Training and Inference Dynamics of PLDR-LLMs: Row-Map Collapse, Renormalization, and Predictive Reduction

Burc Gokden

Source-linked AI summary

Training and Inference Dynamics of PLDR-LLMs: Row-Map Collapse, Renormalization, and Predictive Reduction

Burc Gokden

arXiv:2609.34130v1Sep 28, 2026cs.LGcs.CL

TL;DR

The paper asks what information must be retained when microscopic training computations are replaced by coarser descriptions. It develops finite row-work and blocking analyses, while its reported fluctuation evidence does not identify a thermodynamic critical class.

Problem
The central question is what information a coarse description must retain to represent a microscopic training computation for different tasks.
Source: Guide to the monograph · p. 10
Method
The paper combines finite row-work identities, chronological row blocking, and actual finite matrix increments to analyze reduced PLDR-LLM dynamics.
Source: Principal results and evidence · p. 12
Results
Ten complete cells fail the nominal precision gate, two are face-interrupted, and native batch dependence is not justified by Gaussian calibration.
Source: Principal results and evidence · p. 12
Takeaways & Limitations
A realized row-energy path can use a small exact coordinate, but successor prediction and predictive-distribution preservation require more information, including an emission.
Source: Guide to the monograph · p. 10
Moving finite fluctuation regions and unresolved systematic narrowing do not identify a thermodynamic critical class.
Source: Principal results and evidence · p. 12

Abstract

from arXiv · show

This monograph develops a unified account of training and inference in Power Law Decoder Representation language models (PLDR-LLMs). Exact finite work identities decompose changes in the absolute energy of the row-centered learned map into parameter contributions, signed interactions, and numerical observation defects. Positive affine blocking retains restarts at the row-constant face, while the augmented AdamW state supplies the complete dynamical description. Predictive renormalization acts on the complete conditional training law for a single pass over distinct corpus target blocks, retaining optimizer memory, remaining data, schedule, and numerical policy. Autonomous reductions require closure; approximate reductions carry successor and emission errors. Finite-population covariance, matched physical clocks, matrix fluxes, and signed temporal energy connect row dynamics to model-wide observations. Absolute row collapse, relative row concentration, operator stabilization, and predictive accuracy are distinguished. Experiments reveal observer and optimizer dependence, reject the tested autonomous row-state candidates, and support finite conditional prediction and state-specific operator reduction. Independent single-pass families exhibit moving finite fluctuation regions without establishing a thermodynamic critical class. Conditional symmetry, head limits, covariance flows, and readout error budgets specify assumptions needed to transfer scaling laws to inference. The theory separates exact identities, conditional dynamical claims, and finite empirical findings, with proofs, selected formal checks, and compact numerical evidence.

Guide to the monograph … Finite work, parameter sources, and face restarts

This section connects exact finite row-map and energy accounting—including parameter sources, interactions, and possible reopening from the row-constant face—to conditioned single-pass training laws and predictive reductions. It distinguishes finite identities and observations from sufficient or conditional collapse claims, inference replacement, causal attribution, and stronger limiting claims.

Guide to the monograph

The organizing question is what information a coarser description must retain: a realized energy path, its successor, a predictive distribution, and fluctuation exponents require progressively stronger retention.The monograph organizes these topics across finite mechanics and optimizer dynamics, chronological blocking, consuming-corpus laws, predictive inference, conditional scaling, and appendices.Source: Guide to the monograph · p. 10, Organization of the theory · p. 14

Assumptions and conditioning at a glance

The primary training law is single-pass RefinedWeb; fixed-corpus and newly drawn-corpus ensembles differ, and changing the clock does not remove corpus exhaustion.Exact identities apply on stated domains, whereas closure, all-future attraction, and predictive coverage require their own retained state and conditions.Source: Guide to the monograph · p. 10, Assumptions and conditioning at a glance · pp. 13–14

Principal results and evidence

Chronological row blocking separates mean collapse from mean-square collapse, which additionally requires the exact transported variance to vanish; the stated summed budget is sufficient but not necessary.The row-RG procedure reports ten complete cells failing the nominal precision gate, including two face interruptions, and does not justify native batch dependence by Gaussian calibration.Source: Principal results and evidence · p. 12
Single-pass width refinement finds moving finite fluctuation regions, while unresolved systematic narrowing does not establish a thermodynamic critical class.The reported refinement uses four trajectories of 196,608 updates; its result is limited to the stated finite family and sampling units.Source: Principal results and evidence · p. 12
Conditional predictive tubes provide finite future-path coverage, but not simultaneous or outer-model coverage; initial-cache transfer fails in all six positive-control cells.Exceptions reach six of 64 contexts, so the reported fidelity is fixed-panel fidelity rather than latency or uniform generated-history accuracy.Source: Principal results and evidence · p. 13
State-specific cache reduction passes in 18 recalibrated cells, while physical-readout claims require conditional head-sign invariance, convergence, and errors small relative to the reference fluctuation scale.A transferred second-moment slope does not by itself establish connected variance, generated-law, or fourth-moment conclusions.Source: Principal results and evidence · p. 13

1.2. The chain of conclusions

Finite mechanics, complete-state dynamics, closure and emission tests, and conditional scaling form a chain in which no stage substitutes for the next.Finite continuation tubes and recalibrated caches yield predictive results only at specified states and error budgets; limiting laws require additional assumptions.Source: §1.2 · p. 16

1.3. The row quotient and its finite work

For fixed finite r, vanishing centered row energy is equivalent to vanishing maximum pairwise row distance, but does not force the common row or downstream operator to vanish.Contraction requires signed work Wt to exceed the nonnegative finite-step charge Qt; the charge retains every pairwise source interaction.Source: §1.3 · p. 17
The complete source increment determines whether a restart preserves the face; small native energy does not establish exact real row equality.The numerical observer must preserve that distinction when reporting gains or restarts.Source: §1.3 · p. 17, §2.1 · p. 23

1.4. A path coordinate and a predictive state

The canonical energy edge composes chronologically, but a block crossing the row-constant face can erase dependence on incoming energy, so endpoint ratios lose source history.This loss concerns only the homogeneous scalar-energy coefficient; parameters, optimizer moments, common rows, and corpus state can still influence later sources.Source: §1.4 · p. 17

1.5. The single-pass resource and the choice of clock

A consuming single-pass corpus induces finite-population dependence, while optimizer memory and physical time require matched clocks for meaningful finite comparisons.Repeated-corpus results do not automatically transfer to single-pass families, and fixed-checkpoint context samples differ from independently trained checkpoints.Source: §1.5 · p. 18

1.6. From geometric concentration to inference

Absolute row energy and relative row concentration are distinct measurements, and PLGA can generate row variation even from constant-row input.Transferring a row theorem to predictive output requires controlling response and defect terms and transporting them through the decoder; finite probes do not uniformly cover every inference prefix.Source: §1.6 · p. 18
Predictive reduction can succeed without exact row collapse: an operator cache approximates context-conditioned means, while risk retains context scatter and training-state displacement.Nested vocabulary projections preserve a specified probability-vector resolution but do not establish an economical autonomous training algorithm.Source: §1.6 · p. 19

1.7. What the finite evidence supports

Finite single-pass families show moving fluctuations without systematic narrowing, and frozen-inference calibration supports finite conditional prediction; criticality and economical autonomous training remain unresolved.Matched interventions reject the tested reduced closure states, while recorded continuations verify finite matrix transport and signed temporal interference.Source: §1.7 · p. 19

1.8. Hypotheses behind the limiting summaries

Limiting head and readout claims require explicit symmetry, dimension, boundedness, convergence, and error-transfer premises; finite symmetry checks or absolute accuracy thresholds alone do not suffice.Readout error can transfer an existing second-moment slope under the stated condition, but does not establish that slope; imposed Ising/Potts source laws provide calibration, not evidence of native self-organization.Source: §1.8 · p. 19

2.2. The conditioned single-pass experiment

The conditioned experiment retains parameters, optimizer clocks, remaining block identities, and program state; its prediction uses a proper input prefix and excludes the external target.The query Gram uses every supplied token, so causal masking does not justify including the target in the model call.Source: §2.2 · p. 24
A thermodynamic sequence specifies model size, initialization, horizon, batch and source sizes, consumed fraction, learning-rate groups, and both Adam memory clocks; a consuming finite source has no nonterminal stationary law without refill.Completed measurements remain tied to their recorded finite family rather than being relabelled as another joint limit.Source: §2.2 · pp. 24–25

2.3. Relation to optimizer dynamics and collapse theories

Training-time selection of a row-constant quotient is not an inference-replacement theorem [31, 32, 33, 37]; unlike depth-driven rank collapse [22] [62] or attention-entropy collapse [77], it tracks optimizer time, with AdamW memory and decay affecting successors.Replacing AdamW by gradient flow, freezing its denominator, or deleting moment coordinates can change the normal successor.Source: §2.3 · p. 25
The model-specific analysis identifies the row-collapse stratum, constructs the implemented successor, retains its invariance defect, and links its normal coordinate to a physical row map; its radial/tangential split and zero-face restart coordinate are exact constructions.The radial/tangential split projects an executed finite secant onto the current observable state; it is not an optimizer-Jacobian eigendecomposition or continuous-time Lyapunov derivative.Source: §2.3 · p. 25
The exact block criterion gives collapse if and only if anchor energies and block excursions vanish; positive variation is sufficient but not necessary.This criterion characterizes the declared orbit, including paths that oscillate inside a shrinking envelope.Source: §2.3 · p. 25, §3.1.4 · p. 29

2.3.1. Probabilistic reduction and the architecture-specific contribution.

Predictive renormalization retains the native state under elimination, row-face bookkeeping, and tests of predictive closure; architecture-specific calibration coverage averages over the calibration draw, not every realized tube at every state.The cache analysis concerns state transport, recalibration, signed risk accounting, and fresh-context fidelity; finite evidence does not establish a diverging correlation length or autonomous critical attractor.Source: §2.3.1 · p. 26

3.1.2. Physical row quotient.

For fixed finite row count, energy decay, centered-map collapse, and row-diameter decay are equivalent; physical collapse holds exactly when |γk,t| √υk,t →0 for every layer k, although neither factor must converge separately.The centered quotient removes common-row translations, so the learned common row can retain nontrivial geometry at collapse.Source: §3.1.2 · p. 28, §3.1.3 · p. 28

3.1.5. Exact face and energy strata.

The exact face is the zero centered map, while positive energies divide into floor-censored and resolved states; a certified enclosure distinguishes a face, positive state, and unresolved near-face state.An upper bound Ut →0 certifies energy convergence, but a fixed tolerance Ut ≤ϵ∗ does not.Source: §3.1.5 · p. 30, §3.2.1 · p. 30

3.2.2. Arithmetic resolution and the native estimand.

Native centered rows and their energies are the primary endpoint estimand; replay disagreement alone does not certify exact energy error, and unresolved relative gains or zeros should not be reported.The factorized reconstruction is a finite-precision diagnostic because evaluation order can change rounding; a separate error analysis is needed for a certified exact-energy bound.Source: §3.2.2 · p. 31

4.1. Exact direct observable-energy dynamics.

The direct observable-energy balance uses executed row-map increments without Taylor truncation, optimizer-state closure, or assuming the row-constant face is invariant.The complete-state cocycle separately describes how hidden optimizer coordinates transport perturbations.Source: §4.1 · p. 32

4.1.1. The exact physical work ledger.

For every implemented update, the exact finite-step identity is Et+1 − Et = −Wt + Qt, and a complete source ledger retains cross-Gram terms.Energy decreases, stays equal, or increases exactly when W exceeds, equals, or falls below Q; at the row-constant face, E+ = Q, so contraction is impossible and the face persists only if D = 0.Source: §4.1.1 · pp. 32–33

4.1.2. Exact gate, shape, interaction, and arithmetic-defect secants.

Native energy change decomposes exactly into gate, shape, interaction, and arithmetic-defect sources; factorized collapse plus vanishing defect suffices for native collapse, though neither condition alone is necessary.Omitting the bilinear interaction makes the ledger only a first-order approximation; factorized and native terms can also cancel.Source: §4.1.2 · pp. 34–35

4.1.3. Endpoint-exact parameter sources under AdamW.

An integrated implemented-parameter secant assigns the executed physical increment exactly to declared parameter displacements, including AdamW sources, but does not itself identify a causal intervention.A causal claim requires a paired successor; the secant applies under piecewise continuous differentiability along the parameter segment.Source: §4.1.3 · p. 35

4.1.4. Exact paired-intervention contrast.

The sign of a fixed paired intervention depends on the current physical state and natural increment, not solely on the source label or the intervention’s norm.The contrast includes state projection, natural-increment interaction, and a nonnegative intervention charge; a source-resolved ledger retains its linear and quadratic cross terms.Source: §4.1.4 · p. 36

4.1.5. Exact radial–tangential normal form.

For positive Et, contraction occurs exactly when 0 < αt < 2 and tangential charge stays below the remaining radial margin.Failure is exhaustively due to a noninward or overshooting radial coefficient, or tangential charge exceeding that margin; the source-resolved radial ledger retains tangential cross-Gram terms.Source: §4.1.5 · pp. 36–37
The exact radial–tangential contrast separates radial work from tangential charge and interaction; radial work predicts change only when it dominates the tangential terms.For a positive-energy excursion, collapse is equivalent to the exact chronological product of energy ratios vanishing.Source: §4.1.5 · p. 38
Along an infinite positive-energy excursion, collapse occurs exactly when the chronological product of the energy ratios vanishes.Source: §4.1.5 · p. 38

4.1.6. A direct nonautonomous attraction envelope.

If the work margin satisfies the direct energy bound, a nonautonomous envelope yields collapse when its propagated initial-energy and forcing terms vanish; no pointwise contraction is required.The complete state may be expansive even when the observable work margin closes, and the converse route through observable cancellation need not imply that margin.Source: §4.1.6 · p. 38

4.1.7. Reopenings and the row-constant face.

Arbitrarily late small energies imply energy convergence to zero when absolute positive reopenings are summable.Source: §4.1.7 · p. 39
At zero energy, the exact affine cocycle represents a restart without a relative gain; preserving the row-constant face requires a zero physical increment.At the face, linear source work vanishes, so preservation depends on the complete increment and its quadratic charge; a zero gate is preserved only if the implemented optimizer yields no effective physical increment.Source: §4.1.7 · pp. 39, 41
In each of 18 executed dense windows, expanding endpoint pairs occur at every measured horizon from 1–32; at horizon 32, 58,363 of 143,424 expand, falsifying a simple fixed-window uniform-contraction description.Failure of this sufficient premise falsifies neither the exact cocycle nor row-map collapse.Source: §4.1.7 · p. 40

4.1.8. Comprehensive direct collapse theory.

For a finite registry, row-diameter collapse, vanishing stacked energy, the radial-face cocycle criterion, and the cumulative work-charge ledger are equivalent characterizations.The cocycle characterization requires extinction of both propagated initial energy and transported zero-face restarts.Source: §4.1.8 · p. 41
A vanishing direct envelope, summable reopenings after arbitrarily late small energies, or qualifying anchor-block conditions each suffices for collapse; the characterization does not assert AdamW supplies these closures.Establishing a sufficient closure requires an all-future analytical bound.Source: §4.1.8 · p. 41

Complete-state AdamW dynamics

This section develops complete-state AdamW dynamics for row-map behavior, retaining optimizer memory and all coordinates that can affect the selected layer’s row-map normal. It derives exact transport identities and scheduled-norm tube bounds while distinguishing local variation, forcing, nonlinear observation effects, and collapse conditions, and states why vanishing observations or finite-horizon behavior alone do not establish state convergence.

