PaperlayerView exact revision on arXiv

Source-linked AI summary

From Redundancy to Minimality: Fixed-Point-Guided Hierarchical Reduction of Learned Piecewise-Linear Dynamics

Hiroto Tamura, Gouhei Tanaka

arXiv:2610.01369v1cs.LGnlin.CD

TL;DR

Trajectory reconstruction alone may not provide a parsimonious representation of nonlinear dynamics, and direct training at minimal nonlinear capacity can be unreliable. The paper develops fixed-point-guided reduction and parent-guided retraining; on 3-scroll Chua, this strategy achieves approximately 71% seed-macro success for high-fidelity minimal realizations versus 20% for direct training at the same final capacity.

  • Problem

    Trajectory reconstruction alone may not provide a parsimonious account of dynamics, while fixed-point analysis alone does not describe transitions between local dynamics.

    Source: §1 · pp. 1–2

  • Method

    The method progressively linearizes selected ReLUs to merge regions without merging distinct fixed-point symbols, then retrains reduced candidates under guidance from parent dynamics.

    Source: §1 · p. 3, §5 · p. 9

  • Results

    Approximately 71% seed-macro success for high-fidelity minimal realization was achieved by learn–reduce–retrain, versus 20% for direct training at the same final nonlinear capacity.

    Source: §1 · p. 3

  • Takeaways & Limitations

    Higher-capacity AL-RNNs can serve as scaffolds for identifying and realizing minimal dynamical representations, with reduction providing a structural scaffold rather than fixing the final transition topology.

    Source: §5 · p. 9

    The approach depends on the parent correctly capturing relevant fixed points, and not every successful parent reduces to both symbolic-complexity and nonlinear-capacity lower bounds.

    Source: §5 · p. 9

Abstract

from arXiv · show

Understanding a nonlinear dynamical system from time series requires not only reproducing its trajectories, but also identifying a simple representation that preserves its essential dynamical structure. Almost-linear recurrent neural networks (AL-RNNs) are piecewise-linear RNNs in which only a subset of units use ReLU nonlinearities, so that nonlinear capacity is explicitly controlled by the number of ReLU units. Their activation patterns define linear regions, represented as symbols, whose observed transitions form a symbolic transition graph. However, directly training AL-RNNs with few ReLU units to realize minimal dynamical representations can be unreliable. We ask whether an AL-RNN with more ReLU units can instead be trained first and systematically reduced to a minimal dynamical representation. We introduce a fixed-point-guided hierarchical reduction procedure that progressively linearizes selected ReLU units, merging neighboring linear regions and graph nodes while preserving distinct symbols containing fixed points (FPs). The resulting reduction tree defines a hierarchy of progressively simpler candidates. Each reduced candidate is initialized from the parent parameters and retrained under guidance from the parent dynamics. We also prove that reproducing $Q$ distinct fixed points requires at least $Q$ FP-containing symbols, providing a certificate of symbol-level minimality when this bound is attained. On the 3-scroll Chua system, direct training with the theoretical minimum of three ReLU units achieves high-fidelity minimal realizations in only 20% of seeds, whereas our learn-reduce-retrain strategy increases the seed-macro success rate to approximately 71% at the same final nonlinear capacity. These results show that redundant nonlinear capacity can serve as a scaffold for discovering and realizing minimal dynamical representations.

1 INTRODUCTION

Trajectory reconstruction alone may not yield a parsimonious, structurally informative account of learned dynamics, and direct training of minimally nonlinear AL-RNNs can be unreliable. The paper uses fixed-point-guided hierarchical reduction and parent-guided retraining to identify and realize simpler representations.

