A complete one-state error–delay law for asymmetric three-node routing

Strengths

• The finite quadratic formulation is exact for arbitrary port count because extra coordinates only strengthen the lower-bound inequalities and every feasible quadratic pair has an explicit two-port realization.
• The inactive certificate identifies precisely when its equality residual ceases to be positive, rather than silently extrapolating an unattainable quartic formula.
• The active branch has a genuine positive lower certificate with strictly positive endpoint weights and a nonnegative activation multiplier, covering complex coefficients and unused ports.
• The compiler supplies an outer scalar factor and a paraunitary completion whose determinant has the claimed degree and exact peak delay.
• The inverse theorem reduces the unrestricted one-parameter feasibility family to an isolated cubic root and handles zero error, first-cap, boundary, symmetric, and active cases explicitly.

Weaknesses

• Bibliographic novelty is deliberately not established; this review did not access external primary sources, so only the internal mathematical distinctness of the stated contributions was assessed.
• The model excludes degree at least two, nonorthogonal targets, fixed output phases, losses, physical time calibration, and larger interpolation tables, limiting immediate engineering generality.
• The classical degree-one Blaschke–Potapov structural fact is summarized rather than reproved from first principles, although the specialized spectral factor and completion are explicit.

Potential errors

None reported.

Strong novelty candidates

• The global-positivity phase transition and its explicit active lower certificate appear substantially stronger than a routine conic reformulation.
• The exact inverse law with a single cubic isolation, coupled to an explicit lossless router, is a strong constructive addition to the forward minimax result.

Unresolved material objections

None reported.

Strengths

• Section 2, (5)-(7): the denominator eigenvalue ratio gives the original angular delay ellipse; nonnegative quadratics capture complex numerator coefficients exactly. Aggregating unused ports yields a necessary lower-bound problem that a two-port factorization attains.
• Section 3, (8)-(11): interpolation and ellipse support give the inactive lower certificate. The determinant factor F(u-B)(B+v)/(F+2) correctly identifies when its otherwise sharp candidate becomes globally infeasible.
• Section 4, (12)-(16): the sign comparison is restricted to its valid interval, U_c decreases to 3v, and the retained unsquared sign selects the larger transition root. Symmetry and the q<=v case are not lost in radical manipulations.
• Section 5, (18)-(21): lambda_-, lambda_+ and lambda_0 are strictly positive, nu>=0, and the universal quadratic identity yields a contradiction for epsilon<e. Equality conditions fix H and Q and exclude an inactive-node competitor.
• Section 6, (22)-(23): |h_0|^2-|h_1|^2=4Delta>0 places the pole outside the disk. The sharp-polynomial identity gives a lossless completion with determinant degree one and the exact peak; constants are separately implemented at T<1.
• Theorem 2, (31)-(33): endpoint equalities force H, residual positivity gives |y|<=d/g, and the cubic derivative has one relevant root. Clipping at d/g correctly expresses the active inverse and retains the squared-error convention.
• Sections 7-8: exact active and transition fixtures, the obstruction gap, the T=1 law, zero-target impossibility and forward/inverse round trips were reproduced. The supplementary inverse program completed 74 exact Sturm cases.

Weaknesses

• Theorem 1 depends on the classical degree-one Blaschke-Potapov structure. Direct inspection of Alpay–Jorgensen–Lewkowicz, arXiv:1410.0283v2, supports this ingredient after the disk-convention change; it is not a new structural theorem of this paper.
• Section 9 compares against local development manuscripts and discusses review history. Those materials were excluded and were not accessed; their claimed comparison cannot establish novelty in this assessment.
• The mathematical scope is narrow: three prescribed nodes, phase-free chordal error, orthogonal targets, losslessness and degree at most one. No component-level synthesis or calibrated laboratory delay follows from the rational-inner compiler.
• The fresh SLSQP diagnostic is not a continuum certificate: one start reported a line-search failure, and the degenerate T=1 ellipse admitted an apparent 6.4e-9 improvement within numerical constraint tolerance. Exact T=1 analysis gives 1/3 in that symmetric case; this is not a valid counterexample.

Potential errors

• Section 6 says the inactive root has algebraic degree at most four for algebraic u,v,T. This is valid over the field generated by the parameters; read as degree over Q it is too broad when those parameters themselves have high degree. Clarifying the base field would remove this minor ambiguity.

Strong novelty candidates

• Theorem 1: explicit positivity-active minimax law, pinned denominator coefficient B=-v, and a positive dual certificate valid with arbitrary extra output ports.
• Theorem 2: exact minimum-delay inversion by one isolated cubic root with a positivity clipping rule and elementary active branch.
• Equation (27): the explicit positive T^-4 gap between the infeasible inactive bound and the true active optimum.

Unresolved material objections

None reported.

