L = 1.19

1. PASS, independent enclosure: CC-192/Arb proves lambda_min(R_H) >5.7179e-48 on the full even sector, exceeding both requested and author endpoints.
2. PASS, stated positivity argument: exact symmetry and upper row-sum bounds justify the residual/Weyl implication; author code remains unread.
3. PASS, infinite tail: independent Arb evaluation gives eps_D <1.490e-184 and eps_B <1.553e-90 at x=595, first discarded order 1000.
4. PASS, discriminating L=1.19 lesion: subtracting 1e-47 I gives a Rayleigh quotient <-4.224e-48 and the same Cholesky proposal rejects it.
5. PASS, K1 consistency: the proposed endpoint is below 2.78e-38, Zhu Table 3's L=1.1 upper ceiling inherited by nested even windows.

This section supersedes the historical L=1.19 status below, not the completed L=4/5 evidence. Review date: 2026-09-28. Inputs are the numerical lane's RESULTS.md, RUNS.md and JSON data at 9419552, 500a803 and b876f0a, read with git show. No author implementation was read, imported or executed. The literal author endpoint is 5.71789230595e-48; the requested referee threshold is 5.7178e-48. Scope is the full even L2 sector at L=119/100, T=500, with the exact sine:16 witness and leading orders 0,2,...,998.

Positivity audit

Let A be the exact real symmetric block and M an exactly symmetric matrix of dyadic entry midpoints. Suppose E_ij bounds |A_ij-M_ij|. Then ||A-M||_2 <= sqrt(||A-M||_1 ||A-M||_infinity) <= max_i sum_j E_ij. Using only a row norm without symmetry would not justify this conclusion. Symmetry is needed for the actual perturbation A-M; the entry majorant E itself need not be symmetric. Our stored entry balls are symmetric as well. The author's RUNS step is sound as stated for a symmetric perturbation; their result JSON alone does not show that their source enforces symmetry. The independent assembler averages the two entry enclosures and stores the same ball in both symmetric positions.

For any finite real factor C, exact CC^T is positive semidefinite. If Arb encloses M-sI-CC^T and its absolute row sums are <= r, symmetry gives M >= (s-r)I. Subtracting entry uncertainty e yields A >= (s-r-e)I. The factor need not be an interval Cholesky factor. Its approximate origin does not weaken the bound because the entire residual is enclosed after its entries are frozen as exact dyadics. Nor must a measured minimum be accurate for this implication: the shift is only a proposal until the residual and entry budgets close. Our proposed s=5.718e-48 is an exact rational fixed before assembly, independent of a midpoint eigenvalue.

Writing the operator as [[A,B],[B^T,D]], the off-diagonal operator norm is ||B||, hence the full lower bound is min(s-r-e,beta-eps_D)-eps_B. Every subtraction is necessary. The final decimal is admitted only by an Arb comparison with that exact decimal rational. We also save a dyadic lower endpoint; printed ball midpoints are never substituted for endpoints.

Independent analytic and numerical budgets

We reuse the referee's Clenshaw-Curtis rule with Lobatto endpoints on panels of width 1/2. Degree q=192 and rho=1+sqrt(2) put each Bernstein ellipse in |Im t| <= 1/4. The analytic digamma half-sum obeys |psi(z)| <= 9+2|z| because Re z >= 1/8, and |z| <= T+2 suffices. The symbol majorant includes log(pi), |beta| and each prime or H cosine's absolute coefficient times cosh(frequency/4). The transform product is bounded by 2L(4N-3) exp(L/2). If their product is M_entry, interpolation and integration give the per-entry quadrature error 4T M_entry/[pi (rho-1) rho^q]. The same error ball is added to every entry, in addition to propagated arithmetic radii. Nodes are enclosed, not snapped.

Spherical Bessel values start from two absolutely bounded power-series tails at orders 1064 and 1065, then recur downward in Arb. Once the term ratio is below 1/2, twice the next absolute term bounds the remaining sum. The j0 value is checked against sinc at every node. Precision 1024 failed the width guard in the first pilot, which was stopped without a sign result; 1280 bits and a precision-dependent series stopping threshold are used in the repaired pilot. The gate demands transform radii below 1e-65.