1 INTRODUCTION

  • Parsimonious representations are sought because trajectory reconstruction alone can leave models with different accounts of underlying dynamics, while reduced structural complexity can aid interpretation, analysis, and control.The introduction situates this goal within scientific machine learning.Source: §1 · p. 1
  • Directly training smaller AL-RNNs can be unreliable near the capacity boundary, while independent training across nonlinear capacities does not explicitly relate their regions or transitions.The paper frames structural minimality and successful optimization as distinct problems.Source: §1 · p. 2
  • Reproducing Q distinct fixed points requires at least Q fixed-point symbols, certifying symbol-level minimality when exactly Q symbols preserve all target fixed points.Distinct isolated fixed points require distinct FP symbols under the stated conditions.Source: §1 · pp. 2–3
  • The reduction procedure linearizes selected ReLUs to merge regions and graph nodes, while terminating branches that would merge distinct fixed-point symbols.It enumerates distinct linearization sequences to build progressively coarser representations.Source: §1 · pp. 2–3
  • Reduced candidates are built from higher-capacity parent parameters and retrained under guidance from parent dynamics to test whether graph-derived structures yield faithful autonomous dynamics.This scaffold is intended to help realize compact models that may be difficult to learn directly.Source: §1 · p. 3
  • Approximately 71% seed-macro success for high-fidelity minimal realization was achieved by learn–reduce–retrain, versus 20% for direct training, at the same final nonlinear capacity.The 3-scroll Chua system has five target fixed points and a theoretical minimum of three ReLU units.Source: §1 · p. 3

2 AL-RNNS AND FIXED-POINT STRUCTURE

The paper represents AL-RNN trajectories as sequences of visited linear regions and studies minimality in the number of those regions. It derives a fixed-point-based lower bound and uses it to constrain hierarchical reduction.

2.1 AL-RNNS AND SYMBOLIC DYNAMICS

  • AL-RNNs restrict ReLU nonlinearities to P units, making P an explicit control on nonlinear complexity while the latent dimension remains fixed.The chapter reviews the AL-RNN formulation and symbolic representation before deriving the fixed-point constraints used by reduction.Source: §2.1 · p. 3, §2 · p. 3
  • Visited activation patterns define symbols, and their observed transitions form a directed symbolic graph; symbolic complexity is the number of visited symbols, |Σ|.Minimality here concerns visited linear regions, not the number of nonlinear units P.Source: §2.1 · p. 4

2.2 FIXED-POINT THEORY

  • A symbol is FP or FP-free according to whether its corresponding linear region contains a valid fixed point.The fixed-point count Q refers to distinct target fixed points the learned representation must reproduce.Source: §2.2 · p. 4
  • A faithful representation preserving Q distinct target fixed points requires at least Q FP symbols, assuming each fixed point lies in a visited region.The proposition assumes distinct target fixed points and their corresponding symbols are represented.Source: §2.2 · p. 4
  • If a representation preserving the fixed-point structure visits exactly Q symbols, it is minimal in visited regions, although additional FP-free regions may still organize global flow.The reduction preserves the fixed-point structure while testing which remaining regions can be removed.Source: §2.2 · p. 5
  • The necessary capacity bound P ≥⌈log2 Q⌉ is not sufficient, because enough activation patterns alone do not ensure training realizes the required fixed points or target dynamics.The bound follows because P ReLUs generate at most 2^P activation patterns.Source: §2.2 · p. 5

3 METHODS: FIXED-POINT-GUIDED HIERARCHICAL REDUCTION AND RETRAINING

The method learns a higher-capacity AL-RNN, builds progressively simpler symbolic representations while preserving distinct fixed-point symbols, then retrains selected reduced models from parent parameters. Evaluation tests whether these candidates realize minimal, high-fidelity autonomous dynamics.

