Loading history and window geometry bound compact-state slip ranking during granular shear startup

Ruixin Zhou; Boliang Yu

cond-mat.softSep 9, 2026

Loading history and window geometry bound compact-state slip ranking during granular shear startup

Authors: Ruixin Zhou, Boliang Yu

Organizations: School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China

Abstract

Granular slip forecasting can conflate material state, loading progress, and the geometry of event-centered sampling. We separated these contributions in slowly sheared two-dimensional frictional disks using a compact neural score of stress, pressure, coordination, non-affine motion, and force-network observables. The model was developed on 36 trajectories and frozen efore testing on 18 new trajectories under two nested stress-drop definitions. Inspection of held-out results revealed post-event sampling asymmetry; recovery-aware analyses are therefore descriptive. With trajectories weighted equally, the compact score ranked near-slip windows above both prevalence and within-trajectory circular-phase controls under both definitions (representative average precision 0.310 versus prevalence 0.173; phase-null upper bound 0.257). Loading-history coordinates ranked more strongly, reaching 0.534 for causal elapsed strain. The recovery-aware rule retained 78.4% of activity-gated events and preferentially selected longer preceding intervals; ranking by time since the previous catalogued event remained compatible with a count-conditioned geometry null. Compact observables thus contain temporally aligned slip information, but stronger loading-history baselines and window-geometry sensitivity bound that evidence. These startup data do not isolate a state-specific short-horizon precursor beyond loading history or support a renewal interpretation of elapsed-strain ranking.

Figures & tables

Figure 1: Simulation and evaluation design. (a) Bidisperse configuration; displayed edges satisfy

f_{n}\geq 1.2\langle f_{n}\rangle

and periodic-crossing edges are omitted from the drawing only. (b) Instantaneous and five-frame-smoothed stress, one accepted onset, its forecast interval, and the original event-exclusion interval. (c) Schematic of the observed original post-event asymmetry for

H=4

. Applying the original negative-window buffer to positive and negative windows gives a common onset-based re-entry at offset 25; the lookback-aware rule can move it later until the complete

D^{2}_{\min}

lookback follows trough and recovery. (d) Each

(\phi,\mu)

cell contributes four development and two held-out trajectories

Definition	Rule	Windows (positive)	Mean positive fraction	Compact AP	Cumulative- strain AP	Causal elapsed AP
Gated	Original	6,166 (1,230)	0.208	0.339	0.409	0.539
Gated	Common exclusion	5,935 (999)	0.177	0.311	0.388	0.519
Gated	Lookback-aware	5,742 (939)	0.173	0.310	0.398	0.534
Ungated	Original	5,945 (1,396)	0.247	0.359	0.443	0.534
Ungated	Common exclusion	5,609 (1,060)	0.202	0.326	0.406	0.499
Ungated	Lookback-aware	5,439 (1,006)	0.199	0.325	0.420	0.508

Table 1: Mean per-trajectory AP on the held-out cohort. “Common exclusion” applies the same previous-onset separation to positive and negative windows. “Lookback-aware” also requires every frame used by the four-frame

D^{2}_{\min}

history to follow the previous trough and recovery. Elapsed AP uses the causal prefix replay in Eq. ( 9 ).

Definition	Rule	Compact $-$ prevalence	Compact $-$ strain	Compact $-$ elapsed
Gated	Original	0.131 [0.076, 0.187]	-0.070 [-0.127, -0.013]	-0.200 [-0.260, -0.140]
Gated	Common exclusion	0.133 [0.081, 0.186]	-0.077 [-0.143, -0.010]	-0.208 [-0.270, -0.146]
Gated	Lookback-aware	0.138 [0.083, 0.192]	-0.088 [-0.162, -0.013]	-0.223 [-0.288, -0.159]
Ungated	Original	0.112 [0.065, 0.158]	-0.085 [-0.133, -0.036]	-0.175 [-0.217, -0.134]
Ungated	Common exclusion	0.124 [0.081, 0.167]	-0.080 [-0.138, -0.022]	-0.173 [-0.216, -0.131]
Ungated	Lookback-aware	0.126 [0.080, 0.171]	-0.095 [-0.161, -0.029]	-0.184 [-0.224, -0.143]

Table 2: Paired mean per-trajectory AP differences. Brackets are exact 95% intervals from all 19,683 condition-stratified resampling compositions. The elapsed score is the causal prefix replay.

