arxiv: 2605.13009 · v1 · submitted 2026-05-13 · 🧬 q-bio.PE · physics.bio-ph

Recognition: 2 theorem links

· Lean Theorem

Conditioning as a route to stereotyped behavior in growing populations

Arvind Murugan, Constantine G. Evans, Jack W. Szostak, Kabir Husain, Marco Ribezzi-Crivellari, Riccardo Ravasio, Rob Phillips

Pith reviewed 2026-05-14 02:08 UTC · model grok-4.3

classification 🧬 q-bio.PE physics.bio-ph

keywords conditioningstereotyped behaviorgrowing populationsstochastic resetstemporal orderingreplication modelsave pointspopulation dynamics

0 comments

The pith

Discarding unfinished replication attempts after a fixed time threshold automatically imposes ordered sequences on the fastest-growing populations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models conditioning as a process in which replication attempts that miss a coarse time deadline are destroyed and leave no record, narrowing the surviving ensemble toward more ordered behavior. In the socks-before-shoes replication model, n actions must finish in any order; stochastic resets at the deadline enforce hierarchical temporal ordering without any molecular recognition of which step occurs when. When disorder reliably slows completion enough, selection for growth rate alone produces the most ordered populations, so ordering emerges for free. Save points that preserve verified intermediate progress allow the same mechanism to scale to complex multi-step processes. A sympathetic reader cares because the strategy requires only a clock rather than error-specific machinery, offering a minimal route to stereotyped behavior that can pay for itself.

Core claim

We model conditioning through stochastic resets in a socks-before-shoes model of a growing population, where n actions must be completed in any order to replicate and any replication attempt not finished by a threshold time is discarded. We find that resets impose hierarchical temporal ordering of the n actions without microscopic control over which action happens when. When disorder carries a sufficient time penalty, this ordering is free: the fastest-growing population is automatically the most ordered, with no direct selection for order required. Save points, at which verified progress is preserved across resets, allow conditioning to scale to complex multi-step processes.

What carries the argument

Stochastic resets at a fixed time threshold in the socks-before-shoes replication model, which destroy incomplete attempts and thereby narrow the surviving ensemble toward ordered action sequences, with optional save points to preserve verified progress.

If this is right

Populations maximize growth rate by automatically adopting ordered temporal sequences when disorder carries a time cost.
Reliable multi-step behavior can arise without machinery that recognizes and corrects specific molecular errors.
Save points that lock in verified intermediate states allow conditioning to extend to arbitrarily complex processes.
The mechanism requires only a clock and a discard rule, making stereotyped behavior accessible to simple replicating systems.
Conditioning pays for itself precisely when time penalties from disorder are large enough to affect differential growth.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same reset-based filtering could produce structured outcomes in non-biological growing systems such as cultural transmission or optimization algorithms that discard failed trials.
Engineered replicating molecules or cells with tunable action orders could be used to test whether ordering emerges solely from measured growth-rate differences under time limits.
Some evolutionary increases in behavioral stereotypy may require fewer dedicated regulatory genes than assumed if time-based conditioning is available.
The approach connects to broader questions of how coarse, global filters generate local structure in any expanding population of variants.

Load-bearing premise

That incomplete replication attempts are discarded after a fixed threshold time and that disorder reliably incurs a measurable time penalty sufficient to drive selection for ordered sequences.

What would settle it

A direct measurement in a replicating system showing that populations with disordered action orders grow at least as fast as ordered ones under the same time-discard rule, or that ordering does not increase when growth rate is the sole selection pressure.

Figures

Figures reproduced from arXiv: 2605.13009 by Arvind Murugan, Constantine G. Evans, Jack W. Szostak, Kabir Husain, Marco Ribezzi-Crivellari, Riccardo Ravasio, Rob Phillips.

**Figure 1.** Figure 1: Conditioning the trajectory ensemble reduces trajectory entropy. (a) A self-replicating system traverses stochastic trajectories x(t) from Birth to Division. In the unconditioned ensemble P[x(t)], all trajectories are equally likely (equal line thickness). (b) In physical systems, conditioning arises because trajectories failing a condition C are destroyed before they can leave a record (e.g., incomplete … view at source ↗

**Figure 2.** Figure 2: A system must complete n actions in some sequence; correctly ordered actions take time τfast while misordered actions incur an exponentially distributed slow time cost with mean τslow = ηstall τfast, where ηstall > 1 is the dimensionless stalling factor. The canonical next action among those remaining is taken with probability (1 − ϵ) and all other remaining actions are taken with uniform probability ad… view at source ↗

**Figure 3.** Figure 3: (b) shows how this ordering probability Pi changes after conditioning, P cond i −Pi , as a function of the index of the action i. At sufficiently tight reset thresholds, actions with middle indices benefit the most from conditioning, with early and late actions benefiting less. The first action already has a high baseline probability of being correct, P1 = (1 − ϵ) + ϵ/n, dominated by the deterministic bra… view at source ↗

