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EMAN: Optimization-Driven Capacity Growth through Path Emergence in Multi-Task Learning

Chenlei Fang, Jingchen Li, Hongzong LI, Qingyao Li, Yixuan Zhang, Huarui Wu, Haobin Shi, Chunjiang Zhao

arXiv:2608.16930v1cs.LGcs.AI

TL;DR

Multi-task networks must decide when fixed shared capacity no longer suffices without paying for independent paths from the outset. EMAN uses optimization evidence to certify persistent demand before materializing a second path, achieving competitive final performance with reduced path-training computation across controlled and natural-image experiments.

  • Problem

    Fixed one-path sharing can lack capacity, whereas always-dual designs incur two-path computation from the start, motivating selective growth after persistent demand appears.

  • Method

    EMAN begins with exact single-path computation, probes a latent antisymmetric direction, and uses fixed evidence certification to trigger independent path materialization only after persistent evidence.

  • Results

    EMAN demonstrates exact sharing under sufficient rank, capacity recovery under deficits, adaptive release under changing demand, competitive final performance, and reduced path-training computation.

  • Takeaways & Limitations

    Path emergence is most valuable when additional capacity becomes useful after a meaningful shared-training stage and demand persists.

  • Takeaways & Limitations

    A strict persistence rule can miss mild bottlenecks, and both paths remain active after materialization and during inference.

Abstract

from arXiv · show

Existing multi-task learning methods rely on hard sharing, multiple paths or experts, adaptive sharing, and dynamic expansion. However, their capacity changes are usually constrained by predefined structures or triggered by task boundaries and conflict signals. This raises a fundamental question: can a network start from exact single-path computation and grow a new independent path only when persistent optimization evidence appears? We propose the Emergent Modular Atomic Network (EMAN), an optimization-driven framework for exposing an antisymmetric growth direction through latent relative phases without instantiating a second path, and for monitoring multiple decision signals during training to transform local optimization evidence into a structural decision. EMAN materializes two equal-capacity independent paths only after certification. EMAN adaptively allocates shared and task-specific representation capacity to accommodate varying task requirements. Extensive experiments on controlled rank settings, PASCAL-Context, and NYUv2 validate its effectiveness, achieving improved performance at a competitive computational cost.

1 Introduction

The introduction frames a capacity-timing problem in multi-task learning: fixed sharing can bottleneck heterogeneous tasks, whereas always-dual paths incur excess computation. EMAN addresses this by using optimization evidence to grow an independent path only when persistent demand emerges.

  • Motivation: Hard sharing is efficient but fixed capacity can cause negative transfer or bottlenecks under heterogeneous task demands.Soft-sharing, task-interaction, and adaptive-sharing approaches provide greater flexibility.
  • Motivation: Always-dual paths avoid capacity bottlenecks but maintain two-path computation from the start, motivating selective capacity growth.The desired design retains exact sharing while one path is sufficient and adds independent capacity after persistent demand appears.
  • EMAN: EMAN starts with one physical forward path and exact shared computation, then uses a latent relative phase to probe optimization geometry without changing physical capacity.The fixed odd phase interaction exposes a candidate antisymmetric growth direction.
  • EMAN: Optimization Evidence Certification evaluates the candidate direction before triggering zero-sum materialization into two independently trainable paths.The stated evidence includes strength, cross-probe consistency, finite-step descent, and dynamic safety.
  • Mechanism: Near the shared solution, latent phase dynamics can expose an antisymmetric descent direction before physical path coordinates separate.After phase release, the cross term ϕcᵀd induces the antisymmetric force cϕ while the second physical path remains absent.
  • Validation: In task-arrival experiments, EMAN preserves one-path sharing initially and releases after saliency and surface-normal tasks enter training.In static PASCAL-Context, all four tasks begin together and EMAN releases early; across three seeds, final performance is comparable to the always-dual baseline.

2 Related Work

Prior multi-task learning methods address incompatible objectives through hard or soft sharing, task-dependent policies, expert routing, modularity, and expansion. EMAN instead uses phase-structured optimization evidence and transverse geometry to make capacity growth an endogenous structural event.

