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Fiber Fingerprints of Hidden Learning-State Dynamics

Qinyou Wang

arXiv:2608.15976v1cs.LG

TL;DR

Execution states that are indistinguishable under present-behavior readouts may respond differently to future training. The paper formalizes controlled future-training probes as fiber fingerprints and reports that present behavior is not sufficient for declared future learning.

  • Problem

    Checkpoint summaries capture present behavior, leaving open whether execution states indistinguishable under declared readouts must learn in the same way.

    Source: §1 · p. 1

  • Method

    The paper formalizes controlled future-training response laws on present-behavior equivalence classes, yielding a predictive quotient and a minimal recursively sufficient representation.

    Source: §1 · p. 1

  • Results

    Transformer studies support structured local action response but find longer-horizon non-closure and fresh visible-relative completion with output-range reuse and a low-rank irreducible sector.

    Source: §10 · p. 21

  • Takeaways & Limitations

    Present behavior is not a sufficient statistic for declared future learning within the reported support-, scale-, metric-, and context-resolved boundaries.

    Source: §10 · p. 21

    The re-anchored path study rejects one empirical transport membership but does not prove global path independence, while finite response frames need not identify stable ambient linear transport.

    Source: §9.3 · p. 20

Abstract

from arXiv · show

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.

1 Introduction

Present behavior does not fully determine future learning: execution states that agree under declared readouts can respond differently to controlled training. The paper defines this distinction first at a finite, set-level scale, then develops conditional mathematical realizations and tests their boundaries in frozen Transformer studies.

1 Introduction

  • Execution states can match on present loss, accuracy, or outputs yet differ in future training because optimizer and other runtime state also shape the trajectory.The paper frames controlled future training as the experiment for distinguishing states within a declared present-behavior equivalence class.Source: §1 · p. 1

1.1 Contributions

  • The contributions define a finite predictive quotient and fiber, then add visible-relative completion and a history-reachability bridge under a finite-dimensional Hilbert realization.The predictive construction includes congruence, quotient descent, Nerode-type minimality, finite-horizon typing, and an exact Transformer–LoRA–AdamW instantiation; completion separates visible, visible-mode-reuse, and irreducible-new response sectors.Source: §1.1 · p. 1

PRESENT EQUIVALENCE

  • Chronology has an exact graph-coboundary-plus-interaction decomposition; under a separately declared crossing regime, cycle response has conditional RMS scale √p·η^(3/2) with graph-cyclomatic prefactor and finite-grid corrections.For Adam, an exact moment section preserves the immediate adaptive field, while common future gradients can locally reveal hidden moment differences. Preregistered Qwen and Mistral studies retain positive, negative, and mixed outcomes.Source: PRESENT EQUIVALENCE · p. 2

1.2 Scope and organization

  • The finite predictive quotient and set-level fiber require only a declared probe doctrine, whereas completion, Hodge, switching, Adam, and smooth-global results require additional assumptions.The paper treats experiments as tests of conditional structures, not premises of the core theory; negative results delimit their support, scale, metric, and context.Source: §1.2 · p. 2

2 Finite controlled experiments and predictive quotients

The section defines hidden learning-state distinctions through future-probe responses among states with the same declared present behavior, then constructs predictive quotients and exact prefix-depth fibers. These are set-level, doctrine-relative representations; a Transformer–LoRA–AdamW system instantiates the framework when primitive execution is single-valued.

2.1 Primitive finite experiment

  • A hidden learning-state distinction is a difference in declared future-probe responses between states sharing the same present readout.The operational fingerprint is restricted to a present-behavior fiber and a declared probe family; it does not identify a latent coordinate or physical model dimension.Source: §2.1 · p. 3

2.2 Complete and finite-horizon equivalence

  • The complete predictive quotient is the coarsest recursively sufficient representation for declared responses and protocols, and predictive equivalence descends functorially [19].Its canonical global finite-scale fiber remains defined when quotient sizes or local ranks jump, contexts are singular, protocols are irreversible, or no global chart exists.Source: §2.2 · pp. 3–4

