Certified Localized Weil Positivity Through Support 0.72: Multiband Schur Complements and a Complete Stieltjes Hierarchy
Admitted after checks by named AI models and editor approval. Not peer reviewed. Screening record
Suzuki's localized Weil form is represented by a lower-bounded self-adjoint operator A_a on L2(-a,a); nonnegativity for every a > 0 is equivalent to the Riemann hypothesis, while positivity at one fixed support is an unconditional and strictly weaker problem. We give a source-level interval certificate proving A_0.72 >= 5.890 x 10^-17 I > 0. Nested-core monotonicity gives the same scalar lower bound for every 0 < a <= 0.72. The proof decomposes the exact prime-power translation graph through n = 4 into thirteen intervals and bounds its infinite complement by a mode-sensitive Schur estimate. It isolates degrees 12 through 23 before controlling [24,176) and [176,infinity); both parity Schur matrices have 78 certified positive directions and no unresolved direction at 512-bit Arb precision. We also prove an exact Gauss-Stieltjes hierarchy for the logarithmic boundary potential, complete for strict positivity at every fixed support, and a multiband Loewner majorant requiring only O_epsilon(log log M) bands through degree M. The revised manuscript prints a formula-level source-to-Gram specification. A separately written program importing neither project modules nor python-flint reconstructs the prime-power graph and parity maps and independently reassembles the exported Schur balls with positive Weyl margins in both sectors. This is a bounded-support theorem, not a proof of the Riemann hypothesis.
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Cite this paper
Lluis Eriksson (2026). Certified Localized Weil Positivity Through Support 0.72: Multiband Schur Complements and a Complete Stieltjes Hierarchy. AIRR.SCIENCE, ARR-2026-7XT7AB8WJP9QPTGT, v1. https://airr.science/papers/ARR-2026-7XT7AB8WJP9QPTGT/versions/v1/
Use this version link to cite the manuscript you read. A Working paper label means the version has not completed AIRR screening. AIRR screening is not journal peer review. Import BibTeX, RIS or CSL JSON into your reference manager to apply a citation style.
Version history
The paper identifier remains stable. Each version has its own immutable release, timestamp and version identifier.
- v1 · source snapshot available · viewing
Verification record
- Frontier-model screening
- Not assessed
- Source integrity
- Pass
- Bibliographic integrity
- Not assessed
- Reproducibility
- Partial
- Lean 4
- Not applicable
Recorded under ARR-VERIFY-1.0. AIRR verification and screening are not peer review.
AI assistance statement
AI tools assisted with literature discovery, adversarial proof checking, software inspection, and manuscript preparation. The author selected the claims, reviewed the mathematical arguments and executable evidence, and is responsible for the final text and any remaining errors.
Frontier-model screening
Status: not_assessed. Any listed reports correspond to this exact version under ARR-SCREEN-1.0; no absent assessment is represented as a pass.
Model assessments and review context
- Distinct models
- 2
- Range
- 2.20–2.40
- Method
- bounded capability-weighted mean · exact version only
Limited evidence · 1/2 benchmark matches. This is an evidence description, not a probability of correctness.
One provider; shared blind spots possible; Fresh blinded review context not documented for every model; Benchmark match incomplete; Adverse reports or unresolved objections require editorial disposition
OpenAI · gpt-5.6-sol · High · unknown · review context: unknown2.20
Recommendation: Major Revision · Material objections: 3
The noncomputational framework is promising: the localized operator normalization agrees with current primary sources, the Gauss-Stieltjes hierarchy is correctly derived, and the abstract multiband Schur bounds are sound. The central claim A_0.72 >= 5.890e-17 I is nevertheless not independently assessable from this batch. Its two interval NPZ proof objects and source archive are absent, the release verifier fails before any Arb adjudication, and the supplied theorem JSON does not match the registered digest. Partial recomputation confirms the printed partition, increasing denominators, and final scalar transport formula, but those are downstream calculations, not a proof of the interval matrices. Major revision is required before the main theorem can receive an accept recommendation.
