A finite geometry of low-rank three-list Weyl probes in ternary dimensions

Strengths

• Lemma 1 supplies both directions of the operational sparse-positive-factorization equivalence, including normalization and an explicit covariant POVM construction.
• Lemma 3 turns a rank inequality on every positive summand into a rigid order-three commuting algebra; positivity then makes the global classification possible.
• The affine-plane argument distinguishes crossing pairs, parallel pairs, and nonconcurrent triangles and supports uniqueness through private cells rather than spectrum alone.
• Theorem B cleanly separates two-list midpoints from all other low-rank states and proves the 4d/9 versus 5d/9 exact-rank reversal.
• Fresh runs checked 4,095 line subsets, exact matrices in dimensions 9, 27, and 81, 6,320 ordered label pairs, and an explicit normalized POVM fixture.

Weaknesses

• The main novelty claim is responsibly bounded in the paper, but this review performed no external priority search, so the exact bibliographic position remains unverified.
• The result is restricted to the single cyclic Weyl representation, exact zeros, k≥2 ternary powers, and ranks below 7d/9; approximate stability and higher-rank strata remain open.
• The stored certificate is not provenance-clean for the current source because its embedded manuscript hash is 7ff52d… rather than the manifest-verified 725210….

Potential errors

• The stale embedded manuscript hash in the stored ternary certificate could mislead a reader into treating it as a certificate for the current source. Regeneration corrects the hash and all checks pass, so this does not presently undermine the proofs.

Strong novelty candidates

• Conditional on external priority, the complete positive-sum low-rank ternary stratum with unique line-projector decompositions and exact sparse-square lengths is a strong contribution.
• The explicit exact-rank list-complexity pattern, including list size four at 4d/9 but three at 5d/9, is a sharp and conceptually useful nonmonotonicity example.

Unresolved material objections

None reported.

Strengths

• Lemma 1, p.3: the partial-trace transpose and 1/d twirl normalization are consistent. Fresh crossing and triangle fixtures with generic complex ancilla bases yield 162 and 243 outcomes, lists of size three, and normalization residuals below 2.4e-14.
• Lemma 2, p.4: the valuation normal form gives |R|=d/q and q-dimensional central blocks. Nonzero shift weights ensure nullity one per singular block, while the two-circle equation permits at most two singular characters.
• Lemma 3, pp.4-5: right multiplication preserves CC*, sparse rank below 7d/9 excludes noncommuting trinomials, and commuting groups of order at least nine have too few zero characters. This forces the actual square, not merely its rank, into at most two parallel line projectors.
• Lemma 4 and Theorem A, pp.5-6: support containment and private cells establish uniqueness of positive decompositions. Exact enumeration gives 12 vertices, 12 parallel edges, 54 crossing edges and 72 triangles, with the stated ranks and sparse-square lengths.
• Theorem B, p.7: order-three singular binomial squares are flat complements, leaving exactly twelve two-list states. (I-U)(I-V) supplies rank 4d/9 with four labels, while Theorem A excludes that rank with three labels.
• Corollary 5 and Theorem C, pp.7-8: kernel intersections rule out two labels at rank seven in dimension nine. The order-nine polynomial kills exactly two eigenvalues and has a nonzero T^2 square coefficient outside the fixed algebra, proving sharpness of the strict cutoff.
• Both supplied mathematical programs passed. The primary run checked 4095 line subsets, actual projector matrices at d=9,27,81 and 6320 ordered label pairs at d=9; the second checked an explicit 81-outcome POVM. Fresh sampled subgroup checks extended to d=243.

Weaknesses

• The supplied stored ternary_certificate.json records manuscript SHA-256 7ff52dd2210f7b63ab0060c8fa021d1e352a54fd6a91e3b851faaa171d107700, not the supplied ternary_low_rank.md hash 725210b87e576ba5f0658e579bac18f4050cc3208353d8b51a97c7da2b9abe0f. Fresh execution changes only this parsed field. Refresh the certificate binding; stored status alone is not exact-source verification.
• The original wrapper failed only after both mathematical subprocesses passed, when hashing the packet-missing public-code/paper.pdf. The authenticated supplement supplies that genuine path and the complete replay now passes. This was a packet-layout limitation, not verified absence from the public repository.
• The verifier proves finite identities and incidence counts, not the arbitrary-coefficient and arbitrary-k rigidity assertions. Those remain analytic dependencies examined here, without proof-assistant formalization.
• No stability estimate is given when exact zeros become small errors; rank and exact list complexity can change discontinuously. The manuscript appropriately does not extrapolate to approximate decoding or ranks at/above 7d/9.
• The general learning-width/factor-width antecedent was checked directly, but the separately published trinomial predecessor was not retrieved. Novelty is therefore a bounded assessment of the stated positive-sum contribution, not a verified priority claim.

