Exact Branch Rigidity and Unique Crossing in Grassmann Matrix--Bingham Free Energies: Finite Certification, All-Field Gr_C(3,6) and Gr_C(4,8) Two-Block Order, and a Gr_C(2,5) Exchange Theorem

Strengths

• Pages 3-4, Theorem 3.1: the defect factor r squared minus q equals (r-k) squared, denominator ranges and exceptional ranks are handled explicitly. The 220 replay fixtures passed, and I independently matched all 74 printed Appendix B coefficients.
• Pages 5-6, Proposition 4.1/Theorem 4.3: the endpoint exponent is kr, and the balanced matrix-cube event has real dimension r squared and probability (4r)^(-r squared). The ratio base exceeds one, so the stated finite sufficient certificate really terminates.
• Pages 6-7, Theorem 5.1: normalized Hankel formulas and decreasing polynomial/exponential ratios prove all-degree Gr_C(3,6) order. The exact induction identities ran successfully; finite samples alone are not being used as the proof.
• Page 8, Corollary 6.1: independent conditional-unitary covariance and Jacobi integration by parts reproduce the constrained Hessian sign. Explicit all-j coefficient formulas show instability of k=1,2 and stability of k=3; this closes the r=3 dependency locally.
• Pages 9-11, Theorem 7.1/Lemmas 7.2-7.3: equal moments through order two consume three Laplace zeros; four density sign changes leave exactly one simple crossing. Orientation symmetry and the separate odd-density zero budgets exhaust all four multiplicities.
• Pages 11-12, Theorem 8.1: the Fraction determinant replay completed m=2,...,134 for all three losing branches. The exact tail ratio (378/5)40^(-16)(361/240)^m exceeds one at m=134, so there is no uncovered degree.
• Pages 12-13, scope ledger: two-level dominance is expressly separated from full-spectrum optimality. I did not credit the paper with the unproved arbitrary-spectrum or all-distortion RDF conclusions.

Weaknesses

• Page 8 cites reference [4] for essential Hessian structure without reproducing its derivation. Direct public PDF retrieval attempts failed. My independent r=3 calculation supports Corollary 6.1, but the broader no-spinodal background, including the r=4 statement in section 9, was not independently audited.
• Page 13 advertises SHA256SUMS and a differently organized source package; that checksum file is absent from the supplied batch. PUBLIC-CODE-MANIFEST.json verifies this batch, but cannot verify absent release-archive bytes.
• The low-degree checker reconstructs positivity and tests finite Schur-Weyl identities; it does not itself compare the complete printed Appendix B arrays. My separate exact comparison closes that particular audit gap, but the advertised checker coverage should be described accurately.
• The source-specific novelty claims are plausible, not established priority. I checked relevant primary abstracts on Jacobi traces and majorization, and did not treat the manuscript assertion of a targeted search or a hostile audit as independent evidence.
• Theorem 4.3 is a finite sufficient decision procedure for coefficientwise dominance, not a proof that every rank passes. A negative coefficient would not alone disprove full Laplace order; the paper correctly retains this limitation.

Potential errors

None reported.

Strong novelty candidates

• Theorems 5.1 and 8.1: strict all-field, all-degree normalized two-block order on Gr_C(3,6) and Gr_C(4,8).
• Theorem 4.3: an explicit matrix-cube event converting each fixed-rank infinite coefficient comparison into a finite sufficient rational certificate.
• Theorem 7.1 and Lemma 7.3: a unique simple canonical exchange on Gr_C(2,5), together with strict orientation dominance closing its complete two-level family.

Unresolved material objections

None reported.

Strengths

• Theorem 3.1 exposes the multiplicity defect r^2-q=(r-k)^2 and reduces degrees four through ten to rational expressions with positive-coefficient certificates, including separately listed low-rank exceptions.
• Theorem 4.3 gives a real finite certificate rather than an asymptotic slogan: a translated and scaled event in the uniform Hermitian matrix interval supplies a lower moment bound whose exponential base eventually dominates the maximal unbalanced support bound.
• Theorem 5.1 compares every Gr_C(3,6) two-level multiplicity coefficientwise. The discrete inequalities in Equations (5.10)-(5.13) correctly convert polynomial/exponential terms into monotone normalized sequences with positive base cases.
• Theorem 8.1 combines exact rational Hankel coefficients through m=134 with a strict tail ratio already above one at m=134; the overlap of finite and analytic ranges closes every even Taylor coefficient without floating-point sign decisions.
• Theorem 7.1 uses the correct zero budget: equal moments through order two give a triple zero at the origin, four density sign changes permit only one further real zero, and the larger positive support endpoint forces that zero to exist and be simple.
• Fresh execution of run_all_replays.py passed all five theorem-bearing checks, including 220 Schur-Weyl fixtures, the r=4 exact coefficient range, and exact Q(sqrt(6)) Sturm counts; exact_moment_screen.py found no counterexample in 351 additional cases.

Weaknesses

• Corollary 6.1 imports the Jacobi-Stein constrained-Hessian operator, its sign convention, and balanced Morse-Bott stability from ARR-2026-6F8XRSBM0J9Q2R2B. The public primary PDF corroborates those formulas and theorems, but the dependency is not self-contained here.
• Lemma 7.2 cites the extended-complete-Chebyshev interpolation theorem behind multiplicity-counting variation diminution. The proof sketch is credible, but a theorem number and hypotheses covering piecewise-polynomial signed densities would make the crossing certificate easier to audit.
• The full-spectrum global maximizer remains open. Consequently the branch order cannot by itself select the true matrix-Bingham envelope; the manuscript states this boundary correctly, but it substantially limits operational consequences.
• The claimed targeted primary-literature search is not reproducibly documented. Exact-title and distinctive-phrase searches found no competing theorem, but priority remains provisional.

Potential errors

• REPLAY_MANIFEST.json assigns d5r2_canonical_crossing_exact.json hash 880a2179... and replay_results.json hash 8f5eab53..., while the supplied and freshly regenerated files hash to 2b84943b... and 6a8aa9f... respectively. The outer PUBLIC-CODE-MANIFEST.json is correct, so the inner freeze is stale or inconsistent.
• REPLAY_MANIFEST.json also lists paper.pdf, source/main.tex, source/references.bib, and SHA256SUMS-style release materials that are absent from the supplied public-code subtree. Their hashes were not and could not be verified; no substitute files were used.
• Theorem 4.3's universal cutoffs are sufficient, not necessary. A failed finite coefficient would not refute all-field Laplace order; the paper says this later, but readers could otherwise overinterpret the certificate as an equivalence.
• The Gr_C(2,5) replay verifies densities and Sturm counts, not strict total positivity itself. The unique-crossing conclusion therefore still depends on the cited classical Chebyshev-system theorem rather than on execution alone.

Strong novelty candidates

• The all-degree coefficientwise two-level Laplace order on Gr_C(3,6), including the separated-value comparison across all multiplicities.
• The finite exact certification architecture at fixed half rank and its complete execution for Gr_C(4,8) through degree 268 plus analytic tail.
• The exact unique simple crossing and orientation dominance for the complete two-level family on Gr_C(2,5), proved through compact densities, Sturm counts, and variation diminution.

Unresolved material objections

None reported.