- Correctness confidence
- The analytic Stieltjes and abstract Schur lemmas check out, but Theorem 1.1 depends on interval matrices that are absent. The supplied scalar record is internally consistent yet cannot establish the claimed operator inequality by itself.
- Rigor
- Sections 4-5 state sound abstract mechanisms and Appendix A exposes many signs and constants. However, the decisive source-to-Gram enclosures and complement bounds are delegated to unavailable proof objects and incomplete source imports.
- Novelty
- If certified, support 0.72 and the source-faithful multiband integration would be plausible advances beyond the cited fixed-support literature. The literature check was targeted rather than exhaustive, so priority remains provisional.
- Significance
- A rigorous coercivity radius crossing the n=2,3,4 prime-power channels would be a notable unconditional result adjacent to Weil positivity, while still being explicitly far short of RH.
- Reproducibility
- The release verifier exits 1 because the aggregate NPZ is missing; the band NPZ, source ZIP, provenance inputs, and shadow record are also absent. The supplied JSON has a different SHA-256 from the paper's registered theorem record.
Strengths
- Proposition 2.1 correctly transfers any fixed-support lower bound to smaller supports by zero extension without claiming an operator order between different Hilbert spaces.
- Theorem 4.1 gives a credible exact Markov remainder, endpoint identity, Jacobi-resolvent formula, and positive Schur-square increment for the logarithmic boundary potential.
- Theorem 5.1 correctly uses inverse antitonicity to avoid assuming that the complement operator preserves each degree space, and Lemma 5.3 transparently transports a Schur lower bound to the full block.
- The manuscript sharply distinguishes fixed-support positivity from RH and explains why positive Ritz values alone cannot certify coercivity.
- Independent arithmetic using the supplied JSON confirmed the thirteen-interval ordering, the registered band boundaries, strict denominator growth, and both Table 3 full-block lower bounds to floating precision.
Weaknesses
- The proof of Theorem 1.1 is computer-assisted, but the batch omits the aggregate and degreewise NPZ matrices on which every final Gershgorin and Schur sign depends.
- Appendix A calls itself normative yet leaves several adjacent and separated Legendre-Green integrals and analytic remainder constructions referenced only by generator names; the corresponding complete source modules are not supplied here.
- The claimed 512-bit Arb environment was not available: this runtime has Python 3.12.14, NumPy 2.3.5, and SciPy 1.17.0, with mpmath and python-flint missing.
- The very small even coercive margin, 5.890e-17, is orders of magnitude below the source and coupling scales, making exact availability and re-adjudication of every interval radius indispensable.
Potential errors
- The PDF registers SHA-256 63b0dd91... for theta-schur-a072-multiband-to24-by-degree-v3.json, whereas the supplied file hashes to af75bfb5...; the public-code manifest locks the latter, so the paper's exact release verifier must reject this copy.
- Relative to public-code/src, MANIFEST_SHA256.txt has 22 matching entries, ten missing paths, and one mismatched path; the mismatch is the final theorem JSON and the missing paths include both decisive NPZ objects and paper17-v3-source.zip.
- Running verify_release.py on the supplied directory exits 1 at the first missing aggregate NPZ. Running independent_reference_audit.py exits 1 at import because mpmath is absent and would in any case lack the raw component balls.
- The supplied theorem JSON records only downstream scalar outputs. Recomputing formula (20) from s, d, and b reproduces the displayed floors, but it cannot validate the unprovided interval matrices that supposedly certify s and b.
Strong novelty candidates
- A source-level interval proof of positivity through support 0.72, if reproduced from the registered objects, would be a strong concrete novelty candidate.
- The exact Gauss-Stieltjes lower hierarchy specialized to the logarithmic boundary potential, including positive increments and a terminating fixed-support criterion, is a plausible reusable contribution.
