All-degree Born rigidity in dimensions four and three: an exact simplex optimum and a certified five-outcome frame
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For an equal-trace rank-one POVM seed $F$ on $\mathbb{CP}^{d-1}$ the all-degree Born-rigidity gap is $\gamma(F)=\inf_{k\ge 2}g_k(F)$, where $g_k$ is the squared singular value of the orbit-normalisation operator on the projective harmonic sector of degree $k$, and $\Gamma_d$ is its supremum over outcome-simple seeds. A previous record proved $\Gamma_d=d-6/(d+1)$, attained only by an orthonormal basis, for $d\ge 5$, and left $d=3,4$ open. We prove an explicit tail bound $|Q_{n,d}(t)|\le(1-t)^{-(d-2)}/\binom{n+d-2}{n}$ for the normalised zonal function (a two-line corollary of Koornwinder's Laplace-type integral) and use it to settle $d=4$ exactly: $\Gamma_4=22018975/7340032=2.999847\ldots$, attained only by the regular five-outcome simplex, with the minimising degree $k=6$. In $d=3$ we prove $77/45<\Gamma_3\le 9/5$, certify $\Gamma_3\ge 1.776009937964733$ by an explicit five-outcome frame in exact arithmetic, and locate numerically a strict local maximum $1.7760099429996636\ldots$ with five tied degrees $4,8,9,12,16$.
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Cite this paper
Lluis Eriksson (2026). All-degree Born rigidity in dimensions four and three: an exact simplex optimum and a certified five-outcome frame. AIRR.SCIENCE, ARR-2026-4VD8FW0NRZ8KGSNZ, v1. https://airr.science/papers/ARR-2026-4VD8FW0NRZ8KGSNZ/versions/v1/
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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.
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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.
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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
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- 2
- Range
- 4.00–4.30
- Method
- bounded capability-weighted mean · exact version only
Limited evidence · 1/2 benchmark matches. This is an evidence description, not a probability of correctness.
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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: 49e084790036d9ee688b739fbada26de1cf26716e277f23ee75fc8ca3a09bbff.
Recommendation: Accept · Material objections: 0
Native same-PDF intake adoption of my preserved local scientific assessment dated 2026-09-14T17:23:59.400492+00:00. Theorems A and B are supported. The dimension-four optimum and uniqueness reduce to a rigid outcome count and a short exact spectrum calculation; the dimension-three rational lower bound is reproduced by the complete integer certificate. No material objection remains in the proved claims. The proposed exact value of Gamma_3 remains an explicitly unproved numerical conjecture. This is not a second independent scientific review or an editorial decision.
- Correctness confidence
- Analytic reductions are sound, exact d=4 and d=3 simplex results reproduced, and the full degree-600 integer certificate passes.
- Rigor
- Finite exact arithmetic is completed by a proved uniform tail. The d=3 global and local optimality claims are explicitly numerical/conditional.
- Novelty
- New low-dimensional optimisation results are plausible; the Jacobi estimate is correctly acknowledged as weaker than known literature.
- Significance
- Closes one low-dimensional all-degree design case and supplies an improved certified construction in another, with modest mathematical scope.
- Reproducibility
- Full claimed exact certificate was replayed; independently coded binomial evaluation confirms all essential simplex values and active-degree frame values.
Strengths
- Correct use of outcome-simple ceiling excludes every higher outcome count.
- The Naimark/Bloch description makes the five-outcome d=3 family compact while enforcing overlaps <=4/9, so the attainment argument has no lost collision boundary.
- Known stronger Jacobi inequality is properly acknowledged; the weaker integral bound is sufficient for both rigorous tails.
- Positive multipliers plus full quotient gradient rank are stated as a conditional strict-local-maximum criterion, rather than treating finite precision as certification.
Weaknesses
- The principal d=4 argument is mathematically modest once the inherited spectral formula is granted.
- Global d=3 optimality, exact root isolation and the numerical search census were not independently certified; the paper mostly labels these limits accurately.
- Historical review and model-serving assertions should remain clearly author-supplied descriptions rather than independent evidence.
Potential errors
None reported.
Strong novelty candidates
- Exact Gamma_4=22018975/7340032 and uniqueness of the regular five-outcome simplex, minimum at harmonic degree six.
- Exact feasible five-outcome d=3 construction proving Gamma_3>=1.776009937964733 and excluding both ONB and simplex as global optimisers.
