bound m is the missing lemma.

Evidence: hunts/support_baf4cde6/RESULTS.md, Section 6 'Where the constraint actually lives' (lines 185-223) Prior art: cited: Evertse-Schlickewei-Schmidt unit-equation bound; Erdos-Stewart-Tijdeman lower bound exp(c (k/log k)^{1/2}) Reused in: none recorded Why it travels: Precise transport of a pairwise-sum smoothness condition into unit-equation counting, with the direction and the fiber gap stated so a later run can attack m directly. Index note: Same reduction as support_8ea74995 and support_d5d5ccae, reached independently.

Two-base-element injectivity lemma reducing Erdős #126 to a two-variable S-unit equation count

lemma | hunts/support_d5d5ccae/ | grade: ordinary derivation, self-reviewed (two-line proof); counts measured (exact enumeration in a box, lower bounds on N_S for d > 1, oracle-checked against Lehmer's Størmer table for d = 1)

Let S be a set of k primes and A a finite set of distinct positive integers with every off-diagonal sum a + b S-smooth (admissible); g(k) = max |A|. Lemma: for any a > b in A with d = a - b, the map x -> (a + x, b + x) injects A \ {a,b} into Sol_S(d) = {(U,W): U

max_{d>=1} N_S(d), finite by Mahler. Implication runs the useful way (bound on S-unit solutions => bound on g(k) => Erdős #126); the converse fails. Free facts: gcd(a+x, b+x) | d so Sol_S(d) decomposes over S-smooth divisors g | d as g times primitive solutions of u - w = d/g; d = 1 is the Størmer case counted exactly by Lehmer. Quantitative accounting: Evertse 1984 (ax + by = 1, at most 3·7^{d+2s}) gives g(k) <= 2 + 3·7^{2k+3} ~ 1029·49^k versus the elementary 2^k, so the theorem, not the route, is loose by ~24.5^k (measured: at k = 7 the lemma's value at the extremal witness is 96, 2^k = 128, Evertse 7e14). Obstruction for higher arity: fixing three base elements gives (b-c)(a+x)

coefficients are not S-smooth, forcing rank-based theorems (Beukers-Schlickewei 2^{16k+16}) instead of Evertse's arbitrary-coefficient accounting; cross-ratios of four elements reintroduce the same defect. Named missing lemma: a constant c and phi(k) = exp(o(k)) such that every admissible A with |A| > c has a pair with N_S(a-b) <= phi(k).

Evidence: RESULTS.md §1 (lines 24-53, lemma, corollary, direction), §2 (lines 55-79, three-element downgrade), §3 (lines 81-111, Evertse vs Beukers-Schlickewei table), §4 (lines 113-160, exact N_S counts to 10^14 with Lehmer oracle check), §7 (lines 193-203) Prior art: searched-and-absent, with the hunt explicitly not claiming originality: 'very likely in Győry-Stewart-Tijdeman 1986 in some form'; Evertse 1984, Beukers-Schlickewei 1996, Erdős-Stewart-Tijdeman 1988 cited Reused in: support run for hunts/r_186989 (0897a5a7 arm 'sunit-equations'); none else stated Why it travels: A clean bridge from any 'all pairwise sums are S-smooth' clique problem to counting solutions of U - W = d in S-units, plus the exact statement of why gadgets with three or more base elements lose (coefficients leave S).

Unit-equation reduction for sets with S-smooth pairwise sums

lemma | hunts/support_f3ab3e34/ | grade: proved (elementary), self-reviewed; checked as exact rationals on the witness A = {1,2,3,5,7,13}, S = {2,3,5,7}

Let S be a set of k primes and A a set of n >= 3 distinct positive integers with every off-diagonal a + b S-smooth. Lemma 1: d = gcd A is S-smooth and A/d is again valid. Lemma 2: fix distinct a_1, a_2 in A, D = a_1 - a_2, and Gamma = <-1, p_1, ..., p_k, D> <= Q^*, rank <= k + 1; then c -> (X_c, Y_c) = ((a_1 + c)/D, -(a_2 + c)/D) injects A minus {a_1, a_2} into {(X, Y) in Gamma^2 : X + Y = 1}. Corollary: g(k) <= 2 + N(k+1) where N(r) is the maximal number of solutions of x + y = 1 in a rank-r subgroup, so N(r) = exp(o(r)) implies Erdos #126, and unconditionally g(k) <= 2 + 2^{8k+16} via Beukers-Schlickewei. The implication is one-directional: lower bounds on N (Erdos-Stewart-Tijdeman) cannot refute. Elementary lower bound: A = {1..m} is valid for S = {p <= 2m - 1}, so g(k) >= (1 + o(1)) k log k / 2. Audit of r_186989's loose thread 3: the pigeonhole '|A| <= p - 1 when p not in S' is false for odd p (A = {1,3,7,13}, S = {2,5,7} avoids 3 with |A| = 4).

Evidence: RESULTS.md sections '2. The lemma I attacked, in full: the unit-equation reduction' (Lemmas 1, 2, Corollary, 'Direction check'), '4. Audit of r_186989' (4b, 4c); probe.py checks.unit_equation_reduction Prior art: searched, not found written down; the hunt says it is almost certainly folklore inside the Gyory-Stewart-Tijdeman method; literature rows marked 'cited, not verified at the source' (erdosproblems.com returned 403) Reused in: none recorded Why it travels: Localises the difficulty of a family of additive-smoothness problems in a single named counting theorem, and the more-base-points variant (chain B: m base points give m - 1 simultaneous twisted unit equations) is a concrete, unexamined door.

Counting-horn threshold: log Psi(2N,S)/k -> 0 iff log 2N = o(log rad S), two-sided

bound | hunts/support_95bb5cb7/ | grade: 'both directions are rigorous and measured' (hunt's words): Rankin and simplex bounds are ordinary derivations, self-reviewed; the table is measured; witnesses re-verified by trial division

Notation: S = k primes, rad(S) = prod p, theta = log rad(S), Psi(x,S) = number of S-smooth integers <= x. Normalisation (N1): if A is admissible and lambda is S-smooth then lambda A is admissible, and A/gcd(A) is admissible with gcd(A) S-smooth, so WLOG gcd(A) = 1; (N2) translation is unavailable (A + t shifts sums by 2t), so max A is a genuine invariant. Lemma 1: with a_n = max A the n-1 sums a_j + a_n are distinct S-smooth integers in (N, 2N], so |A| <= 1 + Psi(2N, S). Lemma 2 (threshold): writing log 2N = c theta, log Psi(2N,S)/k = Theta_c(1) and -> 0 iff c -> 0, uniformly in k, with S = first k primes the worst case. Upper: Rankin Psi(x,S) <= x^sigma prod_{p in S}(1 - p^{-sigma})^{-1} minimised over sigma. Lower: lattice points of {e >= 0: sum e_i log p_i <= log x} are S-smooth integers <= x and number at least the simplex volume (log x)^k/(k! prod log p_i). Measured table: at c = 0.5 the lower bound is 0.32 k and at c = 1 it is 1.01 k, so for max A >= rad(S)^{1/2} the counting horn is false, not merely unproven; at c -> 0 the Rankin bound -> 0. Consequence: the controlled-interval horn yields log g(k) = o(k) iff max A = rad(S)^{o(1)}, and no refinement of the counting step (dyadic, short interval) moves this. Companion measurement: primitive admissible triples for S = {2,3} enumerated through their sums (the constrained objects, giving an exhaustive universe) have height tracking the sum cutoff linearly over 10^4..10^15, so sub-extremal admissible sets have unbounded height and any descent argument fails.