2.3 Exact prefix-depth fibers

  • Exact prefix-depth fibers match present behavior and a declared initial action sequence before comparing longer futures.They remain set-level objects; tangent-kernel formulas require additional regularity assumptions [15].Source: §2.3 · p. 4

2.4 Transformer execution-state instantiation

  • A frozen Transformer–LoRA–AdamW learner satisfies the finite-scale axioms when each primitive step is single-valued given its complete execution state and control.The mutable state includes parameters, optimizer moments and step, scheduler, buffers, random streams, data cursor, and mutable execution variables that can alter future behavior.Source: §2.4 · p. 4, COMPLETE TRANSFORMER–LORA–ADAMW… · p. 5

COMPLETE TRANSFORMER–LORA–ADAMW EXECUTION STATE

  • Predictive quotient, bridge, and completion ranks are operational dimensions, not identifiable physical neural-state dimensions.Adding an auxiliary controlled system ignored by declared responses can increase implementation-state dimension without changing the operational fingerprint or predictive quotient.Source: COMPLETE TRANSFORMER–LORA–ADAMW… · p. 5

2.2 Complete and finite-horizon equivalence

  • Approximate present-behavior matching is separate from the exact quotient theory because tolerance relations need not be transitive or transition-preserved.The finite-horizon maps must also account for the primitive steps consumed by each protocol.Source: §2.2 · p. 4, COMPLETE TRANSFORMER–LORA–ADAMW… · p. 5

3 Visible-relative predictive completion

A conditional finite-dimensional Hilbert realization decomposes future response relative to declared visible inputs and outputs, while history weighting limits irreducible completion to distinctions generated on naturally occupied directions. The resulting predictive representation is metric- and stability-dependent, and its completion rank describes response modes rather than physical state dimension.

3.1 Stable finite-dimensional realization

  • Under a stable finite-dimensional realization with fixed metrics and separated ranges, operator-based quantification uses pseudoinverses, Hilbert–Schmidt norms, traces, and spectral tails.The finite-scale response contrast remains primary when a differential limit is unresolved.Source: §3.1 · p. 6

3.2 Visible, reuse, and irreducible-new sectors

  • The response splits orthogonally and uniquely into visible-input response, input-invisible reuse of the visible output range, and irreducible response outside that range; the new sector vanishes exactly when Ran(KQ) ⊆ Ran(KP).These sectors are defined relative to the declared visible input and output ranges.Source: §3.2 · p. 6

HISTORY-SUPPORTED RESPONSE COMPLETION

  • The optimal rank-m addition of output-orthogonal modes gives a finite-horizon irreducible completion dimension, with rank S measuring required response modes rather than physical state dimension.The result is relative to the declared realization.Source: HISTORY-SUPPORTED RESPONSE COMPLETION · p. 7

3.3 History-reachability bridge

  • History weighting retains only irreducible responses generated on naturally occupied directions, and its covariance-supported quotient is the minimal reachable completion.A geometric response direction may exist without being generated by natural histories; low completion rank does not identify neural-state dimension.Source: §3.3 · pp. 7–8

3.4 Cumulative horizon filtration

  • For nested cumulative response horizons, the irreducible filtration is monotone: its kernels can only shrink and finite-dimensional completion ranks cannot decrease.Source: §3.4 · p. 8

3.5 Complete prediction versus irreducible novelty

  • The predictive exact sequence separates the complete reachable predictive tangent from the irreducible bridge quotient, with kernel distinctions expressed entirely through visible-mode reuse.Source: §3.5 · p. 8

3.6 Gauge and stability boundary

  • Orthogonality, spectra, pseudoinverses, and shorted operators depend on declared metrics and stable ranges; finite responses remain primary because projectors and pseudoinverses may fail to converge as a singular gap closes.Nonzero spectra are invariant under unitary gauge changes; general coordinate changes require transporting the metrics.Source: §3.6 · p. 8

4 History generation, chronology, and switching

The section separates which states natural training histories generate from how task order organizes hidden-state directions. It gives a conditional graph-projected switching law and distinguishes generation from future observability.