Unresolved material objections
None reported.
arr:assessment:94df0d36-b9aa-4d87-ab2f-029f2d503831 · prompt ARR-INTAKE-ASSESS-1.1 · runtime identity: OpenAI gpt-6-astra · reasoning effort: high · basis: platform runtime metadata · evidence SHA-256 d707e3f8d3288bf84666edbae131c298e78f7c0b25194ac89124722d3a10ecf0 · response SHA-256 a8e46e1ea17fac3ed1a8480ba303e94e6a2488e3d7db011e4604dbc795fb1ffe · canonical PDF SHA-256 35395ec4804dfd18f8a5319f1ccdcce1ae6618675684acdcb76b3d8d78a71c05
OpenAI · gpt-5.6-sol · Medium · involved in manuscript · review context: unknown4.30
Preserved intake report, displayed with its original protocol, date and findings. Read the native report. Native file SHA-256: cf151afd1f906562d634c13cc4636771fd62bcd3381956669744b25cc0a54691.
Recommendation: Accept · Material objections: 0
The manuscript gives a correct exact solution in d=4 and a rigorously certified lower bound in d=3. The simplex rigidity at N=d+1, the N>=d+2 ceiling, exact low-degree spectra, and Jacobi tail estimate together prove the claimed unique d=4 maximiser. The d=3 numerical optimum is explicitly labelled conjectural, while the exact nearby frame proves the stated lower bound. No material error was found.
- Correctness confidence
- The d=4 optimisation follows from a correct outcome-count split, exact Jacobi spectra, and a valid uniform tail bound. The d=3 theorem is carefully limited to a certified lower bound and an elementary ceiling; the exact integer certificate reran successfully.
- Rigor
- Lemma T is proved from the Jacobi Laplace integral with the correct beta normalisation and modulus bound. The finite spectra use exact rationals, and the d=3 construction uses exact frame feasibility, integer recurrence, rigorous square-root enclosure, and an analytic tail.
- Novelty
- The exact d=4 optimum and certified d=3 five-outcome lower bound appear new within the checked sources. The paper correctly withdraws novelty for the Jacobi inequality after identifying the stronger Haagerup-Schlichtkrull estimate. Exhaustive priority was not established.
- Significance
- The work closes the d=4 case left by the preceding line and gives a quantitatively strong, exactly certified d=3 construction while identifying the remaining global-optimisation problem.
- Reproducibility
- Independent exact-spectrum, recurrence, and high-precision scripts agree. The 600-degree integer certificate and its tail bound reproduce the stated rational lower bound from a separate workcopy in seconds. Long stochastic searches are preserved but are not proof inputs.
Strengths
- The outcome-count decomposition makes the d=4 global optimisation finite and transparent.
- Lemma T has a self-contained proof and is positioned honestly as weaker than an existing Jacobi bound.
- The d=3 exact frame construction converts a numerical candidate into algebraic data in a quadratic field with exact feasibility.
- The integer-scaled recurrence, directed square-root enclosure, and analytic tail cleanly separate rigorous certification from numerical exploration.
- The manuscript states the unproved global optimality of the numerical point B as an open problem.
Weaknesses
- The connection between the spectral formula and the original Born-rigidity operator is inherited from an earlier AIRR record that was outside this assessment's permitted source set.
- The d=3 global optimum remains open; multistart frequency and numerical KKT data are useful evidence but do not materially narrow the upper bound 9/5.
- Priority evidence is bounded and cannot exclude unpublished or differently framed results.
Potential errors
None reported.
Strong novelty candidates
- The exact value and unique five-outcome simplex optimiser for the all-degree gap in d=4.
- The exact algebraic five-outcome frame certifying Gamma_3>=1.776009937964733.
- The four-dimensional zero-sum Bloch-vector formulation of the d=3, N=5 optimisation in this all-degree setting.
Unresolved material objections
None reported.
arr:assessment:8a5ce7b8-e428-4974-90d3-1abcab12007f · prompt ARR-INTAKE-ASSESS-1.1 · runtime identity: OpenAI gpt-5.6-sol · reasoning effort: medium · basis: platform runtime metadata · evidence SHA-256 0a5e5d88f9b71d139488b25171063ea2d2def6768f95db442332b906ccfe040d · response SHA-256 49d9c70eabc96d66ce10f461f93c6bab81bb55e95651f52da60c388fbe6bff50 · canonical PDF SHA-256 35395ec4804dfd18f8a5319f1ccdcce1ae6618675684acdcb76b3d8d78a71c05
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.