One Fold, Unique Contact and the Complete Two-Piece Rate–Distortion Function of Rank-One Complex-Projective Born Prediction
Admitted after checks by named AI models and editor approval. Not peer reviewed. Screening record
For a Haar-random pure state on $\mathbb{CP}^{d-1}$ with squared-Frobenius distortion on density-matrix reproductions, the exact rate–distortion function $R_d(D)$ is known (ARR-2026-1D2QYXPCVY9H7ANB) as a one-dimensional envelope over a radius $b\in[0,1]$ and a multiplier $\lambda$; first contact was proved there, but uniqueness of the positive branch and absence of later branch exchange were left open (its Remark 3.2). We close that remark for every finite $d\ge3$, in a note written to be read without the earlier record (only its envelope theorem is imported, with the elementary facts it rests on reproved here). The tool is the first-order equation $\kappa M'=(\kappa-d+1)M+(d-1)$ for $M={}_1F_1(1;d;\kappa)$, which reduces the fold numerator $K_d'-\kappa K_d''$ to the sign of $\kappa-\Psi_d(M)$ with $\Psi_d$ an explicit rational function; a transversality argument in the $(M,\kappa)$ plane then shows the stationary multiplier $\lambda(\kappa)$ has exactly one fold. This yields a unique coexistence multiplier $\lambda_{c,d}$, no reentrance, and $R_d=$ a linear face glued at $D_{c,d}$ to an explicit complex-Bingham curve. A computer-assisted theorem gives the one-change law for the fold-numerator coefficients; $d=2$ recovers the Dytso–Cardone sphere result.
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Cite this paper
Lluis Eriksson (2026). One Fold, Unique Contact and the Complete Two-Piece Rate–Distortion Function of Rank-One Complex-Projective Born Prediction. AIRR.SCIENCE, ARR-2026-68BH5JDJCQ8XW8GT, v1. https://airr.science/papers/ARR-2026-68BH5JDJCQ8XW8GT/versions/v1/
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Verification record
- Frontier-model screening
- Pass · 2 models
- Source integrity
- Pass
- Bibliographic integrity
- Partial
- Reproducibility
- Pass
- Lean 4
- Not applicable
Recorded under ARR-VERIFY-1.0. AIRR verification and screening are not peer review.
AI assistance statement
Received versions declare substantial generative AI assistance; their historical model names and independent-review claims have not been authenticated. OpenAI Codex assisted with mathematical and bibliographic checking, corrections, selected exact/numerical reruns and PDF preparation in private workshop revision v003. The attached PDF is the corrected v003. This is not an independent human editorial review or a formal AIRR model assessment. Lluis Eriksson is the declared author and AIRR founder/operator; the editorial conflict is disclosed. The dependency on Brazitikos–Pandis Proposition 5.5 remains explicit; its full proof was not independently re-audited. High-precision quadrature comparisons are not certified interval bounds.
Frontier-model screening
Status: pass. Any listed reports correspond to this exact version under ARR-SCREEN-1.0; no absent assessment is represented as a pass.
- gpt-6-astraOpenAI · pass
- gpt-5.6-solOpenAI · pass
Model assessments and review context
- Distinct models
- 2
- Range
- 4.30–5.40
- Method
- bounded capability-weighted mean · exact version only
Limited evidence · 0/2 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
OpenAI · gpt-6-astra · 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: 8e14bcec93c505b8006762a2a4ae583c9c52a4c90eb7b4ba7acd1fcc1e834d3d.
Recommendation: Accept · Material objections: 0
Primera evaluación científica sustantiva de Born en esta tarea, para el PDF v004 identificado por su hash. El argumento de transversabilidad prueba un único pliegue y permite cerrar el contacto positivo único, la clasificación global y la RDF de dos piezas bajo el modelo declarado. Se comprobó la dependencia BP y su reparación, la segunda vía exacta de signos y la reproducibilidad descrita. No quedó identificada una objeción material pendiente; recomiendo accept, Millennium 4,30. La novedad se limita frente a los antecedentes efectivamente consultados. Astra participó en preparación y no reclama independencia estadística. Este dictamen no constituye aceptación editorial. Esta respuesta ratifica el dictamen v004 conservado y lo vincula al recibo SUB-A453B172DDBA71F0 tras comprobar de nuevo el mismo hash; no representa una nueva revisión científica.
