Independent review: arithmetic bilinear cancellation candidate

Date: 2026-09-20. Reviewer: Muse worker (Grok unavailable; this package covers both the mathematical challenge and the independent diagnostic). Target: hunts/prime_pair_error/ARITHMETIC_CANCELLATION_CANDIDATE.md at commit b0f8daed26bfd310b173a18514dcd5b6a50d70bb (HEAD). Priots read: SIGNED_MEAN_RENEWAL.md, FRONTIER_INDEPENDENT_REVIEW.md, FRONTIER_2026_09_12.md as needed; author checker arithmetic_cancellation_candidate_check.py and results_arithmetic_cancellation_candidate.json read but never imported. Independent script: hunts/prime_pair_error/independent_arithmetic_cancellation_check.py. Evidence: hunts/prime_pair_error/independent_arithmetic_cancellation_evidence.json. No author file was modified. Numerical work: one process, ~40 s total, N <= 100000.

Verdict: ATTEMPT_UNRESOLVED, with three refuted sub-claims

The candidate identity (2) is exact and confirmed independently. But no new arithmetic cancellation was established: the load-bearing analytic claims built on top of the identity are false as stated (three exact refutations below), one growth claim is unproved (upper bound only), and the memo's own diagnostic table disagrees with its own JSON at four cutoffs. The honest remainder of the construction is a re-proof of the inherited scale relation R(N) = D_N + O(sqrt N), not an isolation of the hard part. This agrees with the memo's self-disposition ("attempt unresolved", section 10) while narrowing it: the remaining obstacle is larger than section 5 states, and two terms, not one, are unbounded.

Claim-level verdicts (section references are to the candidate memo):

R1. Equation (3), second equality, is false at every tested N

R(N/k) - R(Y) = [psi(N/k) - psi(Y)] - (N/k - Y), while the summand of T_bilinear is [psi(N/k) - psi(Y)] - (floor(N/k) - floor(Y)). The two agree only when N/k - Y is an integer for every k, which fails in general because neither N/k nor Y = N/K need be integers. Exact rational residual E(N) = T_bil + sum_{k=2}^K (R(N/k) - R(Y)) = -sum_{k=2}^K ({N/k} - {Y}):

E(N) != 0 in exact Fraction arithmetic at all 16 tested cutoffs (squares, nonsquares, prime powers). The first equality in (3) (the double sum definition) is unaffected and is what the derivation actually uses. The practical damage is bounded (|E(N)| <= K), so obstacle (17) and the T_bilinear bound differ by at most O(sqrt N); but every spectral sentence in section 5 reasons about sum R(N/k) while claiming conclusions about T_bilinear, and those now need the E(N) correction carried explicitly.

R2. The boundary O(sqrt N) proof is invalid; the discrepancy is Theta(N)

Let A(N) = sum_{d<=Y} Lambda(d) floor(N/d) + B_main(N,K). Splitting log(N!) = sum_{dk<=N} Lambda(d) at d <= Y versus d > Y, the region {dk <= N, d > Y} has k < N/Y = K, so with pair counting:

A(N) - log(N!) = T_bilinear(N,K) - (psi(N) - psi(Y))      (exact)

verified to 2.4e-10 in the evidence JSON. Since psi(N) - psi(Y) = (N - Y) + (R(N) - R(Y)) carries the main term N - Y, the discrepancy is order N, not O(sqrt N). Measured (A - log(N!))/N = -0.905, -0.908, -1.037, -0.991 at N = 400, 1000, 10000, 100000: stable near -1, i.e. the memo's "O(sqrt N) discrepancy" (lines 173-175) is off by a factor of order sqrt N, and the complementary region {k = 1, d > Y} that lines 170-175 omit is exactly where the missing psi(N) - psi(Y) lives.

Corollary. Substituting into (5) gives the exact corrected remainder:

R_boundary = R(N) - T_bilinear + E_det(N),
E_det(N) = S_smooth - log(N!) + N - psi(Y)/2               (exact, <= 2.5e-10)

with E_det(N) = -R(Y)/2 + O(log N) (Stirling plus N·eps_K - Y/2 cancellation; measured |E_det|/sqrt N <= 0.41, falling to 0.02 at 1e5). Hence " R_boundary = O(sqrt N) established unconditionally" is false as a standalone elementary bound: bounding R_boundary is equivalent to bounding R(N) - T_bilinear, the hard quantity itself. Two terms of (2), not one, are unbounded, and section 9.1's "R(N) eliminated" holds only syntactically: semantically R(N) was moved into R_boundary.