4.1 History-natural occupancy

  • History-natural occupancy maps a declared law on rooted histories to a reachable-state law; its commuting relation is primary, while differentiation requires regular local charts.The local chart also defines a centered history-coordinate covariance.Source: §4.1 · pp. 8–9

4.2 Task-graph chronology and Hodge sectors

  • On a finite task graph, the hidden-state edge field separates into exact and cycle sectors, whose mechanism-specific meaning concerns optimizer memory.The decomposition is standard Hilbert-valued graph Hodge theory; the paper’s contribution is interpreting the sectors under optimizer memory.Source: §4.2 · p. 9

4.3 Common-affine memory isolates interaction

  • With a common bounded operator, cycle projection removes common-affine single-task potential memory while retaining the state-dependent interaction obstruction.For task-dependent operators, an additional contraction/commutator obstruction remains, so the exact decomposition requires the common-operator specialization or explicit retention of that term.Source: §4.3 · p. 9

4.4 Smooth and switching chronology

  • A 3/2 norm exponent holds for graph-projected cycle responses only under the regular crossing assumptions, and is not universal across architectures.The stated regime requires regular crossing counts, post-update marks, finite second moments, and sufficiently weak aggregate dependence.Source: §4.4 · p. 10

4.5 Second-order finite-scale correction

  • Second-order concentration conditions yield a finite-scale effective-exponent correction; the effective event count pη gives joint-limit fluctuations of order (pη)^-1/2.The concentration statement is an explicit assumption; bounded dependency graphs with suitable fourth-moment control are one sufficient route.Source: §4.5 · p. 10

4.6 Generation and future observability

  • Generation and future observability are distinct: generated directions may lie in the future-response operator’s kernel, while revealing directions may be rarely generated.Weak visibility alone therefore does not identify rapid dissipation.Source: §4.6 · p. 11

5 Adam execution fibers, accessibility, and future revelation

Adam moment states can share the same current parameter, present behavior, and adaptive field, yet common future gradients can reveal their differences. The section distinguishes optimizer-level reachability from neural accessibility and future revelation from cross-bank coherence.

5.1 Exact adaptive field and execution fiber

  • Different Adam moment states can share the same current parameter, present behavior, and immediate adaptive field along an exact matched section.The cancellation follows because the matched scaling multiplies both the numerator and the square-root denominator by the same factor.Source: §5.1 · p. 12

5.2 Moment reachability versus neural accessibility

  • Local neural accessibility follows when the control-to-moment derivative is surjective, but algebraic reachability, metric stability, and finite-panel membership are distinct.At the optimizer level, for K≥2 the reachable terminal injections form an epigraph, filled by nullspace components of the first-moment functional; for K=1, equality is required. Real neural gradients are coupled and model- and data-constrained.Source: §5.2 · pp. 12–13

5.3 Future-gradient de-cancellation

  • Common future gradients can first-order resolve the matched section at a horizon and coordinate when the theorem’s right-hand side is nonzero.At horizon n=0 the derivative vanishes; with all future gradients zero and epsilon=0, unequal decay rates alone do not break the cancellation.Source: §5.3 · p. 13

5.4 Revelation is not coherence

  • Positive revelation by every bank does not imply positive cross-bank directional coherence, which can range from −1 to 1.A bank-invariant direction requires an additional homogeneous condition; the general theory therefore uses a context-indexed observability field, and revelation remains future-gradient and context conditioned.Source: (a) EXACT MATCHED SECTION (b)… · p. 14

6 Global transport and resolution-supported geometry

The section treats the predictive quotient as a canonical directed transport system while distinguishing set-level results from additional smooth or globally fixed realizations. Empirical transport is restricted to resolved supports, so unresolved loop directions remain undefined rather than being treated as identity.

