A Finite-Dimensional Nonanalytic Spectral Transition and Exact High-Fidelity Rate–Distortion for Rank-r Born Prediction on Complex Grassmannians

Strengths

• The weak-field selector uses the first genuinely shape-sensitive invariant and handles the balanced case separately through the negative quartic coefficient.
• The appendix supplies the uniform two-derivative chart, exterior, annular, and Gaussian estimates needed to upgrade Ky Fan convergence to exact eventual orbit uniqueness.
• The rate–distortion reduction correctly allows arbitrary standard-Borel memories and mixed posterior means in the full capped-density body, avoiding a source-orbit restriction.
• The covariant Bingham channel satisfies the Shannon converse equality conditions and its report equals its posterior mean, giving operational attainment rather than only a dual formula.
• The Jacobi–Selberg calculation yields an exact multiplicative normalizer and therefore the additive high-rate constant, with explicit complementary-rank rescaling.

Weaknesses

• The thresholds u*(d,r), λ*(d,r), and D*(d,r) are existential; the paper does not provide computable numerical bounds or the intermediate spectral-phase classification.
• The global analytic theorems rest on concise asymptotic estimates rather than formal proof-assistant verification; the supplied programs check constants and examples, not universal global optimality.
• The reproducibility statement names a frozen replay ZIP that is not present as a single archive in this curated packet, although every manifest-listed replay source and stored JSON input needed for the stated commands is present.
• External bibliographic priority was not checked in this review, so the historical distinctness of the finite-d transition and unrestricted high-fidelity formula remains an honest limit.

Potential errors

• The finite Monte Carlo spectral diagnostic selects random sampled spectra over both canonical candidates at fields 5 and 20. Because intermediate phases are explicitly open and high-field importance sampling is noisy, this neither contradicts nor verifies the theorem's unspecified eventual threshold.
• Regenerated floating-point diagnostic JSON differs from the stored files in last-bit values, while exact identities and the d=5,r=2 crossing JSON agree. Consumers should not demand byte identity for those numerical outputs.

Strong novelty candidates

• The incompatibility of exact weak and exact eventual strong optimizers as a mechanism forcing nonanalyticity at fixed finite dimension is a strong conceptual contribution.
• The unrestricted high-fidelity RDF over the entire physical posterior body, with exact Bingham attainment and a sharp Selberg additive constant, is a strong information-theoretic contribution.

Unresolved material objections

None reported.

Strengths

• Theorems 3.2 and 4.3: the cubic selector and Ky Fan equality select genuinely different orbits for 1<r<d/2. Within-block symmetry makes each candidate exactly stationary, preventing the perturbative argument from proving only convergence.
• Lemma 4.2 and Appendix A: the maximizing graph plane remains unique under internal eigenvalue collisions. Fixed cutoffs, the positive cross-gap and polynomially weighted Gaussian domination support the required uniform C2 remainder.
• Theorem 4.4: equality with the analytic one-spike branch on an open interval, followed by a strictly superior strong branch, correctly forces finite nonanalyticity. No first-order or unique transition is inferred.
• Theorems 5.2 and 6.1: optimization ranges over 0<=sigma<=I/r, not just projectors. Localization, eventual spectral uniqueness, radial concavity and posterior-mean self-consistency supply the missing operational steps.
• Proposition 5.1 and Corollary 6.2: Jacobi-to-Laguerre scaling gives n=r(d-r), K_5,2=144 and the stated additive high-rate constant. Replayed quadrature agrees with (30) to about 3e-14 at kappa=20,40,80.
• Proposition 6.3 and (31): complementary-rank distortion rescales by r^2/(d-r)^2; the Haar query second moment gives the d(d+1) Born-loss conversion. Both respect the source normalization.

Weaknesses

• Theorem 6.1 supplies no effective D_*(d,r). Accordingly Table 2 cannot certify that its finite kappa values lie on the unrestricted frontier; the manuscript correctly labels this limitation.
• The replay checks finite moments, constants and selected spectra. Its weak/strong numerical diagnostics neither bound the entire spectral sphere nor prove the differentiated Laplace theorem.
• Bibliographic priority remains bounded. I directly inspected the sphere RDF comparator but did not conduct an exhaustive search or access unpublished drafts or assessment pages.
• The named frozen replay ZIP on p.11 was not supplied as that exact archive, so its advertised archive digest was not independently reproduced. The 10 supplied public-code files for this paper were individually authenticated; this is not evidence of public repository absence.

