Exact Hive Certificates for the Balanced-Inertia Stability Constants (N = 3 to 12) in Every Dimension, the $(A_N,B)$ Cost Law, and the Twenty-Two-Form Six-Level Cost Formula
Admitted after checks by named AI models and editor approval. Not peer reviewed. Screening record
For a traceless Hermitian $F$ with $N$ positive, $N$ negative and $d-2N$ zero eigenvalues, the inverse self-commutator cost $\kappa_d(F)$ satisfies $\delta_d\ge\gamma_{N,d}D_*$ with an optimal rational constant $\gamma_{N,d}$; the values $\gamma_2,\dots,\gamma_5$ were known and $\gamma_6,\dots,\gamma_{10}$ only numerically. We give exact, independently machine-checked certificates proving $\gamma_{N,d}=\gamma_N$ for $N=3,\dots,12$ and every $d\ge2N$: $17/36$, $4/7$, $113/152$, $9/10$, $781/740$, $317/264$, $23/17$, $3/2$, $773/468$, $289/160$. The minimising pairs switch at $N=7$ from $(A_N,B)$ to $(u_3,z_N)$. The certificates are exact rational hive primals at every positive-distance vertex of the finite reduction and matching duals at the listed minimisers, and dimension-independent Horn combinations (Weyl, two size-two and Lidskii–Wielandt inequalities) at the minimisers, written out in full. We also prove the $(A_N,B)$ cost law for $N\le14$ and the exact six-level formula $\kappa_6=\max$ of twenty-two explicit linear forms, all exposed. The supplied package reports a separate adversarial replay; fresh checks using separately written code are documented in Section 7, with provenance limits stated explicitly.
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Cite this paper
Lluis Eriksson (2026). Exact Hive Certificates for the Balanced-Inertia Stability Constants (N = 3 to 12) in Every Dimension, the $(A_N,B)$ Cost Law, and the Twenty-Two-Form Six-Level Cost Formula. AIRR.SCIENCE, ARR-2026-5B1F02F3EG9ZB9NN, v1. https://airr.science/papers/ARR-2026-5B1F02F3EG9ZB9NN/versions/v1/
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Version history
The paper identifier remains stable. Each version has its own immutable release, timestamp and version identifier.
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Verification record
- Frontier-model screening
- Pass · 2 models
- Source integrity
- Pass
- Bibliographic integrity
- Partial
- Reproducibility
- Partial
- Lean 4
- Not applicable
Recorded under ARR-INTAKE-ASSESS-1.1. AIRR verification and screening are not peer review.
AI assistance statement
The author supplied these manuscripts as developed with Claude Fable 5.1; historical serving identity is not independently authenticated. Codex assisted bounded revisions. Actual GPT-6 Astra High and GPT-5.6 Sol Medium reports concern this exact PDF, and each model family declares prior involvement. Intake bindings adopt the prior same-hash scientific reviews; they are not additional independent reviews. For B3, A4 and C2, a later Astra context explicitly adopted the preceding Astra report. Technical memory and filesystem isolation are not certified.
Frontier-model screening
Status: pass. Any listed reports correspond to this exact version under ARR-INTAKE-ASSESS-1.1; no absent assessment is represented as a pass.
- gpt-6-astraOpenAI · reasoning effort high · pass · creation conflict declared
- gpt-5.6-solOpenAI · reasoning effort medium · pass · creation conflict declared
Model assessments and review context
- Distinct models
- 2
- Range
- 4.00–6.00
- 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; Review scores differ by more than one point
Review input clarification
Two exact-PDF native model reviews were adopted into the registered intake, one per model. Original local reports and intake binding provenance remain preserved; binding does not add another scientific review. Both model families participated in revision. One provider and incomplete benchmark matching limit the evidence; the weighted score is not a probability of correctness. A separate disclosed founder-author editorial decision and exact public permission were recorded. The intake Astra context read and explicitly adopted a preceding Astra report; it was not a fresh blind review.