Strengths

• R-S1 | Sections 1-2, pp. 1-4: for a nonconstant degree-one inner matrix, the determinant contributes a single Poisson-kernel trace delay with minimum possible peak one. This explains the jump at T=1 and the constant optimum 1/2 below it. The Cayley denominator ellipse is equivalent to the eigenvalue-ratio constraint, not an uncalibrated surrogate for angular delay.
• R-S2 | Section 2, equations (5)-(7), pp. 3-4: every extra-port router yields nonnegative Q and P=H-Q satisfying the necessary node inequalities. Factoring a feasible pair and completing two scalar numerators gives a two-port realization, so the lower certificates apply to arbitrary N>=2 without a hidden truncation assumption.
• R-S3 | Sections 3-4, pp. 4-5: interpolation gives the inactive lower bound; (A prime D-A D prime) prime=A double-prime D+2A>0 isolates its minimizer. Equation (10) reduces residual PSD to B>=-v. The support comparison at tau and decreasing U_c account for the single finite transition exactly when u>3v, including cases where the square root defining U_c is unavailable.
• R-S4 | Section 5, pp. 5-7: the active construction has Q=k(s-t)^2, P=e(s-r)^2 and H=(s-v)^2+L^2. The positive endpoint weights and polynomial identity (19) yield a contradiction for every smaller candidate error. Equality fixes H, Q and all three active nodes; at the boundary nu=0 the remaining rank-one and leading-coefficient conditions still establish uniqueness.
• R-S5 | Exact independent check of the p. 9 active fixture: T=2,v=9/20,u=1399/180 yields lambda_minus=3/16, lambda_plus=13/16, nu=2/123 and lambda_0=182077/17280. Coefficient matching in (19), the first-moment identity (20), and all node equalities hold in rational arithmetic. The transition fixture u=165/28 gives t=tau=-15/16 and e=9/25 exactly.
• R-S6 | Section 6, pp. 7-8: |h_0|^2-|h_1|^2=4*Delta>0 places denominator zeros outside the closed disk; the sharp-polynomial completion is unitary on the boundary with determinant h_z sharp/h_z of degree one, including h_1=0. Across 400 independent parameter cases (253 inactive,147 active), 17 circle evaluations each gave maximum unitarity residual 4.45e-15 and node residual 2.0e-15. These samples support the algebra, not a replacement for its continuum proof.
• R-S7 | Section 7, pp. 8-10: u=3,v=1/3,T=2 gives active e=0.250257935326986 versus inactive lower bound 0.250029973086878, confirming that the latter is unattainable. The symmetry, T=1 jump and fixed-gap asymptotic regimes are consistent with the formulas; the fourth-order discrepancy has the claimed positive coefficient d^2(u-3v)^2/4.
• R-S8 | Theorem 2, pp. 10-12: (31)-(32) show that |y|<=d/g is exactly residual positivity, with H positive definite. The cubic derivative sign and endpoint truncation identify the minimum of the delay parameter. At u=2,v=1,epsilon=35/123, G(1/4)=0, B=-9/70 and C=223/70 were verified exactly. Independent scalar minimization of the inverse family agreed with the forward cap over all 400 cases within 1.08e-7 relative error, including the T=1 endpoint.
• R-S9 | Main public replay, before its final missing-file failure: 9 symbolic groups, 2 exact rational fixtures, an outward rational enclosure separating the obstructed example, 14 high-precision constructor cases, 3 constant-cap cases, 9 phase comparisons, 7 inverse symbolic groups, 4 inverse exact fixtures, 6 target boundaries and 12 numerical round trips completed. Recovered scientific fields agree with the frozen certificate; the largest scaled numeric-string difference is 2.44e-45, consistent with its 45-digit serialization.
• R-S10 | Additional supplied checkers: independent_review_checks.py solved dual moment equations afresh and completed 18 exact active/boundary fixtures, 3 inactive fixtures and a constant-completion determinant regression before its metadata failure. inverse_independent_checks.py completed symbolic endpoint/derivative identities, 74 exact Sturm cases and 4 exact fixtures before the same failure.
• R-S11 | Successful supplied CLI executions: --u 2 --v 1 --inverse-certificate --error-squared 35/123 returns cubic 352*y^3-1282*y+315, exact root 1/4 and minimum delay (sqrt(15529)+9*sqrt(53))/106. --u 3 --v 1/3 --T 0.5 --construct returns an explicitly degree-zero constant router, peak 0 and squared error 1/2.