Evidence: RESULTS.md §1 (N1/N2, lines 23-38), §2 Lemma 1, Lemma 2 and table (lines 40-88), §3.1 (lines 92-117, sum-bounded enumeration and unbounded height), §5 'Added' (lines 192-199, bounding sums is exhaustive for the property) Prior art: cited: Rankin's trick, Lehmer/Størmer; the threshold lemma and the sum-bounded enumeration are the lab's, unsearched Reused in: audit of hunts/r_186989/RESULTS.md (corrects its claim that every optimal witness has all elements < 50: the extremal 5-set {5,11,25,245,475} for S = {2,3,5} has height rad^{1.81}) Why it travels: Gives the exact regime in which 'count smooth numbers in an interval' arguments can and cannot deliver subexponential bounds, and the enumeration-through-sums trick makes exhaustive searches honest for any 'all pairwise sums lie in a sparse set' problem.

Control: cancellation-free cross-route for a high-cancellation Epstein value, and the ambient-precision argument trap

control | hunts/support_e6241336 | grade: measured (first rung for the value, second rung for the agreement of independent routes)

epstein_completed at s = 0.8 + 85.7i splits the theta Mellin transform at t = 1 and loses about 55 digits to cancellation (pole terms ~1e-3 against an answer ~1e-58), so a dps ladder only shows convergence, not correctness. Three checks against routes that do not share the cancellation: (A) normalisation at s = 4 where the lattice sum converges, (B) the Dedekind class-number identity sum_Q Lambda_Q(s) = 2 (sqrt23 / 2pi)^s Gamma(s) zeta(s) L(s, chi_{-23}), which reaches 1e-58 with no cancellation because the smallness comes from Gamma directly (agree to 4.0e-17 at dps 60, 6.3e-76 at dps 120), (C) the functional equation across the inverse class. Trap found: an mpc built at the ambient default dps arrives short of digits and workdps inside the routine cannot repair it (2.1e-15 shift at this point since d/ds log Lambda ~ log|s|); the same trap (reference strings parsed at ambient dps) produced two uniform floors in rogue_frontier/weil_trunc, detected by the 'identical floor across unrelated cells = artifact' reflex and by checking floors for c-dependence.

Evidence: RESULTS.md 'How much of that number is real', lines 22-47; 'The three checks', lines 49-83; 'One defect found', lines 85-103; hunts/rogue_frontier/weil_trunc/RESULTS.md section 6, lines 204-215 Prior art: cited: Dedekind zeta factorisation for class number 3 discriminant -23 (standard); reproduces hunts/dps_cap (Hunt #12) Reused in: reproduces hunts/dps_cap value digit for digit; trap recorded in HANDBACK.json Why it travels: Any Epstein/Dedekind evaluation at height where the Mellin split cancels can be validated through the class-group identity; the ambient-precision trap is a recurring lab defect with a named detector.

Order-blindness barrier: the selection lemma is equivalent to the target bound; corrected selection inequality (star) and the primitivity requirement

obstruction | hunts/support_517b887f | grade: proved (ordinary derivation, self-reviewed); exhaustive check of the repaired form over 104,183 admissible sets with zero primitive violations (measured)

Define c(A) = max size of a subset of A all of whose distinct pairwise sums are powers of 2. Then c(A) in {1,2} for every admissible A (archimedean base case), so for any phi, [forall admissible A: c(A) >= |A|/phi(k)] iff [g(k) <= 2 phi(k)] up to a factor 2. c(A) is defined by a maximum over subsets and mentions no prime order or grouping, so no reordering or joint processing of primes changes what the Erdos-Turan architecture proves. Two defects in the standard sketch: the per-prime halving |B'| >= |B|/2 is false (A = {1,3,15,21,33}, q=3 gives 2 < 5/2); the correct form is |B'| >= (|B| - n_0(q))/2 + min(n_0(q),1) and the assembled bound c(A) >= n/2^s needs primitivity (3*{1,3,7,17,47} has c = 1 < 5/4). The CRT configuration x_eps = eps_i mod q_i realises the 2^s loss exactly, jointly or one prime at a time.

Evidence: RESULTS.md section 3 (defects 3.1, 3.2), lines 77-116; section 4 Theorem (order-blindness) and Corollary, lines 118-160 Prior art: cited: Erdos-Turan 1934, Erdos-Suranyi, Gyory-Stewart-Tijdeman 1986 as the two-set tool the architecture does not use Reused in: none recorded Why it travels: Shows how to prove that 'reorder/combine the steps' cannot help: find the order-blind quantity the architecture reduces to and show the refined lemma is the target restated.

Refutation of the mod-p pigeonhole size bound, with the repaired residue-support lemma

obstruction | hunts/support_60982bf6/ | grade: unconditional (explicit witnesses re-verified); the structural argument is an ordinary proof

Proposed (r_186989): |A| ≤ p − 1 whenever p ∉ S, by pigeonhole on residues mod p against r ↔ −r. False for every odd p, structurally: p ∉ S forces a + b ≢ 0 (mod p) for distinct a, b ∈ A, so the occupied residue set R satisfies R ∩ (−R) ⊆ {0} and at most one element of A lies in class 0; but for odd p and r ≢ 0, 2r ≢ 0, so arbitrarily many elements may share one nonzero class. The pigeonhole bounds the number of occupied classes by (p−1)/2

parity argument works there and nowhere else. Counterexamples (re-verified by trial division): p=3, S={2,5,7,11,23}, A={5,45,65,95,155,395}, |A|=6; p=5, |A|=8; p=7, |A|=8; smallest: S={2,5,7}, A={1,3,7,13}, 3 ∉ S. Repaired Lemma 3: for p ∉ S, A meets at most one of {r,−r} for each r ≢ 0 and |A ∩ pℤ| ≤ 1; hence |A| ≤ 2 when 2 ∉ S, and no size bound when p is odd.

Evidence: RESULTS.md §3 'Refutation: the mod-p pigeonhole lemma is false. Unconditional.' (lines 126–180) Prior art: none Reused in: audit table §6; door ranking (2 ∈ S is the one proved binding constraint) Why it travels: Closes the class of residue-pigeonhole size bounds for this problem with an argument, and records the one prime where it works; the residue-support form is the correct starting point for any counting argument.

Residue relaxation (R1)+(R2) has optimum exactly 2^k: no residue/parity/deletion recurrence beats loss 2 per prime

obstruction | hunts/support_7ddfee4b | grade: proved (ordinary derivation, self-reviewed)

A relaxed instance of order k is a finite set V, odd primes p_1..p_{k-1}, maps r_i : V -> (Z/p_i)^*, and a graph G on V with (R1) every distinct pair either has r_i(u)+r_i(v) = 0 mod p_i for some i or is an edge, and (R2) G triangle-free. Every primitive admissible pair with 2 in S yields such an instance (G = pairs summing to a power of 2). Theorem: max |V| = 2^k exactly (upper: the sign vector epsilon_i(u) = [r_i(u) in H_i] has fibres that are cliques, so size <= 2; lower: V = {0,1}^{k-1} x {0,1} with residues 1 or p_i-1 and a perfect matching). Hence any exp(o(k)) bound must read something the relaxation discards: actual S-smoothness (not just divisibility by some p_i), CRT rigidity of residues of the same integer, or the ordering of Z.

Evidence: RESULTS.md section 3 Theorem 5 and the 'Discarded' table, lines 131-184 Prior art: unsearched Reused in: support_517b887f's order-blindness barrier is the same wall from the selection side Why it travels: A template for proving that a whole family of cheap arguments is capped: define the relaxation they all factor through, compute its exact optimum, list what it discards.