Potential errors

• Lemma 2.3, p.4: Gibbs posteriors and constant Q_lambda on the support are insufficient for equality in (5); that constant must equal sup Q_lambda. Counterexample: d=2,r=1,lambda=10 and the sole report A=0 satisfy both displayed conditions, yet I+lambda D=5, whereas the RHS is at most log(20/(1-exp(-20)))<3. Theorem 6.1 uses an actual maximizing orbit, so its application remains valid.
• Equation (37), p.12: with dW ordinary Lebesgue measure as stated in (33), the complex Gaussian integral contributes pi^n, n=r(d-r). For d=2,r=1 and A=diag(1/2,-1/2), c_2,1=1/pi but Z=2sinh(s/2)/s. Thus (37) misses a factor pi. This changes an A-independent constant, not Lemma 4.2 derivatives; (22) and the high-rate constant are independently correct.
• Theorem 5.2 states uniqueness of the positive self-consistency solution more globally than its localized proof spells out. The omitted exclusion is available: b(kappa)/kappa is positive and bounded away from zero on each compact interval including its positive limit at zero, so large lambda forces any positive root to large kappa. This is an exposition gap, not an outstanding counterexample.

Strong novelty candidates

• Theorem 4.3: exact eventual uniqueness at fixed radius, rather than merely convergence to the Ky Fan maximizer, using uniform differentiated asymptotics through internal block collisions.
• Theorem 6.1: exact high-fidelity Shannon RDF over the entire physical posterior body, attained by a covariant matrix-Bingham channel with self-consistent depolarized-projector reports.
• Theorem 4.4: finite-dimensional nonanalyticity caused by optimizing the external spectrum, distinguished from fixed-spectrum large-dimensional spherical-integral transitions.

Unresolved material objections

None reported.

Strengths

• G-S1 | Lemmas 2.2 and 2.3, pp. 3-4: conditional Hilbert-Schmidt least squares and data processing justify restricting arbitrary reports to 0 <= sigma <= I/r, tr(sigma)=1. The relative-entropy inequality itself is correct and does not assume that reconstructions remain pure orbit points.
• G-S2 | Lemma 3.1, Theorem 3.2 and Proposition 3.3, pp. 4-5,13: beta moments give the stated quadratic/cubic coefficients; the cubic extremum has one positive spike, and the balanced negative quartic selects equal positive/negative blocks. Exact rational checks covered 304 beta identities for d=3,...,18 and balanced Schur coefficients for even d=4,...,18. Within-block symmetry keeps these spectra stationary under the invariant perturbation.
• G-S3 | Lemmas 4.1-4.2 and Theorem 4.3, pp. 5-6,11-12: the strict cross-gap supplies a unique maximizing plane even with internal eigenvalue collisions. Fixed cutoffs, exponentially small exterior/annular contributions and Gaussian polynomial domination support C2 control. On the labeled trace-zero sphere, Hess(h_r)=-h_r/R^2 times the metric; this makes the large-s optimizer exactly two-block, beyond mere asymptotic convergence.
• G-S4 | Theorem 4.4 and Proposition 4.5, pp. 6-7: incompatible weak/strong unique orbits and the analytic identity theorem force failure of real analyticity at finite field. The first-exit coexistence-or-degeneracy alternative follows from compact separation and local strictness. The physical-radius bound satisfies both eigenvalue constraints. No claim of a first-order or unique transition is needed.
• G-S5 | Theorem 5.2 and Theorem 6.1, pp. 7-8: radial localization keeps maximizers away from zero; exact fixed-radius optimality reduces the complete physical-body dual to b S_Q. The inward boundary derivative and large-lambda radial concavity produce an interior maximizer. Haar covariance gives the correct source marginal, and equation (25) yields the posterior-mean report. Thus the stronger, corrected equality condition in G1 is met here.
• G-S6 | Proposition 5.1, Corollary 6.2 and Proposition 6.3, pp. 7-9: independent exact Hankel-determinant ratios verified all 81 (d,r) cases with 2<=d<=18 and r<=d/2. K_5,2=144. Direct Jacobi quadrature reproduced the crossing s=4.8585790529165 and all three Table 2 rows. The coefficient r(d-r), distortion rescaling r^2/(d-r)^2, and Born-query factor 1/[d(d+1)] are consistent.
• G-S8 | Fresh public-code execution: verify_grassmann_crossover.py completed its 1558 unbalanced parameter pairs through d=80 and cubic complement checks. verify_grassmann_spectral_switch.py --d 5 --r 2 completed its exact Schur coefficient comparisons and bounded seeded Monte Carlo.
• G-S9 | Fresh public-code execution: generate_d5r2_example.py completed with quadrature orders 96/144; its parsed JSON equals the supplied canonical-branch JSON. verify_high_fidelity_grassmann_rdf.py completed 81 low-rank and 72 complement cases through d=18. Across the supplied high-fidelity JSON, the maximum difference divided by max(1,absolute values) was 3.41e-15; the seeded switch JSON difference was at most 2.2e-16. No discrepant exact scientific fields were found.