Figure 2: Loading-history comparison and window-rule sensitivity. (a) Lookback-aware mean per-trajectory AP and pooled-window point estimates (

\times

; omitted for the phase null). Error bars are exact 95% resampling intervals except for the circular-phase null, where they show the central 95% randomization range about its mean. (b) Mean per-trajectory compact AP minus mean per-trajectory prevalence. (c) Compact minus cumulative-strain AP. (d) Compact minus causal prefix-replay elapsed-strain AP. Original denotes the initial window set; common exclusion applies the same previous-onset separation to both window types. The lookback-aware rule additionally places the complete

D^{2}_{\min}

history after previous-event recovery. Every comparison is paired on identical windows within a trajectory

Appendix figures & tables23 assets

Supplementary material from the paper’s appendix.

Appendix

Quantity	Definition or value
Preparation / production step	$10^{-4}$ / $4\times 10^{-4}$
Contact-scale regime	Across 26,955,862 stored contacts, overlap divided by pair diameter had median 0.011, 97.5th percentile 0.045, and maximum 0.141. The range of $k_{n}$ divided by trajectory-median pressure was 38.4–286.9.
Damped contact times	Small–small / small–large / large–large restitution: 0.568/0.520/0.447; damped collision duration in production steps: 71/82/101.
Loading and drag times	$1/\zeta=0.500$ ; one saved-frame interval $=0.333$ ; the forecast horizon spans 10.000 drag times.
Compression and consolidation	Compression from $\phi=0.58$ in 16 stages; zero-shear convergence checks every 250 preparation steps; velocities and angular velocities multiplied by 0.5 every 500 steps.
Normalized kinetic-energy criterion	$(K/N)/(k_{n}\bar{d}^{2})\leq 10^{-7}$ , where $\bar{d}$ is mean diameter.

Appendix

Table 3: Reproducibility details for simulation, labels, inputs, and independent-test evaluation.

Quantity	Count
Activity-gated onsets	88
Activity-ungated onsets	106
Additional activity-ungated onsets	18
Activity-gated onsets matched exactly	88
Activity-gated onsets matched within tolerance	88
Trajectories with changed event list	11

Appendix

Table 4: Independent-test event counts and onset overlap. “Within tolerance” uses the same

\Delta\gamma=0.010

separation used to merge nearby candidates.

Definition	History	Windows	Positive windows	Events
Activity-gated	$H=4$	6166	1230	88
Activity-gated	$H=8$	5856	1163	84
Activity-ungated	$H=4$	5945	1396	106
Activity-ungated	$H=8$	5586	1287	97

Appendix

Table 5: Independent-test window, positive-window, and distinct-event counts. Differences between

H=4

and

H=8

reflect complete-history availability; differences between definitions additionally reflect their event lists and exclusion regions.

Definition	Model	Seed 3407	Seed 7349	Seed 9109	Ensemble AP	Prevalence
Activity-gated	Compact MLP, $H=4$	0.2812	0.2792	0.2585	0.284	0.199
Activity-gated	Stress–pressure–ratio MLP, $H=4$	0.2725	0.2420	0.2662	0.259	0.199
Activity-gated	Compact MLP, $H=8$	0.2107	0.2386	0.2420	0.234	0.199
Activity-gated	Graph GNN–LSTM, $H=8$	0.2489	0.2451	0.2204	0.244	0.199
Activity-ungated	Compact MLP, $H=4$	0.3029	0.2805	0.3138	0.305	0.235
Activity-ungated	Stress–pressure–ratio MLP, $H=4$	0.3242	0.3361	0.3248	0.337	0.235

Appendix

Table 6: Fixed seed-level AP, probability-ensemble AP, and ensemble prevalence on the independent test set.

Difference	Common windows	Estimate	95% interval	Fraction $>0$
Activity-gated compact $H=4$ $-$ control	6166	0.025	[-0.016, 0.072]	0.873
Activity-ungated compact $H=4$ $-$ control	5945	-0.032	[-0.071, 0.016]	0.085
Activity-gated compact $H=8$ $-$ graph	5856	-0.010	[-0.046, 0.028]	0.295
Activity-ungated compact $H=8$ $-$ graph	5586	0.009	[-0.013, 0.031]	0.762

Appendix

Table 7: Independent-test paired AP differences. The final column is the descriptive fraction of valid condition-stratified bootstrap draws above zero.