Potential errors

• The stale manuscript hash in the stored certificate is a reproducibility metadata error, not a counterexample to the classification. All its other parsed fields are reproduced by the freshly executed primary verifier; the new wrapper also verifies the locked canonical PDF.

Strong novelty candidates

• Theorem A: rigidity of every sparse positive summand below 7d/9 yields a unique, finite simplicial description of all feasible reduced states and exact sparse-square lengths.
• Theorem B: the complete fixed-state two/three-list distinction within that stratum and exact-rank nonmonotonicity, with a four-term construction at rank 4d/9.
• Theorem C: a concrete order-nine square shows that the universal commuting-algebra conclusion fails exactly at the proposed cutoff.

Unresolved material objections

None reported.

Strengths

• W-S1 | Lemma 1, p. 3: the vectorization convention and partial-trace transpose give sum B_j^*B_j=d I_r in the necessity direction, hence C_j=A B_j^*/sqrt(d). Conversely A^+C_j and the 1/d Weyl twirl yield a normalized POVM with the specified support lists. Label inversion/phases preserve sparsity; the channel interpretation is not inferred from rank alone.
• W-S2 | Lemma 2, p. 4: the 3-adic coordinate reduction makes the central subgroup order d/q, so each central sector has dimension q rather than an unaccounted repeated multiplicity. Nonzero shift weights give nullity exactly one in a singular sector. With a,b,c nonzero, the determinant-zero condition is an intersection of distinct circles, allowing at most two central-character pairs. The crucial subgroup count passed for all 5616 noncommuting ordered distinct nonzero label pairs in dimension nine.
• W-S3 | Lemma 3, pp. 4-5: rank<7d/9 excludes noncommuting trinomials because d-2>=7d/9 for d>=9. A commuting group of order h>=9 similarly gives rank>=d-2d/h. Thus the relative support must reduce to an order-three algebra, and CC^* is a nonnegative combination of at most two parallel line projectors. Right multiplication preserves CC^*, so the resulting fixed ambient algebra is not lost in normalizing the sparse factor.
• W-S4 | Lemma 4 and Theorem A, pp. 5-6: positivity makes support exactly a union of character cells. Independent enumeration of all 4095 nonempty line subsets gives 12 singletons,54 crossing pairs,12 parallel pairs and72 triangles with union size below seven, and verifies both no hidden lines and a private point on every constituent. Private-cell values recover the weights uniquely, including when unordered eigenvalues coincide.
• W-S5 | Theorem A, pp. 2,6: each individual sparse square can use only parallel lines. Crossing pairs and triangles therefore require respectively two and three squares, whereas a parallel pair can be combined into one factor. This establishes the stated minimal lengths, not just upper bounds. Exact integer-pair arithmetic verified all nine scaled cell projectors, twelve scaled line projectors and 66 pair intersections per dimension in d=9, 27, 81.
• W-S6 | Theorem B and Corollary 5, pp. 7-8: singular order-three binomial squares are flat complements, forcing the twelve equal-weight parallel midpoints as the only two-list states below the cutoff. The four-term (I-U)(I-V) has square nine on exactly four cells, rank4d/9 and trace4d; the square/trace identity was checked exactly in all three dimensions. The rank-seven two-list obstruction in d=9 correctly uses the common kernel of positive sums.
• W-S7 | Theorem C, p. 8: (T-I)(T-xi I) with T=Z^(d/9) has exactly two zero order-nine eigenvalues and hence rank7d/9. Its nonzero T^2 coefficient lies outside the order-three label subgroup, proving failure of the fixed-algebra conclusion at equality. This is a valid sharpness example for the structural statement, not a classification of every state at or above the cutoff.
• W-S8 | Fresh public replay: ternary_verify.py completes exact Q(omega) matrix and integer checks for all 4095 line subsets, 150 low-rank supports, spectral probes, twelve lines/nine cells in dimensions 9, 27, 81, projector intersections, explicit mixtures, the four-term construction and 6320 ordered dimension-nine label pairs. Its parsed output matches the supplied certificate exactly apart from the omitted missing-Markdown hash.
• W-S9 | Additional checker before its metadata failure: 6309 assertions completed. These include a collinear-triple reconstruction of the geometry, exact four-term matrix rank in dimensions 9 and27, the 1/d twirl normalization of an actual rank-three probe in a 27-dimensional joint space, all 81 three-label outcome supports, and 5616 noncommuting central-subgroup counts. This directly exercises Lemma 1 normalization in a physical finite fixture, beyond the original assessment's projector-only computations.