Amplification-refutes theorem and enumeration-is-refutation-only quantifier obstruction

obstruction | hunts/support_8ea74995 | grade: proved (ordinary derivation, self-reviewed)

(5.1) If g(k+C) >= lambda g(k) for constants C >= 1, lambda > 1 and all k >= k_0, then log g(k) >> k and #126 is false; supermultiplicativity is the special case C=1, lambda=g(1)=2, no Fekete needed. (5.4) #126 is a forall S forall A statement; every bounded computation (clique search in a box, sweep over a finite family of S, witness table) establishes an exists S exists A statement, so no enumeration can contribute to a proof, only refute. Companion table of which WLOG normalisations are valid for upper bounds (gcd A = 1, 2 in S) versus invalid (1 in A, max A <= N, S = first k primes), the S-unit scaling orbit making every box unjustified for upper bounds.

Evidence: RESULTS.md section 5 Theorems 5.1, 5.2, 5.4, lines 196-248; section 3 normalisation table, lines 127-150; sharpened in support_eccd5f5e section 3 (composition law would pin g(k) in [2^k, 1.5*2^k]; g(3) <= 7 refutes it) and 'Correction B' (report max(g_N, ceil((p_k+1)/2))) Prior art: cited: Erdos-Turan 1934; Fekete noted as unnecessary Reused in: support_eccd5f5e, support_7ddfee4b, support_517b887f (direction-safe target list) Why it travels: A one-line test that sorts any proposed lemma or instrument into prove/refute/settles-nothing before budget is spent; the quantifier argument applies to every extremal problem attacked by search.

Obstruction: admissible cliques yield only four-term S-unit relations, for which no height theorem exists

obstruction | hunts/support_95bb5cb7/ | grade: ordinary derivation, self-reviewed (argument from cited theorems plus the measured threshold table)

Height bounds for S-smooth numbers (Baker-Győry effectively, abc conjecturally) are theorems about x + y = z with all three terms S-smooth or two smooth and one fixed. An admissible clique (all pairwise sums a + b S-smooth) produces no such relation; its only relations are (a+b) + (c+d) = (a+c) + (b+d), four-term S-unit equations, for which only the Evertse-Schlickewei-Schmidt bound on the number of non-degenerate solutions is available, which is exp(O(k)) and provably not improvable to exp(o(k)) in general (Erdős-Stewart-Tijdeman exhibit S with more than exp(c (k/log k)^{1/2}) solutions of x + y = 1). A count is not a height. Even the best imaginable outcome, a three-term relation plus abc, gives max A <<_eps rad(S)^{1+eps}, i.e. c -> 1 in the threshold lemma where the lower bound already reads 1.01 k, one full power of the radical above what the counting horn needs. Hence the size dichotomy for Erdős #126 cannot close from inside the data (integers and smoothness of pairwise sums); the missing object is a height theorem for x_1 + x_2 = x_3 + x_4 in S-units with a clique-suppliable non-degeneracy hypothesis.

Evidence: RESULTS.md §3.2 (lines 119-139), 'The doors' §3 information class (lines 236-243), loose thread 'Four-term S-unit heights' (lines 247-252) Prior art: cited: Baker-Győry, abc, Evertse-Schlickewei-Schmidt, Erdős-Stewart-Tijdeman 1988 Reused in: agrees with hunts/r_186989's conclusion 'from the other side', now quantified as one power of rad(S) Why it travels: Closes every descent/height route to Erdős #126 that stays inside smoothness-of-sums data, and states the exact literature request (a four-term S-unit height theorem) that would reopen it.

Lean arm: formalised elementary number theory

Kernel-checked Mertens bands, Abel summation by direct induction, and the transfer patterns that make an aggregate inequality formalisable.

Zero-admission sandwich for formalised extremal counts

lemma | hunts/r_186989/ | grade: proved (elementary), self-reviewed; the finite table is exhaustive over [0,40]

Let f(n) be the minimum over n-sets of positive integers and f_0(n) the minimum over n-sets of nonnegative integers (the Finset N formalisation) of the number of primes dividing some off-diagonal sum a + b. Then for n >= 2, f(n-1) <= f_0(n) <= f(n). Proof: every positive set is a nonnegative set (upper); if an f_0-optimal set contains 0, removing it leaves a positive (n-1)-set whose off-diagonal sums are a subset (lower). Hence f(n)/log n -> inf iff f_0(n)/log n -> inf, so the formalised limit statement is faithful, while pinned finite values differ (f(2) = 1 via {1,2}, f_0(2) = 0 via {0,1}). Also proved: 2 not in S implies |A| <= 2 (parity), and g(1) = 2 (three elements with pairwise sums powers of two are impossible).

Evidence: RESULTS.md section '2. Arm 0: the Formal Conjectures positivity mismatch. Settled.' (lines 27-61); section 4 opening paragraph for g(1) = 2 and the parity remark Prior art: unsearched Reused in: hunts/support_f3ab3e34/RESULTS.md Why it travels: The same one-step sandwich settles the 0-versus-positive mismatch for any extremal problem over integer sets whose constraint is monotone under deletion, before anyone repairs a formal statement unnecessarily.

Pointwise-from-density transfer for normal-order theorems via the sqrt(N) split

lemma | hunts/r_233abe/ | grade: kernel-checked (lake build ZetaLean.HardyRamanujantheorem, zero sorry, axioms [propext, Classical.choice, Quot.sound])

Given the density form of Hardy-Ramanujan at scale N (for every eps > 0, #{n <= N : |omega(n) - log log N| > eps log log N}/N -> 0), the pointwise-normalised form (deviation measured against each n's own log log n) follows with no new arithmetic: split (0, N] at n^2 <= N; the low range has at most sqrt(N) elements and sqrt(N)/N -> 0; on the high range N < n^2 gives log log N - log 2 < log log n <= log log N, so |omega n

log N eventually; hence the pointwise exceptional set sits inside the low range union exceptional(N, eps/2) and the density theorem at eps/2 finishes. The pointwise form inherits every sharpening of the density form for free. Also: omega n = ArithmeticFunction.cardDistinctFactors n is rfl (primeFactors.card vs primeFactorsList.dedup.length unfold to the same term), giving a Mathlib-facing statement hardy_ramanujan_cardDistinctFactors.

Evidence: RESULTS.md sections '(a) The bridge' and '(b) The pointwise form' (steps 1-4: card_low_range_le, tendsto_natSqrt_div, loglog_band, exceptionalPointwise_subset); 'What the kernel said' Prior art: unsearched (the split is a standard device; the formal transfer is the hunt's own) Reused in: none recorded Why it travels: The same split-and-slack transfer converts any scale-N normal-order or density statement into its pointwise-normalised form in a formal library, at the cost of one constant (log 2) and a sqrt(N) exceptional set.

One-sided Mertens band accounting and the log t <= t/e majorant

bound | hunts/r_4218d4/ | grade: kernel-checked (Lean 4 + Mathlib v4.33.0-rc2, zero sorry, standard axioms), as the hunt states

Three local tightenings move the kernel-checked Turan variance constant from 5855 to 275 with no statement reshaped: (1) log t <= t/e for t > 0 (ZetaLean.Mertens.log_le_div_exp_one, from log x <= x - 1 at x = t/e), giving 2 sqrt(x) log x <= (4/e) x < (3/2) x and psi(x) <= x log 4 + (3/2) x for x >= 1 (psi_le_const_mul_self'), against Mathlib's log 4 + 4; (2) sum_{n <= N} (log n)/n^2 <= 3/2 by starting the telescope at n = 2 (sum_{2 <= n <= N} n^{-3/2} <= 2 - 2/sqrt N) with the same majorant, so the prime-power tail in Mertens I is 3 not 12; (3) keep the two halves of the von Mangoldt form apart: the lower half loses only 1 and the upper half loses the Chebyshev constant, and the prime form loses the tail on the lower side only, so the honest band is max(c_psi, 1 + tail) = 4 instead of the symmetric c_psi + tail = 5.886 (new one-sided lemma log_sub_one_le_sum_vonMangoldt_div). Resulting constants: mertens_first_theorem band log 4 + 3 = 4.3863 (was log 4 + 16), mertens_second_theorem band 16 (was 76; assembly needs 15.698 with 1/log 2 <= 1.443), sum_sq_dev_le constant m^2 + m + 3 = 275.