5.1. Complete-state forced AdamW cocycle

The selected layer’s complete state retains every parameter and AdamW-moment coordinate that can return to its row-map normal, while the changing environment makes its update nonautonomous.Conditioned on the minibatch, data cursor, scheduler, clipping cell, decay mask, random state, and intervention branch, each update is deterministic.Source: §5.1 · p. 43, §5.1.1 · p. 43

5.1.2. Implemented smooth-cell derivative.

The implemented smooth-cell AdamW Jacobian includes the live denominator response and actual-batch upstream gradient paths, so freezing the preconditioner gives a different successor.A differentiable post-update intervention modifies the applicable position block; a shared additive perturbation cancels between affine responses.Source: §5.1.2 · pp. 43–44

5.1.3. The gate-zero stratum is a dynamical condition.

A zero gate persists exactly when its updated first moment vanishes; from zero first moment this requires zero clipped gradient, while weight decay cannot reopen the gate.The retained second moment changes the magnitude of reopening but not its zero criterion, and physical row-face invariance can be weaker than invariance of the full gate-zero parameter face.Source: §5.1.3 · p. 44

5.1.4. Transverse charts and the invariance defect.

On an admitted chart itinerary, the realized normal coordinate has an exact forced linear-plus-remainder recurrence; its force may be nonzero, and the coordinate is not thereby autonomous or Markov.The force vanishes when the reference successor lies in the next selected row-constant stratum; batch imbalance, tangent-base movement, data drift, branch changes, or numerical evaluation can contribute.Source: §5.1.4 · p. 45

5.1.5. Exact observable transport and cancellation.

Physical row-map collapse over a finite registry is equivalent to convergence of the full observable transport vector, and opposing terms can cancel even when component norms do not vanish.Requiring every component to vanish is sufficient but not necessary; replacing the signed identity with a triangle bound removes cancellation and yields a stronger certificate.Source: §5.1.5 · p. 47
Finite-increment state and observation balances remain exact even when paired trajectories take different routes, but their correction terms do not establish asymptotic or operator-norm behavior.Large cancellation indices and negative inner products are measurements, not consequences of the identity.Source: §5.1.5 · p. 48
The exact recurrence transports observables across route and chart changes, but differentiability is needed for Jacobian identification or higher-order remainder bounds.At exact-zero AdamW variances, the square-root map is not two-sided smooth; directions leaving the variance cone require separate stratum analysis.Source: §5.1.5 · pp. 48–49

5.1.6. Scheduled metrics and variable tubes.

Under the stated itinerary and forcing conditions, scheduled-norm tubes remain invariant and collapse without requiring comparisons among the scheduled metrics.The theorem uses chronological products and transported forces; individual gains may exceed one, and matrix factors need not commute.Source: §5.1.6 · pp. 49–50
A finite uniform physical-cover bound transfers scheduled-state collapse to registered row-diameter collapse, while fixed-norm transfer additionally requires a uniform lower metric bound.No upper bound on the scheduled metrics is needed for either stated transfer.Source: §5.1.6 · p. 50
Finite-horizon tube inequalities establish bounds only through the declared horizon and do not imply asymptotic collapse..Source: §5.1.6 · p. 50

5.1.7. Continuous-time specialization.

In continuous time, observable collapse is equivalent to the full transported expression tending to zero, but this specialization alone proves nothing about discrete AdamW.A discrete-to-continuous passage must preserve features including clipping strata, bias-correction clocks, finite learning rates, and coordinatewise second moments.Source: §5.1.7 · p. 51

5.1.8. Stochastic forcing.

For deterministic propagators and centered martingale forcing, mean-square collapse requires both the transported mean and variance to vanish; mean collapse alone does not control fluctuations.Random state-dependent propagators need additional control because they may depend on the noise they transport.Source: §5.1.8 · p. 52

5.1.9. Necessary obstructions.

Under boundedness and remainder conditions, convergent full state requires vanishing realized invariance defect, while collapsed observations can persist with a nonzero state.Scheduled-norm convergence need not imply Euclidean convergence without a uniform metric lower bound, and finite observed prefixes do not establish asymptotic collapse.Source: §5.1.9 · p. 53

From row geometry to PLGA and predictive emission … Positive affine blocking through the row-constant face

These parts connect PLDR row geometry and row-energy transport to PLGA behavior and predictive outputs, distinguishing absolute row-energy collapse from normalized concentration and predictive fidelity. They combine finite experimental tests and exact affine-blocking ledgers—including face-hit restarts—while keeping observer- and optimizer-dependent findings separate from asymptotic guarantees.

6.1.1. Five sufficient structural conditions.

Five structural conditions—row-constant W1, b, p, and ba, plus a1 in the span of the all-ones vector—suffice for PLGA to preserve row constancy.Their quotient norms diagnose the sufficient conditions, while the direct defect at the registered constant input is the definitive test; the conditions are not necessary.Source: §6.1.1 · p. 55

6.2. PLGA and complete-state transfer

The fixed-parameter PLGA map is downstream of PLDR, but upstream row collapse need not be inherited automatically.Its implemented positive base makes arbitrary real learned powers well defined and the map continuously differentiable for finite inputs.Source: §6.2.1 · p. 55

6.2.1. Constant-input invariance defect.

The five structural conditions are jointly sufficient, and the last two become necessary when the first three hold, the intermediate response is nonconstant, and the face is preserved.Without nondegeneracy, annihilation or parameter cancellation can preserve the face despite violating those conditions.Source: §6.2.1 · p. 56

6.1. PLGA transfer as a quotient-defect problem

For every input, PLGA’s output quotient decomposes into its response to the occupied quotient segment plus its defect at the row-constant projection.At a row-constant input the response vanishes, so the output quotient equals the defect; upstream collapse alone does not ensure downstream collapse.Source: §6.1 · p. 54, §6.2.1 · p. 56

6.2.2. Embedding the defect in the affine RG.

In the positive-affine RG, PLGA response acts as a gain and its constant-input defect as a source; optimizing the bound cannot remove a nonzero defect.Sequential downstream collapse follows when response is uniformly bounded and tends to zero while the defect also tends to zero.Source: §6.2.2 · p. 57

6.2.3. Complete optimizer state.

A complete optimizer state supports positive-affine Lyapunov comparisons that imply row-energy collapse, but the converse fails when hidden optimizer modes can later re-enter the row quotient.Complete-state and observable RGs may therefore have different gains, sources, fixed sets, and universality classes.Source: §6.2.3 · p. 58

6.3. Absolute collapse, relative concentration, and predictive error

Absolute row-energy collapse and normalized row concentration are equivalent only under both an upper and a positive lower bound on total energy; finite decoder error propagates through layer Lipschitz factors and local errors.A row-to-decoder bound may contribute both a term proportional to the square root of row energy and a constant-input or replacement defect.Source: §6.3 · p. 59

6.3.1. A worked chain from row energy to predictive fidelity.

Energy alone does not determine a successor: the same incoming matrix yields energies 1/2 or 9/2 under different realized contrast increments.Predictive fidelity is conditional: a row-energy-based logit bound requires justified downstream sensitivity and a defect term, while small row energy alone does not bound that defect or justify a shared cache.Source: §6.3.1 · p. 60, §6.3 · p. 60

7.1.3. Float64 endpoints and mixed closure.

Six of nine trajectory-layer medians meet E < 10^-3, while A2, C1, and C2 remain far from the row-constant face.Five of the six small endpoints show stronger normalized-shape closure than effective-gate closure; B1’s shape reopens by 2.228 decades despite 1.884 decades of gate closure, leaving a 0.336-decade energy reopening. Pooling layers would erase this layer resolution.Source: Table 7.1 · p. 62, §7.1.3 · p. 62

7.1.1. Logical role and analysis unit.

At A1, median float32 relative energy error is 2.66 and maximum error is 12.35; 6,685 coordinate energies are zero in float32 but positive in float64.The real-arithmetic factorization residual remains below 10^-12 in every float64 record, so the false zeros reflect observer resolution. The statistical unit is a trajectory-layer pair, not each of the 96 shared-weight map observations.Source: §7.1.1 · p. 61, Table 7.2 · p. 62, §7.1.3 · pp. 62–63

7.1.4. Lifted final-gate operators.

At update 4096, none of nine lifted final-gate operators has induced norm or spectral radius below one, ruling out instantaneous contraction of that substate.Induced norms range from 8.58 to 791.81 and spectral radii from 1.009 to 1.471; this does not rule out the complete-state theorem or characterize a long chronological product.Source: Table 7.3 · p. 63, §7.1.4 · p. 64

7.1.5. Same-device component interventions.

In every unit, removing the upstream row-program write has a larger one-step response than removing the final-gate write, while the joint response is not assumed additive.The same-device placebo is at most 0.00822 of the smallest main removal response. A first-order upstream predictor passes only for A0 and B1, with residuals 0.181 and 0.230; the other seven residuals range from 0.314 to 1.081.Source: Table 7.4 · p. 63, §7.1.5 · p. 64, §7.1.6 · p. 65

7.1.7. Direct PLGA quotient defect.

At A0, median input quotient 1.31×10^-7 accompanies median PLGA defect 81.77; none of nine units preserves the row-constant face at registered constant inputs.None of 180 tested sufficient-condition residuals falls below 10^-10, rejecting automatic inheritance of upstream PLDR row collapse by downstream PLGA, not PLGA as an attention mechanism.Source: Table 7.5 · p. 63, §7.1.7 · p. 65

7.1.8. Matrix-free complete-state tangent response.

For the complete-state tangent predictor, 14/27 cells qualify at one update, but none qualifies at horizons two, four, eight, or sixteen.On held-out paths B and C, 30/54 signed-response tests pass, but only 7/54 visible gains are below one; directional medians are 1.3401, 34.0398, and 4.0044 for row adjoint, optimizer displacement, and balanced full state. These float32 observations do not establish operator norms or asymptotic contraction.Source: Table 7.7 · p. 66, Table 7.9 · p. 66, Table 7.10 · p. 66

7.1.9. Retained physical-energy and reopening replay.

Across 1,056 captured update edges, 131,748 of 304,128 stored scalar energy increments are positive, and every registered map identity reopens at least once.The narrower full-map archive has 474 positive increments among 864 pairs (54.86%); its independently replayed work-charge residual is at most 1.5×10^-12. Dense records lack physical arrays, so signed work and finite-step charge cannot be evaluated for them.Source: §7.1.9 · p. 67, Table 7.11 · p. 68
Six cancellation-dominant held-out units occur in the replay, concentrated as five on one path and four in layer 2, but do not establish an asymptotic collapse theorem.The replay establishes finite reopenings and exact-ledger compliance, not summability of r_t or a future work margin.Source: §7.1.9 · p. 68

7.1.10. Exact finite-increment observable balance.

Only 6 of 54 held-out units are cancellation-dominant, refuting universal cancellation; state correction balances an unstable multistep tangent selectively, not uniformly.All six cancellation-dominant units are row-adjoint directions, while optimizer-displacement and balanced directions have observed responses close to their homogeneous responses.Source: §7.1.10 · pp. 68–70

7.1.11. Precision-resolved paired source experiments.

The temporal replication refutes persistent positive sign in locked layer 2 on both held-out paths, although the exact paired-contrast state-projection term matches 96.8% of sign-eligible maps.The state-projection term is largest in 2,115 maps; interaction and charge still matter in the cancellation tail.Source: §7.1.11 · pp. 71–72

7.1.12. Exact radial and tangential edge analysis.

Among 4,608 eligible arm maps, 2,452 contract and 2,156 reopen; paired radial contributions match 2,236 of 2,304 raw contrast signs.Of the reopenings, 2,015 fail the radial interval and 141 violate the tangential-charge condition.Source: §7.1.12 · p. 72

7.1.13. Fresh-context radial holdout.

Fresh-context radial prediction yields 96.75% pooled sign agreement among 1,632 eligible maps, but the all-cells claim remains insufficient because one cell has no policy-eligible maps.No powered cell falls below 70% agreement; the excluded cell’s 96 maps are positive but below the frozen observer floor.Source: §7.1.13 · p. 73

7.1.14. Observer-stratum and finite-block analysis.

No exact source-energy zeros occur among 5,184 maps, and every measured dense window contains expanding pairs, falsifying fixed-window uniform contraction on this sample.The finite analysis does not test the sufficient premises of Theorem 4.16 or rule out collapse through another exact route.Source: §7.1.14 · p. 74

7.1.15. Controlled gate-zero invariance test.

In all nine controlled gate-zero tests, natural clipped gate-gradient writes reopen the row face, while zeroing that selected gradient preserves it bitwise.Other gradients match, verifying the stated invariance condition by intervention at these states.Source: §7.1.15 · p. 75

7.1.17. What the finite record establishes.

The finite record shows layer-selective row-map collapse, requires the full nonlinear optimizer state, and rejects generic cancellation and fixed-window uniform contraction.Six of nine float64 endpoints are small; the source-written upstream predictor fails in seven of nine units, and every dense window contains expanding endpoint pairs at every measured horizon through 32.Source: §7.1.17 · pp. 75–76
Finite observations show PLGA does not automatically inherit the upstream row-constant face, but they establish no infinite-time premise for physical row-map collapse.The record supports the observer-resolved work-charge theory, positive-energy radial normal form, and exact-face branch, without establishing a complete-state tube, closing block sequence, or vanishing invariance-force convolution.Source: §7.1.17 · p. 76

8.1.1. Radial and tangential edge geometry.

On every positive-energy edge, energy decreases exactly when radial motion is inward and tangential charge stays below its radial margin.The decomposition into radial and tangential components is unique; failure occurs through a noninward or overshooting radial step, or excessive tangential charge.Source: §8.1.1 · p. 78

8.1.2. Canonical face completion.

Canonical face completion preserves the exact face only when the complete centered increment is zero; an intermediate face hit can make endpoint-only recanonicalization lose path information.For example, the path 1, 0, 1 composes to (0, 1), whereas recanonicalizing only its endpoints gives (1, 0).Source: §8.1.2 · p. 79

8.1.3. Ordered affine blocking.

Chronological affine edges form an associative semigroup, and their block formulas remain valid through exact face hits, where transported energy is erased and a restart enters as a new source.Aligned blocking composes exactly, even when fine edges differ or do not commute.Source: §8.1.3 · p. 80

8.1.4. Interval-affine observer RG.

Interval-affine edges provide monotone enclosures for exact compositions, but floating-point regrouping can change valid intervals and cannot certify a near-face state as an exact source event.When an energy interval includes zero but its successor interval permits positive energy, the intervals alone imply no finite upper gain bound.Source: §8.1.4 · p. 81

8.1.5. Endpoint-gauge covariance.

Endpoint-gauge changes commute with exact finite blocking, but physical energy convergence is preserved only under bounded gauges or otherwise controlled transfer.Normalization must therefore be recorded rather than silently absorbed.Source: §8.1.5 · p. 82

8.1.6. Finite-registry matrix extension.

Nonnegative matrix comparison edges form an associative affine semigroup that composes valid registry bounds and can encode proved cross-registry transfers.Diagonal gains recover independent scalar comparisons; off-diagonal entries can represent physical covers, shared optimizer comparisons, or layer-to-layer transfer.Source: §8.1.6 · pp. 82–83

8.1.7. Exact two-sector collapse criterion.

Total energy tends to zero exactly when both transported initial energy and accumulated restart energy tend to zero; for a finite registry, this is equivalent to collapse of every registered row diameter.This is a characterization, not an optimizer guarantee: a collapse theory must close both the memory of initial transverse energy and every restart after later transport.Source: §8.1.7 · p. 83

Stratified flow and orbitwise collapse criteria · Closed states, gauge fibers, and conditional universality · Executed closure and observer tests

This section develops criteria for stratified flow and collapse, distinguishing orbit histories, pathwise blocking, and autonomous reduction across complete-state fibers, including gauge and optimizer coordinates. It sets out conditional temporal and universality claims and reports executed closure, observer, and intervention tests, whose conclusions are limited to the tested domains.

9.1.1. Homogeneous flow in the comparison cone.

The homogeneous comparison flow fixes the memoryless line and (1, 0), but only (0, 0) is absorbing; positive-source critical recurrence diverges.Homogeneous blocking of a pair with qζ > 0 is a comparison flow, not a realized fine-edge trajectory.Source: §9.1.1 · p. 84

9.1.2. Relevant fields and clean exponents.

The exact linearized dimensions are yτ = yζ = 1 and ϕ = 1, not fitted trajectory exponents; the declared correlation-length exponent is ν∥ = 1.There is no path-independent order-parameter exponent: if ζ scales as τ^p, collapse requires p > 1.Source: §9.1.2 · p. 85

9.1.3. Face-free and finitely reopened orbits.

On a positive tail, energy collapses, diverges, or tends to L exactly when cumulative log gain tends to −∞, +∞, or log(L/Ek); a nonzero mean rate determines exponential decay or growth.After finitely many face hits, this criterion applies from the final restart; an orbit remaining on the face is already collapsed.Source: §9.1.3 · pp. 86–87

9.1.4. Recurrent faces and the exact excursion criterion.

With recurrent positive restarts, collapse occurs exactly when excursion peaks rjMj vanish; vanishing restart levels alone is insufficient, and χexc separates collapse from peak divergence at thresholds below or above one.At χexc = 1, subleading terms decide the outcome.Source: §9.1.4 · p. 88

9.1.5. Uniform comparison closure.

Uniform contraction with vanishing forcing implies decay, and vanishing maxima over an unbounded block cover imply collapse along the full orbit.Source: §9.1.5 · p. 89

9.1.6. The stratified critical set.

Collapse criteria depend on orbit stratum: positive tails use cumulative log gain, recurrent restarts use rjMj, and the full comparison cone requires both homogeneous memory and transported source to vanish.No scalar gain coordinate alone is complete; typical contraction, uniform contraction, restart extinction, and orbit collapse are distinct.Source: §9.1.6 · pp. 89–90

10.1.1. Standard complete-state blocking and scale-indexed lumpability.

A b-step observable transition is autonomous for every initial law exactly when states in each observable fiber have identical b-step pushforward laws; closure then holds at multiples of b but need not hold at one step.Row energy alone need not be Markov when equal-energy states have different next-energy laws.Source: §10.1.1 · p. 91

10.1.2. Gauge fibers and optimizer-coordinate closure.

Under an equivariant gauge action, consistent parameter-and-moment transformations are null controls, while unequal successor energies under paired parameter-only and moment-only changes reject raw (e, m, v) closure on that domain.This does not rule out a gauge-covariant moment coordinate, optimizer-state quotient, or another sufficient statistic.Source: §10.1.2 · p. 94
A fixed finite probe cannot globally close a domain containing unobserved model coordinates that later affect training and change the readout; this does not settle closure on a restricted reachable orbit.Changing an embedding row absent from every probe input leaves the probe unchanged, but a later batch can read that row and alter the post-update readout.Source: §10.1.2 · pp. 94–95