6.1 Canonical directed transport

  • On a connected context groupoid with isomorphic quotient transports, endpoint-only transport, identity loop holonomy, and path-independent trivialization are equivalent.The set-level theorem does not itself provide a smooth connection; bundle and differential geometry require additional regularity, and rank jumps yield only stratumwise geometry.Source: §6.1 · p. 14

6.2 Resolution-supported empirical transport

  • On a declared resolved source support, transport is exact and unique, its zero extension minimizes Hilbert–Schmidt norm, and it is locally Lipschitz when retained rank and spectral gap persist.Stability constants worsen as the resolved singular scale approaches zero, typically as δ^-1 for target perturbations and δ^-2 for source-pseudoinverse perturbations; no uniformly stable ambient extension exists as retained singular values vanish.Source: §6.2 · p. 15

6.3 Partial monodromy

  • Composed empirical edges yield partial monodromy only on the common domain surviving their resolutions; an undefined unresolved direction does not establish identity action.The quotient functor supplies directed transport along controlled protocols, while loop endomorphisms form a monodromy semigroup; group-valued holonomy requires a reversible subgroupoid.Source: §6.1 · p. 14, §6.3 · p. 15

6.4 Context-independent propagation as an additional factorization

  • A context-indexed predictive quotient remains valid without fixed action propagators, smooth bundles, differential connections, or nontrivial holonomy; these are additional realizations or empirical memberships.Same-label discrepancies or label-null loop effects can obstruct a fixed action-labelled propagator without invalidating the context-indexed quotient.Source: §6.4 · p. 15

7 Transformer studies and empirical membership

Across frozen Qwen2.5-7B and Mistral-7B-v0.3 studies, present behavior and even an initial learning action can fail to determine longer-horizon response, while controlled future gradients can reveal hidden optimizer-state differences. The evidence supports specific conditional mechanisms but leaves stronger transport, switching, accessibility, and revelation claims unresolved or unsupported.

7.1 Execution contexts and evidentiary discipline

  • The confirmatory studies fixed model and data revisions, probes, selection rules, and CPU verdict scripts before protected outcomes, without repairing failed criteria after inspection.Frozen Qwen2.5-7B and Mistral-7B-v0.3 models used low-rank adapters and AdamW execution states across three text domains and controlled task histories.Source: §7.1 · pp. 15–16

7.2 Overview of the evidence chain

  • The evidence chain progresses from a local action backbone and longer-horizon non-closure to replicated visible-relative completion, then tests stronger transport, switching, accessibility, and revelation claims.The studies test membership in conditional layers rather than serving as premises for the finite predictive quotient.Source: §7.2 · p. 16, §7.1 · p. 16

7.3 Local action backbone and first-return non-closure

  • Fresh Qwen and Mistral histories support an action-specific local response backbone, but ordered two-step response includes first-return terms and does not close as a finite-dimensional cocycle.The full 9 × 9 first-return system has local r90 and r95 near four over H3–H5, with shifted-H6 low-rank replication; a fixed context-independent propagator is unsupported.Source: §7.3 · p. 16

7.4 Fresh visible-relative completion replication

  • At H5 and shifted H6, the irreducible-new fraction has one-sided 95% lower bounds of 13.64% and 10.87%, respectively, with median r95(S) of three at both horizons.The replicated finite-sector picture combines a declared three-dimensional visible action core, structured visible-output reuse, one or two dominant irreducible modes, and a weaker residual tail; it does not measure physical neural-state dimension.Source: §7.4 · pp. 16–17, Figure 6 · p. 17

7.5 Re-anchored path-transport boundary

  • The re-anchored path-transport panel does not support nontrivial transport, but also cannot establish path independence because effects are unresolved and relative transport is unstable on most units.Mean cross-bank energy was 3.57 × 10^-6 with one-sided 95% lower bound −3.00 × 10^-6; only one of twelve reconstructions passed the principal conditioning criterion.Source: §7.5 · p. 17

7.6 Controlled hard-switching scaling and finite-scale bias

  • The controlled hard-ReLU study gives strong prospective support for the Hodge–3/2 mechanism, but its full preregistered finite-grid conjunction is unmet and native SiLU membership is unclaimed.Initial-grid cycle scaling is approximately p^0.5113η^1.5163 versus η^2.0095 for the softplus control; refined-grid future-bridge η exponents are approximately 1.5061 versus 2.000554 for softplus.Source: §7.6 · p. 17

