{"id":"8d1b808a-c1d8-40f2-9791-ecf6b251d9c2","arxiv_id":"2607.18005","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"FIRE-2 galaxies show apparent ergodic convergence in star-forming-main-sequence deviations by the Thirumalai–Mountain metric, but block-scrambling reveals this is not true ergodicity.","lead":"In simulated FIRE-2 galaxies, deviations from the star-forming main sequence look as if they become ergodic over time — individual galaxies' histories seem to match the ensemble average. But a block-shuffling test shows this is only apparent: the convergence comes from decreasing variance, so the star formation histories are not truly ergodic.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Block-scrambling control is unvalidated for nonstationary finite series and internally contradicted by Table 2; the central non-ergodicity claim rests on it.","rationale":"The reader identified the block-scrambling validity as the weakest assumption; my read agrees. The paper's logic is: (i) TM metric decays for original data; (ii) it does not decay after block-scrambling; (iii) therefore original convergence was apparent, not true ergodicity. Step (ii) is only meaningful if block-scrambling is a validated control for the exact data type used. The cited literature (Kelty-Stephen & Mangalam 2022) does not obviously cover finite, heteroscedastic, nonstationary astrophysical series with variable membership. A concrete failure mode exists: a deterministic amplitude modulation of a stationary ergodic process is nonstationary but still has time averages converging to the ensemble mean; block-scrambling destroys the modulation and could suppress TM decay. That would make the control false-negative for a truly ergodic (in the paper's sense) process. The Table 2 result (Sample, 1 Gyr, sSFR7 α=1.13±0.09) is an internal inconsistency that further undermines the universal claim. The proposed synthetic calibration directly settles whether the concern lands. If the calibration shows scrambled exponents <1 for known-ergodic nonstationary series, the paper's strongest claim (not truly ergodic) must be downgraded or re-derived with a different test. If the calibration separates cleanly, the original verdict stands. Since the paper currently lacks this calibration and exhibits at least one contradicted data point, the CONDITIONAL verdict is appropriate and not changed by my read.","tokens_in":22365,"tokens_out":10307,"duration_ms":112203,"concrete_test":"Run a synthetic-control calibration matched to the FIRE-2 sample: N=20 independent realizations of an ergodic-but-nonstationary process, e.g. X_i(t)=g(t)Y_i(t), with Y_i stationary Gaussian (unit variance, short correlation) and g(t) decreasing according to the observed bursty-to-smooth variance trend (fit to Fig. 7). Resample at the FIRE-2 snapshot cadence, apply the same SFMS-deviation construction, compute the TM metric, and perform the same 500 Myr/1 Gyr block-scrambling with identical power-law fits (Table 2 protocol). If scrambled exponents fall below unity for this known-ergodic process, the block-scrambling test is invalid as a discriminator in this regime, and the central non-ergodicity claim is unsupported. Include a non-ergodic control (e.g. per-galaxy static variance) to confirm the test can separate the cases.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central conclusion—FIRE-2 SFMS deviations are only apparently ergodic—depends on the block-scrambling control (Section 3.4, Table 2). The premise is that block-scrambling preserves TM-metric convergence in truly ergodic systems, citing Kelty-Stephen & Mangalam (2022). That result has not been established for ~14 Gyr, N=20, heteroscedastic nonstationary series with missing snapshots and acknowledged edge effects (Figure 11). For a process X_i(t)=g(t)Y_i(t) with deterministic decreasing amplitude g(t) and stationary ergodic Y_i, the time average still converges to the ensemble mean, so the process is ergodic in the paper's operative sense; block-scrambling removes the amplitude schedule and can flatten or suppress TM decay. The cited control does not address this regime. Moreover, the paper's own Table 2 contains an internal counterexample: the sample-constructed SFMS with 1 Gyr blocks gives α(sSFR7)=1.13±0.09, i.e. in the 'ergodic' regime after scrambling, contrary to the claim that scrambling removes convergence 'regardless of the time block size.' If block-scrambling artificially suppresses convergence for nonstationary-but-ergodic finite samples, the inference that true ergodicity is absent does not follow; only the weaker statement that decreasing variance produces TM convergence remains.