10.1.3. The renormalized random variable.

Multiplicative universality requires observer-certified positive excursions, aligned exact-cocycle blocking, specified temporal and dependence limits, sample-independent centering and scaling, and convergence to a shared nondegenerate law.The definition also requires matching drift scaling dimensions, persistence across held-out seeds and the declared architecture family, and treatment of face restarts.Source: §10.1.3 · p. 95
A gain-only universality class describes a quotient observable, not the complete training state or model outputs, unless closure or an explicit comparison to those quantities is established.The final source condition must also be closed for the gain-only class to describe the complete row-collapse process.Source: §10.1.3 · p. 95

10.1.4. Clean multiplicative class.

For constant log gain Yt=µ, fluctuations vanish; the sign of µ determines the clean phase, µ=0 is source-free criticality, and the e-fold scale has exponent one.This zero-variance case is distinct from the normalized Gaussian class because square-root-variance normalization is singular at zero variance.Source: §10.1.4 · p. 95

10.1.5. Finite-variance temporal disorder.

Under independent, identically distributed log gains with finite positive variance, the normalized block law converges to N(0, 1), and the standardized drift has scaling dimension 1/2.The stated crossover exponent is derived from the declared drift-to-fluctuation balance, not separately fitted empirically.Source: §10.1.5 · p. 96

10.1.5. Finite-variance temporal disorder

Under iid finite-variance increments, the typical energy e-fold scale is asymptotically 1/|µ|, whereas deterministic drift dominates square-root fluctuation at scale σ^2/µ^2.For stationary mixing sequences, the corresponding fixed point requires a central limit theorem with positive long-run variance; those assumptions require testing.Source: §10.1.5 · pp. 96–97

10.1.6. Stationary and nonautonomous probability RG domains

A nonautonomous probability RG must retain block origin, cumulative drift, and location-dependent centering and scaling; a stationary universality label requires a controlled limiting domain.The deterministic drift sector may be asymptotically closed while the centered fluctuation sector still depends on block origin.Source: §10.1.6 · p. 97

10.1.7. Other conditional subclasses

For conditional correlated-sum normalization b^H L(b), the crossover exponent is 1/(1−H); stable attraction gives α/(α−1), but need not give a two-sided pure-power bound.The proposition requires a diverging scale satisfying its balance and does not assert a unique finite crossover or limiting fluctuation law; the logarithmic-tail example has an unbounded multiplier.Source: §10.1.7 · pp. 97–98

10.1.8. Reopening is a relevant affine field; 10.1.9. Recurrent-face affine classes

Gaussianization of positive-excursion log gains is insufficient for row-map collapse without source closure; recurrent-face laws must retain excursion timing and peak, and no fixed distribution is asserted without restart assumptions.A full universality claim requires face invariance, vanishing transported restart effects, or a joint gain-and-source fixed law.Source: §10.1.8 · pp. 98–99, §10.1.9 · p. 99

10.1.10. What identifies an empirical class; 11.1.3. Registered four-trajectory PLDR experiment

Empirical class identification requires raw energies, certified observer intervals, aligned blocks, a stable temporal domain, dependence tests, convergence evidence, held-out seeds and architectures, and source irrelevance or a fitted reopening law.The registered decision procedure was qualified before the evidence-producing run; the four-trajectory design used a 131,072-update holdout as its resource-capped horizon.Source: §10.1.9 · p. 99, §11.1 · p. 100, §11.1.3.1 · p. 102

11.1. Executed computational results

Ten positive-domain cells failed the nominal precision gate, while two cells met face interruptions across four trajectories; nested measurements do not add independent trajectories or establish a universality class.The registered run is also a design-precision record for its nominal confidence-width target.Source: §11.1 · p. 100

11.1.1. Deterministic and stochastic qualification

At block size 256, the largest non-Gaussian Kolmogorov distance was 0.02834 and the largest pairwise empirical Wasserstein distance was 0.03309 for controlled finite-variance independent inputs.These experiments qualify the reducer on known inputs, not stationarity, mixing, or a domain of attraction for PLDR training; finite-difference checks agreed with q log q to maximum relative discrepancy 7.025 × 10^-7.Source: §11.1.1 · p. 101

11.1.2. Guarded reduction of retained trajectories

All six aggregate medians were below one, but captured cells had heterogeneous flow: four of nine were guaranteed to decrease by at least ten orders of magnitude, and none was guaranteed to increase.Every captured window contained expansion at every horizon; the largest retained relative excursion was 1.39097 × 10^4, despite typical endpoint contraction.Source: §11.1.2 · p. 102

11.1.3.1. Experimental design and statistical calibration.

Zero of the 10 complete cells met the design-only precision target, while two cells were labeled face-interrupted from recorded bitwise row equality.Native batch dependence was not established, so the procedure is reported with nominal calibration; precision failures do not establish zero drift or a critical phase.Source: §11.1.3.1 · p. 107, §11.1.3.2 · p. 109

11.1.3.2. Flow, temporal domain, and finite-scale laws.

No complete cell met the registered finite-scale criterion or precision target, and no cell reached confirmatory held-out prediction.Eleven of 30 matched-normal tests rejected; only two cells passed the uniform-negative screen, neither passed all fit gates, and zero predictions passed.Source: §11.1.3.2 · p. 109

11.1.3.3. Observer regime and stage-resolved represented zeros.

All 20 first-coalescing float32 LayerNorm cases had strictly positive exact-output lower endpoints, so represented equality did not certify an analytical face hit.Every represented excursion peak remained below the observer floor despite maximum restart amplification of 6.872 × 10^10; exact evaluation applied to rounded arrays does not certify the ideal real computation.Source: §11.1.3.3 · p. 110

11.1.3.4. Source-remeasured matched-fiber test.

The tested moment interventions rejected autonomous energy-vector and raw-moment closure, while consistent gauge actions preserved successors exactly.Zeroing first moments changed all 96 successor energies at every scale; the full consistent gauge action had zero successor defect. Raw-moment augmentation was rejected on the tested gauge domain, without deciding uniqueness or minimality.Source: §11.1.3.4 · p. 112

11.1.3.5. Executed staged predictive-state closure gate.

Every prespecified reduced state was rejected on the intervention domain, while the consistent complete-state gauge quotient was not rejected.All 48 successor maps differed at each of five scales for every candidate in design and held-out trajectories; the quotient control had zero differing maps at all five scales.Source: §11.1.3.5 · p. 113

11.1.3.6. Implemented PLGA checkpoints.

All 192 trained PLGA maps exceeded the 10^-6 constant-input defect tolerance at every checkpoint, while constrained heads had a maximum defect of 8.468 × 10^-15.The smallest final trained-map defect was 3.202×10^-5; the results reject pointwise row-constant face preservation at the measured checkpoints, not the conditional transfer theorem.Source: §11.1.3.6 · p. 114

11.1.3.7. Implementation and numerical integrity.

Chronological replay, positive-tail telescoping, and PLGA decomposition residuals were at most 2.218 × 10^-16, 1.039 × 10^-15, and 1.389 × 10^-16, respectively.These were integration-test results, not experimental support for mathematical identities.Source: §11.1.3.7 · p. 114

11.1.3.8. Empirical decision ledger.

The empirical ledger supports no uniform PLDR phase or asymptotic universality class on this domain, and the closure tests motivate retaining a complete conditional law.The exact affine state organization applies to every exact branch, while interval RG retains unresolved observer states without inventing a source.Source: §11.1.3.8 · pp. 114, 118
At b=1, 2, 4, and 8, consistent full Cg was 0/96 while parameter-only Pg changed successors in 96/96, rejecting raw (e, m, v) closure.The parameter-only action preserves source energy and raw first and second moments; the consistent action is the exact null control.Source: Table 11.23 · p. 115, §11.1.3.8 · p. 115

11.1.3.5. Executed staged predictive-state closure gate.

Every reduced candidate was rejected in both splits, leaving no licensed reduced flow, critical surface, critical exponent, or universality fit.The staged closure decision was terminal, not a pending confirmation program; it does not rule out closure on a smaller invariant reachable manifold.Source: §11.1.3.5 · p. 113, §11.1.3.8 · p. 114, Table 11.26 · p. 116

Complete laws and consuming-corpus training · The law retained under elimination · A single pass as a finite data resource

The section specifies training and inference laws on complete augmented states, including optimizer, execution, and—under single-pass training—the remaining corpus and its sampling convention. It distinguishes exact conditional-memory descriptions from autonomous or approximate reductions, setting out their closure and error requirements while accounting for finite-corpus uncertainty and the limits of distribution-averaged descriptions.

The law retained under elimination

The retained model-wide law links complete training state to inference emissions and supports exact maps as well as explicitly qualified predictive reductions.The complete state includes parameters, optimizer memory, schedule and clocks, remaining corpus, and stochastic-program state.Source: The law retained under elimination · p. 120, §12.1 · p. 120

12.1. The object being renormalized.

A renormalization description must specify its scale, retained variables, map, and acted-on law; separate scalar head norms do not specify a model-wide predictive state.PLDR attention heads interact through output mixing, residual paths, normalization, and feed-forward networks.Source: §12.1 · p. 120

12.1.1. From a complete law to a predictive theory.

Exact scale maps act on one joint conditioned law, retaining shared randomness, source insertions, and aligned block boundaries; a separately trained smaller network needs an additional model-matching hypothesis.Conditioning on a realized corpus and order is distinct from averaging over corpora.Source: §12.1.1 · p. 120, §12.1 · p. 120

12.2. Exact model-wide renormalization maps

Temporal, decoder, document, and linear observation maps act on the same declared experiment, but commuting constructions require compatibility assumptions and changing native width is not exact projection to a smaller PLDR network.The complete law provides the common foundation for the section’s model-wide construction.Source: C6: Predictive and cache reduction · pp. 122–123

C6: Predictive and cache reduction.

The primary training law is a single pass through specified RefinedWeb data, so the remaining corpus itself must be retained because its block count generally does not identify its law.Approximate predictive reductions need a successor law; their projected-law error obeys et+1 ≤ Lt et + dt, with initial-law and emission errors also accounted for.Source: C6: Predictive and cache reduction · pp. 122–123

12.1.2. Contributions and related work.

The contribution connects PLDR row dynamics to finite-corpus transport, optimizer memory, measured training selection, and quantitative reduction errors, extending learned-operator caching [33] with causal-prefix and answer-margin analysis.Exact compatibility, conditional reductions, and finite empirical evidence have distinct evidential roles; an economical reduced training law and critical-surface claims remain unestablished.Source: §12.1.2 · pp. 124–125

12.2.1. Decoder elimination with retained source insertions.

Decoder blocks compose associatively by composing boundary maps and pulling later source insertions back through earlier maps, preserving source partition functions and terminal predictions.The resulting effective block need not be another PLDR decoder with fewer weights.Source: §12.2.1 · p. 125

12.2.2. Joint-law coarse observation and data blocking.

Compatible aligned-block observations compose exactly when the complete joint block dependence is retained; replacing dependent fields with marginal products changes the renormalization.The graph-map and law-map route gives the same retained fields as observing and blocking the fine computation.Source: §12.2.2 · p. 126

12.2.3. What head or state elimination retains.

Eliminating unresolved state variables generally leaves memory kernels, so head norms alone cannot replace the full state in a predictive recurrence.The kernels vanish only under additional structure.Source: §12.2.3 · p. 126

12.3.1. An augmented state and two different scale coordinates.

Exact temporal blocking composes transition kernels on the augmented state and preserves terminal inference laws; optimizer time, block length, and document scale remain distinct coordinates.The augmented state includes parameters, optimizer variables, sampler state, and the step counter; source insertions and linear coarse observations are also compatible with regrouping.Source: §12.3.1 · pp. 127–128

12.3.2. When a smaller training state closes.

A smaller autonomous training state requires equal next-step retained laws for fine states sharing its retained coordinate; differing AdamW moments can violate this condition.The exact alternatives are the full augmented state or the projected path law with induced memory.Source: §12.3.2 · p. 128

12.4. Predictive reduction and training selection of inference regimes

The exact predictive object couples the training transition kernel with an inference emission, while replacing it with collective variables introduces a separate approximation.The emission may retain the full graph trace, generated operators, and source derivatives jointly.Source: §12.4 · p. 129

12.4.1. Compatible approximate kernels.

Total-variation and Wasserstein closure bounds propagate one-step and emission defects, but uniform domain-wide defects are distinct from finite-trajectory prediction errors.Source-weighted transfer operators need their own norm factors, and unresolved optimizer coordinates or earlier coarse states can instead be retained as conditional memory.Source: §12.4.1 · pp. 129–130

12.4.2. Closure under the evolving training law.

Under the evolving training law, successor defects are averaged against the fine law and propagated through coarse transport factors; empirical pairing bounds need no independence assumption.Estimating expected defects requires reset branches to sample the specified conditional successor law, and a fitted coarse map must be independent of paths used to estimate defects; a full-vocabulary risk coordinate still requires a native forward pass.Source: §12.4.2 · pp. 130–131, . The empirical pairing cost satisfies · p. 131

12.4.3. Exact closure with retained conditional memory.

Retaining the conditional law of the fine state given retained history yields an exact closed update and emission, but does not establish a finite-dimensional training reduction.A useful finite approximation also requires accurate vocabulary emission, collective covariance in fixed units, retained successor laws, and a measured acquisition budget.Source: §12.4.3 · p. 132

12.4.4. Required accuracy along an increasing horizon.

For an increasing horizon, the stated stability result assumes zero initial error and bounds each successor error by L times the prior error plus defect d.The supplied passage states the recurrence conditions but does not include the displayed accumulated bound.Source: §12.4.4 · p. 132
O(N^ζ) total error requires d_N = o(N^((κ−1)/2−ζ)); matching uncentered laws also requires mean error o(a_N), and a law-averaged W1 bound alone gives no L2 coupling.The sufficient transfer condition is d_N = o(a_N) with emission error o(a_N); this asymptotic requirement is distinct from the experiments’ fixed finite operating target.Source: §12.4.4 · p. 132

A single pass as a finite data resource

The chapter treats a single training pass as consumption of a finite corpus resource, defining its sampling state and deriving finite-population covariance and document-dependence consequences.Retaining the remaining resource allows chronological training predictions to respect the single-pass constraint.Source: §12.4.4 · p. 133, A single pass as a finite data resource · p. 134

13.1. The pretraining data resource in the scale flow.

The underlying text distribution, sampled corpus, and corpus-presentation rule are distinct parts of the training law; data-dependent claims require conditioning on that law.Repeated sampling from a finite empirical distribution and a single pass through a larger corpus need not select the same inference regime.Source: §13.1 · p. 134

13.1.1. An exact state for sampling without replacement.

The augmented sampling state includes remaining block indices: omitting them generally destroys the Markov description, while inference uses the trained-weight marginal and native emission.Under unrevealed uniform shuffling, the next batch is drawn from distinct remaining indices; conditioning on a realized shuffle instead makes its choice deterministic.Source: §13.1.1 · p. 134
For a uniformly sampled batch, Cov(bh_B|U) = (r−B)/(B(r−1))Σ_R; masked gradients are unbiased only with equal positive target counts, and final-batch sampling covariance vanishes.The covariance identity applies to additive observables at a frozen state, not directly to clipped Adam increments; optimizer memory and trained-state randomness can remain at exhaustion.Source: §13.1.1 · pp. 135–136

13.1.2. The supervised source position as an exposure unit.

For the executed repeated corpus, all-target and last-target objectives expose at most 3072 · 512 and 3072 · 449 distinct supervised source positions, respectively; occupancy is not an effective-sample-size formula.Different crops can supervise overlapping target positions, and reused context positions can affect gradients even when their targets are new.Source: §13.1.2 · p. 137

13.1.3. When resampling and a single pass are close.

In the recorded branch family, 64-update batch-32 histories had index-law distances 0.413255 and 0.486490; these are not prediction-law distances or proofs against local stationarity.The values correspond to 8,192 and 32,768 incoming updates, respectively; the whole-path comparison supplies no useful equivalence guarantee at a fixed positive consumed fraction.Source: §13.1.3 · p. 138

13.1.4. Temporal data aggregation and the response cutoff.

Averaging consecutive batches equals directly forming a larger batch; a complete pass removes constant temporal source fluctuation, while unequal response weights can retain ordering variation.For low-frequency exponential response, a complete pass has a quadratic small-window cutoff; this is a source-constraint result, not an identified PLDR critical exponent.Source: §13.1.4 · pp. 139–140
A random corpus’s same realization persists throughout its path, so averaging corpus-conditioned laws differs from redrawing data at each transition, which can introduce memory when corpus information is discarded.A complete-pass sum is permutation-invariant for a fixed corpus but can vary across corpora; dropping the distribution-level uncertainty term would remove that variation.Source: §13.1.4 · p. 140, §13.1.5 · p. 141

13.1.5. Averaging over corpora and retaining document dependence.

With independent blocks, the recorded causal training coordinates through T batches, where BT ≤ M, have the same averaged path law as a fresh independent stream, including adaptive gradients and optimizer memory.This equality excludes unrevealed corpus coordinates and does not equate corpus-conditional exhaustion or finite-corpus resampling with fresh independent draws.Source: §13.1.5 · p. 141
Grouped blocks retain common-document covariance under permutation, so unique positions are not independent-document or independent-gradient counts, and fixed-corpus experiments do not estimate new-corpus ensembles.The stated example gives distinct positions from the same document probability (m −1)/(M −1); its covariance model is not asserted for RefinedWeb.Source: §13.1.5 · p. 141

13.1.6. Loss drift and the distinction between fitting and prediction.

Bounded-Hessian risk drift depends on the actual native increment, while fitting empirical frequency can lower training risk and raise evaluation risk without a critical singularity.Contraction of row contrast does not determine the sign of risk change under a different evaluation law; train, held-out, and preceding online losses are distinct observations.Source: §13.1.6 · p. 142
A loss decrease requires target-directed logit movement to outweigh the nonnegative predictive KL; small predictive distance alone does not determine loss change.The finite log-score identity is invariant to constant logit shifts and needs neither a small parameter displacement nor a differentiable weight-to-logit map.Source: §13.1.6 · p. 142

Training sources, potential activity, and predictive transport

This section develops finite transport and observation tools connecting parameter and positive-base motion, training updates, and chronological forcing to potential activity and predictive observables, retaining signed interactions, optimizer state, and the imposed clock. It distinguishes conditional bounds, source-based inference response, covariance transport, and finite-sample prediction from claims requiring autonomous row states or stationary limits, while stating the assumptions and limits of operator approximation, response estimation, and fluctuation scaling.

