{"id":"0b19c46e-8ebb-45ad-b2bf-b30fb51e2111","arxiv_id":"2608.13462","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A finite-time measurement of the return probability of an initial-state subspace is shown to be sufficient to certify macroscopic quantum thermalization at almost all times.","lead":"The paper proves a theorem: if a collection of out-of-equilibrium quantum states rarely returns to its starting point within a fixed time window, then almost all those states will spend almost all their time at equilibrium for any macroscopic observable. This converts an easily measured return probability into a finite-time certificate of thermalization, which previous theory could only address at infinite times.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.1 as printed inverts the link between aperiodicity and the explored Hilbert-space fraction: substituting Eq. (45) into Eq. (42) gives mu_w = (D_I/D) A_T, the reciprocal of Eq. (46).","rationale":"The reader's weakest_assumption correctly identifies the algebraic error in Lemma 3.1 as the most load-bearing weakness for the paper's central claim. The central claim, as covered by the strongest_claim, is the thermalization bound in Theorem 2.4(2). That bound is proven through Lemmas 3.1-3.4, and Lemma 3.1 is the exact step where aperiodicity (measured by A_T) is converted into the explored Hilbert-space fraction mu_w. The printed Eq. (46) has the sign of the exponent flipped: it says mu_w = (D_I/D) integral p_I c_w, whereas the correct relation is mu_w = (D_I/D) / integral p_I c_w. The theorem statement itself already uses the correct direction (D_I A_T in the denominator), confirming that the error is a typo and not a fundamental flaw in the argument. After correction, the chain from Lemma 3.1 through Lemmas 3.2-3.4 does establish the stated bound, so the central claim is plausibly correct and the appropriate verdict is CONDITIONAL, unchanged from the reader's. I also noted a potential overclaim in Theorem 2.4(1) (the statement D_M >= p_M D_I A_T appears stronger than what Lemma 3.2(1) proves, namely D_M >= p_M^2 D_I A_T), but that part is not used in the proof of the thermalization bound, so it is secondary to the central claim.","tokens_in":38397,"tokens_out":25070,"duration_ms":225346,"concrete_test":"Re-derive Eq. (46) independently from Eq. (42) and Eq. (45), without consulting the paper's conclusion. Compute P_w = (1/D_I) integral dt p_I(t) [integral dt' w(t+t')w(t')] and substitute into mu_w = 1/(D P_w). Confirm that the resulting expression is the reciprocal of the printed Eq. (46). Then verify that Theorem 2.4(2)'s proof uses D_I A_T in the denominator, consistent with the corrected mu_w = (D_I/D) A_T, so that correcting the lemma restores the proof. If the printed formula is used instead, the bound direction reverses and Theorem 2.4(2) fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central thermalization claim is Theorem 2.4(2), whose proof passes through Lemma 3.1 (Eq. 46). Following the paper's own definitions: Eq. (42) defines mu_w = 1/(D P_w); Eq. (45) gives P_w = (1/D_I) integral dt p_I(t) c_w(t), where c_w(t) = integral dt' w(t+t')w(t'). Substituting yields mu_w = D_I / (D integral p_I c_w), not the printed (D_I/D) integral p_I c_w. For uniform w on [0,T], the double time average in Definition 2.3 gives A_T = 1/(integral p_I c_w), so the correct relation is mu_w = (D_I/D) A_T. The theorem statement itself uses D_I A_T in the denominator of the bound, which is the corrected direction, indicating a typo rather than a conceptual failure. Nevertheless, the proof as written is invalid: applying the printed Lemma 3.1 would produce D mu_w = D_I / A_T and reverse the inequality in Lemma 3.3, destroying the bound. Because this lemma is the bridge from aperiodicity to Hilbert-space exploration used by the proof of Theorem 2.4(2), the manuscript requires this correction before the central claim is supported.