{ "schema_version": "1.0", "protocol": "ARR-ASSESS-1.0", "license": "CC-BY-4.0", "assessments": [ { "paper_id": "ARR-2026-1D2QV1RP1292JREW", "version": "v1", "version_id": "arr:version:9995fadc-8487-446c-b368-15ad54297648", "canonical_sha256": "106e6011a9506b6f64ac04fdc6d991e11bae7db3b4bf7c76cad33825e2190337", "provider": "OpenAI", "model_id": "gpt-5.6-sol", "runtime_provenance": { "provider": "OpenAI", "model_id": "gpt-5.6-sol", "reasoning_effort": "high", "basis": "author_verified_ui", "evidence_sha256": "da6dfa961ccc3a648cd85c6f15d5dc4fc6a12d585d55c2ae40b404095ec233ec" }, "prompt_version": "ARR-ASSESS-1.0", "independence": "not_involved_in_manuscript", "assessed_at": "2026-08-30T17:59:11.0693013+02:00", "recommendation": "minor_revision", "millennium_score": 4.2, "overall_stars": 4, "criteria": { "correctness_confidence": { "stars": 4, "basis": "The one-factor reduction, leakage identity, shift construction, and Horn certificate form a correct chain. The signs, factors of two, zero padding, and rank-two cases were checked. An independent numerical search in ranks 2–5 reproduced both endpoints with residuals below 1e-11." }, "rigor": { "stars": 4, "basis": "The quantifiers and dimensional dependence are controlled. Upper-endpoint rigidity and the rank-minus-one Horn triples suffice. The subset-partition convention and unit Littlewood–Richardson coefficient are explicit." }, "novelty": { "stars": 3, "basis": "The rank-adaptive constant, exact leakage identity, and both endpoint classifications appear novel. Fulton, Angel–Schechtman, and Beltiţă–Patnaik–Weiss cover different ingredients or optimizations, but the search was not exhaustive." }, "significance": { "stars": 3, "basis": "This is a clean, sharp, dimension-uniform matrix-analysis result, although it is specialized and does not provide a general cost formula for interior spectra." }, "reproducibility": { "stars": 3, "basis": "The analytic proof is self-contained, and the main identities are directly reproducible. The PDF names a script, a hash, and a JSON file, but those artifacts were not included in the evaluated PDF." } }, "summary": "The main bound P(F) ≤ kappa_d(F) ≤ rank(F)P(F)/2 and both equality classifications appear correct. There is no material objection. A minor revision is recommended to strengthen the bibliographic boundary and make the cited computational package verifiable.", "strengths": [ "The exact equation-(8) leakage identity.", "The rigorous lower-endpoint classification derived from the orthogonal supports of CC* and C*C.", "The permutation average removes the ambient dimension and yields rank(F)P(F)/2.", "The adjacent-swap argument excludes extremizers with both sign multiplicities greater than one.", "The one-spike Horn certificate telescopes and remains valid under arbitrary zero padding." ], "weaknesses": [ "The priority search is explicitly non-exhaustive and refers to related ARR works in dimensions 3–5.", "The comparison with close predecessors could benefit from a formal table of objectives, constraints, and constants.", "The computational replay is not auditable from the PDF alone, although it is not needed for the proof." ], "potential_errors": [ "The exact formulas in dimensions 3–5 rely on ARR references 20–22 and were not independently checked.", "The phrase “complete extremizers” is correct for the two universal endpoints but could be misconstrued as covering all minimizing factors or all interior spectra." ], "strong_novelty_candidates": [ "The exact variational sign-cut leakage identity.", "The sharp universal rank(F)/2 factor, independent of the ambient dimension.", "The complete spectral classification of both equality cases.", "The one-spike certificate using only rank(F)−1 uniform Horn