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper uses 20 FIRE-2 zoom-in simulations to test whether deviations from the star-forming main sequence (SFMS) are ergodic, i.e. whether ensemble averages of galaxy SFMS deviations reproduce individual time-averaged histories. Three SFMS definitions are considered (Speagle et al. 2014, Popesso et al. 2023, and a fit to the simulated sample itself), together with short-timescale (~20 Myr) and long-timescale (800 Myr) SFR estimators. The central claim is that the SFMS-deviation time series display apparent ergodic convergence of the Thirumalai-Mountain (TM) metric at late cosmic times, but that this convergence is not true ergodicity: it is driven by decreasing variance accompanying the bursty-to-smooth transition, and block-scrambling the time series removes the convergence. The paper carefully restricts its claims to star-forming galaxies, repeats the analysis with external and sample-based SFMS definitions, and explores mass and morphology splits.","tokens_in":22624,"tokens_out":7939,"duration_ms":90223,"significance":"If the central claim is correct, the paper has a useful cautionary message for galaxy-evolution studies: apparent TM-metric convergence in SFMS deviations can be a nonstationarity artifact, not evidence that ensemble averages can be substituted for individual SFHs. The paper's strengths include the use of high-resolution FIRE-2 simulations, multiple SFMS definitions and SFR estimators, explicit treatment of morphology, cross-checks with archaeological SFHs, and an explicit toy-model demonstration (Section 3.3.1) that the TM metric alone cannot distinguish stationary from decreasing-variance processes. The final conclusion is appropriately hedged: the authors state that TM-metric convergence is necessary but not sufficient. However, the quantitative case for the stronger 'not truly ergodic' claim rests on a block-scrambling diagnostic whose validity for these short, nonstationary, unevenly sampled series is not demonstrated, and the current Table 2 contains an internal inconsistency with the text.","major_comments":[{"comment":"The block-scrambling control is the main evidence for the claim that the observed convergence is only apparent. The paper cites Kelty-Stephen & Mangalam (2022) for the premise that block-scrambling preserves TM convergence for ergodic systems, but that premise is not established for the present regime: ~14 Gyr, N=20, heteroscedastic, nonstationary series with missing snapshots and visible edge effects (Figure 11). A concrete validation is needed, e.g. applying the same 500 Myr/1 Gyr block-scrambling to stationary ergodic surrogates and to the g(t)Y(t) amplitude-modulated process discussed in the text, with the same N and sampling pattern. Without this, the inference from 'scrambling suppresses convergence' to 'the original series is not ergodic' does not follow; a nonstationary but finite-sample ergodic process could in principle behave similarly. The internal consistency of the reported","section":"Section 3.4, Table 2"},{"comment":"The fitted power-law exponents in Table 2 are presented as supporting the ergodic classification, but many have error bars so large that the classification is not meaningful. Examples include alpha(sSFR9)=5.25±17.14 for the Sample SFMS complete sample, alpha(sSFR7)=8.89±5.61 for the P23 high-mass group, and alpha(sSFR9)=1.31±5.30 for the P23 complete sample. The statement in Section 3.4 that block-scrambled TM metrics 'asymptotically approach a nonzero value' is also not supported by Eq. (7), in which alpha<1 corresponds to slower power-law decay toward zero, not a nonzero asymptote. Either fit a model with an explicit offset and report it, or revise the interpretation. The quantitative convergence claims should be revisited with bootstrap/confidence intervals and a clear statement of which alpha values are statistically distinguishable from the ergodic threshold of unity.","section":"Section 3.3.4, Eq. (7)"},{"comment":"The Sample SFMS is fitted to the same FIRE-2 snapshots whose deviations are subsequently analyzed, so the zero-mean property of the residuals is partly inherited from the fit rather than from galaxy physics. The authors mitigate this by repeating every analysis with the Speagle et al. (2014) and Popesso et al. (2023) SFMSs, and the central visual trends are indeed present for those external definitions. However, Table 2 shows that the Sample-SFMS fits produce extreme exponents (e.g. 8.36±2.53 and 13.80±8.57) that behave very differently from the external relations. The abstract's claim that apparent convergence is seen 'regardless of the SFMS definition adopted' should therefore be qualified with the caveat that the sample-constructed SFMS is not an independent test. This is not a fatal flaw, but the current wording overstates the uniformity across definitions.","section":"Section 2.5, Section 3.2, Table 2"},{"comment":"The number of galaxies contributing to the TM metric changes with time (rightmost column of Figure 9), especially at early epochs. Changes in Ngal and the associated changes in the ensemble mean can themselves produce artificial TM-metric evolution. The text acknowledges this qualitatively, but the main conclusion would be strengthened by a fixed-sample or completeness-corrected analysis, or by an explicit test of how much of the early-time TM behavior is driven by sample membership changes rather than by intrinsic SFMS-deviation dynamics.","section":"Section 3.3.2, Figure 9"}],"minor_comments":[{"comment":"The notation for time averages and ensemble averages is introduced only verbally. Please define <...