{"id":"2e9abaeb-9d9e-4ad8-9618-cb56d34228ab","arxiv_id":"2504.16857","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A single time-dependent symmetry transformation is claimed to explain the known scaling phenomenology of physical ageing and to predict new finite-size plateau scalings.","lead":"After a sudden temperature change, many materials relax very slowly and show a pattern called ageing. The paper proposes one mathematical rule that explains most of the known patterns and predicts new ones.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"λ_C=λ_R is not derived from (2.5a,b) alone: the proof requires the extra assumption δ̃=δ, so the central claim overstates what follows from generalized time-translation-invariance plus dynamical scaling.","rationale":"The reader's weakest assumption is the physical validity of the intertwining postulate (2.1). My concern is downstream of that: even granting (2.1), the derivation of λ_C=λ_R needs the additional identity δ̃=δ, which is a separate assumption about the response operator and is not contained in generalized time-translation-invariance plus dynamical scaling. This is more specific than the reader's concern and directly targets the strongest claim that the whole generic phenomenology follows from two symmetries. The paper itself flags the condition in (3.5), but the propositions and abstract state the equality as derived. The mathematical core is otherwise transparent, and the exact spherical-model checks and the 1D Glauber-Ising response provide real independent support. The concern does not invalidate the framework; it narrows the provenance of one headline result. Therefore the appropriate verdict remains CONDITIONAL, unchanged from the reader, but the paper should be asked to state explicitly that λ_C=λ_R follows from generalized time-translation-invariance together with the local-scale-invariance assumption δ̃=δ, not from (2.5a,b) alone.","tokens_in":52817,"tokens_out":24466,"duration_ms":221096,"concrete_test":"Recompute Section 3.2 with independent (δ̃,ξ̃) for the response operator: substitute into (2.6) and verify that λ_R/z=δ+δ̃−ξ and that λ_C=λ_R iff δ̃=δ. Then check whether the local-scale-invariance Ward identities used in Section 4 (e.g. the X_1 condition for z=2) force δ̃=δ. A direct model test is to fit the exact 1D Glauber-Ising correlator C=(2/π)arcsin(√(s/t)) and response R=s^{−1}(y−1)^{−1/2} to the covariance solution with independent δ,δ̃,ξ,ξ̃. If no single set of parameters satisfies both, or if the fit requires δ̃≠δ, then Proposition 2 must be restated as conditional on δ̃=δ rather than as a derivation from (2.5a,b).","verdict_should_be":"UNCHANGED","load_bearing_attack":"Proposition 2's equality λ_C=λ_R is not a consequence of the two stated symmetries (2.5a,b) alone. For the response R=⟨φ(t)φ̃(s)⟩ the response operator φ̃ can carry an independent pair (δ̃,ξ̃). Solving the covariance equations via the lemma (2.6) gives, for y=t/s≫1, R∼s^{−δ−δ̃+ξ+ξ̃} y^{ξ−δ−δ̃}, hence λ_R/z=δ+δ̃−ξ. The autocorrelator gives λ_C/z=2δ−ξ. Equality holds only if δ̃=δ. This equality is imposed in eq. (3.5) by an appeal to local scale-invariance ('If that is admissible'), not by generalized time-translation-invariance plus dynamical scaling. Thus the abstract's claim that the equality λ_C=λ_R is a consequence of 'these two dynamical symmetries' is an overstatement: an extra input about the response operator's scaling dimension is required. The exact 1D Glauber-Ising case illustrates the subtlety: the leading asymptotic forms of C and R can be reconciled with δ=δ̃=1/2, but only by fixing ξ,ξ̃ and δ̃ through the response exponents; the equality is not forced by (2.5a,b) alone. The paper's own wording in (3.5) is conditional, but Propositions 1-2 and the abstract present the equality as derived, not as an assumption.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes that the generic two-time phenomenology of physical ageing in classical systems can be derived from two dynamical symmetries: generalized time-translation invariance and dynamical scaling. The central postulate, Eq. (2.1), replaces the equilibrium generator of time translations by an intertwined form X = e^{ξ ln t} X_equi e^{-ξ ln t}, producing the modified generators (2.2). Solving the resulting covariance equations for two-point functions yields the exact form (2.6), from which the paper derives algebraic autocorrelation and autoresponse scaling, the equality λ_C = λ_R, the Janssen-Schaub-Schmittmann relation, extensions of that relation to T < T_c, finite-size plateau scalings in fully finite systems, and global two-time observables. Section 4 translates the same representation into a criterion for the irrelevance of nonlinear terms in the equation of motion, and