Fiber Fingerprints of Hidden Learning-State Dynamics

Qinyou Wang

cs.LGAug 17, 2026

Fiber Fingerprints of Hidden Learning-State Dynamics

Authors: Qinyou Wang

Abstract

A learning system can occupy execution states that are indistinguishable under every declared present-behavior readout yet respond differently to future training. We formalize this through fiber fingerprints: controlled future-learning response laws restricted to present-behavior equivalence classes. Prefix-compatible finite probes induce a predictive quotient functor, a Nerode-type minimal recursively sufficient representation, and a canonical set-level predictive fiber without assuming smoothness, reversibility, finite rank, or a manifold. Under an explicit finite-dimensional Hilbert realization, response decomposes into visible, visible-mode-reuse, and irreducible-new sectors; a history-reachability bridge retains only distinctions generated by natural training histories. Conditional mechanism results then identify a graph-Hodge chronology decomposition, a regular switching class with root-mean-square scale $\sqrt{p}η^{3/2}$ and finite-scale corrections, and an exact Adam moment section whose immediate adaptive field is constant while common future gradients can reveal hidden moment differences. Frozen Transformer--LoRA--AdamW studies with Qwen2.5-7B and Mistral-7B-v0.3 support a local action backbone, longer-horizon first-return non-closure, and fresh visible-relative completion with output-range reuse and a low-rank irreducible sector. Stronger claims remain bounded by preregistered negative or mixed results: re-anchored transport is unresolved above its measurement floor; the strict finite-grid Hodge-- $3/2$ conjunction is unmet despite prospective contraction; Qwen accessibility is not established in the frozen raw moment chart; and Mistral revelation is future-context dependent rather than bank invariant. Within these support-, scale-, metric-, and context-resolved boundaries, present behavior is not a sufficient statistic for declared future learning.

Figures & tables

Panel of Figure 1 — Figure 1: Finite controlled core and conditional realizations.

Panel of Figure 2 — Figure 2: Complete Transformer–LoRA–AdamW execution state.

Panel of Figure 3 — Figure 3: Visible-relative completion on history-supported directions.

Panel of Figure 4 — Figure 4: Chronology interaction and the regular switching class.

Panel of Figure 5 — Figure 5: Adam matched-section cancellation and future-gradient revelation.

Empirical object	Evidence supported by the frozen studies	Status
Local action and finite return	Action-specific one-step backbone; longer-horizon first-return non-closure.	Supported, local
Visible-relative completion	Fresh histories reproduce visible response, output-range reuse, and a low-rank irreducible-new sector in both models.	Supported
Re-anchored path transport	No path effect is resolved above the independent-bank floor; most ambient relative transports are ill conditioned.	Not resolved
Hard-switching chronology	Crossing, Hodge, bridge, and smooth-control diagnostics support the mechanism and prospective contraction.	Strict conjunction unmet
Adam accessibility	Mistral matched sections are accessible in the frozen chart; joint Qwen–Mistral panel membership is not established.	Context/metric dependent
Future-gradient revelation	Nonzero responses, zero-gradient controls, amplitude scaling, and analytic recurrence support de-cancellation.	Supported in Mistral

Table 1: Empirical membership of the conditional theory.

Figure 6: Fresh visible-relative completion at H5 and shifted H6.

Figure 7: Contraction of finite-grid exponent errors after prospective refinement.

Figure 8: Matched-section response scale and cross-bank direction; panel (b) is exploratory.

Premise		Not implied
Present-behavior equality	$\not\Rightarrow$	future-learning equality
Set-level quotient transport	$\not\Rightarrow$	stable ambient linear transport
Asymptotic $3/2$ membership	$\not\Rightarrow$	exact finite-grid exponent
Algebraic moment reachability	$\not\Rightarrow$	neural-data accessibility
Surjective control differential	$\not\Rightarrow$	arbitrary raw-coordinate condition threshold
Positive future observability	$\not\Rightarrow$	positive cross-bank directional coherence

Table 2: Key non-implications retained by the theory.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Formal study name	Role in the paper
Local action and return structure	Establishes an action-specific local backbone, first-return non-closure, low local return rank, and the boundary of a fixed context-independent propagator.
Fresh visible-relative completion replication	Independently replicates visible/reuse/irreducible-new response completion in both models.
Re-anchored path-transport study	Tests a finite linear transport/monodromy membership after exact endpoint re-anchoring.
Controlled hard-switching scaling study	Tests crossing counts, marks, graph-Hodge cycle scaling, future bridges, smooth controls, and prospective finite-scale convergence.
Joint matched-section accessibility screen	Tests natural tangent alignment and joint Qwen–Mistral membership in a frozen two-step, three-coordinate control chart.
Conditional matched-section revelation study	Tests future-gradient de-cancellation in the empirically accessible Mistral subclass.

Appendix

Table 3: Formal study names and roles in the paper.