3 METHODS: FIXED-POINT-GUIDED HIERARCHICAL REDUCTION AND RETRAINING

  • The pipeline learns a higher-capacity parent, reduces its symbolic structure, and retrains compact models from the parent parameters under parent-dynamics guidance.Graph reduction identifies structurally admissible candidates; retraining tests whether they are realized as actual dynamical systems.Source: §3 · p. 5

3.1 FIXED-POINT-GUIDED GRAPH REDUCTION

  • The graph represents visited activation patterns as nodes, classified as fixed-point or fixed-point-free symbols using the parent model’s recovered fixed-point structure.The procedure does not require ground-truth fixed-point annotations or a prescribed fixed-point count.Source: §3.1 · p. 6
  • Linearizing selected ReLUs removes their switching boundaries and projects symbols into quotient graphs, while forbidding merges between distinct fixed-point symbols.Admissible single-unit linearizations are enumerated into a reduction tree that records the units linearized along each path.Source: §3.1 · p. 6
  • Candidates are characterized by remaining nonlinear capacity and symbolic complexity; Chua experiments select those reaching |ΣD| = 5 and PD = 3.These are the two theoretical lower bounds used for candidate selection.Source: §3.1 · p. 6

3.2 REDUCTION-GUIDED RETRAINING

  • Each selected reduced model starts from parent parameters with chosen ReLUs permanently linearized, then retrains using the autonomous parent’s full latent state as teacher.Periodic state synchronization also uses the autonomous parent trajectory as its reference.Source: §3.2 · pp. 6–7

3.3 EVALUATION

  • For the 3-scroll Chua system, each model uses a 10,000-step autonomous rollout with the first 1,000 steps discarded; success requires visiting all Q=5 admissible fixed points and attaining the fixed-point lower bound in post-transient symbolic realization.The stringent high-fidelity criterion is Smin = 1 and Estsp < 2.0.Source: §3.3 · p. 7

4 EXPERIMENTS AND RESULTS

Experiments on the 3-scroll Chua system compare direct training at minimal capacity with a learn–reduce–retrain pipeline. Reduction often reaches both theoretical lower bounds, and retraining improves high-fidelity minimal-realization success at matched final capacity.

4.1 MINIMAL NONLINEAR CAPACITY IS DIFFICULT TO OPTIMIZE DIRECTLY

  • At P = 3, 6/30 models (20%) achieve high-fidelity minimal realization, while P = 10 models recover the five-fixed-point structure in 30/30 runs.Only 7/30 direct P = 3 models achieve minimal realization; reliability increases as P increases.Source: §4.1 · p. 7

4.3 LEARNING LARGE FIRST PROVIDES A MORE RELIABLE ROUTE TO MINIMAL MODELS

  • Approximately 71% high-fidelity minimal-realization success with reduction-guided retraining versus 20% for direct P = 3 training, a roughly 3.5× increase at the same final capacity.Seed-macro success rates were approximately 73%, 71%, and 71% for FP recovery, minimal realization, and high-fidelity minimal realization, versus 23%, 23%, and 20% for direct P = 3 training.Source: Figure 2 · p. 8, §4.3 · p. 9

4.2 A HIERARCHY OF REDUCED REPRESENTATIONS IS EMBEDDED IN THE PARENT

  • Among 30 P = 10 parents, 26/30 yield a reduced graph with |ΣD| = 5, and 25/30 yield one with both |ΣD| = 5 and PD = 3.The hierarchy progressively merges neighboring regions while preserving distinct fixed-point symbols; accessibility of both bounds depends on the learned parent representation.Source: Figure 2 · p. 8, §4.2 · p. 8

4.3 LEARNING LARGE FIRST PROVIDES A MORE RELIABLE ROUTE TO MINIMAL MODELS

  • The main comparison uses 126 of 141 selected five-symbol candidates with PD = 3, matching direct P = 3 training; candidates are retrained using the reduction-guided protocol.Success indicators are first averaged within each parent seed, then across 30 parent seeds; failed training and parents without qualifying candidates count as failures.Source: §4.3 · pp. 8–9
  • Retrained PD = 3 models have substantially lower Estsp than direct P = 3 models, approaching the geometric fidelity of P = 10 parents.High-fidelity minimal realizations need not reproduce the quotient transition graph exactly.Source: §4.3 · p. 9

5 DISCUSSION AND CONCLUSION

The discussion presents higher-capacity models as scaffolds for reducing and retraining minimal dynamical models, while noting that fixed-point bounds do not fully characterize minimal dynamics. The paper also reports computational costs, limits its empirical study to the 3-scroll Chua system, and documents AI use and reproducibility materials.