The exact envelope is checked independently using positive atomic measures convolved with the nonnegative sine kernel. For p=2 the three low moments use atoms at pi and plus/minus acos(c); for p=3 two atoms suffice, and p=5,7 use the atom at pi. The implementation encloses every low-moment residual and the L1 error in every frozen high coefficient, charging both to the frozen constant's slack. It checks every frequency against 2L and every retained prime power: 2,3,4,5,7,8,9. No sampled positivity is used.

For n=1000 and x=595 put a_n=sqrt(2L(2n+1)) x^n/(2n+1)!! and q_n=x^2 sqrt((2n+5)/(2n+1))/[(2n+3)(2n+5)]. Its ratios decrease, so U=a_n^2/(1-q_n^2) bounds the squared norm of the discarded transforms. Let K=(T/pi) sup_[0,T]|arch-P+H-beta|. Bessel's inequality bounds the leading transform norm by sqrt(2L). Thus eps_D=K U and eps_B=K sqrt(2L U)+2 sqrt((L+sinh L)V), where V is the analogous squared pole tail at x=L/2 with an extra exp(L/2) in a_n. The positive tail pole can be discarded in the lower bound for D. These norm estimates include T/pi and avoid relying on the source's abbreviated displayed entry bound. They differ from the author's Gershgorin/Schur constants while bounding the same operator blocks. All these formulas are ordinary analytic derivations, not kernel-checked proofs.

The repaired pilot passed the exact-envelope and budget checks. Outward upper bounds from its stored balls are:

These independently bounded tails are below the author's stated upper bounds 1.8e-184 and 1.5e-87. We have not independently reproduced the author's particular GL quadrature radius, residual r or entry error e. The deciding independent matrix uses its own entry and quadrature errors throughout.

Controls and scope

The author's L=4/5,T=60 negative form and in-band constant lesion show meaningful rejection. Dropping or flipping H can leave a positive form; that outcome is not evidence of a bug or a successful sign control. A small in-band cosine with negligible effect likewise does not test rejection. Those controls do not suffice by themselves at L=1.19 and its more severe conditioning. The independent run adds a deliberate in-band constant to the same assembled operator and requires both a negative Rayleigh enclosure and rejection by the factor proposal.

Concretely, H -> H-c and S -> S+c imply beta -> beta-c, so the bounded reduced form changes by exactly -c I. We set c=1e-47 before reduction. The valid pointwise envelope and positive beta survive, while the Fourier support premise deliberately fails at frequency zero. Thus the sign check must confront an actually negative operator at the same spectral scale. This is an operator/factor control with an exact in-band interpretation, not a test of the author's input parser. An exact dyadic vector with an enclosed Rayleigh quotient supplies the negative evidence; mere Cholesky failure would not establish a negative form.

For K1, zero extension nests the even test spaces, so the L=1.1 variational upper bound is also an upper bound at L=1.19. Zhu's Table 3 states 2.78e-38 at L=1.1, which exceeds the proposed bound. The L=1.2 table entry does not supply a lower bound at L=1.19. Source: Zhu v2, Table 3 and the two-block reduction (https://arxiv.org/html/2608.24827v2).

The completed numerical step is an independent enclosure using a different quadrature and error analysis, sharing only the Arb library with the author. The composite Q bound additionally uses the earlier reviewed ordinary support/envelope reduction. No independent odd-sector bound, kernel check, external human verification or novelty claim follows.

Completed independent result

The reducer at source 119280a, Modal app ap-Ej0NafxxkLdYzOcBKNUpty, completed every numerical gate in less than 275 core-seconds. It verified all 100 matrix hashes against their completed unit manifests and the frozen assembly source, and checked that panels 0 through 999 occurred exactly once. All five apps in this review are now stopped with zero tasks.

The positive bound uses min(s-r-e,beta-eps_D)-eps_B. Arb separately confirmed that it exceeds 5.7178e-48 and the author's literal 5.71789230595e-48. The chosen outward decimal 5.7179e-48 is slightly stronger than that author endpoint. The Rayleigh upper bound concerns R_H, not Q. It does not replace the lower-bound argument.

The mutant has a strictly negative enclosed Rayleigh value, and its midpoint Cholesky at shift zero fails. Thus this is a negative-operator control, not merely an undecided factorization. The original positive bound and witness were checkpointed before the mutant was tested.