Evidence: RESULTS.md sections 'Before and after' (table), 'What was chosen, and why' items 1-4; PrintMertensAxioms.lean; lean/ZetaLean/Mertensstheorems.lean Prior art: cited: Mathlib Chebyshev.psi_le; sharp values 4/e and classical bands 2 and 4 quoted Reused in: hunts/r_233abe/RESULTS.md; lean/ZetaLean/Mertensstheorems.lean; lean/ZetaLean/HardyRamanujantheorem.lean Why it travels: The majorant and the one-sided accounting are drop-in tightenings for any formalised Chebyshev/Mertens-type chain; the quadratic propagation m^2 + m + 3 tells you where upstream slack pays most.

Direct-induction discrete Abel summation and explicit Mertens bands in Lean

computational technique | hunts/r_3c1cbb/ | grade: kernel-checked (lake build exit 0, zero sorrys by grep, no warnings)

Mertens's theorems kernel-checked against pinned Mathlib v4.33.0-rc2 with explicit constants: |Σ_{n ≤ N} Λ(n)/n − log N| ≤ log 4 + 4; |Σ_{p ≤ N} (log p)/p − log N| ≤ log 4

(mertens_second_theorem), all deliberately slack, with the slack decomposed into named local steps (tightening the first band to the classical 2 would bring 76 to ≈ 9.8). Technique that closed the second theorem: the discrete Abel identity Σ_{p ≤ N} 1/p = A(N)/log N + Σ_{n=2}^{N−1} A(n)(1/log n − 1/log(n+1)), A(n) = Σ_{p ≤ n} log p/p, proved by induction from N = 2 via Finset.sum_Ioc_succ_top (increment 1/(N+1) if N+1 prime, else 0, one field_simp; ring per case) instead of reindexing Mathlib's range-indexed sum_range_by_parts, killing the estimated 150–250-line reindexing labour; termwise comparison via (v−u)/v ≤ log v − log u ≤ (v−u)/u at u = log n, v = log(n+1), with overshoot ≤ 4/n² and Σ_{n ≥ 2} 1/n² ≤ 1 from sum_Ioc_inv_sq_le_sub. Also built: Σ_{n ≤ N} 1/(n√n) ≤ 3 − 2/√N (absent from Mathlib PSeries, which has only s = 2).

Evidence: RESULTS.md 'The statements that landed' (lines 23–47); RESULTS-second.md 'How the block was closed' (lines 53–88), Loose threads; lean/ZetaLean/MertensSecond.lean sum_inv_primes_eq (line 149), sum_Ico_telescope (line 128) Prior art: searched-and-absent in pinned Mathlib (grep -ril mertens hits only Dedekind–Mertens; SumPrimeReciprocals has divergence with no rate) Reused in: hunts/r_8c3b94 (Erdős–Kac pricing uses mertens_second_theorem as the O(1) band and finds the constant is not needed); the direct-induction Abel pattern flagged for extraction Why it travels: The induction-on-N Abel pattern closes any partial-summation argument with a monotone weight f on [2,∞) without Finset reindexing; the explicit bands are the constants downstream Lean work must consume.

Supermultiplicative composition refutes rather than proves (Fekete direction check)

obstruction | hunts/r_186989/ | grade: proved (elementary), self-reviewed; the search data are lower bounds only

For Erdos #126 in the inverse form g(k) = max{n : f(n) <= k}, the conjecture is g(k) = exp(o(k)). Any rigorous composition law g(k_1 + k_2) >= g(k_1) g(k_2) makes log g superadditive, so by Fekete lim g(k)^{1/k} = sup g(k)^{1/k} >= g(1) = 2, hence g(k) >= 2^k and the conjecture is false. Therefore a 'find the gadget' lane is a refutation search; the problem asks for an anti-composition theorem. Bounded evidence against any small gadget: supermultiplicativity at (1,2) needs g(3) >= 8, exhaustive search to N = 60000 gives 5; all 11 tested (k_1, k_2) pairs fail by wide margins. Confirmed and extended by the independent-architect arm: transferring Erdos-Stewart-Tijdeman's exp((4+o(1))(s/log s)^{1/2}) construction would confirm, not refute.

Evidence: RESULTS.md section '4. Arm 3: the composition gadget points the wrong way' (lines 108-143); support_f3ab3e34/RESULTS.md section '5. Routes that point at refutation' Prior art: unsearched Reused in: hunts/support_f3ab3e34/RESULTS.md Why it travels: A direction check to run on any proposed 'structure theorem' for an extremal function before funding it: if the structure is a semigroup or product law with a positive seed value, Fekete turns it into a refutation.

Certificates, verifiers and exact arithmetic

Exact rational acceptance of published witnesses, fault-injection ladders for verifiers, and the controls that separate an instrument reading from a mathematical claim.

Signed-lattice binomial/trinomial sieve on a support torus

lemma | hunts/r_044dd2/ | grade: exact (integer certificates, independently audited); the hunt labels the algebraic sieve 'open' overall

For a polynomial system restricted to a candidate support (all supported variables nonzero, a torus), every equation whose support has exactly two monomials x^a + x^b = 0 yields the signed lattice relation x^{a-b} = -1. If an odd integer combination of these relations equals the exponent difference of two terms of an equation whose support has exactly three monomials, those two terms cancel identically on the torus, forcing the remaining supported monomial to vanish, which is impossible; hence the support is infeasible. Each exclusion is an exact integer certificate (l1 norms 1, 3, 1 for three successive orbit-18 supports of the Krenn-Gu 8x3 system, with 81/240/42 zero-binomial relations and 771/792/294 zero trinomials), independently verified by audit_laurent.py (binomial and trinomial supports, exponent identity, odd parity, nonzero forced monomial).

Evidence: RESULTS.md 'Result 2: three successive orbit-18 supports have exact contradictions' (statement, table, audit paragraph); laurent_sieve.py; audit_laurent.py; CHECKSUMS.sha256 Prior art: unsearched Reused in: none recorded Why it travels: A cheap exact pre-filter for any support-enumeration attack on a sparse polynomial system (matching polynomials, design equations): it kills branches with a small integer certificate before any Groebner or numerical solve.

Branch-and-bound prune soundness under nonnegative pair charges, with planted-bite control

lemma | hunts/r_401bbf/ | grade: measured to double precision (both solvers float; residual 1e-16), plus an ordinary derivation, self-reviewed, for the soundness claim

Consider maximising a quadratic trade over multiplicity vectors m (sum m <= budget) with objective val = sum_i cap_i m_i - sum_{j<i} 2 q_ji m_j m_i - (diagonal terms), zones sorted by descending cap. Claim: if every pair charge q_ji >= 0 and every cap >= 0, then the inner prune 'descend into branch m only if v > val - 1e-18 or m == 0' removes nothing and the outer bound 'return if val + rem*caps[idx] <= best' is admissible, so the search returns the exact maximum. Proof: F(ms, idx, rem) (best completion) is nondecreasing in rem and nonincreasing in each prefix multiplicity (a prefix atom enters only through -2 q m_j m_i <= 0); a non-improving branch's best completion v + F(ms+[m], idx+1, rem-m) <= val + F(ms+[0], idx+1, rem), which the always-taken m = 0 branch reaches. The hypothesis holds by construction when q = Kpair(.)/200 with Kpair = Re(ghat)^2, a square, for any cap vector. The 1e-18 tolerance is in the permissive direction. Planted-bite control: break the hypothesis with one negative off-diagonal charge and the four cut configurations give 0.184 / 0.146 / 0.226 / 0.226 against exhaustive 0.226, showing the probe detects a bite and that disabling only the inner prune (0.146) is worse than leaving both cuts (half-audit worse than none). Audit method: since components have at most 8 charged zones, enumerate all C(Z+10,Z) <= 43758 vectors as one vectorised quadratic form (exact maximum with no cuts) and difference against the published search cell by cell on the same zone data: 13,200 cells, max |delta| 1.1e-16, 1467/965 negative deltas proving the residual is summation order.