5 DISCUSSION AND CONCLUSION

  • Higher-capacity AL-RNNs serve as scaffolds for minimal modeling, with reduction providing a hierarchy rather than fixing the final transition topology.Successful retraining can reorganize transitions into the five-node path-like organization even when the quotient candidate is non-path-like.Source: §5 · p. 9
  • Fixed-point preservation gives a lower bound, not a complete characterization: FP-free regions may still organize global flow, and reduction depends on the parent recovering relevant fixed points.Not every successful parent can be reduced to both symbolic-complexity and nonlinear-capacity lower bounds.Source: §5 · p. 9
  • Exhaustive reduction scales combinatorially with Pparent, taking median 0.2 s at Pparent = 10 and 10.2 s at Pparent = 14 in the studied regime.Enumeration is not required for a compact realization, and the combinatorial dependence is on nonlinear switching variables rather than ambient dimension.Source: §5 · p. 9
  • Empirical evaluation is limited to the 3-scroll Chua system; other attractor organizations and noisy or partially observed real-world data remain future directions.Source: §5 · p. 9

AI USE STATEMENT

  • The authors used generative AI tools across drafting, literature work, experimental design, software, mathematical formulations, and figure preparation, with author review and validation.They state that reported experimental results came from their implementations and experiments rather than AI generation.Source: AI USE STATEMENT · p. 10

REPRODUCIBILITY STATEMENT

  • The paper reports that code, scripts, datasets, and trained checkpoints are available to regenerate reported results and figures without retraining.The appendix documents experimental settings, hyperparameters, seeds, fixed-point evaluation procedures, and additional analyses.Source: REPRODUCIBILITY STATEMENT · p. 10

A.1 ASSUMPTIONS AND PROOF OF THE FIXED-POINT LOWER BOUND

Under the stated assumptions, distinct target fixed points require distinct fixed-point-containing symbols, yielding a lower bound on symbolic complexity. The reduction procedure preserves that fixed-point skeleton while exploring simpler quotient graphs, but structural admissibility alone does not establish faithful dynamics.

A.1 ASSUMPTIONS AND PROOF OF THE FIXED-POINT LOWER BOUND

  • A region contains at most one fixed point when target fixed points are isolated, interior to their regions, and I − Fσ is nonsingular; therefore Q distinct fixed points require at least Q FP symbols.This gives a symbol-count lower bound for representations preserving the target fixed points.Source: §A.1 · p. 12

A.2 CONSEQUENCES FOR SYMBOLIC AND NONLINEAR COMPLEXITY

  • Meeting the fixed-point symbol lower bound certifies minimality in visited-region count, not nonlinear-unit count or sufficiency for reproducing global dynamics.A necessary capacity bound follows from at most 2^P activation patterns, but satisfying it does not guarantee realizability or learning.Source: §A.2 · p. 12

A.3 SCOPE AND POSSIBLE NON-TIGHTNESS OF THE CERTIFICATE

  • The fixed-point bound may be non-tight for full attractor dynamics, because additional FP-free regions can organize global flow and the absence of a fixed point does not make a symbol removable.Accepted quotient graphs must still be retrained and evaluated for faithful autonomous dynamics.Source: §A.3 · p. 12

B.1 RELU LINEARIZATION AND SYMBOL PROJECTION

  • Linearizing a ReLU removes its switching boundary without removing a latent-state dimension, and quotient nodes result from projecting symbols onto retained activation bits.Transitions are projected at both endpoints; quotienting may create self-transitions that can be omitted from visualizations.Source: §B.1 · p. 13

B.2 FP-SYMBOL SEPARATION CONSTRAINT

  • A structurally admissible reduction may merge FP-free symbols with each other or one FP symbol, but cannot merge distinct FP symbols; this protects only the fixed-point skeleton.Faithful dynamical realization is assessed separately through retraining and autonomous evaluation.Source: §B.2 · p. 13