Weaknesses

• W-W2 | References [2]-[6], p. 9: the general learning-width/factor-width correspondence is explicitly present in Johnston et al., Lemma 9 and Corollary 11 [https://arxiv.org/html/2510.20789v1]. Character-space size and Weyl block structure have the cited precedents [https://arxiv.org/pdf/1101.1519; https://arxiv.org/pdf/2101.00129]. These sources support the ingredients, not priority of the 150-family positive-sum classification. Gross and Veitch et al. were checked only at abstract level, not at all cited sections.
• W-W3 | pp. 1-2,8-9: the commuting rank-d/3 projectors belong to a single cyclic d-dimensional representation. They cannot be identified as operators with twelve pure qutrit stabilizer states or carried unchanged to a tensor-product Pauli sparsity problem. The paper states this correctly; it materially narrows possible generalizations.
• W-W4 | pp. 3-8: finite enumeration certifies the nine-point incidence geometry, not all complex sparse coefficients or every k. The latter depend on the analytic circle and central-sector arguments. No formalization or numerical search over all coefficients was performed, and no stability statement under approximate zeros is proved.
• W-W1-S | Source packaging: the programs and frozen numerical certificate are now supplied. research/ternary_low_rank.md remains absent. The main verifier treats its hash as optional and completes without it; independent_ternary_review.py requires it and fails after its calculations. replay.py invokes that checker and fails too; its final paper.md/paper.pdf hash stage is not reached. No missing manuscript was fabricated or substituted.
• W-W5-S | Certificate binding: the frozen ternary_certificate.json source_sha256 equals the supplied ternary_verify.py hash. Its separate manuscript_sha256 is for missing Markdown, not the canonical PDF. The exact geometry/matrix/arithmetic content is fully reproduced, but that Markdown/PDF lineage is not verified.
• W-W6-S | Execution of a program named independent_ternary_review.py is additional computation, not evidence of external reviewer independence. No earlier referee reports or ratings were consulted. The source reference/reading-ledger limitation in the original assessment is not resolved by the numerical archive.

Potential errors

None reported.

Strong novelty candidates

• Lemma 3 plus Theorem A: forcing every summand of an arbitrary low-rank positive sparse factorization into one fixed order-three algebra, then classifying all positive sums.
• Theorem A: unique decomposition into line states together with exact minimal three-sparse-square length for each family.
• Theorem B: the exact fixed-state two/three-list distinction and ell_min_exact(d,4d/9)=4 versus ell_min_exact(d,5d/9)=3. The incidence geometry and operational factor-width principle themselves are antecedents.

Unresolved material objections

None reported.

Strengths

• Lemma 1 states and proves an operational if-and-only-if factorization, with normalization and an explicit covariant POVM rather than only a necessary condition.
• Lemma 3 uses positivity to constrain every summand simultaneously and converts the analytic rank cutoff into a finite affine-plane classification.
• Theorem B distinguishes sparse-square count from list size and supplies explicit witnesses for l_min^exact(d,5d/9)=3 and l_min^exact(d,4d/9)=4.
• The rerun programs use exact rational/Q(omega) arithmetic for the central finite geometry and matrix identities; no numerical rank threshold is used.

Weaknesses

• The proof that the radical is generated by q u and q v with the asserted order, and hence that every singular noncommuting trinomial has nullity at most two, is much terser than the rest of the paper.
• The exhaustive label-pair computation is only for d=9; arbitrary k still rests entirely on the written p-adic normal-form proof.
• Novelty is only boundedly searched, and the same-author predecessor cited as reference 1 was not among the supplied artifacts or located in the public checks.
• The structural theorem is intentionally narrow: it does not address rank at or above 7d/9, approximation, or tensor-product qutrit Weyl bases.

Potential errors

• Lemma 2's circle-intersection argument assumes the central-character pairs are distinct unit-modulus pairs and that the two circles cannot coincide. Nonzero c makes coincidence implausible, and I found no counterexample, but the manuscript does not spell out this exceptional-case exclusion algebraically.
• The code verifies finite incidence and representative matrix identities, not the quantification over arbitrary complex coefficients in every 3^k dimension; passing it therefore cannot close a hypothetical gap in Lemma 2 or Lemma 3.

Strong novelty candidates

• The complete 150-relative-interior description of the low-rank three-list state set for the cyclic ternary Weyl basis.
• The exact fixed-rank list-size nonmonotonicity 3,4,3,2 across ranks d/3,4d/9,5d/9,2d/3.
• The uniqueness and minimum sparse-square length of every listed affine-line mixture.

Unresolved material objections

None reported.