  • Sharing in multi-task learning: Hard sharing is efficient but can entangle incompatible objectives, motivating softer sharing mechanisms and task-dependent sharing policies.Examples include cross-stitch, sluice, attention, multi-scale interaction, and AdaShare methods.
  • Experts, modularity, and expansion: Sparse mixture-of-experts and multi-gate models route inputs to specialized computation, while modularity provides room for specialization shaped by data structure and optimization.These approaches use routers or modular structures rather than starting from exact single-path computation.
  • Experts, modularity, and expansion: Phase-structured evidence determines whether a second independent path should emerge, making capacity growth endogenous to optimization.This frames structural expansion as a decision driven by optimization evidence rather than a preset event.
  • Symmetry and local curvature: Symmetry-breaking analysis supplies symmetric and antisymmetric coordinates, while Hessian-vector products characterize local curvature near the exactly shared state.In EMAN, these tools describe the transverse geometry of the shared branch, with relative phase exposing a growth direction.

3 EMAN: Optimization-Driven Path Emergence

EMAN begins with exact single-path computation and uses latent relative-phase probing to expose an antisymmetric growth direction without adding physical capacity. Optimization Evidence Certification converts persistent optimization evidence into irreversible materialization of two equal-capacity, independently trainable paths.

  • Phase-first probing: EMAN starts with one physical path and uses latent relative phases to expose a candidate antisymmetric direction without instantiating a second path.Logical path coordinates remain exactly tied, so parameter count, memory cost, and executed path computation match a shared model.
  • Optimization Evidence Certification: OEC certifies path growth using direction strength, cross-probe consistency, finite-step descent, dynamic safety, and temporal persistence.The criteria distinguish numerical noise, subset-specific directions, weak local utility, unstable probes, and transient events.
  • Certification and transition: Persistent and numerically safe evidence triggers a zero-sum split, while insufficient evidence retains exact shared computation.Certification requires instantaneous criteria to pass for a fixed number of consecutive probe epochs before the structural transition.
  • Path materialization: After certification, EMAN materializes two equal-capacity paths through opposite perturbations and optimizes them independently.The initialized paths preserve the mother tensor’s arithmetic mean, copy optimizer state symmetrically, and then become independently trainable.

4 Experiments

Experiments evaluate EMAN under controlled rank constraints and natural-image multi-task benchmarks, linking optimization evidence to certification, released representations, task performance, and computation. EMAN releases and materializes capacity when persistent evidence satisfies its criterion, while improving selected benchmark aggregates at competitive computational cost.

  • Experimental setup: The controlled benchmark uses six latent factors, three regression tasks, joint ranks of 3, 4, and 6, and physical paths of rank 3.NYUv2 includes semantic segmentation, depth estimation, and surface-normal prediction; standard capacity is 64 channels and Capacity-Stress uses 8 channels, or 0.125× path width.
  • Controlled capacity and representations: In the Medium rank-4 regime, EMAN remains shared and matches Hard-sharing at 0.7086, while PhaseOnly reaches 0.7081 without satisfying frozen persistence.Under rank-6 demand, released paths have principal angles of 58.3◦to 72.0◦, and concatenation recovers all six latent factors with R2 ≥0.999892; permutation control reduces mean recovery to 0.0243.
  • Natural-image benchmarks: On standard-capacity NYUv2, EMAN has the best mean mIoU, depth error, and G, reaching G = 0.0078 ± 0.0152, while Recon leads normal error and joint loss.Under Capacity-Stress, EMAN achieves the highest G at 0.0306±0.0049, compared with 0.0237±0.0086 for MeanPaths and 0.0115 ± 0.0048 for PhaseOnly.
  • Cross-dataset evaluation: On PASCAL-Context, EMAN releases at the epoch-21 demand change in all three seeds, reaches G = 0.0043±0.0042, and uses 75% of always-dual MeanPaths’ path-training compute.MeanPaths reaches 0.0027±0.0018 and Hard-sharing reaches 0; earlier arrival moves the same frozen release from epoch 21 to epoch 11, with EMAN reaching G = 0.0037.

5 Discussion and Limitations

EMAN’s central result is that capacity should be added when optimization evidence becomes persistent, separating structural diagnosis from structural commitment. Its certificate favors reliable expansion, while revealing sensitivity and post-release computation as limitations.

  • Capacity timing and certification: EMAN distinguishes a useful local growth direction from a persistent capacity deficit before committing to new parameters.Phase probing identifies a possible growth direction, while OEC evaluates strength, consistency, descent, safety, and persistence.
  • Controlled and natural-image evidence: One path remains sufficient at joint rank three, whereas joint rank six creates a deficit that materialization recovers through complementary latent subspaces.The released paths occupy complementary latent subspaces rather than duplicating the shared representation.
  • Controlled and natural-image evidence: Delayed demand produces delayed release, while earlier demand shifts the same frozen decision from epoch 21 to epoch 11.Natural-image experiments support the same timing principle observed in controlled rank settings.
  • Limitations: A strict persistence rule avoids premature growth but can miss a mild bottleneck near the capacity boundary.The Medium regime exposes the trade-off between reliable evidence and aggressive expansion.
  • Limitations: After materialization, both paths remain active during path training and inference, leaving post-release computation as a separate limitation.Reported savings measure path-training evaluations before release; future work may reduce post-release cost through compression.