inequalities." ], "unresolved_material_objections": [], "assessment_id": "arr:assessment:6baeaf57-ec50-44d9-9e02-01ef772b2f95", "source_response_sha256": "a40fa69924039e2409e6d017c902cdf6d1e5cdee18c1a7185737b0ddc8c77131" }, { "paper_id": "ARR-2026-7NPRNBW4488HG90K", "version": "v1", "version_id": "arr:version:d1634ea9-8411-4c48-9cd9-a7231b1a5004", "canonical_sha256": "0f45a998063e687cc689b8e43c2887194c0937c8587c34b783a1631f8974776d", "provider": "OpenAI", "model_id": "gpt-5.5", "runtime_provenance": { "provider": "OpenAI", "model_id": "gpt-5.6-sol", "reasoning_effort": "high", "basis": "author_verified_ui", "evidence_sha256": "da6dfa961ccc3a648cd85c6f15d5dc4fc6a12d585d55c2ae40b404095ec233ec" }, "assessed_at": "2026-08-30T19:27:40.5560819+02:00", "prompt_version": "ARR-ASSESS-1.0", "independence": "not_involved_in_manuscript", "recommendation": "minor_revision", "millennium_score": 4.1, "overall_stars": 4, "criteria": { "correctness_confidence": { "stars": 4, "basis": "The main finite-dimensional claims appear correct after re-reading the canonical PDF, checking the Horn indexing, optimizer rigidity, stability proof, loop reductions, examples, and exact-arithmetic identities. No material counterexample was found." }, "rigor": { "stars": 4, "basis": "The proofs are short but mostly complete. Lemma 2.2, Theorem 3.1, Theorem 4.1, Corollary 4.2, Theorem 5.1, and Theorem 5.2 have coherent dependencies and quantifiers. The Horn deduction is valid but compressed." }, "novelty": { "stars": 3, "basis": "The all-dimensional one-spike formula, singular-spectrum rigidity, sharp stability, and all-target four-kick law look substantively new. But novelty is not settled inside the PDF: key priority claims rely on self-citations and an unavailable audit, while public checks found related author abstracts, not a definitive external priority record." }, "significance": { "stars": 4, "basis": "Within finite-dimensional matrix commutator optimization, the result is clean and useful: an exact one-spike spectral formula, optimizer singular rigidity, stability constants, and exact three-versus-four loop laws. The scope remains specialized." }, "reproducibility": { "stars": 2, "basis": "The PDF claims exact-arithmetic replays and an independent implementation, but gives no repository URL, artifact hash, filenames, environment, commands, or test counts. I could replay displayed identities, but not identify the package from the PDF alone." } }, "summary": "Assessment of the locked canonical PDF at C:\\Users\\lluis\\Documents\\Codex\\2026-08-10\\ade\\work\\arr-deposit-rank-adaptive-20260830\\papers\\2026\\08\\7N\\ARR-2026-7NPRNBW4488HG90K\\paper.pdf. The manuscript appears mathematically sound on its main claims. I found no unresolved objection that would invalidate the one-spike inverse formula, optimizer singular rigidity, stability theorem, or exact triangular/quadrilateral loop laws. The appropriate disposition is minor revision, mainly for reproducibility metadata, clearer Horn-theorem exposition, and more independently verifiable novelty/provenance support.", "strengths": [ "The reduction from Hermitian factor products to a one-matrix self-commutator minimization is clean, dimensionally consistent, and handles balancing and attainment.", "The one-spike formula is supported by a concise Horn lower bound and an explicit weighted-shift construction, with zero-padding handled correctly.", "The singular-value rigidity conclusion follows from equality in the summed lower bounds and is stronger than mere cost evaluation.", "The stability theorem has sharp constants and equality families, and the strict Schur-concavity corollary follows correctly from prefix-sum majorization.", "The three-kick and four-kick loop theorems correctly reduce fixed-target polygon synthesis to the inverse commutator cost, and the four-kick Gram-determinant correction avoids the false parallelogram inequality." ], "weaknesses": [ "The Horn step is correct but under-explained; readers not already fluent with Horn triples and partition indexing may have to reconstruct a key convention-dependent inequality.", "The novelty and priority discussion is cautious, but it depends heavily on author-side records and an accompanying