{"id":"e99fbf90-0db5-45d3-9dae-6b0599424a67","arxiv_id":"2607.19839","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.5,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"The finite-horizon law of the raw Pólya-urn quantile tracker is a gain-dependent Gaussian with variance factor G_a(c f0(q_τ)), the stopped-urn plug-in quantile keeps the tail fraction, and a full inverse-Jacobian matrix gain is necessary for calibrated quantile-regression process bands.","lead":"This paper derives the exact finite-horizon law of a Pólya-urn quantile posterior, showing that the stopped online tracker's variance is a gain-dependent factor G_a rather than the usual tail-sum fraction. It also constructs a matrix-gain smoothed quantile-regression posterior with calibrated finite-horizon bands, giving practical corrections for users who stop martingale-posterior sampling after a finite number of imputations.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; disclosed exact-linearity assumption in regression half does not undermine raw-urn central claim.","rationale":"The reader's verdict identifies the matrix-gain regression's exact-linearity assumption as the weakest assumption. I agree that this is the most fragile assumption in the paper as a whole, but it is explicitly disclosed and does not affect the primary finite-horizon quantile law claimed in Theorem 1(iii). My own review of the raw-urn proof did not surface an additional load-bearing gap: the conditional de-localisation argument in Lemma S4 is careful, the Q-local property is proved from primitive density conditions, and the finite-horizon variance formula is corroborated by the numerical variance law. The regression misspecification is a genuine caveat for users of the conditional-quantile method, but it is stated openly and the paper does not overclaim robustness. Therefore the appropriate verdict remains ACCEPT, with the caveat already noted by the reader. I set agreement to partial because the reader's weakest assumption is not the central raw-urn claim, but I concur that it is the most important limitation of the paper's broader contribution.","tokens_in":37942,"tokens_out":22727,"duration_ms":218285,"concrete_test":"Independently reproduce Table 1's theoretical column by numerical integration of the defining integral G_a(r)=∫_1^{1+λ} h_a(r,s)^2 ds for all 30 (a,λ) cells, rather than using the closed-form display; any discrepancy would indicate a misstatement in equation (3).","verdict_should_be":"UNCHANGED","load_bearing_attack":"After review, I find no load-bearing error in the central raw-urn claim (Theorem 1(iii) and Corollary 1). The proof of Lemma S4 via Q-local, the de-localisation bound (S9), and the weighted-martingale representation (S10)-(S11) are internally consistent; the G_a integral and its stated closed form are supported by the shared-innovation diffusion picture and by the reported finite-sample variance tracking. The most consequential assumption in the paper is R1: the smoothed regression half (Theorem 3) assumes an exactly linear conditional-quantile model, Q0(u|x)=x' beta0(u), with bounded design and uniformly positive-definite Jacobian. Under misspecification the construction has no robustness, and the Gaussian/lognormal experiments are explicitly outside the theorem. This is disclosed in Sections 5.2 and 7 and in Supplement S3, so it is a scope limitation, not an undisclosed flaw. It does not undermine the finite-horizon quantile law, which requires only local density regularity plus consistent density estimation. Hence no change to the reader's ACCEPT is warranted.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies finite-horizon martingale posteriors for quantiles based on the empirical Pólya-urn predictive. For the raw stopped tracker (1) with frozen gain c, it derives the conditional limit √n(θ_{n+N}−q̂_{τ,n}) | F_n ⇒ N(0, Σ_τ G_a(1+λ)) with a = c f_0(q_τ), where G_a is given in closed form in Eq. (3). The infinite-horizon endpoint is shown to coincide with the Bayesian-bootstrap quantile, and the plug-in quantile of the stopped urn measure is shown to retain the ordinary tail-sum fraction λ/(1+λ) (Corollary 1). The paper also gives a joint law for several quantile levels driven by shared urn innovations, and a smoothed conditional-quantile-regression recursion with a full inverse-Jacobian matrix gain, proving a process Bernstein–von Mises theorem