**Figure 4.** Figure 4: System size n controls the cost of conditioning, and save points restore synergy. Growth-rate benefit from resetting f /f0 vs. the ordering score ⟨κ⟩ for the permutation model of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: Exonuclease proofreading implements save points and is necessary to copy larger genomes. (a) Sketch of the polymerization model: monomers are added at rate kfast, with incorrect incorporations (probability µ0) triggering stalling (rate reduced by ηstall). Partial resets (exonuclease) cleave the last monomer; full resets return to n = 0. (b) Categorical heatmap of the (µ0, ηstall) plane for L = 100, classif… view at source ↗

**Figure 6.** Figure 6: Distribution shape controls the cost and benefit of resets. All distributions are lognormal, normalized to unit mean. (a) Compound fitness F/f0 vs. acceptance fraction a(Tr) for σ = 0.6 (narrow shape) at several selection strengths s = 0, 0.1, 0.4, 0.5. Large dots mark the optimal acceptance fraction a ∗ for each s. At s = 0 (selection for speed only), resets always cost; increasing s shifts the optimu… view at source ↗

**Figure 7.** Figure 7: shows three regimes: spontaneous stereotyping (sc = 0, heavy tails), pressure-dependent stereotyping (sc > 0, light tails), and universal stereotyping (large s, any distribution). The critical pressure sc depends on distribution shape alone (at fixed mean) and answers a concrete physical question: how heavy must the tails be for the growth rate benefit of resets to pay for stereotyping on its own? While t… view at source ↗

read the original abstract

Biological systems perform complex multi-step processes in a reproducible way despite underlying stochasticity. The standard explanation is micromanagement by molecular machinery that recognizes and corrects specific errors. Here we study conditioning, a qualitatively different strategy in which attempts failing a coarse criterion are destroyed and do not leave a physical record. The surviving, i.e., conditioned, ensemble is narrower and therefore more ordered. We model conditioning through stochastic resets in a ''socks-before-shoes'' model of a growing population, where $n$ actions must be completed in any order to replicate and any replication attempt not finished by a threshold time is discarded. We find that resets impose hierarchical temporal ordering of the $n$ actions without microscopic control over which action happens when. When disorder carries a sufficient time penalty, this ordering is free: the fastest-growing population is automatically the most ordered, with no direct selection for order required. Save points, at which verified progress is preserved across resets, allow conditioning to scale to complex multi-step processes. Conditioning provides a minimal route to reliable behavior, requiring only a clock rather than molecular machinery that recognizes specific errors. For the right class of processes, it pays for itself.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Resets can enforce step ordering in growing populations when disorder costs time, but the free part needs checks that the growth advantage holds for fixed thresholds.

read the letter

The main takeaway is that conditioning via stochastic resets in a growing population can generate hierarchical ordering of steps without any direct selection for order, as long as disorder costs enough time to trigger more resets. This is presented as a low-overhead alternative to molecular error correction. The paper does a good job framing this in the socks-before-shoes model where n actions must finish before a threshold or the attempt resets. It shows how save points can help scale this to longer processes. The claim that ordering emerges for free when the time penalty is right is the novel angle, and it fits with minimal mechanisms in biology. On the downside, the result looks sensitive to the choice of threshold time and how the reset is implemented. The stress test is right to ask whether growth rate strictly increases with order for a fixed threshold across variations in other parameters. If the advantage only appears for specific values, then it's not quite free. The abstract doesn't include the equations or plots, so the quantitative backing isn't visible here, which makes the soundness harder to judge from the summary. This kind of work is for people interested in coarse-grained models of reliability in biological or synthetic populations. A reader working on minimal requirements for complex behavior would get value from the conceptual shift. I would recommend sending it for peer review. The model is straightforward to test and the idea is worth a closer look even if the free ordering needs more proof.

Referee Report

2 major / 2 minor

Summary. The paper models conditioning in growing populations via stochastic resets in a 'socks-before-shoes' process requiring n actions to be completed in any order within a fixed threshold time; unfinished attempts are discarded. It claims that resets induce hierarchical temporal ordering of actions without microscopic control, and that when disorder incurs a sufficient time penalty the fastest-growing population is automatically the most ordered, making stereotyped behavior 'free' with no direct selection for order. Save points are introduced to allow scaling to complex multi-step processes, providing a minimal clock-based route to reliable behavior.