compares the resulting response forms with a large body of exact and numerical results. The paper is explicit in Section 5 that the intertwining form W(t)=ξ ln t is a postulate whose physical origin is not derived.","tokens_in":53216,"tokens_out":7033,"duration_ms":69629,"significance":"If the central hypothesis is accepted, the paper offers a valuable unifying perspective: a single representation choice (2.1) with W(t)=ξ ln t reproduces a wide set of known ageing results and produces new, testable predictions, notably the finite-size plateau scalings (3.15), (3.18), (3.21), (3.43), (3.46) and the extension of the JSS relation to all T ≤ T_c. The covariance lemma (2.6) is a simple but correct exact statement, and the paper backs its claims with exact checks in the spherical model, the 1D Glauber-Ising model, and the fully connected spherical spin glass, together with an extensive compilation of numerical results. The significance is moderated by the facts that the central postulate is not derived, that the equality λ_C = λ_R requires an extra assumption about the response operator that is not among the two stated symmetries, and that the response-plateau finite-size scalings are conditional on a convergence assumption that fails in at least one known model. These issues do not invalidate the framework but they do mean the paper's stated claims exceed what is logically established.","major_comments":[{"comment":"The equality λ_C = λ_R is not a consequence of the two stated symmetries (2.5a) and (2.5b) alone. Solving (2.6) for the response R = ⟨φ(t)φ̃(s)⟩ with independent parameter pairs (δ,ξ) and (δ̃,ξ̃) gives λ_R/z = δ + δ̃ − ξ, whereas the autocorrelator gives λ_C/z = 2δ − ξ. Equality holds only if δ̃ = δ. Eq. (3.5) imposes δ̃ = δ by an appeal to local scale-invariance ('If that is admissible'), which is an additional dynamical input not present in (2.5a,b). The abstract and Proposition 2 present λ_C = λ_R as a consequence of generalized time-translation-invariance combined with dynamical scaling; this overstates the logical content of the derivation. The authors should either state the extra assumption explicitly whenever the equality is claimed, or prove δ̃ = δ from the stated symmetries.","section":"§3.2, Eq. (3.5), Prop. 2"},{"comment":"The central postulate (2.1) with W(t) = ξ ln t is not derived, and Section 5 explicitly states that its physical origin is open. All propositions that follow are therefore conditional on this specific logarithmic intertwining form; a different W(t), or a time-dependent ξ, would change the scaling forms and exponent relations. The paper should state this conditionality prominently in the abstract and in Propositions 1–7, and should soften the claim that the whole generic phenomenology of ageing is derived from the two dynamical symmetries, because one of those symmetries is itself an unproved representation choice rather than a symmetry that has been shown to hold for the systems under study.","section":"§2, Eq. (2.1); §5"},{"comment":"The finite-size plateau scaling for the auto-response is an assumption, not a consequence of (2.5a,b). Proposition 4 is phrased conditionally ('if the auto-response function converges to a plateau'), and Appendix D shows that in the fully connected p = 2 spherical spin glass the response does not plateau (D.11). The same conditional structure propagates to Corollaries 3 and 6. The paper should clearly separate the derived scaling (3.17) from the additional hypothesis of plateau convergence whenever these finite-size plateau scalings are advertised as consequences of the generalised time-translation-invariance programme.","section":"§3.4, Prop. 4; §3.7, Cor. 6"},{"comment":"Appendix B shows that generalized time-translation covariance cannot accommodate single-time correlators: it forces FC(u) ∼ u^{−2δz}, in contradiction to the known scaling behaviour. The paper excludes single-time correlators (Section 5, assumption 4), but the abstract and Section 1 claim that the whole generic phenomenology of ageing is derived. Since single-time correlators are part of the standard phenomenology of ageing, the claim of scope is overstated. The authors should qualify the claim to the two-time sector, or justify why the failure of the covariance requirement for single-time correlators does not cast doubt on the use of the same requirement for two-time correlators.","section":"Appendix B; §5"}],"minor_comments":[{"comment":"The abstract contains the typo 'celebrate' for 'celebrated'; this should be corrected.","section":"Abstract"},{"comment":"There are several non-native spellings, e.g., 'littérature' in §3.2 and 'Forth' in Appendix C; a careful language edit is needed.","section":"Throughout"},{"comment":"The uniqueness claim in the proof