Exact Memory of Finite Spectral Routing Tables: The Direct-Sum Occupancy Law and Its Singular Strata

Strengths

• Lemma 2.1 and Theorem 1.2, pp.2-3: exterior powers of rank-one Potapov factors remain affine in each scalar Blaschke coordinate. Clearing d denominators gives numerator degree at most d, not kd. The wrong-node determinant multiplicity is k, yielding the sharp occupancy lower bound.
• Section 3 and Lemma 3.1, pp.3-4: injectivity of the stacked frames keeps F0 full rank on the entire boundary. Outer matrix factorization and a shared k-vector delay line give degree at most k(L-n*). Degree-preserving lossless completion is supported by Alpay et al., Theorem 2.1: https://arxiv.org/pdf/1410.0283.
• Theorem 5.2 and Corollary 5.3, pp.5-6: admissibility ensures a determinant is nonzero somewhere, and a rank-q endpoint contributes at least k-q zeros. This avoids counting zeros of an identically zero minor and produces the weighted hyperplane interpretation for lines.
• Lemma 5.4 and Theorem 5.5, p.6: the Bloch-sphere circle and oriented disk-automorphism construction explains why three distinct coplanar lines cost one state, while a repeated line can cost two. Independent diag(1,z) and diag(1,z^2) fixtures confirmed these contrasting cases.
• Corollaries 4.1-4.2 and 6.1, Theorem 6.2, pp.4,6-7: openness/density, determinant winding and the collision discontinuity follow within the stated comparison class. The rank certificate uses a strict eigenvalue threshold and is conditional on the declared operator-norm error bound.
• The full producer completed with maximum scalar inner residual 1.5984412853442542e-05 and minimum sampled density eigenvalue 7.822126783096168e-09. These pass its bounds, although they do not match every stored floating-point value or prove positivity between sample nodes.

Weaknesses

• Section 7 and Table 2, pp.7-8: both verifier programs require src/replay/results/unified_routing_table_memory/*.json, but this ZIP places the certificates directly in src/replay/results/. Unchanged executions failed there. No files were substituted or relocated to manufacture a verifier pass.
• Theorem 5.1, p.5: give the precise tangential boundary-interpolation theorem, the off-diagonal blocks and the regularity/degree argument for a singular Pick completion. Fixed-derivative and upper-bound-derivative versions have different solvability conditions; the latter is the relevant comparison here.
• The scalar producer computes an inner column, but the k=2,3 campaign checks only the precursor density and ranges. No fresh executable witness constructs a completed square block network and independently measures its minimal McMillan degree.
• The independent routing verifier recomputes occupancy examples and one dual fixture before reading stored aggregates. The detector verifier checks stored expected values and a digest. Their independence from producer imports should not be read as independent reproduction of the complete numerical campaign.
• The code does not pin a numerical environment. Fresh floating-point JSON differs from the stored certificate while passing the same tolerances; a content digest establishes byte consistency, not cross-environment numerical identity.
• Section 8 priority claims and the relationship to the three predecessor releases were not independently established. Only necessary external primary mathematical sources were used; development history, CI claims and other assessments were not consulted.

Potential errors

None reported.

Strong novelty candidates

• Theorem 1.2: least occupancy as the exact state statistic for repeated direct-sum target planes with arbitrary nonorthogonal Gram matrix, together with the matching matrix spectral compiler.
• Theorem 5.2: occupancy-weighted rank deficiency over admissible constant detectors, with the sharp three-line singular-stratum classification as a concrete consequence.

Unresolved material objections

None reported.

Strengths

• Equations (4)-(8) give a clean gauge-invariant lower bound: each dual block detector is invertible on one target, zero on the others, and its determinant spends k boundary zeros per wrong occurrence.
• Equations (9)-(13) share one degree budget across all targets: at each node only the requested polynomial block survives, direct-sum injectivity makes F0 full rank on the circle, and Fejer-Riesz normalization yields an inner column.
• The cited rectangular paraunitary realization theorem supports adding channels without enlarging the state matrix, which is the exact degree-preservation needed by Lemma 3.1.
• The detector-rank bound records occupancy-weighted rank deficiency for arbitrary target arrangements and reduces correctly to weighted hyperplane occupancy for lines.
• Lemma 5.4 uses the Bloch-sphere circle and triple transitivity of disk automorphisms to give a credible explicit one-state construction for three distinct coplanar lines.
• The collision example honestly distinguishes discontinuity of exact McMillan degree from numerical conditioning, and Theorem 6.2's Weyl-eigenvalue test fails closed.

