Strictly Scalable Exterior Decoders for Quantum Lists: Exact Full-Spark Widths and Fixed-Probe Weyl Bayes Curves

Strengths

• Theorem 2.1 correctly translates sparse Gram factors into a cone of physical annihilator projectors; positivity justifies spectral refinement, and surjectivity of the synthesis map supplies the converse coordinate identity.
• Lemma 3.1 follows directly from the exterior power of the tight-frame operator, and Theorem 3.2 pairs it with the sharp full-spark obstruction to obtain both the exact threshold and an explicit POVM.
• Theorem 6.1 uses orthogonal shift sectors and a Vandermonde full-spark phase frame without silently optimizing over probes; Remark 6.4 correctly shows that Schmidt rank alone does not determine the threshold.
• Corollary 6.5 has a self-contained dimension converse, 1 <= ell*dim(K)/d^2 <= ell*r/d, and the arithmetic-support construction attains equality whenever r divides d.
• Theorem 6.3 proves the polygon root-sum maximum and matches the resulting spectral upper bound with a translated equatorial covariant POVM for every list size.

Weaknesses

• The novelty case rests on a targeted but necessarily incomplete literature comparison; the claimed formulas were not found in the checked primary sources, which is evidence of search scope rather than proof of priority.
• The reproducibility certificate includes an unstabilized floating value despite comments elsewhere in the checker about avoiding environment-dependent last-bit serialization.
• The supplied package_release.py explicitly expects a pre-deposit directory tree not included here, so its --check path fails at output/release/manifest.json and cannot validate this ARR layout.
• A TeX source and bibliography are supplied, but the local rebuild could not start because the installed pdfTeX lacks pdflatex.fmt; this is an environment limitation and not evidence of source-PDF mismatch.

Potential errors

• The frozen-certificate claim is false in this execution: every assertion ran, but byte comparison failed solely because 0.010000000000000002 regenerated as 0.010000000000000004.
• Proposition 3.3 is numbered 3.3 in the PDF while the checker docstring calls it Proposition 3.4; this is a minor cross-reference defect in the code comments, not a mathematical inconsistency.

Strong novelty candidates

• The exact N-r+1 learning width of every strictly scalable full-spark ray ensemble and its weighted Hodge POVM.
• The exact d-r+1 fixed-probe Weyl frontier and the d/r optimum over all pure rank-r probes in the divisor branch.
• The complete one-shot rank-two regular-polygon list-success curve for all 1 <= ell <= d.

Unresolved material objections

None reported.

Strengths

• Theorem 2.1, p.3: positivity forces each positive spectral component of an effect to annihilate all omitted states. Surjectivity of the synthesis matrix justifies the reverse Gram-to-physical identity resolution.
• Lemma 3.1/Theorem 3.2, pp.4-5: the antiunitary Hodge identification correctly conjugates complex cofactors; exterior powers of TT*=alpha I produce the alpha^(r-1) normalization. Full spark independently supplies the sharp converse.
• Proposition 3.3, p.5: a real nullspace dependence among nonzero PSD projectors has both signs. The minimum positive ratio update preserves nonnegative weights and completeness while eliminating an outcome.
• Theorem 4.1, pp.5-6: Q_L >= gamma_l I gives the prior-weighted Bayes floor, and finiteness plus full spark makes it strictly positive. The omitted-prior perturbation budget is correctly distinguished from the looser total budget.
• Theorem 6.1, p.7: shift-sector orthogonality and pinching justify both directions for the consecutive-support probe. Consecutive Fourier rows give Vandermonde full spark for composite d as well as prime d.
• Lemma 6.2/Theorem 6.3, pp.7-8: replacing a subset by largest directional projections yields the polygon bound; the equatorial translated top eigenvectors resolve I and attain the spectral upper bound for every list size.
• Remark 6.4/Corollary 6.5, p.9: arithmetic support produces d*r orthogonal classes of d/r identical outputs. The dimension converse applies to every pure Schmidt-rank-r probe, closing the claimed divisor branch.
• Section 7, pp.9-10: duplicated orthogonal states and the strictly positive Bloch-hemisphere example genuinely show why full spark and scalability cannot simply be dropped from the universal statement.
• The supplied 15 harmonic-frame fixtures and 54 rank-two (d,list-size) cases complete. Referee-created rational complex Parseval frames at (N,r)=(5,3),(6,2),(6,4) independently satisfy full spark and the exact Hodge identity.

Weaknesses

• Section 9 and README frozen replay: --check exits 1 after the mathematical calculations, in both tested NumPy versions. Only /robust_floor_fixture/perturbation_budget differs, by about 1.7e-18. Use a reproducible serialization or a separately specified numerical comparison without concealing this failed byte check.
• Theorem 6.1 includes r=d but invokes Theorem 3.2, whose stated hypothesis is N>r. At r=d the phase vectors are an orthonormal Fourier basis and list size one is immediate; the proof should explicitly separate this endpoint.
• Theorem 3.2 assumes existence of strictly positive scaling weights; no general efficient or well-conditioned algorithm for finding them is claimed. The exterior construction starts with combinatorially many wedges before r^2 compression.
• Section 8 priority boundaries are appropriately narrow, but the author-companion comparison and every recent exclusion/dense-coding antecedent were not independently checked here. The result is not a new general learning-width criterion.
• The supplied package_release.py is a pre-deposit packaging helper, not a working build path for this flattened batch. Its expected build PDF, release manifest and other original-tree files are absent; I inspected but did not execute a manuscript-rebuilding helper or reconstruct that history.

Potential errors

• Theorem 6.1 endpoint r=d: this is a repairable proof-reference mismatch, not a counterexample to the formula; orthogonality proves the endpoint directly.

Strong novelty candidates

• Theorem 3.2: the exact learning-width evaluation N-r+1 and weighted complementary Hodge POVM for arbitrary strictly scalable full-spark ray ensembles.
• Theorems 6.1 and 6.3: the all-(d,r) consecutive-support zero-error law and the complete one-shot rank-two list-valued Bayes curve.
• Corollary 6.5: the matching arithmetic-support construction closes the exact optimization over pure rank-r probes when r divides d, beyond a dimension-only obstruction.

Unresolved material objections

None reported.