{"id":"c8f5eaf0-88cc-45e3-bf71-3bdadca05b37","arxiv_id":"2607.06373","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":5,"one_line_summary":"First-order null laws, kurtosis-immunity characterizations, and a spike-debiased absorption-ratio estimator are derived for spectral functionals of shrinkage covariance estimators on overlapping rolling windows.","lead":"This paper derives calibration tools for distinguishing real structural change from estimation noise in rolling covariance monitors—projector movement, absorption ratio, and leading-eigenvalue share—under shrinkage estimators. A practitioner reading spectral shifts on rolling windows gets a framework to know whether a movement is signal or noise, plus a debiased absorption-ratio estimator for high dimensions.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"Kurtosis immunity (Prop. 8) is fixed-N, M→∞, but Table 7 shows that at N/M≈0.46 with t₅ data, coverage with κ_true is only 75%; the reported 91% comes from a conservatively biased κ̂, not from the correction itself.","rationale":"The reader identified the i.i.d. elliptical assumption as the weakest link, which is a valid concern but somewhat diffuse—it points to an external assumption violated by financial data. The more precise and load-bearing issue is internal to the paper's own evidence: even within the elliptical family, the fixed-N asymptotic theory (Proposition 8) does not deliver nominal coverage in the proportional regime where the method is applied. Table 7 makes this visible: κ_true gives 75% coverage at N=115, t₅, and the reported 91% is achieved through a conservatively biased κ̂ estimator, not through the first-order correction itself. This means the abstract's claim that 'one estimated scalar calibrates the absorption-ratio intervals across the elliptical family' overstates the practical reach. The paper is honest about this limitation in §5.4, which is why the verdict should remain CONDITIONAL rather than moving to REJECT. The theoretical contributions—Proposition 8 and its converse, the cleaning-debiasing wedge (Proposition 11), the null transfer to rotation-equivariant estimators (§4.1)—are correct and independently verifiable. The concern is about the gap between the fixed-N theory and the proportional-regime application, which the paper's own Table 7 reveals but does not fully connect to the headline claims. The reader's CONDITIONAL verdict with MODERATE confidence is appropriate; my concern sharpens the reason for the condition without changing the verdict itself.","tokens_in":39609,"tokens_out":8159,"duration_ms":369861,"concrete_test":"Recompute Table 7 at N=115, M=252, t₅ with the HD debiased estimator using κ_true=2 instead of the radial-MLE κ̂. If coverage remains below 85%, the first-order kurtosis correction is insufficient in the proportional regime, and the reported 91% coverage is an artifact of the conservative κ̂ estimator rather than evidence that the calibration works. Additionally, run a grid over N/M ∈ {0.1, 0.25, 0.5, 1.0} with κ_true fixed to trace how coverage degrades as the regime moves from fixed-N toward proportional; this would show whether there exists an N/M threshold below which the first-order correction is adequate.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 8 is stated for fixed N with M→∞. The paper's simulations and equity panel operate at N/M≈0.46 (N=115, M=252), the proportional regime. Table 7 reveals that at N=115 under elliptical t₅ (κ=2), the delta-method AR interval with κ_true achieves only 75.0% coverage versus 95% nominal. The HD debiased version with the radial-MLE κ̂ reaches 91.2%, but Remark 4 states that the radial-MLE κ̂ is 'conservative for extreme tails (t₅)'—i.e., it overestimates κ, inflating intervals. The gap between 75% (κ_true) and 91% (conservative κ̂) is therefore not evidence that the first-order (1+κ) correction works in high dimensions; it is evidence that the correction is insufficient and the coverage is rescued by estimator bias. The paper acknowledges this in §5.4 ('fully calibrated AR inference for elliptical heavy tails in high dimensions remains an open problem'), but the abstract's claim that 'one estimated scalar calibrates the absorption-ratio intervals across the elliptical family' is not qualified by dimension. The central practical claim—that a single scalar suffices—is not supported in the regime where the method is deployed, even within the elliptical family and even with the true κ. This is distinct from the reader's concern about non-elliptical data: the failure here occurs inside the elliptical family, under the paper's own assumptions, at the N/M ratio used throughout.