{"id":"8fb47fe7-7a1f-4645-9598-ed11d3807f8e","arxiv_id":"2607.13761","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Under isotropy, the rank-r top-stratum chi-bar-square limit stochastically dominates every lower-rank stratum and every local rank transition in PSD bounded-rank likelihood-ratio tests.","lead":"This paper derives limit distributions for likelihood-ratio tests of whether a positive-semidefinite matrix has rank at most r, showing that lower-rank null points need a different calibration than the usual chi-bar-square law. It proves that, under an isotropy condition, the top-rank critical value is the most conservative reference for all lower-rank strata and local rank transitions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader's ACCEPT verdict is supported. Theorem 3.3 is a clean reduction, and Theorem 3.10's dominance follows from interlacing and Poincaré separation; the unresolved anisotropic corank≥2 case is explicitly stated as an open problem rather than hidden. The sensitivity result is honestly conditional on Assumption 4.2, with a family-specific check. The reader's weakest assumptions (isotropy and Assumption 4.2) are the most restrictive conditions, and I agree they are the least secure; however, because they are explicit conditions rather than unstated premises, I do not regard them as load-bearing objections to the stated claims. The only unreviewed item is the supplied R code, whose detailed protocol (seeds, file manifest, analytic cross-checks) makes it likely correct but not machine-verified.","tokens_in":35407,"tokens_out":24887,"duration_ms":259565,"concrete_test":"Run the supplied R script code_Geometry_LR_calibration.R from a clean R session and independently verify the quantitative tables, especially Table 3's finite-sample rejection probabilities and the analytic top-stratum quantile 5.4845131865391; this would certify the numerical artifacts, the only component not machine-checked.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims—Theorem 3.3's active-block reduction, the isotropic least-favourable Theorem 3.10, the corank-one anisotropic Proposition 3.12, and the conditional sensitivity Theorem 4.3—are internally coherent and correctly scoped. The proofs of the tangent-cone structure, the affine profiling identity (3.3), and the interlacing/compression arguments in Theorem 3.10 check out. The isotropy assumption is explicit and the anisotropic corank≥2 case is declared open; Assumption 4.2 is likewise declared conditional with an explicit verification family. These are honest scoping statements, not hidden gaps. I did not find a load-bearing mathematical error or an unstated assumption that would overturn the main calibration result.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies likelihood-ratio tests for the positive-semidefinite rank hypothesis H0: rank(Σ) ≤ r against the unconstrained PSD alternative. Under uniform LAN and local set convergence, it develops a stratified active-block reduction: after profiling regular nuisance parameters, the limiting statistic depends only on the kernel block of the true matrix. The main reduction (Theorem 3.3) represents the limit as a distance-difference between projections onto S^{-1/2}(S^k_+) and S^{-1/2}(K^k_m). On the top stratum this recovers the classical chi-bar-square law. Under isotropy, interlacing and compression establish that the top-stratum law stochastically dominates every fixed lower stratum and every admissible local null rank transition (Theorem 3.10); for arbitrary anisotropy the same transition dominance is proved when the active corank is one (Proposition 3.12). A conditional positive-level Hadamard shape derivative for the top-stratum distribution and quantile is given in Theorem 4.3, with an explicit verification family. Numerical experiments cover nuisance profiling, lower-stratum rank transitions, anisotropic lower-stratum laws, and derivative/ascent checks. The paper is explicit about the open anisotropic corank-at-least-two interface problem and about the conditional nature of Assumption 4.2.","tokens_in":35553,"tokens_out":32844,"duration_ms":285938,"significance":"Assuming the results hold, this is a significant contribution to constrained likelihood-ratio inference. It provides a unified treatment of the stratified PSD rank null, identifies precisely where chi-bar-square calibration fails, and gives a least-favourable calibration result under isotropy and in the corank-one anisotropic case. The geometric proofs — tangent-cone structure, local Hausdorff convergence, interlacing/compression dominance — are careful and self-contained. The assumptions are stated openly, and the numerical validation is reproducible, with exact finite-sample likelihood calculations and no parameters fitted to the target claims. The honest scoping of the unresolved anisotropic interface and the conditional sensitivity theorem is a strength rather than a defect: the main least-favourable theorem is explicitly conditional on isotropy, and the sensitivity theorem is explicitly conditional on Assumption 4.2.","major_comments":[],"minor_comments":[{"comment":"The symbol C is used both for the cone C_{P_s} in (3.4) and for an admissible drift C in Proposition 3.9 and Theorem 3.10. This is locally clear but can be confusing; consider using a different letter (e.g., M) for the drift.","section":"§3.6"},{"comment":"The transition from the q=3 top-stratum example to the q=2 lower-stratum experiment is abrupt. A sentence stating explicitly that the same elliptic cone family Cγ is being reused for