Weaknesses

• R-W1 | pp. 1,13: substantial narrative concerns local derivation editions and their internal review status. Those statements provide no independent evidence for the present assessment and were not followed into project history. Novelty over those local predecessors remains unverified; the standalone formulas and proofs were assessed directly.
• R-W2 | Section 9, p. 13: the cited structural and convexification precedents are real and relevant. Alpay-Jorgensen-Lewkowicz, Theorems 2.3 and 3.1, cover degree-preserving completion and rank-one factors [https://arxiv.org/html/1410.0283v2]; Wu-Boyd-Vandenberghe describe magnitude-square convexification [https://web.stanford.edu/~boyd/papers/magdes.html]. These established ingredients should not contribute to an unsupported claim of broad novelty.
• R-W3 | Section 9, p. 13: the Dong et al. QSP comparator was verified only at primary abstract/search-record level; direct full-text retrieval failed. The Zhang et al. comparison was checked only against its primary abstract. Neither check excludes earlier results equivalent to the exact routing phase law, and no exhaustive priority search was performed.
• R-W5 | Model and Section 9: phase-free orthogonal targets and exact degree one are essential. The explicit matrix compiler is not a physical component synthesis, and the high-delay expansions require fixed positive gaps. The manuscript acknowledges these restrictions, which constrain the result's significance.
• R-W4-S | Default replay failure: verify_active_delay.py, independent_review_checks.py and inverse_independent_checks.py each raise FileNotFoundError for active_delay.md at the final source-hash stage. That Markdown file is absent from the supplied batch. No dummy manuscript was created and no hash requirement was bypassed. A diagnostic runner retained already-computed scientific state from the exceptions; this is not successful completion of the original entrypoints.
• R-W6-S | Certificate binding: the frozen active_delay_certificate.json correctly identifies the supplied main script by SHA-256, but its manuscript_sha256 refers to unavailable Markdown bytes, not the canonical reviewed PDF. The report is useful numerical evidence; its source-to-PDF binding cannot be independently completed from this bundle. The PDF was separately verified and inspected.
• R-W7-S | The public filenames and docstrings calling checks independent do not establish external referee independence. Only their computations were used here. No referenced earlier review documents were opened, and their scientific status labels were not substituted for this assessment.

Potential errors

None reported.

Strong novelty candidates

• Equations (2)-(4),(17)-(21): the explicit positivity-active phase, sharp threshold u>3v and matching positive certificate in the original angular-delay coordinate.
• Theorem 2: inversion of the complete law using one isolated cubic root plus an endpoint test, with a simpler radical active branch.
• Equation (27): quantification of the inactive certificate's failure at fourth order. These are plausible priority candidates, not established first-in-literature claims.

Unresolved material objections

None reported.

Strengths

• The active branch is not inferred from a solver: equations (18)-(21) give a positive lower certificate valid for complex coefficients and arbitrary extra ports.
• Section 6 provides a concrete two-port spectral factor and inner completion whose determinant has degree one and whose peak equals the prescribed cap.
• Theorem 2 inverts the forward law globally, handles the T=0 and T=1 boundaries, and isolates at most one cubic root.
• The exact rational active and boundary fixtures, symbolic identities, and independent Sturm checks substantially reduce the risk of a sign or branch-selection error.

Weaknesses

• The external novelty search is explicitly bounded, so the priority of the complete law and inverse remains unverified.
• The initial reduction imports the classification of degree-one rational-inner matrices and degree-preserving completion rather than proving that general theorem from first principles.
• The physical interpretation is limited to an abstract rational-inner routing model; no component, calibration, or loss realization is supplied.
• The source/replay package referred to in Section 9 was not part of the PDF batch, although the later public-code batch supplied the relevant verification programs.

Potential errors

• The universal N-port converse uses only the e2 intensity Q and aggregates all other outputs into P=H-Q. This is valid because the repeated e1-node constraints imply the necessary Q endpoint bounds and the constructed two-port solution attains them, but this reduction is easy to misread as preserving every original extra-port error equality.
• The radical formula for the transition requires retaining the unsquared sign and larger root. The supplied symbolic and exact boundary checks support the manuscript's branch rule, but arbitrary floating-point use without those retained conditions could select a spurious root.

Strong novelty candidates

• The exact positivity-driven phase boundary separating inactive and active minimax certificates.
• The positive active-branch dual certificate together with an operational two-port compiler for all N>=2.
• The exact error-target-to-minimum-delay inverse requiring at most one isolated cubic root.

Unresolved material objections

None reported.