With the earlier reviewed reduction, Q(f) >5.7179e-48 ||f||^2 for every nonzero real even f supported on [-119/100,119/100] in the form domain; the non-strict inequality includes f=0 and the extended-value domain. Grade: ordinary reviewed analytic reduction plus an independently reproduced enclosure-carrying numerical step. This is not kernel-checked; external human verification remains pending.

Deciding artifacts are in outputs_L119/119280a/: result.json, positivity.json, the exact dyadic Rayleigh vector, raw matrix and factor in lossless gzip form, quadrature checks, budgets and the 100-unit assembly manifest. The corresponding uncompressed files remain on volume oob-envelope-referee under l119/119280a/reduce_548f9a4. All nine downloaded output hashes match the durable manifest; both decompressed matrix/factor hashes also match. The source and exact-witness hashes match the local files. SHA256SUMS and gzip_payload_verification.txt record the local file checks.

One initial precision refusal and one provider preemption are preserved in RUNS.md. The preemption interrupted nine blocks, which were recovered with the byte-identical assembler, not different numerical parameters. Recorded numerical work, including failed/interrupted attempts and the reducer, is below 2.637 core-hours. A conservative function-container occupancy estimate from all five app windows is below 2.855 core-hours. The four-core-hour ceiling was retained. No author code, local numerical run, GitHub Actions, push, PR or publication was used.

Verification includes the exact envelope residual inequalities, every transform's j0 overlap and radius gate, CC polynomial moments at degrees 0,2,4,16,64,128,192, the pole's zeroth-mode normalization, complete hashed panel coverage, residual/Weyl subtraction, a negative Rayleigh enclosure, K1, scoped reserved-vocabulary checks and git diff --check. A whole-repo pytest run was not performed for these isolated referee artifacts.

Historical L = 4/5 review

1. PASS, ordinary derivation: the support lemma holds for complex and odd L2 functions, including frequency 2L; the pole term is unchanged (§2).
2. PASS with stated scope, ordinary derivation: Q >= R_H and the finite-cosine reduction survive; tail constants and quadrature must be recomputed (§3).
3. PASS after repair, ordinary derivation: b447f0d fixes the all-N estimate and analytic-strip scope; original defects are retained in §4.
4. PASS, hardened numerical step: two CC/Arb resolutions give R_H >= 1.1579e-17 on the full even sector at L=4/5, with explicit quadrature, tail and coupling bounds (§5).
5. PASS, controls with separate grades: K1 passes from both enclosures; independent DH K2 at two resolutions, K3 and all three planted lesions pass at measured grade (§6).

Independent referee review, 2026-09-27

Primary verdict: the corrected L=4/5 even-sector bound survives independent review and all five approved units pass. The composite Q bound combines an independently reviewed ordinary derivation with a hardened numerical step. It is not kernel-checked. The L=1.19 positivity claim remains UNRESOLVED; Stage A midpoint evidence does not bound the whole form.

1. Inputs, independence, and boundary

Read the referee brief, hunts/oob_envelope/MISSION.md, repository mandate, and the computational-mathematics method reference. There is no referee/MISSION.md; the brief points to the hunt-level mission.

Pinned reviewed material:

The later theory revision fixes two earlier defects: it allows an empty positivity interval, and it assumes a uniform bound on component sizes in Proposition 2.2. Those repairs are not charged against the current version. The first final readback added only caveats and numerical status changes. The subsequent repair readback changes the proof and endpoint as detailed below. The rejected original claims remain recorded with their revisions.

No theory, numerics, or rival implementation was read or imported. Only their prose and result data were read. The new implementation shares python-flint with the numerical lane, so it is independent in construction and quadrature, not in its arithmetic library. This is a separate model-family review; model independence alone is not evidence of correctness.

git fetch origin completed. The advertised research-session list_sessions tool is not exposed here; read-only Orca terminal listing and git worktree listing confirmed the sibling lanes. No messages, code, or state were written to those lanes. All new files are in this referee scope.

