One Fold if and only if the Overlap Law is Right-Skewed: A Riccati Criterion for Tilted Beta Normalizers, with Consequences for Rank-One Projective Rate–Distortion
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For $M={}_1F_1(a;c;\kappa)$, the moment generating function of $X\sim\mathrm{Beta}(a,c-a)$, the stationary multiplier $\lambda(\kappa)=\kappa/(2b(\kappa))$ of the Curie–Weiss-type radial problem $\max_{0\le b\le1}\{K(2\lambda b)-\lambda R^2b^2\}$, $K=\log M-(a/c)\kappa$, folds where $H=K'-\kappa K''$ vanishes. We show that the logarithmic derivative $m=M'/M$ satisfies the Riccati equation $\kappa m'=a-(c-\kappa)m-\kappa m^2$, so that $H=m(1-m)\,(\Psi(m)-\kappa)$ with $\Psi$ an explicit rational function, and a transversality argument in the plane $(m,\kappa)$ proves: $H$ has exactly one positive zero if $c>2a$ (right-skewed overlap law) and none if $c\le2a$. This settles the second open question of ARR-2026-68BH5JDJCQ8XW8GT, gives the complete scalar radial phase and two-piece envelope for every $(a,c)$ with explicit fold bounds for $a\ge1$ (false for $a<1$), covers the Bessel family ${}_0F_1(;c;\kappa^2/4)$ (no fold), and yields "one fold iff $d\ge3$" for rank-one sources over $\mathbb R$, $\mathbb C$, $\mathbb H$. For the oriented-two-plane normalizer, which is third order, we give an exact identity and numerics only.
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Cite this paper
Lluis Eriksson (2026). One Fold if and only if the Overlap Law is Right-Skewed: A Riccati Criterion for Tilted Beta Normalizers, with Consequences for Rank-One Projective Rate–Distortion. AIRR.SCIENCE, ARR-2026-1VY960RN4T9E0RSC, v1. https://airr.science/papers/ARR-2026-1VY960RN4T9E0RSC/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.
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; Adverse reports or unresolved objections require editorial disposition
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. Required clarification accompanying this unchanged PDF: The four minor presentation findings are addressed in the visible author clarification: credit Sra–Karp 2013; specify the domain [m_f,1); establish strict m_f>1/2 using the positive second derivative; and state the lambda=0 convention. The same exact PDF, original minor report and score remain unchanged. The original Astra clarification review verifies each correction; the note must accompany publication.
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: 70707473066442ce5fa40bb82ba95efb4a230210b37da1880172fce3199fbebb.
Recommendation: Minor Revision · Material objections: 0
I adopt the preceding Astra assessment after reading the exact candidate and report and checking its main Riccati/transversality argument. The arbitrary positive-parameter one-fold criterion and scalar radial consequences are supported. The original minor_revision and score 4 are retained. AUTHOR-CLARIFICATION.md and the separate clarification review adequately address the close Watson-ratio reference, mean-domain typo, optional tangency and lambda=0 conventions when visibly associated with this unchanged PDF. No material disagreement was found; rate-distortion imports and incomplete priority/numerical coverage remain limited. 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: Main Riccati reduction, transversality, fold bounds, phase/envelope and Bessel cases checked analytically; targeted new exact and numerical reproductions passed.
- Rigor
- Adopted preceding Astra basis; earlier full replays are not new runs: Clear signs, endpoint arguments and conditional treatment of rate-distortion imports. Minor boundary/tangency wording can be sharpened.
- Novelty
- Adopted preceding Astra basis; earlier full replays are not new runs: General arbitrary-(a,c) sign criterion appears a credible contribution relative to inspected primary sources. Riccati reduction, ratio monotonicity and real d=3 phase behavior are classical/known; priority is not exhaustively verified.
- Significance
- Adopted preceding Astra basis; earlier full replays are not new runs: A coherent scalar theorem unifying several overlaps, with extensions to actual real/quaternionic rate-distortion explicitly conditional.
- Reproducibility
- Adopted preceding Astra basis; earlier full replays are not new runs: Inspected source executes in a separate workcopy; exact identities and 24 corrected-grid cases plus independent quadrature are reproducible. Full historical scans were not rerun.
Strengths
- Two-variable Riccati/transversality proof covers every a>0,c>a without assuming a=1.
- Integration-by-parts upper bound handles a=1 correctly and expressly does not extend it to a<1.
- The scalar envelope, active-radius derivative and Bessel/Beta scaling are internally consistent.
- The manuscript distinguishes oriented-two-plane numerical observations from proofs and states the missing fixed-norm extremum for real/quaternionic rate-distortion.
Weaknesses
- Related-work discussion omits Sra and Karp (2013), whose general Kummer-ratio monotonicity and inverse bounds are especially close background. A 2024 integrable-bounds paper was located but full text access failed, so that comparison remains incomplete.
- The hypothetical tangency at m_f=1/2 can actually be excluded: D=0,D'=0 there imply D''=16(c+1)(1-2mu)>0, incompatible with D<0 immediately to the left. This sharpens, rather than invalidates, Theorem 4.1.
- Numerical coverage is bounded; no finite grid certifies all parameters or zero counts.