Study	Frozen outcome	Status
Local action and return structure	Fresh Qwen/Mistral histories support action-specific local maps. The one-step law extends, but a fixed closed three-dimensional two-step cocycle is not supported. The full $9\times 9$ first-return system has $r_{90},r_{95}\approx 4$ ; shifted-H6 low rank replicates.	Supported / boundary
Fresh visible-relative completion replication	At H5, median response energy is 60.29% visible, 23.99% reuse, and 15.72% irreducible-new. At shifted H6 it is 67.15%, 19.73%, and 13.13%. Both models satisfy the frozen replication without pooling rescue.	Supported
Re-anchored path-transport study	Mean cross-bank energy $3.57\times 10^{-6}$ ; one-sided lower bound $-2.996\times 10^{-6}$ ; synchronized sign $p=0.171875$ ; median SNR $0.001651$ ; one of twelve principal relative-transport condition checks passes.	Not supported
Controlled hard-switching scaling study	Initial cycle approximately $p^{0.5113}\eta^{1.5163}$ and softplus $\eta^{2.0095}$ . Refined cycle $p^{0.497334}\eta^{1.506284}$ , future bridges near $\eta^{1.5061}$ , and softplus $\eta^{2.000554}$ . Primary cycle equivalence holds, but four narrow conjunction criteria remain outside their frozen bands.	Mixed
Joint matched-section accessibility screen	Natural tangent alignment is high. Mistral satisfies the frozen first-slot screen for all six candidate seeds; Qwen satisfies none under the declared raw $(m,v)$ Euclidean conditioning criterion. No future outcome is computed.	Criterion not met
Conditional matched-section revelation study	Six Mistral chronology slots are dynamically reconstructed; twelve of twelve full-amplitude responses are nonzero; full zero-gradient controls are zero; median full/half norm ratio $1.99894677$ ; median full/half cosine $0.999998942$ . A bank-invariant direction is not supported.	Mixed

Appendix

Table 4: Frozen outcomes stated in formal language.

Horizon	Visible	Reuse	Irreducible-new	Median $r_{95}(S)$
Source H5	60.29%	23.99%	15.72%	3
Shifted H6	67.15%	19.73%	13.13%	3

Appendix

Table 5: Median sector fractions in the fresh completion replication.

Topic	Licensed interpretation	Not licensed
Predictive state	Present behavior is not a sufficient statistic for declared future-learning responses.	A finite fingerprint recovers the complete physical neural state.
Global fiber	The predictive quotient functor has a canonical set-level category of elements; regular bundles are conditional.	The measured Transformer system is unconditionally a smooth fiber bundle.
Completion	Fresh response contains visible, reuse, and irreducible-new sectors relative to a frozen projector and metrics.	The irreducible rank is the true neural hidden dimension.
Transport	Quotient transport is canonical; empirical finite linear transport is support- and resolution-conditioned.	The negative path study proves path independence or identity monodromy.
Switching	The controlled hard-ReLU system supports the proposed Hodge– $3/2$ mechanism and second-order approach while missing its full finite-grid conjunction.	Native Qwen or Mistral training universally obeys a $3/2$ law.
Adam	An exact algebraic matched section exists; accessible Mistral sections show future-gradient-conditioned revelation.	Generic natural reachability or a bank-invariant revelation direction.

Appendix

Table 6: Interpretive boundaries.

Result	Required	Not required
Predictive quotient and global opfibration	Complete composition and prefix-compatible probes	Topology, probability, smoothness
Prefix-depth filtration	Declared prefix action readouts	Differentiability
Visible-relative completion	Finite-dimensional Hilbert metrics and a declared visible projector	Global bundle
History bridge	History chart and covariance	Stationarity
Hodge interaction isolation	Finite task graph and common-affine memory channel	Switching assumptions
Graph-projected $3/2$ law	Crossing covariance, centering, and aggregate dependence conditions	Transformer or ReLU by name

Appendix

Table 7: Dependencies of the principal results.

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Dynamics of Hidden Learning Structure

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Neural Network Training Dynamics Dynamic Neural Networks Continual Learning for LLMs

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A learner's present predictions do not fully determine how it will learn next. We study the dynamics of hidden learning structure through two related operations. Fiber Regulation Theory (FRT) describes protected repair of existing function and the learning responses left available after repair. Fiber Acquisition Theory (FAT) describes the formation and renewal of trainable structure, distinguishing neutral installation, access to a learning signal, and additional functional directions. The two operations also constrain one another. A repair obligation can force a change in a learning channel, while acquired features can make new protected repairs feasible or reduce the minimum coefficient cost of existing ones. We characterize joint admissibility and response reachability, and give a finite supervised-learning construction in which the two operations together produce a favorable response absent from either operation alone. We implement protected head repair and fixed-capacity LoRA renewal in Qwen2.5-32B with AdamW, under replay and output distillation. Across two training trajectories differing in seed and within-task example order, the coordinated policy achieves higher teacher-forced Exact than either separate policy in all four comparisons. It also has the lowest macro NLL under both learning rules on the second trajectory. The implementations realize structure-conditioned self-regulation within specified information and action constraints: accessible state readouts and arrived evidence guide repairs and structural renewals, and ordinary learning continues from the modified state.

Abstract

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