6 Conclusion

EMAN starts with exact single-path computation and uses phase-first optimization evidence to decide when independent capacity should emerge. Its experiments support adaptive capacity growth under deficits and changing task demands, positioning expansion as part of optimization itself.

  • 6 Conclusion: EMAN starts from exact single-path computation and uses phase-first evidence to decide whether independent capacity should emerge.OEC converts a local antisymmetric direction into a persistent structural decision.
  • 6 Conclusion: The zero-sum transition preserves accumulated optimization state when independent capacity is released.This links local optimization evidence to the resulting structural expansion.
  • 6 Conclusion: Controlled experiments show exact sharing with sufficient rank, capacity growth under clear deficits, and complementary representation recovery.The PhaseOnly comparison shows that direction discovery alone does not provide missing representation capacity.
  • 6 Conclusion: NYUv2 and PASCAL-Context demonstrate adaptive capacity release under changing task demand with competitive final performance.These results support capacity growth as part of optimization and provide a route from fixed hard sharing to evidence-driven structural expansion.

A Extended Theory and Method Details … Theory Claim Boundary

EMAN distinguishes latent logical coordinates from physical paths, using phase exposure and conservative optimization evidence to decide when to materialize independent capacity. Its theory establishes local mechanisms and continuity properties while explicitly limiting claims about global convergence and universal growth.

  • Physical Paths, Logical Coordinates, and Structural States: Before release, EMAN evaluates one physical path while logical duplication enables symmetry probes without a second encoder, doubled path memory, or independent representation parameters.Exact-shared and phase-released states use one encoder evaluation; independent representation capacity appears only after path release.
  • State invariants and accounting: The monotone Exact-Shared → Phase-Released → Path-Released sequence keeps failed certificates unchanged and forbids fallback, validation-driven, or task-ID-triggered splits.Phase release changes interaction geometry but cannot add an independently trainable encoder; path release is the only stage creating independent representation capacity.
  • Exchange Symmetry and Coordinate Decomposition: Exchange symmetry makes tied path labels equivalent, so antisymmetric probe directions have sign ambiguity and require absolute-cosine consistency across probes.Phase release activates a mixed phase–path block whose coupled derivative can expose an antisymmetric force despite zero first-order antisymmetric gradient at the tied state.
  • Norm Calibration Preserves the Local Geometry: NormCal stabilizes forward amplitude and preserves local coefficient geometry, but it neither creates an antisymmetric direction nor certifies growth.Stop-gradient prevents derivatives through RMS ratios that could rotate or attenuate the measured direction.
  • Local Phase–Path Expansion: Locally, nonzero phase probes expose opposite orientations of one separation axis, while phase–path coupling can produce a descending mode even when restricted path and phase blocks are stable.The result is a sufficient local mechanism, not a claim that phase is necessary, global convergence is guaranteed, or specialization persists after release.
  • Formal Optimization Evidence Certification: OEC tests finite-step descent under frozen and dynamic calibration, direction strength, cross-probe consistency, safety, and temporal persistence before materialization.Every frozen component must pass throughout the persistence window; failure causes no structural action and can yield false negatives.
  • End-to-end decision sequence: The implementation releases phase first, probes deterministic population partitions, materializes two paths only after certification, and copies parameters and optimizer state symmetrically.Zero-sum splitting preserves the mother parameter and cancels first-order change under symmetric fusion, establishing local rather than exact function continuity.
  • Deep-Network Local Bridge: The local feature-to-parameter bridge and numerical checks support the mechanism: restricted blocks can have minimum eigenvalue 2.0 while the coupled system has eigenvalue −1.1231.These results remain checkpoint-local; the theory supports local sufficiency, whereas experiments test consequences without establishing universal growth, global convergence, or persistent complementarity.

B Extended Experimental Results · Reporting Protocol and Evidence Organization · Implementation and Decision Settings

The extended experiments organize evidence by the question each protocol can answer, while reporting preserves native task metrics alongside directional aggregates. Implementation settings are transcribed from archived files and distinguish standard OEC requirements from Capacity-Stress population settings.

