Three-term Weyl operators: central multiplicities, rank laws, and exact list decoding

Strengths

• Theorem 2 cleanly separates the standard cyclic Weyl-pair block form from the new rank application: each singular central character contributes exactly m=d/(qh), and the circle argument bounds genuine-trinomial singular sectors by two.
• Theorem 3 treats one-, two-, commuting three-, and noncommuting three-term supports separately and supplies explicit attainments for every listed prime-power corank; the dyadic exclusion of new individual coranks is logically complete.
• Lemmas 5-6 and Theorem 7 use a sharp dimension-intersection argument: below 3d/4, every sparse square has one of thirty half-dimensional ranges, and two such ranges inside the same smaller support must coincide.
• Theorem 8 correctly exploits positivity: the three kernels have exactly the characters (0,0),(0,1),(1,0), giving corank 3d/16, while the divisibility proof rules out list size two.
• Executed quantum_weyl_verify.py passed, checking 714,096 universal arithmetic pairs, 1,948,590 noncommuting prime-power pairs, the d=8 and d=16 projector incidences, rank-13d/16 cases, and exact composite matrices.

Weaknesses

• The claimed negative literature search is necessarily bounded. The inspected primary antecedents support the cyclic-block, stabilizer-multiplicity, Horn-style, and factor-width background, but they cannot establish worldwide priority for the new rank lists.
• Section 10 refers to quantum_sources.md, but that file is not listed in the supplied public-code manifest. This is a packet/documentation limitation, not evidence that the file is absent from the public repository.
• The original quantum_independent_checks.py completes its mathematical function calls but then fails while hashing manifest-unlisted legacy paths under work/cycle3/quantum. A referee wrapper reran those functions unchanged against the current supplied sources, but the original main program did not fully pass.
• The finite programs test many dimensions and examples but do not prove arbitrary-coefficient or arbitrary-dimension quantifiers; correctness still rests on the written representation, circle-intersection, character, and positivity arguments.

Potential errors

• The opening phrase that a noncommuting pair in prime-power dimension has m=1 must be read with the later hypotheses: distinct nonidentity labels in the fixed cyclic Weyl representation. The formal statements supply those restrictions, so this is expository rather than a counterexample.
• The lift-independent gcd formula (4c) is compressed. I rederived G/g and checked the cancellation using gcd(q,delta/g)=1; the exhaustive arithmetic replay found no violating lift in its tested range.
• The POVM normalization in equations (8)-(9) depends on the stated vectorization convention and transpose placement under partial trace. Tracing the dimensions and Weyl twirl recovers the asserted normalization; no scaling error was found.

Strong novelty candidates

• The explicit common central multiplicity m=d/(qh)=dq/|H| and exact sector-count formula for arbitrary cyclic composite dimension.
• The complete individual rank lists for Weyl sparsity at most three in every prime-power dimension.
• The thirty-state classification below rank 3d/4 and the exact three-label rank 13d/16 construction for all dyadic d divisible by 16.

Unresolved material objections

None reported.

Strengths

• Lemma 1 and Theorem 2, pp.2-5: commuting with both generators forces coefficients divisible by q, so R=qH and |H|=q^2h. Trace character averaging gives equal sector dimensions; cyclic blocks give determinant [c^q-(-1)^q(a^q s+b^q t)]^m and exactly m nullity per singular sector.
• Theorem 2: the two-circle argument needs abc!=0 and bounds distinct central pairs, not multiplicity. Exact composite ranks include d=36 rank 30 with m=3, and d=12 rank 6 when c=0. These correctly refute the tempting multiplicity-free and zero-constant extensions.
• Corollary 2.1 and Theorem 3, pp.5-6: p-adic valuation makes m=1 for noncommuting prime-power supports; commuting character multiplicities give the remaining coranks. The binomial and quadratic constructions attain the whole stated list, including p=2 and p=3 endpoints.
• Lemmas 5-6 and Theorem 7, pp.7-9: low rank forces flat half-dimensional sparse squares. Distinct half-ranges intersect in at most d/4, so a positive sum below 3d/4 has a single common range. Exact checks reproduce 30 projectors and the d=8 and d=16 incidence counts.
• Theorem 8, pp.10-11: the three joint kernel cells are (0,0),(0,1),(1,0); positive summation gives rank 13d/16 and trace 12d. The two-label common-fixed-space argument forces corank to divide d, excluding two labels for this family.
• Equation (9), p.9: an independently written exact d=8 check verifies 1,024 POVM completeness entries and 4,096 outcome/input support relations for a rank-four probe. This confirms the 1/d normalization and translated reported lists in that finite case.
• Theorem 9, p.11: the order-eight cubic has four nonzero coefficients, exactly three zero eigenspaces, and squared norm 4d(2+sqrt(2)). Theorem 7 supplies the lower bound. Exact replays cover d=8,16,32,64.
• Main replay checks 714,096 general label pairs, 1,948,590 noncommuting prime-power pairs, nineteen determinant orders, all 435 projector pairs at d=8,16, and seven composite matrix fixtures. A separate supplied route adds 605 label pairs, six exact matrix cases, and nine determinant orders.