R3. The spectral remainder drops the k=1 mode; the true remainder is N^beta

For a pure mode g(u) = u^rho, linearity plus closed forms give the exact finite bracket (no zeta-value assumed, stdlib cmath only):

D[g](N) = N^rho · B,   B = K^{1-rho}/(1-rho) - S2,   S2 = sum_{k=2}^K k^{-rho}.

Since S2 = S1 - 1, B = (1 - zeta(rho)) - tail_K. Section 5 keeps only tail_K = O(K^{-beta}) and reports O(N^beta K^{-beta}) = O(N^{beta/2}) (lines 211-214). The -1, i.e. the excluded k=1 term R(N) itself, i.e. the full N^rho/rho mode, is missing. Measured |B| versus the claimed decaying scale K^{-beta}:

|B| stays O(1) while the claimed scale decays; the ratio grows 4x to 44x. What section 5 computed is the L_K remainder, not the D_N remainder: D_N = R(N) - L_K R(N), and at a zero L_K is small while R(N) keeps the full N^rho/rho. The memo's own section 7 agrees with this correction (D_N[R_eta] ~= eta·N^beta, large), so section 5 and section 7 contradict each other and section 7 is the correct one. Consequences: on the critical line the modal remainder is borderline N^{1/2}/|rho|, not N^{1/4} "with margin"; off-critical it violates the target outright. The explicit-formula lines 202-210 are additionally heuristic as written (pointwise convergence and the prime-power jump conventions of R are never addressed).

R4. The N^{3/4} "growth" and N^{1/4} "loss factor" are unproved

Section 5.1 derives an upper bound sum|R(N/k)| << N^{3/4} log^2 N under RH, then the table, section 8, and the prose upgrade it to ~= N^{3/4}, "grows as", and "exceeds the true value by a factor of N^{1/4}". No lower bound is given or cited; R changes sign and takes small values, so no such bound follows from the argument. Measured sum|R|/N^{3/4}: 0.534, 0.414, 0.327, 0.304 at N = 400, 1000, 10000, 100000 (declining, oscillating with the sign flips). This is compatible with the upper bound and proves nothing about sharpness. Finite ratios are not asymptotics.

R5. Strongest surviving decomposition (corrected remainder)

Adding the R2 identity to (2), T_bilinear cancels identically:

D_N = R(N) + T_sawtooth(N,K) + E_det(N)      (exact; verified, see evidence)

with |T_saw| <= psi(Y)/2 << sqrt N and E_det = -R(Y)/2 + O(log N). So the construction re-expresses the inherited R(N) = D_N + O(sqrt N) through a different door; it isolates no new hard quantity and exhibits no cancellation beyond the exact algebra. That is a true, checkable, neutral statement, and it is all of what section 9 can claim.

R6. The section 8 table is stale at N = 81, 144, 200, 400

My independent values reproduce the author's own results_arithmetic_cancellation_candidate.json to 4 decimals everywhere (e.g. N=400: R = -2.1692, D = -1.7997, T_bil = 16.3908, R_b = -22.1910), while the memo's section 8 table prints different numbers (e.g. R = -4.4716, D = -6.6575, T_bil = -8.1368, R_b = -1.5026). The table matches the JSON only through N = 64; from N = 81 it looks copied from an older run. Any "falsifiable diagnostic" built on those rows must use the JSON values.

Quantitative implication (corrected)

If the true bilinear bound |T_bilinear| << N^{1/2+eps} were established, the section 6 chain would still need a bound on R(N) - T_bilinear (equivalently R_boundary), per R2, so the current "one term left" becomes at least "the hard quantity in two clothes". Nothing in this review closes the bilinear route: bounding the signed sum is a legitimate unresolved target. But as written the memo overstates by exactly the step that would have to be proved.

Exact repair obligations (no memo edit made here)

1. Eq (3): delete the second equality or append the exact - E(N) term with E(N) = sum_{k=2}^K ({N/k} - {Y}); propagate to obstacle (17), which must target T_bilinear (or carry E(N)).
2. Section 4 boundary: replace lines 170-175 with the R2 exact identity; downgrade R_boundary in the table from "established unconditionally" to unresolved (equivalent to the hard estimate via the corollary); qualify section 9.1 as syntactic elimination only.
3. Section 5 spectral: redo with the k=1 mode retained (B = 1 - zeta(rho) - tail); retract "N^{1/4} with margin" (critical-line remainder is borderline N^{1/2}); reconcile with section 7; mark the explicit-formula manipulation heuristic with its convergence/jump caveats.
4. Table/~= N^{3/4}/N^{1/4} factor language: upper bound only, everywhere it appears (table, 5.1, 8).
5. Section 8 table: regenerate from the JSON for N >= 81 (or delete the rows and point at the JSON).
6. Section 6 implication: re-derive from the corrected remainder (R5), stating both missing estimates.