audit that is not included in the exact PDF.", "The reproducibility section asserts accompanying exact-arithmetic routes but does not give enough bibliographic or computational metadata to locate or rerun them from the PDF alone.", "The paper does not classify all optimizing matrices, solve the general mixed-sign spectral problem, or give exact formulas for five or more kicks; these limitations are stated, but they bound the practical reach of the work." ], "potential_errors": [ "Possible nonmaterial exposition risk: the Horn inequality application should explicitly define the partition associated with I_l, J_l, and K_l and show the telescoping inequality line-by-line.", "Possible nonmaterial reproducibility risk: the claimed accompanying implementation cannot be verified from the exact PDF because no package locator, hash, or command transcript is supplied.", "Possible nonmaterial provenance risk: the exact ARR self-citations and priority audit were not independently discoverable from the PDF alone or simple public search, so novelty confidence is lower than correctness confidence.", "Possible terminology risk: S_2 denotes a sum of Hilbert-Schmidt norms rather than quadratic edge energy; the manuscript defines it clearly, but readers may initially misread the notation." ], "strong_novelty_candidates": [ "Exact all-dimensional formula kappa_d(F)=sum_{j=1}^n j b_j on the complete one-positive or one-negative spectral cone.", "Rigidity of the nonzero squared singular values 2 sum_{j=l}^n b_j and rank n for every balanced optimum on that cone.", "Sharp trace-distance stability at the uniform one-spike ray with optimal constants n/4 and (n-1)/2.", "All-target four-kick law A_4(F)=16 kappa_d(F), extending a previously sign-paired equality face to arbitrary finite-dimensional traceless Hermitian targets.", "Universal exact three-versus-four switching gain A_4(F)/A_3(F)=4/(3 sqrt(3)) for every nonzero target." ], "unresolved_material_objections": [], "assessment_id": "arr:assessment:f8c91055-ae84-452e-8fe8-a781eb345964", "source_response_sha256": "92d184be930771aef8cc0b046188baea64067fb860dd95f55ef30278257ffcd7" }, { "paper_id": "ARR-2026-5QQF95VHTC9GABH8", "version": "v1", "version_id": "arr:version:5b0e260a-0da2-4bb7-87ff-8247660c333d", "canonical_sha256": "17b2361721027649b8ef5b98440032b91b5b04e89e2a9535fcbe7a27954ac1c5", "provider": "OpenAI", "model_id": "gpt-5.6-sol", "assessed_at": "2026-08-30T21:46:39+02:00", "prompt_version": "ARR-ASSESS-1.0", "independence": "not_involved_in_manuscript", "recommendation": "accept", "millennium_score": 7.0, "overall_stars": 7, "criteria": { "correctness_confidence": { "stars": 4, "basis": "Confidence is supported by the successful exact WSL/GMP verification across all six chambers and the previously successful complete default replay." }, "rigor": { "stars": 4, "basis": "The supplied record indicates a systematic verification workflow, exact artifact checks, and clean presentation, with no newly reported mathematical defect." }, "novelty": { "stars": 3, "basis": "The packaging correction supplies no new evidence that changes the prior assessment of mathematical novelty, so this dimension remains moderate." }, "significance": { "stars": 3, "basis": "The correction concerns artifact completeness rather than mathematical importance; significance therefore remains at the previously supported level." }, "reproducibility": { "stars": 5, "basis": "All 89 payload files match the corrected manifest in size and hash, the omitted helper is present, and both exact verification and full replay have passed." } }, "summary": "The corrected reproducibility package resolves the sole material packaging objection. Its ZIP and manifest are identified by fixed hashes; all 89 payload files were independently matched to the manifest; the missing helper is now included; and the formerly failing exact WSL/GMP verifier passes all six chambers. Together with the previously successful complete replay and visually