B.3 REDUCTION-TREE ENUMERATION

  • The reduction tree enumerates single-ReLU linearizations and terminates a branch when distinct FP symbols collide, since further linearization cannot undo that collision.Each distinct linearization set is stored once, even when different orders produce it.Source: §B.3 · p. 13

B.4 REDUCTION COMPLEXITY AND CANDIDATE SELECTION

  • For Pparent = 10, exhaustive tree construction took a median 0.2 s per parent, while Pparent = 14 took 10.2 s despite a combinatorial mask space of 16,384 possibilities.The search has at most 2^Pparent masks and prunes branches that violate fixed-point separation; targeted search can stop once qualifying candidates are found.Source: §B.4 · pp. 14–15, Table A1 · p. 15

B.5 CONSENSUS ESTIMATION OF SYMBOLIC GRANULARITY

The consensus procedure estimates symbolic granularity across parent models at a prescribed reduced nonlinear capacity. Its Chua estimate is diffuse, so the modal layer is informative but not a certificate of global minimality.

B.5 CONSENSUS ESTIMATION OF SYMBOLIC GRANULARITY

  • The procedure estimates symbolic granularity conditional on a chosen reduced capacity, using admissible reductions from each parent and parent-balanced votes rather than pooling all nodes.Selecting the reduced capacity itself is a separate model-selection problem, and the parent-inferred fixed-point count need not suffice for faithful full dynamics.Source: §B.5 · p. 15
  • When consensus support is diffuse, neighboring well-supported layers should also be retained for retraining and dynamical evaluation rather than treating the mode as a sharp minimum.For Chua, the modal estimate k = 5 can be checked against the known five-symbol lower bound; that external reference is generally unavailable for unknown systems.Source: §B.5 · p. 16
  • For the Chua hierarchy at PD = 3, the consensus mode is k = 5, with H(5) = 0.370, H(6) = 0.221, and H(7) = 0.320.The k = 5, 6, and 7 layers together account for 91.1% of support, so the distribution is relatively diffuse.Source: §B.5 · p. 16
  • The consensus estimate is an empirical compact-granularity estimate, not a global-minimality certificate, because the fixed-point bound may be non-tight and candidates still require autonomous fidelity evaluation.The estimate is also conditional on the prescribed reduced nonlinear capacity.Source: §B.5 · pp. 15–16

C.1 3-SCROLL CHUA DATA GENERATION

The experimental pipeline uses the 3-scroll Chua system and higher-capacity AL-RNN parents to construct reduced candidates. Candidates inherit parent parameters and are retrained using parent-dynamics guidance, with phase-aligned autonomous targets and several guidance variants.

C.1 3-SCROLL CHUA DATA GENERATION

  • The target Chua fixed-point set contains five points within the region occupied by the attractor, setting Q = 5 for the analysis.The trajectory is generated from the continuous-time system using a numerical integration step of ∆t = 0.01.Source: §C.1 · p. 17

C.2 PARENT AND DIRECT-TRAINING CONFIGURATIONS

  • Parent and direct-training configurations use the same latent dimension, optimization schedule, sparse teacher forcing, and checkpoint rule, varying only ReLU count for direct-training baselines.Checkpoint selection includes the number of visited admissible learned fixed points, although no fixed-point term enters the training loss.Source: §C.2 · p. 17

C.3 REDUCED-MODEL INITIALIZATION

  • Each reduced candidate copies the parent’s trainable parameters and imposes its fixed linearization mask by replacing selected ReLUs with identity activations.Candidates from a parent therefore begin with identical trainable values and differ in which nonlinear units have been linearized.Source: §C.3 · pp. 17–18

C.4 AUTONOMOUS TEACHER TRAJECTORY

  • Teacher-guided retraining uses targets and stored latent states from a single continuous autonomous free-run of the frozen parent model.The rollout stores states, readouts, and activation patterns for sharing across reduction candidates from that parent.Source: §C.4 · p. 18