  • Reporting Protocol and Evidence Organization: The directional aggregate uses positive signs for higher-is-better metrics and negative signs for lower-is-better metrics, while all native task metrics remain visible.The aggregate convention is defined by metric direction rather than by a universal improvement sign.
  • Reporting Protocol and Evidence Organization: Table 7 maps each evaluation protocol to the question it can answer, preventing findings from one protocol from answering another protocol’s question.Controlled benchmarks isolate known capacity, natural-image experiments test external task behavior, Capacity-Stress narrows paths, and task-arrival protocols separate periods before and after additional demand.
  • Reporting Protocol and Evidence Organization: Controlled benchmarks isolate capacity using a known joint target rank, whereas standard-capacity natural-image experiments test task behavior without guaranteeing a capacity bottleneck.Capacity-Stress deliberately narrows the path and records the full population certificate.
  • Reporting Protocol and Evidence Organization: For G, Hard-sharing is the per-seed reference with value zero by definition; positive G indicates directional improvement and negative G indicates directional decrease.Task-level columns are reported alongside G because averages can conceal compensating improvements and degradations.
  • Implementation and Decision Settings: All implementation values are transcribed from archived configuration or implementation files, and missing fields are not inferred.This reporting rule applies to the implementation and decision settings summarized below.
  • Implementation and Decision Settings: Table 10 separates the standard OEC contract from population settings used only by Capacity-Stress, while Table 11 records baseline path configurations.Baselines are categorized by whether they begin shared, remain shared, or expose two trainable paths.

Rank-Controlled Results for Every Seed … Detailed Representation Geometry

Across controlled-rank experiments, EMAN preserves exact sharing when one path suffices and materializes independent capacity when persistent rank demand requires it. Diagnostic and geometry analyses associate the gains with complementary released subspaces that jointly recover latent factors, while documenting certificate and archive boundaries.

  • Rank-Controlled Results for Every Seed: EMAN exactly matches Hard-sharing on every High run, while every Low run releases phase and materializes an independent path, recovering larger macro-R2.Each physical path has rank-three capacity; High has joint rank three, whereas Low has joint rank six.
  • Rank-Controlled Results for Every Seed: Across all three seeds, High remains exact-shared and Low releases phase and path, while Low macro-R2 rises from 0.4639 to 0.7518 and from 0.4823 to 0.8466.The structural decision is stable, but predictive recovery varies by seed.
  • Rank-Controlled Results for Every Seed: PhaseOnly releases latent phase without materializing a path, keeping Low macro-R2 within 0.0008 of Hard-sharing; EMAN’s gains therefore track independent path capacity.Phase freedom alone does not supply the missing rank-six representation or create a task-specific loss trade-off.
  • Rank-Controlled Results for Every Seed: For Medium joint rank four, EMAN and Hard-sharing both obtain 0.7086 macro-R2, while PhaseOnly obtains 0.7081; the frozen certificate retains this as a conservative false negative.The setting lies near the single-path boundary, and no matching three-seed table is archived for these aggregate values.
  • Archived Structural Events and Diagnostic Boundaries: Archived NYUv2 static matched-adapter runs release phase at epoch 1 and path at epoch 2, but lack OEC probe traces and therefore establish performance comparison rather than persistent-evidence decisions.Static PASCAL uses dual-path ratio 39/40 and path-training ratio 79/80 in each archived run.
  • Coalition Diagnostics and Specialization Boundary: The K=2 coalition audit supports complementary released subspaces, not task-wise expert specialization, because the retained Low result has near-zero CSS and assigns all tasks to one path.Pair synergy is reported separately and is not interpreted as path specialization.
  • Detailed Representation Geometry: In the released Low model, neither individual path recovers every latent factor, whereas concatenation recovers each factor with R2 ≥0.999892.The three principal angles are 58.2795°, 59.3178°, and 71.9900°; normalized projector overlap is 0.210805, with conditional gains 0.500812 and 0.497777.

(a) Parameter and protocol matching … NYUv2 Task Arrival at Epoch 21

Across controlled latent-representation and NYUv2 evaluations, EMAN grows independent capacity only when persistent optimization evidence is certified, yielding complementary paths, seed-dependent release, and competitive performance with delayed path allocation.