Weaknesses

• Theorem 5.1's universal upper bound compresses a boundary matrix Nevanlinna-Pick construction into one paragraph without writing the Pick matrix or citing a precise theorem that guarantees regular square inner interpolation of rank k(L-1).
• The reproducibility table names results/unified_routing_table_memory/certificate.json and detector_rank_certificate.json, but the exact ZIP contains only flat results/certificate.json and results/detector_rank_certificate.json.
• The numerical certificate is deterministic at the seed level but not byte-stable across this environment: the fresh certificate SHA-256 was 5adfdd74fdab8d2a91cb6a79f025be67c860a35be08b74f2cc193c511defb71d instead of the stored 999735f96358ab08b47e042f5ef9dc07bfa0cadebf576666b2166cc6e6764b68, although all thresholds passed.
• The literature-positioning search is described as targeted and cannot support an exhaustive priority claim; the paper appropriately narrows its wording, but novelty confidence remains below theorem confidence.

Potential errors

• Lemma 2.1's exterior-power proof assumes a rank-one Blaschke-Potapov factorization with exactly the McMillan-degree budget in the disk-analytic convention. The primary source supports the factorization, but the manuscript could state the convention transfer more explicitly.
• Theorem 5.1's span lower bound is straightforward, but the upper bound's claimed Pick-matrix rank and regular boundary interpolation are not independently executable and are the least fully documented proof step.
• Corollary 6.1 is valid within the preceding existence of mutually orthogonal k-frames E0,...,Em; readers could miss the implicit ambient requirement N >= (m+1)k because it is not repeated in the corollary statement.

Strong novelty candidates

• The exact direct-sum occupancy identity d_min=k(L-min_a n_a) for repeated nonorthogonal target subspaces with arbitrary distinct boundary nodes.
• The constant-detector hierarchy d_min >= max{r-k, Delta(Y,n)} and its weighted hyperplane-occupancy specialization for line targets.
• The exact three-line phase diagram and the algebraic collision discontinuity separating exact degree from conditioning.
• The fail-closed noisy rank certificate that converts a spectral error bound on the summed projectors into a guaranteed McMillan lower bound.

Unresolved material objections

None reported.

Strengths

• The lower bound isolates each target with a block row of a left inverse. Passing to the kth exterior power correctly turns every wrong target node into at least k zeros while keeping the total numerator degree within the McMillan budget.
• The upper construction uses polynomials vanishing at all wrong nodes, and direct-sum injectivity guarantees full boundary rank. Matrix Fejer-Riesz factorization then normalizes the column without changing its endpoint ranges.
• The detector-rank hierarchy extends the same determinantal mechanism beyond direct sums, and for lines it reduces transparently to weighted hyperplane occupancy.
• The three-line phase diagram withstands case analysis: coincident lines admit degree zero, three distinct coplanar lines are realized by a disk automorphism and one Potapov factor, and all other configurations meet the universal degree-two upper bound.
• The paper states the exact comparison class and separates algebraic degree discontinuity from numerical conditioning, avoiding an experimentally misleading interpretation.

Weaknesses

• Lemma 3.1 compresses the realization-degree argument into a recurrence and a cited rectangular completion theorem. The citation does support unitary state-space augmentation, but the disk/exterior-disk paraconjugation and preservation of the designated column should be written explicitly.
• Theorem 5.1's boundary Nevanlinna-Pick upper bound is sketched at a high level. The freedom in diagonal blocks, regularity at every node, and the conversion of Pick rank to McMillan degree deserve a precise theorem statement or a fuller derivation.
• Lemma 5.4 omits the harmless but necessary constant input unitary that maps the arbitrary input line X to the chosen vector x; the construction can be repaired without adding degree.
• The executable-artifact table names results/unified_routing_table_memory paths, but the supplied certificates are one directory higher. Both independent verifiers therefore stop with FileNotFoundError and never validate the stored digests.
• The seeded floating producer is not byte-reproducible in this environment: its fresh certificate hash is 5adfdd74... while the supplied certificate hash is 999735f9..., with different reported numerical maxima although both satisfy tolerances.

Potential errors

• The phrase 'degree-preserving lossless completion' is stronger than mere existence of a square para-unitary extension; it depends on using a minimal realization and augmenting its colligation without new states. The cited Alpay-Jorgensen-Lewkowicz theorem supports this, but the manuscript should identify the exact branch of that theorem and conventions.
• The universal arbitrary-target upper bound assumes the boundary interpolation theorem permits the stated diagonal-block choice while retaining regular finite rational-inner values. I found no contradiction, but the proof is too compressed to verify every boundary hypothesis from the manuscript alone.
• The verifier failure is not merely cosmetic: the unified verifier completes its degree examples and a fresh dual-frame calculation before failing, but it never reads or hashes the stored source; the detector verifier fails before any certificate check.

Strong novelty candidates

• The exact direct-sum occupancy law d_min=k(L-n_*) for dense nonorthogonal repeated targets.
• The occupancy-weighted detector-rank lower hierarchy on arbitrary equal-rank target arrangements.
• The complete three-line singular phase diagram and fail-closed span-rank certification tied to exact degree.

Unresolved material objections

None reported.