Evidence: RESULTS.md §0 table (lines 11-36), §1 (lines 38-66, exhaustive enumeration as a fortiori bound), §2 'Claim' and proof (lines 74-115), §3 controls table and planted bite (lines 119-145); test_prune_discharge.py Prior art: unsearched (standard branch-and-bound monotonicity; the hunt does not claim novelty) Reused in: discharges the load-bearing assumption recorded in hunts/r_a97060/RESULTS.md §4 for hunts/frontier_math/k2_closure.py zone_trade; recommends the exhaustive trade replace the heuristic Why it travels: Whenever a heuristic pruning rule in an LP/branch-and-bound feeds a published table, the pattern is: prove the prune sound from a sign hypothesis that holds by construction, enumerate exhaustively where components are small, and plant a hypothesis-breaking instance to show the audit can see a bite.

Exact rational dual certificate by directed rounding of irrational LP coefficients

computational technique | hunts/overlap_lower/ | grade: VERIFIED (exact arithmetic, no floating point in the value)

For White's simplified program (4.1)-(4.4), the dual is: maximize (N/4) lam - z/3 subject to sum_j y_j <= 1, lam <= y_j + sum_m a_{m,j} u_m + s_j z for all j, y,u,z >= 0. The float solver only proposes (u, z). Then in fractions.Fraction: replace each trigonometric envelope alpha^-_{j,2m} by a rational lower bound (float cosine, floored at denominator 10^7, minus one unit, minus the Lipschitz term as an exact rational), which enlarges the primal feasible set and can only weaken the bound; form theta_j exactly; find the largest feasible lam exactly by sorting theta. At N, R = 5000, 20: float optimum 0.3739966049729149, exact dual value 0.37399241331212807... (29-digit numerator over 29-digit denominator), sum y_j = 1 exactly, dual feasible exactly, cost of directed rounding -4.19e-6. The exact step is also the only guard that catches a dropped envelope (front F): monotonicity in N and R does not.

Evidence: hunts/overlap_lower/RESULTS.md, Section D3 'Exact rational dual certificate (VERIFIED)' (lines 257-287) and Section F 'The guard, and what it misses' (lines 406-434) Prior art: White arXiv:2201.05704 program cited; exact directed-rounding acceptance unsearched Reused in: none recorded Why it travels: Template for turning any float LP dual into a rigorous certificate when constraint coefficients are irrational: round each coefficient in the direction that relaxes the primal, then verify feasibility and value in exact rationals. Index note: See the r_8539dc entry: same family.

Semi-infinite LP repair: subtract the sup to get a valid bound and a bracket

computational technique | hunts/r_6f0f63/ | grade: measured (float, rung 1); the repaired value would be enclosure-carrying only after recomputing sup in ball arithmetic and rationalising coefficients, not done

A Delsarte-type LP (Gegenbauer basis, f_0 = 1, coefficients f_k ≥ 0, f(t) ≤ 0 on [−1,1/2]) solved on a finite node set is a relaxation, so its optimum can sit strictly below the true LP value and is not a bound: at m = 600 nodes in dimension 24 it reports 196505.76 for the exactly-tight 196560 (dimension 8: 239.9930 for 240), at every m tested up to 20000. Repair: compute M = sup_{[−1,1/2]} f (here via Chebyshev interpolation and companion-matrix roots); if M > 0 and f_0 > M, then f − M satisfies the sign condition exactly, leaves f_k (k ≥ 1) untouched, and gives the valid value (f(1) − M)/(f_0 − M). The true LP value is bracketed: node optimum ≤ truth ≤ repaired value, the repaired value converging from above. Measured: the one-sided error decays like m⁻² with Chebyshev-clustered nodes; degree saturates by d ≤ 14 in every dimension 3–24, so node count, not degree, is the binding parameter.

Evidence: RESULTS.md §2 'The soundness read' (lines 55–103), §3; probe.py sup_on_interval (line 91), repaired_bound (line 114); HANDBACK.json core_candidates Prior art: unsearched; hunt says 'built by hand'; Delsarte LP and the tight E8/Leech certificates are the literature's Reused in: HANDBACK.json names zeta/weil.py positivity probe as first consumer (not done) Why it travels: Every semi-infinite programme in the lab (Delsarte, Cohn–Elkies, Bachoc–Vallentin, Weil-positivity probes) is discretised in the same unsound direction; the repair prices the coarseness instead of hiding it.

Exact rational acceptance of a primal step-function witness with mass repair, and the rounding-denominator calibration

computational technique | hunts/r_828c8b/ | grade: VERIFIED for the acceptance step and D1-D2; MEASURED for D3 witnesses

The value of an m-piece step function f for Erdos's overlap functional is a finite sum of products of its pieces. Round the pieces to multiples of 1/D, repair the mass constraint in Q so the object is exactly feasible, then evaluate max_j h_j with fractions.Fraction; no floating point survives into the value. Calibration: at the natural D = 2m the rounding penalty is 2.4e-4 to 6.8e-4 (larger than the remaining gap to the published constant); D = 20m costs nothing in runtime and drops the penalty to 1.1e-5. Best exact object: C <= 9990167/26214400 = 0.381094627... (128 pieces, D = 2560). Also the vacuity obstruction for single-weight averaging: for any probability density w on the shift axis, sup_t h_f >= <f, w*1> - <f, w*f>, but the naive Fourier bound <f, w*f> <= sup what * ||f||_2^2 <= 1 since what(0) = 1, so the bound collapses to <= 0 for every admissible weight (200 random weights: 0 positive), and explicit f-witnesses cap the averaging skeleton at 0.2526 (uniform), 0.2500 (tent), 0.1909, 0.1701.

Evidence: hunts/r_828c8b/RESULTS.md, Section C 'The acceptance step (VERIFIED)' (table lines 96-103, remarks 1-2) and Section D 'The lower-bound side' D1-D3 (lines 134-181) Prior art: White arXiv:2201.05704 and Haugland cited as reference values; acceptance step unsearched Reused in: none recorded Why it travels: Generic exact acceptance for any upper-bound witness given by a finite parameter vector, with a measured warning that coarse rounding throws away more than float optimisation gained; the D2 argument kills any 'average against a probability weight, bound the quadratic term by Parseval' route in one line. Index note: See the r_8539dc entry: same family.

Exact rational evaluation of autoconvolution functionals on step-function witnesses (knot lemma)

computational technique | hunts/r_8539dc/ | grade: measured (the hunt's own words: every arithmetic step is exact rational arithmetic on the published decimals; what is not exact is the published truncated witnesses themselves)

For a step function f on [−1/4, 1/4] with n equal steps of width h = 1/(2n) and heights a_0…a_{n−1}, f*f is supported on [−1/2, 1/2], is piecewise linear with knots at multiples of h, and takes the value h·b_k at the k-th knot where b = a⋆a is the discrete autoconvolution. A piecewise-linear function attains its extrema at knots, so with ∫f = hΣa: A := max_t (f*f)(t)/(∫f)^2 = 2n·max_k b_k/(Σa)^2 and B := max_t |f*f(t)|/(∫f)^2 = 2n·max_k |b_k|/(Σa)^2. Since ∫f*f = (∫f)^2 > 0 forces max_t f*f > 0 for every admissible f, an abs() outside the max is a no-op, and A ≤ B always, so a bound under B is a bound under A but not conversely. Ingest published ten-place decimals as Fractions with denominator 10^10 so the entire chain is integer arithmetic, and self-check the discretisation by reproducing a published figure on which both readings must agree (1.4688 at n = 150, where max|b| = max b). Results: height_sequence_3 (n = 400) gives A = 1.455642795374540… but B = 4.334046524387984… (extreme knot negative at index 215); height_sequence_4 (n = 150) gives A = B = 1.468762069741021…; so 1.4557 was stated against the wrong functional in November 2025, and the corrected statements (arXiv v2, colab commit 39d0c63) survive with no number moving.