- The mode-sensitive O_epsilon(log log M) band count is a useful structural observation for harmonic complement denominators, though the asymptotic estimate itself is elementary once the majorant is set up.
Unresolved material objections
- The main coercivity claim is not verifiable from the supplied artifacts because theta-schur-a072-d12-p47-tail8192-v3.npz and theta-near-band-a072-d12-to24-by-degree-p512-v3.npz are absent; without their outward-rounded matrices the two 78-direction positivity decisions are unsupported.
- The exact theorem record fails the manuscript's own byte identity: the supplied JSON SHA-256 is af75bfb59b3bc72514bac1e9a398defbd2e6e2a6d37f3afbd553ce93bb0364ea rather than the registered 63b0dd91dab9fe7a00b734644c7a0400c2240a51cc25e6dde42751208dfc4f08.
- The release's fail-closed verification cannot proceed past missing files, and the complete source archive and provenance objects needed to regenerate or audit the matrices are also absent. Consequently no successful Arb run, provenance replay, or independent shadow reconstruction was performed.
arr:assessment:a8d1fbda-fc42-4eb6-95b7-9a2cc70edc06 · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-5.6-sol · reasoning effort: high · basis: operator verified ui · evidence SHA-256 92703c3c3d72e9347295c9be5ba97924b67ac85672a1724b468bcb11111112de · response SHA-256 c96e3bbbd844fd78b1eea4edb7ebd46f05a5e42d4f0c3d064af17f8e25ffe87c · canonical PDF SHA-256 e4fd12894f2d15bd503370a016f259aa0e2f619810ba309108ca6a1150a02935
OpenAI · gpt-6-astra · High · unknown · review context: unknown2.40
Recommendation: Major Revision · Material objections: 2
This exact 10-page PDF presents a plausible finite-support Weil-positivity strategy with sound analytic components, but the supplied evidence does not establish its principal numerical theorem. I checked the source normalization against Suzuki, the thirteen-cell graph, Stieltjes identities, Schur inequalities and printed full-block arithmetic. The release verifier stops at a missing NPZ; the second NPZ, frozen source archive and 22 imported project modules are also absent. Independently hashing the supplied theorem JSON gives af75bfb5...bb0364ea, not the registered 63b0dd91...dfc4f08, despite its matching the batch manifest. The complement floor remains a load-bearing unverified assertion, and the shadow code differs from the Arb denominator accounting. These findings do not disprove localized positivity, but preclude accepting the claimed certificate on this record. No RH solution or successful full replay is asserted.
- Correctness confidence
- Propositions 2.1 and 3.1, Theorem 4.1 and the Schur inequalities are credible and survived individual checks. Theorem 1.1 cannot be confirmed: load-bearing NPZ balls and source generators are absent, and the supplied theorem JSON misses its registered byte hash.
- Rigor
- The Stieltjes remainder and block-coercivity derivations are sound. Sections 6/A list numerical complement floors and software interfaces without enough supplied evidence to establish the full operator lower bound. The shadow implementation is not an exact independent assembly of the stated denominators.
- Novelty
- A verified complete-operator certificate at support 0.72 would be a substantive specialized contribution. Gaussian quadrature, Schur antitonicity and compact-resolvent monotone convergence are classical; the priority comparison with all earlier explicit radii was not independently established.
- Significance
- Exact prime-power geometry and degree-dependent tail control could be reusable. A single bounded-support inequality, even if confirmed, neither settles RH nor provides a support-uniform method; the very small conservative margin demands reliable enclosure accounting.
- Reproducibility
- Release verification failed at the missing aggregate NPZ. With installed mpmath/flint, the shadow passed graph/parity then failed at that file; pytest had three import-collection errors and multiband import failed. One isolated outward-scalar test passed. No full Arb certificate was regenerated.
Strengths
- Section 2: the polar cross term, von Mangoldt weight for n=4 and scalar normalization agree with the inspected Suzuki global formula. Extension by zero proves Proposition 2.1 without comparing operators on different Hilbert spaces.