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes that aperiodicity of an initial subspace, defined as the reciprocal of its time-averaged return probability over a finite interval, is sufficient for finite-time thermalization of macroscopic observables. The central result, Theorem 2.4, claims that aperiodicity implies partial ergodic exploration of Hilbert space and, for any concentrated observable, that almost all states in any orthonormal basis of the initial subspace spend almost all times in the equilibrium subspace over arbitrarily long intervals. The proof is built from a purity calculation for a weighted time-averaged density operator (Lemmas 3.1-3.4), and the corollaries connect the criterion to energy-shell delocalization, pure-state limits, and multiple-observable thermalization.","tokens_in":38583,"tokens_out":10308,"duration_ms":114219,"significance":"If established, the result would be a substantial advance: it gives an observable-independent, finite-time, finite-resolution criterion for macroscopic thermalization from a measurable return probability, with no fitted parameters and with the thermodynamic and infinite-time limits taken afterwards rather than assumed. The paper also recovers, as special cases, standard infinite-time effective-dimension bounds, and it is explicit about the limitations of the approach for individual pure states (Corollary 4.2) and about the operational assumptions needed for the preparation protocol (Section 5.1.2). The derivation is a chain of inequalities from definitions, and I found no circular reliance on the results being proven; the dependence on the author's prior works is citation of independent derivations. However, the printed proof contains a load-bearing algebraic error in Lemma 3.1 and a false quantitative claim in Theorem 2.4(1), which must be corrected before the central claim is fully supported.","major_comments":[{"comment":"Lemma 3.1 is stated with the reciprocal of the correct expression. Substituting Eq. (45) into Eq. (42) gives mu_w = (D_I/D) / [integral dt p_I(t) c_w(t)], where c_w(t) = integral dt' w(t+t')w(t'); for the uniform weight on [0,T] this is mu_w = (D_I/D) A_T[H_I]. The printed Eq. (46) instead has (D_I/D) times the integral, i.e. the reciprocal. Because Lemma 3.1 is the bridge from aperiodicity to the explored Hilbert-space fraction used in Lemmas 3.2-3.4 and in Theorem 2.4(2), the proof as written is invalid: applying the printed lemma would give D mu_w = D_I/A_T and would reverse the inequality in Lemma 3.3. The theorem statements and the corollaries consistently use the corrected direction, so I regard this as a typographical error rather than a conceptual failure, but it must be fixed and the subsequent uses of mu_w re-verified.","section":"Sec. 3.1, Eq. (46)"},{"comment":"The quantitative claim D_M >= p_M D_I A_T[H_I] is false as stated. Lemma 3.2(2), which is the stated basis for part (1), gives only D_M >= p_M^2 D_I A_T[H_I] (setting 1-epsilon = p_M). The stronger printed inequality fails for general density operators: e.g. for rho = diag(0.9, 0.05, 0.05) in D=3, one has D mu = 1/0.815 ~ 1.227; taking M to be the rank-one subspace spanned by the dominant eigenvector gives p_M = 0.9 and p_M D mu ~ 1.104 > 1 = D_M. The qualitative statement -- that any subspace containing most of the history must have size Omega(D_I A_T) -- survives, but Eq. (34) should be replaced by the p_M^2 bound or by a correctly stated weaker formulation. This does not affect Eq. (36), whose proof uses D_I A_T directly, but Theorem 2.4(1) as printed is a mathematical claim of the paper and must be corrected.","section":"Theorem 2.4(1), Eq. (34)"}],"minor_comments":[{"comment":"The summand in Eq. (92) should carry the observable-subset index m, i.e. f_eq[BI, A_m](RT, epsilon), rather than the index-free f_eq[BI, M](RT, epsilon); as written the bound is not clearly the termwise union bound described in the text.","section":"Sec. 4.3, Eq. (92)"},{"comment":"The existence of a dissipative preparation D with finite overlap p_0 and the state-wise bound p_k >= p_0/D_I is an additional operational assumption; the text should present it as an explicit assumption rather than as something 'taken for granted,' since Theorem 2.4 itself does not require this protocol.","section":"Sec. 5.1.2, Eqs. (103)-(106)"},{"comment":"There are a few presentational slips: 'Dneq,,max' in the paragraph before Corollary 4.3 should read D_neq,max, and the notation mu_Theta(t in [0,T])/T in Corollary 4.1 is typeset in a way that makes the subscript hard to parse; a local definition of the subscript would improve readability.","section":"Sec. 4.2 and Cor. 4.