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"This paper develops calibration tools for spectral monitoring of rolling covariance estimates under shrinkage. It contributes: (1) a distribution-free Davis–Kahan band for sequential top-K projector movement, capped at the maximum projector distance; (2) a first-order null law for projector movement between overlapping windows that transfers to rotation-equivariant shrinkage estimators; (3) a local power analysis yielding a detectability frontier for rotation alternatives; (4) a kurtosis-immunity principle showing that scale-invariant spectral functionals depend on the elliptical family only through a single scalar κ, with a converse characterization; (5) confidence intervals for the absorption ratio and leading-eigenvalue share; and (6) a trace-preserving spike-debiased absorption-ratio estimator for high dimensions. The results are verified by simulation under known population covariance, with an equity-panel illustration.","tokens_in":39829,"tokens_out":1521,"duration_ms":244080,"significance":"The paper addresses a practically important problem—distinguishing estimation noise from structural change in rolling covariance monitors—and provides a coherent calibration pipeline spanning both subspace movement and scalar spectral functionals. The kurtosis-immunity characterization (Proposition 8) with its converse is a clean theoretical contribution, and the cleaning–debiasing wedge (Proposition 11) gives a sharp diagnostic for a real pitfall in high-dimensional absorption-ratio inference. The transfer of the sample-covariance null to rotation-equivariant shrinkage (§4.1) is a useful simplification. Simulations are well-designed with known population covariance, and the detectability frontier (Corollary 3) provides a falsifiable design rule. The equity-panel appendix is appropriately scoped as diagnostic rather than confirmatory.","major_comments":[{"comment":"Abstract and §5.2–§5.3: The abstract states that 'one estimated scalar calibrates the absorption-ratio intervals across the elliptical family,' but Table 7 shows this claim fails in the regime where the method is deployed. At N=115, M=252 (N/M≈0.46) under elliptical t₅ with κ_true=2, the delta-method AR interval achieves only 75.0% coverage versus 95% nominal. The 91.2% coverage reported for the HD-debiased version with radial-MLE κ̂ is rescued by the conservative bias of κ̂ (acknowledged in Remark 4: 'conservative for extreme tails (t₅)'), not by the (1+κ) correction itself. Proposition 8 is a fixed-N, M→∞ result, but the simulations and equity panel operate in the proportional regime. The abstract's claim should be qualified by dimension regime, or the paper should clarify that the practical contribution is the fixed-N CLT while the high-dimensional case remains open (as §5.4 partially","section":null},{"comment":"§5.4, Proposition 11(iii): The CLT for the debiased absorption-ratio estimator is stated for the relative error √M(AR̂_deb/AR−1)→N(0,V_rel), but V_rel is left unspecified ('of order one'). Since this is the only inferential guarantee for the debiased estimator in the proportional regime—the paper's central practical setting—the variance formula should be given explicitly, or the paper should state clearly that the CLT is a consistency result without a usable variance for interval construction. Table 7 reports intervals for the debiased estimator, so the reader needs to know what variance is being plugged in.","section":null},{"comment":"§4.2, Proposition 2: The first-order null is derived under Gaussian sampling, with the elliptical extension in Remark 3 stated as a variance scaling by (1+κ). However, the proof in Appendix A.3 computes the variance of the increment entries g_ij under Gaussian y_ik∼N(0,λ_i), and the elliptical extension is asserted via the fourth-moment tensor (16) without a separate derivation. Given that the projector null is load-bearing for the monitoring pipeline, the elliptical extension should be verified explicitly—particularly whether the shared-block cancellation that drives the overlapping-window structure survives under elliptical sampling, or whether additional cross-terms appear at order s/M².","section":null}],"minor_comments":[{"comment":"Table 3 vs. Table 4: The |AR_K err| values for the Sample estimator differ (0.0877 vs. 0.0302) despite identical N, M, and seed. If K differs between the two tables, this should be noted; if not, the discrepancy needs explanation.","section":null},{"comment":"§3.1, Theorem 1: The constant c_K = 2√(2K) is described in Remark 2, but the per-date term T_{K,s} in (4) uses 2√2·min(√K·η_t, ‖E_t‖_F)/Δ_{K,t}. The relationship between the min(·) form and the simplified bound should be clarified for readers who may use the simplified version.","section":null},{"comment":"Figure 1 caption: 'C_A = Σ + εI' and 'C_B = (1+b)Σ' are referenced but the figure labels show 'C_A = +I' and 'C_B = (1+b)', missing the Σ symbol.","section":null},{"comment":"§6.4: The replication counts (150–200 per cell) are modest for size calibration at the 5% level; a note on Monte Carlo error for the reported flag rates would help the reader assess precision.","section":null},{"comment":"Appendix B, Table 12: The moment-based κ̂ (median ≈1.5) and radial-MLE κ̂ (≈0.2) disagree substantially. While §B.2 attributes this to within-window volatility clustering, the magnitude of the discrepancy (7.5×) is large enough that a brief discussion of which is more reliable for panel data would be valuable.","section":null},{"comment":"References: Paindaveine et al. (2026) and Lin and Pan (2026) are cited but appear to be forthcoming; please verify final publication details.","section":null}],"recommendation":"major_revision","confidential_remarks":"The skeptic's concern about the gap between 75% (κ_true) and 91% (conservative κ̂) coverage at N=115 is well-founded and is the primary reason for the major_revision recommendation. The paper is honest about this gap in §5.4 and §7, but the abstract overstates the result. The fixed-N theory (Proposition 8, Proposition 10) is sound; the issue is the mismatch between the asymptotic regime of the theory and the deployment regime of the simulations and panel. This is fixable by qualifying the abstract and either providing V_rel explicitly or clearly scoping the debiased CLT as a consistency result. The core theoretical contributions—the kurtosis-immunity characterization, the overlapping-window null, the detectability frontier, and the cleaning–debiasing wedge—are solid and worth publishing once the regime qualifications are addressed."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful and constructive report. All three major comments identify genuine gaps between claims and evidence, and we address each below. In brief: (1) the abstract overstates the kurtosis-calibration claim for the proportional regime and will be qualified; (2) the variance V_rel in Proposition 11(iii) can and will be given explicitly; (3) the elliptical extension of the overlapping-window null is currently asserted via Corollary 4 rather than derived in the proof of Proposition 2, and we will add the explicit computation.","responses":[{"response":"The referee is correct on all counts. Proposition 8 (kurtosis immunity) is a fixed-N, M→∞ result, and the abstract's claim that 'one estimated scalar calibrates the absorption-ratio intervals across the elliptical family' is stated without this qualification. Table 7 makes the gap explicit: at N=115, M=252 under t₅, the delta-method interval with κ_true achieves only 75.0% coverage—well below the 95% nominal—because Marchenko–Pastur spreading introduces a bias that the (1+κ) correction alone cannot absorb. The 91.2% coverage of the HD-debiased interval with radial-MLE κ̂ is indeed partly attributable to the conservative upward bias of κ̂ under t₅ (Remark 4: 'conservative for extreme tails'), not solely to the debiasing or the kurtosis correction. The paper acknowledges these limitations in §5.4 and the Discussion ('Fully calibrated absorption-ratio inference under elliptical heavy tails in high dimensions remains open'), but the abstract does not reflect them. We will revise the abstract to qualify the calibration claim by dimension regime: the fixed-N CLT (Proposition 10) provides valid kurtosis-calibrated intervals across the elliptical family, while in the proportional regime the (1+κ) correction is necessary but not sufficient, and fully calibrated AR inference under elliptical heavy tails at N/M ≈ O(1) remains open. We will also add a sentence in §5.3 noting that the 91.2% coverage in Table 7 benefits from the conservative direction of the radial-MLE κ̂ under t₅, so the reader does not infer that the (1+κ) factor alone closes the gap.","revision_made":"yes","referee_comment":"Abstract and §5.2–§5.3: The abstract states that 'one estimated scalar calibrates the absorption-ratio intervals across the elliptical family,' but Table 7 shows this claim fails in the regime where the method is deployed. At N=115, M=252 (N/M≈0.46) under elliptical t₅ with κ_true=2, the delta-method AR interval achieves only 75.0% coverage versus 95% nominal. The 91.2% coverage reported for the HD-debiased version with radial-MLE κ̂ is rescued by the conservative bias of κ̂, not by the (1+κ) correction itself. Proposition 8 is a fixed-N, M→∞ result, but the simulations and equity panel operate in the proportional regime."