Weaknesses

• Corollary 8.1, p.11, should explicitly say that there exists a probe of rank r. Its abstract and proof classify attainable ranks; the current wording can be read as sufficient for a fixed arbitrary probe. The distinction matters because Theorem 7 enforces flatness at rank eight.
• Standard components dominate the universal reduction: repeated Weyl blocks and central characters are prior representation theory, while general list decoding/factor width is prior quantum classification. The strongest novelty candidates are the specific rank and state consequences.
• The full set of higher-rank three-list states for general dyadic d and the global composite-dimension individual rank list remain unclassified. Exact-rank conclusions do not automatically transfer to rank budgets, approximate decoding, mixed probes, or multiple uses.
• evidence/cycle3/quantum_independent_checks.py completed its five mathematical functions but failed hashing the absent work/cycle3/quantum/quantum_dyadic_trinomials.md path. Full execution therefore failed. A separate referee wrapper preserves the function results; no source path was filled.
• quantum_sources.md is referenced but excluded from this packet. Public repository absence was not established, and no prior screening material was sought. The present primary-PDF comparison is bounded and does not certify worldwide priority.

Potential errors

• Corollary 8.1 has a false literal fixed-probe reading: diag(2/9,1/9 repeated seven times,0 repeated eight times) has rank 8 but is not flat, so Theorem 7 forbids a three-list decoder. Replacing the opening by an explicit existence statement resolves this local wording issue; the abstract and constructive proof already use that quantifier.

Strong novelty candidates

• Theorem 3: complete individual corank lists for cyclic prime-power Weyl sparsity at most three, obtained by combining central multiplicities and a two-circle count.
• Theorem 7: exactly thirty flat half-rank states form the entire three-list stratum below 3d/4 in every dyadic dimension d>=8.
• Theorem 8: the 13d/16 exact three-label family from three positive sparse squares, separating individual sparse rank laws from feasible ranks of their sums.

Unresolved material objections

None reported.

Strengths

• Theorem 2 identifies the radical R=<qu,qv>, proves |H|=q^2|R|, and obtains the common block multiplicity m=d/(q|R|); the cyclic determinant and one-dimensional kernel recurrence then justify the exact nullity formula sector by sector.
• Theorem 3 cleanly separates binomial, commuting trinomial, and noncommuting trinomial cases and supplies explicit attainments of every listed corank in prime-power dimension.
• Lemmas 5-6 and Theorem 7 turn individual-rank rigidity into an exact low-rank positive-factorization stratum; the intersection-dimension argument correctly forces all half-rank ranges in a sub-3d/4 support to coincide.
• Theorem 8 explicitly demonstrates why sums of three-sparse squares have richer ranks than individual three-sparse matrices, and the two-label divisibility converse rules out a smaller list for corank 3d/16.
• Fresh execution of quantum_weyl_verify.py, composite_matrix_checks.py, and quantum_composite_review_checks.py completed successfully with exact arithmetic and without using the supplied stored certificates as executions.

Weaknesses

• The novelty discussion depends partly on a companion quantum_sources.md record that is mentioned in Section 10 but is not present in the supplied batch, so the scope of the manuscript's own negative search cannot be audited.
• Several standard phase-removal and character-representation steps are stated tersely; the arguments are recoverable and appeared correct, but fuller definitions of phased representatives and label-group characters would reduce the burden on readers.
• The finite replay is extensive but cannot certify the quantified statements over arbitrary dimensions and arbitrary complex coefficients; those remain dependent on the analytic proofs, as the manuscript itself correctly acknowledges.