a different active-space problem (q=2, r=1, s=0) would help the reader.","section":"§5.4"},{"comment":"The sentence 'The same active-dimension-two law is the γ=1 member of the analytic chi-bar-square family below' is potentially confusing because the preceding sentence refers to the rank-one top stratum. Clarify that the top-stratum law, not the lower-stratum law, is the γ=1 member.","section":"§5.3"},{"comment":"In the paragraph before Table 7, 'T wo runs satisfy' should read 'Two runs satisfy the gradient tolerance directly.'","section":"§5.5"},{"comment":"In Assumption 4.2(2), the notation 'whenever (R̃,t)→(R,c)' is fine, but it may be worth writing 'where R and c are fixed' to avoid confusion with the evaluation point on the right-hand side of (4.5).","section":"§4.2"}],"recommendation":"accept","confidential_remarks":"The paper is ready for publication in my assessment. The anisotropic corank≥2 gap and the conditional nature of Assumption 4.2 are clearly and repeatedly stated; there is no hidden circularity or overfitting. I recommend acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nYou should know this paper is a solid, field-specific advance, not a reshuffle. The central move is Theorem 3.3: after profiling nuisance parameters, the LRT limit at every rank stratum and along local rank transitions reduces to a single active-block Gaussian experiment. That reduction is clean and the proof holds together. The genuinely new piece is Theorem 3.10: under isotropy, the top-stratum chi-bar-square law is least favourable over all fixed lower strata and all admissible local rank transitions, via interlacing and compression. That directly contrasts with Chen–Fang's unrestricted-rank warning, and it is a useful calibration guarantee. I also credit the authors for proving the tangent-cone formula they need rather than just citing it, and for shipping detailed R code with seeds and file maps.\n\nThe soft spots are real but proportionate. The global least-favourable claim stops at isotropy; for anisotropic information the paper only settles active corank one (Prop. 3.12) and leaves corank ≥2 as an open problem. That is not hidden — it is stated repeatedly, including in the abstract and discussion — but it does limit the scope of the headline calibration claim. The sensitivity analysis in §4 is deliberately conditional on Assumption 4.2, which is verified only for an explicit family. That is honest, but it means the quantile derivative result is a conditional statement with a family-specific check, not a general theorem. I did not find a load-bearing error. The numerical experiments look careful; I did not run the code, but the protocol is unusually transparent.\n\nWho should read this: anyone working on likelihood-ratio inference for covariance components, low-rank PSD matrices, or constrained testing at singular boundaries. It will be cited for the active-block reduction and the isotropic dominance. If I were refereeing it, I would ask for a clean statement of what remains open in the anisotropic corank≥2 case and maybe a sharper check of Assumption 4.2 in a second non-trivial family, but I would not block acceptance on those. The paper deserves a serious referee.\n\nRecommendation: send it to peer review.","headline":"A careful, genuinely useful paper on stratified LRT calibration under PSD rank constraints; the core reduction and isotropic dominance are new and the paper is honest about its anisotropic gap.","tokens_in":36002,"tokens_out":774,"would_cite":true,"duration_ms":50583,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F05","62E20","62H15","90C22"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper establishes that likelihood-ratio tests for the hypothesis that a positive-semidefinite matrix has rank at most r are governed, after nuisance profiling, by a single reduced Gaussian experiment: the limit is a difference of squar","keywords":["likelihood-ratio test","rank constraint","positive semidefinite","chi-bar-square","stratified null","semidefinite cone","rank transition","least favourable distribution"],"falsifier":"Take an anisotropic reduced covariance with active space of dimension three and rank budget m=1 (active corank two), for instance the elliptic-cone family with S=diag(1,1,γ) in adapted coordinates; compute the 95th percentile of the fixed-stratum and transition laws Δ_{0,1}(C;Y) for several drifts C and compare with the top-stratum chi-bar-square 95th percentile. If any of those quantiles exceeds the top value, the isotropic dominance theorem does not extend to this anisotropic case.","tokens_in":35302,"feed_emoji":"📊","tokens_out":7218,"duration_ms":68946,"temperature":0.7,"pith_summary":"The paper studies likelihood-ratio tests of the hypothesis that a positive-semidefinite matrix has rank at most r. Because the null set is a union of rank strata, the usual chi-bar-square calibration—correct at matrices of rank exactly r—does not by itself control the test at lower-rank matrices or along sequences where the rank changes at the n^{-1/2} scale. The paper shows that after profiling all regular nuisance parameters, every fixed null rank and every local null rank transition is described by one reduced Gaussian experiment: the limit is the difference of squared Frobenius distances to a rank-constrained semidefinite cone and to the semidefinite cone. On the top stratum this recovers the chi-bar-square law; on lower strata it is generally a nonconvex projection law. The main calibration result is that, under isotropy, the top-stratum law dominates all fixed strata and