Definition	Rule	Positive frac.	Compact	Strain	Causal elapsed	Fraction diag.
Activity-gated	Original	0.199	0.284	0.331	0.329	0.306
Activity-gated	Common exclusion	0.168	0.248	0.287	0.286	0.252
Activity-gated	Lookback-aware	0.164	0.245	0.290	0.290	0.246
Activity-ungated	Original	0.235	0.305	0.364	0.344	0.375
Activity-ungated	Common exclusion	0.189	0.245	0.296	0.284	0.301
Activity-ungated	Lookback-aware	0.185	0.241	0.301	0.286	0.297

Appendix

Table 8: Pooled-window AP under the window sensitivities. Elapsed strain is reconstructed by causal prefix replay. The positive-fraction diagnostic uses evaluation labels and is shown only as a nuisance check.

Definition	Original events	Lookback events	Coverage	Median preceding interval, original $\rightarrow$ lookback
Activity-gated	88	69	78.4%	48.0 $\rightarrow$ 78.0
Activity-ungated	106	74	69.8%	39.0 $\rightarrow$ 72.5

Appendix

Table 9: Event selection by the lookback-aware window rule. Coverage is relative to events represented by the original positive windows. Preceding intervals are onset-to-onset saved-frame intervals with every retained event weighted once.

Definition	Compact AP	Circular-phase null	Phase upper tail	Catalog-age AP	Hard-core onset null
Activity-gated	0.310	0.211 [0.171, 0.257]	0.00005	0.527	0.461 [0.383, 0.538]
Activity-ungated	0.325	0.239 [0.196, 0.287]	0.00051	0.504	0.503 [0.428, 0.572]

Appendix

Table 10: Post-test temporal-null audit. Brackets are central 95% randomization ranges, not trajectory-resampling confidence intervals. The phase null uses 100,000 cohort compositions; the hard-core onset null uses 2,000 surrogate cohorts.

Definition	Compact AP	Strain AP	Compact $-$ strain	Time increment	Time+condition increment
Activity-gated	0.405	0.296	0.109	0.029	0.012
Activity-ungated	0.476	0.326	0.150	0.051	0.029

Appendix

Table 11: Development-test ordering and nested-stacker increments. The time-plus-condition stacker used one validation trajectory per condition for fitting.

Definition	Window set	Windows	Compact AP	Prev.	$\gamma_{\rm obs}$ AP	Compact $-$ prev. (95% interval)	Compact $-$ $\gamma_{\rm obs}$ (95% interval)
Activity-gated	Original	6166	0.284	0.199	0.331	0.085 [0.038, 0.152]	-0.047 [-0.101, 0.013]
Activity-gated	Post-yield	4492	0.314	0.258	0.338	0.055 [0.018, 0.102]	-0.025 [-0.065, 0.015]
Activity-ungated	Original	5945	0.305	0.235	0.364	0.070 [0.035, 0.130]	-0.059 [-0.111, -0.004]
Activity-ungated	Post-yield	4285	0.338	0.310	0.371	0.028 [-0.007, 0.082]	-0.033 [-0.079, 0.010]

Appendix

Table 12: The all-frozen compact-minus-prevalence entries reproduce the original prespecified primary checks. Cumulative-strain comparisons and all post-yield entries were specified after the independent outcomes were inspected. Intervals use 10,000 condition-stratified trajectory draws.

Figure 3: Loading-progress audit. (a) Prevalence, compact-model AP, and zero-parameter cumulative-strain AP on the original

H=4

endpoints. (b) Compact AP minus prevalence on all endpoints and on the post-yield subset. The all-endpoint compact-minus-prevalence checks were prespecified; cumulative-strain and post-yield comparisons were introduced after the frozen summary was inspected. (c) AP increment obtained by adding the fixed compact score to development-fitted loading-time stackers, without and with condition indicators. Error bars are percentile 95% intervals from 10,000 condition-stratified trajectory draws. Panel (c) is post-test

Definition	Endpoint subset	Causal elapsed AP	Compact $-$ causal elapsed (95% interval)
Activity-gated	Lookback-aware	0.534	-0.223 [-0.288, -0.159]
Activity-ungated	Lookback-aware	0.508	-0.184 [-0.224, -0.143]
Activity-gated	Post-first	0.652	-0.332 [-0.402, -0.261]
Activity-ungated	Post-first	0.593	-0.251 [-0.298, -0.204]

Appendix

Table 13: Post-test causal-availability audit on lookback-aware windows. “Post-first” removes endpoints before the prefix replay has detected its first event. Brackets are exact 95% intervals from the finite condition-stratified resampling design.