- Proposition 3.1: endpoint differences are positive in the stated third window, and the listed shifts give the 7+4+2 components. The supplied graph/parity function actually completed, checking all eleven edges and the 78+78 parity decomposition.
- Theorem 4.1: the Markov quadrature remainder has the stated squared numerator; Jacobi block inversion gives the positive increment. Independent high-precision checks for m=1,...,5 at x=0.2,0.7,0.99 confirmed the remainder and R_m(1)=H_m identities numerically.
- Corollary 4.2: monotone forms plus the common compactness supplied by H_n -> infinity justify eigenvalue convergence and eventual strict positivity at fixed support. This does not handle a zero ground state or give uniform control in a.
- Theorem 5.1 and Lemmas 5.2-5.3: inverse antitonicity needs a lower form bound, not degree invariance. Congruence transport and weighted Cauchy-Schwarz correctly turn a genuine Schur margin and coupling bound into full-block coercivity.
- Equation (24) and Table 3: the isolated supplied outward-scalar test passed; the two-rounding remainder is 9.2437306756331e-16. Recalculation from the stored s,d,b gives approximately 5.890068275105137e-17 and 1.652995907846963e-13; this validates arithmetic, not the input bounds.
Weaknesses
- Section 7: neither registered NPZ, paper17-v3-source.zip, predecessor/replay NPZ nor shadow output is supplied. Missing imported modules include interval_inertia, arb_third_window_source and third_window_pointwise_floor. I did not obtain another release or substitute source files.
- Theorem 1.1 depends on the complement inequality D >= sum d_n P_n. Equation (22) prints floors but does not derive the pointwise comparison establishing d12 or the complete error budget. The absent floor generator prevents auditing that step.
- Appendix B overstates the precision of independent assembly: registered_mid, registered_rad and their residual are accumulated/subtracted as NumPy binary64 arrays before mpmath conversion. This shadow diagnostic is not a directed-rounding proof, as the paper also acknowledges.
- The actual environment used Python 3.12.14, NumPy 2.3.5 and SciPy 1.17.0, not the frozen versions. mpmath 1.3.0 and python-flint 0.9.0 were installed locally. Missing project files, rather than those version differences, caused the observed import failures.
- Theorem 4.1 specializes classical Gauss/Markov theory; Theorem 5.1 is a useful packaging of standard operator order. Novelty rests mainly on the unverified source-to-certificate application, not on these general ingredients.
- Reference [7] has author-name initials inconsistent with the inspected primary record, which lists Bart Rosenzweig and Jonathan Stanfill. Their displayed Legendre/harmonic identity supports equation (7), but the bibliography should be corrected.
Potential errors
- Section 7 registered JSON hash mismatch: actual af75bfb59b3bc72514bac1e9a398defbd2e6e2a6d37f3afbd553ce93bb0364ea; expected 63b0dd91dab9fe7a00b734644c7a0400c2240a51cc25e6dde42751208dfc4f08. Both are exact byte claims. The verifier run stopped earlier at the absent NPZ; this mismatch was found by a separate static hash check.
- Appendix B / independent_reference_audit.py: the far denominator uses complement_floor + H176-H12, whereas the Arb adjudicator debits 2*smooth_remainder. The shadow denominator is larger by about 1.84874613512662e-15, reducing its subtraction. Without the Grams I cannot bound the effect; its quoted margins are not an exact replay of equation (37).
- Theorem 5.1 is stated for a general self-adjoint block with only d_n>0. Uniform positivity of the comparison, boundedness of B and an appropriate form-domain formulation should be explicit when inverses and infinite sums are used. The intended finite-source application has a positive d12, so this general-statement omission is not a demonstrated counterexample to Theorem 1.1.
Strong novelty candidates
- A source-verified complete localized Weil-operator certificate at a=0.72, crossing the n=2,3,4 channels, is the strongest novelty candidate; it remains conditional on repairing the evidence gap.
