Exact Rank Transitions through p=32 and a Half-Integral Optimum at p=53
Admitted after checks by named AI models and editor approval. Not peer reviewed. Screening record
For F_(p,q)=diag(q repeated p times, -p repeated q times), this paper studies the least rank among factors minimizing one half of the squared Hilbert--Schmidt norm subject to CC^*-C^*C=2F. On q=2p+1, exact rational hive primal-dual certificates determine the complete finite cost-and-rank frontier for 4<=p<=32: the rank excess above q is 0 on p=4..7, 1 on p=8..14, 2 on p=15..26, 3 at p=27, and 4 on p=28..32. The resulting consecutive rank transitions at p=27 and p=28 are followed by a distinct cost-slope transition at p=29. Separately, exact certificates prove kappa(F_(53,107))=8847 with minimum attaining rank 115; its optimum is half-integral, while an earlier integer candidate of trace 8843 is refuted by an integral Farkas certificate. The conclusions are finite and conditional on the classical Horn--Klyachko/hive theorem; no all-parameter recurrence, classification of all minimizers, exhaustive priority result, proof-assistant formalization, or independent peer review is claimed.
Cite this paper
Lluis Eriksson (2026). Exact Rank Transitions through p=32 and a Half-Integral Optimum at p=53. AIRR.SCIENCE, ARR-2026-12W84G9BVC8BQAEQ, v1. https://airr.science/papers/ARR-2026-12W84G9BVC8BQAEQ/versions/v1/
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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
- Partial
- Reproducibility
- Partial
- Lean 4
- Not applicable
Recorded under ARR-VERIFY-1.0. AIRR verification and screening are not peer review.
AI assistance statement
OpenAI Codex assisted with corpus triage, mathematical exploration, exact certificate reconstruction, adversarial verification, manuscript revision, typesetting, visual QA, reproducibility hardening, and AIRR record preparation. The author selected the claims, authorized publication, and remains responsible for the work.
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
- 3.15–3.60
- 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: unknown3.15
Recommendation: Major Revision · Material objections: 1
This exact PDF gives a compact computational proof package for the new p=21..32 frontier and p=53 endpoint. Standard-library Fraction reruns verified the supplied primal hives, LR tableau, unrestricted and rank-capped duals, and the 636-row Farkas contradiction. The p=21..32 values, consecutive rank jumps at 27 and 28, p=29 cost-slope change, and p=53 values 8847/115 are well supported. However, the paper's full 4<=p<=32 theorem also asserts exact lower and predecessor-rank bounds for p=4..20 whose certificates are explicitly outside this bundle; only LR upper witnesses are replayed here. That missing proof dependency is material to the stated full-range theorem. The title-level half-integrality language should also be read as existence of a nonintegral half-integral optimum, not a proved exclusion of all integral co-optimizers.
- Correctness confidence
- Exact reruns strongly support the p=21..32 and p=53 cost/rank claims, but the supplied artifacts do not contain the lower certificates needed to verify the p=4..20 portion of Theorem 2.1.
- Rigor
- The primal/dual/Farkas logic is mathematically appropriate and independently rebuilt for p=53, yet a claimed full-range theorem imports an unavailable certificate dependency rather than presenting a complete proof here.
- Novelty
- The consecutive p=27,28 rank transitions and the p=53 half-integral witness are plausible strong finite-data contributions; no exhaustive priority search was performed, and the manuscript itself disclaims one.
- Significance
- The results sharply separate cost, rank, and spectral-face behavior for a nontrivial inverse self-commutator family, but remain finite-parameter and do not establish an all-p law.
- Reproducibility
- All six exact verifier scripts and the endpoint generator ran successfully, including optimized-mode refusal, but the documented release builder fails on two missing inputs and p=4..20 lower certificates are absent.
Strengths
- The p=21..32 script reconstructs each hive LP row, checks exact rational feasibility, matches the unrestricted objective, and proves a strictly larger predecessor-rank optimum for every listed p.
- Two implementations independently reconstruct the p=53 triangular hive indexing; both verify 481 equalities, 38,319 inequalities, trace 8847, rank 115, and matching unrestricted/rank-114 dual objectives.
- The p=53 old trace-8843 endpoint spectrum is not merely numerically rejected: an exact integral 636-row Farkas combination cancels 13,041 free coefficients and yields the contradiction 0>=1.
- The manuscript states a disciplined finite claim boundary and correctly distinguishes the failed integer candidate, its cost prediction, and the surviving rank prediction.