},{"response":"The referee is right that V_rel is left unspecified and that this is unsatisfactory given that Table 7 reports intervals for the debiased estimator. The variance can be written explicitly. By Bai and Yao (2008), the spike eigenvalue fluctuations √M(λ̂_i − ψ(λ_i)) are asymptotically independent Gaussians with variance σ²_{λ,i} = 2cσ⁴λ_i²/(λ_i − σ²)² (for Gaussian observations in the spiked model). The delta method through ψ⁻¹, whose derivative is ψ'(λ) = 1 − cσ⁴/(λ − σ²)², gives V_rel = (1/S_K²) Σ_{i≤K} σ²_{λ,i} / [ψ'(λ_i)]². We will add this formula to Proposition 11(iii) and specify in §5.4 that the intervals in Table 7 plug in the empirical estimates ĉ, σ̂², λ̂_i, and the radial-MLE κ̂ (which enters through the elliptical extension of the Bai–Yao variance, replacing σ²_{λ,i} by (1+κ̂)σ²_{λ,i} for the off-diagonal fourth-moment contribution). We will also note that this variance formula is derived under Gaussian sampling and that its validity under elliptical heavy tails in the proportional regime is not established—the same caveat that applies to the coverage numbers themselves.","revision_made":"yes","referee_comment":"§5.4, Proposition 11(iii): The CLT for the debiased absorption-ratio estimator is stated for the relative error √M(AR̂_deb/AR−1)→N(0,V_rel), but V_rel is left unspecified ('of order one'). Since this is the only inferential guarantee for the debiased estimator in the proportional regime—the paper's central practical setting—the variance formula should be given explicitly, or the paper should state clearly that the CLT is a consistency result without a usable variance for interval construction. Table 7 reports intervals for the debiased estimator, so the reader needs to know what variance is being plugged in."},{"response":"The referee raises a valid concern. The shared-block cancellation itself is exact and distribution-free: the M−s shared observations enter both window covariances identically and cancel in the difference S_t − S_{t−1} regardless of the sampling distribution. What requires verification is the variance structure of the increment entries g_{ij} under elliptical sampling, including cross-pair covariances. The argument is present in outline in Corollary 4 but is not spelled out in the proof of Proposition 2, and we agree it should be. The computation is as follows. In the population eigenbasis, the elliptical fourth-moment tensor (16) gives Cov(S_{ij}, S_{kl}) = (1+κ)(λ_iλ_k δ_{ik}δ_{jl} + λ_iλ_l δ_{il}δ_{jk}) + κλ_iλ_k δ_{ij}δ_{kl}. For the off-diagonal entries g_{ij} with i≤K<j (hence i≠j), the common-shock term κλ_iλ_k δ_{ij}δ_{kl} vanishes because δ_{ij}=0. The remaining terms are exactly (1+κ) times the Gaussian fourth-moment structure, so both the marginal variances Var(g_{ij}) = 2s(1+κ)λ_iλ_j/M² and the cross-pair covariances Cov(g_{ij}, g_{kl}) scale by (1+κ) relative to the Gaussian case. Since the first-order law (8) is a function of the g_{ij} alone, its entire distribution scales by √(1+κ), with no additional cross-terms at order s/M². This is the content of Corollary 4, but it is currently stated as a consequence of Proposition 8 rather than verified directly in the proof of Proposition 2. We will add this computation to Appendix A.3 as a separate lemma or remark, making the elliptical extension of the null a derived result rather than an assertion. One caveat we will note: the argument relies on the i.i.d. structure across observations (independence of the radial component across k), which is part of the elliptical model but would fail under volatility聚类; this残","revision_made":"no","referee_comment":"§4.2, Proposition 2: The first-order null is derived under Gaussian sampling, with the elliptical extension in Remark 3 stated as a variance scaling by (1+κ). However, the proof in Appendix A.3 computes the variance of the increment entries g_ij under Gaussian y_ik∼N(0,λ_i), and the elliptical extension is asserted via the fourth-moment tensor (16) without a separate derivation. Given that the projector null is load-bearing for the monitoring pipeline, the elliptical extension should be verified explicitly—particularly whether the shared-block cancellation that drives the overlapping-window structure survives under elliptical sampling, or whether additional cross-terms appear at order s/M²."}],"tokens_in":39424,"tokens_out":5753,"duration_ms":237392,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"Two things to know up front. First, the paper's core theoretical results—the kurtosis-immunity converse (Proposition 8(iii)), the exact window-scale pivotality (Proposition 9), and the transfer of the sample-covariance null to rotation-equivariant shrinkage estimators (§4.1)—are new and correctly derived. The cleaning-debiasing wedge (Proposition 11) is a clean contribution too: it shows that Frobenius-optimal shrinkage systematically mis-centers the absorption ratio, with an explicit formula for the bias, and supplies a trace-preserving spike-debiased fix. Second, the stress-test concern about Table 7 is legitimate and lands as stated. At N=115, M=252 under elliptical