Positive circuit obstructions and rigidity for lattice-fiber log-concavity
Citable version that has not completed AIRR screening. Not peer reviewed. Screening record
We characterize the full-row-rank integer matrices whose lattice-point counts satisfy midpoint log-concavity in every fixed integer coordinate box. Under the promise that all three integer fibers are nonempty, the property holds exactly for equimodular matrices, whose nonzero maximal minors have one common magnitude. The same classification holds coefficientwise and at positive external fields. Necessity has an explicit certificate: a primitive minimal kernel circuit with a coefficient of magnitude at least two gives counts (1,1,2) in one box supported on at most r+1 coordinates. Its defect is one negative monomial, so the same triple violates the inequality at every positive field. If empty fibers are allowed for arbitrary ambient integral right-hand sides, the common minor magnitude must additionally be odd. Classical circuit and lattice-normalization results supply the structural tools, while sufficiency uses Leake--Mohammadi Yekta’s coefficientwise inequality. We retain equality by integral translations via squarefreeness of complete lattice enumerators, and a test using 6n linear optimizations. The rational-box boundary and the distinction from global Jensen remain explicit. Exact finite controls have a byte-preserving reproduction supplement.
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Cite this paper
Lluis Eriksson (2026). Positive circuit obstructions and rigidity for lattice-fiber log-concavity. AIRR.SCIENCE, ARR-2026-0BQ1X4SGFJ8AJ8N1, v2. https://airr.science/papers/ARR-2026-0BQ1X4SGFJ8AJ8N1/versions/v2/ Research preprint; ARR-2026-0BQ1X4SGFJ8AJ8N1 v2. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.
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Revision statement
Major revision. Adds the equimodular classification with positive-count circuit obstructions, the odd-index empty-fiber boundary, equality transfer and its own exact-PDF full review.
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
The named author directed substantial OpenAI Codex assistance in derivation, writing and numerical work. Full requested GPT-6 Astra xhigh review861 assessed this exact ten-page PDF once: scientific importance4.0/10 and separate technical quality8.5/10, minor_revision. Original complete report, all R1-R4 recommendations, seven adverse reservations and reading/context limits are retained. Binding869 is later format-only serialization, not a second vote or a regrade. The reviewer found no invalidating mathematical flaw in the stated scope; scoped reading does not certify priority. Backend, effective effort and independence are unverified; shared filesystem and prior Codex participation are disclosed. The PDF statement that full assessment was pending predates the now-completed report. External author clarifications leave PDF bytes unchanged. The18 exact LP certificates are a post-review author supplement and were not assessed by861. Classical structural and Newton tools and the essential LMY inequality are credited. No global Jensen theorem, general counting algorithm, Tardos implementation, formal verification, independent human review, certified worldwide originality or importance greater than7 is claimed. Founder-author/operator conflict disclosed. Working paper only; native two-model admission gate pending, no Accepted status or author-editor acceptance.
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
- 1
- Range
- 4.00–4.00
- Method
- bounded capability-weighted mean · exact version only
Limited evidence · 0/1 benchmark matches. This is an evidence description, not a probability of correctness.
Only one distinct model; 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
Review input clarification
Original full861 assessed this exact ten-page PDF once: importance4.0/10 and separate technical8.5/10, minor_revision, diagnoses4/4/3/3/5. Binding869 serializes the completed report without another vote or regrade. ARR-ASSESS-1.0 is a later serialization format, not the literal original prompt. Requested GPT-6 Astra xhigh; effective backend, effort and independence unverified, shared filesystem and prior Codex participation disclosed. All R1-R4 and seven adverse reservations remain visible in the full report and serialization. External author response clarifies terminology/attribution and preserves PDF bytes. The18 LP certificates are later author evidence outside the original assessment. The PDF pending-assessment statement predates861. Ordinary author revision2, clean scanner, exact server PDF, exact author CC-BY-4.0 grant and founder-author/operator conflict recorded. Working paper only; native two-model admission gate pending, no Accepted status, author-editor acceptance or independent human review. Scientific importance greater than7 is not established.