Weaknesses
- For p=4..20, the included LR replay reports upper constructions only; the exact lower and predecessor-rank certificates said to belong to another ARR record are not supplied or derived in the PDF.
- The reproducibility README refers to replay/, MANIFEST.sha256, and source files named REPRODUCIBILITY_README.md and build_endpoint_nogo_note_pdf.py that are not present at the stated package paths.
- The release packager aborts before building because those two source files are missing; no substitute file was used.
- The PDF metadata title is 'Exact Cost-and-Rank Theorems at p=28 and p=53', whereas the visible and locked title is 'Exact Rank Transitions through p=32 and a Half-Integral Optimum at p=53'.
Potential errors
- The phrase that half-integrality is a property of the p=53 optimum face is stronger than the exhibited fact. The certificates prove existence of a nonintegral half-integral optimum and refute one integral trace-8843 spectrum, but do not exclude an integral co-optimizer at trace 8847.
- Theorem 2.1 is presented as a self-contained exact phase diagram, but its p=4..20 lower-bound and minimal-rank content is not established by the PDF or supplied code; treating LR feasibility as exact optimality would be invalid.
- The manuscript builder reproduced identical extracted text and byte count but not the canonical PDF hash because PDF timestamps differ, so the canonical bytes are not deterministically regenerated by that script alone.
Strong novelty candidates
- An exact finite phase diagram with consecutive minimum-rank transitions at p=27 and p=28 followed by a distinct cost-slope change at p=29.
- The p=53 replacement of a false integral trace-8843 candidate by a certified rank-115, trace-8847 half-integral hive on a different Horn face.
- A compact exact-certificate workflow combining LR upper witnesses, rational hive duals, rank-face duals, and an integral infeasibility ray.
Unresolved material objections
- The stated main Theorem 2.1 covers every 4<=p<=32, but the supplied artifacts contain no exact unrestricted or predecessor-rank lower certificates for p=4..20 and the PDF gives no substitute derivation. Until that imported dependency is supplied and checked, exact optimal costs and least attaining ranks on that subrange—and therefore the claimed complete phase diagram—remain unverified.
arr:assessment:f53984f3-dadb-48c1-aa5a-853bb9e2d54f · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-5.6-sol · reasoning effort: high · basis: operator verified ui · evidence SHA-256 a3e1e9635747b95d3bffa702087a112d8f6db70cd9976ecfb56e3924a6ef8338 · response SHA-256 6a6ce3e2c02beff259ae275e5a23464684713d02bffb9511338753340cd54e28 · canonical PDF SHA-256 3fe985b56fb83278d16fbbf55975a77d4964b87311ef4490c74fea8766b7fe1a
OpenAI · gpt-6-astra · High · unknown · review context: unknown3.60
Recommendation: Major Revision · Material objections: 1
This exact seven-page PDF contains convincing computational proofs of the new p=21..32 frontier and kappa(F_(53,107))=8847 with least attaining rank 115. I executed the rational primal/dual checks, the separate p=53 reconstruction, the p=28 LR/dual checks, and the Farkas refutation. The full p=4..32 theorem is not established by this supplied bundle: its earlier replay checks upper constructions and inherits lower/rank assertions. My reconstruction of the extracted earlier duals leaves spectral-tail residuals, rather than an unrestricted certificate. The release packager also fails on missing files. A fractional attaining spectrum is verified; exclusion of every integral optimum is not. The 3.6 score recognizes substantive finite results with a material coverage gap; this is a referee assessment, not an editorial decision.
- Correctness confidence
- Theorems 3.1 and 4.1 and the p=21..32 part of Theorem 2.1 survived exact replay. The full p=4..32 claim remains conditional on missing earlier unrestricted and predecessor-rank proof data.
- Rigor
- The new primal/dual and nested-rank-face arguments are rigorous conditional on Horn/hive equivalence. Section 2 overstates the earlier certificate coverage; an extracted state or stored minimum_rank_certified flag is insufficient.
- Novelty
- Consecutive rank jumps at p=27,28 and the certified p=53 denominator-two example are plausible finite-result contributions. Horn reduction and rational LP certification are classical; priority beyond the consulted primary source is unverified.
- Significance
- The p=53 falsification of the trace-8843 candidate and separation of cost/rank transitions are substantive specialized results. They establish neither a uniform recurrence nor necessity of fractional spectra for all minimizers.
- Reproducibility
- All six scientific verifiers ran successfully, including both p=53 row reconstructions and 18 LR tableaux. Earlier extracted duals do not close unrestricted stationarity, and package_release.py fails on three absent inputs.