  • (a) Parameter and protocol matching: Concatenating the two paths recovers every latent factor with minimum factor score 0.999892, whereas neither path alone achieves full recovery.Four canonical correlations are one, with remaining correlations 0.954043 and 0.940091, supporting preservation of all six latent directions.
  • (a) Parameter and protocol matching: The paths are complementary rather than duplicated or task-specialized: permutation control drops mean recovery from 0.999899 to 0.024310, while all tasks receive the same assignment.EMAN’s HPS is 1.2757, and task-wise pair synergies are 2.7079, 3.6472, and 3.8095.
  • NYUv2 Capacity-Stress: Complete Per-Seed Performance: Under Capacity-Stress, EMAN probes a frozen eight-channel protocol with delayed surface-normal demand and releases in Seeds 1 and 3, while Seed 2 remains exactly shared.The protocol uses 80 epochs, full training data with flips, deterministic shards, a 10−3 path-strength floor, and separate persistence rules.
  • NYUv2 Capacity-Stress: Complete Per-Seed Performance: Seeds 1 and 3 pass curvature, safety, persistence, strength, and consistency gates, whereas Seed 2 fails the confidence rule and never enters path probing.Phase releases occur at epochs 32 and 11, followed by path releases at epochs 49 and 19; Seed 2 has no release.
  • NYUv2 Capacity-Stress: Complete Per-Seed Performance: 69.6% of MeanPaths path-training evaluations and 97.7% of its estimated total FLOPs are used by EMAN, while certification raises mean wall-clock to 2.24×.EMAN has the highest mean G under the registered Capacity-Stress protocol; population probes create current systems overhead.
  • Standard-Capacity NYUv2: Seed Dependence and Systems Cost: EMAN’s standard-capacity NYUv2 result is competitive rather than uniformly superior: G is positive in Seed 1, near-zero in Seed 2, and negative in Seed 3.Across task-level plots, EMAN stays close to the leading method without catastrophic task collapse; AdaShare is the unstable baseline.
  • Standard-Capacity NYUv2: Seed Dependence and Systems Cost: EMAN uses 31.44M parameters versus 26.20M for Hard-sharing and 28.82M for MeanPaths, with wall-clock matching Hard-sharing and remaining close to MeanPaths.Its mean peak memory is below AdaShare and Recon, while Recon and AdaShare have higher runtime in this implementation.
  • NYUv2 Task Arrival at Epoch 21: G = 0.0226 ± 0.0081 for EMAN under task arrival at epoch 21, exceeding Scheduled split’s 0.0202±0.0076 and MeanPaths’ 0.0114 ± 0.0121.Scheduled split and EMAN release simultaneously with the same 0.75 path-FLOP ratio, isolating delayed allocation rather than sparse inference.

PASCAL-Context as a Cross-Dataset Boundary

PASCAL-Context is a static cross-dataset scope check in which four tasks are active from epoch 1. It evaluates final multi-task behavior but does not directly test EMAN’s timing advantage from delayed path materialization.

  • Protocol: PASCAL-Context simultaneously activates semantic segmentation, human-part segmentation, saliency, and surface-normal prediction from the beginning.Because all four tasks are present from epoch 1, the protocol is a cross-dataset scope check rather than a delayed-growth setting.
  • Aggregate results: Recon has the highest mean aggregate, G = 0.0158±0.0069, while EMAN has G = −0.0098±0.0007.The aggregate is consistently small and negative for EMAN, while native task metrics remain mixed.
  • Native task results: EMAN records the strongest semantic score and joint loss in Seed 1, the strongest saliency and joint loss in Seed 2, and near-reference semantic and joint-loss values in Seed 3.These mixed native-task results narrow the performance scope of the setting.
  • Aggregate decomposition: The negative G is mainly associated with human-part segmentation and normal prediction, while semantic, saliency, and joint-loss values remain competitive.The aggregate decomposition makes this association explicit.
  • Timing limitation: Since all tasks are present from epoch 1, PASCAL-Context is a demanding test of final multi-task behavior rather than a direct test of EMAN’s timing advantage.There is no pre-demand interval in which delayed materialization can save path training.

PASCAL-Context Task Arrival at Epoch 21 … Evidence hierarchy and interpretation

Across discrete task-arrival protocols, EMAN releases at the demand epoch with competitive performance and little measured timing overhead, while gradual ramps do not show a clear benefit for added capacity. The evidence hierarchy supports conditional capacity allocation and timing adaptation, but not universal superiority, specialization, sparse inference, or general speedups.