14.1.1. Endpoint allocation and the native operator.

Symmetric endpoint allocation splits the mixed corner equally, with discrepancy bounded by CP Cb h^2/2 under the stated step bounds, but it is not an independent parameter intervention.In the native program, the positive base may depend on parameters included in the exponent, so mixed observation corners need not be realizable training states.Source: §14.1.1 · p. 145

14.1.2. A signed-area completion for component blocking

Exact component blocking retains a signed area that composes associatively; endpoint displacement or a scalar activity trace alone loses signed temporal-order information.The area quantifies component transport, not spatial propagation or a thermodynamic exponent.Source: §14.1.2 · p. 146

14.1. Potential activity, training clocks and endogenous relaxation

Learning-rate-normalized potential activity diagnoses imposed step-amplitude changes but does not remove optimizer memory or make training stationary; the positive base can keep moving after row collapse.Warm-up and annealing can change threshold excursions even at constant underlying velocity.Source: §14.1.2 · p. 147

14.1.2. A signed-area completion for component blocking

A positive base offset bounds the local regularized amplification, whereas zero offset with density near zero yields a heavy tail; this local approximation is not a closed native-training law.The passage cautions that broad amplitudes do not by themselves imply a collective critical point.Source: §14.1.2 · pp. 147–148
An exponential exponent distribution yields a power-law marginal potential, and independent exponents can yield exceedance-run durations without temporal propagation, so marginal power laws do not establish critical dynamics.The section distinguishes a learned power operation, a heavy-tailed observable, and an interacting critical process.Source: §14.1.2 · p. 148
With new loss gradients set to zero, AdamW moments decay but can still drive parameter updates; passive optimizer motion does not emulate relaxation under the native loss.A self-organization interpretation would additionally require an identified state, redistribution and dissipation law, and evidence of feedback toward a critical window.Source: §14.1.2 · pp. 148–149

14.1.3. Transport under actual training interventions

Conditional coupled-path bounds transport offset interventions into emissions while retaining optimizer-state perturbations; resetting only the first Adam moment can produce nonzero initial distance despite matching parameters and forward tensors.The bounds require assumed update and emission stability constants; they are not inferred from activity correlation.Source: §14.1.3 · p. 149

14.1.4. Finite categorical transport of an intervention

Finite categorical transport decomposes a logit intervention contrast using the changing predictive covariance, whose integrated quadratic term is nonnegative while the other terms need not have fixed signs.The identity is invariant to independent time-dependent constant-logit shifts in either branch.Source: §14.1.4 · pp. 149–150

14.1.4. Finite categorical transport of an intervention.

Finite logit segments quantify paired predictive responses, not native training paths or causal mediation; predictive states must retain signed increments, categorical geometry, source, and optimizer coordinates.The resulting observation budget uses categorical logit oscillation, is invariant to additive logit gauges, and bounds metric replacement rather than forecasting unobserved logits.Source: §14.1.4 · p. 150

14.1.5. A finite budget for metric replacement

The finite metric-replacement bound brackets KL curvature K using initial-Fisher quantity Q and logit-increment oscillation r, without requiring a lower bound on token probabilities.Its small-r relative error is η(r) = r/3+O(r2); the results are real-arithmetic bounds, not rigorous floating-point certificates.Source: §14.1.5 · pp. 150–151

14.2.1. The complete derivative cocycle

Instrumental sources perturb the forward computation, while probability sources reweight observations; their full context-dependent derivative jets compose chronologically and must be averaged only after composition.Differentiating then blocking agrees with blocking then differentiating, but the context-dependent matrices generally do not commute.Source: §14.2 · p. 152, §14.2.1 · p. 152

14.2.2. A source metric at the predictive output

The source metric Hx is a positive-semidefinite, logit-gauge-invariant pullback of categorical Fisher information, and an ensemble average does not replace context-specific prediction.Source spectra depend on coordinate units; the evaluated 70 fractional-gain sources define a specified intervention family, not arbitrary operator changes.Source: §14.2.2 · pp. 153–154

14.2.3. Inference response along a realized optimizer update

For a realized optimizer update, the full-parameter logit direction gives a quadratic categorical-KL response without differentiating through the optimizer; conditioning on state fixes the next AdamW displacement law.This predicts an update after it is calculated, not an unseen training gradient.Source: §14.2.3 · p. 154, §14.4 · p. 156

14.2.4. An invariant deductive sector

If every generated operator equals a fixed matrix on the specified context support, replacing the operators preserves predictions and yields a zero-variance centered operator sector.This exact implication is associated with PLGA [37]; the experiment instead measures finite-set replacement error, and context-dependent attention and predictions may still vary.Source: §14.2.4 · p. 154

14.3.1. Training variability and conditional fluctuation classes

With one shared training state across a block of b contexts, Cov(Φ) = At + Bt and Cov(Zb) = At + bBt; redrawing the state independently for each context leaves covariance At + Bt.At is within-checkpoint covariance, whereas Bt is between-state covariance of inference means.Source: §14.3.1 · p. 155
Finite checkpoint mixtures retain conditional independent-context Gaussian limits after checkpoint-specific whitening, while global centering retains the bBt sector; this does not imply a Gaussian training-path fixed point.Training may change means, covariances, and predictive response without changing the conditional class.Source: §14.3.1 · p. 155

14.4.1. The conditional drift of the inference law.

Taylor expansion predicts local inference-observable drift from a distribution of unseen updates, with a cubic remainder under smoothness and finite-third-moment conditions, but does not make the drift autonomous.Training-block resolution, optimizer time, document-block size, decoder grouping, and architecture size remain distinct scales.Source: §14.4.1 · p. 156

14.4.2. One perturbation identity for coupled observations.

The exact covariance change is Cov(Φ, δΦ) + Cov(δΦ, Φ) + Cov(δΦ), so its sign is not fixed and distinct components can yield contraction or growth.For scalar coupled observations, preserving the same leading asymptotic requires 2 Cov(X, e) + Var e = o(Var X); signed cancellation can preserve variance despite nonvanishing relative error.Source: §14.4.2 · pp. 156–157

14.4.3. PLGA attenuation and the error of an effective operator.

Operator replacements yield predictive-error bounds only under stated Lipschitz assumptions: layerwise operator discrepancies propagate to logits, and predictive KL is at most half the squared logit error.A small norm coefficient of variation alone does not establish those constants or an effective-operator approximation.Source: §14.4.3 · p. 158

14.4.4. A curvature-dependent response window.

A quadratic predictive-response estimate is reliable only within a curvature-dependent window: cubic error bounds apply, but numerical error dominates at very small amplitudes and nonlinear error can grow at larger ones.When curvature is zero or unresolved, use absolute error bounds; independent calibration means are unbiased with variance Var(qv)/n, while context heterogeneity remains.Source: §14.4.4 · pp. 158–159

14.4.5. Three fluctuation sectors and contrasts.

Hierarchical fluctuations separate training-state, document, and residual covariance sectors, while zero-sum contrasts cancel the two shared components exactly.A surviving common component has H = 1 under global centering, independent finite-variance fluctuations have H = 1/2 after conditioning on shared state, and summable residual covariance does not itself imply a Gaussian limit.Source: §14.4.5 · pp. 159–160

14.4.6. A predictive coordinate for regime selection.

With vocabulary probabilities bounded below by ϵ, predictive context dispersion satisfies 2 tr Σp ≤ Ictx ≤ ϵ^-1 tr Σp, and Ictx = 0 exactly when predictions are almost surely constant.Finite-cohort measurements use all vocabulary probabilities and can distinguish nearly context-insensitive outputs from active prediction even when operator dispersion is small.Source: §14.4.6 · p. 160

14.5. Chronological covariance and conditional prediction.

Chronological covariance is conditional on declared external information and finite second moments; a complete incoming state together with the data law is one possible conditioning choice.Source: §14.5 · p. 161

14.5.1. A common chronological covariance identity.

A common chronological covariance identity decomposes the conditional covariance into transported initial and forcing covariances, including cross terms, and remains valid under regrouping when the full forcing and joint law are retained.Source: §14.5.1 · p. 161
A deterministic incoming coordinate removes initial covariance and initial–forcing cross terms, but correlations between distinct forcing intervals remain unless the joint conditional law retains them.Sampling q0 and the full forcing sequence jointly yields a finite-horizon Markov representation only for the enlarged state containing the remaining sequence and time index; it does not establish that qj alone is Markov. A predictive construction must specify information available at the incoming time, and native tangent approximations must retain their nonlinear remainder.Source: §14.5.1 · pp. 161–162

14.5.2. Covariance blocking and finite predictive memory.

A second-order state of joint finite-emission fluctuations supports linear covariance transport and optimal affine prediction, but need not determine the nonlinear successor law.The best affine predictor is KX with residual covariance S; nested linear observation spans cannot increase oracle mean-square error, but fitted-history errors also include estimation and regularization error.Source: §14.5.1 · p. 162, §14.5.2 · p. 162

14.5.3. Finite prediction sets for a conditional path law.

n/(n+1) is the conditional lower bound on coverage for a simultaneous prediction set for one future risk path, given exchangeability of calibration and future paths.Coverage averages over calibration and future paths, not a realized calibration set; the unit is an entire reset branch, conditioned on the incoming model, optimizer, and remaining corpus, with no stationary or critical limiting assumption.Source: §14.5.3 · p. 163

Shared collectives, sign symmetry, and training selection · Physical clocks and finite matrix flux

This section connects single-pass training observations to inference through shared and transverse collectives, sign-invariant observations, optimizer memory, matched physical clocks, and finite matrix-flux coordinates. It distinguishes exact finite identities and clock-dependent fluctuations from conditional results requiring limiting assumptions, without establishing a critical exponent or an autonomous predictive state.

15.1. Collective variables linking training and prediction.

Exact row-moment transport separates the common metric coordinate from transverse row covariance, so local contraction can suppress row contrast while the common coordinate changes.A useful training hierarchy retains metric and exponent paths, row covariance, exceptional rows, predictive sources, optimizer information, drive, and remaining data; inadequate finite coordinates require conditional memory.Source: §15.1 · p. 164

15.2. Signed and invariant sectors of the single-pass flow.

For conditional covariance C(r) ∼ M r^−γ, 0 < γ < 1, the second-moment normalization is b^−(1−γ/2), while independent finite-variance blocks fluctuate at b^−1/2 and shared fields at b^−1.The selected law is conditional on the fixed corpus, realized block order, shared initialization, and remaining-data and training clocks.Source: §15.1 · p. 164, §15.2 · p. 165

15.2.1. A native head-sign quotient for training and inference.

Native head-sign flips preserve metrics, predictions, and the complete predictive law while inducing an equivariant optimizer update; invariant observations therefore define a quotient of the complete state.Under the stated hypotheses, the signed mean operator converges to zero in mean square, but invariant quadratic and row fields can retain shared-state correlations.Source: §15.2.1 · pp. 165–167

15.2.2. Predictive law fluctuations.

The square-root embedding of predictive probabilities yields a finite susceptibility bounded by 4N that contracts under any shared column-stochastic observation channel.This measures variation across training initializations, not propagation of a specified state displacement; an observation channel can erase a common sector without a visibility condition.Source: §15.2.2 · p. 168

15.3. Training selection under a single-pass corpus law

Independent endpoint row-susceptibility peaks occur at g = 1.25, 1.5, 1.25, and 1.25, with half-height widths 0.617, 0.558, 0.593, and 0.705.The peak has a large common covariance component, and its control window does not narrow systematically with width.Source: Table 15.3 · p. 170, §15.3.1 · p. 170

15.3.1. A finite kinetic control scale.

Frozen square-root-clock forecasts have relative RMS errors of 7.09%, 2.03%, 4.11%, and 1.90% across widths, supporting a finite kinetic scale rather than an identified critical exponent.At g = 3, centered metric energies fall to 1.013×10−3, 7.444×10−5, 1.229×10−4, and 2.390×10−5 while total energies remain between 1.019 and 1.027, indicating absolute row-contrast suppression.Source: §15.3.1 · p. 170, §15.3.2 · p. 170

15.3.2. What training concentrates and what it retains.

At g = 2, endpoint per-coordinate variances were 1.070, 1.105, 1.123, and 1.103 times 10^-3, accounting for 99.92–99.96% of displacement; signed cross-block terms contributed 2.55–3.61%.The accumulation law retains signed cross-block terms and within-block dynamics; approximate quadratic-rate, linear-time accumulation does not establish independent one-update forcing.Source: §15.3.2 · p. 170
Held-out next-token NLL fell from approximately 10.38 initially to 6.32–6.41 at g = 0, 6.55–6.66 at row peaks, and 6.70–6.78 at g = 3; row peaks did not select the smallest measured risk.Predictive improvement spans several row regimes on the fixed 128-context training-observation panel.Source: §15.3.2 · p. 170

15.4.1. Finite-time continuity and limiting control fields.

For a finite native architecture and fixed finite single-pass stream, state, conditional field moments, and finite-vocabulary probabilities vary continuously with control, with no finite-control divergence in fixed-width susceptibilities.A critical control can emerge only in a specified limit; finite steepness or noncommuting limits alone do not specify a thermodynamic family.Source: §15.4.1 · pp. 171–172

15.4.2. Compact shared paths at matched single-pass clocks.

At matched single-pass clocks with rates bounded by c/N, shared paths, empirical head measures, and finite-panel predictive vectors have jointly convergent subsequences.These results establish existence, not uniqueness, a closed finite successor kernel, or convergence rates.Source: §15.4.2 · pp. 172–173

15.4.3. Complete common fields.

Row-centered energy, the common centroid, and the head-mean operator retain distinct information; concentration of a centroid’s norm does not imply concentration of its vector.The recorded experiments specialize the common-field formulas to d = 64.Source: §15.4.3 · p. 174

15.4.4. Creation of a shared fluctuation by the first adaptive update.

Under AdamW’s first adaptive update, shared-parameter fluctuation depends on the gradient sign distribution and its size relative to the denominator offset, not gradient magnitude alone.The first-step identity makes no stationarity assumption; fluctuation accumulation depends on temporal covariances.Source: §15.4.4 · p. 175

15.4.5. Exact accumulation of shared fluctuations.

The exact shared-path covariance budget retains signed temporal cross-terms; η^2T is a variance clock only under additional stationarity and summability conditions.A Gaussian readout construction also gives a nonconstant shared gradient with the same finite positive variance at every width, showing that its scale must be measured from the native computational graph.Source: §15.4.5 · pp. 175–176

15.4.6. Separating timing fluctuations from a critical control.

Timing-label conditioning decomposes covariance into within-class and between-class terms, while regular kinetic evolution can yield an O(N) peak and an N^-z/p window without identifying a native critical flow.The kinetic construction is an observation-law example, not an identification of the native PLDR update with scalar logistic flow.Source: §15.4.6 · pp. 176–177

15.4.6. Separating timing fluctuations from a critical control

Smooth row trajectories can exhibit a finite nonzero variance maximum, but these powers do not establish a positive limiting critical control or intrinsic relaxation exponent.Assess size scaling alongside peak-location drift, both optimizer clocks, source depletion, and resolved response; universality concerns limiting joint laws and predictions under specified microscopic changes.Source: §15.4.6 · p. 177
At fixed generator/body-rate ratio, keeping the quadratic control bounded away from zero requires T_N of order N^2 and a source population of order N^2.Both accumulated learning-rate clocks are then order N; the measured finite-g^2T approximation supplies no native long-horizon theorem at this scaling.Source: §15.4.6 · p. 177

16.1. A uniform force envelope for matched optimizer memory

Bias-corrected Adam with matched memory has a uniformly bounded normalized force, and the stated decayed coordinate remains in an invariant interval.Finite checks found maximum extremizer discrepancy 4.45 × 10^-16, excess below 3.34 × 10^-16, and limiting envelope approximately 1.165107; the bound does not establish compact network derivatives or a critical limit.Source: §16.1 · pp. 178–179, Table 16.1 · p. 179

16.2. Physical response and the finite resource

For h_N ~ N^-a and κ_N ~ N^-b, scalar relaxation is order N^(a+b) updates and N^b physical time; with innovation variance N^-(a+v), stationary variance is order N^(b-v).These response scalings assume stationary scalar forcing and stability; nonstationary or nonnormal processes retain their chronological propagator and full forcing covariance.Source: §16.2 · pp. 179–180
For fixed batch and window N^ζ, the sufficient independent-source coupling condition holds in the power-law comparison only when population growth exponent m exceeds 2ζ.Failure does not rule out other approximations, but does not justify stationary independent forcing; the complete law retains the consuming source.Source: §16.2 · p. 180

16.3. Chronological forcing from a consuming finite population

Uniform-permutation sampling from a centered finite population yields a positive-semidefinite chronological covariance, with zero variance for the unweighted fully consumed population average.At fixed consumed fraction, the physical-clock covariance retains an order-one correction; the 1-q = 3/4 value for M_N h_N = 128, B = 32, τ = 1 is a reference-law prediction, not native gradient-noise estimation.Source: §16.3 · p. 181

16.4. Finite row flux and the collective clock

The finite row-map increment decomposes exactly into a linear term and a quadratic term, while compatible blocking preserves cross-time quadratic interactions.Replacing the finite cross term with a differential approximation incurs a bounded error, with an additional generator-remainder bound when the derivative is Lipschitz.Source: §16.4 · pp. 182–183
Matched physical-clock row increments are (¯ℓk,N + ¯qk,N)/hN, with block rates 0.5376g for generator coordinates and 0.0768 for remaining coordinates.The optimizer force envelope alone does not control generated matrix fluxes; model response also depends on width, incoming state, and source block.Source: §16.4 · p. 184
Finite fluxes are transition variables, not an autonomous incoming state; prediction requires conditional dependence, unresolved memory, and source-consumption law to be retained.Reconstructing fluxes from an observed successor verifies the balance but does not test forecastability; an autonomous row theory needs closure and emission control.Source: §16.4 · pp. 184–185
Finite row-fraction transitions compose associatively as (q, r), but coarse-graining must retain their joint law or bound quotient error; at zero total energy, retain unnormalized energies.The physical action domain is r > 0 and 0 ≤ u + q ≤ r.Source: §16.4 · p. 185

16.4.1. Signed temporal energy and compatible scale summaries.

For m increments, the signed temporal term X satisfies −S ≤ X ≤ (m − 1)S, and its sign alone does not diagnose centered forcing covariance or a critical mode.Temporal summaries combine associatively through (V, Q, H) ⊙ (eV, eQ, eH) = (V + eV, Q + eQ, H + eH + 2⟨V, eV⟩); merged blocks use incoming energy for normalization.Source: §16.4.1 · p. 186

Single-pass training and potential observations … Predictive resolution and conditional operator risk

This section studies finite single-pass training through predictive, row-dynamic, and operator observations, tracing their dependence on data, control, optimizer state, schedule, training age, and measurement choices. It develops conditional predictive and operator reductions with finite error certificates, while limiting conclusions to tested states and paths rather than asserting closure, universal scaling, or broad guarantees.