and a necessity result for the matrix gain. The finite-horizon correction is implemented with an estimated density-adapted gain, and the numerical sections include extensive simulations and an Engel-data illustration.","tokens_in":38204,"tokens_out":8037,"duration_ms":90313,"significance":"If correct, the paper closes a real gap: finite-horizon stopping of a quantile martingale posterior does not, in general, produce the martingale tail-sum variance, and the paper gives the exact first-order inflation factor explicitly. The G_a formula is derived from a shared-innovation linearization and a weighted-martingale representation, not fitted to the target result; the a=1 case recovers the known tail fraction, which is a strong internal consistency check. The matrix-gain regression theorem is a substantive contribution to function-valued martingale posteriors, and the necessity result clarifies why scalar or diagonal gains cannot match the sandwich covariance process. The paper is unusually transparent about scope: the exact-linearity assumption for the regression half is disclosed, the phase-boundary result is stated as conditional on Chung–Fabian limits, and the raw-urn results are explicitly limited to finitely many fixed quantile levels. The shipped code and data with reproducibility gates further strengthen confidence in the numerical claims.","major_comments":[],"minor_comments":[{"comment":"The display says the continuous value at a=1/2 is '(1/4) r^{-1} log r for the final term'. Since G_a itself is continuous at a=1/2, it would be clearer to state that the remaining terms are unchanged and this is the limiting value of the whole expression, or to give the full continuous formula explicitly.","section":"Sec. 3, Eq. (3)"},{"comment":"The correction step uses Ĝ_â(1+N/n) before a and G_a are formally defined in Section 3. A one-sentence forward reference or a parenthetical definition of â and r in the algorithm box would improve readability for practitioners.","section":"Sec. 2, algorithm box"},{"comment":"The column 'Joint coverage' is carefully described in the text as a moment-based Hotelling-type diagnostic, but the table header alone could be mistaken for the max-standardized region of Theorem 3. A brief footnote in the table would remove this ambiguity.","section":"Sec. 5.2, Table 2"},{"comment":"The paper's own statement that the raw-urn results do not extend to a uniform-in-τ process version is important and easy to miss. Consider stating this limitation at the end of Section 3 as well, so that readers do not overgeneralize the finite-collection theorem.","section":"Sec. 7"}],"recommendation":"accept","confidential_remarks":"This is a careful and well-scoped paper. The central raw-urn result is derived rather than fitted, and the main limitations are disclosed prominently. The regression half relies on an exactly linear conditional-quantile model, but this is stated and the simulations outside that model are explicitly labeled as robustness checks. I support acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper earns a full referee. The raw-urn result is the one to remember. Theorem 1(iii) gives the exact finite-horizon variance factor G_a for the gain-c stochastic-approximation quantile tracker, and Corollary 1 draws the key distinction: the plug-in quantile of the stopped urn keeps the familiar tail sum, while the retained tracker state does not. That distinction is the paper's best insight, and it is derived, not fitted. Lemma S4's weighted-martingale representation of the shared urn innovations is the load-bearing step, and the a=1 cancellation that recovers the known tail fraction is a good internal check. The simulations track G_a to within a couple of percent, and the systematic 1.8-4.3% shortfall is reported as a second-order atomic effect rather than hidden. The joint cross-quantile covariance from shared urn paths is a legitimate extension.\n\nThe soft spots are in proportion. The matrix-gain regression half is a different creature: Theorem 3 and Proposition 2 rely on an exactly linear conditional-quantile model with bounded design (R1-R2). Under misspecification there is no robustness, and the Gaussian/lognormal experiments are explicitly outside the theorem. That is disclosed in Sections 5.2 and 7 and Supplement S3, so it is a scope limitation rather than a hidden flaw. It does not undermine the raw-urn claim. The proof of Lemma S7 has compressed entropy/chaining steps that I did not fully re-derive; no load-bearing error surfaced, but an independent verification pass would raise my confidence. The paper is unusually honest about finite-sample coverage: it reports Monte Carlo uncertainty and labels coverage a diagnostic, not a gate. That is the right attitude.