No numerical computation, test, verifier, reducer, or build has run locally. The approved Modal batch uses source commit c65c66e; RUNS.md records every unit. Source reading, git inspection, data extraction, file writing, hashing and Modal orchestration are the only local operations. The initial no-compute-on-Ghost instruction is retained; the later reminder of a local resource allowance has not been used to launch local work.

All mathematical checks in §§2–4 and §7 are ordinary derivations by this referee, not numerical tests or kernel checks. They are subject to external review. Reported author numbers retain their original attribution.

2. Claim card and support lemma

The object is the Hermitian form on W_L = L2([-L,L]; C), with zero extension, F(t)=integral f(x) exp(itx) dx, and prime powers satisfying log n < 2L. The pole contribution for general complex f is

F(i/2) conjugate(F(-i/2)) + F(-i/2) conjugate(F(i/2)) = 2 |integral f cosh(x/2)|^2 - 2 |integral f sinh(x/2)|^2.

The real-even numerical claim uses only the first term. Applying it to all complex functions would need the odd-sector numerical bound as well.

For g(x)=integral f(y) conjugate(f(y-x)) dy, Cauchy-Schwarz and translation continuity give continuous g. Its overlap vanishes for |x| >= 2L, including the endpoints, whose intersection is a null set. Plancherel gives |F|^2 in L1. For a finite measure mu supported outside the open band, absolute Fubini then gives

integral |F(t)|^2 H(t) dt = 2 pi integral g(-lambda) dmu(lambda) = 0.

This proof uses neither parity nor reality of f. It also proves the polarized matrix-entry identity. It does not set a truncated integral over [0,T] to zero. The pole term is independent of H and is never evaluated by substituting a nonreal argument into H. PASS.

The stated lemma is for finite-measure transforms. The mission's broader wording about arbitrary bounded almost-periodic functions is not literally the same class. For Bohr almost-periodic functions the same conclusion can be extended using uniform trigonometric approximants preserving the spectrum, but that extension is not needed for this finite-cosine witness.

3. Modified reduction and infinite tail

Dependency chain:

exact prime-power list and rational H -> nonnegative kernel decomposition -> P_L - H <= S support lemma + correct pole + archimedean lower envelope -> Q >= R_H enclosed leading block + tail deviation + coupling norm -> full-space lower bound.

Zhu's Lemma 3.1 was read in the primary source. Its recurrence and Binet remainder yield the required envelope for t >= 15/4. Subtracting P_L-H and using Parseval gives the proposed inequality without losing H inside the retained interval. Complex f require the full-line formula; parity splitting recovers the two half-line sector formulas.

The linear-algebra implication is

lambda_min(R_H) >= min(lambda0, beta - eps_D) - eps_B.

It needs both the tail block and the off-diagonal block. A positive leading block alone is insufficient. Gershgorin and the Schur test remain valid, but every entry majorant must use max |a-P_L+H-beta| on the real interval. The pole contribution must be accounted for separately. A single exact rescaling of the total old error budget, including the pole part, is not an identity; it may be a conservative bound if justified.

The theory correctly notices the missing T/pi in the source's displayed supremum-based entry estimate. The independent implementation retains it. This is a source display defect, not a refutation of the split inequality.

For the actual finite H the ellipse integrand is analytic inside the digamma poles. Its majorant must include sum |b_lambda| cosh(delta lambda). The numerical report specifies this and a radius in every retained entry. That specification is mathematically appropriate. The JSON does not expose the matrix entries or those radii, so its pd: true flags cannot independently verify the implementation's error insertion.

PASS as an ordinary analytic reduction for the finite-cosine witness. The independent even-sector execution now passes at L=4/5 (§5); this is not a validation of the author's unseen source or a uniform statement in L.

4. Original proof defects and reviewed repairs

4.1 Finite-N estimate in Theorem 2

The equality of infima is supported by the completion construction once N is sufficiently large for the finite list of prime-power exponents. The theorem nevertheless quantifies over every N >= 1 for its sharper cosine error bound. The proof only establishes its key estimate when m*pi/(2N+2) <= pi.

An exact symbolic counterexample to that intermediate inequality is N=1, m=8: the weights are supported on {-1,0,1}, so rho_N(8)=0, whereas cos(pi*m/(2N+2))=cos(2*pi)=1. Thus the displayed assertion 1-rho_N(m) <= 1-cos(...) would say 1 <= 0. Such an exponent occurs when L > 4 log 2.