- The thirteen-cell arithmetic graph integrated with complete source-to-tail Gram rows and mode-sensitive bands is a plausible reusable computational construction.
- The explicit match between the boundary potential and a nested positive Gauss-Stieltjes hierarchy is a useful specialization, with classical quadrature ingredients and no established broad priority claim.
Unresolved material objections
- Theorem 1.1 / Section 7: the two load-bearing interval NPZ objects and complete frozen source are missing from this batch, and the present theorem JSON is not the hash-registered object. The observed release/import failures leave both 78-dimensional interval signs and their source provenance unverified. Stored scalar claims and printed hashes cannot replace those objects or an execution.
- Theorem 1.1 / equation (22) / Appendix A: the full complement lower form inequality and analytic tail-enclosure bounds have not been established from the accessible proof material. The numbers d12,d24,d176 alone do not supply that proof, and the relevant pointwise-floor/source/inertia modules are absent. Until this source-to-infinite-operator step is auditable, positivity of an asserted finite Schur matrix cannot certify the complete operator.
arr:assessment:6f2ee6e7-f42f-4bb2-866f-5c28fcf98759 · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-6-astra · reasoning effort: high · basis: operator verified ui · evidence SHA-256 d6795eeac3911d1340112c3a81067c06d55b17243edbe75932bb156f1258b1ee · response SHA-256 9ba108899f865c2e7c44fbb829c1c1f7c3bd2720d8d833a09f9e0126cd7fa4e2 · canonical PDF SHA-256 e4fd12894f2d15bd503370a016f259aa0e2f619810ba309108ca6a1150a02935
OpenAI · gpt-5.6-sol · High · unknown · review context: unknown2.35
Recommendation: Major Revision · Material objections: 2
This exact 10-page v1 paper proposes a potentially strong computer-assisted theorem: coercivity of Suzuki's localized Weil operator at support 0.72, propagated to smaller supports, together with an exact Stieltjes hierarchy and multiband Schur bounds. The analytic lemmas are plausible and limited independent calculations agree with the printed partition and block-transport arithmetic. However, the central interval proof is not verifiable from the supplied artifacts: both NPZ proof objects and much of the source are missing, and the only supplied theorem JSON fails the manuscript's locked hash. Because that unresolved packaging/provenance failure bears directly on Theorem 1.1, acceptance is not supportable for this exact batch.
- Correctness confidence
- The analytic monotonicity, Stieltjes, and abstract Schur arguments are credible, and independent arithmetic checks reproduced the printed graph and final transport formula. The decisive 156-dimensional interval sign and infinite-tail bounds cannot be verified from the supplied wrong-hash/incomplete payload.
- Rigor
- The paper states fail-closed conditions and a detailed source-to-Gram map, but the main coercivity theorem ultimately rests on computed matrices and tail enclosures not derivable from the ten-page text alone. Missing proof objects and missing imported modules therefore leave the load-bearing sign uncertified.
- Novelty
- If independently reproduced, a source-faithful coercivity proof through support 0.72 and the specialized exact Gauss-Stieltjes hierarchy would be strong contributions beyond the cited numerical work. The limited primary-source check did not establish exhaustive priority.
- Significance
- A rigorous improvement of localized Weil positivity across active prime-power channels would be important, though still strictly weaker than RH as the paper correctly emphasizes. Present significance is conditional on restoration and successful replay of the exact certificate.
- Reproducibility
- The release verifier stopped at the first missing NPZ. Ten files named by the internal manifest are absent; 22 imported source modules are absent; the supplied theorem JSON has SHA-256 af75...64ea instead of the locked 63b0...4f08; mpmath and python-flint are also unavailable.
Strengths
- Proposition 2.1 correctly derives nested-core monotonicity from zero extension of the same global quadratic form, without asserting a false operator order across different Hilbert spaces.
- Theorem 4.1 gives explicit positive Gauss-Stieltjes remainders and Schur-square increments for the logarithmic boundary potential; the cited primary source confirms the Legendre/harmonic-number diagonalization of A2.
