Projective Memory and Resonant Bottlenecks in Passive Spectral Routing

Strengths

• Theorem 2.3 gives matching bounds: Fejer-Riesz normalization plus same-state lossless completion attains a projective interpolant's degree, while d<L and a common realization denominator force every minimum routed column into the actual target span.
• The block law correctly exposes the full-support condition that ordinary kernel tests miss and derives exact shifted maximum costs; the generic rank calculation uses a concrete Hall-Vandermonde witness rather than parameter counting alone.
• Theorem 4.1 separates exact interpolation from its closure through ker(A_d), and the base-point deletion plus primitive-pencil construction explains how deficient kernel vectors yield arbitrarily accurate routers without fabricating exact feasibility.
• The singular-value error floor is dimensionally and algebraically consistent, uses normalized quotient maps and coefficient norms, and includes a calibrated operator-norm perturbation margin.
• Theorem 5.1 turns nonattainment into a compactness statement under bounded trace delay; the rank-one Blaschke-Potapov parameterization and Poisson-kernel peak bound justify divergence along every zero-error approximating sequence.
• The planar binary, generic, and four-node strata provide explicit coprime witnesses and exact lower certificates, making the abstract projective invariant testable on nontrivial collision patterns.

Weaknesses

• The supplied requirements.txt omits SymPy even though both exact verifiers import it. The stated environment therefore fails immediately until an undeclared dependency is added.
• The manuscript build source global_projective_memory.tex named by the supplied README is absent; only machine-readable PDF renditions are included, so the canonical PDF cannot be rebuilt from the stated source path.
• The PDF and submission metadata state planar certificate SHA-256 09e3aa7d..., but the manifest-verified and freshly regenerated byte-identical certificate hashes to 0d90d877....
• The generic full-support and projective-smoothing arguments are compact and would benefit from a more explicit scheme/resultant formulation, especially for readers assessing behavior at infinity in dimensions above two.
• The priority conclusion remains bounded: the checked primary sources verify the classical completion and representation ingredients, while negative searches cannot establish that the combined memory and delay laws are first.

Potential errors

• The published planar-certificate file hash is stale: 09e3aa7dc2b2a80ce2222bae0a4a85357d0d36d09e257595b992ceb900db5c47 is not the supplied file's SHA-256; the actual value is 0d90d877135cb5f566764890b88a93520d0dd13d4a5074720fd0c260370a5f82.
• The reproducibility claim that the accompanying archive contains the manuscript source is not satisfied by this batch's declared public-code paths; global_projective_memory.tex is missing and no substitute was used.
• The exact verifier environment is incomplete as declared because sympy is absent from requirements.txt; after an explicit referee-added sympy==1.14.0 installation, both verifiers passed.

Strong novelty candidates

• The equality of minimum passive McMillan degree with minimum projective interpolation degree for every finite one-input-line routing table.
• The exact maximum/minimum split between multiband exact memory and border memory, together with the base-point deletion formula.
• The theorem that every nonattained fixed-degree zero-error sequence must develop divergent peak Wigner-Smith trace delay under the stated rational-inner model.

Unresolved material objections

None reported.

Strengths

• Lemma 2.2/Theorem 2.3, p.3: scalar spectral factorization and the reflected coisometric completion preserve state dimension; the L>d annihilator argument correctly prevents an optimal routed column from escaping H.
• Lemma 3.1/Theorem 3.2, pp.4-5: the capacitated coordinate assignment supplies maximal-rank witnesses. The full-support condition and the k versus max(0,k-s_i) dimension comparison address the generic nonvanishing-at-nodes issue.
• Theorem 4.1, pp.5-6: deleting common node factors gives the lower degree accounting; the pencil with lambda(A)=1 supplies a primitive degree-D perturbation. Exact and border memory legitimately use different block extrema.
• Theorem 5.1, pp.7-8: positivity of the Poisson-kernel trace delay confines every zero under a finite cap. Compactness then rules out every bounded-delay subsequence in the nonattained phase.
• Theorem 7.3/Example 11.2, pp.10,18: the normalized incidence estimate handles projection outside H. The reproduced Sylvester minors certify the strict one-state error bound sqrt(3/67), rather than a numerical rank guess.
• Theorems 9.2, 10.4, 10.9 and 11.1, pp.12-17: fiber counts, homogeneous smoothing and the explicit quadratic witnesses support the binary phase diagram, universal closure threshold and four-node classification.
• Referee-created exact checks cover all 256 four-node tables over four specified rays in C^3; full-support exact degrees and first-kernel border degrees agree with the deletion minimum, including 192 nonattained cases.
• Corollary 12.1, p.18: determinant winding gives the integrated-delay law under the stated dimensionless convention. The independent family [epsilon:z-1] exhibits decreasing error and peak delays from about 4.12 to 2000.00.

Weaknesses

• Reproducibility record, p.19: the printed planar certificate SHA-256 is 09e3aa7dc2b2a80ce2222bae0a4a85357d0d36d09e257595b992ceb900db5c47; the supplied file and fresh producer output both hash to 0d90d877135cb5f566764890b88a93520d0dd13d4a5074720fd0c260370a5f82. The PDF record is incorrect for this batch.
• Section 12 replay: requirements.txt omits SymPy although all three mathematical programs import it. Initial executions failed at import; installing SymPy 1.14.0 enabled successful full runs. This is a dependency-packaging issue, not a failed mathematical assertion.
• Figure 1(b), p.7, plots r>L while calling r the actual target-span dimension; that span cannot exceed L. Figure 2, p.17, labels L as the number of distinct targets while also plotting a binary repeated-target family. Clarify these parameter labels.
• Theorem 8.1, p.11: the cap identity and three-active-target necessity are convincing, but a reproducible implementation for the stated antipodal/degenerate candidate enumeration is not supplied. The code tests the two-target special case only.
• Section 12 priority comparisons include an author companion not supplied here. I did not access that project history. The detector comparison is supported by the stated annihilator definition, but the companion-specific priority assertions remain unaudited.
• Theorem 5.1 proves qualitative peak-delay divergence, not its optimal rate as error vanishes. The positive-degree minimax values are bounded rather than solved; the manuscript correctly limits those claims.

Potential errors

None reported.

Strong novelty candidates

• Theorems 2.3 and 4.1: an all-table correspondence between minimum passive state count and projective degree, coupled to an intrinsic incidence-rank characterization of zero infimum.
• Theorems 3.2 and 4.1: full-support block exactness versus one-block border feasibility produces the explicit max/min split for mixed incidence constraints.
• Theorem 5.1: forcing peak-delay divergence along every degree-bounded nonattained routing sequence is a meaningful consequence beyond merely exhibiting a resonant example.

Unresolved material objections

None reported.