Rank plus Weyl sparsity is at least p + 1: an operator form of the Chebotarev-Tao uncertainty principle in prime dimension
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Let $p$ be prime and let $W_{(a,b)}=X^aZ^b$ be the cyclic Weyl operators on $\mathbb C^p$. We prove that every nonzero linear combination $C$ of $\ell$ Weyl operators satisfies $\dim\ker C\le\ell-1$, i.e. $\operatorname{rank}C+s_W(C)\ge p+1$, where $s_W$ counts the nonzero Weyl coefficients. The collinear case is Tao's uncertainty principle on $\mathbb Z/p$ (Chebotarev's theorem on Fourier minors). The general case follows from the reduction-modulo-$(1-\omega)$ mechanism of Frenkel and Tao after two further observations: Weyl combinations stay integral under the non-unimodular Clifford conjugations $F'Q_s$, which realise all $p+1$ line directions of $\mathbb F_p^2$, and, in the nonvacuous sparsity range, a pigeonhole over those directions produces a nonzero fibre sum. The bound is sharp for every $\ell\le p$ and fails in every composite dimension. As a self-contained corollary, for one use of the $p^2$ Weyl channels the minimal perfect list length over probes of Schmidt rank $r$ is exactly $p-r+1$. An exact certificate refutes a natural "line conjecture".
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Cite this paper
Lluis Eriksson (2026). Rank plus Weyl sparsity is at least p + 1: an operator form of the Chebotarev-Tao uncertainty principle in prime dimension. AIRR.SCIENCE, ARR-2026-3WVWP20DY09HFASA, v1. https://airr.science/papers/ARR-2026-3WVWP20DY09HFASA/versions/v1/
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Version history
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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. Weyl normalization, domain and proof corrections are documented.
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
- 4.40–5.20
- 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
OpenAI · gpt-6-astra · involved in manuscript · review context: unknown4.40
Preserved intake report, displayed with its original protocol, date and findings. Read the native report. Native file SHA-256: 3def74fc74bba1a244647ca1780321499f568a225647907452f2cc9f4fdcac88.
Recommendation: Accept · Material objections: 0
Evaluación del PDF v004 con el hash indicado. Las correcciones resuelven los puntos menores de mi dictamen v003: significado de la optimización numérica, referencia de valoración y exposición de la especialización. El resultado principal y el corolario de listas siguen respaldados por la prueba inspeccionada y los controles propios acotados reutilizados. Recomiendo accept sin objeción material pendiente identificada; Millennium 4,40 conserva la valoración del aporte, no es una nota escolar. Astra participó en la preparación: esta evaluación no es independiente de autoría ni estadísticamente independiente. No constituye aceptación editorial. Esta respuesta ratifica el dictamen v004 conservado y lo vincula al recibo SUB-1D9853A95C42CD40 tras comprobar de nuevo el mismo hash; no representa una nueva revisión científica.
- Correctness confidence
- El núcleo valuativo y de comparación de rangos permanece intacto; se reutilizan expresamente 4.023 identidades exactas y los certificados propios v003. Los cambios de especialización, contracción de valoración y normalización de Weyl fueron comprobados en v004, sin error material identificado.
- Rigor
- La especialización conserva un punto no nulo y permite perder soporte en la dirección correcta. La referencia de valoración se precisa y los mínimos numéricos se distinguen ahora de cotas certificadas. La prueba general no depende de optimización local.
- Novelty
- Frente a las fuentes primarias contrastadas en la revisión propia y a la normalización de Bu comprobada de nuevo, la ley aditiva para operadores arbitrarios y la optimalidad de listas siguen siendo candidatos específicos. No se comprobó prioridad exhaustiva.
- Significance
- La desigualdad óptima en dimensión prima y la curva p-r+1 de listas perfectas son resultados concretos y reutilizables en el modelo exacto de un solo uso. No se infieren extensiones a ruido o composición.
- Reproducibility
- Texto y suplemento son inspeccionables; las salidas propias v003 se conservan y su reutilización se declara. En v004 se verificaron diffs y etiquetas numéricas. No se repitieron todos los barridos de órbitas, Gröbner ni primo escindido.
Strengths
- Desigualdad aditiva óptima para operadores arbitrarios en dimensión prima, con tratamiento explícito de la integridad tras conjugaciones no unimodulares.
- La forma valuativa explica cómo las cancelaciones y la geometría del soporte afectan las cotas.