Evidence: hunts/r_8539dc/RESULTS.md §1 The two functionals, and why the published abs() is a no-op; §2 Both functionals on both published sequences, exact; probe.py (exact_heights, autoconvolve, functionals); MISSION.md kill_conditions Prior art: unsearched for the knot lemma (elementary); the functionals and prior bounds are Matolcsi–Vinuesa 2010 and Vinuesa 2009 as published Reused in: none stated; HANDBACK.json names it a Core candidate shared with r_2ac05f, r_a7c12f and overlap_lower, each of which hand-rolled its own exact evaluator Why it travels: Any published constant defined as a max of a convolution of a step-function witness can be recomputed exactly under each candidate reading of the inequality, on the same witness, with the discretisation validated by a figure both readings share. Index note: Same technique as the exact rational acceptance in overlap_lower and r_828c8b; three hunts each hand-rolled it, which is exactly the reuse this index exists to stop.

Soundness read of a certificate verifier: target-encoding prunes, duplicated constants, binary64 goal comparisons, cap semantics

control | hunts/bloch_ceiling/ | grade: measured/verified code reading; no finding overturns a published run

Checklist applied to third-party rigorous verifiers, each item with a finding: (1) does any prune's justification encode the target (the Hunt #79 zeta pattern)? Bloch: no, the away subdivision discards a box only against goal = sqrt(3)/4 + target with target a run parameter. (2) Are load-bearing constants duplicated across source and stored data and read by different routines with nothing asserting agreement? Bloch: ETA and LARGE_RAD live in three places; verify_positivity uses the source LARGE_RAD while verify_near_moment integrates to the data's large_rad (latent, bit-identical as shipped). (3) Does a constant silently encode another certificate's result? Bloch: C = 3.2888 is valid only because the fine fixed-radius certificate proves |a_3| <= 3.28877762819, asserted nowhere in the variable-radius programs. (4) Are Arb bounds compared against a goal computed in binary64? Bloch: float(sqrt(3)/4 + 0.0153) is 4.5e-17 below exact, so accepted sectors prove target - 4.5e-17. (5) Does the cap refuse or accept when hit? Bloch: refuses (RIGOROUS INCOMPLETE), soundness untouched but the documented command cannot reproduce the run. Field audit of trmdy: acceptance one-sided against the target rounded up, tables clamped nonnegative before unsigned products, sign-aware product used where the table can be negative, float pre-filter can only decline to prune.

Evidence: hunts/bloch_ceiling/RESULTS.md, Section 3 'Soundness read of the verifier' (findings 1-6, lines 117-180); hunts/field_audit/RESULTS.md Section 6 'Soundness read of trmdy's verifier' (lines 187-259) Prior art: lab-internal pattern (Hunt #79 / Gohms target-wiring); unsearched externally Reused in: hunts/field_audit/RESULTS.md Section 6 applies the same read to trmdy/zeta-simple-zeros-673137 Why it travels: A concrete, repeatable audit list for any interval/Arb branch-and-bound certificate; distinguishes latent from load-bearing defects and names the sound failure direction for each.

Theorem-universe instrument validation: point the diffraction instrument at a proved quasicrystal before pointing it at zeta

control | hunts/golden_control/ | grade: measured (numpy float64; exact integers for the Pisano stage); 'instrument validation against a proved answer, in the same spirit as zeta/finitefield.py'

Before trusting a tapered-transform 'atoms at log prime powers' measurement on zeta's zeros, run the identical transform architecture on objects whose spectrum is a theorem, with predictions pre-registered: P1 calibration on Z (Poisson summation peaks at 2pi, 4pi, 6pi to 6.4e-14, fixing the instrument's one constant, taper mass T sqrt(2pi), by an elementary identity); P2 the golden cut-and-project set with window [0,1), density 1/sqrt5, whose Fourier module and intensity law |W|/covol * |sinc(k*|W|/2)| are derived in code from the embedding lattice and reproduced to 8.7e-9 in position and 2.5e-8 relative in amplitude; P3 silence off the module (200 random frequencies, median 1.0e-5 of scale vs weakest peak 9.7e-2, 9717x separation against a 30x bar; zeta's prime-power gate measured 26.8x); P4 lesions: Gaussian jitter suppresses peaks by the Debye-Waller factor (log-slope vs -sigma^2/2 within 1%), a Poisson set of matched density shows max response 14x below the weakest true peak; P6 precision response monotone under doubling extent/taper (8.6e-9 -> 1.8e-9 -> 1.0e-10). P5 recorded a pre-registered Pisano claim that the exact-integer stage falsified at p = 3 (correct statement: pi(p) | p-1 when chi5(p) = 1, pi(p) | 2(p+1) when chi5(p) = -1), kept on the books as the derive-never-remember rule doing its job.

Evidence: RESULTS.md 'What was measured' P1-P6 (lines 15-52), 'What this buys the tree' (lines 54-62); probe.py, results.json; MISSION.md pre-registered predictions Prior art: cited: Poisson summation, cut-and-project diffraction theory, Debye-Waller; the control design is the lab's Reused in: KAPPA-CLOSED-FORM.md §5 in hunts/lambda_dh_bounds (conductor-5 golden arithmetic connection); the quasicrystal gate on zeta inherits the control Why it travels: Any spectral instrument aimed at an arithmetic object should first reproduce a proved spectrum, a proved silence and a quantitative lesion law; the pre-registered bars and the recorded failed prediction are the reusable discipline.

Planted-fault guard audit against provable monotonicity, with the recorded miss

control | hunts/overlap_lower/ | grade: measured

When a program is provably monotone in a parameter (adding constraints cannot lower a minimum; refining a grid cannot lower a relaxation), plant faults in the builder and check which are caught by monotonicity in N and in R. At N, R = 2000, 10: negative control 0.372105479; drop the pi m L/4 envelope: +2.84e-3, unsafe direction, NOT caught (the defective program is still monotone, it is simply a different program); constrain every other mode: -1.33e-2, safe, not caught; constrain cos(pi m x) at odd m: infeasible, unsafe, caught. Conclusion stated for reuse: on this program the guard is the exact acceptance step, and monotonicity is not a substitute for it. Companion from r_828c8b: grid_sufficiency_defect (8x-refined reference supremum minus reported maximum) over 40 random step functions catches mis-strided enumeration (defects 0.104, 0.067) but not truncation of the shift range (0.000), a miss written into the guard ledger.

Evidence: hunts/overlap_lower/RESULTS.md, Section F 'The guard, and what it misses (MEASURED)' (table lines 416-421); hunts/r_828c8b/RESULTS.md Section E 'The guard, and its power (MEASURED)' (table lines 207-212) Prior art: unsearched Reused in: r_828c8b records its miss in harness/departments/guard_ledger.py Why it travels: A cheap power measurement for any monotonicity-based sanity check: it tells you which fault classes the check is blind to, so the blind class can be covered by an exact step.