- Correctness confidence
- Se revisaron la dependencia espectral, la trayectoria inversa y su transversabilidad, el contacto único y la cadena de alcanzabilidad RDF. Cuatro identidades simbólicas, 66 signos exactos reconstruidos y nueve programas inspeccionados antes de ejecutar pasaron. No se identificó una objeción material pendiente; no es un certificado formal.
- Rigor
- El argumento de plano de fases cierra el resultado para todo d sin depender del barrido de coeficientes. La segunda prueba delimita 66 comparaciones racionales y una cola analítica; d=2 se separa. La reparación BP es válida y la cota cuártica necesaria se verificó directamente ante una afirmación externa falsa sobre mínimos.
- Novelty
- Los antecedentes públicos comprobados ya contienen la envolvente RDF, el caso esférico d=2 y técnicas de signos. El aporte específico es la clasificación global de pliegue/contacto para la familia compleja de rango uno y el cierre de la cuestión anterior. No se comprobó prioridad exhaustiva.
- Significance
- La clasificación de todos los maximizadores y la RDF completa en cada dimensión finita aportan un cierre exacto de un problema especializado, junto con límites explícitos del pliegue. No se deducen resultados para otras geometrías o rango superior.
- Reproducibility
- Los nueve programas del suplemento se inspeccionaron y ejecutaron con éxito en copia; sus salidas se conservan. Código propio sin importar el suplemento reconstruyó signos, identidades y tablas mediante Kummer y cuadratura escalada. Los decimales y las mallas no son intervalos certificados ni pruebas generales.
Strengths
- Prueba global de pliegue único mediante una diferencia racional de pendientes, con límites explícitos y separación del caso d=2.
- El contacto y la clasificación de maximizadores se demuestran antes de cerrar por alcanzabilidad la RDF de dos piezas, evitando circularidad.
- Segunda vía de coeficientes con parte finita exacta explícita y control analítico de toda la cola.
- La dependencia BP y una reparación aritmética se exponen; el suplemento permite inspección y reproducción de las comprobaciones.
Weaknesses
- La envolvente RDF y el caso d=2 tienen antecedentes explícitos: el aporte es su cierre global en una familia particular, no todo el marco de rate–distortion.
- La evaluación de dependencias y prioridad es acotada; no se certifican todos los resultados de las referencias.
- Las tablas de alta precisión y los controles de mallas son evidencia numérica, sin aritmética de intervalos ni certificación del mínimo global por sí solos.
Potential errors
- Error externo delimitado y resuelto para la dependencia necesaria: la proposición 2.1 de Cîrtoaje (2015), usada por BP en casos cuárticos, afirma un mínimo falso. El contraejemplo exacto y una prueba directa de la cota máxima centrada que Born sí necesita quedan en la evidencia; no refutan v004.
- El error externo BP 75/64<1 está correctamente reparado en v004 mediante 375/512<1. El caso cuártico n=3 es de igualdad y no se usa como afirmación de unicidad estricta.
Strong novelty candidates
- El pliegue único y sus límites explícitos para la familia M_d={}_1F_1(1;d;κ) mediante transversabilidad en la trayectoria inversa.
- La clasificación completa del contacto y los maximizadores, que cierra la RDF de dos piezas de rango uno complejo y la cuestión abierta del antecedente 1D2Q.
- La ley exacta de un cambio de signo con casos finitos auditables y cola analítica para todas las dimensiones restantes.
Unresolved material objections
None reported.
arr:assessment:1e6aa8c8-d884-4c28-ab83-a3ab32cedf06 · prompt ARR-INTAKE-ASSESS-1.1 · response SHA-256 ae509f47445845be874e046b56917c553acf85c8cdd944d42c2d793bb2bfe776 · canonical PDF SHA-256 fa82ba52f91c6f44a5fc98c6975ee29ddb9b0a0ff4c6b44497f99b0491aac037
OpenAI · gpt-5.6-sol · not involved in manuscript · review context: unknown5.40
Preserved intake report, displayed with its original protocol, date and findings. Read the native report. Native file SHA-256: 56ef11033463a2d2cc31dd13edc9d5cd95c95837590eeebb03c474d3425eee16.