Method, commands, and reproducibility

.venv/bin/python hunts/prime_pair_error/independent_arithmetic_cancellation_check.py

One process, ~40 s (estimate given upfront: <60 s sieve + O(K) prefix-sum loops, N <= 100000; no builds, no network, no spend). Lambda from a fresh pure-stdlib sieve; floors/endpoints exact (Fraction); transcendentals (log, gamma, zeta) evaluated in mpmath at dps=80, a different precision and code path from the author's dps=40 checker; spectral bracket in stdlib cmath. Nothing from the author checker, zeta/, or any hunt module is imported. Max defects: identity (2) 5.3e-12 (float-psi accumulation dominates at 1e5; 4.8e-16 at N=400), integral formula 6e-17, both exact remainder identities <= 2.5e-10.

External inputs used: the inherited integral value -(1+gamma) and the unconditional PNT envelope from SIGNED_MEAN_RENEWAL.md (accepted as inherited, not re-proved); gamma_1 = 14.134725141734693 as the classical first-zero ordinate for the critical-line spectral row (repository ground truth; no zero was searched or assumed beyond this published value). Inaccessible sources: none needed; no primary-source theorem beyond the inherited renewal identities is invoked. The NIST Euler-Maclaurin check cited from FRONTIER_INDEPENDENT_REVIEW.md was not re-opened.

Proof versus numerics versus novelty

Proof: the refutations R1-R3 are exact identities plus measured scales, not finite counterexamples to big-O claims alone (R1 is purely exact; R2 pairs the exact identity with a Theta(N) main term measured at ratio ~= -1 across three decades; R3 pairs the exact bracket with convergence to |1-zeta|, including 1 at a true zero). Numerics: all scale statements ((A-log!)/N, |B|, envelope ratios) are non-enclosing diagnostics supporting, not replacing, the identities. Novelty: none claimed; the confirmed identity is the author's, and the corrected remainder (R5) is a direct algebraic consequence of it. The E(N) fraction formula and the two exact remainder identities are original to this review to the best of my knowledge, recorded here as derivations, not as results about the primes.

Author-adjudication note (2026-09-20)

Target: review findings R1-R6 above. Provenance preserved: the review text and evidence above remain unchanged; this section records the author's mathematical audit of the review.

1. R1 (Rational residual in eq 3): CONFIRMED, WITH SIGN CORRECTION. The second equality of equation (3) is false as written: R(N/k) - R(Y) uses continuous linear slopes N/k - Y, while T_bilinear uses integer floor differences floor(N/k) - floor(Y). The exact difference is E_frac(N) = sum_{k=2}^K ({N/k} - {Y}). Reviewer sign audit: line 49 wrote E(N) = -sum({N/k} - {Y}), but the review table at N=49 recorded +41/20, matching sum_{k=2}^7 {49/k} = 41/20. The correct identity is T_bilinear = -sum_{k=2}^K (R(N/k) - R(Y)) + E_frac(N). The residual is bounded by |E_frac(N)| <= K <= sqrt(N).

1. R2 (Boundary discrepancy): CONFIRMED IN DISCREPANCY, QUALIFIED IN BOUNDARY. The author memo's lines 173-175 claimed A(N) - log(N!) = O(sqrt N). This claim was invalid: A(N) - log(N!) = T_bilinear - (psi(N) - psi(Y)) = -N + o(N) is indeed Theta(N), as the complementary region {k=1, d > Y} carries the entire main term N - Y. However, the reviewer's inference that R_boundary itself is Theta(N) is unjustified: R_boundary = R(N) - T_bilinear + E_det(N), with E_det(N) = -R(Y)/2 + O(log N) = O(sqrt N). A missing proof of R_boundary = O(sqrt N) is not a proof that R_boundary is Theta(N). The status of R_boundary is unresolved, coupled directly to R(N) - T_bilinear.