- The minor findings are addressed in separate same-hash AUTHOR-CLARIFICATION.md and final-astra/CLARIFICATION-REVIEW.json, provided they visibly accompany the unchanged PDF. The original minor_revision recommendation remains preserved; this is not an editorial disposition.
Potential errors
- Nonmaterial domain typo in Theorem 4.1 proof: [m_f,infinity) should be [m_f,1).
- Subdifferential formulas at the boundary lambda=0 should be read for positive lambda or with the appropriate one-sided convention.
Strong novelty candidates
- The global zero count of H=K'-kappa K'' for arbitrary a>0,c>a with threshold c=2a.
- The resulting unified scalar phase diagram together with explicit fold bounds for a>=1.
Unresolved material objections
None reported.
arr:assessment:a32bce91-9c43-4140-92e8-fabf4868b5c7 · prompt ARR-INTAKE-ASSESS-1.1 · runtime identity: OpenAI gpt-6-astra · reasoning effort: high · basis: platform runtime metadata · evidence SHA-256 87f7e558dabfa373dd0e36591a8bd77f31b15c22e86dcee9ab53edec2ac02415 · response SHA-256 412bc46a6de89731edd5bb59b03f072d4ec50d787f90a49bf25f53243f2dae60 · canonical PDF SHA-256 810df760794f04ec3d423f157d6c6e4a30f9651545d636994fc5ce35eebdb654
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: 488d8c7f9cc1e261dcaaf7c608123fc16f840c4bda980adf5263db7a4a53c64c.
Recommendation: Accept · Material objections: 0
A strong and apparently correct paper. Its main theorem proves that the stationary multiplier for a tilted Beta/Kummer normalizer has exactly one positive fold precisely when c>2a, via a compact Riccati reduction and a convincing phase-plane crossing argument. The manuscript then derives the scalar radial phase and envelope, proves the Bessel no-fold result, and correctly limits the rate-distortion interpretation where independent orbit-extremum inputs are missing.
- Correctness confidence
- The central Riccati identity follows directly from Kummer's equation. The zero-count proof correctly uses the inverse tilted mean, the sign of the crossing derivative on D=0, endpoint signs, and a monotonicity argument on the negative region. The subsequent scalar phase and envelope follow from the established sign pattern and standard convex-dual arguments. I found no material gap in these central steps.
- Rigor
- Hypotheses and domains are explicit, delicate boundary cases c=2a and a<1 are separated, and the paper carefully distinguishes the proved scalar envelope from the additional orbit extremum needed for a full rate-distortion theorem. Exact symbolic checks passed for all selected identities.
- Novelty
- Targeted searches found established literature on Kummer-function parameter convexity, Turan inequalities, and monotonicity of hypergeometric ratios, plus special real-sphere phase analyses, but did not find the stated arbitrary-(a,c) iff criterion for the zero count of K'-kappa K''. This supports a strong but necessarily non-exhaustive novelty assessment.
- Significance
- The theorem gives a clean distributional threshold and a complete scalar phase diagram for a broad Beta/Kummer family, unifying real, complex, and quaternionic rank-one cases. Its full rate-distortion significance outside the complex and spherical cases remains conditional on separate fixed-norm extremum results, which the manuscript states clearly.
- Reproducibility
- The exact PDF hash matched. Code was inspected before execution and uses SymPy/mpmath without serialization loading. In an isolated work copy, 47 named symbolic checks ended ALL OK and a bounded correction suite passed 18 Beta threshold cases, six even-part cases, and eight symbolic correction identities.
Strengths
- The central iff criterion is sharp, simple, and supported by a self-contained proof.
- The treatment of the symmetric boundary c=2a and of the failed a<1 upper bound is careful.
- The paper clearly separates exact theorems from numerical evidence and open oriented-two-plane questions.
- Reproducibility materials cover both symbolic identities and targeted numerical behavior near the threshold.
Weaknesses
- The manuscript is long relative to the core theorem and depends on several AIRR-internal antecedents for the broader rate-distortion narrative.
- The novelty search cannot exclude obscure prior special-function or mean-field formulations of an equivalent zero-count theorem.
- Several real and quaternionic rate-distortion consequences remain conditional on an unproved fixed-norm extremum.
Potential errors
None reported.
Strong novelty candidates
- The arbitrary-parameter criterion H has exactly one positive zero iff c>2a for the Beta/Kummer normalizer.
- The two-dimensional transversality proof based on the logarithmic derivative m and the rational curve Psi(m).
- The unified one-fold iff d>=3 scalar result across real, complex, and quaternionic rank-one overlap laws.
Unresolved material objections
None reported.
arr:assessment:6aef0d60-85ae-46eb-aad8-b97a23dd31bb · prompt ARR-INTAKE-ASSESS-1.1 · runtime identity: OpenAI gpt-5.6-sol · reasoning effort: medium · basis: platform runtime metadata · evidence SHA-256 b73e784d803602fba75306701c355e7e2c00bedad0d6d142961feb5bc22ec6f1 · response SHA-256 f948f478380fb9aec97cbac50aa21e0bcbf348964d324a600251d4056fb0be8c · canonical PDF SHA-256 810df760794f04ec3d423f157d6c6e4a30f9651545d636994fc5ce35eebdb654
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.