  • PASCAL-Context Task Arrival at Epoch 21: EMAN releases at epoch 21 in every seed, with a three-seed mean G of 0.0043 ± 0.0042 versus 0.0027 ± 0.0018 for MeanPaths and 0 for Hard-sharing.Seed 3 has the strongest aggregate and best semantic, parts, and saliency values among the listed methods.
  • PASCAL-Context Task Arrival at Epoch 21: Measured release-run times are 14.2, 15.7, and 16.2 minutes for EMAN versus 14.1, 15.8, and 16.4 minutes for MeanPaths.These hardware-specific values do not establish a general speedup.
  • Earlier Arrival at Epoch 11: With task arrival moved ten epochs earlier to epoch 11, EMAN releases at epoch 11 and obtains G = 0.0037, supporting timing adaptation rather than superiority to always-dual training.MeanPaths has G = 0.0054 and the strongest aggregate, while EMAN has the best semantic, human-part, and saliency scores.
  • Gradual Demand: Ramp from Epochs 11 to 15: In the ramp protocol from epochs 11 to 15, EMAN and Scheduled split release at epoch 11, while Conflict Trigger does not release in the archived runs.Mean G values are 0.0068 ± 0.0022 for Conflict Trigger and −0.0039 ± 0.0053 for EMAN.
  • Gradual Demand: Ramp from Epochs 11 to 15: The ramp evidence does not show a clear benefit for additional path capacity: EMAN and matched Scheduled split have nearly identical means, and Seeds 1 and 2 are negative for both methods.The current phase-first rule is most clearly supported under discrete demand onset, not every temporal demand profile.
  • Replay, Provenance, and Reproducibility Boundaries: Replay audits reproduced recorded metrics for retained task-arrival run families within recorded tolerances, while exact cross-platform bitwise identity was not claimed.Large natural-image checkpoints are not redistributed.
  • Consolidated Evidence Boundary: The consolidated evidence is strongest for structural selection under explicit capacity control and timing adaptation, while natural-image outcomes remain task- and seed-dependent.The experiments do not establish universal wall-clock gains, uniform superiority, task-wise expert specialization, sparse inference, or statistical significance from three seeds.
  • Evidence hierarchy and interpretation: The evidence hierarchy progresses from rank-controlled capacity deficits, through certificate acceptance and complementary representations, to natural-image performance–computation trade-offs under realistic task mixtures.Controlled Low and High regimes isolate conditional capacity allocation from generic multi-path overparameterization, while natural-image protocols evaluate structural decisions rather than treating favorable performance as proof of a certificate event.

Reading non-release and weak-result cases · Metric-by-metric reading guide

The section explains how to interpret non-release runs, weak results, and diagnostic metrics without conflating missing states with zero measurements. It also identifies the conditions under which EMAN’s evidence is strongest and how aggregate and system metrics should be read.

  • Reading non-release and weak-result cases: A non-release run means the structural rule retained a shared state because persistent population evidence failed, not that an experiment or force measurement was missing.Seed 2 under Capacity-Stress failed the phase certificate, so no path-force probe ran and no second path existed.
  • Reading non-release and weak-result cases: The rank-four false negative reflects conservative frozen-persistence behavior near the capacity boundary, narrowing sensitivity claims without contradicting verified rank-six recovery.Static PASCAL-Context’s mixed aggregate is a cross-dataset boundary because all tasks are active from the first epoch and no delayed compute interval exists.
  • Reading non-release and weak-result cases: Conflict Trigger achieves the strongest ramp aggregate without releasing a path, while EMAN and matched Scheduled have nearly identical means at the same release epochs.This result indicates clearer evidence under discrete demand onset and suggests smooth demand may not require additional capacity.
  • Metric-by-metric reading guide: A negative phase-curvature estimate alone does not trigger release; uncertainty, finite-difference, safety, and persistence checks jointly determine whether phase is released.Path-force strength is evaluated only after phase release and the adaptation window, so a missing trace does not mean measured force was zero.
  • Metric-by-metric reading guide: Cross-half and shard cosines measure separation-axis stability, whereas frozen and dynamic descent tests assess objective reduction and dynamic safety verifies registered tolerances.Persistence requires this complete conjunction to survive across probe epochs.
  • Metric-by-metric reading guide: Path distance is a post-release diagnostic whose magnitude depends on release epoch and remaining optimization horizon, not a certificate of success or failure.Representation complementarity is assessed separately using latent-factor recovery, canonical correlation, principal angles, and permutation controls.
  • Metric-by-metric reading guide: The aggregate G summarizes native task directions relative to Hard-sharing but cannot replace task-level metrics, while parameter count, memory, path evaluations, FLOPs, and wall-clock time remain distinct.A positive G may coexist with a weak task, and a negative G may coexist with competitive native metrics.
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