Strengths

• W-S1 | Lemma 1, p. 3: the vectorization convention and partial-trace transpose give sum B_j^*B_j=d I_r in the necessity direction, hence C_j=A B_j^*/sqrt(d). Conversely A^+C_j and the 1/d Weyl twirl yield a normalized POVM with the specified support lists. Label inversion/phases preserve sparsity; the channel interpretation is not inferred from rank alone.
• W-S2 | Lemma 2, p. 4: the 3-adic coordinate reduction makes the central subgroup order d/q, so each central sector has dimension q rather than an unaccounted repeated multiplicity. Nonzero shift weights give nullity exactly one in a singular sector. With a,b,c nonzero, the determinant-zero condition is an intersection of distinct circles, allowing at most two central-character pairs. The crucial subgroup count passed for all 5616 noncommuting ordered distinct nonzero label pairs in dimension nine.
• W-S3 | Lemma 3, pp. 4-5: rank<7d/9 excludes noncommuting trinomials because d-2>=7d/9 for d>=9. A commuting group of order h>=9 similarly gives rank>=d-2d/h. Thus the relative support must reduce to an order-three algebra, and CC^* is a nonnegative combination of at most two parallel line projectors. Right multiplication preserves CC^*, so the resulting fixed ambient algebra is not lost in normalizing the sparse factor.
• W-S4 | Lemma 4 and Theorem A, pp. 5-6: positivity makes support exactly a union of character cells. Independent enumeration of all 4095 nonempty line subsets gives 12 singletons,54 crossing pairs,12 parallel pairs and72 triangles with union size below seven, and verifies both no hidden lines and a private point on every constituent. Private-cell values recover the weights uniquely, including when unordered eigenvalues coincide.
• W-S5 | Theorem A, pp. 2,6: each individual sparse square can use only parallel lines. Crossing pairs and triangles therefore require respectively two and three squares, whereas a parallel pair can be combined into one factor. This establishes the stated minimal lengths, not just upper bounds. Exact integer-pair arithmetic verified all nine scaled cell projectors, twelve scaled line projectors and 66 pair intersections per dimension in d=9,27,81.
• W-S6 | Theorem B and Corollary 5, pp. 7-8: singular order-three binomial squares are flat complements, forcing the twelve equal-weight parallel midpoints as the only two-list states below the cutoff. The four-term (I-U)(I-V) has square nine on exactly four cells, rank4d/9 and trace4d; the square/trace identity was checked exactly in all three dimensions. The rank-seven two-list obstruction in d=9 correctly uses the common kernel of positive sums.
• W-S7 | Theorem C, p. 8: (T-I)(T-xi I) with T=Z^(d/9) has exactly two zero order-nine eigenvalues and hence rank7d/9. Its nonzero T^2 coefficient lies outside the order-three label subgroup, proving failure of the fixed-algebra conclusion at equality. This is a valid sharpness example for the structural statement, not a classification of every state at or above the cutoff.

Weaknesses

• W-W1 | Section 7 and reference [1], pp. 8-9: ternary_verify.py, ternary_certificate.json and the predecessor reading ledger are not in the supplied ZIP. The predecessor was not sought in project history or other contexts. Its claimed rank-law priority and the manuscript-specific hash binding of the original replay remain unverified, even though the needed argument is reproduced in Lemmas 2-3.
• W-W2 | References [2]-[6], p. 9: the general learning-width/factor-width correspondence is explicitly present in Johnston et al., Lemma 9 and Corollary 11 [https://arxiv.org/html/2510.20789v1]. Character-space size and Weyl block structure have the cited precedents [https://arxiv.org/pdf/1101.1519; https://arxiv.org/pdf/2101.00129]. These sources support the ingredients, not priority of the 150-family positive-sum classification. Gross and Veitch et al. were checked only at abstract level, not at all cited sections.
• W-W3 | pp. 1-2,8-9: the commuting rank-d/3 projectors belong to a single cyclic d-dimensional representation. They cannot be identified as operators with twelve pure qutrit stabilizer states or carried unchanged to a tensor-product Pauli sparsity problem. The paper states this correctly; it materially narrows possible generalizations.
• W-W4 | pp. 3-8: finite enumeration certifies the nine-point incidence geometry, not all complex sparse coefficients or every k. The latter depend on the analytic circle and central-sector arguments. No formalization or numerical search over all coefficients was performed, and no stability statement under approximate zeros is proved.

Potential errors

None reported.

Strong novelty candidates

• Lemma 3 plus Theorem A: forcing every summand of an arbitrary low-rank positive sparse factorization into one fixed order-three algebra, then classifying all positive sums.
• Theorem A: unique decomposition into line states together with exact minimal three-sparse-square length for each family.
• Theorem B: the exact fixed-state two/three-list distinction and ell_min_exact(d,4d/9)=4 versus ell_min_exact(d,5d/9)=3. The incidence geometry and operational factor-width principle themselves are antecedents.

Unresolved material objections

None reported.