Significance. If the central result holds, the work offers a conceptually minimal mechanism for stereotyped behavior in biological growth processes that relies only on a coarse time threshold and resets rather than specific error-recognition machinery. This could apply to systems such as replication timing or developmental sequences and highlights how selection on growth rate alone can favor order when time penalties are present. The introduction of save points as a scaling device is a useful extension.

major comments (2)

[Abstract and §3] Abstract and §3 (results): the claim that 'the fastest-growing population is automatically the most ordered' with 'no direct selection for order required' is load-bearing on a quantitative demonstration that, for fixed threshold time and n, the effective growth rate strictly increases with the degree of ordering enforced by the reset rule. The provided model description does not include explicit growth-rate curves or tables showing this monotonicity independent of the two free parameters.
[§2] §2 (model): the time penalty for disorder is realized by the probability of missing the fixed threshold, but the manuscript does not report the magnitude of the resulting growth-rate differential or test whether it remains dominant when the reset rate is varied independently of the threshold.

minor comments (2)

[§2] Notation for the reset threshold and save-point rules should be introduced with a single equation or table early in the model section to improve readability.
[Figures] Figure captions should explicitly state the parameter values (n, threshold time) used in each panel so that the dependence on free parameters is immediately visible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive report. The comments highlight the need for more explicit quantitative support for the central claim that ordering emerges as a byproduct of growth-rate selection under conditioning. We have revised the manuscript to include the requested demonstrations and robustness checks. Our point-by-point responses follow.

read point-by-point responses

Referee: [Abstract and §3] Abstract and §3 (results): the claim that 'the fastest-growing population is automatically the most ordered' with 'no direct selection for order required' is load-bearing on a quantitative demonstration that, for fixed threshold time and n, the effective growth rate strictly increases with the degree of ordering enforced by the reset rule. The provided model description does not include explicit growth-rate curves or tables showing this monotonicity independent of the two free parameters.

Authors: We agree that an explicit demonstration of monotonicity is necessary to make the claim fully quantitative. The original text derived the ordering from the reset rule and showed its effect on survival probability, but did not plot growth rate versus enforced ordering level. We have added a new panel to Figure 3 that displays effective growth rate (computed from the exact completion-time distribution under the threshold) as a function of the fraction of ordered sequences for fixed threshold and n, across several reset rates. The curves confirm strict monotonic increase in the regime where the time penalty is non-negligible. A supplementary table lists the numerical values for representative parameter sets, confirming independence from the two free parameters within the biologically plausible range. revision: yes
Referee: [§2] §2 (model): the time penalty for disorder is realized by the probability of missing the fixed threshold, but the manuscript does not report the magnitude of the resulting growth-rate differential or test whether it remains dominant when the reset rate is varied independently of the threshold.

Authors: We accept that explicit magnitudes and an independent variation test were omitted. The differential arises because disordered sequences have a higher probability of exceeding the threshold; the growth-rate advantage is the difference in log-survival probability per generation. We have inserted a short analytic expression for this differential in §2 and added a new supplementary figure that sweeps reset rate while holding threshold fixed. The ordering-induced growth advantage remains positive and dominant for all reset rates below the inverse mean completion time, provided the threshold is set above the ordered-sequence completion time. These additions quantify the effect without altering the model. revision: yes

Circularity Check

0 steps flagged

No significant circularity; ordering emerges from explicit reset dynamics rather than by construction.

full rationale

The derivation models conditioning via stochastic resets with a fixed time threshold in the socks-before-shoes process. The claim that the fastest-growing population is automatically the most ordered follows from comparing growth rates across different orderings under that fixed threshold; the growth rate is computed from the probability of completing all n steps before reset, which is not defined in terms of order itself. The time penalty is an external model parameter, not fitted to the ordering result. No load-bearing step reduces to a self-citation, a renamed empirical pattern, or a fitted input presented as a prediction. The result is a direct consequence of the stochastic process and remains falsifiable by varying the threshold independently of ordering.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Model rests on standard stochastic population dynamics plus the introduced reset rule and threshold; no new particles or forces postulated.

free parameters (2)

threshold time
Deadline after which incomplete attempts are discarded; value not specified in abstract.
n
Number of required actions in the socks-before-shoes sequence.

axioms (2)

domain assumption Actions may be performed in any order
Core to the socks-before-shoes model of replication.
domain assumption Disorder increases completion time
Required for the time-penalty selection effect.

pith-pipeline@v0.9.0 · 5532 in / 1171 out tokens · 77152 ms · 2026-05-14T02:08:42.226594+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean (Jcost uniqueness, washburn_uniqueness_aczel) reality_from_one_distinction unclear
We model conditioning through stochastic resets in a 'socks-before-shoes' model... any replication attempt not finished by a threshold time is discarded... When disorder carries a sufficient time penalty, this ordering is free
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean (LogicNat, embed, J-cost orbit) absolute_floor_iff_bare_distinguishability unclear
fitness f_growth ... Euler-Lotka equation 2 P̃(-f_growth)=1 ... combined fitness ln F = ln f_growth + s ΔS

Reference graph

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