7.7 Matched-section accessibility and future revelation

  • Mistral passes the frozen accessibility screen for six candidate seeds, while Qwen passes none under the declared raw (m, v) Euclidean criterion; this is not coordinate-free inaccessibility.In the accessible Mistral subclass, all six chronology slots reconstruct matched sections and all twelve full-amplitude future-bank responses are nonzero, but bank-invariant revelation direction is unsupported.Source: §7.7 · pp. 17–18

7.8 Empirical synthesis

  • The studies support longer-horizon non-closure, visible-output reuse with a small irreducible completion on natural-history support, and future-gradient revelation of optimizer-state differences.They do not establish a globally smooth empirical bundle, universal finite-rank state, native Transformer 3/2 membership, nontrivial path monodromy, coordinate-free Qwen inaccessibility, or bank-invariant revelation.Source: §7.8 · p. 18

8 Related work and novelty boundary

The paper’s novelty claim is the coupling of present-behavior equivalence with controlled future learning, not the invention of its mathematical ingredients. It frames the resulting distinctions as a state-theoretic question about which differences within present-behavior equivalence classes future-learning operators can reveal.

8.1 Observability and predictive state

  • Fiber fingerprints specialize observability and predictive-state ideas to complete training states, present-readout quotients, and future-training protocols, with a predictive quotient defined before smooth rank conditions.Relative to loss-invisible overlaps in recurrent networks [5], the framework allows arbitrary declared present readouts and adds quotient minimality, history-reachable completion, and a Transformer–AdamW instantiation; it also relates to Nerode quotients and category-of-elements assembly.Source: §8.1 · p. 19

8.2 Function-preserving gauges and hidden learning dynamics

  • Function-preserving parameterizations can alter optimization trajectories, as shown by ReLU rescaling [14] and hidden-gauge effects [30], but Fiber Theory targets protocol-relative quotients of complete execution states.Its target includes cases where parameters and immediate adaptive fields are matched, rather than claiming the broader same-function/different-training-dynamics observation as new.Source: §8.2 · p. 19

8.3 Balancing, shorting, and predictive completion

  • Classical balancing, dissipativity, pseudoinverses, shorted operators, and low-rank approximation are applied relative to a frozen visible-action backbone.The completion separates input-invisible response into visible-output reuse and irreducible-new directions, then weights the factorization obstruction by naturally reachable-history covariance; it is not a state-space identification theorem.Source: §8.3 · p. 19

8.4 Task order, Lie brackets, and optimizer memory

  • Graph Hodge decomposition, Lie-bracket analyses [23] [26], and optimizer-memory order effects [27] are established; the paper’s narrower chronology result isolates state-dependent interaction under a common-affine memory channel.Under a separately declared crossing regime, the cycle field has a graph-projected 3/2 rare-event law with a cyclomatic prefactor; neither Hodge decomposition nor smooth Lie brackets are claimed as new.Source: §8.4 · p. 19

8.5 Adam, adapters, and execution-state geometry

  • The optimizer contribution is an exact Adam moment-space section that preserves the immediate adaptive field while allowing future gradients to reveal hidden execution-state differences.It also gives a scalar moment-reachability region, separates neural from metric-stable accessibility, and derives a first-order future-gradient de-cancellation formula; Adam/AdamW [12] and LoRA variants [2] [8] are standard ingredients.Source: §8.5 · p. 19

8.6 Plasticity and future-learning probes

  • Work on future optimization gain [29], constrained LLM probing [6], and few-shot stability/plasticity evaluation [9] motivates future-learning probes, while this paper asks which present-equivalent distinctions future training can reveal.The focus is the state-theoretic question that precedes deployment decisions: how such distinctions arise and which future-learning operators reveal them.Source: §8.6 · pp. 19–20

8.8 Precise novelty statement

  • The claimed novelty is a coupled architecture of behavior-matched learning states, future-training probes, predictive quotients, history occupancy, response completion, chronology, and optimizer revelation—not an unqualified worldwide-first claim.The strongest theorem-level contributions include quotient minimality, the history-reachability–visible-completion bridge, the common-affine decomposition, the graph-projected switching law, and exact Adam cancellation and future-gradient results.Source: §8 · p. 18, §8.8 · p. 20

9 Limitations, non-implications, and open boundaries

The theory is intentionally modular: its mathematical and empirical results do not license claims beyond their declared assumptions, measurement semantics, and study conditions. Several transport, switching, reachability, generalization, and verification boundaries remain open.