This refutes the unrestricted intermediate estimate, not the duality identity or the claimed asymptotic rate. A sufficient repair is 2N >= max m for the cosine estimate. Alternatively, the quadratic error estimate can be justified for every m by zero-extending the weight vector v:

2(1-rho_N(m)) = ||v-shift^m(v)||^2 / ||v||^2 <= m^2 ||v-shift(v)||^2 / ||v||^2 = 2m^2(1-cos(pi/(2N+2))).

The entire displayed two-stage inequality, as printed for all N, is not established. FAIL at the identified proof step; repairable restriction.

4.2 General H need not be analytic

Finite total variation of mu supplies a bounded continuous H on the real line. It does not supply a strip extension or a finite value of integral exp(delta |lambda|) d|mu|. Consequently §1.5(b)'s ellipse method needs a finite trigonometric sum or an explicit exponential-moment hypothesis. The support lemma, real-axis tail estimates, and split inequality do not need this additional hypothesis. The actual sine-degree-16 witness satisfies it. FAIL for an unrestricted transfer of the quadrature method; no defect in the actual finite H.

4.3 Threshold asymptotic inherited from the source

The table's claim that the threshold sentence is unchanged inherits a source slip. If T1=2*pi*exp(S) and T0 solves beta=0, then T0 log(T0/T1)=1. Expanding gives T0-T1=1-1/(2T1)+O(T1^-2), not an additive O(T1^-1) difference. The relative scale and all uses of the exact beta formula survive.

4.4 Extended-value notation and remaining theory

Proposition 1.3 writes Q - integral_tail on all W_L. Both terms can be infinite. Use its bounded multiplier/time-operator expression to define R', and define B directly by its capped multiplier, or initially restrict to the finite form domain and extend. The domination argument survives this repair; infinity-infinity is not a definition.

The completion induction, weak duality, continuous-measure tail argument, and model-operator asymptotic proof were inspected. No additional decisive gap was identified, subject to the above corrections and the stated PNT input. This is not a numerical reproduction of their tables. The sampled finite-component census does not prove component exhaustion. Novelty, record comparisons, and the exact Liu-source translation are not independently cleared by this review; their external-source status remains conditional.

The theory worker's later fa6f450 progress note independently acknowledges the extended-value definition problem, an even/full-space sector mismatch in the essential-spectrum argument, the missing odd-sector citation for the T=150 bracket, and an overstatement that a better joint H can help only when the envelope threshold binds. These are valid caveats. In particular, R_H <= R' <= B_T gives a common lower limit on operating heights, not a proof that optimizing H cannot help above that limit. The essential-spectrum argument on the even subspace needs the symmetrization/modulation step written out; it is not supplied by a complex modulation that leaves that subspace. None of these ancillary claims is needed for the L=0.8 split bound.

Repair readback at b447f0d

The revised theorem uses 1-rho_N(m) and the shift-norm quadratic bound for every N, restricting the cosine comparison to m <= 2N+1. That closes the N=1, m=8 defect. The strip hypothesis now distinguishes finite measures from finite cosine sums or exponential moments; the actual witness qualifies. The threshold expansion and pole-versus-symbol rescaling are corrected. The bounded-symbol definitions remove the infinity subtraction, and the even symmetrization of the modulation sequence supplies the missing weakly null sequence. The two-sector citation and sampled component census caveat are now explicit. These changes pass ordinary analytic review. The phrase that every difference has a bounded integrand should exclude the displayed Q-B_T, which the revised proof correctly treats as extended nonnegative. No numerical or kernel-checked grade is claimed for these repairs.

5. Numerical claim at L=4/5

The target is the exact rational sine:16 witness, T=100, and the first 96 even Legendre modes. A different basis of the same size would usually be a different Ritz problem, so this review keeps the subspace and changes the quadrature and transform evaluation instead.

The inspected author control JSON reports:

The original numerics RESULTS at ed52a32, lines 17–20 and 22–24, assert the exact full-space endpoint 1.158e-17. The displayed two-block proof only yields 1.158e-17 - eps_B when the leading shift is that same number. The JSON's decimal lower_bound rounds away this loss; its lower_bound_arb is a ball, not a lower endpoint equal to its midpoint. It could be possible to prove an extra leading-block margin and recover the claimed endpoint, but that extra margin is not given by a boolean LDL pass.