Definition	Score	AP	Compact AP minus score AP (95% interval)
Activity-gated	Final-catalog elapsed score	0.281	0.003 [-0.057, 0.068]
Activity-ungated	Final-catalog elapsed score	0.295	0.010 [-0.033, 0.067]
Activity-gated	Loading-and-event- history MLP	0.326	-0.042 [-0.112, 0.032]

Appendix

Table 14: Earlier retrospective temporal baselines on the independent set. Every paired difference is computed on exact common endpoints; intervals use 10,000 condition-stratified trajectory draws.

Definition / endpoints	Paired difference	Bootstrap estimate (95% interval)	Exact central 95% interval
Activity-gated / $H=4$	Compact $-$ strain	-0.047 [-0.101, 0.013]	[-0.100, 0.012]
Activity-ungated / $H=4$	Compact $-$ strain	-0.059 [-0.111, -0.004]	[-0.112, -0.004]
Activity-gated / $H=8$	Compact $-$ strain	-0.100 [-0.160, -0.045]	[-0.161, -0.044]
Activity-ungated / $H=8$	Compact $-$ strain	-0.075 [-0.142, -0.011]	[-0.143, -0.013]
Activity-gated / $H=8$	Graph $-$ strain	-0.090 [-0.150, -0.029]	[-0.152, -0.028]
Activity-ungated / $H=8$	Graph $-$ strain	-0.083 [-0.146, -0.024]	[-0.146, -0.025]

Appendix

Table 15: Loading-progress comparisons under Monte Carlo bootstrap and exact enumeration. Negative values favor cumulative strain.

Definition	Events	Median drop/ $P$	Median duration $\Delta\gamma$	Median interevent $\Delta\gamma$	Event $q_{D^{2}}$	Quiet $q_{D^{2}}$	Event $-$ quiet (95% interval)
Activity-gated	88	0.028	0.010	0.048	0.108	0.178	-0.073 [-0.086, -0.060]
Activity-ungated	106	0.025	0.009	0.039	0.108	0.179	-0.073 [-0.084, -0.061]

Appendix

Table 16: Exploratory event and non-affine concentration statistics. A participation ratio

q_{D^{2}}=1

denotes uniform activity, with smaller values denoting stronger concentration. Event and quiet columns are pooled medians; the final column is a paired mean difference formed after within-trajectory averaging and therefore is not their arithmetic difference.

Definition	Events	Matched	Coverage	Trajectories	Activity-maximum $q_{D^{2}}$ difference	Onset $q_{D^{2}}$ difference
Activity-gated	88	60	68.2%	18	-0.044 [-0.056, -0.031]	0.040 [0.018, 0.062]
Activity-ungated	106	64	60.4%	18	-0.042 [-0.053, -0.030]	0.026 [0.003, 0.049]

Appendix

Table 17: Exploratory duration- and selector-matched quiet-window control. Differences are event minus matched quiet values; brackets are condition-stratified trajectory-bootstrap 95% intervals. Before the post-test spatial contrasts were computed, the adequacy target was set to at least 80% event coverage with all 18 trajectories represented.

Figure 4: Illustrative matched pair under the activity-gated definition at

\phi=0.85

and

\mu=0.3

. The pair was selected before map inspection as the activity-gated match whose event-minus-quiet participation difference lay closest to the median. (a) Event onset. (b) Within-release activity maximum. (c) Matched quiet-window activity maximum. Colors show

ND_{i}^{2}/\sum_{j}D_{j}^{2}

on one logarithmic scale. Smaller

q_{D^{2}}

denotes stronger concentration. The maps illustrate the exploratory matched subset and do not enter its estimates

Comparison	Pooled-window AP difference	Equal-trajectory mean $\pm$ SD	MDE
Activity-gated compact $H=4$ $-$ control	0.025	0.036 $\pm$ 0.171	0.120
Activity-ungated compact $H=4$ $-$ control	-0.032	-0.009 $\pm$ 0.149	0.105
Activity-gated compact $H=8$ $-$ graph	-0.010	0.017 $\pm$ 0.183	0.128
Activity-ungated compact $H=8$ $-$ graph	0.009	0.061 $\pm$ 0.108	0.076

Appendix

Table 18: Estimand audit for descriptive two-sided MDE calculations (

\alpha=0.05

, 80% power,

n=18

trajectory differences).