9.1 Probe relativity and measurement

  • Predictive equivalence depends on the declared probe family, response gauge, horizon, and measurement semantics; finite banks and approximate equality do not necessarily define a complete or exact quotient.Population response, finite readout, and implemented floating-point response are distinct objects.Source: §9.1 · p. 20

9.2 Regularity and representation

  • The set-level fiber need not be a manifold, and geometric or spectral tools require regularity, rank, range, and metric assumptions; low response rank does not identify physical neural-state dimension.A smooth global bundle or connection additionally requires local triviality and smooth path transport; otherwise the unconditional global object is the category of elements of the predictive quotient functor.Source: §9.2 · p. 20

9.3 Transport

  • The re-anchored path study rejects one specific transport membership, not global path independence; finite empirical response frames need not identify stable ambient linear transport.Resolution-supported partial transport is the appropriate finite estimand when source directions approach the measurement floor.Source: §9.3 · p. 20

9.4 Switching and the 3/2 class

  • The 3/2 switching exponent is conditional on crossing, mark, centering, variance, dependence, and joint-limit assumptions, and the preregistered finite-grid conjunction was not met.The controlled studies use a hard-ReLU head over frozen Transformer features, not native Qwen or Mistral SiLU membership; second-order theory does not change the empirical verdict.Source: §9.4 · p. 20

9.5 Optimizer reachability and revelation

  • An algebraic matched section need not be visited by natural neural training, and optimizer-level moment reachability is not neural-data reachability; Qwen inaccessibility is not established coordinate-free.The conditional Mistral study supports dynamic accessibility and future-gradient-conditioned revelation, but not a stable bank-invariant response direction; its closed derivative formula assumes exogenous common gradients.Source: §9.5 · pp. 20–21

9.6 Generalization

  • Empirical evidence is limited to the declared Qwen and Mistral revisions, LoRA, AdamW conventions, text domains, horizons, and numerical contracts; cross-context claims require replication.CUDA and TPU/XLA are distinct execution contexts absent an exact or tolerance-controlled intertwiner.Source: §9.6 · p. 21

9.7 Open questions

  • Open questions include finite-probe criteria beyond finite-rank settings, all-horizon rank stabilization, native Transformer switching membership, supported cross-context transport, and replication across model and optimizer families.These are extensions of the theory, not missing premises of the finite predictive quotient.Source: §9.7 · p. 21

9.8 Audit status

  • The theorem chain, counterexamples, code, manifests, and frozen analyses received internal adversarial review and computational checks, but not external peer review or proof-assistant certification.The release is a formal manuscript and reproducibility archive, not a claim of mechanized proof completeness.Source: §9.8 · p. 21

10 Conclusion

Relative to declared future-training probes, present-behavior-equivalent learning states can carry distinct predictive response laws, represented by a canonical quotient and fiber without smoothness or fixed-dimension assumptions. Transformer studies support this hierarchy while identifying limits in horizon closure, transport, switching, and context invariance.

10 Conclusion

  • Finite controlled probes induce a canonical predictive quotient, a minimal recursively sufficient representation, and a global set-level fiber without smoothness or fixed dimension.These laws are defined over present-behavior equivalence classes.Source: §10 · p. 21
  • Explicit realizations separate visible action response, visible-output reuse, and irreducible-new modes, while history reachability restricts geometry to directions generated and occupied by training histories.Graph-Hodge chronology distinguishes task-potential memory from state-dependent interaction; a conditional crossing regime yields a graph-projected 3/2 scale with finite-scale corrections.Source: §10 · p. 21
  • Adam admits an exact execution-state fiber: distinct moment states can share an immediate adaptive field, yet common future gradients can locally reveal their difference.Source: §10 · p. 21
  • Transformer studies find structured local action response but longer-horizon non-closure, fresh completion with reuse and a low-rank irreducible sector, unresolved re-anchored transport, and a switching mechanism short of its full finite-grid conjunction.Accessible Mistral matched sections are future-gradient revealing but not bank-invariant; the resulting geometry is support-conditioned, scale-resolved, metric-resolved, and context-indexed.Source: §10 · p. 21
  • Present behavior is not a sufficient statistic for declared future learning.The conclusion is scoped to the declared future-learning probes.Source: §10 · p. 21
  • Fiber fingerprints operationalize missing predictive structure; deciding what structure to acquire and use for production is a downstream decision-theoretic problem, not a prerequisite for the state theory.Source: §10 · p. 21