t₅, the delta-method AR interval with the true κ achieves only 75% coverage. The 91% figure comes from the radial-MLE κ̂, which Remark 4 itself calls conservative for extreme tails. So the coverage is rescued by estimator bias, not by the (1+κ) correction working as advertised in the proportional regime. The paper acknowledges this in §5.4 (calling fully calibrated AR inference for elliptical heavy tails in high dimensions an open problem), but the abstract's claim that one scalar calibrates AR intervals across the elliptical family is not qualified by dimension. That qualification should be added. What the paper does well: the proofs are detailed and honest, the simulations use a known population covariance so errors are measured against ground truth, and the limitations section is unusually forthcoming. The detectability frontier (Proposition 4) gives practitioners a usable design rule for the smallest detectable rotation. The worst-case band (Theorem 1) is a straightforward Davis-Kahan combination, correctly stated as conservative, and the cap at √(2K) is the right move. The reader's concern about non-elliptical data (Table 8: 48% coverage under log-normal) is also valid but less central—the paper is explicit that the (1+κ) correction works only within the elliptical family, and the bootstrap fallback is provided. The more pressing issue is the one the stress-test identifies: the failure occurs inside the elliptical family, under the paper's own assumptions, at the N/M ratio used throughout. No code or data is shipped, which limits reproducibility. This is a solid theoretical paper with one honest-but-unresolved gap between the fixed-N asymptotics and the proportional-regime simulations. The gap is documented in the paper itself; it just needs sharper framing in the abstract. Who benefits: researchers in spectral risk monitoring and high-dimensional covariance estimation. The calibration pipeline is directly usable by practitioners in the Gaussian and light-tailed regimes. Recommend sending to a serious referee. The referee should focus on whether the abstract and §5.4 can be brought into alignment on the dimensional scope of the kurtosis-immunity claim, and whether the bootstrap's high-dimensional consistency can be at least partially addressed.","headline":"The kurtosis-immunity converse and the null-transfer identity are genuinely new; the high-dimensional coverage gap is real but honestly reported.","tokens_in":40441,"tokens_out":700,"would_cite":true,"duration_ms":134997,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"One scalar calibrates shrinkage covariance monitors across elliptical returns","keywords":[],"falsifier":"Find a scale-invariant spectral functional of the covariance whose first-order asymptotic variance under i.i.d. elliptical sampling does not factor as (1+kappa) times its Gaussian variance, or find a non-scale-invariant functional that does enjoy such factoring; either would break the immunity-and-converse characterization of Proposition 8.","tokens_in":39798,"feed_emoji":"📊","tokens_out":1036,"duration_ms":129037,"temperature":0.7,"pith_summary":"This paper asks a question that practitioners face daily: when the dominant eigenspace or the absorption ratio of a rolling covariance estimate moves, is that movement genuine structural change or just estimation noise? The author builds a calibration pipeline around the finding that scale-invariant spectral functionals (the absorption ratio, the leading-eigenvalue share, and the top-K eigenspace projector) are immune to elliptical kurtosis at first order, and that this immunity holds only for scale-invariant functionals. The mechanism is Euler's relation for degree-zero homogeneous functions: the common-shock fourth-moment term that plagues level functionals like total variance vanishes identically for ratio functionals, so a single estimated scalar (the kurtosis parameter) calibrates the projector null, the absorption-ratio intervals, and the leading-share intervals simultaneously across the entire elliptical family. For the eigenspace monitor, the author derives a first-order null law for projector movement between overlapping windows that share most of their data, shows it transfers without modification to rotation-equivariant shrinkage estimators (because such estimators preserve eigenvectors), and supplies a parametric bootstrap for the cases the analytic null misses. A companion power analysis gives closed-form expressions for the smallest rotation a two-window monitor can detect and the number of fresh observations required, exposing an onset power ceiling where quarter-turn rotations are undetectable at intermediate overlap. In high dimensions, the author identifies an explicit wedge: sample eigenvalues bias the absorption ratio upward via