Algebraic Query Support for Unitary Oracles: Exact Hilbert Laws, Harmonic Spectra, and General-Tester Bounds

Strengths

• Theorem 3.1, p.4: the annihilator of span{U^tensor k} is exactly I(X)_k; quotient dimension therefore gives H_X(k). Vectorized parallel Bell outputs expose that entire span, so the maximum causal-support assertion is an equality, not merely an upper bound.
• Theorems 5.1 and 6.1, pp.6-8: the quadric, traceless projective plane and central point give distinct correct branches. For selective-phase qudits L_c(X)=Tr(X)I+cX has eigenvalues c+d and c; the physical exceptional singularity c=-d occurs only at d=2,c=-2. The Segre Hilbert count and two-query defect follow.
• Theorem 8.1 and Proposition 8.2, pp.9-10: the zonal kernel (a+b n.m)^k gives the stated Legendre moments and multiplicities. Integrating its square yields the purity formula and linear participation dimension for a fixed interior angle; endpoint parity and rank losses are consistent.
• Theorem 9.1, p.11: equal top harmonic eigenvalues force a/b=1/(2k+1), after which the third-sector ratio is k(2k+3)/(2k+1)>1. Sector-trace invariance also justifies the positive-reweighting obstruction; this is stronger than checking only uniform designs.
• Theorem 7.1, p.9, correctly separates rank from attainable success through positive scalability KCK=K. The manuscript also correctly warns that antipodal traceless axes are identical channels and that necessary query bounds do not construct attaining protocols.

Weaknesses

• Sections 4 and 12, pp.5,13: the causal-versus-GEN boundary is unnecessarily restrictive. Put v_i=|U_i>>^tensor k and w_i=W^(1/2)v_i. Equation (5) makes w_i unit vectors; W^(-1/2)T_iW^(-1/2) is a POVM on supp W. Their span has dimension at most H_X(k), giving the same H_X(k)/M bound for arbitrary families. This weakens the claimed special role of orbit leverage but does not falsify Theorem 4.2.
• Novelty boundary, pp.2-3,13-14: Bavaresco-Murao-Quintino Theorem 5 and Appendix F already contain the ambient cap and group-average domination (https://arxiv.org/html/2105.13369v3). The credible additional content is the exact promise geometry and spectra. The special 24-echo predecessor [11] was not retrieved, so its overlap and priority remain unverified.
• Section 12 calls the verifier range configurable, but the supplied main() hard-codes its ranges and has no command-line range option. Its leverage check sums harmonic multiplicities rather than constructing an orbit average or a tester. This is narrower evidence than a direct operational verification.
• The supplied reproducibility README refers to verification/ and paper/ layouts and manuscript sources absent from this batch. The exact checker runs directly, but the release rebuild and package_release.py --check were not reproduced. No failure of the supplied canonical PDF hash is implied.
• Sections 11-12: exact rank is discontinuous at singular promises and does not quantify noisy discrimination or the attainable success of generic finite subsets. The manuscript acknowledges these limits; they cap the practical significance of the asymptotic support exponent.

Potential errors

• Lemma 4.1, p.5: transitivity of an abstract action on a set of unitaries alone does not imply a unitary representation on their Choi-vector span. State the linear/unitary covariance assumption used in the proof. The conjugation actions in all concrete applications satisfy it, so this is a formulation issue rather than a demonstrated application failure.
• Section 11, p.13: the necessary query scale is said to drop when the support exponent decreases. For fixed d and large M, M^(1/(2d-2)) grows faster than the ambient M^(1/(d^2-1)); the promise strengthens (raises) the necessary query count. The displayed formula is correct.

Strong novelty candidates

• Theorem 6.1: the projectively transformed Segre description supplies an exact all-dimension, all-query support law with the physically constrained singular branch; this is more specific than the classical Hilbert-function identity.
• The combined Theorems 8.1, 9.1 and 10.1: exact sphere/circle spectra, endpoint cascades and the all-k Bell-tightness obstruction form a potentially useful package for constrained oracle discrimination. Priority is a candidate assessment, not an exhaustive literature conclusion.

Unresolved material objections

None reported.

Strengths

• Theorem 3.1 correctly identifies the annihilator of the Veronese span with the degree-k vanishing ideal, yielding the exact Hilbert function and a causal discrimination cap rather than only an ambient count.
• Lemma 4.1 and Theorem 4.2 use constant orbit leverage correctly: averaging the pseudoinverse leverage gives rank D_k, and tester normalization applies to each repeated channel tuple and hence to the orbit average.
• The fixed-angle qubit, Segre selective-phase, Funk-Hecke, purity, endpoint, and great-circle formulas agree algebraically; the supplied exact verifier passed all of its finite rational checks.

Weaknesses

• The proof of Theorem 6.1 states, rather than develops, why physical rank-one projectors are Zariski dense in the full Segre variety and why the exceptional d=2, c=-2 projection has closure P^2; these are central closure identifications.
• Operational significance is limited by design: Hilbert rank is only a converse, and Theorem 7.1 is a scalability test for a fixed effective state family rather than a construction attaining the general-tester cap.
• The public-code README describes a larger release tree than the files supplied here, so two advertised commands are not replayable from this batch without reconstructing missing directories and assets.

Potential errors

• Equation (29)'s density argument is concise enough that a reader could miss the projective scaling step: a rank-one matrix with nonzero trace becomes an idempotent after scaling. The conclusion appears correct, but the omitted step should be explicit.
• The novelty boundary relies partly on a targeted literature search and a cited 2026 ARR record. The checked primary sources establish important antecedents but do not prove priority for the exact combined theorem package.

Strong novelty candidates

• The promise-sensitive replacement of the ambient general-tester numerator by the exact Hilbert rank for compact transitive unitary orbits is a strong candidate, subject to a broader priority search.
• The complete branch-resolved fixed-angle support laws together with the all-k sphere and circle spectra form a specific, technically substantial candidate contribution.

Unresolved material objections

None reported.