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely salvageable and the contribution, if corrected, would be of genuine interest. The Lemma 3.1 reciprocal error appears typographical because the theorem statements and the final thermalization bound use the corrected direction. I recommend major revision rather than rejection: the authors should fix Eq. (46), replace or weaken Eq. (34), and re-check every subsequent display that uses mu_w."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is a genuine result worth engaging, but the manuscript as printed has a reciprocal error in Lemma 3.1 that breaks the proof chain as written. I checked the stress-test note and it lands: substituting Eq. (45) into Eq. (42) gives mu_w = (D_I/D) A_T, not the printed (D_I/D) times the integral. Since A_T is the reciprocal of that integral, the printed lemma is inverted. The theorem statement itself uses the correct direction, so this looks like a typo, but it is load-bearing: Lemma 3.1 is the bridge from aperiodicity to Hilbert-space exploration, and Lemma 3.3 uses it. The proof needs a corrected Eq. (46) before the theorem as stated is supported.\n\nWhat is genuinely new: Theorem 2.4 gives a finite-time, observable-independent sufficient condition for macroscopic thermalization in the thermodynamic limit, using only the return probability of an initial subspace over [0,T]. That addresses the gap Tasaki explicitly called out. The bound is clean: f_eq >= 1 - (1/epsilon) sqrt(D_neq/(D_I A_T)). The proof strategy (Cauchy-Schwarz, Markov, time-translation stitching) is sound in outline, and Corollaries 4.1-4.3 are useful, especially the connection to energy-shell delocalization and the recovery of infinite-time effective-dimension results. The paper is also honest about limitations: Corollary 4.2 correctly states that a pure state needs exponentially long time for the theorem to certify thermalization of all concentrated observables.\n\nThe soft spots: the Lemma 3.1 error is the main one. Also, the preparation protocol in Sec. 5.1.2 is heuristic--it sketches a dissipative cooling argument but does not rigorously establish that the required p0 is finite for a concrete process. That is not fatal to the theorem, but it means the 'operational' claim rests on an assumption. The paper is long and does a lot of positioning; the technical core could be presented more tightly.\n\nWho is this for? Quantum statistical mechanics researchers working on thermalization and equilibration bounds. It deserves a serious referee; this is exactly the kind of paper editors should send out, with a request to fix the lemma. After correction, I would expect it to be citable. My verdict is conditional, not skeptical: the central argument holds up, but the printed proof does not.","headline":"Real finite-time macroscopic thermalization result with a clean bound, but the printed Lemma 3.1 has a reciprocal error that must be fixed before the proof as written works.","tokens_in":39190,"tokens_out":2008,"would_cite":true,"duration_ms":21205,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81Q50","82C10","37A60"],"pacs":["05.30.-d"],"model":"deepseek-v4-flash","headline":"A single measurable quantity—the return probability—is enough to guarantee macroscopic thermalization.","keywords":["macroscopic thermalization","aperiodicity","return probability","partial ergodicity","quantum equilibration","operational statistical mechanics","concentrated observables","effective dimension"],"falsifier":"Directly substitute Eq. (45) into the purity definition and compare with Eq. (46): the printed formula gives $\\mu_w=(D_I/D)\\int dt\\, p_I(t)\\int dt'\\, w(t+t')w(t')$, while the substitution yields the reciprocal of that integral, so a numerical random-Hamiltonian check of $P_w$ distinguishes the two immediately. A physical check would measure the return probability of a prepared nonequilibrium subspace over $[0,T]$, compute $D_I A_{[0,T]}[H_I]$, and compare the predicted equilibrium fraction with the time spent by basis states in the equilibrium subspace of a concentrated observable.","tokens_in":38053,"feed_emoji":"⚛️","tokens_out":10245,"duration_ms":98238,"temperature":0.7,"pith_summary":"The paper claims that macroscopic thermalization—the settling of coarse-grained observables such as charge densities onto equilibrium values—follows from