17.1.2. Optimizer memory and finite predictive reduction.

With incoming logits held equal, changing optimizer moments produced one-step absolute antisymmetric risk responses of 2.86×10−5–8.03×10−4 nats, supporting retention of optimizer state in the model-wide kernel.Four frozen-coordinate forecasts missed their uniform 0.01-nat criterion; the rank-four adaptation's technical-test NLL change was −0.00227 nats, with family-adjusted paired interval [−0.00381, −0.00076].Source: §17.1.1 · p. 187, §17.1.2 · p. 187, §17.1.3 · p. 188

17.1.3. Inference-law transport and its conditioning.

Physical adaptations improved aggregate held-out next-spin loss and accuracy, but transfer varied by class and seed and coexisted with an almost input-invariant metric generator.These physical source laws test conditional inference-transfer bounds, not native PLDR critical exponents.Source: §17.1.3 · p. 188

17.2.2. Measured supervised-position reuse.

At 65,536 updates, repeated all-target sampling reused 98.83% of target events, while every single-pass target event used a distinct source position.The repeated and single-pass comparison changes both corpus content and presentation; overlapping crops mean distinct crops can still draw targets from nearly the same finite position set.Source: §17.2.2 · pp. 189–190

17.2.3. Constant-rate comparison under a single pass.

Single-pass held-out NLL was lower in 21 of 36 constant-rate checkpoint pairs, yet every long path worsened by 131,072 updates under both data laws.At that endpoint, mean single-pass NLL was 8.5225 at N = 4 and 7.5736 at N = 14, modestly below repeated sampling, despite the late loss increase.Source: §17.2.3 · p. 190

17.2.5. Row contrasts, common fields and predictive risk.

Compact endpoints span 0.0429 ≤ R ≤ 0.694 and 1.007 ≤ Neff ≤ 1.047, while the common sector remains strongly correlated across heads.During the positive floor, absolute row energy fell in all eight source-recipe/head-count conditions even as normalized row contrast rose in six; retaining the absolute numerator and common field is necessary to interpret normalized collapse.Source: §17.2.5 · pp. 192, 194
Prediction rankings depended on observation law: Near 1 beat Below 1 on external-target loss at all six matched endpoints, but at N = 8 their proper-panel losses were 13.677 and 7.672 nats.The positive floor reduced external-target loss in nine of ten conditions but increased proper-panel risk in seven, including all four Near conditions.Source: §17.2.5 · p. 194

17.2.6. Paired constant and cosine rate histories.

Cosine lowered external-target held-out loss in all 16 checkpoint pairs, with endpoint mean gains of 0.2772 nats (N = 4) and 0.1979 nats (N = 14), alongside higher row contrast and absolute row energy.Cosine also produced higher paired-context predictive KL; the comparison changes the complete rate history, including accumulated parameter motion and optimizer feedback.Source: Table 17.6 · p. 195, §17.2.6 · p. 195

17.2.7. Temporal observations at the positive rate floor.

During the positive floor, the consumed fraction rose from 1/4 to 5/16, and the two halves removed 1/24 and 1/23 of their incoming remaining source populations.These are exact total-variation changes for uniform laws on remaining block indices, not measured changes in emitted predictions; sampling-resource and model-dynamics stationarity require separate controls.Source: §17.2.7 · p. 196
At the positive rate floor, linear drift explains 0.291–0.998 of common-centroid temporal variance, while detrended covariances establish neither stationarity nor a diverging relaxation time.Threshold intervals depend on threshold choice and censoring; the measurements support finite transient dynamics under the specified drive, not an avalanche distribution, critical exponent, or endogenous transition.Source: §17.2.7 · pp. 196–197

17.2.8. Complete matched data-law comparison.

At 65,536 updates, single-pass last-target NLL is 5.094 nats (Near) and 5.177 nats (Below), versus 9.136 and 8.893 nats under repeated data.Both single-pass paths improve from 32,768 updates on this held-out cohort, but at 32,768 the Below setting has last-target NLL 7.420 with single-pass data versus 5.979 with repeated data; a single pass is not always better.Source: §17.2.9 · p. 198, Table 17.11 · p. 198

17.2.9. Prediction improves at the longer matched horizon.

At the longer single-pass horizon, Near and Below have similar external-target loss but mean row contrasts of 3.578 × 10−14 and 0.0302, respectively.Single-pass train–held-out gaps are 0.044 and 0.023 nats on their specified cohorts; repeated-data training risk falls as held-out risk rises, supporting data-law-conditioned regime selection without identifying all causes or unseen population risk.Source: §17.2.9 · pp. 198–199, §17.2.10 · p. 199, Table 17.12 · p. 199

17.2.10. Row concentration and proper-prefix prediction.

At 65,536 updates, Near and Below single-pass proper-prefix risks are 5.382 and 5.745 nats, with proper-minus-full gaps of 1.15 × 10−8 and 0.395 nats, respectively.Their maximum measured forward KL values are 1.98 × 10−12 and 3.991; repeated-law states show that even small row contrast and prefix defect can coexist with large held-out risk, so concentration does not replace predictive fit.Source: §17.2.10 · p. 199, Table 17.13 · p. 200

17.2.11. Proper-prefix risks under constant rates and completed annealing.

At the compact-schedule cosine endpoint, Near and Below proper-prefix risks are 17.5955 and 12.8919 nats, with proper-minus-full gaps of 13.9129 and 8.9201 nats.The gaps remain large after 8,192 rate-floor updates; shortening the horizon changes matched-update proper risk, showing that recipe name and update count alone do not specify inference behavior.Source: §17.2.11 · pp. 201–202, Table 17.15 · p. 201

17.2.12. The learned row map over the inference input domain.

On selected local points, every Near Jacobian contracts, while at 65,536 updates all 18 decoder-four Below points remain expansive, with norms up to 3.313.Below endpoint prefix secant ratios reach 0.573–0.583 in decoder four, locating a weakly attenuated route; neither finite-point derivatives nor finite-cloud secants establish a uniform Lipschitz bound or training critical eigenmode.Source: §17.2.12 · pp. 202–203, Table 17.16 · p. 202

17.2.13. Decoder interventions distinguish prefix consistency from fit.

Fixing Below decoder four reduces maximum prefix KL to 1.11 × 10−10 and the absolute mean risk gap below 7.8 × 10−8 nats, but raises full-window risk from 5.3503 to 5.8331 nats.The interventions locate a dominant measured prefix-dependent route, not a unique route; forcing row concentration can reduce prefix defect without preserving predictive fit or guaranteeing useful autoregressive prediction.Source: §17.2.13 · pp. 203–204, Table 17.18 · p. 204

17.2.14. Conditional native risk transport.

Conditional risk changes depend on incoming state: the constant-rate N = 4 full mean increments are −5.14959×10−4 and −2.80621× 10−5 nats, while both tested N = 14 states increase risk.At the Below endpoint, a 4.22079 × 10−3-nat mixed term reverses the negative sum of partial means; at Near, generator learning improves prediction while row concentration persists. Frozen conditional means cannot replace the evolving weights, optimizer memory, and remaining-data law.Source: §17.2.14 · pp. 204–206
The six selected states support a model-wide risk law retaining data conditioning, optimizer memory, and coupled parameter sources, but their balances do not support common component dominance or endogenous criticality.These observations concern selected incoming states and conditional update laws, not thermodynamic exponents.Source: §17.2.14 · p. 206

17.2.15. Task margins under the single-pass training laws.

Across 27 native conditions, strictly correct means were 119.5–128 of 320, but only 0–3 of 96 counterfactual pairs were correct on both variants.Near 1 had better external-target loss than Below 1 at every compact endpoint but lower ARC-Challenge accuracy at all six; these fixed-cohort differences do not rank population reasoning ability.Source: Table 17.22 · p. 207, §17.2.15 · p. 207
Operator replacements changed 67.25–115 answers at compact Near 1 endpoints, while the source-horizon Near endpoint preserved all 320 decisions under calibration and row projection.Constant-rate conditions changed at most 0.5 decisions under calibration and one under row projection; at the source-horizon Near endpoint, permutation changed 41 decisions, showing that harmless calibrated replacement did not make learned coordinates irrelevant.Source: §17.2.15 · pp. 207–208, Table 17.23 · p. 208, Table 17.24 · p. 209

17.3.2. Base motion dominates the measured potential decomposition.

Across 18 early paths, base motion supplied 0.99843–0.99985 of measured log-potential increments, while small bases accounted for 45.7–63.7% of sampled squared activity.The paths used 2,048 updates on distinct corpus blocks; passive reconstruction with a 10^-6 base offset retained 45.6–65.7% of sampled activity, indicating sensitivity to small positive bases without identifying a native zero-offset limit.Source: §17.3.1 · pp. 209, 211, §17.3.2 · pp. 211–212

17.3.3. Excursions and correlated controls.

Early excursion fits and late tensor-tail observations remain finite, state- and clock-dependent descriptions rather than evidence for a shared critical exponent or calibrated excess persistence.Only seven early raw paths admitted fits, with exponents 1.466–1.877; all 18 raw paths outlasted permutation controls, but the smallest adjusted descriptive rank was 0.81. In the annealed strongly collapsed branch, raw tensor–KL correlations of 0.757–0.891 fell to −0.096–0.056 after rate normalization.Source: §17.3.3 · pp. 212–213, §17.3.5 · p. 215

17.3.4. Activity after long pretraining and row collapse.

After long training, a state with incoming row-contrast ratio 2.31 × 10^-14 stayed at native-precision row contrast under both drives, while base motion still supplied 0.99252–0.99278 of measured activity.Small bases accounted for 81.8–88.9% of late sampled squared activity, but passive offset reconstruction retained only 9.4–18.7%; the states are described as row-collapsed without assigning a criticality class.Source: §17.3.4 · p. 213

17.3.6. Residual activity carried by the optimizer.

With zero new gradients, retained Adam moments produced 17.57–46.06 times the integrated log-potential activity of reset moments; signed-area retention also preserved exact component blocking over tested blocks.At 256 updates the area norm was 3.3–16.8% of the blocked total-increment norm in early paths and 2.6–5.5% in continuations, so omitting it loses path-order information over longer intervals.Source: §17.3.6 · p. 215, §17.3.7 · p. 216

17.3.8. Native offset, optimizer memory and the source suffix.

The native offset and optimizer-memory study crossed 10^-9 and 10^-6 base offsets with retained or reset first moments, using two distinct 4,096-block source suffixes per incoming state.Each branch started from the same incoming state within its suffix comparison, and no reserved block was used; the suffixes share a finite corpus and are not independent initializations.Source: §17.3.8 · p. 217
Raising the offset suppressed integrated log-potential activity on both suffixes, but primary-probe KL responses varied by source and checkpoint and could reverse after first-moment reset.Retained-moment activity ratios were 0.295–0.488 on suffix A and 0.278–0.711 on B; the partially concentrated-state KL ratio was 0.719 on A and 1.058 on B, while strongly concentrated-state ratios were about 0.996. Offset-contrast interactions after reset were +0.234399 on A and −0.075348 on B.Source: §17.3.8 · pp. 217–218, Table 17.31 · p. 218, Figure 17.7 · p. 218

17.3.9. Resolving the finite predictive contrast.

At the partially concentrated state, the predictive contrast reflected a negative cross term, a positive squared-increment term, and signed metric change, without identifying causal mediation or a closed successor law.Across 128 scale cells, signed-area component residuals were at most 3.82 × 10−17; metric-replacement errors were summarized over 16,320 spans, with no population intervals.Source: §17.3.9 · pp. 219–220, Table 17.33 · p. 220

17.3.10. Finite metric budgets on both predictive probes.

Local-Fisher error reached 11.27% on one-step probe-1 spans and 72.71% on probe-2 128-step spans, while probe contrasts depended on source and moment arm.With retained moments, probe 1 changed sign across suffixes while probe 2 remained negative; after reset, both changed from positive to negative. The finite bounds and refinements concern observed logits, not autonomous prediction.Source: §17.3.10 · p. 220, Table 17.34 · p. 221, Figure 17.8 · p. 221

17.4. Matched single-pass training across width and schedule

Cosine produced lower final external-target NLL in all 12 width/seed comparisons, improving by 0.0204–0.3173 nats per target, without a consistent direction of row contrast.Late potential block activity was smaller under cosine in 11 pairs, but learning-rate-normalized velocity ratios were 1.543–9.808, consistent with a clock-conditioned activity interpretation rather than a universal restoring drift.Source: Table 17.36 · p. 222, §17.4 · pp. 222–223, Figure 17.9 · p. 223

17.4.1. Predictive metric transport on newly trained paths.

On 192 full-horizon endpoint pairs, 441–527 segments per job enclosed every direct KL, with maximum relative midpoint error 1.994 × 10−6 within the proved 1% envelope.These finite endpoint observations quantify metric-holding cost on the same trajectories; the fixed horizon, four small widths, and one corpus do not establish stationarity or a native critical exponent.Source: §17.4.1 · pp. 223–224, Table 17.37 · p. 223

Independent width families and chronological continuations

The independent-width continuation examines conditional fluctuation profiles, shared-parameter accumulation, sign-orbit budgets, and finite row transport over matched physical durations.The passage introduces these measurement targets without reporting their results in this section part.Source: Independent width families and… · p. 225

18.1.1. Row contrast, collective directions and training age.

At 2,048 updates, row-susceptibility peaks occurred at g = 2, with heights increasing from 0.0751 to 0.455 while intensive variances stayed 0.0325–0.0412.The fitted peak-height slope of 0.948 cannot be assigned a critical exponent because the intensive variance is compatible with a nonvanishing common component.Source: §18.1.1 · p. 226, Figure 18.1 · p. 226
At g = 4, centered metric energy fell from 5.1×10−6–2.58 × 10−3 despite total metric energy of 1.029–1.034, so denominator growth did not explain row-contrast suppression.The half-mean row crossing moved toward smaller generator multipliers with training age; fitted control-versus-time powers ranged from −0.554 to −0.522, and are conditional drift estimates rather than universal exponents.Source: §18.1.1 · pp. 226–227
At g = 2, earlier conditional signed-operator Gaussian-mixture errors are 0.003629, 0.002306, 0.001395, and 0.000861 across increasing head count, with effective-head fractions 0.710, 0.567, 0.462, and 0.414.These checks apply to exact sign-orbit averages at trained endpoints and support the stated regular conditional symmetry law, not an invariant row or predictive critical law.Source: §18.1.1 · p. 228

18.1.3. Adaptive shared force and predictive observations.

Held-out NLL is 6.472, 6.453, 6.406, 6.385 at g = 0 versus 6.789, 6.806, 6.753, 6.713 at the row-fluctuation maximum g = 2.The initial full-vocabulary predictive variance is about 0.00796 at every width, while trained g = 4 endpoint values are 0.0303, 0.0318, 0.0416, and 0.0515; the row maximum is not a task optimum.Source: §18.1.3 · p. 229