> and the overbar explicitly in Eq. (6), including the time dependence of Ngal.","section":"Eq. (5), Eq. (6)"},{"comment":"Table 2 is titled 'Power law exponents derived from TM metric convergence fit', but Section 3.4 refers to 'convergence values'. Use the same terminology throughout to avoid confusing alpha with a terminal offset.","section":"Section 3.4, Table 2"},{"comment":"The caption of Figure 1 says 'Error bars show the accepted scatter of 0.3 dex', but it is unclear whether the plotted error bars represent the observed scatter of FIRE-2 galaxies or the input scatter used in constructing the SFMS. Please clarify.","section":"Section 3.1, Figure 1"},{"comment":"The statement that archaeological SFHs are 'inherently smooth due to the high sampling rate' is confusing. A 1 Myr binning is fine, but smoothness is not guaranteed by sampling rate alone; clarify the smoothing or interpolation procedure.","section":"Section 2.3"},{"comment":"The SciPy reference is incorrectly formatted: 'Nature Medicine, 17, 261' should be 'Nature Methods, 17, 261'.","section":"References"},{"comment":"The TM-metric and block-scrambling codes are said to be 'available from the author upon reasonable request'. Given the increasing reproducibility standards in astrostatistics, please deposit the analysis code in a public repository.","section":"Data Availability"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Smith and Thacker have written a paper that does something genuinely useful: it checks whether ensemble averages of SFMS deviations are representative of individual galaxy histories in FIRE-2, and it shows the answer is basically no — the TM metric converges over cosmic time, but that convergence is driven by decreasing variance, not true mixing. That distinction matters for how people interpret scatter about the star-forming main sequence. The paper is careful to define apparent ergodicity, flags the metric's limitations in Sec. 3.3.1, and repeats the analysis with three SFMS definitions, which mitigates concerns about the sample-fitted SFMS being circular. The block-scrambling control is a good idea in principle.\n\nThe main soft spot, which the authors themselves partially acknowledge, is that the block-scrambling test is imported from statistical-physics settings and is never validated for these short, nonstationary, heteroscedastic series. The stress-test note points out a concrete internal inconsistency: Table 2 lists α(sSFR7)=1.13±0.09 for the sample SFMS with 1 Gyr blocks — above the threshold the paper itself uses for ergodic-like behavior — while the text says scrambling removes convergence 'regardless of the time block size.' That is a genuine contradiction and should be fixed, either by reclassifying that case or by softening the claim. The deeper worry is that block-scrambling can suppress TM decay for any finite nonstationary series even if the underlying process is ergodic in the time-average/ensemble-average sense, because it removes the amplitude schedule. I don't think that kills the central claim — the decreasing-variance explanation is independently demonstrated with the toy model in Sec. 3.3.1 — but it means the block-scrambling result is suggestive rather than decisive.\n\nOther soft spots are more minor: the analysis code is 'available from the author upon reasonable request' rather than archived; there is no bootstrap uncertainty on the TM curves themselves, only on the power-law fits; and the sample is selected and small (N=20), which the authors acknowledge. The power-law exponents in Table 2 have large error bars, so the quantitative convergence rates should be treated as indicative.\n\nWho is this for? Anyone working with simulated or observed galaxy populations who wants to avoid overinterpreting ensemble averages as individual histories. It deserves a proper referee; the central nuance is real, the presentation is honest, and the issues are addressable in revision. I would accept it for peer review, and after the block-scrambling inconsistency and validation are dealt with, it would be a solid contribution.","headline":"Useful and honest paper: apparent TM-metric ergodicity in FIRE-2 SFMS deviations is really decreasing variance, but the block-scrambling control has a validation gap and one internal contradiction.","tokens_in":23215,"tokens_out":2718,"would_cite":true,"duration_ms":28408,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"In FIRE-2 galaxies, star-formation ergodicity is only apparent","keywords":["ergodicity","star-forming main sequence","star formation histories","Thirumalai-Mountain metric","block scrambling","FIRE-2 simulations","galaxy formation","star formation variability"],"falsifier":"Take a long, stationary, genuinely ergodic time series of the same length as the FIRE-2 SFMS deviations but engineered to have decreasing variance, block-scramble it, and measure the TM exponent: if scrambling pushes the exponent below unity, the block-scrambling test is a false positive for non-ergodicity. Alternatively, construct a synthetic galaxy population with constant variance in SFMS deviations and check whether TM convergence survives scrambling; survival would contradict the paper's variance-decrease explanation.","tokens_in":22156,"feed_emoji":"🌌","tokens_out":4490,"duration_ms":37522,"temperature":0.7,"pith_summary":"The paper asks whether the star-formation history of one simulated galaxy can stand in for the average of many: whether ensemble averages over galaxies match time averages of individual galaxies. Tracking deviations from the star-forming main sequence across 20 FIRE-2 galaxies, the authors find that the Thirumalai-Mountain metric drifts toward zero over cosmic time — the signature usually read as ergodic convergence. But when the time series is block-scrambled to destroy temporal correlation, that convergence disappears. The paper concludes that the apparent ergodicity is