a single dynamical property of the initial state: aperiodicity, meaning the initial subspace of states has a small return probability over a finite time window. If that return probability is small enough, the subspace necessarily explores a correspondingly large fraction of the Hilbert space, and almost every state in any orthonormal basis of the subspace spends almost all times in the equilibrium subspaces of every sufficiently concentrated observable. This turns thermalization into a finitely computable, observable-independent prediction over finite time intervals, not just in the infinite-time limit. The result matters because eigenstate-based arguments cannot make finite-time predictions in the thermodynamic limit, while this mechanism can.","feed_headline":"Low return probability alone guarantees macroscopic thermalization","feed_subtitle":"A single finite-time measurement on initial states certifies equilibrium for almost all states and times.","key_machinery":"The load-bearing object is the aperiodicity $A_{\\mathcal{T}}[H_I]$, defined as the reciprocal of the double time-average of the subspace return probability $p_I(t,t')=\\mathrm{Tr}[\\hat{\\Pi}_I(t)\\hat{\\Pi}_I(t')]/D_I$. Low return probability makes the time-averaged density operator $\\hat{\\rho}_w=\\frac{1}{D_I}\\int dt\\, w(t)\\hat{\\Pi}_I(t)$ low-purity, so the nominal explored fraction $\\mu_w(\\hat{\\Pi}_I)=1/(D\\,\\mathrm{Tr}[\\hat{\\rho}_w^2])$ is large (Lemma 3.1). Lemma 3.2 then uses Cauchy-Schwarz to show that any small subspace can capture only a small part of that history; Lemma 3.3 uses Markov's inequality to conclude that most basis states spend most weighted time in equilibrium; and Lemma 3.4 extends the bound from $[0,T]$ to arbitrarily long intervals by time-translation invariance.","core_discovery":"The paper's central claim is Theorem 2.4: for an initial subspace $H_I$ of dimension $D_I$ and a time interval $[0,T]$, let $A_{[0,T]}[H_I]$ be the reciprocal of the time-averaged return probability of the subspace. Then every concentrated observable with nonequilibrium dimension $D_{\\mathrm{neq}}$ satisfies\n$$f_{\\mathrm{eq}} \\ge 1-\\frac{1}{\\epsilon}\\sqrt{\\frac{D_{\\mathrm{neq}}}{D_I A_{[0,T]}[H_I]}}$$\nfor the fraction of time spent within $\\epsilon$ of equilibrium, uniformly over arbitrarily long intervals $[0,RT]$ including $R\\to\\infty$. In words, aperiodicity over a finite interval forces a partially ergodic exploration of the Hilbert space of size $D_I A_{[0,T]}[H_I]$, and that exploration forces almost all states in every orthonormal basis of $H_I$ to sit in the equilibrium subspace at almost all times, for every observable whose nonequilibrium subspace is small compared with the explored space. The same chain of lemmas also yields delocalization of the initial state over energy shells of width $\\Delta E\\sim 1/T$, and reduces to standard effective-dimension equilibration bounds in the infinite-time limit.","pith_inferences":["Because the probe is a single subspace return probability, the criterion could be adapted to certify thermalization in classical dynamical systems by tracking the cycle length of a phase-space cell, a link the paper only offers as intuition.","The same forgetfulness measure may unify state-dependent macroscopic thermalization with observable-dependent autocorrelator and OTOC criteria, which would imply a single operational signature for all rows of the paper's Table 1; the paper raises this possibility but does not prove it.","A corrected version of Lemma 3.1 would change quantitative predictions for any application that uses Eq. (46) verbatim, so downstream numerical protocols should re-derive the explored-fraction formula from the purity rather than quoting the printed expression.","The bound in Theorem 2.4 is likely not tight: permutation caricatures give a linear dependence on return probability, while the quantum bound has a square root, so tighter quantum bounds may exist for special classes of initial subspaces."],"forward_implications":["A single finite-time return-probability measurement on an initial subspace certifies thermalization of every sufficiently concentrated observable, for almost all states in every orthonormal basis, over arbitrarily long intervals.","The same finite-time data imply the initial state is delocalized