A Proof details for the finite predictive core … C.2 Second-order switching law

The appendix derives the finite predictive core using sets, functions, finite protocols, and prefix-compatible responses, without topology or rank assumptions. It also gives conditional mechanisms and resolution boundaries for completion, chronology, optimizer revelation, and finite-scale switching.

A Proof details for the finite predictive core; A.1 Proof of predictive descent and functoriality; A.2 Grothendieck construction; A.3 Nerode minimality

  • Prefix-compatible response semantics make predictive transitions descend uniquely to the quotient, whose functoriality and Nerode minimality require no topology or rank assumptions.The induced opcartesian lift gives the discrete opfibration construction; any recursively sufficient representation factoring all responses maps uniquely through the quotient.Source: §A.1 · p. 22, §A.2 · p. 22, §A.3 · p. 22

A.4 Finite-horizon loss of depth; A.5 Probe enrichment; A.6 Depth filtration

  • Consuming ℓ steps can increase a probe’s horizon by ℓ; adding probes or increasing cumulative horizon refines the predictive quotient, with depth filtration requiring no differential argument.The finite-horizon equivalence classes therefore need not retain equal horizon labels across a transition.Source: §A.4 · p. 22, §A.5 · p. 22, §A.6 · p. 22

A.7 Regular tangent specialization; A.8 Finite operator data are not a complete ontology; A.9 Non-identifiability of implementation dimension

  • At regular points, present-equivalence level sets have tangent kernel, but singular points may admit only one-sided tangent cones; first-order operator data cannot determine unrestricted finite-scale responses, nor implementation dimension.Centered finite probes require an admissible two-branch realization; auxiliary controlled state can have arbitrary complexity without changing the predictive quotient.Source: §A.7 · pp. 22–23, §A.8 · p. 23, §A.9 · p. 23

B.1 Visible-relative decomposition; B.2 History bridge; B.3 Monotone cumulative shorting

  • The visible-relative decomposition is Hilbert–Schmidt orthogonal and yields optimal rank-m completion; the history bridge gives minimal squared error, and cumulative shorting increases monotonically with horizon.The history bridge applies the preceding result after weighting by the history-reachability terms.Source: §B.1 · p. 23, §B.2 · p. 23, §B.3 · p. 23

B.4 Common-affine chronology; B.5 Graph-projected rare-event law

  • Under Assumption 4.2, graph-Hodge projection removes the common-affine coboundary; the rare-event law has off-diagonal plus remainder o(pη^3), and equal q yields q b1(G).The scaling interpretation links N ≍ pη centered crossing events to post-update displacements of O(η).Source: §B.5 · p. 24, §B.4 · p. 24

B.6 Exact Adam moment epigraph

  • For K ≥ 2, moment pairs can realize every second-moment value above the boundary while holding the first moment fixed.A nonzero direction orthogonal to the first-moment weights leaves M_K fixed and continuously increases V_K.Source: §B.6 · p. 24

B.7 Matched-section cancellation and future derivative; B.8 Endogenous feedback boundary

  • For matched states, a common exogenous future gradient can reveal moment differences; this is the first resolving event, while endogenous divergence requires the full tangent recursion.The conditional study freezes the first future-gradient comparison to test the closed mechanism before endogenous feedback dominates.Source: §B.7 · pp. 24–25, §B.8 · p. 25