produced by decreasing variance as galaxies transition from bursty to smooth star formation, not by true ergodic exploration.","feed_headline":"Star-formation histories only look interchangeable","feed_subtitle":"TM-metric convergence vanishes under block-scrambling, so a galaxy's past can't be read from the ensemble average.","key_machinery":"The Thirumalai-Mountain (TM) metric, which measures the spread of individual time averages around the ensemble-averaged time average, is the main diagnostic. The paper uses its power-law convergence exponent to classify systems, and block-scrambling of the time series into 500 Myr or 1 Gyr blocks as a control; in a genuinely ergodic system, shuffling blocks should preserve convergence. SFMS deviations are the quantity tracked, defined relative to three main-sequence fits.","core_discovery":"The central discovery is that SFMS deviations in FIRE-2 galaxies display apparent, not true, ergodicity. By the TM metric, both short-timescale (10^7 yr) and some long-timescale (10^9 yr) deviations converge toward zero over roughly 10 Gyr, across three SFMS definitions and for disk- and spheroid-dominated morphologies. However, block-scrambling the time series removes the convergence, yielding TM metric exponents below unity. The authors attribute the apparent convergence to decreasing variance with time — the bursty-to-smooth transition and absence of late mergers — so the ensemble average is not constant in the strong sense required for ergodicity.","pith_inferences":["Because the sample is a z=0 disk-selected set with no AGN feedback, the apparent-ergodicity result may not extend to quenched systems or high-redshift populations; adding AGN feedback could turn apparent convergence into genuine non-ergodicity in massive galaxies.","The paper leaves open the possibility that alternative ergodicity metrics — designed for nonstationary, finite-length series — could distinguish true from apparent ergodicity more cleanly than block-scrambling alone; a natural test is to apply such metrics to the same FIRE-2 time series.","A direct observational analogue would be to reconstruct SFHs for a sample of local galaxies and apply the same TM-metric-plus-scrambling procedure; if observed data also lose convergence under scrambling, the FIRE-2 result is a generic property of galaxy SFHs rather than a simulation artifact."],"forward_implications":["TM-metric convergence to zero alone is not sufficient to establish ergodicity of star-formation deviations; stationarity tests and temporal-scramble checks are needed.","Observational estimates that treat a galaxy population's scatter about the main sequence as a proxy for individual galaxy variability may conflate ensemble spread with time variation.","The bursty-to-smooth transition in massive galaxies is the likely cause of the apparent convergence, so samples dominated by massive, late-time disks will look artificially more ergodic.","Short-timescale SFR indicators converge faster than long-timescale ones, so conclusions about ergodicity depend on the averaging window of the SFR tracer.","For currently star-forming galaxies in FIRE-2, one cannot infer an individual galaxy's full star-formation history from the ensemble SFMS at a single epoch."],"fun_headline_variants":["FIRE-2 galaxies: apparent ergodicity in star formation is misleading","Block-scrambling kills apparent ergodicity in galaxy star formation","Star-formation histories in FIRE are not truly interchangeable","Galaxy star formation only spuriously matches the ensemble average","Apparent ergodicity in FIRE: a statistical mirage"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The conclusion depends on block-scrambling being a valid way to expose non-ergodicity in short, nonstationary, heteroscedastic time series: if scrambling destroys TM-metric convergence for a process that is genuinely ergodic but has decreasing variance, the paper's distinction between apparent and true ergodicity would not be established.","fun_headline_variants_meta":{"raw":{"variants":["FIRE-2 galaxies: apparent ergodicity in star formation is misleading","Block-scrambling kills apparent ergodicity in galaxy star formation","Star-formation histories in FIRE are not truly interchangeable","Galaxy star formation only spuriously matches the ensemble average","Apparent ergodicity in FIRE: a statistical mirage"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000962,"raw_usage":{"total_tokens":3925,"prompt_tokens":728,"completion_tokens":3197,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":472,"completion_tokens_details":{"reasoning_tokens":3118}},"tokens_in":472,"tokens_out":3197,"duration_ms":19339,"temperature":1.0,"reasoning_tokens":3118,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T16:22:38.373581+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a long, stationary, genuinely ergodic time series of the same length as the FIRE-2 SFMS deviations but engineered to have decreasing variance, block-scramble it, and measure the TM exponent: if scrambling pushes the exponent below unity, the block-scrambling test is a false positive for non-ergodicity. Alternatively, construct a synthetic galaxy population with constant variance in SFMS deviations and check whether TM convergence survives scrambling; survival would contradict the paper's variance-decrease explanation.","supporting_citations":[],"review_version":1}