Finite-Sample-Valid Tests for Structured Matricial Hausdorff Moments: Joint Gaussian Grams, Strict Half-Time Separation, and Sharp Tangent-Cone Power

Strengths

• Theorem 3.1, p.3: the even conditions H_r>=0 and C_{r-1,s}>=0, and odd conditions G_r>=0 and L_{r,s}>=0, match the classical matricial interval characterization, including singular sequences. The closed-measure-cone separation argument has the correct positivity pairing.
• Theorem 4.1, p.4: the true structured covariance is feasible on one simultaneous Loewner event. Searching all moment directions inside that band costs no extra multiplicity factor, and the congruence proof does not invert a singular K.
• Theorem 5.1, pp.4–6: K is positive definite for the stated two-atom family, the old diagonal margin is positive, and sM1-M2 is negative. The rational fixture gives 23/128, -3/128 and W^2=261/16384. Its comparison is explicitly to the defined relaxed predecessor.
• Theorem 6.1 and Theorem 8.3, pp.6–8: the structured cone is closed, and the shrinking band gives consistency only outside the exposed covariance cone. At positive-definite K0, Gaussian KL is quadratic, supporting the n^(-1/2) local separation rate.
• Theorem 7.1, p.7: a positive matrix density on (0,s) supplies a strict localizer direction. This justifies the tangent preimage; positivity of H is inactive at K0>0. The additional equality-subspace part of the unrestricted LRT is explicitly acknowledged on p.8.
• Proposition 9.1, p.9: dual signs and adjoint anti-diagonal sums are correct. The sandwich bounds approximate moment sequences, so a vanishing-residual infeasible sequence would have a feasible convergent subsequence. The collapsed-band rational dual evaluates to -3/128.
• Fresh q=1,c=1 simulation at n=9421: in 20,000 Wishart draws from the strict alternative, the full band test rejected every draw and the old test rejected none. Two null cases, including a singular covariance, each had zero rejections. These are diagnostics with binomial uncertainty, not a proof of uniform level.

Weaknesses

• Theorem 8.1/Corollary 8.2 describe a likelihood test different from Algorithm 10. The Fisher whitening, active kernel and critical law depend on the fixed boundary K0. A practical plug-in rule and its validity across changing faces are not supplied; the demonstrated guarantee is pointwise local theory.
• Section 10 provides algebraic fixture code, not a general SDP solver, dual recovery/rationalization pipeline, or constrained likelihood optimizer. The regular tangent replay checks one scalar halfspace, not noncommuting active-kernel cone projections or singular-face numerical certification.
• Section 11 does not define a common sampling budget or ordering of independent-moment experiments. Its broad phrase strictly less informative should be confined to discarding the particular half-time cross block in Theorem 5.1.
• The sharp normal envelope in Eq.8.10 applies to a smooth halfspace and alpha<1/2. A general spectrahedral cone projection is a likelihood-limit law, not a demonstrated uniformly most powerful test; the text mostly respects that distinction.
• The Gaussian joint sketch, known zero mean, fixed finite dimension and covariance compatibility null are substantive restrictions. Alternatives with a null-representable exposed moment sequence remain undetectable by this method. The paper states these limits clearly.
• Primary-source checks confirmed the matricial parity criterion and Gaussian singular-value inequality. The Geyer publisher page did not expose usable full text, and the literature search was not exhaustive; general novelty or all citation dependencies were not independently authenticated.

Potential errors

• Corollary 8.2, p.8: Eq.8.7 uses strict thresholding. If the critical law has an atom, the subsequently mentioned randomized rule needs an added equality-probability term in its power formula. This is a presentational mismatch at the stated atom case, not an error in the alpha<1/2 smooth-halfspace formula.
• Section 11: independent moment estimates are not intrinsically less informative without fixing acquisition and precision. The proven strict comparison is to the explicitly relaxed integer-time test on the same S; it should not be extrapolated to arbitrary independent-oracle designs.

Strong novelty candidates

• Theorem 5.1 isolates a clean same-data separation caused solely by retaining the half-time cross moment, and converts it to a finite-sample power guarantee with an exact rational witness.
• The integrated parity-complete joint-Gram feasibility problem, active-kernel tangent description and ordinary bounded-band Farkas certificate provide an auditable workflow. The component theories themselves are established.

Unresolved material objections

None reported.

Strengths

• Theorem 3.1 states the correct parity-dependent matricial Hausdorff conditions, including rank-deficient sequences, and the paper uses the even criterion appropriate to the acquired H_q covariance.
• Theorem 4.1 obtains level control from one Gaussian singular-value event: the true structured covariance is feasible throughout that event, so adaptive inspection of all moment witnesses adds no multiple-testing penalty.
• Theorem 5.1 is a genuine same-acquisition separation. For the two-atom measure, s^2M0-M2 is positive while sM1-M2 is negative, and the nuclear-norm support calculation yields an explicit sample threshold.
• Theorem 7.1 correctly treats H_q positivity as inactive at K0 positive definite and compresses the active localizer to its kernel; the positive-density direction supplies the needed linearized Slater condition.
• Proposition 9.1 avoids treating floating-point infeasibility as proof. Injectivity of the Hankel map plus the bounded Loewner sandwich rules out weak infeasibility and supports an ordinary separating certificate.
• The exact replay passed every advertised algebraic fixture, including noncommuting 2x2 weights, the -3/128 dual objective, strict old/full margins, and the smooth-face local-power illustration.

Weaknesses

• The operational model requires iid, zero-mean Gaussian joint exponent sketches. The paper does not provide a corresponding validated acquisition for sub-Gaussian, dependent, or raw Monte Carlo observations.
• The finite truncation tests only the exposed covariance cone; alternatives sharing all acquired moments with a supported measure remain undetectable, as the manuscript acknowledges.
• The LAN and cone-likelihood proof compresses stochastic uniformity, scaled-set convergence, and the Fisher whitening identification into a short appeal to standard theory; readers must reconstruct these conditions from Chernoff/Shapiro-style results.
• The public-code replay checks algebraic fixtures but does not simulate coverage, solve a range of numerical SDPs, or validate scientific data, limiting empirical reproducibility claims.
• Targeted primary-source checks confirmed classical matricial Hausdorff and inequality-constrained likelihood antecedents but were not exhaustive enough to prove first priority for the integrated test.

Potential errors

• Theorem 8.1 reuses closely similar K0 notation for the base covariance and the whitened tangent cone; this is editorially hazardous in the displayed likelihood limit and should be disambiguated, although the intended geometry is recoverable.
• Theorem 8.3's detection statement is stronger than the fixed-distance consistency theorem cited in its proof. It is supportable by choosing a high-probability covariance radius o(dist(K_n,C)), but that vanishing-distance argument should be written explicitly.
• The local likelihood conclusion depends on positive definiteness and linearized Slater at the active face; more singular intersections are outside the theorem, so the phrase sharp tangent-cone power should not be generalized beyond those regular points.

Strong novelty candidates

• The single joint half-time Gaussian acquisition whose complete covariance enforces every anti-diagonal identity and uses cross-exponent blocks in one parity-complete support test.
• The explicit two-atom family proving finite-sample power separation between the full joint test and the precisely defined predecessor integer-grid test on the same observations.
• The combination of an exact active-kernel tangent preimage, Gaussian cone-projection local law, and ordinary auditable SDP rejection certificate for the structured moment cone.

Unresolved material objections

None reported.