Weaknesses

• G-W1 | pp. 9-11: D_*(d,r) and the large-field threshold are existential. The listed kappa=20,40,80 values are verified channel values, not certified points inside the unrestricted-optimality interval. The paper appropriately acknowledges this, but it substantially limits numerical deployment of the exact theorem.
• G-W2 | pp. 1-2,10,13-14: the checked prior work supports the distinction between fixed-d spectrum optimization, orbit-restricted coding and ambient reconstruction. In particular, Dytso-Cardone already solve ambient squared-error sphere rate-distortion [https://arxiv.org/html/2401.04248v1], while Harremoes uses the group as both alphabets [https://arxiv.org/pdf/0901.0015]. These comparisons do not establish that no earlier Grassmann spectral-optimization theorem exists.
• G-W3-S | Public code supplement: the four source files are now verified and executed. The differently packaged code ZIP does not reproduce the manuscript-named Eriksson_Unified_Grassmann_Transition_RDF_Replay_v1.zip byte stream, so that original aggregate digest is still not verified. The available files are bound instead to the supplemental manifest.
• G-W4-S | verify_grassmann_spectral_switch.py: its controlled_log_mgf modifies a sampled log MGF by empirical mean/variance corrections at every field, including s=20. In the reproduced 12000-projector/300-random-spectrum run, the best candidate is index 210 at s=5 and index 164 at s=20; these are not the two canonical spectra. These noisy, corrected scores cannot certify a global phase or refute the theorem. They also do not establish that s=20 exceeds the existential threshold. The script correctly labels itself diagnostic.
• G-W5-S | Replay scope: the crossover script uses rational arithmetic for the moment identities, but floating-point square roots for cubic candidate values and Ky Fan slopes, despite its aggregate output saying parameter pairs were checked exactly. The high-fidelity sweep mainly checks consequences of inserted formulas; the independent moment-determinant calculation from the original assessment provides a separate normalization check. Neither program tests the analytic uniform derivative estimate.

Potential errors

• G1 | Confirmed local error, Lemma 2.3, p. 4: constancy of Q_lambda on report support is insufficient for equality in (5); it must equal the global supremum there. Counterexample: d=2,r=1,lambda=10 and constant A=0. The stated Gibbs posterior is Haar and Q is constant, yet I+lambda D=5. A source-orbit report has Q=(1-exp(-20))/20, so the right side of (5) is at most 2.9957322756, strictly below 5. Theorem 6.1 is not invalidated because Theorem 5.2 separately establishes that its reports maximize Q.
• G2 | Confirmed local normalization error, Appendix A.3 equation (37), p. 12: with dW explicitly ordinary real-coordinate Lebesgue measure in (33), each complex Gaussian integral is pi/(a_i-a_j). Equation (37) needs pi^[r(d-r)] multiplying the same c_d,r. For d=2,r=1, c_2,1=1/pi and A=diag(a,b), the exact normalizer is (exp(sa)-exp(sb))/(s(a-b)); (37) is smaller by pi at leading order. This shifts the unspecified constant in (13), not spectral derivatives. Proposition 5.1 independently supplies the correct RDF normalization, so the main high-rate constant survives.
• G3 | Local proof omission, Theorem 5.2, p. 8: localization plus concavity proves uniqueness of the maximizing radius in the localized annulus, whereas the statement calls b the unique positive solution of (21). To exclude other stationary roots, one additionally needs to use that lambda=r^2*kappa/[2*b(kappa)] is bounded on every bounded kappa interval, including its continuous positive limit at zero; thus large lambda forces kappa large and b near one. The positivity of b prime and the stated asymptotics supply this missing step. No contrary stationary branch or main-theorem counterexample was found.