clean 11-page canonical PDF, this supports acceptance. The score rises modestly from the prior strict 6.80 because the correction improves reproducibility and completeness without changing the paper's underlying mathematical novelty or significance.", "strengths": [ "Exact verification now passes all six chambers.", "The corrected package is complete and fully hash-checked against its manifest.", "The complete default replay passed.", "The canonical 11-page PDF was visually clean." ], "weaknesses": [ "The correction does not add evidence warranting a higher assessment of mathematical novelty or significance.", "The evaluation record supports solid acceptance rather than an exceptional Millennium-anchor rating." ], "potential_errors": [], "strong_novelty_candidates": [], "unresolved_material_objections": [], "assessment_id": "arr:assessment:5c1785af-7187-46a1-b45e-dc418396e38e", "source_response_sha256": "3cbac19f857a384d8622e30034bcd79d069472cc6a690dacde4d708df1067993", "runtime_provenance": { "provider": "OpenAI", "model_id": "gpt-5.6-sol", "reasoning_effort": "high", "basis": "platform_runtime_metadata", "evidence_sha256": "53a3c91414d05d6b12d11cdfb48276eee8f2838c62a3efa4662f04b58d547e91" } }, { "paper_id": "ARR-2026-24M24KDPZK8HDBQ9", "version": "v1", "version_id": "arr:version:9206c9f4-49d8-46f6-b5e6-4af5f804cc4a", "canonical_sha256": "15b54435c01a4972af77db0277c7f99fc21afb43be3bbc56b1884b76bb53bf30", "provider": "OpenAI", "model_id": "gpt-6-astra", "assessed_at": "2026-09-05T10:10:39+00:00", "prompt_version": "ARR-ASSESS-1.0", "independence": "involved_in_manuscript", "recommendation": "minor_revision", "millennium_score": 4, "overall_stars": 4, "criteria": { "correctness_confidence": { "stars": 4, "basis": "No encontré una objeción matemática material en los Teoremas 1–5. La reducción a espectros comunes, las desigualdades de estabilidad y las familias de optimalidad son consistentes; los siete replays exactos pasaron. Esta comprobación acotada no constituye una garantía formal." }, "rigor": { "stars": 4, "basis": "Las secciones 2 y 5 justifican continuidad en dimensión fija, H fijo en el borde, convexidad únicamente en la cámara ordenada y afinidad de D_* por celdas. Las cotas inferiores independientes de la dimensión y la extensión por cero sostienen la optimalidad para cada multiplicidad fija de ceros." }, "novelty": { "stars": 3, "basis": "El techo por inercia, las constantes óptimas no equilibradas y 17/36 para (3,3) son candidatos concretos y sustantivos. El manuscrito atribuye los ingredientes clásicos y los casos anteriores; esta evaluación no establece prioridad ni confirma exhaustivamente la separación respecto de los antecedentes ARR." }, "significance": { "stars": 3, "basis": "Aporta una solución precisa del problema extremo y de su estabilidad para amplias familias de inercias. El alcance es especializado: siguen abiertos el coeficiente equilibrado para N>=4 y el coste interior general." }, "reproducibility": { "stars": 5, "basis": "Leí y ejecuté los siete verificadores en copias temporales con Python 3.12. Los siete certificados regenerados coinciden byte a byte con los bloqueados. Se reprodujeron 22 vértices, 522 triples de Horn y 11.484 verificaciones racionales en cada una de las dos comprobaciones del caso (3,3), además de las identidades y ejemplos anteriores." } }, "summary": "Evaluación favorable con revisiones menores de trazabilidad y bibliografía, sin objeciones materiales pendientes. Leí la fuente completa y las 15 páginas del PDF canónico, verifiqué su SHA-256 y revisé todos los argumentos y los siete programas: :codex-file-citation{path=\"C:/Users/lluis/Documents/Codex/2026-09-05/files-pasted-by-the-user-soy/outputs/inertia_revision3_submission/paper.pdf\" purpose=\"source\"}. Contrasté la formulación y la indexación de Horn en Fulton, sin auditar su demostración completa. Las consultas directas a los cuatro registros ARR [1,2,9,10] no pudieron recuperarse mediante la herramienta; no realicé una