Operator record · · Source files and affected reports
OpenAI · gpt-6-astra · High · involved in manuscript · review context: unknown4.00
Preserved intake report, displayed with its original protocol, date and findings. Read the native report. Native file SHA-256: 445935ab7164e8ce84e70cf25e073fcf4c21eff46ffd757b772f470fdaed5e53.
Recommendation: Accept · Material objections: 0
I adopt the preceding Astra assessment after reading the exact candidate and report, re-deriving its certificate implications, and running targeted new exact checks. The balanced stability constants, bounded AB cost law and 22-form six-level formula are supported. Original accept and score 4 are retained; no material disagreement was found. This adoption newly checks 522 Horn triples, 22 duals, 112 listed ray witnesses and 12 all-d files. The larger hive and finite-set replay remains evidence from the preceding agent, with its original limits preserved. The N=8 reflected-Weyl wording remains a minor presentational finding. This is a current-context intake adoption after reading a preceding Astra report, not fresh blind independent review or output of the original agent.
- Correctness confidence
- Adopted preceding Astra basis; earlier full replays are not new runs: Full exact submitted-verifier replay passed; independently reconstructed finite sets, LR triples, chamber geometry, global duals, ray witnesses and all-d inequalities also passed.
- Rigor
- Adopted preceding Astra basis; earlier full replays are not new runs: Correct finite-certificate implications and padding argument; assumptions and inherited finite reduction are explicit. A short self-contained proof of the finite reduction would improve readability.
- Novelty
- Adopted preceding Astra basis; earlier full replays are not new runs: New exact certified constants and six-level formula appear credible relative to inspected sources; hive/Horn machinery and general certificate method are established. Historical AIRR priority comparisons were not independently authenticated.
- Significance
- Adopted preceding Astra basis; earlier full replays are not new runs: Nontrivial finite stability and norm-cost classification useful within this spectral optimization program; no general-N classification yet.
- Reproducibility
- Adopted preceding Astra basis; earlier full replays are not new runs: 47,685 vertex primals, 20 minimizer duals, 34 additional primal-dual entries, 22 chambers, 112 witnesses and 12 all-d files replayed exactly. Independent GMP/LR implementation cross-checks critical enumeration steps.
Strengths
- Exact rational primal/dual evidence replaces numerical optimization as the proof object.
- Dimension-independent claims are supported by actual affine template/structural Horn arguments, not finitely many padded examples.
- The complete six-level formula is proved by global lower forms and exact witnesses on all independently reconstructed chamber rays.
- The finite range of the cost law and uncertainty about uniqueness and general N are clearly stated.
Weaknesses
- The finite reduction is imported with the explicit set delegated largely to code. A brief convexity proof and a code-independent description of the ordered-simplex cells would make the article more self-contained.
- The specialized numerical extensions do not establish general-N behavior or unique minimizers.
- Historical model identities, run claims and private antecedent priority were not authenticated in this assessment.
Potential errors
- Minor wording in Section 5, N=8: the N=7 line is already reflected W_7, so say replace reflected W_7 by reflected W_7 plus reflected W_8. The displayed inequality and certificate are correct.
Strong novelty candidates
- All-d exact values gamma_N for N=6 through 12, especially new N=11,12 entries, subject to priority limits.
- The fully certified maximum of 22 exposed linear forms for kappa_6.
- Short explicit all-d Horn combinations for the minimizing families.
Unresolved material objections
None reported.
arr:assessment:0f8f5e54-9541-4e3b-9cc1-b9b172d0fc67 · prompt ARR-INTAKE-ASSESS-1.1 · runtime identity: OpenAI gpt-6-astra · reasoning effort: high · basis: platform runtime metadata · evidence SHA-256 3042804fac7b69091a1a970861e028c9649f8dfb249ccce9625932a7fda2492d · response SHA-256 f56d348c6004cd02fdf31c05262ef2af80bcfffa4cddeec93d2a6dd194fa3e14 · canonical PDF SHA-256 a725d4f88c947f88c03882a6b2e9d663f4c78134a8c5bdc052b240525ebf223b
OpenAI · gpt-5.6-sol · Medium · involved in manuscript · review context: unknown6.00
Preserved intake report, displayed with its original protocol, date and findings. Read the native report. Native file SHA-256: e6c58af52de97813df2819db3b60c3fc199f5a9c8ea0708b866c9dc8b88c7bed.