18.2. Independent size and training-age comparison; 18.2.1. A reproduced finite-time fluctuation region; 18.2.2. Frozen time predictions

At T = 4096, sampled row-susceptibility maxima occur at g = 1.25, 1.5, 1.25, and 1.25 across increasing widths, but four widths do not identify a limiting critical window.The independent family uses 216 complete paths from a disjoint partition of one master corpus; doubling training age moves and narrows the finite control region, while half-height widths show no systematic narrowing. Frozen square-root time predictions are useful over this finite range but do not establish a critical exponent or self-organized attraction.Source: §18.2 · p. 230, §18.2.1 · pp. 230–231, §18.2.2 · p. 231

18.2.3. Complete shared-parameter accumulation.

At T = 4096 and g = 2, shared-parameter variance accounts for 99.92–99.96% of decay-corrected displacement, while the net cross-block covariance contributes 2.55%–3.61%.At g = 3, T = 4096, mean centered energies fall from 1.013 × 10^-3 to 2.390 × 10^-5 across increasing widths while total energies remain between 1.019 and 1.027, supporting absolute row-contrast suppression. The measured accumulation does not identify an autonomous diffusion law.Source: §18.2.3 · p. 233, §18.2.4 · p. 233

18.2.4. Absolute contrast and inference observations

At g = 3, the complete common-vector susceptibility reaches 24.42 at N = 24, while row susceptibility is only 4.27 × 10^-7 near collapse; predictive variation also occurs outside row peaks.At row peaks, full-vocabulary predictive intensive variances are 0.0979, 0.0979, 0.1399, and 0.1267, versus 0.1118, 0.1018, 0.0933, and 0.1025 at zero generator control. Mean held-out NLL is 6.32–6.41 at g = 0, 6.55–6.66 at row peaks, and 6.70–6.78 at g = 3, within the fixed panel and short single-pass horizons.Source: §18.2.4 · pp. 233–234

18.2.5. Independent native sign-orbit checks

At the endpoint, the signed operator susceptibility divided by mean one-head power remains between 0.9583 and 1.0592 across 36 independent-family cells, while projected sign quadrature reconstructs orbit second moments to 1.78 × 10^-15.These finite symmetry checks add no training realizations and support regular signed-operator normalization without identifying a unique limiting covariance law.Source: §18.2.5 · p. 235

18.2.5. Independent native sign-orbit checks; 18.1.2. A regular conditional signed-operator class

At g = 2, mean conditional characteristic-function errors against the variance-matched Gaussian at standardized frequency 1/2 were 0.002433, 0.001384, 0.000854, and 0.000523 across increasing widths.At N = 24, g = 2, mean excess was −0.0314, combining a positive variance-mixture contribution 0.2065 and a negative sign contribution −0.2380. These exact finite symmetry averages supply no new training realizations and do not identify a unique limiting covariance law.Source: §18.2.5 · p. 235

18.3. Measured single-pass dynamics on a matched physical clock

At physical age one, row coordinates at g = 1.5 are 0.0347, 0.4081, 0.7518, and 0.8959 across increasing widths, so matched physical clocks do not collapse the finite curves.At g = 2 the corresponding coordinates are 0.00022, 0.0463, 0.3687, and 0.7723; the table reports completed matched-clock endpoints.Source: §18.3 · p. 236, Table 18.1 · p. 236, Figure 18.8 · p. 237
External-target NLL fell from 10.3604–10.3805 to 7.3739–7.8536 nats; g = 0 had the lowest endpoint mean at each measured width, despite weaker row suppression.Across the state-calibrated cache cells, relative centered RMS ranged from 0.00945 to 0.19509 and mean KL was at most 0.000696 nats; individual-context errors remained heterogeneous, so the evidence supports state-conditioned reduction, not row suppression as its sole cause.Source: §18.3 · p. 237, Figure 18.9 · p. 238

18.4. Finite row-energy transport in recorded native continuations

The finite row-energy identity reconstructed all 4,000 head/decoder/context pairs within 4.57 × 10−16; its quadratic contribution was positive in every averaged cell and could exceed the signed cross term.For N = 4, g = 1.5, the cross term was −0.00392 and the quadratic term 0.01118, yielding a 0.00726 net increase; these are recorded transitions, not a closed successor law or limiting scaling result.Source: §18.4 · pp. 238–239

18.5. Executed multistep chronological transport

Across 16 recorded paths, the accumulated quadratic term was positive in every path, and the matrix-derivative sum had the opposite sign from the exact increment in 7 of 16 paths.For N = 4, g = 1.5, initialization 2, the cross and quadratic contributions were −0.101882 and 0.110077, giving an increment of 0.0081953; signed cross-time energy is part of blocked displacement, not a sum of squared individual increments.Source: §18.5 · p. 240

18.6. Equal physical-duration finite transport

At equal physical duration, all 16 paths satisfied finite transport identities; the derivative sum had the opposite sign from the exact increment in 4 of 16 paths, with absolute error 1.5285×10−5 to 0.40378.The signed temporal contribution was positive in 53 cells and negative in 11, spanning −0.57208 to 0.3561; it includes coherent drift and does not estimate stationary forcing covariance or establish critical exponents or an autonomous flux law.Source: §18.6 · pp. 241–242

Predictive resolution and conditional operator risk

Predictive reduction is defined at a declared observation resolution, separating lost predictive information, estimation error, and the cost of replacing native operator generation.The resolution, context law, and conditional risk specify the setting for assessing cached or coarsened operators.Source: Predictive resolution and conditional… · p. 244

19.1. Nested predictive resolutions and a finite error budget

Nested vocabulary resolutions use a shared reference law; retaining more tokens reduces the KL reconstruction error and contracts predictive fluctuation measures.These composition identities concern acquired predictive observations, not a transition kernel for a smaller optimizer state or a separately trained smaller architecture.Source: §19.1 · p. 244, §19.1.1 · p. 245

19.1.1. Conditional density and compatible time transport

With a fixed reference and partition, projection commutes with conditional expectation and signed increments, but the projected prediction alone does not determine the complete chronological successor law.Using the complete state as conditioning preserves the optimizer and remaining-resource state; a reduced successor still requires closure bounds, and refitting the reference or partition can break the fixed-map identities.Source: §19.1.1 · pp. 244–245
Centered residual variance is bounded by four times mean KL error, with an additional S/(S −1) factor empirically; the certificate can be conservative because it includes mean reconstruction bias.For coupled fluctuation-scale limits, EDKL(pN∥ΠkN pN) = o(vp,N) suffices for the relative residual variance to vanish, but does not identify a native fluctuation law or critical exponent.Source: §19.1.1 · p. 246

19.2. The context law of a predictive observation

Context-resolved relative error is summarized under declared context weights; its tail bound controls weighted context mass above a threshold, not the largest error or unweighted context count.Relative error is undefined when native variance is zero, so absolute variances must be retained there.Source: §19.2 · pp. 246–247

19.3. Equivariant operator elimination at frozen inference

A fixed, assessment-disjoint prefix panel supplies weighted per-head operator caches while the decoder recomputes other blocks; this extends [33] [37] to single-pass ensembles and measures model-call cost.The calibration panel contains neither assessment documents nor their targets.Source: §19.3 · p. 247
The cache transforms equivariantly under the head-sign action, while decoder-error bounds require same-input defect and Lipschitz conditions; small row contrast alone is not a premise.The finite experiment measures emission errors directly rather than estimating uniform decoder Lipschitz constants.Source: §19.3 · pp. 247–248

19.4. Conditional risk of a calibrated operator

Conditional calibration risk separates evaluation scatter and mean displacement from cache sampling variance; Vν/m applies to uniform independent calibration, while dependence retains cross terms.The fixed-panel identity requires no sampling independence, but distinct sampled documents alone do not establish conditional independence.Source: §19.4 · pp. 248–249
Transporting a cache across states adds evaluation scatter, squared mean shift, squared initial cache offset, and a signed cross term, so operator drift alone does not predict cache error monotonically.Predictive-risk transport additionally requires the stated uniform sensitivity hypotheses.Source: §19.4 · pp. 249–250
Native-emission fidelity differs from external-target NLL: an intervention can lower target NLL while moving away from the recorded native law.The target-law NLL difference equals KL(r∥q) − KL(r∥p), and equals nonnegative fidelity loss KL(p∥q) only when r = p.Source: §19.4 · p. 250

Executed predictive resolution and cache transfer · Finite optimizer transport and autoregressive prediction

This section reports finite cache-transfer and predictive-reduction results and develops conditions for relating optimizer transport and fixed-weight input contraction to inference observations and autoregressive task performance. It distinguishes input variation from training response and specifies how predictive laws, retained coordinates, and training fluctuations bear on prefix risk, answer margins, and autoregressive fidelity.

20.1.1. Time-conditioned predictive scale.

At 8,192 tokens, the centered-RMS target holds in 32 of 36 cells at T = 2048 and all 36 cells at each later time.At T = 3072, actual relative centered RMS is 0.124–0.227, whereas the sufficient KL-based bound is 0.838–1.389; resolution accuracy therefore depends on training age and observation/source law.Source: §20.1.1 · p. 251

20.1.2. A fixed dictionary on disjoint documents.

On the fixed 32-document panel, all four 8,192-token cells meet the 25% target, retaining 96.1–97.2% of predictive variance with relative centered RMS 0.176–0.197.At 16,384 tokens, retained variance is 97.9–98.5% and relative centered RMS is 0.129–0.144; the shared dictionary tests this finite panel, not uniform prompt accuracy.Source: §20.1.2 · pp. 251–252, Figure 20.1 · p. 252

20.1.3. Transfer to an additional untouched document panel.

On the additional untouched panel, all four 8,192-token cells meet target, retaining 96.35–97.14% of predictive variance with relative centered RMS 0.175–0.193.This supports a finite observation scale under the declared context law, not uniform prompt fidelity or task-level accuracy; acquiring the reduced observation still requires full native predictions.Source: §20.1.3 · pp. 252–253

20.1.4. Context-resolved risk on a frozen 128-document panel.

Across 128 contexts, all four 8,192-token aggregates meet target with relative centered RMS 0.182–0.197 and retained variance 96.07–96.82%; exceedance counts are 24, 22, 23, and 12.Exceedance counts share trained replicas and are not independent-trial estimates; the native-variance weight of exceeding contexts is 4.86–11.10%, and the deterministic bound does not guarantee the maximum.Source: §20.1.4 · pp. 253–254, Table 20.1 · p. 253, Figure 20.2 · p. 254

20.2. Executed elimination of native operator generation

All four operator-cache conditions meet the frozen accuracy and latency targets: relative centered RMS is 0.042–0.124, mean KL is at most 0.000712 nats, and median latency falls 62.96–65.12%.Cached/native predictive variance ratios are 0.99961–1.00497. The timing covers batch-eight final-token prefill at length 64, not autoregressive decoding; context errors measure fidelity to the native predictor, not improved language capability.Source: Table 20.2 · p. 255, §20.2 · p. 255

20.3. Conditional calibration risk and prefix transfer

All 112 calibration/prefix cells meet the aggregate targets, with RMS 0.0396–0.1582 and mean KL at most 0.001497 nats, despite individual-context errors reaching 0.8514.Assessment scatter accounts for 74.42–93.26% of operator risk, with calibration displacement supplying the remainder; fixed panels do not establish a population 1/m risk term or an autonomous optimizer successor.Source: §20.3 · p. 256, Table 20.3 · p. 257

20.4. Training selects the cache state

All 18 recalibrated state/length cells meet both targets, while positive-control initial-cache RMS is 1.05765–1.20826, distinguishing state-specific accuracy from cache portability.All six zero-control initial-cache cells also meet both targets; initial predictive error does not establish training-created operator invariance or an equally competent initial model.Source: §20.4 · pp. 258–259, Table 20.4 · p. 258, Table 20.5 · p. 259
On disjoint contexts, all 18 recalibrated cells met both fidelity targets, while positive-control initial caches exceeded aggregate targets; fixed-cache transfer also failed in both positive-control cells.The recalibrated cells had centered RMS 0.03250–0.14382 and mean KL at most 0.001154 nats; on 16 further documents, positive-control fixed-cache RMS was 1.11646 and 1.16609, with all contexts exceeding 0.25.Source: §20.4 · pp. 260–261, Table 20.6 · p. 260, Table 20.7 · p. 261

20.5. Predictive visibility under categorical elimination

At 8,192 retained tokens, all 36 cells met the finite target, preserving 95.2–98.0% of native predictive variance with relative centered RMS 0.150–0.226.At 16,384 tokens, variance retention was 97.5–98.9% and relative centered RMS was 0.107–0.163; this is finite predictive visibility, not an autonomous optimizer successor law or compute-saving result.Source: §20.5 · p. 262, Figure 20.6 · p. 262

Finite optimizer transport and autoregressive prediction

The chapter treats finite optimizer and source interventions as changes to inference successors, retaining optimizer memory, prefix consistency, task margins, and coupled adaptation sectors.Source: Finite optimizer transport and… · p. 263

21.1. Finite native pulses and their response domain

Finite native pulses decompose each paired observation path exactly into an unperturbed component, an even response, and an odd response, whose covariance retains signed cross terms.Taylor-based derivative approximations require additional smoothness and derivative bounds; finite-amplitude agreement alone does not establish them or identify a scalar relaxation gap.Source: §21.1 · pp. 263–264

21.2. Finite optimizer memory as a source for inference

When β1 > 0, an incoming first-moment displacement can produce different outgoing parameters under the same first batch, so parameter-only projection is not exactly lumpable on domains containing those states.The incoming inference emission remains unchanged, but the successor can differ; this intervention result does not establish an autonomous multi-update moment law.Source: §21.2 · pp. 264–265

21.3. From operator stabilization to autoregressive inference

Operator stabilization, row concentration, and task correctness are distinct observations; reasoning here means performance on specified inference tasks whose answers depend on supplied premises.Source: §21.3 · p. 265

21.3.1. The global metric and prefix consistency.

A causal mask alone does not guarantee that full-window logits equal proper-prefix logits, because the global metric can depend on suffix inputs; inference transfer also needs operator and logit-response control.Small row contrast alone does not establish task competence, which requires correct margins on the specified task law.Source: §21.3.1 · pp. 265–266
Input contraction can suppress row variation while leaving training-sensitive common modes predictively visible; a unit multiplier alone does not establish criticality.The contrast uses fixed-weight input derivatives versus a training linearization that must include optimizer memory.Source: §21.3.1 · pp. 267–268
With equal prefixes and token-independent fixed metrics, causal decoding gives identical prefix logits, while collapsed rows can retain nontrivial, context-dependent operators.Row concentration alone is insufficient: the common-row field and its predictive transport may still vary with the suffix.Source: §21.3.1 · pp. 268–269

21.3.2. The training objective and the inference law.

Low parallel risk need not imply low proper-prefix risk; their gap combines conditional suffix information with model-error terms and can conceal cancellation.The exact risk decomposition is distinct from a claim that row projection or fixed-operator replacement improves proper risk.Source: §21.3.2 · pp. 269–270
Proper conditional excess log losses sum to bound continuation KL and event-probability error, but transfer requires matching support and conditional laws, and decoder changes can invalidate the bound.Nucleus truncation may assign zero probability to target-supported tokens; even a fixed retained nucleus needs threshold-margin control for stability.Source: §21.3.2 · pp. 270–271

21.3.3. Task margins and the retained critical coordinates.

A reduced computation changing every candidate score by at most e(x) preserves correctness whenever the original signed margin exceeds 2e(x).For opposite-answer questions, response must overcome average answer preference; contraction alone gives no sign or lower bound on that response.Source: §21.3.3 · pp. 271–272
Deductive tensors alone cannot determine token-label predictions: the conditioned training law must retain the predictive readout and contextual computation.Permuting the untied vocabulary projection changes predictions while leaving upstream deductive tensors unchanged.Source: §21.3.3 · p. 272
Under a native-margin tail condition, the probability that freezing turns a strictly correct answer incorrect vanishes as conditional operator variance tends to zero.The bound combines small-margin probability with a tail bound on score error; its task law and constants must match the operators and score collection being analyzed.Source: §21.3.3 · p. 272
Freezing a finite reference is useful when conditional operator variation and calibration error are small in predictively visible coordinates and native answers have adequate margins.Candidate scores include every proper prefix, with score error bounded by summing log-probability errors; differing candidate lengths use common slots whose unused reference operators contribute zero error.Source: §21.3.3 · pp. 272–273
At a fixed checkpoint, zero conditional operator variance makes the operator constant under the input law, yet it may differ across independently trained checkpoints.Thus input variation can vanish while training-seed susceptibility persists; the common metric remains part of training state even if its inference realization is frozen.Source: §21.3.3 · p. 273
For a specified data, drive, time, and model-size family, the retained inference observation is represented as a deterministic offset plus a linear transform of centered training fluctuations and a residual.The supplied passage also assumes convergence of the scaled training-fluctuation covariance and convergence of the deterministic linear maps; it does not state the resulting inference limit here.Source: §21.3.3 · p. 273

N). Then · Conditional moments, memory, and observation cost · Executed forecasts on fresh continuations

This section develops and evaluates finite, state-conditioned forecasts of training continuations and inference observations, accounting for optimizer memory, numerical approximation, covariance structure, and observation coupling. It distinguishes conditional prediction and calibration from autonomous or portable laws, inherited critical behavior, and uniformly linear responses, with claims bounded by finite-sample and observation-domain conditions.

