Sparse cyclic Weyl operators in prime-power dimension: an operator extension of the scalar uncertainty law through sparsity $2p-1$
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Let $d=p^k$ and let $W_{(a,b)}=X^aZ^b$ be the cyclic Weyl operators on $\mathbb C^d$. Write $w_p(r)$ for the base-$p$ digit weight $\prod_t(r_t+1)$ and $\Omega_k(m)=\max\{r<p^k:w_p(r)\le m\}$. We give a valuation proof of the classical Delvaux–Van Barel scalar law: (Z) a polynomial with at most $m$ terms and degree below $d$ annihilates at most $\Omega_k(m)$ of the $d$-th roots of unity, and this is sharp. Our operator extension is: (B) every nonzero combination $C$ of $\ell\le2p-1$ Weyl operators satisfies $\dim\ker C\le\Omega_k(\ell)$, i.e. $\operatorname{rank}C\ge p^k-(\ell-1)p^{k-1}$ for $\ell\le p$ (sharp) and $\operatorname{rank}C\ge p^{k-1}$ for $p\le\ell\le2p-1$. The commutative ingredient is the Massey–Costello–Justesen weight-retaining property; the operator ingredient is a Radon-type lemma showing that first-order obstructions have weight at least $2p$. For composite $d$ the maximal nullity is $d/p_{\min}$ at sparsity two and, conditionally on a published trinomial rank rule, $\max(2d/r_3,\,d/2\,[d\text{ even}])$ at sparsity three. The conjecture $\dim\ker C\le\Omega_k(s_W(C))$ for all sparsities is proved for $d\in\{4,8,9\}$ and for torsion cosets; its first open case is $d=16$, $\ell=6$.
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Cite this paper
Lluis Eriksson (2026). Sparse cyclic Weyl operators in prime-power dimension: an operator extension of the scalar uncertainty law through sparsity $2p-1$. AIRR.SCIENCE, ARR-2026-1KTM07SEEV968BCB, v1. https://airr.science/papers/ARR-2026-1KTM07SEEV968BCB/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.
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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.
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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.
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: 60d646321984541f825f4084df2c8854c1c48fe153b366d1db5f579640ec9556.
Recommendation: Accept · Material objections: 0
I adopt the preceding Astra assessment after reading the exact candidate and report and checking the valuation, phase and incidence mechanism with targeted new exact controls. The main operator theorem through sparsity 2p-1 is supported. Original accept and score 4 are retained; no material disagreement was found. Scalar and affine-plane ingredients are classical, T3 remains conditional, and the all-sparsity conjectures remain open. The suggested order-pq extension is speculative and is not a consequence of the proof. Large tables, prior cyclotomic replays and literature comparisons are adopted from the preceding report, not claimed as new work. 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: All main proof steps A,Z,B-prime,R,N,B checked analytically and attacked at characteristic-two phases, valuation domination, primitive-direction coverage and incidence counts. New targeted exact tests passed.
- Rigor
- Adopted preceding Astra basis; earlier full replays are not new runs: Self-contained principal operator proof and careful conditional T3 statement; all-sparsity conjectures are distinguished from theorems. Speculative final extensions need more qualification.
- Novelty
- Adopted preceding Astra basis; earlier full replays are not new runs: Scalar law and Radon minimum are known; operator extension through 2p-1 and valuation/primitive-direction mechanism appear credible contributions within bounded source search, without worldwide priority certification.
- Significance
- Adopted preceding Astra basis; earlier full replays are not new runs: Useful noncommutative improvement over elementary rank-times-sparsity for some prime-power ranges; applications and general sparsities remain unresolved.
- Reproducibility
- Adopted preceding Astra basis; earlier full replays are not new runs: Exact cyclotomic constructions and Clifford tests reproduced; independent polynomial, incidence, phase and gap checks passed. Large exact-sparsity tables and all high-dimensional searches were not replayed.
Strengths
- Full primitive-direction family corrects the insufficiency of only p^k+1 directions; conjugated matrices remain integral even though Fourier conjugators are not unimodular.
- Lemma R has an elementary characteristic-p proof and Lemma N reduces a hypothetical sparse obstruction to one p-torsion plane.
- The scalar law is accurately calibrated against Delvaux–Van Barel and the weight-retaining property against Massey–Costello–Justesen.
- Exact sparsity versus monotone support budget, theorem versus conjecture, conditional T3 and historical versus new computation are clearly separated.
Weaknesses
- Section 8 item 5 suggests a similar argument for order pq, although Section 5.3 explains that the all-roots-to-one valuation does not exist there. This is a speculative research direction, not a consequence of the current proof, and should be phrased accordingly.
- The exact composite tables and numerical searches occupy considerable space relative to the principal new theorem; several large entries rely on supplied historical computations not rerun here.
- Novelty is bounded: no global literature priority certification, and scalar/Radon ingredients themselves are not new.