Independence-by-mutation: shared-layer lesion for 'independent' verification routes

control | hunts/r_065f29/ | grade: measured

Two verification routes are independent only on the layers they do not share. Test: mutate a layer both routes import and check whether their outputs diverge; if both move by the identical amount they are one route. Applied to the urms2-0.51 window functional, mutating the shared layer moved both the primary route and the audit's 'JSON fixture and rebuilt tail sum' route by exactly 4.426081703885579e-27, the tail terms are bit-identical exact rationals, and the fixture C2_EXTENDED.json reproduces corrected_coefficients(40) index for index: the audit route is a second arithmetic assembly of the same numbers. Companion formulation in the compiler department: enumerate each route's layers and report the independence radius (shared prefix length), e.g. 1 of 6 with only 'LLVM IR text' shared; agreement is evidence only about the distinct layers. The director's ledger states the same for zeta/rigor.py's two backends (shared _exact, contour, grid policy) and predicts the next correlated failure will be a shared input.

Evidence: RESULTS.md table row A2 and section 'A2: gate 6's independent route is arithmetic, not independence'; director_run/CLAIMS.md C-RIG-01 and C-RIG-03 and GRAVEYARD.md 'Standing predictions' item 2; r_e2ee73/RESULTS.md section '6. Verification Path Independence' Prior art: unsearched Reused in: hunts/director_run/CLAIMS.md; hunts/r_e2ee73/RESULTS.md; harness/departments/review_ledger.py Why it travels: A mechanical check to run before any 'two independent routes agree' sentence is written, in Python or Lean or C++: plant one fault in the suspected common layer and watch both routes.

Axis-separated sweep and hypothesis check of a falsification control's family

control | hunts/r_065f29/ | grade: measured (numpy and sympy, no enclosures)

A control offered as evidence for a step must (a) run on a family satisfying that step's hypothesis and (b) vary the quantity the step claims is invariant while holding the others fixed. The urms2-0.51 record's section 9 ladder moves x = e^{0.51 l} and W = e^l together along one ray, so its decreasing sequence (0.409, 0.282, 0.185) is consistent both with the claimed W-independence and with W^{0.825} growth; sweeping W alone at fixed x on the record's own frozen coefficient family shows the upper-range sum growing 32.1x, and that family violates the hypothesis A(y) << y log y (A(y)/(y log y) climbs by a factor 88 from 0.0231 at y = 400 to 2.036 at y = e^{10}), whereas a surrogate built to satisfy the law (|b(n)|^2 = log n) saturates (441.6 -> 984.7 over a 67x sweep). The load-bearing step itself survives when attacked correctly: the exact block second moment int_U^{2U} |sum c_n n^{-it}|^2 dt saturates to four figures (17.2964 -> 17.3642) as W/U grows 7.5x. Also found: the recorded rational witness does not select alpha = 51/100 (admits 257/500 at the published (delta, gamma, eps), and 0.9 with them free).

Evidence: RESULTS.md table rows A1, A3, A5 and sections 'A3: the load-bearing step holds where it was attacked', 'A5: the record's own control does not satisfy its own hypothesis'; probe.py; results.json Prior art: unsearched Reused in: harness/departments/review_ledger.py Why it travels: Two questions to ask of any numerical falsification table before citing it: does the family obey the hypothesis, and does the sweep move the claimed-invariant axis alone; both are cheap and both were failed by a record that looked supported.

Control: planted-fault lesion with two separated verdict classes

control | hunts/r_0dfb8d | grade: VERIFIED (re-derived from pinned artifacts), per the hunt's own label scheme

For a two-verdict reproduction (A: aggregate recount equals the published integer; B: each per-instance verdict is consistent with the stated rule, required lists taken from the split not the entry), lesion a copy of one unit's archive with four faults chosen so that the expected response pattern differs per class: flip one verdict (A mismatch, B inconsistent), add a failure under a positive verdict (B only), drop required successes (B only), delete one report file (A only). All four must be detected and the two classes must move independently. Supplement: a finer published artifact (12 per-repository sub-counts) is cross-checked so one aggregate agreement cannot come from cancelling errors. Recorded self-catch: a '+' in an object key read as a space produced a false absence; a procedure that reports the target's absence when the fault is its own fetcher is the failure mode this class of check exists to catch.

Evidence: RESULTS.md 'The check can fail: four planted faults, four reds', lines 71-86; per-repo cross-check, lines 55-60; section 6 on the self-inflicted retrieval bug, lines 187-194 Prior art: unsearched; label scheme inherited from Hunt #80 Reused in: labels and shape carried from hunts/r_* Hunt #80; probe.py control stage Why it travels: The 'lesion must move exactly the class it should' design is the same discipline the Lean lesion tests use, applied to any recount/reproduction instrument.

Mutation battery for a detector with exit-code, diff and token-in-diff as three separate readings

control | hunts/r_414eed/ (extended by hunts/r_7ad39f/ and hunts/r_cb5ffe/) | grade: measured (17/17 in-scope, 1 genuine miss; 0/299 exhaustive; 5/10 for test_doors.py); 'a mutation battery is a sample, not a census' where stated

To measure a guard's power and scope rather than assert it: (1) copy the tree to a throwaway sandbox (never mutate the live checkout); (2) null control: run the unmutated sandbox through the guard first (rules out catches caused by the copy); (3) after every mutant, undo and re-run the guard (rules out row n+1 inheriting row n); (4) for each mutant record not only the exit code but the number of diff lines in the regenerated artifact and whether the mutant's own new symbol appears in that diff, distinguishing 'caught by comprehension' from 'caught by accounting' (a line-count tell); (5) include boundary probes expected to be silent and a control mutant with no symbol at all (one blank line); (6) apply all escapes simultaneously to one tree to confirm the miss is measured, not inferred; (7) run the expensive tier only where the cheap tier gave 'escaped'. Findings that make it a method: the declared smallest mutant fired for the wrong reason (line count), which predicted and then exhibited the one genuine miss (length-neutral in-place private-to-public rename in an __all__ module, 0/299 detected in the exhaustive census), and the guard's real invariant was restated as 'the artifact is a byte-exact function of the tree'. Result is reported as a proposed ledger amendment (fired, demonstrated_by, scope, known_misses), not applied by the hunt.

Evidence: r_414eed/RESULTS.md 'Controls' (lines 15-26), tables (lines 34-68), 'What the numbers say' items 2-4 (lines 79-101), 'What I chose, and why' (lines 112-127); r_7ad39f/RESULTS.md 'Repository census and exhaustive mutation analysis' (lines 83-96); r_cb5ffe/RESULTS.md 'The table' and combined five-escape tree (lines 21-44), 'Method notes' (lines 105-124) Prior art: unsearched (mutation testing is a standard field; the token-in-diff discriminator and the ledger fields are the lab's) Reused in: r_7ad39f (exhaustive census over 49 __all__ modules, historical git-log scan showing zero incidence); r_cb5ffe (same protocol on tests/test_doors.py with worktrees); proposed amendments to harness/departments/guard_ledger.py Why it travels: The target here was repository tooling, but the protocol (null control, undo re-check, three-level reading, combined-escape tree, blind-region table) is exactly what a planted-fault audit of a mathematical detector needs, and r_cb5ffe's caching caveat (worktrees share the editable install, so assert the module path before trusting a number) is a real pitfall.

Instantiate every enclosure lemma at a recorded box, and parametrise seam denominators as free reals

control | hunts/r_6c7d6a/ and hunts/r_938ab4/ | grade: kernel-checked (Lean 4, zero sorrys, axioms [propext, Classical.choice, Quot.sound]; box0_in_table depends on no axioms)

A Lean interval-enclosure lemma that compiles with zero sorrys can be vacuous: O9Seam.r_comp_mem asked the denominator enclosure E to contain c*c + dOverY*dOverY while the field denAbs2 actually encloses c*c + d*d with d = dOverY*y; the two agree only when y^2 = 1 or s = 0, never on the table's boxes [28/5, 60] x [0, 1/2], so r_comp_mem and rIv_mem were true, zero-sorry and unusable at every box. Control: for every enclosure identification, also prove a *_box instance at a recorded row of the table (e.g. Retention.box0_in_table by membership, point (23/4, 1/8) interior, hypotheses discharged by norm_num), choosing a box that touches the hypothesis' edge (a mode-2 box whose y-range starts at 0), and keep the audit aggregator's #print axioms list complete. Fix pattern: state the seam lemma with the denominator's real as a free variable e (r_comp_mem' : E.mem e -> ... (bOverY*c - a*dOverY)/e), same proof term, one fewer coincidence; leave numerator components abstract so the layer does not couple to the leaves. Companion finding (r_938ab4): imNumOverY encloses Im num / y only for y != 0 (at y = 0 it encloses the removable limit, disequality proved at s = pi), so the y != 0 hypothesis is stated as a disequality rather than hedged.