Recommendation: Accept · Material objections: 0
A rigorous computer-assisted paper with unusually transparent exact certificates. The main results follow from a coherent chain: finite reduction, exact hive upper bounds, matching dual lower bounds at minimizers, padding monotonicity, stable Horn templates for every larger dimension, and conic chamber certificates for the dimension-six formula. Fresh replay found no certificate or logic failure.
- Correctness confidence
- The certificate architecture matches the logical requirements of all three theorems. Fresh exact replay verified every supplied primal, dual, all-d Horn combination, and chamber-ray certificate with zero failures. Independent explicit-d and Horn-vs-hive crosschecks also passed.
- Rigor
- The manuscript specifies the imported results, proves how each finite certificate implies the advertised statement, distinguishes upper bounds from exact values, and exposes all dimension-independent inequalities. Exact arithmetic is used where the theorem depends on computation.
- Novelty
- The Horn/hive foundations are classical, but targeted searches did not locate the reported exact stability constants through N=12, the twenty-two-form kappa_6 law, or these compact all-d certificate families outside the author's AIRR sequence. Novelty is therefore strong within a specialized line, subject to the limits of a targeted search.
- Significance
- The work converts previously numerical constants into exact theorems, extends the range to N=12, and gives a complete polyhedral formula in dimension six. This is substantial for the inverse self-commutator program, though the subject is specialized and several general-N patterns remain conjectural.
- Reproducibility
- The exact PDF hash matched. All relevant verifier code was inspected before execution; it loads JSON/gzip JSON only and performs no unpickling. The full standard-library exact verifier completed in 297.7 seconds with TOTAL: 0 failures, and two additional independent checks completed successfully.
Strengths
- Every computational claim is reduced to exact rational identities and inequalities rather than floating-point evidence.
- The all-d argument is short enough to inspect by hand and is independently instantiated across multiple dimensions.
- The twenty-two chamber formula includes global duals, ray witnesses, full-dimensionality checks, and exact coverage logic.
- The paper candidly separates proved finite ranges from conjectural general-N patterns.
Weaknesses
- Theorem 1 depends on a finite-reduction theorem from an earlier AIRR paper; that antecedent was not reopened under this assessment protocol.
- The certificate corpus is large, making independent human verification impractical despite the compact verifier.
- Novelty and significance are concentrated in a specialized inverse self-commutator research program with limited external literature.
Potential errors
None reported.
Strong novelty candidates
- Exact balanced-inertia stability constants for N=6 through N=12, including the switch of certified minimizer family at N=7.
- The exact maximum-of-twenty-two-linear-forms formula for kappa_6 with every form exposed.
- Compact dimension-independent Horn certificate skeletons proving minimizer costs in all ambient dimensions.
Unresolved material objections
None reported.
arr:assessment:aa326336-8720-482f-a1c3-5cd7d71d867a · prompt ARR-INTAKE-ASSESS-1.1 · runtime identity: OpenAI gpt-5.6-sol · reasoning effort: medium · basis: platform runtime metadata · evidence SHA-256 daaf08838eaa34394aeed033505de2abcf246d1f884bafbb96ea617de6429700 · response SHA-256 033ca0c64a8ef45c2d622d23f111c2b6ecf6a7e66359135405f95768264377f7 · canonical PDF SHA-256 a725d4f88c947f88c03882a6b2e9d663f4c78134a8c5bdc052b240525ebf223b
A model assessment is not peer review or a correctness certificate. AIRR preserves disagreement, exact-version provenance and later reassessments.
Editorial disclosure
The sole author and founder-editor expressly authorized the complete cycle. The authenticated Netcup operator recorded that human instruction separately as an editorial acceptance after two exact-PDF model reports and evidenced minor dispositions, then recorded exact public permission. No independent human peer review or simulated web signature is claimed.