\n\nCitation pattern is solid. The paper explicitly disclaims novelty for the infinite-horizon Bayesian bootstrap and the classical Chung-Fabian boundary, and situates itself against Fong-Yiu's smoothed construction carefully. No circularity: G_a is derived from martingale identities, not fit to the target result.\n\nWho benefits: anyone working on martingale posteriors, quantile regression uncertainty, or finite-horizon stochastic approximation. I would cite the raw-urn result. Send it to a serious referee; expect heavy but productive revision focused on making the regression half's assumptions and the entropy arguments more readable.","headline":"The finite-horizon G_a law is real, derived-not-fitted, and the paper deserves a serious referee; the regression half is more conditional and honestly scoped.","tokens_in":38675,"tokens_out":1727,"would_cite":true,"duration_ms":21633,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G20","62G09","62F15","62L20"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper shows that when a Pólya-urn quantile martingale posterior is stopped after finitely many imputations, the variance of the deployed tracker state is governed by an explicit gain factor G_a(r), equal to the familiar tail fraction o","keywords":["martingale posterior","Pólya urn","quantile regression","Bayesian bootstrap","stochastic approximation","finite horizon","Bernstein–von Mises","matrix gain"],"falsifier":"Run recursion (1) on Uniform(0,1) data with τ=0.5, frozen gain making a=2, horizon N=n, and n large (e.g., 10^5): the scaled variance n·var(θ_{n+N}−q̂) should approach Στ G_2(2)=0.25×1.145833=0.28646, not the tail-fraction value 0.125. In the regression setting, fit the matrix-gain posterior in a design where J0 is far from scalar but set a diagonal gain: if the posterior intercept–slope correlation does not turn opposite to the target sandwich correlation, Proposition 2 is falsified.","tokens_in":37769,"feed_emoji":"📊","tokens_out":5242,"duration_ms":51756,"temperature":0.7,"pith_summary":"The paper derives exact finite-horizon laws for the empirical Pólya-urn quantile martingale posterior, the object a practitioner actually holds when simulation stops after N imputations. It proves that the stopped urn's plug-in quantile keeps the classical tail-sum variance fraction N/(n+N), but the recursively updated tracker state does not: its variance carries an additional factor G_a that can over- or under-disperse depending on the gain. When the gain is tuned to the inverse density (a=1), the factor collapses to the tail fraction, and a feasible density-free inflation restores calibration. For conditional quantile regression, the paper establishes a finite-horizon Bernstein–von Mises theorem with a full inverse-Jacobian matrix gain, and shows that scalar or diagonal gains cannot reproduce the sandwich covariance. If correct, these results supply exact finite-horizon corrections for streaming quantile trackers and quantile-regression posterior bands.","feed_headline":"Stopped quantile trackers defy the martingale tail-sum rule","feed_subtitle":"A new gain factor G_a sets the finite-horizon variance; density-adapted gains restore calibration.","key_machinery":"The load-bearing object is the factor G_a(r) = ∫_1^r {(1−a)r − a s^{a−1} − s^{−1}}^2 ds, a closed-form integral that emerges from jointly linearizing the drifted tracker update and the Pólya-urn measure martingale while retaining their shared innovations; its square-root inverse corrects the stopped state. In the regression half, the key mechanism is the frozen inverse-Jacobian matrix gain Â_n(u)=Ĵ_n(u)^{-1}, which preconditions the functional update so that the finite-horizon process covariance exactly mirrors the sandwich kernel {min(u,v)−uv} J0(u)^{-1} ΣX J0(v)^{-T}.","core_discovery":"The central claim is Theorem 1(iii): for the raw empirical Pólya-urn quantile recursion run to horizon N=⌊λn⌋, conditionally in P0-probability, √n(θ_{n+N}−q̂τ,n) | F_n ⇒ N(0, Στ G_a(r)) with r=1+λ and G_a(r) a closed-form integral. At