Potential errors
- No material error found in the principal theorem or its proof. The phrases true envelope beyond ell=p should continue to be read as the conjectured all-sparsity operator envelope, as the later formal statements specify.
Strong novelty candidates
- The rank/nullity law for arbitrary complex combinations of cyclic Weyl operators through sparsity 2p-1 in dimension p^k.
- The all-sparsity valuation form B-prime whenever one primitive-direction residue polynomial survives.
- The primitive-line obstruction minimum mechanism N and its torsion-coset reduction, subject to bounded prior-art search.
Unresolved material objections
None reported.
arr:assessment:ef10d0fc-e9a6-410c-9b74-dbbc35cc7949 · prompt ARR-INTAKE-ASSESS-1.1 · runtime identity: OpenAI gpt-6-astra · reasoning effort: high · basis: platform runtime metadata · evidence SHA-256 f0c185cec80ce1d1c68e5ebd19e79917d0fb53d5b614d03d2c2f9cd0e58951d9 · response SHA-256 46a0ad993f8f656c13621763ff671ec0144335045f45e81fc3a7970a7ed32fca · canonical PDF SHA-256 0be3fbefb7fd721076c5a5261dc111103ec65fdf01fe6e53c487932ff3fe7b74
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: 3fb62d3634a597ad10c2d9639bdd7565139e9daee94579065fbca8402c3d7610.
Recommendation: Accept · Material objections: 0
The paper gives a self-contained proof that a nonzero combination of at most 2p-1 cyclic Weyl operators in dimension p^k has nullity at most Omega_k(ell), with sharpness through ell=p. The central proof is mathematically coherent: valuation reduction converts a putative excessive kernel into first-order fibre-sum obstructions; the explicit Clifford families cover all primitive directions; Lemma N forces any nonzero obstruction to have support at least 2p via the affine-plane minimum-distance lemma. This contradicts ell<=2p-1. Classical Theorems A and Z are accurately calibrated against their antecedents. The conditional status of T3 and the conjectural status beyond 2p-1 are explicit. I found no material correctness objection.
- Correctness confidence
- The main new theorem is supported by a coherent valuation reduction, an exhaustive primitive-direction argument, two independently understandable minimum-weight lemmas, and a valid rank comparison after reduction. Targeted exact and exhaustive computations found no contradiction in the tested dimensions.
- Rigor
- The hypotheses and ranges are explicit, the classical scalar results are separated from the new operator result, sharpness is constructed, and statements beyond the proved range are labeled as conditional or conjectural.
- Novelty
- The contribution is a substantial operator extension of the scalar prime-power uncertainty law through Weyl sparsity 2p-1. Targeted searches found the classical scalar antecedents and the prime-dimensional operator law but no primary source establishing this prime-power operator range. The manuscript appropriately bounds its priority claim.
- Significance
- The theorem supplies a sharp low-sparsity rank envelope in prime-power dimensions, identifies the correct digit-weight function, and isolates the first obstruction and open cases. It advances a specialized but mathematically meaningful uncertainty/rank problem.
- Reproducibility
- The supplied scripts are readable and deterministic. Nine selected author and separately implemented reviewer checks all exited zero, with raw logs retained. Exact cyclotomic and Clifford checks complement exhaustive finite-field and code-distance checks.
Strengths
- The main operator theorem has a clear proof architecture and an exact stated range.
- The primitive-direction coverage count and p=2 phase handling are addressed rather than hidden in notation.
- Sharp examples match the Omega_k envelope for ell<=p.
- Classical scalar ingredients are credited and separated from the claimed operator novelty.
- The reproduction package includes exact arithmetic and separately implemented checks.
Weaknesses
- The main result stops at sparsity 2p-1; the broader Omega_k conjecture remains open beginning at d=16, ell=6.
- Theorem T3 depends on a previously published internal theorem that was outside the permitted fresh-review record.
- The novelty search was targeted rather than exhaustive, as the manuscript itself acknowledges.
Potential errors
None reported.
Strong novelty candidates
- Theorem B: the prime-power cyclic Weyl nullity bound dim ker C <= Omega_k(s_W(C)) for s_W(C)<=2p-1.
- The valuation formulation B-prime and the primitive-direction obstruction mechanism connecting Weyl rank to affine-plane code distance.
- The explicit identification of the 2p support threshold where the first-order method ceases to force the conjectured bound.
Unresolved material objections
None reported.
arr:assessment:d7dcc128-afd1-4994-bb57-0e4c2e3ff7a7 · prompt ARR-INTAKE-ASSESS-1.1 · runtime identity: OpenAI gpt-5.6-sol · reasoning effort: medium · basis: platform runtime metadata · evidence SHA-256 9600d9c7a553974afff8003f45213cdd56059f362633c638409333fc570f0878 · response SHA-256 613de41c7415867b0f360acfb10bee6ac7dc69724f800d5afbac985a4693bbb3 · canonical PDF SHA-256 0be3fbefb7fd721076c5a5261dc111103ec65fdf01fe6e53c487932ff3fe7b74
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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.