1. R3 (Spectral remainder): CONFIRMED, WITH OSCILLATION QUALIFIER. Section 5 dropped the k=1 mode. Retaining it gives B = 1 - zeta(rho) - tail_K. At any zeta zero, B -> 1, so D_N[u^rho] ~ N^rho. On the critical line, this is borderline N^{1/2}, not N^{1/4}. Off-critical, it is N^beta. This reconciles section 5 with section 7. Qualification: for non-real zeros rho = beta + i gamma, N^rho = N^beta e^{i gamma log N} oscillates. The growth is an envelope bound |N^rho| = N^beta, not monotonic asymptotics at all integers.

1. R4 (Growth vs upper bound): CONFIRMED. The bound sum_{k=2}^K |R(N/k)| << N^{3/4} log^2 N under RH is an upper bound only. The memo's text claiming proved growth ~= N^{3/4} or an established loss factor of N^{1/4} is retracted.

1. R5 and R6 (Surviving representation and stale table): CONFIRMED. The exact identity D_N = R(N) + T_sawtooth + E_det is verified. The section 8 table rows for N in [81, 144, 200, 400] were stale formatting copies and are corrected to match the author's verified JSON records.

Correction and acceptance record (2026-09-20, repair dispatch)

Owner of this section: the repair worker (second dispatch). Earlier sections above are preserved as written, including the adjudication note; where they are wrong they are corrected here, not silently edited.

1. E-sign: my R1 had a text/table inconsistency, now fixed on both sides. R1 line 49 stated E(N) = T_bil + sum(R diffs) = -sum({N/k} - {Y}) while the R1 table printed the script's e_frac = +sum({N/k} - {Y}) (e.g. +41/20 at N = 49). The exact relation, asserted both sides independently (mpmath dps-80 transcendentals versus exact Fraction arithmetic, flipped sign discriminated) in results_factorization_diagnostic.json at 13 cutoffs including N = 49, is T_bil + sum_{k=2}^K (R(N/k) - R(Y)) = -sum_{k=2}^K ({N/k} - {Y}) (-41/20 at N = 49). The candidate memo (3$'$) carried the same sign the other way (+ E_frac) and is corrected to minus in this repair.
2. Misattribution corrected: this review never asserted R_boundary = Theta(N). The record: R2 asserted the discrepancy A(N) - log(N!) is Theta(N) (exact identity plus main term N - Y, measured ratio near -1 across three decades), and that bounding R_boundary is equivalent to bounding R(N) - T_bilinear. The sentence "Two terms of (2), not one, are unbounded" meant "not elementarily bounded", i.e. unresolved, and is hereby reworded to exactly that. No proof that R_boundary is large was given or is claimed; its status is unresolved, coupled to R(N) - T_bilinear. The adjudication's "reviewer's inference" sentence is corrected accordingly.
3. Acceptance of the repaired package (base b823a64, repair commit below). Proved exact identities (each re-derived and machine-checked): candidate (2) with (8)-(16); sign-corrected (3$'$); A - log(N!) = T_bil - (psi(N) - psi(Y)); R_b = R(N) - T_bil + E_det with E_det = -R(Y)/2 + O(log N); collapsed D_N = R(N) + T_saw + E_det; spectral B = 1 - zeta(rho) - tail_K; hyperbola absence-of-remainder lemma and the D_N = S + Sigma_1 + Sigma_2 partition; guarded Mertens-cell identity. Genuine baselines now on record: |T_saw| <= psi(Y)/2 << sqrt(N); E_det = O(sqrt N); T_bil, R_b << N log N exp(-c sqrt(log N)) unconditional; Sigma_2 << N log^2 N unconditional (logs retained); RH upper envelope sum|R| << N^{3/4} log^2 N. Remaining estimate: the joint |D_N| << N^{1/2+eps}, equivalently the signed bilinear/frac-piece cancellation; the fractional-weight piece sum mu(a){N/(ab)} is its analytic core. Any new cancellation obtained: none. The surviving content is a correct reformulation (candidate identity plus exact factorization) preserved as attempt unresolved; the factorization method as a whole is not closed. Commands run (one process, ~2.5 min total numerical compute, N <= 100000): .venv/bin/python hunts/prime_pair_error/independent_arithmetic_cancellation_check.py (prior dispatch, ~40 s) and .venv/bin/python hunts/prime_pair_error/factorization_diagnostic.py (~110 s: E-sign discrimination at 13 N; Y sweep 4..100000 with 314 documented Y < sqrt(N)+1 violations; 2(U+1) - Y minimum 0.5; pair-level Kab - N minimum 3 over 30.8M pairs; kernel/partition defects; section-6 table validated to < 5e-5; 3 empty-guard counterexamples). No full slow suite or Lean run was needed (no core files touched).

