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

Problem

• Dense granular materials release stored load through irreversible rearrangements, making prediction distinctively about ranking near-slip windows and demonstrating information beyond loading-protocol position [3].Average precision above positive-window prevalence alone does not show information beyond loading history; a material-state precursor requires comparison with history variables at the same observation times.Source: §1 · p. 1

Motivation and prior work

• Particle motion and force-network structure motivate state-based ranking, but prior work did not establish improvement over loading history for system-level release timing in homogeneous strain-controlled shear.Relevant observables include D2 min and network topology, with prior studies linking force-network change and microslip to stick–slip [6] and topology to spatial organization [8] [9].Source: §1 · p. 1

Motivation and baselines

• Loading history is a demanding baseline: cumulative strain tracks startup evolution, while causal elapsed strain requires past observations to identify the preceding release and does not establish a renewal mechanism.The event detector is replayed on each observed trajectory prefix; catalog-age ranking can arise from forecast horizon, event exclusion, and right censoring even with randomized event onsets.Source: §1 · p. 2

Evaluation challenges

• Positive windows could occur sooner after slips than negative windows, so sampling asymmetry, trajectory weighting, and event coverage must be audited separately from pooled window performance.The study reports pooled performance separately from mean per-trajectory AP and audits event coverage apart from window counts.Source: §1 · p. 2

Study design

• The held-out study used frozen compact and graph models developed on 36 trajectories, then tested without retraining on 18 new trajectories under two nested stress-drop definitions.The post-test audit applied a shared previous-event exclusion and a stricter lookback-aware recovery rule; AP was averaged over trajectories with condition-stratified resampling.Source: §1 · p. 2

Results

• The compact model ranks near-slip windows above the mean positive-window fraction and circular-phase randomization, while cumulative strain and causal elapsed strain rank them more accurately.The lookback-aware subset preferentially retains events with longer preceding intervals, and final-catalog age ranking remains within a count-conditioned window-geometry null; these post-test analyses are descriptive sensitivities.Source: §1 · p. 2

2.1 Granular system and shear protocol

• 512 disks were sheared across 54 trajectories, with 36 development trajectories and 18 held-out trajectories spanning three densities and three friction coefficients.Production ran to strain 0.50, saving 500 frames per trajectory; model parameters and normalization constants were fitted only on the development cohort.Source: §2.1 · pp. 2–3

2.2 Non-affine motion and slip definitions

• The fixed slip catalogs contained 88 activity-gated and 106 activity-ungated onsets, but both were retrospective because acceptance used centered smoothing, recovery, and candidate merging.The ungated definition removed only the event-interval activity criterion; both catalogs retained the gated onsets at the same frames.Source: §2.2 · p. 3

2.3 Forecast target and window selection

• Positive windows could re-enter nine saved frames after an onset, versus offset 25 for negatives; post-test common-exclusion and lookback-aware rules addressed this asymmetry without recomputing frozen labels.The common-exclusion rule applied the same onset buffer to positive and negative windows; the lookback-aware rule further required the complete D2 min lookback to follow trough and recovery.Source: §2.3 · p. 4, Figure 1 · p. 5

2.4 State models and loading-history scores

• The compact model classified four-frame sequences of six scalar observables, while cumulative- and elapsed-strain scores provided unfitted loading-history comparisons.The compact classifier used two 64-unit ReLU layers and averaged three fixed-seed probabilities; prefix-replay elapsed strain was added after outcome inspection as a causal-availability sensitivity.Source: §2.4 · p. 4

2.5 Evaluation summaries and uncertainty

• Evaluation used mean per-trajectory average precision with equal weight across 18 trajectories and exact condition-stratified resampling, whose intervals did not represent new preparations or training-seed variation.The post-test trajectory-balanced analysis and other sensitivity analyses were descriptive; intervals were not adjusted for multiplicity.Source: §2.5 · pp. 5–6

2.6 Post-test event-selection and temporal-null audit

• Temporal audits compared compact-score alignment with circular-phase shifts and calibrated catalog-age geometry using count-conditioned surrogates, not causal detectors or renewal-process tests.The circular-shift control used 100,000 null cohorts; the separate geometry calibration used 2,000 surrogate cohorts preserving event counts and duration multisets.Source: §2.6 · p. 6

3.1 The original window set contains a post-event sampling asymmetry

• A one-sided post-event region contained only positive windows, while lookback-aware filtering removed asymmetric channels and retained a subset of events with longer preceding intervals.At offsets 9–24 saved frames after the previous onset, 231 activity-gated and 336 activity-ungated windows were all positive. Lookback-aware filtering retained 78.4% and 69.8% of events and shifted median preceding intervals from 48.0 to 78.0 frames and from 39.0 to 72.5 frames, respectively.Source: §3.1 · p. 7

