Lossless Calibration Is Stored Memory: A Topological McMillan-Degree and Wigner–Smith Law for Passive Quantum Networks

Strengths

• Theorem 4.1: One strict stop selects a single nonzero minor valid at every calibration node, avoiding a hidden change of rows or an architecture comparator.
• Lemmas 3.1-3.2 and Appendices A-B: Multiplicity proof and exterior-power denominator argument are explicit; an independent system-matrix proof reinforces the degree budget.
• Theorem 7.1 and Proposition 7.2: Pole-free closed regions, disjoint interiors and strict margins are stated; complex zeros are not confused with persistent boundary calibrations, and approximate samples alone are correctly ruled out.
• Section 8: Exact determinant accounting gives 3S+6, while all 3S distinct directional roots support the universal lower bound. The companion irreducibility hypothesis is mapped to an adequate full-spark subset.
• Sections 2, 9, 10, 12: Physical mode count, Markov interface, historical evidence and limits of numerical degree/contour checks are separated clearly.

Weaknesses

• Novelty boundary in Section 11: The main step is a short consequence of classical transmission-zero/realization machinery; the cited routing paper already shares the minor and delay mechanisms. This constrains novelty rather than invalidating the result.
• Sections 7 and 12: No quantitative bound from separated noisy real-frequency observations is established, so experimental use requires a separately justified passive analytic model and uncertainty margins.
• Section 8 near-optimality: The bracket leaves a six-state gap and does not identify an exact optimizer for the prescribed node-varying directions.
• Section 10 and numerical source: The checker validates a fixed numerical ledger rather than authenticating its generation. Floating-point contour audits and ill-conditioned clustered roots are not interval certificates. The manuscript states these limitations.
• Presentation: Operational provenance occupies substantial space relative to the compact analytic contribution; some material could move to a supplement.

Potential errors

• P1; non-material numerical_conditioning_limit; Historical generator wigner_smith_trace: The denominator 1-2r cos(theta)+r^2 suffers cancellation near r=1. All prescribed-order replays pass; future larger-order work should use the equivalent stable denominator (1-r)^2+4r sin(theta/2)^2. This is not a demonstrated defect of the stated analytic formulas or table.

Strong novelty candidates

• Theorem 4.1: The explicit protected-output formulation with node-varying lossless input subspaces and one all-direction strict stop is the clearest candidate-specific proposition. It uses shared classical machinery and is not established as independent-priority novelty.
• Section 8, equations 25-29: The exact 3S calibration count gives the concrete architecture-independent bracket 3S <= n_min <= 3S+6 and connects it to the explicit trace spectrum. This is the strongest application, rather than a new factorization or delay identity.

Unresolved material objections

None reported.

Strengths

• Theorem 4.1 uses one frequency-independent loss minor selected at the strict stop, so the multiplicities at all calibration nodes can be added without a hidden changing-minor assumption.
• The exterior-power and block-determinant arguments give a credible self-contained route to the degree budget for every minor.
• Theorem 5.1 and Corollary 5.2 correctly connect McMillan degree, winding, integrated trace delay, peak trace, and a peak eigenvalue lower bound.
• The exact saturation family proves coefficient sharpness for every positive m and M, while the six-port construction gives a meaningful near-optimal irreducible application.
• The robustness section correctly separates analytic contour control from pointwise approximate samples and supplies a valid degree-one counterexample to naive sample counting.
• The manuscript is candid about classical ingredients, prior related work, physical assumptions, and what the numerical certificate does not establish.

Weaknesses

• The central lower bound is an elegant corollary of classical inner-function factorization and determinant zero counting; the conceptual repackaging is stronger than the underlying mathematical novelty.
• The irreducibility claim for the six-port application imports a theorem from an earlier manuscript rather than proving that property in this paper, though the dependency and hypotheses are stated precisely.
• Exact boundary calibration is experimentally idealized; the robust result requires an analytic model and contour margin, and the paper correctly shows that unstructured point samples cannot replace them.
• The numerical degree entries are analytic formula values, not independent state-space minimal-realization or Smith-McMillan computations.
• The decoherence-rate interpretation is conditional and does not turn McMillan degree into a universal energy, volume, or thermodynamic cost.

Potential errors

• No material mathematical error was identified. The main residual risk is the convention-sensitive use of minimal Blaschke-Potapov factorization in Lemma 3.2; the stated analytic-disc convention and cited square-case factorization appear consistent.
• The phrase 'independent polynomial root count' describes algorithmic independence inside the supplied computation, not independent provenance or an independent reviewer execution; the manuscript elsewhere makes this distinction explicit.
• The narrow S=19 delay peak makes uniform-grid quadrature unreliable, but both the manuscript and reproduced calculation use adaptive quadrature and disclose the residual.

Strong novelty candidates

• Theorem 4.1's protected signal/loss statement for arbitrary node-varying lossless input subspaces, using one strict stop to select a common nonzero complementary-loss minor.
• The combination of calibration charge with an exact Wigner-Smith phase-volume identity to express the same lower bound as integrated passive memory.
• A coefficient-sharp 2m-port family for every m and M together with a six-port irreducible family of degree 3S+6, at most six states above the unknown optimum allowed by the 3S lower bound.
• The paired positive Rouche certificate and negative degree-one approximate-sampling result, which sharply states what robustness is and is not available.

Unresolved material objections

None reported.