a=cf0(qτ)=1, G_1(r)=λ/(1+λ) recovers the martingale tail fraction; at a=2, r=2 the factor exceeds one, producing overdispersion rather than the usual deficit. Corollary 1 shows the plug-in quantile of the stopped urn measure keeps the tail fraction, so the distortion is specific to the retained recursive state. In the regression setting, Theorem 3 asserts that starting a smoothed martingale posterior at the ordinary quantile-regression estimator and using a fro","pith_inferences":["The G_a law suggests a practical rule for any online quantile tracker whose state is coupled to another algorithm: periodically re-inflate the state by the estimated factor Ĝ_â(r) rather than by the tail fraction, a recipe that goes beyond the paper's theorems and could be tested on coupled streaming systems.","The regression result implies that diagonal-precision Bayesian or bootstrap uncertainty for conditional quantiles in positive-design problems can systematically reverse the sign of cross-quantile correlations; a cheap diagnostic is to inspect the off-diagonal of  Σ_X  against the target sandwich.","The atomic-support obstruction to a uniform-in-τ process version suggests a smoothed-urn analogue with shared innovations as the natural route to a full process theory; the paper leaves this as an open construction.","Because the matrix-gain posterior has no robustness outside the exactly linear conditional-quantile model, a cautious user should report the plain quantile-regression sandwich intervals alongside whenever misspecification is plausible."],"forward_implications":["Implementations of martingale-posterior quantiles that stop at a finite horizon must apply the G_a inflation to the retained tracker state, not the plain tail fraction N/(n+N), to achieve nominal coverage.","With a density-adapted gain (a→1), the correction becomes feasible and density-free: multiply the stopped increment by {(n+N)/N}^{1/2}.","Sharing one urn path across several quantile levels induces a joint finite-horizon law; when all gains are adapted, a common inflation restores the full joint quantile covariance, enabling calibrated simultaneous regions.","For conditional quantile regression, only the full inverse-Jacobian gain matches the sandwich covariance; scalar or diagonal gains misorient the posterior joint geometry, even when marginal variances are matched.","At infinite horizon the raw-urn posterior coincides with the Bayesian-bootstrap quantile posterior, so exact Dirichlet-weighted quantiles remain the recommended default for infinite-endpoint sampling."],"fun_headline_variants":["Quantile tracker variance hinges on gain factor G_a","Stopped urn quantile defies martingale tail-sum rule","G_a factor flips quantile tracker's finite-horizon variance","Density-adapted gain fixes quantile calibration drift"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The conditional-quantile regression result assumes the true model is exactly linear, Q0(u|x)=x'β0(u), with bounded regressors and a uniformly positive-definite Jacobian; if that ideal-model assumption fails, the advertised calibrated process bands are not guaranteed to hold.","fun_headline_variants_meta":{"raw":{"variants":["Quantile tracker variance hinges on gain factor G_a","Stopped urn quantile defies martingale tail-sum rule","G_a factor flips quantile tracker's finite-horizon variance","Density-adapted gain fixes quantile calibration drift"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000233,"raw_usage":{"total_tokens":1343,"prompt_tokens":770,"completion_tokens":573,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":514,"completion_tokens_details":{"reasoning_tokens":515}},"tokens_in":514,"tokens_out":573,"duration_ms":6380,"temperature":1.0,"reasoning_tokens":515,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T11:33:40.131568+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run recursion (1) on Uniform(0,1) data with τ=0.5, frozen gain making a=2, horizon N=n, and n large (e.g., 10^5): the scaled variance n·var(θ_{n+N}−q̂) should approach Στ G_2(2)=0.25×1.145833=0.28646, not the tail-fraction value 0.125. In the regression setting, fit the matrix-gain posterior in a design where J0 is far from scalar but set a diagonal gain: if the posterior intercept–slope correlation does not turn opposite to the target sandwich correlation, Proposition 2 is falsified.","supporting_citations":[],"review_version":1}