C.5 PHASE-ALIGNED STUDENT ROLLOUTS

  • Student segments begin at a sampled parent state and match targets from the same time-aligned parent trajectory, a design motivated by rapid phase divergence in chaotic dynamics.Segments are rolled out autonomously for 200 steps.Source: §C.5 · p. 18

C.6 PERIODIC PARENT-STATE SYNCHRONIZATION

  • The main protocol periodically synchronizes the entire latent state to the parent every 128 steps, while an ablation restricts synchronization to readout and retained-ReLU coordinates.Between synchronization events, the reduced model evolves autonomously; teacher-guided retraining does not overwrite states with raw data.Source: §C.6 · pp. 18–19

C.7 RETRAINING OBJECTIVES

  • Four retraining conditions compare no teacher guidance, full-preactivation guidance, retained-preactivation guidance, and symbol guidance; full-preactivation guidance is the main protocol.The full-preactivation objective matches non-readout latent coordinates, whereas retained-preactivation guidance constrains retained nonlinear coordinates and leaves other linear coordinates free to reorganize.Source: §C.7 · pp. 19–20

D.1 AUTONOMOUS FREE-RUN EVALUATION

The evaluation assesses reduced models through autonomous post-transient dynamics, complementary state-space and spectral measures, and parent-balanced aggregation. Structural outcomes include candidates attaining both Chua lower bounds, while the two complexity measures remain distinct.

D.1 AUTONOMOUS FREE-RUN EVALUATION

  • Final evaluation uses 10,000-step autonomous rollouts without teacher forcing or external correction and discards the first 1,000 steps as transient.The post-transient visited-symbol count measures symbolic complexity, and Qvis counts visited regions containing admissible learned fixed points.Source: §D.1 · p. 21

D.2 STATE-SPACE AND SPECTRAL FIDELITY

  • Autonomous fidelity is measured with state-space KL divergence and spectral Hellinger distance; only state-space error enters the high-fidelity success criterion.Spectral error is reported as an independent diagnostic.Source: §D.2 · p. 21

D.3 CHUA SUCCESS CRITERIA

  • Chua minimal-realization success requires the target five visited admissible fixed points and the certified minimum of five post-transient symbolic regions.A further criterion requires faithful state-space dynamics using the stated Estsp threshold of 2.0.Source: §D.3 · p. 22

D.4 PARENT-BALANCED STATISTICAL AGGREGATION

  • The source parent is the primary statistical unit: each parent contributes equally to seed-macro success regardless of how many candidates it generates.Candidates selected before retraining remain in the parent’s set even if retraining or evaluation fails.Source: §D.4 · p. 22

E ADDITIONAL 3-SCROLL CHUA RESULTS

  • The additional Chua results cover structural outcomes, retraining-protocol comparisons, fidelity-threshold robustness, and changes in symbolic transition structure after retraining.The structural analysis compares reduction candidates against the five-symbol and three-ReLU theoretical lower bounds.Source: E ADDITIONAL 3-SCROLL CHUA RESULTS · p. 23, §E.1 · p. 23

E.1 STRUCTURAL OUTCOMES OF HIERARCHICAL REDUCTION

  • Among selected five-symbol quotient candidates, 126 attain both lower bounds, and another 15 attain the five-symbol bound with PD = 4; the main comparison therefore uses PD = 3 and |ΣD| = 5.A quotient may be symbol-minimal while retaining more nonlinear units than the capacity lower bound.Source: §E.1 · p. 23

E.2 REDUCTION-GUIDED RETRAINING ABLATIONS

Parent-derived guidance improves realization after structural reduction, with full-preactivation guidance performing best across fixed-point recovery, minimality, and high-fidelity criteria. Guided variants also lower state-space error; spectral Hellinger error remains a complementary diagnostic.