The literal endpoint is unsupported by the stated accounting. A candidate safe outward-rounded replacement is 1.1579e-17, conditional on the reported enclosures being valid. This observation does not allege that the true minimum is below 1.158e-17.

Repair accepted at 3132b7d: RESULTS now subtracts eps_B explicitly and states the safe endpoint 1.1579e-17, on the even sector. It also corrects the H=0 calibration to 1.0199e-17. The new endpoint serialization is reported edited but unrun; no author code was read and this review does not validate that edit. The independent replay described next closes the corrected scientific enclosure through a separate quadrature implementation.

The new implementation encloses the same frozen witness with a two-atom kernel decomposition, reassembles the same 96-mode form, and tests both reported shifts. It uses its own independently derived quadrature error bound. It cannot confirm the author's exact quoted quadrature radius merely by obtaining the same final sign. The distinction will be retained in the post-run verdict.

Independent 96-mode results

Both approved units completed on source commit c65c66e, without a code repair or rerun. The exact rational frozen envelope passed the independent two-atom Arb proof in each unit. Both shifted LDL tests decided the same bracket as the author: the exact leading block satisfies 1.158e-17 < lambda_min(A) < 1.1585e-17. The upper test has a negative pivot at index 36, after positive earlier pivots. The measured midpoint Ritz value is 1.1583402660085778616489630627792930825e-17 at the displayed precision; the two result files store the same longer decimal string.

The stored balls justify these outward decimal summaries. Each matrix contains the quadrature radius in every entry and carries series and Arb arithmetic errors. The exact dyadic midpoint/radius lower triangles, preassembly budgets, LDL decisions and results were downloaded from the durable volume and their hashes checked against each manifest. RUNS.md records the app IDs, paths, terminal states and hashes.

The independent full-sector bound follows from min(1.158e-17, beta-eps_D)-eps_B, with the interval comparison against 1.1579e-17 passing in both runs. This is an enclosure-carrying numerical step conditional on the explicit ordinary analytic bounds in §7, which were reviewed here. Together with the reviewed reduction, it yields the same lower bound for Q on real even functions. It supplies no independent odd-sector numerical bound.

The result JSON's top-level grade: measured describes its midpoint eigenanalysis; the separate LDL and full_space_evidence fields carry the enclosure claim. Eigenvalue-string agreement is corroboration, not the proof of the full-sector bound. These runs independently reproduce the author's eigenvalue bracket and corrected endpoint, not every author GL entry or its quoted eps_Q_max: their implementation was not inspected, and a high-precision author matrix is not part of the supplied evidence.

6. Controls and lesions

The brief's instruction that any mutant remaining positive proves a broken pipeline needs a precise interpretation. An altered form can genuinely remain positive. For example, reversing an out-of-band correction does not break orthogonality. The proper failure is rejection of an invalid support, envelope, or defining-coefficient claim. A positive eigenvalue alone is not a lesion detector. These checks target the corresponding broken obligation.

K3 also has an independent frequency-side check on complex polynomial windows with a squared endpoint taper, plus an odd window. Three integrations by parts give |F(t)| <= B/|t|^3, where B=|f''(L)|+|f''(-L)|+integral |f'''|. The omitted two-sided integral, normalized by 2pi, is at most ||H||_infinity B^2/(5pi T^5). The code measures finite integrals at two cutoffs and reports these tail majorants. For the in-band constant-window lesion it uses the slower explicit 2/(pi L T) tail bound. Floating quadrature errors remain unvalidated, so this check is measured rather than an enclosure claim.

For the constant window test, deleting the n=4 term changes the normalized form by log(2)*(1-log(4)/(2L)) > 0. This is an exact derived expression; its numerical value is independently reproduced below. The independent DH control also checks the non-prime-power identity Lambda_DH(6)=(1+kappa^2)log(6).

K2's reference uses width=log(47) and hence this hunt's half-width L=log(47)/2, with 65 even Fourier modes. The target negative value near -0.3163 is old measured evidence, not a result of this review. A refusal caused only by resource exhaustion will not count as a K2 pass.