N). Then

A nonzero limiting observation projection preserves the retained covariance count exponent, but projection, observation units, and uncontrolled remainders can hide or alter inference scaling.Identifying the limit covariance additionally requires second-moment control; these results do not establish a native critical exponent. Under the stated transport and margin conditions, an inherited upper bound concerns loss of already-correct answers, not a reasoning exponent.Source: N). Then · p. 273

21.4. Source selection, coupled sectors and fine-tuning.

Fine-tuning changes the conditional training law and observable directions, so an adapted exponent need not describe the incoming pretrained family.Inherited exponents require additional coupling and visibility conditions; adaptation may reveal a weak projection, generate a slow sector, or move the state away from a critical surface.Source: §21.4 · p. 274

21.4.1. Finite-corpus epochs and parameterized adaptation.

Factorized and merged weights can have identical inference emissions but different training successors, so the merged weight alone does not close factor dynamics.For scalar w = w0 + ab, the simultaneous factor update gives Δw = −ηg(a^2 + b^2) + η^2g^2ab.Source: §21.4.1 · pp. 274–275

21.4.2. The paired operator order parameter.

The paired operator discrepancy depends on the input coupling: it is zero for identical coupled outputs, yet can vanish under regular relaxation without indicating criticality.Paired agreement does not imply row collapse, and its interpretation depends on normalization and the observed operators.Source: §21.4.2 · pp. 275–276

21.4.3. An exact finite source law.

For finite source mixtures, the expected observable is a polynomial of degree at most BT, while finite averaging can remain smooth despite sharp crossovers.This exact law does not make an individual coupled path polynomial or guarantee a good quadratic approximation; singular critical behavior requires a specified joint limit with depletion and optimizer clock retained.Source: §21.4.3 · p. 276

21.4.4. Observable sectors and coupled critical surfaces.

Multiple critical sectors may mix or require additional controls to become marginal together, while source-response rank, local examples, and fitted covariance slopes have distinct meanings.Freezing PLDR’s shared metric generator changes the recursion and need not reveal an independent sector of the full native dynamics.Source: §21.4.4 · pp. 277–278

21.4.5. How mixed critical coordinates can obscure a finite fit.

A finite fitted exponent can drift because mixed positive power-law contributions yield a scale-dependent effective exponent; drift alone does not establish criticality.Observation projections, vanishing amplitudes, and signed cross-covariances can further change what a scalar measurement resolves.Source: §21.4.5 · p. 278

21.4.6. A condition for inherited fluctuation exponents.

Inherited covariance exponents persist when observation transport is bounded below and above and its L2 error is small relative to the weakest sector.The finite source-span certificate separately identifies two resolved current effect directions when λ2(G0) > ϵ.Source: §21.4.6 · pp. 278–279
Fluctuation-exponent inheritance requires controlling errors at the weakest retained fluctuation scale; small parameter displacement or good mean prediction alone is insufficient.A size-dependent observation can suppress a mode, while strong adaptation can replace the incoming fluctuation law; the cited finite experiments measure source response rather than assume the asymptotic inheritance condition.Source: §21.4.6 · p. 279

21.4.7. Representation error and coefficient transport.

A retained dictionary has an irreducible projection residual: no coefficient-acquisition rule can beat the best fit within its span.An assessment-panel oracle fit is a descriptive finite-panel diagnostic, not an out-of-sample prediction or population bound, and does not establish a large-size inheritance premise.Source: §21.4.7 · p. 280

Conditional moments, memory, and observation cost

The chapter develops finite conditional-moment forecasts for training continuations, combining compatible blocking, covariance-aware scoring, state adaptation, refresh costs, and a specified numerical law for smooth-response approximations.Source: Conditional moments, memory, and… · p. 281

22.1. Compatible conditional moment maps and their predictive score

Compatible linear blocking transports conditional means and covariances as (Bµ, BCBT) and composes exactly; the Dawid–Sebastiani score identifies both first moments without assuming Gaussianity.Regularize covariance once before transport; score comparisons require fixed coordinates and do not establish a successor kernel or higher moments.Source: §22.1 · pp. 281–282

22.2. State adaptation, correlation shape and calibrated scale maps

Under conditional exchangeability, split-conformal calibration gives whole-path coverage of at least n/(n + 1), simultaneously for chosen compatible linear observations.Coverage is averaged over calibration and the new path, conditional on the incoming state and adaptation; it does not guarantee a useful width, and missed state drift can widen the tube.Source: §22.2 · pp. 284–285

22.3. Training-state transport and the cost of retaining a forecast

Observation blocking changes forecast resolution at a fixed incoming state, whereas training changes the state, optimizer memory, and remaining corpus and requires a separate transport estimate.Source: §22.3 · p. 285

22.3. Training-state transport and the cost of retaining a forecast.

Forecast reuse across states requires control of complete conditional observation laws; small weight or source changes alone do not certify moment transfer, and a reused parent tube has no automatic successor-state coverage guarantee.A path-law total-variation bound v gives coverage at least n/(n + 1) − v, while fresh adaptation and calibration at the second state restore the n/(n + 1) conditional guarantee. These probabilities average over calibration and one fresh path, not each realized tube or multiple assessment paths.Source: §22.3 · pp. 285–287
Acquisition noise from one fitted mean adds a shared covariance sector across future centered paths, and a fixed acquisition budget can alter the leading coefficient of centered susceptibility.With m fitted paths, two future vectors centered on their common fitted mean have covariance (1 + 1/m)C, while conditioning on that mean leaves covariances C and zero; two independently fitted means have difference covariance 2C/m. The ledger counts restoration, updates, observations, fitting, calibration, and score evaluation, so the cost inequality is accounting rather than a speedup theorem.Source: §22.3 · pp. 287–288

22.4. The numerical law and its smooth approximation.

The implemented numerical policy is part of the finite transition law, while smooth pulse approximations require local regularity and their errors must remain small relative to the inference fluctuation scale.Strict clipping margins, positive Adam second moments and positive observed energies suffice for a local C3 real-arithmetic trajectory, not a uniform certificate for trained models. Under a uniformly contractive reduction with gap 1 − ρ_N ≍ N^-ζ, a sufficient condition for susceptibility χ_N ∼ cN^κ is η_N = o(N^((κ−1)/2−ζ)); numerical, acquisition, and emission errors require control in their respective units.Source: §22.4 · pp. 288–289

Executed forecasts on fresh continuations.

The executed studies test finite forecasts on fresh single-pass paths, including conditional covariance and memory, temporal moment maps, transfer, refresh budgets, and finite pulse response.The experiments use stated numerical and inference observations; frozen predictions and matched interventions distinguish estimation cost, state transfer, and numerical response error.Source: §22.4 · p. 289, Executed forecasts on fresh… · p. 290

23.1. Finite conditional prediction under single-pass continuation.

The 8/9 rank guarantee averages over calibration and one future path, conditional on the incoming state and development fit; it is not simultaneous coverage across models.Source: §23.1 · p. 290

23.1.1. Reduced-coordinate forecasts across incoming initializations.

0.06979, 0.08530, 0.99610, and 0.97406 nats are the largest cell errors for identity, translation, row/predictive affine, and optimizer-augmented affine forecasts; none meets the 0.01-nat target.At horizon 64, empirical mean-risk telescoping bounds range from 0.3814 to 0.9848 for identity, 0.4247 to 0.9963 for translation, 0.7359 to 3.2318 for row/predictive, and 0.7331 to 3.2121 for optimizer-augmented forecasts. These are bounds for retained-coordinate paths, not evidence of useful prediction at the target tolerance or uniform population closure.Source: §23.1.1 · p. 290, Table 23.1 · p. 291

23.1.2. A specified state-conditioned path law.

A specified state-conditioned path law uses eight fresh calibration branches and sixteen validation branches per state; its tube guarantees at least 8/9 joint coverage across four horizons for one new branch, averaged over randomness.The coverage is not 8/9 for each realized tube or simultaneous across all sixteen validation branches; development, calibration, and validation branches are separated under the specified reset law.Source: §23.1.2 · p. 291
Frozen tubes covered 117 of 128 initial fresh paths (91.4%); horizon-64 half-widths ranged from 0.0431 to 0.3472 nats, with maximum center-minus-mean discrepancy 0.0436 nats.Coverage counts whole paths and do not establish an independent eight-initialization population confidence statement.Source: §23.1.2 · p. 291, Figure 23.1 · p. 292, Table 23.2 · p. 292

23.1.3. Measured covariance of the risk-increment history.

Seven of eight states had negative signed cross-interval sums, and cross shares ranged from −3.790 to 0.275, so independent interval marginals would alter endpoint variance in state-dependent directions.The covariance transport is an exact finite reconstruction, distinct from predictive coverage on unseen paths.Source: Table 23.3 · p. 293, §23.1.3 · p. 293

23.2.1. Endpoint variance and a covariance-diagonalization control.

The joint/direct endpoint-variance forecast had smaller absolute error in 7 of eight states and mean absolute error 0.000479 squared nats versus 0.002656 for the diagonal control.Its forecast-to-measurement ratios ranged from 0.890 to 1.547; unchanged tubes covered 229 of 256 further four-horizon paths.Source: Table 23.4 · p. 294, §23.2.1 · pp. 294–295

23.2.2. Predictive memory has an estimation cost.

Endpoint MSEs were 0.002800, 0.002946, and 0.003191 squared nats for state-mean, last-risk, and three-risk-history predictors; the state mean met the 0.01-nat target in 26/32 cells.The largest absolute state-mean cell error was 0.02345 nats, so it did not meet a uniform 0.01-nat criterion; partial-path improvements do not establish autonomous prediction from an initial row summary.Source: Table 23.5 · p. 295, §23.2.2 · pp. 295–296

23.2.3. Native and numerical qualification.

Numerical checks passed for all twelve quadratures, and leading-asymptotic relative errors fell below one percent at gap 10−5; these checks do not establish native stationary critical behavior.Flat-spectrum integrals agreed with 1/(1 −λ2).Source: §23.2.3 · p. 297

23.3.1. Fixed estimation and compatible blocking.

Risk score differences at four intervals, two intervals, and the endpoint were -0.253, -0.085, -0.076; joint risk/entropy differences were -1.249, -0.533, -0.421, favoring regularized covariance overall.Regularized covariance scored lower in 8/8, 7/8, and 6/8 states for risk, and 8/8 states at all three scales for joint risk and entropy; these comparisons condition on eight recorded states, not independent models.Source: Table 23.7 · p. 298, Table 23.8 · p. 298, §23.3.1 · pp. 298–299

23.3.2. Matrix discrepancies and the scope of the reduction.

Fine-resolution regularized covariance had lower observed matrix discrepancy in all eight states, ranging from 0.293 to 0.450 for risk and 0.372 to 0.566 for joint risk/entropy.These discrepancies compare estimates, not population accuracy; the section supports a finite path-moment description, not a Markov kernel, stationary forcing spectrum, or portable law on unseen states. In the finite mixture, between-state variance was 64.38% for risk levels and 7.46% for increments.Source: §23.3.2 · pp. 299–300, Table 23.9 · p. 300, §23.3.4 · p. 300

23.4. Finite prediction across corpus and complete initialization draws

The new-corpus study varies corpus draws, complete initialization identities, and evaluation documents together, so its transfer test cannot isolate any one of those effects.It specifies two finite corpus draws of 65,536 documents each and 64 reserved observation documents; the supplied passages describe the design, not its results.Source: §23.4 · p. 301, §23.4.1 · p. 301

23.4.1. Executed single-pass design.

Risk fell from 10.368–10.386 to 7.288–7.620 nats across all eight trained states, while incoming row ratios ranged from 0.762 to 0.942.The tested incoming regime therefore had substantial row contrast, and its conditional moment and inference predictions did not presume proximity to a collapsed row manifold.Source: §23.4.1 · p. 302, Table 23.11 · p. 302

23.5. Acquisition budget and prediction after a training-state change

Sixteen adaptation paths cost 1,024 native updates per state plus 512 calibration updates; 32 assessment paths added 2,048 evaluation updates, and no smaller-budget candidate passed the declared uncertainty gate.The engineering gate required both maximum empirical mean-path error at most 0.01 nats and primary score excess at most 0.05 at every state and fit cohort; no smaller budget qualified on the declared domain.Source: §23.4.2 · p. 302, §23.5.1 · p. 307, §23.5.2 · p. 309

23.4.2. Frozen transfer and equal-budget adaptation.

For the four-interval risk target, local shrinkage and transferred correlation beat the diagonal control in 4/8 and 6/8 states, with equal-state mean score differences −0.117415 and −0.145059.For joint risk/entropy, both candidates beat the diagonal control in 8/8 states, with mean differences −1.098523 (local) and −0.633682 (transferred); neither candidate dominated at every target and state.Source: Table 23.13 · p. 303, Table 23.14 · p. 303, §23.4.2 · p. 304

23.4.3. Mean accuracy, calibration and inference.

Donor-operator replacement had mean forward KL at most 1.454 × 10−3 nats and row projection at most 8.736 × 10−3, despite individual target-score changes reaching 0.410 and 0.818 nats.The finite empirical endpoint-risk covariance assigned within-state continuation variance 22.8% of the four-head total and 43.3% of the fourteen-head total; two corpora do not estimate a precise population variance.Source: §23.4.3 · pp. 305–306, Table 23.16 · p. 305

23.5.1. Frozen candidates, estimands and costs.

At sixteen paths, the local-minus-diagonal primary score was +0.019 at parents and −0.239 at successors, while four-path local covariance exceeded the diagonal score by 8.126 and 6.308.Transferred correlation was favorable in fifteen of sixteen fit cells at parents and thirteen at successors; these comparisons show a visible covariance-estimation budget cost.Source: Table 23.17 · p. 307, §23.5.2 · p. 309

23.5.2. Measured prediction and the state dependence of the retained law.

At sixteen adaptation paths, successor reacquisition beat parent reuse in all sixteen cells per candidate, with mean reuse excesses of 16.949 (local), 15.645 (diagonal), and 16.021 (transferred).Reacquisition also improved calibration coverage: 222/256 and 230/256 paths versus 92/256 and 125/256 under parent reuse; only three of sixteen renewed means met the 0.01-nat target. The retained object is a family of conditional forecasts and calibrated sets indexed by incoming state and acquisition information, not an autonomous successor kernel.Source: Table 23.19 · p. 309, §23.5.2 · pp. 309–310

23.5.3. Source deletion under a matched sequence coupling.

The matched sequence coupling had exact sequence-law total variation 0.9998973184, despite a derived mean of only 9.142857 replaced slots (variance 9.061426).Observed case means ranged from 9.0537 to 9.1641; the total-variation value came from exact population counts, not estimation from the rare no-edit events.Source: §23.5.3 · p. 311, Table 23.22 · p. 311
A matched sequence coupling used 32,768 source-deletion pairs, generated in 18.3 CPU-worker seconds with zero native training updates.Its risk implications still depend on uniform program-sensitivity and state-drift bounds that the experiment did not measure.Source: §23.5.3 · p. 311

23.6.1. Single-pass design and observation law

The paired study used 288 native continuations and 36,864 updates, restoring eight complete incoming states and retaining optimizer memory during fresh-source pulse interventions.Each continuation used 128 batch-32 updates; observations equally weighted saved times and coordinates, with source-sequence draws conditional on incoming states.Source: §23.6.1 · pp. 311–312

23.6.2. Finite-amplitude response.

At h = 3 × 10−5, all 64 half-amplitude signals were resolved, but none met the whole-path halving condition.Even-to-odd path ratios were 1.006–3.105 for the generator and 0.906–2.699 for the body; finite-pulse reconstruction matched endpoint covariance traces within 5.0 × 10−16.Source: §23.6.2 · pp. 312–313, Table 23.23 · p. 313, Table 23.24 · p. 313

23.6.3. Signed temporal and pulse covariance.

For the zero pulse, interval-diagonal variance overestimated endpoint risk variance by factors 1.181–14.560 across eight states, and every cross-interval sum was negative.Across 64 nonzero endpoint pulses, covariance-trace changes had 35 negative and 29 positive signs, spanning −0.15815 to 0.28702; the complete matrix identity matched within 5.6 × 10−17.Source: §23.6.3 · p. 314, Table 23.25 · p. 315

23.7.1. A fixed small-radius trajectory assessment.

At h = 3 × 10−7, none of the 16 native or 16 arithmetic-control eight-update paths met the complete-path tolerance.Discrepancies ranged from 0.553376–5.356736 for native paths and 0.375120–3.056367 for controls; zero-arm float32/control endpoint differences ranged from 0.00156 to 0.03694.Source: §23.7.1 · pp. 315–316, Table 23.27 · p. 316

23.7.2. A predicted slow mode with zero observed inference response.

The token-zero input-embedding row followed its exact native forecast, while logits and all specified observations remained bitwise identical to baseline.The result is limited to prompts whose prefixes exclude that token; it establishes neither a thermodynamic critical mode nor general invisibility.Source: §23.7.2 · p. 317, Table 23.28 · p. 317

Inference interventions and native adaptation · Conditional scaling and inference visibility · Renormalized laws and conditional criticality

This section examines how finite adaptation and inference interventions affect prediction and representation, alongside operator reductions and conditional renormalization, covariance, and scaling descriptions. It specifies the assumptions and state information these accounts require, distinguishing finite or conditional results from claims about general reasoning, full training dynamics, attraction, or thermodynamic criticality.