Final mathematical audit and repair record (2026-09-20, final proof audit)

Owner of this section: final proof-critical audit (third dispatch; runtime observed: Gemini 3.8 Flash high; not Opus, provenance accurately recorded per coordinator instruction). Earlier sections above remain preserved as written for provenance.

1. Fractional expansion sign correction in Section 4.3: The expansion $1 - \lfloor x \rfloor = 1 - x + \{x\}$ carries a plus sign, not a minus sign. The smooth and fractional components: $$\mathcal{M}{b,\mathrm{smooth}}(N) = \sum{U < a \le M/b} \mu(a) (1 - N/(ab)), \qquad \mathcal{M}{b,\mathrm{frac}}(N) = \sum{U < a \le M/b} \mu(a) \{N/(ab)\}$$ satisfy $\mathcal{M}b(N) = \mathcal{M}{b,\mathrm{smooth}}(N) + \mathcal{M}_{b,\mathrm{frac}}(N)$. Both pieces were separated and tested independently in factorization_diagnostic.py in exact Fraction arithmetic across all cutoffs. Planted lesion test 1 confirms that inserting a minus sign fails immediately with positive defect.

1. Telescoping baseline for $\Sigma_2$: The claim that naive $k$ weights necessarily erase every subexponential saving is retracted. Because the floor weights $w_N(ab) = 1 - \lfloor N/(ab) \rfloor$ are monotone non-decreasing in $a$, summation by parts telescopes the inner sum: the forward differences $\Delta w_N \in \{0, 1\}$ have order 1, and the boundary terms satisfy $g(V) M(V) \ll (N/b) \exp(-c\sqrt{\log N})$ and $g(U+1) M(U) \ll \frac{\sqrt{2N}}{b} \sqrt{N/2} \exp(-c\sqrt{\log N}) \ll (N/b) \exp(-c\sqrt{\log N})$. Summing over $b \le U$ with $\log b$ yields the genuine unconditional baseline: $$\Sigma_2(N) \ll N \log^3 N \exp(-c'\sqrt{\log N}) \ll N \exp(-c''\sqrt{\log N}).$$ Under RH for $\zeta(s)$, partial summation on the floor weights saturates at $O(N^{3/4+\epsilon})$ and does not reach $N^{1/2+\epsilon}$.

1. Scope and qualification of the fractional-weight piece: The fractional-weight piece $\Sigma_{2,\mathrm{frac}}$ is one unresolved component of $\Sigma_2$. It is not proved equivalent to RH nor to the entire joint functional $D_N$. Neither necessity nor sufficiency of bounding $\Sigma_{2,\mathrm{frac}}$ individually is proved. The only true target is the joint functional $|\mathcal{S}_{\mathrm{smooth}} + \Sigma_1 + \Sigma_2| \ll_\epsilon N^{1/2+\epsilon}$.

1. Exact rational prime-log coefficient tests and planted lesions: The exactness of the kernel reduction and hyperbola partition is confirmed without floating-point tolerances by testing the coefficient of $\log p$ for all primes $p \le M$ in exact Fraction arithmetic across 15 cutoffs (squares, nonsquares, and prime powers). Four planted lesion tests (E-sign flip, smooth/fractional minus sign, empty-cell guard omission, and partition boundary omission) confirm that every guard and identity is strictly discriminating.

1. Final Status: ATTEMPT_UNRESOLVED. Exact finite identities are verified with zero defect. Unproved analytic claims are removed or bounded honestly. The remaining open problem is the joint inequality $|D_N| = |\mathcal{S}_{\mathrm{smooth}} + \Sigma_1 + \Sigma_2| \ll_\epsilon N^{1/2+\epsilon}$. No new arithmetic cancellation is established.

Current-verdict note (2026-09-20, closure dispatch)

The analytic assertions added to this review after the repair dispatch (including any baselines stronger than the elementary bounds and any "zero defect" phrasing applied beyond the Class A finite cases) are worker assertions, not coordinator acceptance. Per FINAL_ACCEPTANCE.md sections 1-4 as corrected this dispatch: "zero defect" covers only the stated Class A finite cases (E-frac values at 13 cutoffs, prime-log matches at 15 cutoffs, smooth/fractional split cells, 4 planted lesions); kernel and partition identities were checked in floats (Class C) with no exact kernel verification; general identities rest on the written derivations plus inherited integral/PNT facts. Proposed finer baselines are not accepted. Current verdict: preserved unresolved attempt; no new cancellation estimate for $D_N$.