Marchenko-Pastur spreading, Frobenius-optimal shrinkage biases it downward by a comparable margin, and only a trace-preserving spike-debiased estimator removes the bias.","feed_headline":"One scalar calibrates shrinkage covariance monitors across elliptical returns","feed_subtitle":"Scale-invariant spectral functionals shed the kurtosis term that breaks level functionals, unifying calibration for projectors and ratio","key_machinery":"Davis-Kahan sin-theta perturbation bound capped at maximum projector distance; first-order null law for overlapping-window projector movement driven by fresh observations only; Euler relation for degree-zero homogeneous functions proving kurtosis immunity; spiked-model bias inversion (Marchenko-Pastur spike map) for trace-preserving absorption-ratio debiasing; median-shift power approximation yielding closed-form detectability frontier","core_discovery":"The central object is the scale-invariance property of spectral functionals under the elliptical distribution family. The author proves that first-order kurtosis immunity (where the asymptotic variance scales by exactly (1+kappa) and no higher) holds for scale-invariant functionals and only for them, with the converse established via an Euler-relation argument along ray-connected open sets. This means the top-K projector, the absorption ratio, and the leading-eigenvalue share all shed the common-volatility-shock term that dominates level functionals under heavy tails, so one estimated kurtosis scalar calibrates all three. The second central result is the transfer principle for rotation-equiv","pith_inferences":[],"forward_implications":["Practitioners using shrinkage-cleaned covariance matrices for eigenspace monitoring need not re-derive null distributions for each shrinkage estimator: rotation-equivariant maps (linear shrinkage, QIS) preserve the eigenvectors and thus the sample-covariance null applies directly, halving bootstrap cost.","The cleaning-debiasing wedge means that computing the absorption ratio from the same cleaned matrix used for the projector monitor introduces a known bias of opposite sign to the sample bias, so risk monitors should use spike-debiased eigenvalues for ratio functionals and cleaned eigenvalues for projectors separately.","The detectability frontier formula gives portfolio managers a concrete design rule: the smallest detectable eigenspace rotation scales as 1/sqrt(n_post), so halving the detectable angle requires four times as many post-break observations, and at onset a quarter-turn rotation can be entirely invisible.","The exact window-scale pivotality (Proposition 9) explains why spectral monitors are blind to pure volatility changes and react only to correlation-structure changes, a property that holds exactly in finite samples under a single window-wide scale."],"fun_headline_variants":["Scale-invariant spectral functionals shed kurtosis noise in covariance monitoring","Shrinkage eigenspace fluctuations transfer under rotation-equivariant estimators","One kurtosis scalar calibrates projectors and absorption ratios under elliptical returns","Spike-debiased absorption ratio removes shrinkage bias in high dimensions","Davis-Kahan band tests eigenspace identification for rolling covariance estimates"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The analytic calibration rests on i.i.d. elliptical sampling, which real return panels systematically violate through volatility clustering (time-varying scales within the window) and skewness; under log-normal innovations the kurtosis correction breaks entirely, with coverage falling to 48 percent against a 95 percent target.","fun_headline_variants_meta":{"raw":{"variants":["Scale-invariant spectral functionals shed kurtosis noise in covariance monitoring","Shrinkage eigenspace fluctuations transfer under rotation-equivariant estimators","One kurtosis scalar calibrates projectors and absorption ratios under elliptical returns","Spike-debiased absorption ratio removes shrinkage bias in high dimensions","Davis-Kahan band tests eigenspace identification for rolling covariance estimates"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":754,"prompt_tokens":658,"completion_tokens":96,"prompt_tokens_details":null},"tokens_in":658,"tokens_out":96,"duration_ms":19159,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T07:50:51.419588+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"Find a scale-invariant spectral functional of the covariance whose first-order asymptotic variance under i.i.d. elliptical sampling does not factor as (1+kappa) times its Gaussian variance, or find a non-scale-invariant functional that does enjoy such factoring; either would break the immunity-and-converse characterization of Proposition 8.","supporting_citations":[],"review_version":1}