of the lemma relies on the external reference [144]; stating the method-of-characteristics argument would make the derivation more self-contained.","section":"Eq. (2.6)"},{"comment":"The tables mix literature values with values inferred from the present framework; a sentence clarifying which entries are new determinations and which are taken from the cited references would improve transparency.","section":"Tables 2 and 3"},{"comment":"The axis label 'rL FC' in panel (b) appears garbled; the figure would be clearer if the abscissa were labelled y = t/s and the ordinate simply C(ys,s).","section":"§3.3, Fig. 3"}],"recommendation":"major_revision","confidential_remarks":"This manuscript is largely a synthesis and extension of a long line of local-scale-invariance work by the same author. The mathematical construction (2.1) is a special case of the general representation change given in Lemma A.2, which is already present in the cited literature [178,215]. The genuinely new elements are the finite-size plateau scalings and the all-T extension of the JSS relation. The paper cites the earlier work, but the framing as a new 'generalised time-translation-invariance' postulate should more explicitly identify what is new beyond [126,178,215] to avoid the impression of repackaging. The main technical overstatement is the claimed derivation of λ_C = λ_R from the two stated symmetries alone; this must be corrected before publication. The paper otherwise contains useful, well-tested phenomenology and a clear set of falsifiable predictions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper is a serious attempt to organize the standard phenomenology of classical ageing under one postulate: generalized time-translation invariance via the intertwiner W(t)=ξ ln t. If you work on ageing, it is worth knowing about. The author shows that many well-known results—algebraic two-time scaling, the JSS relation, finite-size plateau scalings—follow from this representation choice, and he extends the JSS relation to all T≤Tc. The spherical-model checks in Appendix D are concrete and reproducible, and the tables comparing the response form (4.26) to data across many models are useful.\n\nThe main soft spot is real but not fatal: the equality λ_C=λ_R is not a consequence of generalized time-translation invariance (2.5a) plus dynamical scaling (2.5b) alone. It requires the extra assumption δ̃=δ, introduced at (3.5) with the phrase “If that is admissible.” Without that input, the response operator carries an independent pair (δ̃,ξ̃), and the covariance equations give λ_R/z=δ+δ̃−ξ, while λ_C/z=2δ−ξ. Equality holds only when you impose the extra input. The abstract and Proposition 2 present the equality as a derived consequence of “these two dynamical symmetries”; that overstates the logical status. The paper itself flags the conditional nature in (3.5), so a careful reader can correct the abstract, but revision should make the extra assumption explicit.\n\nTwo other soft spots are acknowledged by the author: the central postulate (2.1) is not derived, and its physical origin is left open; and single-time correlators are excluded because the covariance equations overconstrain them. Those are honest limitations, not hidden flaws. The genuinely new content—JSS for T<Tc, the finite-size plateau laws, and the irrelevance criterion—is plausible and in part backed by exact spherical-model results. The citation pattern is extensive and appropriate; self-citations are not a problem where the cited results are the author’s own prior work.\n\nBottom line: this deserves a serious referee. It is a coherent organizing framework, not a derivation of the postulate, and the abstract should be toned down. If I were refereeing, I would ask for the δ̃=δ assumption to be clearly separated from the derived consequences. Send it to peer review, with a request for revision on that point.","headline":"A plausible unifying framework for ageing phenomenology, but the equality λ_C=λ_R is not derived from the two stated symmetries alone—an extra assumption about the response operator is needed.","tokens_in":53746,"tokens_out":2176,"would_cite":true,"duration_ms":21889,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82C27","82C31"],"pacs":[],"model":"deepseek-v4-flash","headline":"Classical ageing's standard scaling laws are shown to follow from a single logarithmic change of representation of the time-translation and dilatation generators.","keywords":["ageing","generalised time-translation invariance","dynamical scaling","Janssen-Schaub-Schmittmann scaling relation","two-time auto-correlator","two-time auto-response","finite-size scaling","critical dynamics"],"falsifier":"In a fully finite system quenched to $T\\le T_c$ with $s\\ll N^z$, the