Two-state exact routing: algebraic feasibility and sharp delay-two rigidity

Strengths

• Theorem A gives a concrete necessary-and-sufficient reduction: the routed column has q(t)=t^2-d^2, a stable quadratic h(t), a nonnegative residual quadratic, and the displayed delay formula follows from h*/h.
• Theorem B's equality analysis is sharp: constant peak two forces both determinant zeros to vanish, after which the boundary polynomial's maximum yields exactly a=b at least pi/2; the converse construction verifies the full locus.
• Theorem C tracks the first two Fourier moments of the two Poisson kernels into a uniform denominator perturbation and an explicit positive gap, while Proposition D supplies an exact global quartic-square certificate rather than a sampled optimum.
• The manuscript carefully distinguishes complete feasibility from the unresolved unrestricted acute optimum and states physical and modeling exclusions explicitly.

Weaknesses

• The attainment proof compresses the topology of bounded-degree Blaschke-Potapov realizations and the persistence of exact boundary interpolation in the limit; the conclusion is credible but would benefit from a fully specified compact parameterization.
• The bibliography supports the classical machinery but does not establish priority for the claimed problem-specific deductions; the manuscript acknowledges this limitation.
• The public replay wrapper expects public-code/paper.pdf, which is absent from the supplied batch, so it cannot produce its final report or canonical hash despite both internal audits passing.
• The exploration script requires SciPy although neither research/requirements.txt nor the top-level requirements file declares it; its tolerant real-root filter also emits harmless very-large stationary points near the asymptotic value.

Potential errors

• The pole-exhaustion step assumes without a local proof that equality of column and matrix McMillan degrees forces equality of every pole multiplicity and hence fixes the determinant as h*/h; the cited para-unitary theory appears relevant, but this dependency was not re-proved here.
• The compactness argument ranges rank-one Potapov directions and a constant unitary over compact sets but does not explicitly discuss degenerate factorizations; I found no counterexample, yet this is the least detailed part of the existence proof.
• The numerical exploration's root-imaginary tolerance can classify nearly real remote derivative roots as finite stationary points. The maximum values remained correct in the executed fixtures, so this affects diagnostics rather than the analytic propositions.

Strong novelty candidates

• The exact three-real-parameter classification of all degree-two determinant-delay functions for the phase-free three-node routing table, independent of port count.
• The iff characterization of delay-two attainment together with the explicit geometry-dependent positive gap outside the symmetric right-angle locus.
• The cubic-root acute construction with an exact quartic-square global peak certificate and a strict improvement over the double-pole design.

Unresolved material objections

None reported.

Strengths

• Theorem A, pp. 2-4: three boundary zeros force all extra coordinates to vanish. The repeated-target numerator has degree two and no common factor with the other coordinate; this prevents a hidden degree-one solution.
• Equations (9)-(13), p. 3: Y>0 and V>K^2 characterize upper-half-plane roots; positive residual quadratic power reconstructs a complex linear p and an inner two-port completion with determinant h*/h.
• Pole exhaustion, pp. 3-4: the primitive routed column already uses both allowed pole multiplicities, so additional ports cannot alter the determinant delay. Alpay-Jorgensen-Lewkowicz Theorem 3.1 supports the needed minimal Potapov structure.
• Equations (16)-(17), pp. 4-5: the 3x3 Gram matrix exactly encodes global univariate quartic nonnegativity, including the leading coefficient at infinity. The resulting parameter feasibility is correctly described as polynomial, not jointly convex.
• Theorem B, pp. 5-6: constant peak two forces both Blaschke zeros to vanish through the first two Fourier coefficients. The numerator maximum at z=1 gives exactly a=b>=pi/2, and equation (22) constructs every such case.
• Theorem C and equation (26), pp. 6-7: coefficient bounds on the nonnegative delay deficit yield the stated overshoot gap; the Fubini-Study speed bound independently yields pi/a and pi/b.
• Proposition D, pp. 7-8: Descartes signs, the monotone r-parametrization and R_T=(T-Y)(t^2-x)^2 establish existence, global cap attainment and strict improvement. The 9/sqrt(11) fixture gives the executed 8.52449 percent improvement.

Weaknesses

• Section 6 gives no exclusion of V>d^2 or K!=0 improvements. This is explicitly an open problem, not a defect in Proposition D, but it limits the result to a certified upper bound.
• Theorem A offers exact quantifier-elimination computability rather than a usable complexity bound or general optimization algorithm. Small Gram matrices do not make the nonconvex parameter search an SDP.
• Section 7 lacks a minimal, review-free top-level replay route in the supplied deposit. replay.py invokes research/review/quantum_agent_checks.py; that file was hashed for manifest integrity but its contents and execution were excluded under this task.
• The top-level wrapper also hashes public-code/paper.pdf, which is absent; the exact attachment is at ../paper.pdf. I did not populate the missing path or execute that hash. The standalone verifier successfully hashes its existing research/paper.md.
• Both supplied requirements files omit scipy, although explore_two_state.py imports scipy.optimize. The installed SciPy allowed the exploration to run; a fresh installation from those files alone is incomplete.
• Novelty remains qualified: the full texts of the two same-author routing antecedents and several other cited works were not checked. The primary matrix all-pass comparison was verified at its node-wise delay objective.
• The tests are symbolic identities and finite numerical trials, not formal verification of the analytic compactness or every universal quantifier. The excluded review material contributes no evidence here.

Potential errors

None reported.

Strong novelty candidates

• The complete three-real-parameter delay characterization for every port dimension, equations (3)-(6).
• The exact symmetry/right-angle attainment locus and explicit positive geometry-dependent winding gap, Theorems B-C.
• The attainable cubic-root acute construction with an exact quartic-square certificate and strict improvement over the elementary double pole.

Unresolved material objections

None reported.