Strong novelty candidates

• Theorems 4.3-4.4: exact eventual two-block rigidity combined with the one-spike weak phase to force a finite-d optimized nonanalyticity; not the mere existence of Bingham laws or a thermodynamic-limit transition.
• Theorems 5.2-6.1: eventual optimization over the entire capped density body, followed by an exactly posterior-calibrated covariant channel. This is stronger than optimizing an orbit-restricted quantizer.
• Corollary 6.2: an explicit additive high-rate constant for this unrestricted Grassmann source problem, with the underlying Selberg machinery acknowledged as classical. Priority remains provisional.

Unresolved material objections

None reported.

Strengths

• The paper separates weak-field, strong-field, and unknown intermediate regimes and does not infer the global nonanalytic point from the canonical-branch crossing.
• Appendix A addresses the key need for two differentiated orders, including uniform exterior tails, Gaussian rescaling, and internal block collisions.
• Theorem 6.1 proves unrestricted posterior-body attainment rather than restricting reproduction points to the source Grassmannian.
• The covariant channel satisfies the Gibbs equality kernel and exact posterior-mean self-consistency, closing both converse and achievability.
• The Selberg constant, complement scaling, d=5,r=2 normalizer, and high-rate tables were independently rerun from the supplied public programs.

Weaknesses

• The thresholds u*(d,r), lambda*(d,r), and D*(d,r) are existential; the theorem gives no effective interval for a concrete pair except diagnostic evidence.
• The exact-eventual optimizer proof compresses the passage from spectral-coordinate Hessians to uniqueness modulo the full unitary orbit, especially at within-block eigenvalue collisions.
• The public programs explicitly do not verify the uniform Laplace theorem, exact global optimizer, nonanalyticity, or unrestricted RDF converse; those remain dependent on the written analysis.
• The bounded source check supports adjacency to matrix-Bingham, spherical-integral, Grassmann-quantization, and sphere-RDF work, but does not establish comprehensive bibliographic priority.

Potential errors

• Theorem 4.3 requires uniform C2 control on a normal slice to the conjugacy orbit; Appendix A proves ordinary spectral derivative control, and the manuscript relies on unitary invariance to pass to the quotient. This appears sound but is the point most likely to hide a geometric regularity gap.
• The bounded Monte Carlo spectral diagnostic at s=20 found sampled spectra with a larger estimated score than the canonical rank-two branch. This is not a contradiction because the theorem is only eventual and the estimate is a rare-event-sensitive finite-sample diagnostic, but it shows that the code does not locate the strong-field threshold.
• Proposition 5.1's exponentially small remainder uses the special equal-rank complex Jacobi density with zero x-power. The d=5,r=2 quadrature and constants agree, but the general density normalization remains a mathematical input rather than a program-verified theorem.

Strong novelty candidates

• A fixed-finite-dimensional nonanalyticity forced by optimizing the external matrix-Bingham spectrum rather than by a thermodynamic limit.
• Exact eventual rank-r two-block uniqueness obtained from a differentiated Grassmann Laplace expansion stable under internal block collisions.
• The exact unrestricted high-fidelity rate-distortion parametrization with a posterior-mean-self-consistent covariant matrix-Bingham channel.
• The explicit Jacobi-Selberg additive high-rate constant and complementary-rank distortion transfer.

Unresolved material objections

None reported.

Strengths

• G-S1 | Lemmas 2.2 and 2.3, pp. 3-4: conditional Hilbert-Schmidt least squares and data processing justify restricting arbitrary reports to 0 <= sigma <= I/r, tr(sigma)=1. The relative-entropy inequality itself is correct and does not assume that reconstructions remain pure orbit points.
• G-S2 | Lemma 3.1, Theorem 3.2 and Proposition 3.3, pp. 4-5,13: beta moments give the stated quadratic/cubic coefficients; the cubic extremum has one positive spike, and the balanced negative quartic selects equal positive/negative blocks. Exact rational checks covered 304 beta identities for d=3,...,18 and balanced Schur coefficients for even d=4,...,18. Within-block symmetry keeps these spectra stationary under the invariant perturbation.
• G-S3 | Lemmas 4.1-4.2 and Theorem 4.3, pp. 5-6,11-12: the strict cross-gap supplies a unique maximizing plane even with internal eigenvalue collisions. Fixed cutoffs, exponentially small exterior/annular contributions and Gaussian polynomial domination support C2 control. On the labeled trace-zero sphere, Hess(h_r)=-h_r/R^2 times the metric; this makes the large-s optimizer exactly two-block, beyond mere asymptotic convergence.
• G-S4 | Theorem 4.4 and Proposition 4.5, pp. 6-7: incompatible weak/strong unique orbits and the analytic identity theorem force failure of real analyticity at finite field. The first-exit coexistence-or-degeneracy alternative follows from compact separation and local strictness. The physical-radius bound satisfies both eigenvalue constraints. No claim of a first-order or unique transition is needed.
• G-S5 | Theorem 5.2 and Theorem 6.1, pp. 7-8: radial localization keeps maximizers away from zero; exact fixed-radius optimality reduces the complete physical-body dual to b S_Q. The inward boundary derivative and large-lambda radial concavity produce an interior maximizer. Haar covariance gives the correct source marginal, and equation (25) yields the posterior-mean report. Thus the stronger, corrected equality condition in G1 is met here.
• G-S6 | Proposition 5.1, Corollary 6.2 and Proposition 6.3, pp. 7-9: independent exact Hankel-determinant ratios verified all 81 (d,r) cases with 2<=d<=18 and r<=d/2. K_5,2=144. Direct Jacobi quadrature reproduced the crossing s=4.8585790529165 and all three Table 2 rows. The coefficient r(d-r), distortion rescaling r^2/(d-r)^2, and Born-query factor 1/[d(d+1)] are consistent.