3.3 Compact alignment survives phase randomization, but catalog age is geometry-compatible

• Observed compact AP of 0.310/0.325 exceeded circular-shift null means of 0.211/0.239, supporting temporal alignment beyond that randomization but not eliminating loading-stage information.Observed-minus-null-mean differences were 0.100 and 0.086; plus-one upper-tail probabilities were 0.00005 and 0.00051. The positive-window fraction remains the standard random-ranking baseline.Source: §3.3 · p. 8
• Actual catalog-age AP of 0.527/0.504 fell within count-conditioned hard-core null ranges of [0.383, 0.538] and [0.428, 0.572], constraining elapsed-ranking interpretation without establishing equivalence.The activity-gated value was in the upper part of its null distribution (one-sided upper-tail probability 0.048). This final-catalog score differs from the causal prefix replay used in the main comparison.Source: §3.3 · p. 8

3.4 Pooled AP also rewards differences between trajectories

• Pooled and mean per-trajectory AP differed substantially, and the loading-history variable ordering depended on the summary, so pooled results do not establish a general loading-history advantage.On lookback-aware windows, pooled versus mean per-trajectory compact AP was 0.245/0.241 versus 0.310/0.325. On original windows, pooled cumulative-strain AP slightly exceeded elapsed-strain AP, whereas mean per-trajectory AP gave the opposite ordering; the original pooled compact-minus-cumulative-strain interval included zero for activity-gated events.Source: §3.4 · pp. 8–9

3.2 Equal-trajectory ranking is above the no-skill level but below loading history

• Lookback-aware mean per-trajectory AP was 0.310/0.325 for the compact model versus prevalence 0.173/0.199, while cumulative strain reached 0.398/0.420 and causal elapsed strain 0.534/0.508.Compact-minus-prevalence differences were 0.138 [0.083, 0.192] and 0.126 [0.080, 0.171]; compact-minus-loading-history intervals were below zero. These post-test intervals are descriptive and unadjusted for multiplicity.Source: §3.2 · pp. 7–8, Table 2 · p. 8, Figure 2 · p. 9

3.4 Pooled AP also rewards differences between trajectories

• On lookback-aware windows, compact probability correlated with cumulative strain, while adding the compact score to a time-plus-condition stacker established no increment beyond loading time and condition.Mean within-trajectory Spearman correlations were 0.538 [0.472, 0.603] and 0.593 [0.529, 0.657]. The separate post-test stacker AP changes were -0.003 [-0.029, 0.011] and -0.008 [-0.026, 0.013]; this was not an equivalence test.Source: §3.4 · p. 10

3.5 Secondary model comparisons

• Secondary comparisons detected no increment from the compact model over the bulk-mechanical control, while cumulative strain exceeded compact and graph scores at H = 8.The compact–control intervals established neither a difference nor equivalence. The graph result applies only to the tested pipeline, which omitted bulk stress and pressure, used one contact threshold, and changed the classifier as well as the representation.Source: §3.5 · p. 10
• Release localization was exploratory rather than a precursor result because matched comparisons covered only 68.2%/60.4% of catalogs and concentration appeared after onset.Both coverage values were below the fixed 80% adequacy target.Source: §3.5 · p. 10

4 Discussion

• The compact score exceeded prevalence and circular-phase controls, but ranked below cumulative strain and causal prefix-replay elapsed strain in equal-trajectory held-out comparisons.The comparisons used shared post-event exclusion and recovery-aware non-affine lookback; because they followed outcome inspection, they are descriptive sensitivities rather than new confirmation.Source: §4 · p. 10

4.1 Loading history is a physical baseline during startup

• Cumulative strain and causal elapsed strain ranked above the compact score, while adding the compact score to loading time and condition yielded no detected AP increment.Final-catalog age ranking remained compatible with a count-conditioned geometry null; frame-interval coefficients of variation of 0.753 and 0.837 do not test memorylessness, and equivalence to the null was not established.Source: §4.1 · pp. 10–11

4.2 Window selection changes the comparison

• The lookback-aware rule retained 78.4% and 69.8% of represented events, while compact AP remained above prevalence and its differences from both history scores became more negative.It shifted the event-weighted median preceding interval upward under both definitions, selecting a subset enriched in more widely separated events; this post-inspection sensitivity does not cover every accepted release.Source: §4.2 · p. 11

