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arxiv: 2605.19785 · v1 · pith:AI5HPHNEnew · submitted 2026-05-19 · ❄️ cond-mat.dis-nn

Optimal Persistence Reveals Hidden Topology in Complex Energy Landscapes

LI Zhenpeng This is my paper

Pith reviewed 2026-05-20 01:32 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn
keywords energy landscapesspin glasstopologypersistencecanyon findingdisordered systemsentropic bottlenecks
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The pith

In the p=2 spherical spin glass the canyon-finding rate peaks at an optimal persistence time that reveals the landscape's hidden topology.

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

The paper shows that a search process with a tunable persistence time produces a canyon-finding rate that rises to a maximum and then declines, forming an inverted-U curve whose peak occurs at a finite optimal persistence. Infinite persistence corresponds to a topological transition in the landscape while the finite optimum exposes structure that would otherwise stay hidden. Simulations find that this optimum decreases with system size at low temperature and saturates for large N, with the effective canyon width reaching one, and that temperature enters the picture mainly through thermal velocity. The result is presented as a generic principle for any disordered landscape that contains entropic bottlenecks.

Core claim

Infinite persistence marks the topological transition. For finite persistence, the canyon-finding rate Gamma(tau_p) on the p=2 spherical spin glass forms an inverted-U profile, peaking at an optimal tau_p^*. At low temperature (T=0.05), tau_p^* drops from 10 to 5 as N increases through 128, marking the discrete-to-quasi-continuous GOE crossover. For N>=128, the canyon width saturates at xi_eff=1. At higher temperatures (T>=0.15), tau_p^*=10 and beta(T) scales as 1/T, with temperature dependence entering only through v_th = sqrt(2T).

What carries the argument

Persistence time tau_p that sets how long a trajectory continues in one direction before resampling, yielding the canyon-finding rate Gamma(tau_p) whose maximum locates the hidden topology.

If this is right

  • At T=0.05, tau_p^* drops from 10 to 5 as N grows past 128, signaling the discrete-to-quasi-continuous crossover.
  • For N>=128 the effective canyon width saturates at xi_eff=1, consistent with the measured tau_p^*=5 when beta=0.4.
  • At T>=0.15, tau_p^*=10 and the scaling parameter beta scales as 1/T through the thermal velocity.
  • For N=1024 at low T the peak region is flat between tau_p=5 and 6 within error bars.

Where Pith is reading between the lines

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

  • The same persistence-tuning procedure could be applied to loss surfaces of neural networks to locate regions of flat minima.
  • Measuring optimal persistence times in molecular-dynamics trajectories of glassy materials would provide a direct experimental test.
  • Extending the method to quantum spin glasses or to landscapes with quenched disorder in higher dimensions could show whether the inverted-U shape persists.

Load-bearing premise

That the p=2 spherical spin glass and its observed canyon-finding statistics represent the broader class of disordered landscapes with entropic bottlenecks.

What would settle it

Simulating the canyon-finding procedure on a p=3 spherical spin glass or on a much larger p=2 system and finding that Gamma(tau_p) is monotonic or peaks at a value incompatible with the reported saturation of canyon width at xi_eff=1.

Figures

Figures reproduced from arXiv: 2605.19785 by LI Zhenpeng.

Figure 1
Figure 1. Figure 1: FIG. 1. Inverted-U profile of exploration efficiency for [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Size-driven transition of optimal persistence at [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
read the original abstract

Infinite persistence marks the topological transition. For finite persistence, the canyon-finding rate Gamma(tau_p) on the p=2 spherical spin glass forms an inverted-U profile, peaking at an optimal tau_p^*. At low temperature (T=0.05), tau_p^* drops from 10 to 5 as N increases through 128, marking the discrete-to-quasi-continuous GOE crossover. For N=1024, the peak is flat between tau_p=5 and 6 within statistical uncertainties, preventing a more precise determination. For N>=128, the canyon width saturates at xi_eff=1, consistent with the measured tau_p^*=5 when beta=0.4. At higher temperatures (T>=0.15), tau_p^*=10 and beta(T) scales as 1/T, with temperature dependence entering only through v_th = sqrt(2T). For T=0.10 and N>=128, high-resolution scans give tau_p^*=8.0; for N<=64 at the same temperature, coarse scans place tau_p^* in the range 8-10. Thus, optimal persistence reveals the hidden topology of the landscape-a principle expected to be generic in disordered landscapes with entropic bottlenecks.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript studies navigation of complex energy landscapes via a persistence parameter tau_p in the p=2 spherical spin glass. It reports that the canyon-finding rate Gamma(tau_p) exhibits an inverted-U profile with a finite optimal tau_p^* that depends on system size N and temperature T; infinite persistence is identified with the topological transition. Concrete numerical values are given (e.g., tau_p^*=5 for N>=128 at T=0.05, tau_p^*=8 at T=0.10, saturation of xi_eff=1), and the authors conclude that optimal persistence reveals hidden topology in a manner expected to be generic for disordered landscapes possessing entropic bottlenecks.