all admissible transitions, so its critical value is least favourable for the composite null; for anisotropic information the same dominance is proved only when the active corank is one.","feed_headline":"One reduced Gaussian law sets limits for every rank stratum","feed_subtitle":"Lower-rank nulls and rank-crossing sequences follow the same distance-difference statistic; top-stratum critical values stay valid under iso","key_machinery":"The load-bearing object is the active compression of the true matrix's kernel: for a null matrix of rank s, with kernel dimension k_s=q−s, the active block is A_{P_s}(H)=U_s^T H U_s, and the reduced covariance is S_{P_s}=A_{P_s} I_eff^{-1} A_{P_s}^*. Profiling the nuisance parameters and all off-kernel matrix directions reduces the Gaussian limit to a difference of squared distances between a whitened random matrix Y and the cones C_{P_s}=S^{-1/2}(S_+^k) and D_{P_s,m_s}=S^{-1/2}(K_m^k), where K_m^k is the nonconvex cone of PSD matrices of rank ≤m. The tangent-cone identity T={H:A_{P_s}(H)∈K_m^k} is what makes the reduction exact; it converts a semidefinite-rank problem into a finite-dimensio","core_discovery":"The central claim is Theorem 3.3: at a null matrix of rank s≤r, after nuisance profiling the likelihood-ratio statistic converges to Δ_{s,r}(P_s)=dist²_F(Y,D_{P_s,m_s})−dist²_F(Y,C_{P_s}), where Y is standard Gaussian in the active space, C_{P_s} is the whitened PSD cone, and D_{P_s,m_s} is the whitened set of PSD matrices of rank at most m_s=r−s. This formula holds for every fixed rank stratum, and with a deterministic Gaussian shift it also holds for null paths that cross a rank interface at the local scale. On the top stratum s=r the rank constraint is inactive, D reduces to {0}, and the limit becomes the classical chi-bar-square law. On lower strata the limit is a nonconvex distance-diff","pith_inferences":["If the isotropy assumption can be relaxed, the practical payoff would be large: a single top-stratum chi-bar-square quantile would calibrate the whole composite bounded-rank test in covariance models without knowing the true rank.","The unresolved anisotropic active-corank ≥2 case is the natural place to look for a counterexample; a numerical search over elliptic cones with k=3,m=1 could settle whether the dominance holds more generally or fails.","Because lower-stratum projections are nonunique at eigenvalue ties, extending sensitivity to lower strata would require set-valued derivatives; one testable consequence is that coverage and critical values may jump at ties.","The reduced covariance S_{P_s} functions as a normal-form summary of the model; estimating it directly offers a plug-in calibration route for real-data applications."],"forward_implications":["At a rank-r null point the limiting law is the classical chi-bar-square law, so existing top-stratum critical values remain valid there and only there.","At lower-rank null points the limit is a nonconvex rank-constrained projection law; using the top-stratum critical value is conservative under isotropy, while using the central lower-stratum critical value can under-cover along rank transitions.","Null sequences whose rank changes at the n^{-1/2} scale are covered by the same reduced experiment with a deterministic drift, so no separate asymptotic construction is needed.","Plug-in critical values based on a consistent estimator of the reduced covariance converge by continuity of the stratified law.","On the top stratum the critical value is differentiable at positive levels under the paper's regularity conditions, giving a formula for sensitivity to orientation and nuisance information."],"fun_headline_variants":["One reduced Gaussian law covers every rank stratum","Top-stratum chi-bar-square valid for lower ranks","Nonconvex distance-diff limits for rank transitions","Uniform critical values for anisotropic rank tests","Least-favorable law proven for all rank nulls"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The least-favourable calibration claim assumes the reduced active covariance is proportional to the identity on every stratum; when this isotropy fails and the active corank q−r is at least two, the paper does not prove that the top-stratum law dominates lower strata and rank transitions.","fun_headline_variants_meta":{"raw":{"variants":["One reduced Gaussian law covers every rank stratum","Top-stratum chi-bar-square valid for lower ranks","Nonconvex distance-diff limits for rank transitions","Uniform critical values for anisotropic rank tests","Least-favorable law proven for all rank nulls"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000699,"raw_usage":{"total_tokens":3014,"prompt_tokens":786,"completion_tokens":2228,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":530,"completion_tokens_details":{"reasoning_tokens":2157}},"tokens_in":530,"tokens_out":2228,"duration_ms":17825,"temperature":1.0,"reasoning_tokens":2157,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T03:46:35.512549+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take an anisotropic reduced covariance with active space of dimension three and rank budget m=1 (active corank two), for instance the elliptic-cone family with S=diag(1,1,γ) in adapted coordinates; compute the 95th percentile of the fixed-stratum and transition laws Δ_{0,1}(C;Y) for several drifts C and compare with the top-stratum chi-bar-square 95th percentile. If any of those quantiles exceeds the top value, the isotropic dominance theorem does not extend to this anisotropic case.","supporting_citations":[],"review_version":1}