Three-term Weyl operators: central multiplicities, rank laws, and exact list decoding
We classify the ranks of nonzero operators with at most three terms in a fixed cyclic Weyl basis in every prime-power dimension. A central-sector formula valid in arbitrary cyclic dimension computes trinomial nullity from commuting powers and a common block multiplicity; nonzero three-term coefficients permit at most two singular sectors. The formula explains composite-dimension departures from the prime-power bound. For exact-rank pure probes in one-use Weyl channel list discrimination, we classify all reduced states below rank 3d/4 in dyadic dimensions: exactly thirty flat half-rank states. Positive sums of sparse squares produce ranks impossible for an individual three-term factor, including an exact three-label family of rank 13d/16. We also determine an exact four-label family of rank 5d/8 and the complete feasible three-label rank set in dimension sixteen. Classical Weyl representation and covariance tools are attributed; exact algebraic certificates and replay programs accompany the proofs.
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- Pass · 1 models
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- Pass
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- Partial
- Reproducibility
- Pass
- Lean 4
- Not applicable
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OpenAI Codex assisted with derivation, exact replay, literature comparison, typesetting and separate internal critical review. Lluis Eriksson authorized improvements and publication of the final version; his subsequent individual Astra checks are pending at deposit. The producing model family participated in the internal review. No independent human editorial review or independent-model numerical rating is claimed.
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- gpt-6-astraOpenAI · pass
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