Weaknesses

• G-W1 | pp. 9-11: D_*(d,r) and the large-field threshold are existential. The listed kappa=20,40,80 values are verified channel values, not certified points inside the unrestricted-optimality interval. The paper appropriately acknowledges this, but it substantially limits numerical deployment of the exact theorem.
• G-W2 | pp. 1-2,10,13-14: the checked prior work supports the distinction between fixed-d spectrum optimization, orbit-restricted coding and ambient reconstruction. In particular, Dytso-Cardone already solve ambient squared-error sphere rate-distortion [https://arxiv.org/html/2401.04248v1], while Harremoes uses the group as both alphabets [https://arxiv.org/pdf/0901.0015]. These comparisons do not establish that no earlier Grassmann spectral-optimization theorem exists.
• G-W3 | p. 11: the named replay archive and scripts are not supplied in this batch. Independent calculations here are distinct from that archive; its digest, script contents and claimed coverage were not verified. No formal proof verification or all-parameter numerical certification was performed.

Potential errors

• G1 | Confirmed local error, Lemma 2.3, p. 4: constancy of Q_lambda on report support is insufficient for equality in (5); it must equal the global supremum there. Counterexample: d=2,r=1,lambda=10 and constant A=0. The stated Gibbs posterior is Haar and Q is constant, yet I+lambda D=5. A source-orbit report has Q=(1-exp(-20))/20, so the right side of (5) is at most 2.9957322756, strictly below 5. Theorem 6.1 is not invalidated because Theorem 5.2 separately establishes that its reports maximize Q.
• G2 | Confirmed local normalization error, Appendix A.3 equation (37), p. 12: with dW explicitly ordinary real-coordinate Lebesgue measure in (33), each complex Gaussian integral is pi/(a_i-a_j). Equation (37) needs pi^[r(d-r)] multiplying the same c_d,r. For d=2,r=1, c_2,1=1/pi and A=diag(a,b), the exact normalizer is (exp(sa)-exp(sb))/(s(a-b)); (37) is smaller by pi at leading order. This shifts the unspecified constant in (13), not spectral derivatives. Proposition 5.1 independently supplies the correct RDF normalization, so the main high-rate constant survives.
• G3 | Local proof omission, Theorem 5.2, p. 8: localization plus concavity proves uniqueness of the maximizing radius in the localized annulus, whereas the statement calls b the unique positive solution of (21). To exclude other stationary roots, one additionally needs to use that lambda=r^2*kappa/[2*b(kappa)] is bounded on every bounded kappa interval, including its continuous positive limit at zero; thus large lambda forces kappa large and b near one. The positivity of b prime and the stated asymptotics supply this missing step. No contrary stationary branch or main-theorem counterexample was found.

Strong novelty candidates

• Theorems 4.3-4.4: exact eventual two-block rigidity combined with the one-spike weak phase to force a finite-d optimized nonanalyticity; not the mere existence of Bingham laws or a thermodynamic-limit transition.
• Theorems 5.2-6.1: eventual optimization over the entire capped density body, followed by an exactly posterior-calibrated covariant channel. This is stronger than optimizing an orbit-restricted quantizer.
• Corollary 6.2: an explicit additive high-rate constant for this unrestricted Grassmann source problem, with the underlying Selberg machinery acknowledged as classical. Priority remains provisional.

Unresolved material objections

None reported.