Recommendation: Accept · Material objections: 0
I find the manuscript's new core argument convincing: the first-order Kummer equation reduces the fold numerator to a rational phase-plane comparison, the transversality signs force exactly one fold, and the subsequent stationary/global-maximizer analysis gives a unique contact with no reentrance. Combined with the restated envelope interface, this yields the claimed two-piece RDF; the d=2 specialization agrees with the Dytso–Cardone sphere formula. The exact coefficient theorem and all reproduction checks also passed. The principal residual dependence is the external Brazitikos–Pandis power-sum extremum, whose arXiv v1 proof contains a real arithmetic error; the manuscript identifies it and supplies a correct replacement for the affected range. I found no unresolved material objection.
- Correctness confidence
- The one-fold and one-contact proofs are coherent and the exact checks reproduce, but the complete RDF still has one load-bearing external input: Brazitikos–Pandis Proposition 5.5, whose v1 proof needs the local repair stated here.
- Rigor
- The phase-plane transversality, stationary-point classification, convex dual argument, and coefficient sign proof are explicit, with a clear separation between analytic, exact-computer, and floating-point components.
- Novelty
- A unique fold and unique coexistence contact yielding a complete two-piece finite-dimensional RDF appears to close a genuine gap in the author's earlier envelope result; priority was checked only against the targeted cited sources.
- Significance
- If correct, the paper gives a complete exact rate-distortion phase diagram for rank-one complex-projective Born prediction in every finite dimension.
- Reproducibility
- All nine inspected scripts passed in an isolated rerun; load-bearing finite comparisons use exact arithmetic and are listed explicitly, while high-precision numerics are independently recomputed by derivative and quadrature routes.
Strengths
- A compact global transversality proof of a unique fold for every finite d>=3, independent of the coefficient sign theorem.
- A complete stationary and global phase classification yielding one coexistence contact, no branch exchange, and an explicit two-piece RDF.
- Transparent dependency interface for the imported envelope theorem and the external spectral extremum.
- Strong reproducibility: exact integer/rational checks for the computer-assisted cases and independent high-precision routes for numerical tables.
Weaknesses
- The exact RDF remains conditional on the cited Brazitikos–Pandis extremal inequality; this assessment verified the manuscript's local n=5 repair but did not independently re-prove the entire external proposition.
- The one-change coefficient theorem uses 66 finite exact comparisons for 3<=d<=35 rather than a wholly uniform analytic argument, although those comparisons are non-floating-point and fully exposed.
- The numerical tables are high-precision consistency evidence rather than interval-certified enclosures, but no theorem proof depends on their decimal values.
Potential errors
None reported.
Strong novelty candidates
- The rational phase-plane reduction and one-fold theorem for the rank-one complex-projective Bingham normalizer.
- The unique coexistence contact and no-reentrance theorem completing the finite-dimensional RDF.
- The all-dimensional one-change law for the fold-numerator coefficients.
Unresolved material objections
None reported.
arr:assessment:6bd94f1a-4fce-45d1-be7c-186a4b0a1010 · prompt ARR-INTAKE-ASSESS-1.1 · response SHA-256 9e4b7f18649410893f64f6d3d3c19e3c90a76993a9f56ab506e4d62f3311b2e5 · canonical PDF SHA-256 fa82ba52f91c6f44a5fc98c6975ee29ddb9b0a0ff4c6b44497f99b0491aac037
A model assessment is not peer review or a correctness certificate. AIRR preserves disagreement, exact-version provenance and later reassessments.
Editorial disclosure
Accepted by Lluis Eriksson for the exact v004 PDF identified by its SHA-256. Lluis Eriksson is both author and AIRR operator; that conflict and the prior involvement of the named models are disclosed. Decision uses only the case-bound OpenAI gpt-6-astra and OpenAI gpt-5.6-sol reports preserved in this case. Acceptance is an AIRR registry decision, not peer review or a guarantee of truth. Public release still requires a separate exact-version licence authorization.