C.1 Resolution-supported transport; C.2 Second-order switching law

  • Finite-grid switching exponents can lie systematically above or below 3/2 while converging to the same asymptotic class; resolution-supported transport bounds depend inversely on the resolved singular scale.The ambient transport no-go is demonstrated as a singular scale tends to zero.Source: §C.2 · p. 25, §C.1 · p. 25

C.3 Metric-aware accessibility … E.3 Status of the mathematical package

Metric-aware accessibility and revelation analyses distinguish coordinate-invariant visibility from chart-dependent conditioning, while the paper’s empirical and mathematical claims remain bounded by explicit protocols, assumptions, and unmet criteria. The supplied passages characterize the studies and their status but do not provide the detailed values in the referenced tables.

C.3 Metric-aware accessibility

  • Metric-aware singular values are coordinate-invariant, and the local accessibility proof gives a solution under contraction bounds, whereas raw-chart conditioning is not coordinate-free.The fixed-point argument applies when r ≤ τ/(2L) and the target displacement is at most τr/2; raw (m, v) conditioning can change under coordinate scaling even when rank is unchanged.Source: §C.3 · p. 26

C.4 Revelation–coherence separation

  • Collective visibility is represented by a positive-semidefinite operator, but pairwise coherence can have either sign, so complete visibility need not imply aligned response directions.The rotation example is compatible with complete visibility under every bank and opposite response directions.Source: §C.4 · p. 26

C.5 Four-resolution separation

  • The four-resolution separation theorem rejects the listed implications without their named assumptions and refines coupling as history-generated reachable geometry times resolved observability.The refined observability is resolved by support, scale, metric, and context.Source: §C.5 · p. 26

D.1 General contract; D.2 Models, adapters, and domains; D.6 Artifact integrity

  • The empirical contract specifies complete execution state, frozen selection, separated analysis, independent banks when required, and fail-closed verdicts; studies use frozen-base Qwen2.5-7B and Mistral-7B-v0.3 with LoRA and AdamW.Three text domains use frozen datasets and disjoint training/readout windows. Packages include notebooks, source, logs, raw or recovery archives, frozen CPU analyses, and release audits; large weights and third-party datasets are not redistributed.Source: §D.1 · p. 26, §D.2 · pp. 26–27, §D.6 · p. 28

D.3 Formal study registry; D.4 Detailed empirical ledger

  • The paper uses descriptive study names with a separate release crosswalk, while Tables 3 and 4 respectively cover formal study roles and frozen outcomes.The passages identify the tables’ scope but do not provide their entries.Source: §D.3 · p. 27, Table 3 · p. 27, Table 4 · p. 27

D.5 Selected numerical details

  • The irreducible-new fraction has one-sided 95% lower bounds of 13.64% and 10.87%, with independent projection families and action-specific checks addressing single-sketch artifacts.The passage does not identify which model or condition corresponds to each value.Source: §D.5 · p. 28
  • The strict switching conjunction remains unmet, while conditional revelation shows strong tangent alignment but leaves its lower-bound and coherence criteria unmet.Switching misses four thresholds: crossing-count upper endpoint 1.010357 versus 1.010000, bridge endpoints 1.510310 and 1.510316 versus 1.510000, and cycle estimate 1.506284 versus 1.506000. Revelation alignment has minimum 0.919397 and median 0.981713; full-amplitude response norms range from 5.74 × 10^-9 to 1.41 × 10^-7, and one of 24 reconstruction comparisons narrowly misses the 1% error criterion.Source: §D.5 · p. 28

E.1 Licensed and unlicensed interpretations; E.2 Minimal assumption map; E.3 Status of the mathematical package

  • Tables 6 and 7 organize interpretive boundaries and dependencies; the mathematical package distinguishes theorem-level results from conditional extensions and frozen-context empirical evidence.The finite predictive core and complete-state Transformer instantiation are theorem-level; completion, Hodge, switching, Adam, and regular transport results are conservative extensions under explicit assumptions. The manuscript has not undergone external peer review or mechanized proof certification.Source: Table 6 · p. 28, Table 7 · p. 29, §E.3 · p. 29

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