Strengths

• R-S1 | Sections 1-2, pp. 1-4: for a nonconstant degree-one inner matrix, the determinant contributes a single Poisson-kernel trace delay with minimum possible peak one. This explains the jump at T=1 and the constant optimum 1/2 below it. The Cayley denominator ellipse is equivalent to the eigenvalue-ratio constraint, not an uncalibrated surrogate for angular delay.
• R-S2 | Section 2, equations (5)-(7), pp. 3-4: every extra-port router yields nonnegative Q and P=H-Q satisfying the necessary node inequalities. Factoring a feasible pair and completing two scalar numerators gives a two-port realization, so the lower certificates apply to arbitrary N>=2 without a hidden truncation assumption.
• R-S3 | Sections 3-4, pp. 4-5: interpolation gives the inactive lower bound; (A prime D-A D prime) prime=A double-prime D+2A>0 isolates its minimizer. Equation (10) reduces residual PSD to B>=-v. The support comparison at tau and decreasing U_c account for the single finite transition exactly when u>3v, including cases where the square root defining U_c is unavailable.
• R-S4 | Section 5, pp. 5-7: the active construction has Q=k(s-t)^2, P=e(s-r)^2 and H=(s-v)^2+L^2. The positive endpoint weights and polynomial identity (19) yield a contradiction for every smaller candidate error. Equality fixes H, Q and all three active nodes; at the boundary nu=0 the remaining rank-one and leading-coefficient conditions still establish uniqueness.
• R-S5 | Exact independent check of the p. 9 active fixture: T=2,v=9/20,u=1399/180 yields lambda_minus=3/16, lambda_plus=13/16, nu=2/123 and lambda_0=182077/17280. Coefficient matching in (19), the first-moment identity (20), and all node equalities hold in rational arithmetic. The transition fixture u=165/28 gives t=tau=-15/16 and e=9/25 exactly.
• R-S6 | Section 6, pp. 7-8: |h_0|^2-|h_1|^2=4*Delta>0 places denominator zeros outside the closed disk; the sharp-polynomial completion is unitary on the boundary with determinant h_z sharp/h_z of degree one, including h_1=0. Across 400 independent parameter cases (253 inactive,147 active), 17 circle evaluations each gave maximum unitarity residual 4.45e-15 and node residual 2.0e-15. These samples support the algebra, not a replacement for its continuum proof.
• R-S7 | Section 7, pp. 8-10: u=3,v=1/3,T=2 gives active e=0.250257935326986 versus inactive lower bound 0.250029973086878, confirming that the latter is unattainable. The symmetry, T=1 jump and fixed-gap asymptotic regimes are consistent with the formulas; the fourth-order discrepancy has the claimed positive coefficient d^2(u-3v)^2/4.
• R-S8 | Theorem 2, pp. 10-12: (31)-(32) show that |y|<=d/g is exactly residual positivity, with H positive definite. The cubic derivative sign and endpoint truncation identify the minimum of the delay parameter. At u=2,v=1,epsilon=35/123, G(1/4)=0, B=-9/70 and C=223/70 were verified exactly. Independent scalar minimization of the inverse family agreed with the forward cap over all 400 cases within 1.08e-7 relative error, including the T=1 endpoint.

Weaknesses

• R-W1 | pp. 1,13: substantial narrative concerns local derivation editions and their internal review status. Those statements provide no independent evidence for the present assessment and were not followed into project history. Novelty over those local predecessors remains unverified; the standalone formulas and proofs were assessed directly.
• R-W2 | Section 9, p. 13: the cited structural and convexification precedents are real and relevant. Alpay-Jorgensen-Lewkowicz, Theorems 2.3 and 3.1, cover degree-preserving completion and rank-one factors [https://arxiv.org/html/1410.0283v2]; Wu-Boyd-Vandenberghe describe magnitude-square convexification [https://web.stanford.edu/~boyd/papers/magdes.html]. These established ingredients should not contribute to an unsupported claim of broad novelty.
• R-W3 | Section 9, p. 13: the Dong et al. QSP comparator was verified only at primary abstract/search-record level; direct full-text retrieval failed. The Zhang et al. comparison was checked only against its primary abstract. Neither check excludes earlier results equivalent to the exact routing phase law, and no exhaustive priority search was performed.
• R-W4 | Sections 6,7,9: the software mentioned in the PDF is absent from the batch. The independent checks here reconstruct formulas, not the author implementation. Floating-point sampling does not certify arbitrary algebraic inputs, node-collision limits or all coefficients; exact general validity rests on the printed proofs.
• R-W5 | Model and Section 9: phase-free orthogonal targets and exact degree one are essential. The explicit matrix compiler is not a physical component synthesis, and the high-delay expansions require fixed positive gaps. The manuscript acknowledges these restrictions, which constrain the result's significance.

Potential errors

None reported.

Strong novelty candidates

• Equations (2)-(4),(17)-(21): the explicit positivity-active phase, sharp threshold u>3v and matching positive certificate in the original angular-delay coordinate.
• Theorem 2: inversion of the complete law using one isolated cubic root plus an endpoint test, with a simpler radical active branch.
• Equation (27): quantification of the inactive certificate's failure at fourth order. These are plausible priority candidates, not established first-in-literature claims.

Unresolved material objections

None reported.