Figure 5: Original held-out pipeline comparisons retained as secondary analyses. (a) Event counts and exact onset overlap. (b) Pooled AP minus prevalence for the four-frame compact and stress–pressure–ratio models. (c) Pooled AP minus prevalence for the history-matched eight-frame compact and graph models. (d) Observation-time-matched AP differences. Error bars are 95% intervals from condition-stratified trajectory resampling. These pooled-window comparisons use the original endpoint rule and are not the lookback-aware mean per-trajectory analysis in the main text

Definition	Seed	Best epoch	Best validation AP	Final validation AP	Loss, first $\to$ final
Activity-gated	3407	27	0.381	0.372	0.655 $\to$ 0.074
Activity-gated	7349	19	0.358	0.339	0.662 $\to$ 0.057
Activity-gated	9109	22	0.393	0.377	0.656 $\to$ 0.056
Activity-ungated	3407	10	0.445	0.345	0.659 $\to$ 0.055
Activity-ungated	7349	13	0.382	0.315	0.660 $\to$ 0.054
Activity-ungated	9109	16	0.413	0.371	0.660 $\to$ 0.052

Appendix

Table 19: Saved graph-model training histories. Validation AP selected the checkpoint; “loss” gives the first and final training loss over 30 epochs.

		Activity-gated					Activity-ungated
$\phi$	$\mu$	Windows	Events	AP	Prev.	$\gamma$ AP	Windows	Events	AP	Prev.	$\gamma$ AP
0.84	0.1	594	16	0.399	0.350	0.556	553	20	0.427	0.438	0.568
0.84	0.3	662	10	0.517	0.208	0.298	631	12	0.385	0.242	0.302
0.84	0.5	648	11	0.314	0.235	0.425	639	12	0.308	0.252	0.463
0.85	0.1	643	12	0.398	0.261	0.425	609	17	0.595	0.343	0.641
0.85	0.3	714	9	0.298	0.172	0.389	683	10	0.395	0.202	0.377

Appendix

Table 20: Condition-level results for the

H=4

compact ensemble. Each row contains two trajectories. These values are descriptive; no subgroup inference is made.

Figure 6: Event-aligned compact observables in the development set: 162 cleanly alignable activity-gated slips out of 182 from 36 trajectories. Each event has a complete

[-15,+5]

frame window, is more than 20 frames after the preceding onset, and more than five frames before the next onset. Values are normalized with development-training medians and interquartile ranges. Events are averaged first within trajectory, and bands show 95% trajectory-bootstrap intervals. The dashed line denotes onset. Panel (a) shows oriented shear stress, so an accepted onset is a local maximum.

\langle D^{2}_{\min}\rangle_{i}

is the all-particle mean,

f^{n}_{\max}

is evaluated over all contacts, and

S_{\max}

is the largest-component particle count at

\alpha_{f}=1.2

. The activity-gated event selection conditions the post-onset non-affine profile; it is not evidence of a pre-onset precursor. These profiles are descriptive and do not enter the independent tests

Difference	Development estimate (95% interval)	Independent-test estimate (95% interval)
Activity-gated compact $H=4$ $-$ control	0.020 [-0.064, 0.119]	0.025 [-0.016, 0.072]
Activity-ungated compact $H=4$ $-$ control	0.080 [-0.014, 0.154]	-0.032 [-0.071, 0.016]
Activity-gated compact $H=8$ $-$ graph	0.115 [-0.021, 0.251]	-0.010 [-0.046, 0.028]
Activity-ungated compact $H=8$ $-$ graph	0.160 [0.015, 0.249]	0.009 [-0.013, 0.031]

Appendix

Table 21: Selected development-stage paired AP differences versus their independent-test estimates. Development intervals used nine trajectory clusters and are exploratory.

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