across energy shells of width $\\Delta E\\sim 1/T$, giving a finite-time version of effective-dimension equilibration bounds.","For a single pure state, certifying all concentrated observables requires an observation time $T\\sim D_{\\mathrm{neq}}/\\sigma_E$, exponentially long for macroscopic observables, so useful finite-time predictions are statements about subspaces of initial states.","For any finite set of macroscopic observables, including noncommuting ones, joint thermalization holds for almost all basis states when $D_I=\\Theta(D_{\\mathrm{neq,max}})$.","The infinite-time limit recovers the standard effective-dimension and delocalization-based criteria for thermalization as a special case."],"supporting_citations":[{"why":"Supplies the purity-based measure of ergodic exploration and the identity linking return probability to the purity of the time-averaged subspace.","marker":"[37]"},{"why":"Provides the cyclic-permutation model and spectral-statistics connection that motivate the partial-ergodicity mechanism.","marker":"[38]"},{"why":"Defines the macroscopic equilibrium and nonequilibrium subspace framework and the infinite-time bounds that Theorem 2.4 extends to finite times.","marker":"[21]"},{"why":"Establishes the autocorrelator-based operational framework for finite-time microscopic thermalization that this paper complements for macroscopic observables.","marker":"[23]"},{"why":"Introduces the equilibrium-subspace notion on which Definition 2.1 is built.","marker":"[26]"},{"why":"Gives the infinite-time effective-dimension equilibration bounds recovered as a limit of the finite-time criterion.","marker":"[49–53]"}],"fun_headline_variants":["Aperiodicity alone guarantees macroscopic thermalization","Finite-time aperiodicity certifies equilibrium for almost all states","A single finite-time probe certifies macroscopic equilibrium","Low return probability alone drives finite-time equilibration","Aperiodicity, not effective dimension, drives macroscopic thermalization"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Lemma 3.1's bridge that turns a small return probability into a large explored Hilbert-space fraction; as printed, Eq. (46) states the reciprocal of the correct expression, so the written proof of Theorem 2.4(1) does not deliver its bound unless that formula is corrected.","fun_headline_variants_meta":{"raw":{"variants":["Aperiodicity alone guarantees macroscopic thermalization","Finite-time aperiodicity certifies equilibrium for almost all states","A single finite-time probe certifies macroscopic equilibrium","Low return probability alone drives finite-time equilibration","Aperiodicity, not effective dimension, drives macroscopic thermalization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000872,"raw_usage":{"total_tokens":3839,"prompt_tokens":1076,"completion_tokens":2763,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":692,"completion_tokens_details":{"reasoning_tokens":2686}},"tokens_in":692,"tokens_out":2763,"duration_ms":22219,"temperature":1.0,"reasoning_tokens":2686,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:29:30.586329+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Directly substitute Eq. (45) into the purity definition and compare with Eq. (46): the printed formula gives $\\mu_w=(D_I/D)\\int dt\\, p_I(t)\\int dt'\\, w(t+t')w(t')$, while the substitution yields the reciprocal of that integral, so a numerical random-Hamiltonian check of $P_w$ distinguishes the two immediately. A physical check would measure the return probability of a prepared nonequilibrium subspace over $[0,T]$, compute $D_I A_{[0,T]}[H_I]$, and compare the predicted equilibrium fraction with the time spent by basis states in the equilibrium subspace of a concentrated observable.","supporting_citations":[{"cited_title":"Enhanced entanglement from quantum ergodicity","cited_arxiv_id":"2507.08067","evidence_quote":"Supplies the purity-based measure of ergodic exploration and the identity linking return probability to the purity of the time-averaged subspace."},{"cited_title":"Dynamical quantum ergodicity from energy level statistics","cited_arxiv_id":null,"evidence_quote":"Provides the cyclic-permutation model and spectral-statistics connection that motivate the partial-ergodicity mechanism."}],"review_version":1}