height of the two-time auto-correlator plateau should scale as $N^{-\\lambda}$ at fixed waiting time and as $s^{\\lambda/z-b}$ at fixed size; measuring a different $N$-dependence or a different $s$-dependence would falsify the derived finite-size scaling, and a simultaneous measurement of the global correlator plateau $N^{d-\\lambda}$ would test the extended Janssen-Schaub-Schmittmann relation.","tokens_in":52591,"feed_emoji":"⏳","tokens_out":18006,"duration_ms":144841,"temperature":0.7,"pith_summary":"This paper tries to establish that the standard phenomenology of physical ageing in classical systems is not a set of independent empirical rules but the common consequence of two symmetries: generalised time-translation-invariance and dynamical scaling. The construction is an intertwining change of representation, $X=e^{\\xi\\ln t}\\,X_{\\rm equi}\\,e^{-\\xi\\ln t}$, which turns the equilibrium time-translation generator $-\\partial_t$ into $-\\partial_t+\\xi/t$, softly breaking time-translation-invariance while keeping the Lie algebra intact. From this, the paper derives algebraic two-time scaling functions, the exponent equality $\\lambda_C=\\lambda_R$, the Janssen-Schaub-Schmittmann relation, and new finite-size plateau laws. A sympathetic reader would care because it replaces a list of empirical scaling laws with a single predictive postulate and produces new testable predictions.","feed_headline":"One assumption reproduces all known ageing scaling laws","feed_subtitle":"A logarithmic change of representation turns ageing folklore into derived laws and predicts new finite-size plateaux.","key_machinery":"The central object is the intertwining operator $W(t)=\\xi\\ln t$, used through $X=e^{W(t)}X_{\\rm equi}e^{-W(t)}$ to transform equilibrium symmetry generators into out-of-equilibrium ones. It gives the generalised time-translation generator $-\\partial_t+\\xi/t$ and shifts scaling dimensions by $\\xi$; solving the resulting pair of linear first-order covariance equations (2.5a)-(2.5b) yields the two-time scaling form that carries the argument. The same machinery supplies a criterion $2\\xi>1$ for irrelevance of cubic non-linearities in the equation of motion, which is what justifies applying linear Schrödinger-invariant response forms to a wide class of non-conserved phase-ordering models.","core_discovery":"On the paper's own terms, the central claim is that the whole generic ageing phenomenology of classical systems follows from the two covariance conditions on two-point functions: generalised time-translation invariance $X_{-1}C=(-\\partial_t-\\partial_s+\\xi_1/t+\\xi_2/s)C=0$ and dynamical scaling $X_0C=(-t\\partial_t-s\\partial_s-\\frac{1}{z}r\\partial_r-(\\delta_1-\\xi_1)-(\\delta_2-\\xi_2))C=0$. The unique covariant solution is $C(t,s;r)=s^{-\\delta_1-\\delta_2+\\xi_1+\\xi_2}(t/s)^{\\xi_1}(t/s-1)^{-\\delta_1-\\delta_2}F(r/(t-s)^{1/z})$. Large-argument limits give $f_C(y)\\sim y^{-\\lambda_C/z}$ and $f_R(y)\\sim y^{-\\lambda_R/z}$ with $\\lambda_C=\\lambda_R$, and the same derivation reproduces the Janssen-Schaub-Schmittmann relation $\\Theta=d-\\lambda/z$ at criticality while extending it to all $T<T_c$ through global correlators and responses. It also produces new finite-size plateau scalings for fully finite systems, which the paper verifies in the exactly solvable spherical model.","pith_inferences":["If the intertwining postulate is the true symmetry mechanism, allowing $W(t)$ to be a more general function than $\\xi\\ln t$ should generate logarithmic sub-ageing or multi-scaling regimes; the paper explicitly leaves this generalisation open.","The derivation excludes long-ranged initial correlations, and the paper notes $\\lambda_C=\\lambda_R$ may fail there; a controlled numerical test would quench with power-law correlated initial states and measure $\\lambda_C-\\lambda_R$ as a function of the initial correlation exponent.","In finite samples, the predicted plateau onset offers an alternative explanation for apparent curvature in $f_C(y)$ at large $y$; fits that ignore the plateau could systematically overestimate $\\lambda/z$, so comparing $N$-based and $s$-based plateau scalings provides a consistency check.","The paper restricts to classical dynamics; if the same logarithmic representation describes quantum quenches, the equalities $\\lambda_C=\\lambda_R$ and $\\Theta=d-\\lambda/z$ should appear in quantum ageing as well, which would be a sharp test of the symmetry's physical origin."],"forward_implications":["The equality $\\lambda_C=\\lambda_R$ follows from the covariance conditions rather than being assumed, so it is predicted to hold in every ageing system with short-ranged initial correlations.","The Janssen-Schaub-Schmittmann relation $\\Theta=d-\\lambda/z$ applies