- El corolario de decodificación incluye una prueba directa del criterio de factorización y una construcción normalizada.
- El ejemplo parabólico es un certificado exacto y separa la ley válida de una conjetura geométrica demasiado fuerte.
Weaknesses
- La búsqueda de antecedentes es acotada y no establece prioridad sobre toda la literatura de operadores tiempo-frecuencia.
- Los experimentos de optimización aportan valores observados sin certificar mínimos globales; v004 ya declara correctamente esa limitación.
- La historia de revisiones y verificaciones alarga la presentación de una contribución cuyo núcleo es algebraico.
Potential errors
None reported.
Strong novelty candidates
- La desigualdad rank(C)+s_W(C)≥p+1 para soportes no colineales y coeficientes complejos arbitrarios.
- La cota por sumas de fibras tras elegir una valoración y una dirección.
- La optimalidad global p-r+1, sobre todas las sondas puras de rango de Schmidt r, para listas perfectas de canales de Weyl de un solo uso.
Unresolved material objections
None reported.
arr:assessment:572f947f-3476-4c5d-9d2a-8705cb018068 · prompt ARR-INTAKE-ASSESS-1.1 · response SHA-256 57b1a7f8f16b99b870eae2e1a1b417f5219cec96e37622f6ff5cf62de545e6b7 · canonical PDF SHA-256 75ab15f5b1145b7579de198f99e5c4409ec4a01d9f7c7e52023e5b91b9c91d89
OpenAI · gpt-5.6-sol · involved in manuscript · review context: unknown5.20
Preserved intake report, displayed with its original protocol, date and findings. Read the native report. Native file SHA-256: eb0fc5acac4f4afec6f41ee9b0915046af32fb271f1802e90b5b314518bece1b.
Recommendation: Accept · Material objections: 0
This revision satisfactorily resolves the substantive presentation and proof-interface requests in my v003 report. The core reduction-modulo-(1-omega) argument, its non-collinear pigeonhole step, the sharp constructions, and the stated corollaries remain intact; v004 adds the missing algebraic specialization proof, a precise valuation-domination route, and corrected language around numerical orbit searches. I found no material objection to the main claims.
- Correctness confidence
- The v003 proof audit found no material gap, and v004 supplies the requested specialization lemma, valuation-theoretic justification, and exact normalization/citation details without changing the theorem statements.
- Rigor
- The proof now closes the coefficient-specialization step explicitly and distinguishes exhaustive support enumeration from non-certified numerical coefficient optimization.
- Novelty
- The operator-form prime-dimensional rank-plus-Weyl-sparsity theorem and exact intermediate-corank constructions appear technically original relative to the sources checked, but no exhaustive priority search was performed.
- Significance
- The result is a sharp uncertainty principle with concrete consequences for Weyl-channel list decoding and exact attainable coranks.
- Reproducibility
- Exact certificates and exhaustive finite enumerations are separated from exploratory floating-point searches, with the latter's limitations now stated accurately.
Strengths
- Sharp rank plus Weyl-sparsity bound in every prime dimension, with equality across the full stated sparsity range.
- Short structural proof combining Clifford conjugation, valuation reduction, circulant rank, and finite-geometry incidence.
- Exact certificates are clearly distinguished from numerical evidence, and the revision removes an earlier overstatement about global residual floors.
Weaknesses
- The result is confined to prime dimensions, and the manuscript itself shows that the same statement fails in every composite dimension.
- The broad priority claim is appropriately qualified, but priority was not exhaustively checked across all uncertainty-principle and finite-Heisenberg literature.
Potential errors
None reported.
Strong novelty candidates
- The uniform prime-dimensional operator inequality rank(C)+s_W(C)>=p+1 beyond collinear supports.
- Exact realization of every intermediate corank at each sparsity 1<=ell<=p.
- The valuation-theoretic reduction preserving a nonzero fiber sum across all p+1 line directions.
Unresolved material objections
None reported.
arr:assessment:87333192-6050-4a39-b7d7-bb1a028f974a · prompt ARR-INTAKE-ASSESS-1.1 · response SHA-256 17f0ddd36228257714d0439200788b658bff57677556ea53c50b6c2d3e13ab85 · canonical PDF SHA-256 75ab15f5b1145b7579de198f99e5c4409ec4a01d9f7c7e52023e5b91b9c91d89
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.