4.3 What the model comparison does and does not imply

• The six compact channels are not an exhaustive material state, and the graph comparison cannot establish a representation ablation.The channels omit fabric tensors, contact anisotropy, local free volume, soft spots, and fluidity-like fields; related work considers granular-fluidity models and structural descriptors [25] [26] [27]. The tested graph excludes weak contacts and explicit stress and pressure, and changes representation and classifier together; weighted graphs and alternative readouts remain plausible [28]. Multiple controlled continuations from a fixed configuration could estimate conditional slip probability and separate state insufficiency from single-continuation variability.Source: §4.3 · pp. 11–12

4.4 Scope and limitations

• The findings apply to startup shear at one system size, shear rate, and contact/drag parameter set, not to a statistically stationary or general regime.Only 18 independent test units, two preparations per condition cell, and a fixed three-seed ensemble constrain inference; the nulls do not identify a hazard law or microscopic mechanism. Forecasting was retrospective ranking, not calibration or online alarm control, and release-localization analysis did not meet its event-coverage target.Source: §4.4 · p. 12

• On 18 held-out trajectories, the compact state model ranked near-slip windows above the mean positive-window fraction under both nested event definitions.On post-test lookback-aware windows, it also exceeded the within-trajectory circular-phase control.Source: §5 · p. 12
• Cumulative strain and causal prefix-replay elapsed strain ranked windows more accurately, while the lookback-aware correction retained 78.4% and 69.8% of represented events and selected longer preceding intervals.Final-catalog age AP fell within count-conditioned geometry-null ranges under both definitions, so high elapsed ranking alone does not establish a renewal mechanism.Source: §5 · p. 12

Data and code availability

• The compact data and analysis code supporting reported tables and figures are archived in repository.The archive includes frozen prediction arrays and event and quality summaries.Source: Data and code availability · p. 13
• Repository includes configurations, endpoint-eligibility masks, per-trajectory statistics, resampling and temporal-null results, protocols, final figure assets, and reproduction code.Source: Data and code availability · p. 13
• Larger particle-resolved HDF5 trajectories and model checkpoint binaries are available from the corresponding author on reasonable request.Source: Data and code availability · p. 13

Use of AI-assisted tools

• Generative-AI tools assisted with language drafting and code review.Source: Use of AI-assisted tools · p. 13
• The authors independently verified the source code, calculations, citations, figures, and interpretation and retain responsibility for the manuscript.Source: Use of AI-assisted tools · p. 13
• No generative-AI image was used.Source: Use of AI-assisted tools · p. 13

Competing interests

• The authors declare no competing interests.Source: Competing interests · p. 13

A.1 Cohorts and analysis chronology

• The original 36-trajectory cohort was split into 18 training, nine validation, and nine nominal-test trajectories, whose inspected results are reported as development-stage estimates.The nominal test subset was inspected during iterative refinement of the event audit, controls, and history-matched comparison.Source: §A.1 · p. 16
• Before generating new trajectories, the team fixed simulation settings, event definitions, endpoint construction, normalizers, ensembles, checkpoints, metrics, and sampling; held-out evaluation used no retraining or refitting.The separately reported loading-and-event-history MLP was a development-stage model, not a ninth confirmation-family member.Source: §A.1 · p. 16
• All 18 held-out trajectories passed pre-existing numerical criteria, and none was selected or replaced based on event count, prevalence, average precision, or model ordering.Two new preparations were generated in each (ϕ, µ) cell.Source: §A.1 · p. 16

A.2 Numerical protocol, preprocessing, and quality checks

• Compact-channel normalizers were fitted on complete development-training trajectories and held fixed for independent testing; no optimization used independent-test data.The passage also specifies fixed graph normalization values and development-training-only class weights and training settings.Source: §A.2 · p. 17

B.1 Event-definition comparison

• The independent test reproduced the development-set event-overlap pattern, with event counts not used as a retention criterion.Table 4 reports independent-test event counts and onset overlap; the passage gives development counts of 182 activity-gated and 209 activity-ungated onsets, with 181 exact matches and all 182 matching within tolerance.Source: Table 4 · p. 18, §B.1 · p. 18

B.2 Forecast-window sets

• Table 5 scopes held-out window, positive-window, and distinct-event counts by label definition and history; differences reflect history availability, event lists, and exclusion regions.The finite-tail rule preserves accepted events, and the event column counts events contributing at least one positive observation time at the specified history.Source: Table 5 · p. 18, §B.2 · p. 18