E.2 REDUCTION-GUIDED RETRAINING ABLATIONS

  • The ablation uses 141 selected five-symbol candidates, including 15 with PD = 4, so its success rates differ slightly from the capacity-matched comparison.All methods share parent-derived initialization; full-preactivation guidance is the main protocol because its coordinate-equal weighting needs no additional calibration and performs consistently well.Source: §E.2 · pp. 23–24
  • Symbol guidance yields 74%, 72%, and 71% success, while retained-preactivation guidance yields 74%, 68%, and 68% across the same three criteria.The criteria are fixed-point recovery, minimal realization, and high-fidelity minimal realization, respectively.Source: §E.2 · p. 23
  • Guided retraining shifts post-transient state-space errors lower than unguided retraining, especially for retained- and full-preactivation guidance.The resulting autonomous dynamics are more faithful to the target attractor, not only more likely to recover the desired fixed-point count or symbolic complexity.Source: §E.2 · p. 23
  • Spectral Hellinger error overlaps more across guided variants than state-space error, so it is retained as a complementary diagnostic rather than the binary fidelity criterion.The binary high-fidelity criterion uses Estsp alone.Source: §E.2 · pp. 23–24
  • Full-preactivation guidance achieves 77%, 74%, and 73% success for fixed-point recovery, minimal realization, and high-fidelity minimal realization, versus 19%, 14%, and 14% without trajectory guidance.High-fidelity minimal realization requires Sfid = 1 and Estsp < 2.0.Source: §E.2 · p. 23, Figure A1 · p. 24

E.3 STATE-SPACE FIDELITY THRESHOLD

The advantage of reduction-guided retraining over direct minimal-capacity training persists across a broad range of state-space-error thresholds. The Estsp < 2.0 cutoff is a practical criterion, not a theoretically defined dynamical boundary.

E.3 STATE-SPACE FIDELITY THRESHOLD

  • Estsp < 2.0 is a conservative practical criterion: models below it generally retain well-formed 3-scroll geometry, while larger errors increasingly accompany distorted or incomplete attractors.The cutoff is not a theoretically defined dynamical boundary.Source: §E.3 · p. 24, Figure A2 · p. 25
  • At τ = 2.0, full-preactivation guidance reaches approximately 73% high-fidelity minimal realizations, compared with approximately 20% for direct P = 3 training.Symbol and retained-preactivation guidance yield approximately 71% and 68%, respectively.Source: §E.3 · pp. 25–26, Figure A3 · p. 26
  • Guided retraining remains substantially ahead of direct P = 3 training over a broad range of τ, with guided curves near their plateaus by τ = 2.0.The separation is already evident below 2.0 and persists at more permissive thresholds, rather than arising from models clustered around the cutoff.Source: §E.3 · pp. 25–26, Figure A3 · p. 26

E.4 EMERGENCE OF A COMMON CHAIN-LIKE SYMBOLIC ORGANIZATION

High-fidelity minimal Chua models converge on a common five-node chain-like symbolic organization, even when their selected quotient candidates have different structures. The quotient is an admissible starting point for retraining, not a literal prediction of the final transition topology.

E.4 EMERGENCE OF A COMMON CHAIN-LIKE SYMBOLIC ORGANIZATION

  • The common chain is not explicitly imposed: reduction constrains admissible mergers under fixed-point preservation, while retraining matches continuous trajectories rather than graph topology.Thus, the quotient graph specifies a structurally admissible starting point rather than the final transition topology.Source: §E.4 · p. 26
  • Different non-chain quotient candidates, including two-cycle and one-cycle-plus-tail graphs, reorganize after retraining into the same chain while preserving Qvis = 5 and |bΣ| = 5.The representative retrained models have Estsp = 1.17–1.28.Source: §E.4 · p. 26, Figure A4 · p. 27
  • All 114 observed high-fidelity minimal candidate runs converge to the same five-node chain-like organization.This reveals a common symbolic skeleton of the 3-scroll Chua dynamics.Source: §E.4 · pp. 26–27
Loading 2610.01369v1…