after quenches to all $T\\le T_c$, making $Q(t,0)\\sim t^{\\Theta}$ a practical way to measure $\\lambda$ below criticality.","In fully finite systems with $s\\ll N^z$, plateau heights scale as $N^{-\\lambda}$ and $s^{\\lambda/z-b}$ for correlators, with analogous response laws, giving independent estimates of $\\lambda$, $a$, and $\\lambda/z$ from finite samples.","Whenever the criterion $2\\xi>1$ holds, cubic non-linearities are irrelevant at late times, so the linear Schrödinger-invariant form of the two-time auto-response describes a wide class of phase-ordering and critical models.","At criticality the limit fluctuation-dissipation ratio $X_\\infty$ is finite and expressed through exponents and the amplitude ratio $f_{\\infty,R}/f_{\\infty,C}$, consistent with measured finite values in several experimental systems."],"supporting_citations":[{"why":"Supplies the non-standard conformal-algebra representation with a rapidity that underlies the generalised time-translation generator.","marker":"[178]"},{"why":"Gives the co-variant two-point function of the transformed representation used in the proof of the scaling form.","marker":"[123]"},{"why":"Guarantees uniqueness and generality of the solution of the covariance equations.","marker":"[144]"},{"why":"Fixes the ageing exponents and the dynamical-scaling framework used throughout the derivation.","marker":"[35]"},{"why":"Collects the ageing phenomenology and exponent values that the derived predictions must match.","marker":"[118]"},{"why":"Provides the critical Janssen-Schaub-Schmittmann relation that the paper re-derives and extends to all $T\\le T_c$.","marker":"[138]"},{"why":"Supplies exact spherical-model exponents and fluctuation-dissipation results used as solvable checks.","marker":"[97]"},{"why":"Provides the local-scale-invariance generators and response forms whose applicability the irrelevance criterion justifies.","marker":"[116]"},{"why":"Gives the prior finite-size spherical-model auto-correlator result that the new plateau scalings extend and confirm.","marker":"[124]"}],"fun_headline_variants":["Generalised time symmetry reproduces all ageing scaling","One symmetry explains all ageing scaling laws","Ageing exponents unified by generalised time-translation","Generalised time-translation invariance yields ageing scaling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the intertwining prescription $X_{\\rm equi}\\mapsto e^{\\xi\\ln t}X_{\\rm equi}e^{-\\xi\\ln t}$ with a constant dimensionless rapidity $\\xi$; the paper states in Section 5 that this form is not derived and its physical origin remains open, so if actual symmetries require a time-dependent $\\xi$ or additional corrections the scaling laws and exponent equalities would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Generalised time symmetry reproduces all ageing scaling","One symmetry explains all ageing scaling laws","Ageing exponents unified by generalised time-translation","Generalised time-translation invariance yields ageing scaling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000445,"raw_usage":{"total_tokens":2342,"prompt_tokens":1129,"completion_tokens":1213,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":745,"completion_tokens_details":{"reasoning_tokens":1155}},"tokens_in":745,"tokens_out":1213,"duration_ms":9319,"temperature":1.0,"reasoning_tokens":1155,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:54:12.595866+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In a fully finite system quenched to $T\\le T_c$ with $s\\ll N^z$, the height of the two-time auto-correlator plateau should scale as $N^{-\\lambda}$ at fixed waiting time and as $s^{\\lambda/z-b}$ at fixed size; measuring a different $N$-dependence or a different $s$-dependence would falsify the derived finite-size scaling, and a simultaneous measurement of the global correlator plateau $N^{d-\\lambda}$ would test the extended Janssen-Schaub-Schmittmann relation.","supporting_citations":[{"cited_title":"On the 3-point functions of Aging Dynamics and the AdS/CFT Correspondence","cited_arxiv_id":"1207.0243","evidence_quote":"Supplies the non-standard conformal-algebra representation with a rapidity that underlies the generalised time-translation generator."},{"cited_title":"Dynamical symmetries and causality in non-equilibrium phase transitions","cited_arxiv_id":"1509.03669","evidence_quote":"Gives the co-variant two-point function of the transformed representation used in the proof of the scaling form."},{"cited_title":"Non-equilibrium relaxations: ageing and finite-size effects","cited_arxiv_id":"2211.03657","evidence_quote":"Gives the prior finite-size spherical-model auto-correlator result that the new plateau scalings extend and confirm."}],"review_version":1}