- Theorem 5.1's inverse-antitonicity majorant and Lemma 5.3's full-block coercivity estimate are logically appropriate tools for turning finite interval signs into an infinite-dimensional lower bound.
- The manuscript distinguishes a lower-bound certificate from a positive Ritz computation and explicitly limits the result to bounded support rather than claiming RH.
- Independent calculation at a=0.72 confirmed strict ordering of all 14 cut points and every printed h2, h3, and 2h2 translation pair.
- Independent substitution of both Table 3 triples into Lemma 5.3 reproduced the even and odd lower bounds to floating precision.
Weaknesses
- The numerical theorem has an extremely small even lower bound, 5.890e-17, so exact provenance and directed-rounding replay are essential rather than optional corroboration.
- Appendix A specifies many source, singular, endpoint, self-tail, and other-tail terms, but several decisive bounds are reported rather than proved in enough detail for a hand audit independent of the missing code.
- The claimed improvement over the earlier explicit localized radius was not exhaustively established; accessible primary records confirmed the cited operator and numerical papers but not a complete priority search.
Potential errors
- The supplied theta-schur-a072-multiband-to24-by-degree-v3.json hashes to af75bfb59b3bc72514bac1e9a398defbd2e6e2a6d37f3afbd553ce93bb0364ea, not the 63b0dd91dab9fe7a00b734644c7a0400c2240a51cc25e6dde42751208dfc4f08 locked by the manuscript, internal manifest, and verifier.
- The internal MANIFEST_SHA256.txt names 33 release files, but ten are absent, including both central NPZ objects, the source ZIP, both provenance NPZs, the provenance audit, and the independent audit record.
- The patched-source subset has at least 22 unresolved experiments.theta_pencil imports, so it is not the 'complete source snapshot' described by the paper and cannot regenerate the certificate as supplied.
- The release verifier exited before certificate arithmetic with 'missing theta-schur-a072-d12-p47-tail8192-v3.npz'. The independent shadow audit separately failed at import because mpmath is missing; neither outcome verifies a sign.
- A manual call showed that the wrong-hash JSON passes verify_record's shallow schema and threshold checks, but this post-failure diagnostic does not authenticate its numbers or substitute for the absent interval proof objects.
Strong novelty candidates
- A rigorously replayed full-operator coercivity certificate at support 0.72 that crosses the n=2,3,4 prime-power channels.
- The exact Gauss-Stieltjes lower hierarchy for the boundary logarithm with a positive continuous remainder and explicit Schur-square increments.
- The mode-sensitive multiband Schur majorant with O_epsilon(log log M) harmonic-tail storage.
Unresolved material objections
- Theorem 1.1 is computer-assisted, but its exact interval proof objects are absent and its supplied result record fails the paper's locked SHA-256. Until the correct hash-locked NPZ/JSON set is supplied and the fail-closed verifier completes, the claimed positive sign and 5.890e-17 lower bound remain unverified.
- The complete source needed to justify and regenerate the source-to-tail Grams and analytic remainder bounds is not present. Because those bounds are load-bearing and are not independently derived in the manuscript, the missing 22 imported modules leave a second direct gap in the main certificate.
arr:assessment:6d340fbe-ea93-431e-8eb6-8d2866c792f0 · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-5.6-sol · reasoning effort: high · basis: operator verified ui · evidence SHA-256 7531e97a7baad09cb045dd852e4a088ad369c8df86932feeef1e11f85fb1aa83 · response SHA-256 f4af00b724aa24683991390ba08c5aa4f227bf6d77ef8056b8e8d0d71f1774f4 · canonical PDF SHA-256 e4fd12894f2d15bd503370a016f259aa0e2f619810ba309108ca6a1150a02935
OpenAI · gpt-6-astra · High · unknown · review context: unknown2.40
Recommendation: Major Revision · Material objections: 3
The exact 10-page PDF contains a plausible and useful analytic framework, but its central computer-assisted theorem is not verifiable from this batch. The outer manifests are valid; the internal proof payload is incomplete and the supplied theorem JSON is not the byte-locked record named by the PDF. Source inspection also finds that the shadow implementation does not reproduce every advertised arithmetic step. I checked the graph, Stieltjes identities, full-block transport arithmetic and the standalone outward scalar test, but did not reproduce either positive 78-dimensional Schur sign or the asserted full-operator floor. The unresolved certificate and complement-bound obligations require major revision of the evidentiary package. This is not evidence that the localized inequality is false, and it is not an RH assessment.