B.3 Seed-level and ensemble results

• Seed variation measures optimization variability, while the trajectory bootstrap measures preparation-sampling uncertainty; these are not combined into one interval.Ensemble AP follows averaging seed probabilities and need not equal mean seed AP.Source: §B.3 · pp. 18–19, Table 6 · p. 19

B.4 Exact-observation-time paired comparisons

• Exact (trajectory, tend) keys were aligned before paired resampling, using the same within-condition trajectory draws for both score sets; intervals including zero do not establish equivalence.Table 7 gives the complete paired estimates.Source: §B.4 · p. 19, Table 7 · p. 19

C Sensitivity analyses

• Unless stated otherwise, the sensitivity analyses were specified after inspection of the original held-out summary and should be read as sensitivities.Source: C Sensitivity analyses · p. 20

C.1 Endpoint eligibility and pooled-window diagnostics

• The original endpoint set admitted positive windows nine frames after a previous onset, despite negative endpoints having minimum elapsed offsets of 25 frames under both event definitions.Removing the 231 and 336 positive-only-region windows reduced recency AP from 0.434 and 0.523 to 0.194 and 0.227; this retrospective catalog diagnostic is only a post-test geometry check.Source: §C.1 · p. 20

C.2 Window-geometry and temporal-null audit

• The compact score exceeded the phase-null 97.5th percentile under both event definitions, while catalog-age AP remained within onset-null central ranges (upper-tail probabilities 0.048 and 0.479).The first probability is near the upper tail, not evidence of equivalence; neither comparison tests a Poisson process, memorylessness, or a specific renewal law.Source: §C.2 · p. 21, Table 10 · p. 21

C.3 Temporal activity aggregation replay

• Replaying the stored activity-gated detector reproduced all 88 onsets; changing temporal aggregation yielded 71, 68, or 62 events under the tested alternatives.The alternatives were inclusive mean, mean with the trough excluded, and inclusive median, respectively; none replaced the frozen events or entered model selection.Source: §C.3 · p. 21

C.4 Dependence of label variants and development-cohort reversal

• AP-excess correlation between activity-gated and activity-ungated label results was 0.874 across shared draws and 0.875 under probability weighting, so they are not independent replications.The development-test subset motivating cohort-reversal discussion had already been viewed during model development and is descriptive; Table 11 reports its ordering and nested-stacker increments.Source: §C.4 · p. 21, Table 11 · p. 21

C.5 Loading-progress, post-yield, and threshold results

• 0.599 [0.563, 0.637] and 0.599 [0.568, 0.631] all-window balanced accuracy fell post-yield to 0.517 [0.491, 0.543] and 0.510 [0.471, 0.548] without recalibration.The validation-selected thresholds were 0.069824 and 0.073074, respectively.Source: §C.5 · p. 22
• 0.021 [-0.004, 0.049] and 0.012 [-0.014, 0.041] were AP increments from adding the fixed compact score to loading-time stackers for activity-gated and ungated labels; both intervals included zero.With condition indicators added, the corresponding increments were -0.003 [-0.029, 0.011] and -0.008 [-0.026, 0.013], and every interval included zero.Source: Figure 3 · p. 22, §C.5 · p. 22
• All-endpoint compact-minus-prevalence checks were prespecified, whereas cumulative-strain and post-yield comparisons were introduced after inspection of the frozen summary.Table 12 reports prevalence-referenced and directly paired strain-referenced contrasts, with intervals from 10,000 condition-stratified trajectory draws.Source: Figure 3 · p. 22, Table 12 · p. 22, §C.5 · p. 22

C.6 Elapsed-strain baselines and exact resampling enumeration

• The causal elapsed-strain score replayed the complete fixed detector on each trajectory prefix, and all full-trajectory replays exactly reproduced stored yield, onset, and trough arrays.Table 13 covers lookback-aware and pre-first-event checks; “post-first” removes endpoints before the prefix replay detects its first event, and intervals are exact 95% intervals from the finite condition-stratified design.Source: §C.6 · p. 22, Table 13 · p. 23
• An earlier retrospective score used a stored onset occurring after the final-catalog trough plus two recovery frames, and was not the causal elapsed score used in the main comparison.The development-trained loading-and-event-history MLP was available only for the activity-gated definition; Table 14 reports retrospective temporal baselines with paired differences on exact common endpoints and intervals from 10,000 condition-stratified trajectory draws.Source: §C.6 · p. 23, Table 14 · p. 23
• All 19,683 compositions of the nine-cell resampling design were evaluated, and Table 15 compares the Monte Carlo percentile interval with the probability-weighted central interval over the exact finite distribution.Negative values in the loading-progress comparisons favor cumulative strain.Source: §C.6 · p. 23, Table 15 · p. 23