At b9a25ba, the numerical lane reports a completed Modal K2 run: the legitimate envelope gate returns no positive bound; forced-beta runs are explicitly invalid-envelope lesions. That is a useful gate-rejection control, not an independent validation of the whole enclosure pipeline. This referee's two independent negative-witness and reduction checks pass below, with their measured grade retained.

Independent controls execution, source c65c66e: Modal app ap-NhsIKRPPAvEgmNaQlKQZDY completed the controls unit in 5.6466 seconds. The reconstructed unpadded sine-16 sum is 1.555252826719554395156823327133059661505079595438460442661392472215115356514794825835; the exact frozen sum includes the two constant paddings and passes the independent moment/residual checks. These are measured results; the Arb envelope check belongs to the leading units.

K3 gives zero computed overlap on all ten complex/odd/constant test windows at every out-of-band frequency, including the boundary. The in-band constant-window defect is 0.03125. Both finite-frequency resolutions pass their measured analytic-tail comparison. The one-prime sign flip gives a negative envelope witness -3.163357475943...; dropping n=4 is rejected by the shared support gate and gives defect 0.092580913162... on the constant test. All three lesions are detected. This is a measured PASS for K3 and lesions, not a matrix enclosure. Manifest and result were downloaded from the volume; their source/result hashes agree with the local artifacts.

For DH, Binet's formula at 3/4+it/2 bounds the archimedean deficit by 1/(9t)+3/(2t^2) <= 1/t for t >= 15/4. Thus the proposed H=0 reduction uses beta=log(5T/(2pi))-1/T-sum 2|Lambda_DH(n)|/sqrt(n), without borrowing the zeta pole or prime-power-only coefficient support. The first independent unit dh512 passes: Q on the recovered witness is -0.31630285307629097, R is -0.7314173097418135, and scalar adaptive evaluation gives -0.7314173097417971. The negative beta also blocks a positive whole-form bound. This is measured negative-witness evidence, not just a resource or undecided-enclosure rejection.

The final dh768 unit also passes: Q is -0.31630285307558303, R is -0.7314173097420369, and the scalar adaptive value is -0.7314173097419605. Both resolutions recover the large negative witness and refuse a positive bound. Their saved witness vectors and result hashes were checked against volume readback. No positivity inference relies merely on the runner's hardcoded positive_result: false: the acceptance gate requires the observed negative Q and R values, domination, and scalar/matrix agreement. K2 is a measured PASS, not an interval proof of the DH integrals.

Supervisor's later L=1.19 evidence update

The supervisor reports that app ap-mucAZVkQ7RThKhLbUB9CuN stopped, with nine unit JSON objects and stageA_L119_reduced.json on volume oob-envelope-stages. At N=360, T=500, the midpoint eigenvalue is reported as 5.775648793894534e-48; interval LDL has n_neg=null, undecided=306, and no quadrature, tail, or coupling bounds.

Disposition: UNRESOLVED, measured midpoint only. No support-2.38 positivity result. This is the supervisor's evidence, accepted as reported; no Modal retrieval or reducer was run by this referee. It changes neither the L=0.8 endpoint objection nor the outstanding independent checks.

New reporting defect at b9a25ba: the Stage A discussion says the published upper bounds at L=1.1 and L=1.2 bracket the floor at L=1.19, and that the reduced value must lie between them. This does not follow. Monotonicity and an upper bound U at L=1.2 do not imply lambda*(1.19) >= U. Only the larger-window infimum itself supplies a lower comparison, not its upper bound. The L=1.1 upper bound does give a necessary upper ceiling on a valid L=1.19 lower bound. Remove the claimed two-sided bracket and the words asserting necessity of lying between the two upper bounds. No contradiction with the reported midpoint is claimed. Repair accepted at f779b27: the final numerics text keeps only the one-sided ceiling and explicitly withdraws the false bracket. The independent L=4/5 verdict does not settle L=1.19.

7. Independent numerical budget derivation and execution

The live consumer is the same 96-mode claim. There is no general framework. independent.py contains the implementation; run_modal.py dispatches single units only. The formulas below are ordinary derivations supporting the verifier. Both leading units now carry their interval evaluations; numerical agreement does not replace these analytic derivations.