Potential errors

• The claim that quantum_sources.md records the literature-reading scope is factually unreproducible from this batch because that file is absent; this affects the novelty audit, not the proved algebraic statements.
• No formal proof-assistant or independent exhaustive coefficient check was supplied. I found no concrete counterexample, but the universal coefficient quantifiers are verified here by proof inspection rather than mechanization.

Strong novelty candidates

• The explicit arbitrary-d central multiplicity formula and its use to bound singular sectors of a genuine Weyl trinomial.
• The complete prime-power rank set for nonzero matrices with Weyl sparsity at most three.
• The all-dyadic thirty-state low-rank three-list stratum together with the positive-sum rank 13d/16 family.

Unresolved material objections

None reported.

Strengths

• Lemma 1, p.3: the relation lattice lies in qZ^2, giving |H|=q^2h. Trace orthogonality forces equal central-sector dimensions, and unitary eigenvalue cycling supplies exactly m equivalent q-dimensional copies.
• Theorem 2, pp.3-5: diagonal and full-cycle determinant terms give the correct even/odd sign. Division only by nonzero shift weights establishes one-dimensional singular-block kernels even when c=0; the circle argument is correctly restricted to abc!=0.
• Arithmetic formula (4c)/Corollary 2.1, p.5: the lattice-index gcd yields m, and the prime-power valuation forces m=1 for q>1. The commuting q=1 branch remains included in the exact formula.
• Theorem 3, pp.5-6: the noncommuting and commuting alternatives exhaust supports after right normalization. Binomials realize each p^j corank, and the odd-order quadratic realizes each additional 2p^j corank.
• Proposition 4 and (6a)-(6c), pp.6-7: the d=12 nullity-three case, d=36 nullity-six sharp case and zero-constant control separately test multiplicity, two-sector sharpness and the necessary coefficient hypothesis.
• Lemmas 5-6/Theorem 7, pp.7-9: low-rank squares are flat half-rank projectors; the 30-member count and intersection dimensions force every positive-sum range below 3d/4 to coincide, proving the full state-level stratum.
• Theorem 8, pp.10-11: the three common kernel cells give nullity 3d/16, while the 3,3,9,1 cell multiplicities give the displayed normalized spectrum. The independent two-label divisibility argument proves the lower bound at exact rank.
• Theorem 9/Corollary 8.1, p.11: all four polynomial coefficients are nonzero, precisely three eigenspaces are killed, and the strict low-rank exclusion yields the matching four-label lower bound and complete d=16 attainable rank list.
• Fresh supplied execution covers 714096 arbitrary-dimension label pairs, 1948590 noncommuting prime-power pairs, both 435-pair projector campaigns and direct composite matrices. Additional POVMs at (d,r,list size)=(8,4,3),(8,5,4),(16,13,3) have completeness residual at most 4.13e-14 and wrong-list probability at most 2.04e-30.

Weaknesses

• Section 9, p.11, says to place two Python files together, but the main verifier also hashes quantum_weyl_trinomials.md beside itself. That exact source is included in this batch and the full run succeeds; the short instructions should explicitly name it as a required file.
• Section 6, p.9: the sparse-factor-to-POVM equivalence is terse and the supplied exact programs test rank/character algebra rather than actual measurement probabilities. My additional explicit POVM checks reduce this implementation risk but are finite floating-point tests.
• Section 10, p.13: the priority search and earlier internal-note boundaries are author assertions. I consulted public primary antecedents, but did not access the absent quantum_sources.md, earlier notes, screening files or previous assessments; exhaustive novelty and earlier-version priority are not established.
• The universal central formula does not enumerate every attainable individual rank in composite dimensions. The thirty-state classification stops strictly below 3d/4, and the higher-rank feasible density-matrix set is not classified. These are acknowledged scope limits, not extensions proved by the computational samples.

Potential errors

None reported.

Strong novelty candidates

• Theorem 2 with formula (4c): a coefficient-explicit trinomial rank law in the fixed cyclic Weyl representation with a computable common central multiplicity and sharp two-sector bound.
• Theorem 3: the complete prime-power three-sparse individual rank lists, particularly the dyadic fact that a third individual Weyl coefficient adds no new rank.
• Theorem 7: the all-dyadic state-level rigidity to exactly thirty flat half-rank projectors below the strict 3d/4 cutoff.
• Theorems 8-9: the exact-rank 13d/16 positive-sum three-list family and 5d/8 four-list family, separating sparse-factor rank from probe-rank optimization.

Unresolved material objections

None reported.