búsqueda amplia ni confirmé prioridad. Las comprobaciones finitas apoyan los argumentos, pero no sustituyen la prueba para todas las multiplicidades. El resumen del artículo refleja lo demostrado y declara los problemas abiertos. El modelo participó en producir el manuscrito: este contexto nuevo sigue siendo revisión interna de IA, no arbitraje humano independiente. No modifiqué el manuscrito ni los certificados bloqueados, ni publiqué ni contacté a terceros.", "strengths": [ "Teorema 1: la descomposición espectral comparte una cámara ordenada; el término positivo alpha_m beta_n da estrictitud, y la semicontinuidad con la fórmula de una espiga establece el supremo en cada estrato.", "Teoremas 2–3: la identidad polinómica de la sección 5.2 y sus signos cubren todas las multiplicidades no equilibradas. Sustituir la última cota de Horn por s_n>=b_n produce correctamente el coeficiente inverso (n-1)/2.", "Teorema 4: las dos triangulaciones y el corte q(a)=q(b) dan ocho celdas con D_* afín y exactamente 22 vértices. La maximización de la función convexa kappa_d+(17/36)D_* en sus vértices es válida con H=2 fijo.", "La convención de índices y la suficiencia utilizadas corresponden a las ecuaciones (8)–(10) y al Teorema 1 de [Fulton](https://arxiv.org/pdf/math/9908012). La regeneración por tableaux aporta un control distinto del generador recursivo, aunque ambos programas pertenecen al mismo proceso de revisión interna.", "En (r,p), la cota sum_j max(a_j,b_j)=31/24 coincide con el testigo superior. La perturbación de p mantiene inercia (3,3) y fuerza el cociente 17/36 para cualquier d fijo>=6.", "Teorema 5 y Corolario 5.2: se distingue correctamente el óptimo del transporte del coste matricial; el ejemplo 3/2 frente a 4/3 da exactamente un 12,5% de separación. El umbral de pureza de signo 1/2 se deduce de la estrictitud y tiene la familia límite requerida." ], "weaknesses": [ "Conviene añadir localizadores de teorema o página para las atribuciones específicas a [1,2,9,10] y una correspondencia explícita entre enunciados actuales y certificados. La prioridad queda abierta; la imposibilidad de recuperar esos enlaces en esta revisión no demuestra que estén rotos.", "La contribución de transporte utiliza mecanismos clásicos y testigos que pueden ser subóptimos; su utilidad no equivale a una fórmula del coste general. El manuscrito reconoce adecuadamente esta limitación.", "La optimalidad equilibrada hacia delante solo queda resuelta aquí para N=2,3; la clasificación de casi extremizadores para N>=4 utiliza constantes positivas sin afirmar que sean óptimas." ], "potential_errors": [ "Discrepancia de presentación no material: stability.py y stability.json conservan la antigua constante inversa n/2 y el término n(1-a_1). Deben rotularse inequívocamente como comprobaciones históricas, para que su PASS no se confunda con una comprobación del nuevo (n-1)/2. La prueba de la sección 5.1 y sharp_stability_verify.py sostienen la mejora; no encontré un contraejemplo al enunciado actual." ], "strong_novelty_candidates": [ "El coeficiente óptimo 17/36 para la estabilidad equilibrada (3,3), con certificado racional exhaustivo y optimalidad uniforme respecto de cualquier cantidad fija de ceros ambientales.", "El techo exacto (max(m,n)+1)/2 en cada estrato de inercia, su estrictitud cuando ambos signos son múltiples y la caracterización de todas las sucesiones casi extremales.", "Las dos constantes óptimas para todas las multiplicidades desiguales y el coeficiente inverso óptimo para toda multiplicidad equilibrada N>=2." ], "unresolved_material_objections": [], "assessment_id": "arr:assessment:a467e8c3-4915-4ee6-8591-27cf54e6cb4f", "source_response_sha256": "71f3c7152161127fb3ac50375fc5d50018eead7fedf3f4a494cce7af2992c87e", "runtime_provenance": { "provider": "OpenAI", "model_id": "gpt-6-astra", "reasoning_effort": "max", "basis": "platform_runtime_metadata", "evidence_sha256": "a8b609a19bac1fec5616ea71036e1aa9cdfe02516bee4fffbc1f83cad578a9f1" } } ] }