For a sine kernel of degree D use weights a_j=sin(pi(j+1)/(D+2)), j=0,...,D, and normalized correlations rho_k. At this window, p=2 has two low moments and p=3 has one. Put

u=log(2)/(sqrt(2)*rho_1), v=log(2)/(2*rho_2), M2=(v+sqrt(v^2+8u^2))/2, cos(theta2)=-u/M2, M3=log(3)/(sqrt(3)*rho_1), theta3=pi.

Two equal atoms at plus/minus theta give the required low coefficients after convolution with the nonnegative sine kernel. High cosine coefficients are 2 M_p rho_k cos(k theta_p). Compare each exact rational frozen coefficient with that construction, and charge the L1 difference against the frozen constant's slack. No sampled nonnegativity is used by the proposed zeta enclosure path.

For transforms use the spherical-Bessel power series, with term ratio -x^2/[2(k+1)(2n+2k+3)]. After its absolute ratio is below 1/2, bound the uncomputed tail by twice the absolute next term. Seed two consecutive high orders with such balls and recur downward. The pole uses the same series with positive signs at x=L/2. There is no floating input snap or unbounded recurrence seed error. A separate j0=sinc(x) overlap check is included.

For panels of width h=1/2 choose rho=1+sqrt(2). The ellipse has imaginary height 1/4 and real extent h/sqrt(2). The digamma arguments have real part at least 1/8. From psi(z)=-gamma-1/z+sum_{k>=1} z/[k(k+z)], a conservative majorant follows from |psi(z)| <= 1+8+2|z|. The program uses |z| <= T+2.

If an integrand has ellipse bound M, Chebyshev coefficient bounds and Lobatto aliasing give interpolation error at most 4 M rho^-q/(rho-1). Integrating over a panel gives at most h times this quantity. Each matrix entry gets the sum over all panels as an actual radius. This is Clenshaw-Curtis, independent of the authors' Gauss rule.

For tails let n0=2N and

a_n=sqrt(2L(2n+1))*(LT)^n/(2n+1)!!.

The ratio a_(n+2)/a_n has a decreasing geometric majorant r after the cut. Thus sum_tail a_n^2 <= a_n0^2/(1-r^2) = U. Completeness gives sum_lead |F_n(t)|^2 <= 2L. With K=(T/pi) max_real |a-P+H-beta|,

eps_D <= K U, eps_B <= K sqrt(2L U) + 2 sqrt((L+sinh L) V),

where V bounds the squared pole tail using the same majorant with x=L/2 and an extra factor exp(L/2). The even pole is positive in the tail block. These are alternative operator-norm bounds to the author's Gershgorin/Schur constants. The two independent-quadrature executions close that implication.

The real archimedean multiplier is monotone in |t|, by differentiating its convergent digamma series. Its absolute value on [0,T] is bounded by the two endpoints, giving a sharper real-axis budget than the ellipse budget.

The verifier records the enclosed matrix before LDL, then both shifted decisions, eps_Q, eps_D, eps_B, and an outward-safe full-space endpoint. Failure or uncertainty at any gate remains inconclusive. Both independent resolution units completed and passed before the numerical verdict changed.

8. Approved execution gate

The supervisor approved the exact five bounded Modal units in RUNS.md, sequentially with a $0.15 allowance and a stop on first failure or inconclusive gate. RUNS.md now includes a conservative derivation: CC degrees 160 and 192 give matrix quadrature errors below 2e-49 and 2e-61. The remote units check and save their actual budgets before assembling the matrix. This resolves the degree-selection concern analytically and now by both Arb executions. All five units completed with gate true; every app is stopped, all required artifacts are durable and read back, and there were no failures, inconclusive gates or reruns. See RUNS.md for the complete execution record.

Remaining scope limits: the author GL implementation and its exact quoted quadrature radius were not independently audited; this review proves the corrected L=4/5 result through the new CC route. Symbolic steps remain ordinary reviewed derivations. K2/K3/lesions remain measured controls. The prior-art survey retains the source-verification limits noted in §4. At L=1.19, unresolved interval signs and absent quadrature, tail and coupling bounds still block any whole-form positivity verdict. No new paid unit is requested by this completed review.