Evidence: r_6c7d6a/RESULTS.md §3 (lines 69-126, the defect and the fix), §4 (lines 128-141), §5 (lines 169-183, zero sorrys, audit list); r_938ab4/RESULTS.md §2 (lines 49-83), §5 'Instantiability, checked' (lines 119-137) Prior art: unsearched Reused in: hunts/frontier_math/zeta23ext/Zeta23Ext/EForm3/O9NumShape.lean (27 declarations, imported by EForm3/Main.lean); O9Audit.lean; r_938ab4 explicitly reuses r_6c7d6a's r_comp_mem' rather than r_comp_mem Why it travels: 'A statement that compiles is not thereby a statement with content': every enclosure seam in a Lean interval-arithmetic package should carry a discharged instance at a recorded box, and quantities entering only squared need an external check.

Planted-fault ladder for a polynomial certificate verifier

control | hunts/r_6f0f63/ | grade: VERIFIED (float)

Five items run before any result is read: two controls (E8 and Leech certificates rebuilt from contact structure, f vanishing to order 2 at each realised inner product and order 1 at the interval endpoints, must give sup ≤ 0, min f_k > 0, values 240 and 196560); f_0 scaled by 1.01 must give sup = 0.01·f_0 exactly; f_4 negated must trip the coefficient-sign test while the interval scan stays clean (a certificate can be inadmissible with f ≤ 0 everywhere, so the scan alone is not the verifier); node set starved to 14 points at degree 12 must report below 240 and fail the sign scan. All five behaved as expected.

Evidence: RESULTS.md §0 'The planted-fault ladder, which fires first' (lines 17–32), §1; probe.py planted_faults (line 213) Prior art: unsearched Reused in: none recorded Why it travels: Template for any positivity/sign-condition certificate checker: separate faults for each condition (P2 coefficient signs vs P3 interval sign) so the verifier's blind spots are enumerated. Index note: Planted-fault ladder, polynomial-certificate instance.

Stopping rule for cutting-plane loops: test the true constraint residual, not objective stall; monotonicity separates solver stall from family ceiling

control | hunts/r_828c8b/ | grade: measured (the monotonicity facts are provable; the lesson is stated by the hunts)

Two related rules from the ceiling procedure. (a) A cut loop that stops when the objective stalls can return values that fall when a constraint is added (R = 5 -> 0.37433, R = 10 -> 0.37246, R = 20 -> 0.37504 at N = 2000), which is impossible for the real program; the stopping test must be the residual of the true quadratic constraint, and any remaining non-monotone row is reported as unconverged, not as a measurement. (b) Upsampling lemma: an m-piece step function upsamples to a feasible 2m-piece one with the same value, so the m-piece optimum is non-increasing in m; if a published construction (Haugland 0.380926) sits strictly below the solver's stalled value (0.381083768 identical at m = 32, 64, 128), the stall is the solver's, not the parameterisation's. Stated lesson: a ceiling procedure that cannot tell 'the parameterisation ran out' from 'the search ran out' measures the search.

Evidence: hunts/r_828c8b/RESULTS.md, Section B 'The ceiling on the upper-bound side' (lines 60-87) and loose thread 'The m-piece optimum is not where SLSQP stops'; hunts/overlap_lower/RESULTS.md Section E2 (lines 337-353) and E3 Prior art: unsearched Reused in: hunts/overlap_lower/RESULTS.md Section E2 ('the second time in two days that this family has produced that exact confusion') Why it travels: Applies to every ceiling sweep over a nested family: use a provable monotonicity as the falsifier of the search, and never stop an outer-approximation loop on the objective.

Control: a margin against a prediction drawn at the bound is not evidence; verify the obligation that spends the constant

control | hunts/r_908de5 | grade: measured (exact rational arithmetic over 200 cell obligations; ball and chained-rect enclosures)

If beta_pred was produced by point evaluation minus a sampling slack, the certificate obligation normLower(enclosure) >= beta_pred sits at the line by construction in any arithmetic, and its margin measures where the prediction was drawn, not the enclosure. Remedy: set beta := normLower(B.inflate r) (rounded down to a multiple of 2^-40 so the Lean literal stays small and true), which makes the grid inequality true by construction, and move verification to the obligation that actually consumes beta, eps'

cell fails loudly. Measured: the worst cell obligation moves from 1.3647x to 1.3697x (ball) or 1.3566x (chained rect), not the predicted >= 10x, because 98% of the requirement is L h/2 (gap-limited, not beta-limited). Companion calibration: three enclosure arithmetics coexist (rect non-chain, rect composite chains, ball); non-chain is 2-6x too wide and fails all 24 obligations it carries; the remedy is contingent on emitting composite chains.

Evidence: RESULTS.md 'What the defect was', lines 27-39; 'Before / after' tables, lines 41-69; 'The arithmetic the remedy needs', lines 71-106; 'What changed in the tree', lines 108-128 Prior art: unsearched Reused in: scripts/60_rung3_generate.py and scripts/65_rung3_full_validation.py (generator assertion and validation verdict changed); docs/25 section 4.3 prediction refuted Why it travels: Any certificate pipeline that copies a numerically fitted constant into a proof obligation has this failure mode; the fix (read the constant off the enclosure, assert the consuming inequality at generation) is generic.

Control: compare a sup only on the range the lemma asserts, and price depth-interval inflation before declaring a constant broken

control | hunts/r_a7c12f | grade: enclosure-carrying for the depth ladder on s in [37.0135, 400]; measured for break depths and the asymptotic constant

A reported refutation '0.6636 > 637/1000 at depth 1' was a supremum over s in [8,400] against a lemma (Wt_tail_le, hypothesis w = s^2 - 2 >= 1368, i.e. s >= 37.0135) that carries no depth at all; the depth hypothesis lives in a different lemma (Qim_far_sq, y <= 1/2). Procedure: (1) read the Lean statement to find which lemma carries the hypothesis at issue; (2) restrict the scan to that lemma's range; (3) enclose there (Arb 96 bits, outward cells, enclosure cost 1.00005); (4) when treating depth as an interval, note the normalisation divides by y_lo^2 so a cell of relative width w inflates by (1+w)^2, and with a 0.82% margin cells must be under ~0.4% wide (~174 geometric cells over [1/2,1]). Result: 637/1000 holds at depth 1 with +0.0052; what breaks first is Qim_far_sq, asymptotically iff 4 sinh(y/2)^2 cos(1/sqrt2)^2/y^2 > 5/8, i.e. y > 0.972659.

Evidence: RESULTS.md sections 1-4, lines 21-135; 'Honest scope', lines 155-179 Prior art: unsearched Reused in: harness review ledger claim k2-far-constant-depth1 (white-box attack outcome recorded); input to R-B9552D's k >= 3 pass Why it travels: Generic for auditing any 'constant X is violated' claim produced by a scan against a Lean lemma; the depth-interval inflation arithmetic applies to every ratio-normalised enclosure.

Seen and not admitted

Candidates the sweep surfaced that do not meet the bar yet: measured once, self-reviewed only, not carried anywhere, or hunt-specific. Listed so the next sweep does not re-judge them, and so a hunt that later reuses one knows it has just earned an entry above.