- Correctness confidence
- Proposition 2.1, the thirteen-cell graph, the Stieltjes identities and the Schur transport are credible and individually checked. Theorem 1.1 depends on unavailable source/certificate objects, so its 5.890e-17 full-operator lower bound was not verified.
- Rigor
- Theorems 4.1 and 5.1 and Lemmas 5.2-5.3 use appropriate positive remainders and operator bounds. Section 6 supplies numerical floors without a checkable derivation of the crucial complement inequality; missing generators and interval objects prevent closing this proof obligation.
- Novelty
- A verified support-0.72 coercivity certificate with exact prime-power cells would be a substantive specialized contribution. Gaussian quadrature, monotone forms and inverse antitonicity are classical. The literature search was selective and did not establish endpoint priority.
- Significance
- Full-complement control would improve on positive finite compressions and could be useful for bounded-support Weil positivity. The endpoint and tiny margin are local; the paper correctly provides no support-uniform positivity or RH solution.
- Reproducibility
- All 25 supplied entries match PUBLIC-CODE-MANIFEST, but the required NPZ objects and source ZIP are missing. The theorem JSON fails the PDF's registered digest. The release run aborts; three test modules fail collection. Only graph/parity and scalar checks completed.
Strengths
- Proposition 2.1, p. 2: zero extension on the common global form core proves scalar monotonicity without incorrectly ordering operators on different Hilbert spaces. Suzuki's Corollary 1.2 supports the core-infimum formulation (https://arxiv.org/html/2606.09096v2).
- Proposition 3.1, p. 3: direct endpoint subtraction gives the eleven edges and component sizes 7,4,2. The standalone graph/parity function executed successfully; the n=4 coefficient uses log(2), as required by the von Mangoldt weight.
- Theorem 4.1, p. 4: the Markov identity gives the squared remainder with the stated x^(2m+1) factor, and Jacobi block inversion gives positive increments. Independent 80-decimal diagnostics for m=1,2,3,5 agree with the remainder, resolvent and endpoint formulas to about 1e-80.
- Corollary 4.2, p. 4: monotone form convergence is supplemented by the common compact control A2 with H_n tending to infinity. This supports convergence of fixed eigenvalues and finite detection of strictly positive, but not zero-ground, cases.
- Theorem 5.1 and Lemmas 5.2-5.3, pp. 5-6: inverse antitonicity avoids assuming invariant degree subspaces; congruence transport and the coupling-norm charge are logically appropriate. Rational arithmetic on Table 3 reproduces both conservative public floors, conditional on the input bounds.
- The multiband source retains complete target rows before outer products and subtracts registered bands as Arb matrices. The separately runnable outward smooth-remainder test passed; this checks its scalar export, not the missing component Grams.
Weaknesses
- Section 7: verify_release.py . actually exited 1 at missing theta-schur-a072-d12-p47-tail8192-v3.npz. It never reached interval-schema checks, source extraction, Arb readjudication or the final shadow comparison.
- Appendix B overstates the all-high-precision reconstruction. independent_reference_audit.py accumulates registered_mid and registered_rad and forms residual arrays in NumPy binary64 before converting them to mpmath; no rounding budget is added for those operations.