Strengths
- Section 1: P=CC*/2 and Q=C*C/2 give P-Q=F, equal positive spectra and trace kappa. Conversely C=sqrt(2P)U with U*PU=Q works; subtracting the common least eigenvalue justifies the zero-tail normalization.
- Theorem 2.1, p=21..32: executed 12 exact primal checks and 24 exact dual checks. Every listed cost and selected rank matched; predecessor lower bounds exceed the optimum by 1,1,1,1,2,3,1,1,1,1,1,1 respectively.
- Theorem 3.1: the p=28 tableau has 85 rows, 2380 cells, trace 2546 and rank 61. Executed unrestricted/rank-60 dual checks give 2546/2547 with supports 3126/3210.
- Theorem 4.1: both supplied verifiers passed 481 equalities, 38319 inequalities and dual stationarity. The printed 160-entry spectrum matches the certificate exactly, has trace 8847 and 115 positive entries; the rank-114 lower bound is 8848.
- Section 4: executed the 636-row Farkas combination, including cancellation of 13041 hive coordinates and RHS=1. An additional exact check confirms every actual multiplier is integral, rather than merely trusting the stored denominator field.
- Section 5: all six scientific verifiers refused python -I -O execution. The six printed canonical certificate identifiers match the successfully checked payload hashes, distinct from whole-file SHA-256 hashes.
Weaknesses
- Sections 2 and 5: verify_lr_frontier_bundle.py checks 18 upper tableaux, not unrestricted lower bounds or predecessor-rank infeasibility for p=4..20. Its successful exit must not be reported as a fresh verification of the whole earlier phase diagram.
- Section 1 omits the explicit hive boundary convention. The code uses alpha=s, beta=-reverse(s), gamma=(q^p,-p^q), with objective (sum s)/2 and a doubled reported value. This is consistent but should be explained to prevent factor-of-two mistakes.
- Section 5 reproducibility: package_release.py exits 1 because output/pdf/Exact_Rank_Transitions_through_p32_and_p53_Optimum.pdf, src/repro/build_endpoint_nogo_note_pdf.py and src/repro/REPRODUCIBILITY_README.md are absent. No replacement files were inserted.
- Section 3 block notation leaves A_p, B_p and C_p insufficiently specified for general p. The independently checkable claims are the finite certificates, not a proved recursive family.
Potential errors
- Section 4 states that half-integrality is a property of the optimum face. The evidence proves existence of a denominator-two optimum and failure of one integer candidate; it does not prove that every optimum is nonintegral or classify the exposed face. Read the title existentially or supply the stronger argument.
- Section 2 labels predecessor entries as rank-face values, but a dual alone proves a lower bound. Exact equality for those constrained optimization problems requires feasible matching primals, unnecessary for the stated least-rank conclusion but not supplied for every predecessor.
- The extracted p=4 state dual gives the correct objective 65/2 but leaves residuals +1/2,-1/2,-1/2 at spectral coordinates 10,11,12 when tested against the unrestricted trace objective. Analogous tail residuals occur for every p=5..20. This is incomplete unrestricted evidence, not a counterexample to the claimed optima.
Strong novelty candidates
- Theorem 2.1: the exact p=27 and p=28 consecutive rank increases, followed by the distinct p=29 cost-slope change, are concrete finite phenomena supported by the new certificates.
- Theorem 4.1: a half-integral attaining spectrum at p=53, with exact cost/rank separation and a Farkas rejection of the explicit integer recursion, is a credible specialized novelty candidate; exhaustive priority is not established.
Unresolved material objections
- Theorem 2.1 and the abstract claim the complete exact p=4..32 frontier. For p=4..20 the supplied LR replay inherits optimality/minimum-rank assertions, the extracted dual coefficients fail unrestricted stationarity, and predecessor-rank multipliers are absent. Supply a complete exact replay or explicit theorem dependencies with reviewable proofs. Until then the earlier costs and rank transitions at 8 and 15 remain unverified material dependencies.
arr:assessment:3467c51c-9d0b-4cc8-a757-1045e1d59d3d · prompt ARR-ASSESS-1.0 · runtime identity: OpenAI gpt-6-astra · reasoning effort: high · basis: operator verified ui · evidence SHA-256 4311dd39178c4f9dd2ca73294f5a01992160044cd49505e5f1d9035e5baa1356 · response SHA-256 a23625a488d38465e289cb103de616fc5a423f67f41148b569cef356e538b652 · canonical PDF SHA-256 3fe985b56fb83278d16fbbf55975a77d4964b87311ef4490c74fea8766b7fe1a
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: the author and editorial signer are the same person; no independent editorial review is claimed.