C.7 Event-scale and spatial audit

• The most-active-10% activity shares were 0.704/0.605 at activity-gated event/quiet frames and 0.703/0.603 under the activity-ungated definition.Event frames were activity maxima during onset-to-trough release intervals, so this is not a pre-onset forecast.Source: §C.7 · p. 23
• The matched activity-maximum top-10% share differences were 0.069 [0.053, 0.084] and 0.069 [0.052, 0.084].All 18 trajectories contributed, but complete same-trajectory quiet intervals were available for only 68.2% and 60.4% of events, so this is a matchable-event sensitivity rather than a catalog-wide estimate. Quiet starts were within |∆γ| ≤0.035 of onset, were not reused, and the protocol was fixed before concentration values were computed; positive onset and negative activity-maximum participation differences place the selected concentration change within the release.Source: §C.7 · pp. 23–24, Table 17 · p. 24
• At ϕ = 0.84/0.85/0.86, near-threshold-band fractions were 21.6%/7.4%/29.2% for activity-gated events and 18.2%/15.2%/31.0% for activity-ungated events.The band was 0.010 ≤∆σ/P < 0.015; no alternative drop threshold was used for model evaluation.Source: §C.7 · p. 24
• The yield-following drop exceeded the largest accepted activity-gated drop in only 16 of 54 trajectories, while 38 did not; post-yield search excludes the yield peak but not each trajectory’s largest drop.Slip search begins strictly after yield.Source: §C.7 · p. 24
• In the 15 frames before yield, activity-gated endpoints numbered 264 (69 positive, 195 negative) and activity-ungated endpoints numbered 250 (69 positive, 181 negative).In the retained post-yield subset, 58 gated and 54 ungated four-frame histories still included the yield frame; wider backward-D2 minimum support could cross yield for 169 and 165 endpoints, respectively, making this a post-yield endpoint sensitivity rather than a fully post-yield state-history estimand.Source: §C.7 · p. 24

C.8 Resolution of pipeline comparisons

• Pooled-window AP differences and equal-trajectory means answer different questions; Table 18’s MDE calculation applies to the latter, while original-endpoint pooled comparisons are secondary analyses.The MDE is not used to relabel a pooled comparison as equivalent.Source: §C.8 · p. 24, Table 18 · p. 25, Figure 5 · p. 26

C.9 Graph training history

• 55,235 parameters in the graph network exceeded the compact model’s 7,491 parameters.The compact model was history-matched.Source: §C.9 · p. 27
• No additional retraining was performed.Source: §C.9 · p. 27
• Validation AP selected the checkpoint in the saved graph-model histories.Source: Table 19 · p. 27
• The reported loss values are the first and final training losses over 30 epochs.Source: Table 19 · p. 27

D.1 Descriptive condition-level results

• Table 20 reports H = 4 compact-ensemble results across nine simulated conditions, with two trajectories per row and no subgroup inference.The passage identifies the table's scope but supplies no cell values to report.Source: Table 20 · p. 27, §D.1 · p. 27

D.2 Interpretation boundaries

• The 18 trajectories—not their thousands of windows—are the independent test units; exact enumeration changes resampling integration, not sample size.Source: §D.2 · p. 27
• AP measures retrospective ranking; calibration, prospective false-alarm policy, stiffness and drag sensitivity, finite-size scaling, rate transfer, and experimental transfer remain untested.Source: §D.2 · p. 27
• The event-definition comparison removes only the D2 min activity criterion, while event concentration at the within-release activity maximum does not establish a pre-onset spatial precursor.Other label choices were not tested; graph-versus-compact and control comparisons do not test the physical relevance of excluded variables.Source: §D.2 · p. 27
• The duration- and selector-matched quiet-window control covered 68.2% and 60.4% of the catalogs, supporting only a matchable-event sensitivity.Source: §D.2 · p. 27

D.3 Development estimates retained for transparency

• The development set informed analysis construction, so its nominal test subset is exploratory, and its activity-ungated compact–graph ordering was not reproduced.Source: §D.3 · p. 28
• 162 of 182 activity-gated slips had cleanly alignable development-set windows, but the event-aligned profiles are descriptive and do not enter independent tests.Activity-gated selection conditions the post-onset non-affine profile, so it is not evidence of a pre-onset precursor.Source: Figure 6 · p. 28
• Table 21 compares all four paired development estimates with independent-test estimates; development intervals use nine trajectory clusters and are exploratory.Source: Table 21 · p. 28, §D.3 · p. 28