Significance. If substantiated, the observation of an inverted-U profile in Gamma(tau_p) and the associated optimal persistence would supply a concrete, numerically accessible probe of topological features in a canonical disordered landscape. The reported N-dependent crossover from discrete to quasi-continuous GOE statistics and the saturation of effective canyon width constitute falsifiable, quantitative results that could be checked in related models. The absence of cross-model validation or an isolating analytical argument, however, leaves the generality claim as an untested extrapolation rather than a demonstrated principle.

major comments (2)
  1. [Abstract] Abstract: the assertion that optimal persistence 'reveals the hidden topology of the landscape—a principle expected to be generic in disordered landscapes with entropic bottlenecks' is load-bearing for the headline claim yet rests solely on results for the p=2 spherical spin glass. No numerical checks on p>2 spin glasses, random-energy models, or continuous landscapes with different barrier statistics are presented, nor is an analytical argument given that isolates the topological ingredient from model-specific features such as GOE eigenvalue statistics or the spherical constraint.
  2. [Abstract] Abstract: concrete numerical outcomes (tau_p^*=5 for N>=128 at T=0.05, tau_p^*=8 at T=0.10, flat peak for N=1024) are stated without accompanying error-bar details, data-exclusion criteria, or the precise definition of the canyon-finding rate Gamma(tau_p) that would allow independent reproduction or assessment of statistical significance.
minor comments (1)
  1. The manuscript would benefit from an explicit equation or algorithmic definition of Gamma(tau_p) and xi_eff early in the text so that the reported optima can be directly linked to the underlying topology measure.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their constructive feedback. We address each major comment below and indicate the planned revisions.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that optimal persistence 'reveals the hidden topology of the landscape—a principle expected to be generic in disordered landscapes with entropic bottlenecks' is load-bearing for the headline claim yet rests solely on results for the p=2 spherical spin glass. No numerical checks on p>2 spin glasses, random-energy models, or continuous landscapes with different barrier statistics are presented, nor is an analytical argument given that isolates the topological ingredient from model-specific features such as GOE eigenvalue statistics or the spherical constraint.

    Authors: The manuscript uses the p=2 spherical spin glass as a canonical disordered system with known entropic bottlenecks and a topological transition. The generality is presented as an expectation based on these shared features rather than a fully validated principle across models. We will revise the abstract to qualify the language accordingly and expand the discussion to clarify the basis for expecting broader applicability, without claiming demonstration. revision: partial

  2. Referee: [Abstract] Abstract: concrete numerical outcomes (tau_p^*=5 for N>=128 at T=0.05, tau_p^*=8 at T=0.10, flat peak for N=1024) are stated without accompanying error-bar details, data-exclusion criteria, or the precise definition of the canyon-finding rate Gamma(tau_p) that would allow independent reproduction or assessment of statistical significance.

    Authors: We will incorporate the precise definition of Gamma(tau_p) into the revised manuscript. Additionally, we will report statistical uncertainties or error bars for the quoted tau_p^* values and specify the criteria for data inclusion or exclusion in the numerical scans. These additions will enhance the reproducibility and allow readers to assess the significance of the reported optima. revision: yes

Circularity Check

0 steps flagged

No significant circularity: results from direct numerical scans

full rationale

The paper reports an inverted-U profile for Gamma(tau_p) and a finite optimal tau_p^* obtained via direct numerical scans over persistence times in the p=2 spherical spin glass. These quantities are measured outputs rather than quantities fitted inside equations that then predict the same quantities. The generality statement is presented as an expectation without a self-referential derivation or load-bearing self-citation chain. No equations or steps reduce by construction to their inputs; the derivation chain remains self-contained against the reported simulations.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the representativeness of the p=2 spherical spin glass and on the interpretation of a numerically observed peak as a topological diagnostic; no new particles or forces are introduced.

free parameters (1)
  • beta = 0.4
    Value 0.4 is stated to be consistent with measured tau_p^*=5 and xi_eff=1 at low T.
axioms (1)
  • domain assumption The p=2 spherical spin glass captures the essential topology of generic disordered landscapes with entropic bottlenecks.
    Invoked in the final sentence to extend the numerical findings beyond the specific model.

pith-pipeline@v0.9.0 · 5742 in / 1399 out tokens · 50119 ms · 2026-05-20T01:32:17.086415+00:00 · methodology

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Reference graph

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    is universal across GOE realizations. Third, forT≥0.15 andN≥128,τ ∗ p is constant (τ ∗ p = 10); atT= 0.10, high-resolution scans giveτ ∗ p = 8.0. Fourth, the unsaturated regime (N <128) remains an open question for future investigation. S9. Simulation Parameters and Optimal Persistence Summary Table S2 lists simulation parameters. ForN≤256, 10 disorder se...

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