Operator record · · Source files and affected reports
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OpenAI · gpt-6-astra (requested; runtime unverified) · unknown · review context: unknown4.00
Recommendation: Minor Revision · Material objections: 0
Serialización posterior de la única revisión matrix861, cerrada el 2026-10-27T05:48:31.871992+00:00. No es una nueva evaluación ni un segundo voto. Favorable tras mejoras menores; no se encontró error matemático invalidante en el alcance enunciado. Importancia científica AIRR: 4,0/10. Respuesta inversa completa y certificado explícito con fibras positivas, más igualdad decidible. Dependencias externas y estructura clásica delimitan su profundidad y alcance; no se demuestra una aportación excepcional o transformadora. Calidad técnica separada: 8,5/10; Pruebas coherentes con casos límite, límites racionales y computación reproducible; correcciones menores de terminología y exposición, comparación bibliográfica acotada y cuerpo de Tardos no leído. Los cinco diagnósticos no se promediaron para obtener importancia. ARR-ASSESS-1.0 designa el formato posterior, no el prompt original literal. Se solicitó GPT-6 Astra xhigh; backend y esfuerzo efectivo no atestados, independencia desconocida y filesystem compartido. No implica aceptación editorial o humana.
- Correctness confidence
- Reconstrucción analítica sin defecto invalidante encontrado, dependencia esencial directamente leída y controles exactos. No certificación formal.
- Rigor
- Cuantificadores y fronteras son suficientes; argumentos completos en el alcance declarado. Se recomiendan precisiones de exposición y atribución.
- Novelty
- Aporte específico plausible en certificado positivo y rigidez; criterio de circuitos, normalización, caras de Newton y LP son antecedentes claros. Prioridad no establecida.
- Significance
- Clasificación e igualdad útiles en un ámbito preciso; sin evidencia de alcance transformador, Jensen global o conteo general.
- Reproducibility
- Resultados y certificados nativos reproducidos, entradas inmutables y controles propios exactos documentados. Las tablas no son prueba universal.
Strengths
- La clasificación con fibras enteras no vacías, la obstrucción monomial uniforme y la caracterización de igualdad tienen pruebas que se sostienen bajo las hipótesis declaradas.
- Son seis optimizaciones por coordenada: mínimo y máximo en cada una de las tres fibras prometidas no vacías.
- Los resultados matemáticos de ambos scripts coincidieron con los suministrados; únicamente UTC y tiempo transcurrido difieren conforme al contrato.
Weaknesses
- R1 [minor; Título y lema 2.3]: Positive circuit puede confundirse con signo uniforme; la obstrucción positiva aquí se refiere a las fibras/campos y admite signos mixtos.
- R2 [minor; Sección 3, párrafo bibliográfico]: Precisar el enunciado de soporte completo frente al mecanismo clásico de caras de Newton; la lectura acotada no prueba prioridad ni ausencia de folklore.
- R3 [minor; Sección 8, contexto de auditorías]: Condensar o trasladar historial de modelos; no constituye prueba ni revisión independiente.
- R4 [optional_improvement; Teorema 5.1 y suplemento]: Añadir ejemplo de certificados LP o wrapper de decisión. Las enumeraciones reproducidas no son una implementación de Tardos, como ya reconoce el manuscrito.
- La suficiencia y el paso de igualdad positiva a identidad dependen de LMY 3.1; no se atribuye ese teorema al manuscrito.
- Criterio de circuitos y normalización equimodular son clásicos; no hay nueva clase estructural ni reconocimiento.
- La squarefreeness emplea tecnología clásica de caras de Newton y la prioridad del enunciado concreto no está establecida.
- No se resuelve Jensen global, conteo general, robustez frente a perturbaciones del array ni una brecha uniforme en campos positivos.
- Los controles finitos no prueban teoremas universales y no implementan Tardos.
- El cuerpo completo de Tardos no se obtuvo (403). Se leyó abstract oficial; se declara esta limitación.
- Backend, esfuerzo efectivo e independencia no están atestados; filesystem compartido.
Potential errors
None reported.
Strong novelty candidates
- Aporte específico plausible en certificado positivo y rigidez; criterio de circuitos, normalización, caras de Newton y LP son antecedentes claros. Prioridad no establecida.
Unresolved material objections
None reported.
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A model assessment is not peer review or a correctness certificate. AIRR preserves disagreement, exact-version provenance and later reassessments.
Editorial disclosure
Founder-authored author-authorized Working-paper v2. Original full861 importance4.0/10 and separate technical8.5/10 retained with all objections. Native gate pending; no author-editor acceptance or independent human review.