- Appendix B's shadow tail denominator starts from record["complement_floor"] plus a harmonic increment, whereas the main multiband routine subtracts 2*smooth_remainder first. Thus the shadow uses a larger denominator and a smaller tail penalty; agreement with equation (37) is not exact.
- Theorem 5.1 should state the operator-domain and convergence hypotheses needed for BD^{-1}B*: uniform positivity and bounded coupling make the intended application safe, but the abstract statement only writes d_n>0. Corollary 4.2 is an existence hierarchy, without a quantitative termination bound.
- Endpoint priority remains unverified. Suzuki's operator/core statements were checked and Kim et al.'s abstract confirms a numerical realization, but neither a full Yoshida/Bombieri comparison nor an exhaustive public-source priority search was completed.
- An ancillary routine, certify_second_window_schur, permits one negative Schur direction yet calls _coercive_lower_from_schur with its first positive eigenvalue. Such a value cannot bound the full operator below. The separate main multiband path requires all-positive inertia, so this is not by itself a disproof of Theorem 1.1.
Potential errors
- The PDF, Section 7, registers theorem-JSON SHA-256 63b0dd91dab9fe7a00b734644c7a0400c2240a51cc25e6dde42751208dfc4f08; the actual supplied bytes hash to af75bfb59b3bc72514bac1e9a398defbd2e6e2a6d37f3afbd553ce93bb0364ea. The outer public-code manifest agrees with the latter. The cause and semantic significance of the discrepancy are unresolved.
- The shadow denominator and binary64 residual operations weaken its claimed independence as a wiring check. The missing NPZ files prevent measuring their effect on the quoted 6.3113e-14 even-sector shadow margin. No negative Schur direction was established by this assessment.
Strong novelty candidates
- Theorem 1.1 and Proposition 3.1: an exact prime-power graph integrated with directional tail Grams and a verified full-space lower bound through support 0.72 would be the strongest contribution, if the missing proof obligations are supplied and independently reproduced.
- The combination of the boundary Stieltjes hierarchy and harmonic degree-resolved Schur budgeting may be reusable beyond this endpoint. Its constituent approximation and operator inequalities are classical; novelty would lie in the concrete integration and validated implementation.
Unresolved material objections
- Theorem 1.1, Sections 6-7 and Appendix A require both registered NPZ proof objects and the full source snapshot. These are absent, including the source, flux, singular, self-tail, pointwise-floor and inertia dependencies. The release verifier stops before sign adjudication, and test collection confirms missing project modules. The central certified lower bound therefore remains unverified.
- Theorem 1.1 invokes the complement inequality (15) with floors (22), but the PDF gives no self-contained derivation of the numerical 0.220973215897795 floor or complete tail enclosures. The missing third_window_pointwise_floor and related generators are precisely the source of that inequality. Generic Schur lemmas plus printed positive numbers do not discharge this main proof obligation.
- The supplied theorem JSON does not match the digest frozen in Section 7 or verify_release.py. This must be reconciled against the exact proof objects before a successful byte-locked release replay can be claimed. No file was substituted, regenerated into a missing path or relabelled to bypass this discrepancy.
arr:assessment:f4603818-1185-4ec1-a368-87ee1cc029dc · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-6-astra · reasoning effort: high · basis: operator verified ui · evidence SHA-256 2dc67e969a3ac770c4c0fb93d5a56f8c5d40f49f0714f9b8f056b22f0c6766f6 · response SHA-256 8da3f0a048dbbe057e9053b6fcd0df59df8bda5d2ff782fb03b2167561c99b55 · canonical PDF SHA-256 e4fd12894f2d15bd503370a016f259aa0e2f619810ba309108ca6a1150a02935
A model assessment is not peer review or a correctness certificate. AIRR preserves disagreement, exact-version provenance and later reassessments.
Editorial disclosure
Founder-owned pilot record: Lluis Eriksson is both the author and AIRR's current founder-editor. No independent editorial review, peer review, or frontier-model screening is claimed for this version. Acceptance here records a technically valid deposit, not a finding that the paper is correct.