24.1.2. Predicted state transport and measured visibility.

The incoming predictive response was zero, but first-update native risk responses ranged from 2.857 × 10^-5 to 8.026 × 10^-4 nats after moment-only interventions.All 72 finite first-step predictions met the fixed implementation criterion; later-path responses retained finite pulse sectors and amplitude dependence, without establishing a uniform tangent approximation.Source: §24.1.2 · pp. 320–321

24.2.2. Observation law and finite predictions.

The finite mixture interpolant used three anchor outputs to predict four unused mixture outputs on held-out documents, conditioning on realized source permutations and uniforms.This does not estimate the word-averaged source derivative; each evaluated document also required three anchor-model forward evaluations to acquire the prediction.Source: §24.2.2 · p. 322

24.2.4. Persistence after returning to a common source.

After 256 common-source updates, return transport remained a per-state, conditional observation test, and the seven-contrast dictionary showed only a small held-out improvement over two contrasts.Maps were separately calibrated for each body draw, so the finite certificate does not establish a deterministic map across the initialization ensemble.Source: §24.2.4 · pp. 325–326

24.2.5. Complete fixed-dictionary projection diagnostic.

At least 97.63% of calibrated residual energy was orthogonal to the fixed dictionaries across 24 cells, while oracle relative RMS residuals ranged from 0.7978 to 0.9745.This supports representation error for these finite emissions, not the impossibility of richer or state-dependent representations or the absence of every native critical mode.Source: §24.2.5 · p. 326

24.2.6. A released reference in the concentrated regime.

On held-out documents, the final predictive source Gram had eigenvalues 0.7803 and 0.2849 (ratio 0.3651), while post-adaptation row-fluctuation fractions were 5.65 × 10^-8–2.25 × 10^-7.Proper-prefix risk rose from 3.6362 nats to four terminal risks ranging from 3.7086 to 3.8285, while OG fell from 0.01599 to 0.00320–0.00922. These measurements separate operator invariance from predictive-performance direction.Source: §24.2.6 · pp. 327–328
At 512 updates, the twelve controlled incoming states had held-out predictive Gram ratios λ2/λ1 of 0.264–0.582 and common-field ratios of 0.210–0.612.The family comprised N = 4, 8, and 14 heads with four body initializations per width; these finite source directions do not identify relaxation eigenmodes.Source: §24.2.1 · p. 321, §24.2.7 · p. 328

24.2.7. What the source experiment establishes.

Equal-mixture adaptation lowered balanced held-out risk in all twelve states by 0.00084–0.12334 nats, but finite source responses did not establish native critical exponents.The smallest source-span update cosine ranged from 0.019 to 0.282; three-knot quadratic-rule terminal relative Fisher RMS errors were 0.7926, 0.8188 and 0.8403 at increasing widths.Source: §24.2.7 · pp. 328–329

24.2.8. A qualified released base and finite-corpus adaptation.

Model 5 had lower validation NLL than model 4, with a paired document interval of [−0.1093, −0.0563] nats; full-parameter validation selected the incoming state in all four source arms.The held-out NLL-change intervals were technical [−0.0086, −0.0011], narrative [0.0000, 0.0000], mixture [−0.0021, 0.0004] and general [−0.0017, 0.0000].Source: §24.2.8 · p. 330

24.2.9. A matched single-pass coordinate comparison.

Full-parameter entries were zero, while the factor path’s technical NLL change across prefixes was −0.00227 nats, with adjusted interval [−0.00381, −0.00076].The matched-64 change was −0.00262 nats, but its adjusted interval included zero; other domains did not establish improvement, and the comparison was between specified optimizer laws.Source: §24.2.9 · pp. 330–331, Table 24.14 · p. 331

24.3.1. Native prefix interventions and residual row transport.

Across the two released models, mean predictive KL under suffix replacement was 5.45 × 10−10 and 3.85 × 10−13; largest complete-stack gains were 0.0671 and 2.42 × 10−4.Every sampled local row map contracted infinitesimal perturbations; retaining matrix orientation gave a sharper transport description than multiplying separate-unit scalar gains, without identifying critical training modes.Source: §24.3.1 · pp. 331–332, Table 24.15 · p. 332, Table 24.16 · p. 332

24.3.2. Fixed learned operators and task-margin preservation.

Native external-target losses were 3.77717 and 3.64575 nats; head permutation raised them to 6.10964 and 6.17360 while fixed operators and row projection preserved all 320 decisions.The fixed reference-head norms were preserved under permutation, so the loss differences concern assignment of operators to contextual attention paths; preservation also retains incorrect decisions.Source: Table 24.17 · p. 333, §24.3.2 · p. 333

24.3. Observed operator stabilization and predictive transport.

The observed inference description retains learned common operators, input variation, predictive transport and task margins, but these measurements do not identify a thermodynamic critical surface.The proposed scaling and endogenous-feedback hypotheses remain properties of the joint training law.Source: §24.3.2 · pp. 333–334

24.3.3. Demonstrations and controlled premise changes.

Released models answered both variants correctly on only zero to two of 96 counterfactual pairs at each demonstration count, while fixed calibration and row projection preserved every answer decision.Thus operator stability preserved these weak paired outcomes; reliable composition was not established on this finite task law.Source: §24.3.3 · p. 334

24.3.4. Full ARC test sets and the released score convention.

Native character-normalized accuracies were 36.15% and 22.95% for model 1, and 36.87% and 20.99% for model 2; head permutation changed 1,297 and 1,319 decisions.Fixed calibration operators and row projection preserved every native decision, while head permutation reduced ARC-Easy accuracy but changed ARC-Challenge accuracy in different directions across the models.Source: §24.3.4 · p. 337
The learned operator reduction preserves measured benchmark decisions, but two pretrained checkpoints do not establish a training transition or rule out benchmark contamination.Benchmark knowledge, premise-sensitive composition, and thermodynamic criticality remain distinct properties.Source: §24.3.4 · p. 338

25.1.1. Tensor cumulants and scaling fields.

Under independent-context blocking, the complete cumulant flow makes the mean relevant, covariance marginal, and cumulants of order n > 2 irrelevant at H = 1/2.The covariance transformation retains cross-head and cross-layer entries; the continuous interpolation need not define a probability semigroup for arbitrary finite empirical laws.Source: §25.1.1 · p. 341

25.1.2. The joint Gaussian fixed point.

Independent-context blocking yields a joint Gaussian fixed point for centered finite-dimensional observations, but does not imply that one unblocked output is Gaussian or that training is critical.After centering and covariance normalization, the nondegenerate retained space is standard Gaussian; model covariances and predictive response coefficients can still differ.Source: §25.1.2 · p. 342

25.1.3. The Gaussian effective action under head elimination.

For Gaussian laws, eliminating unresolved fields gives the exact retained precision as the inverse retained covariance, and sequential and simultaneous elimination agree.Replacing the effective precision with the retained precision block instead conditions unresolved fields to zero; non-Gaussian elimination can generate higher interactions.Source: §25.1.3 · pp. 342–343

25.2.1. Conjugate fields and observable susceptibility.

The susceptibility RG is exact without stationarity, while independent contexts keep normalized susceptibility fixed; enhanced finite susceptibility alone does not identify a critical limit.Fluctuation criticality requires divergent dimensionless susceptibility along a specified scale limit, and a universality class additionally requires a limiting joint law and relevant scaling fields.Source: §25.2.1 · pp. 343–344

25.2.2. Conditional long-range scaling.

For stationary covariance with a nonzero symmetric tail proportional to r^-γ, 0 < γ < 1, the nonzero sector has variance-normalizing exponent H = 1 − γ/2, but the covariance condition does not specify a limiting distribution.Finite enhanced susceptibility may instead reflect document heterogeneity or short-range correlations; covariance-shape fits are predictive diagnostics, not thermodynamic exponent estimates.Source: §25.2.2 · pp. 344–345

25.2.3. Thermodynamic families and exponents from an RG.

Thermodynamic exponents require a specified model family, training law, and physical scale; RG-orbit identities yield exponents only under the stated relevant-coordinate and scaling assumptions.A scheduled optimizer is not automatically stationary, and orbit identities alone do not supply a full asymptotic scaling function.Source: §25.2.3 · pp. 345–346
A shared finite training-state mixture can yield a singular limiting generating function through unequal inference means, but its growing document-draw scale does not establish an intrinsic model transition.The block-average potential tends to the maximum conditional potential, producing unequal one-sided derivatives at zero when checkpoint means differ.Source: §25.2.3 · p. 346
Finite-size exponents require an established geometry and critical universality requires the fixed joint law and relevant fields; approaching an optimizer stability boundary alone does not establish a training transition.The proposed finite-size model is conditional rather than a fit to the reported width measurements.Source: §25.2.3 · pp. 346–347

25.2.4. A testable mathematical condition for self-organization

Affine restoring feedback can ensure N^ϕr tends to zero in stationary probability under a sufficient condition, but finite-horizon convergence also requires control of transients.The PLDR trajectories measure finite inference emissions and optimizer interventions, not the critical feedback mechanism; a PLDR transition would need derived dynamics satisfying the conditions.Source: §25.2.4 · p. 347

25.2.5. Closed covariance flows with retained relaxation modes

Geometric covariance blocking composes, and for stationary Gaussian fields the retained covariance specifies the coarse law; lag-zero covariance must remain separate from relaxation modes to preserve memory.The broader hierarchy allows nongeometric and nonstationary residuals, so a large shared sector does not remove that freedom.Source: §25.2.5 · p. 348

25.2.6. A normalized width family and a regular thermodynamic reference

A regular Gaussian fluctuation limit follows under extensive-cumulant bounds, but it is conditional and finite width measurements do not establish those hypotheses or a thermodynamic singularity.Averaging over contexts can produce a Gaussian mixture, and multiplying measured variance by parameter count does not establish criticality.Source: §25.2.6 · p. 349

25.3. Conditional Gaussian limits of signed operators

A signed-operator Gaussian mixture requires complete conditional head-sign invariance, a uniform head bound, fixed observation dimension, and joint convergence; its N^-1/2 rate is regular averaging, not a critical exponent.Conditional sign blocking composes, but a finite covariance matrix alone does not close training evolution; one dominant head can also prevent Gaussian behavior.Source: §25.3 · pp. 350–352

25.4. Conditional head limits and collective stability

Head-count limits condition on initial shared parameters and the complete realized batch sequence while retaining learned shared parameters and optimizer moments as endogenous state coordinates.Conditioning on final shared parameters could remove the collective fluctuation under investigation.Source: §25.4 · p. 352

25.4.1. Head exchangeability and the uniform collective mode.

Head-permutation symmetry makes conditional head observations exchangeable, with Nχq_N = V_N + (N − 1)C_N and bounded susceptibility when V_N and N∥C_N∥ are bounded.The symmetry holds for the real-arithmetic, dropout-free native architecture with permutation-invariant initialization and the specified optimizer; floating-point reduction order may introduce rounding differences.Source: §25.4.1 · p. 353
Compatible covariance blocking composes across aligned head groups, but blocking heads within one trained architecture does not identify a separately trained smaller-width model.At Gaussian block normalization, susceptibility growth with block size probes the common sector at fixed N; the thermodynamic endpoint also changes the law with N.Source: §25.4.1 · p. 354

25.4.2. Initialization units for a nondegenerate head family.

Scaling native PLGA tensors by (d + N)/(d + 2) gives the same one-entry uniform initialization law across head counts as at N = 2.This equalizes an initialization law; it does not establish an infinite-width Adam law [75] [76].Source: §25.4.2 · pp. 354–355

25.4.3. A native initialization limit.

At fixed depth and head dimension, the native initialization limit gives Gaussian residual-coordinate and logit fields while retaining the nonlinear finite-head PLGA operator.The result assumes the uncached real-arithmetic architecture, shape-aware weights, and other conditions in Theorem 25.21; head-mean convergence has no stated rate, and the result is confined to initialization.Source: §25.4.3 · pp. 355–357

25.4.4. A native generator envelope.

Under the stated initialization and AdamW assumptions, the generator envelope is uniform in head count and training time.The bound applies to every head, layer, and finite input in real arithmetic.Source: §25.4.4 · p. 357
The native generator envelope establishes a finite-operator sector under the declared update, but is deliberately conservative and does not uniformly control derivatives, shared-state influence, or the full training limit.Collective susceptibility may still grow through correlations among bounded heads.Source: §25.4.4 · p. 358

25.4.5. Stationary laws of the limiting finite-width update.

A finite native PLDR update under the stated clipped, constant-rate AdamW conditions has an invariant law with finite parameter second moment, supported at zero bias-correction states.The result establishes existence, not uniqueness or convergence of individual-time laws, nor a relaxation rate uniform in width.Source: §25.4.5 · pp. 358–359

25.4.6. Paired dynamics and stationary marginal laws.

The paired training kernel preserves each coordinate’s marginal dynamics and has invariant diagonal laws, but these identities do not imply that displaced pairs contract toward the diagonal.Paired copies use the same training batch; finite pulse separation can therefore probe nonlinear propagation even when stationary marginals exist.Source: §25.4.6 · p. 359

25.4.7. The conditional thermodynamic functional.

Uniformly bounded Q_N yields convex, B-Lipschitz conditional thermodynamic functionals with locally uniform subsequential limits, compatible with fixed linear observations.This does not identify a unique limit, differentiability, or convergence of second derivatives; critical interpretation requires a singular collective sector in fixed units and the appropriate width and training-time limit order.Source: §25.4.7 · p. 360

25.4.8. Shared environments and the order of averaging.

Conditioning on a shared environment separates head-level covariance from covariance of conditional means, whose nonconstant limit can contribute an extensive leading term.Shared initialization, training drive, and document structure are distinct mixture sectors; the factor of N alone does not establish an intrinsic transition among heads.Source: §25.4.8 · p. 361

25.4.9. A regular limit from bounded initialization influence.

A bounded independent-block replacement influence is sufficient to exclude divergent conditional susceptibility, but verifying it for PLDR requires controlling full training and emission transport.The bound is not a uniform certificate from inspecting a few initializations, and the direct-output Adam bound does not control the learned shared generator.Source: §25.4.9 · p. 361

25.4.10. A collective stability mode and its temporal RG.

With independent centered innovations and fixed drift F, head-mean covariance obeys χ_t+1 = Fχ_tF^T + Q and blocks compose; if ρ(F) < 1, the covariance converges to a unique fixed covariance.A mode with eigenvalue λ has conditional mean decay time τ = −1/log|λ| for 0 < |λ| < 1; this does not establish stationarity of native consuming-corpus training.Source: §25.4.10 · p. 362

25.4.11. A finite saturation mechanism and a physical training pulse.

If the largest attention score exceeds every other score by Δ, the softmax remainder δ satisfies δ ≤ (S − 1)e^-Δ, giving a finite saturation mechanism.The passage also considers the head output under a local uniform score gain; saturation is distinct from divergent connected fluctuations.Source: §25.4.11 · p. 363
At fixed head size, a unique top-scoring key yields hard selection as gain rises, while the gain derivative vanishes exponentially; the selected key and value can still vary by context.The bound is 2(S − 1)κe^−κ∆∥Vc∥2∥sc∥, and average concentration does not establish the uniform-gap hypothesis throughout a model.Source: §25.4.11 · p. 363
A measured last-query Jacobian is only a local diagnostic; bounding the full vocabulary response requires control of transport factors, centered scores, and values.Vanishing local factors alone neither bound the whole response nor imply context-independent predictions.Source: §25.4.11 · p. 363
Native PLGA coefficients can be scaled together to realize G 7→(1 + α)G exactly, reproducing the inference head-gain source when downstream decoder states are recomputed.This exact algebraic intervention holds at fixed layer input and extends across layers by induction.Source: §25.4.11 · pp. 363–364
Training pulses keep optimizer moments and counters at pre-pulse values, making matched subsequent updates a state perturbation rather than an Adam coordinate reparameterization.Freezing generator learning rates in paired branches tests a proposed restoring mechanism; recovery does not establish attraction to criticality without approach to the independently identified singular surface.Source: §25.4.11 · p. 364

Colored sources, joint clocks, and selection windows

This section delineates how colored and correlated forcing, chronological covariance, clocks, conditioning, and training selection constrain claims about response, training dynamics, and fluctuation transfer. It distinguishes finite-step effects and conditional predictive or dynamical reductions from conclusions about native training or thermodynamic criticality, emphasizing the assumptions and error controls required for such claims.

26.1. Colored forcing, response gaps and conditional exponents

When s(ω) ∼ s0|ω|^α as ω → 0, with s0 > 0 and −1 < α < 1, δ = 1 − λ, the e-folding matched-forcing response time is τ = −1/log λ ∼ δ^−1.Also, N Var(XN) ∼ h²χN. A depleted low-frequency spectrum with α > 0 weakens the divergence; an enhanced spectrum with α < 0 strengthens it.Source: §26.1 · pp. 365–366

26.2. Chronological covariance before a stationary limit

A deterministic chronological response retains initial-state and forcing covariance sectors, but an exact residual identity does not provide a predictive residual law.Native Jacobians may depend on the random path, and the stationary scalar limit χN ≍ N^(ζ(1−α)−υ), τN ≍ N^ζ requires separate assumptions from finite conditional path descriptions and sampling control.Source: §26.2 · pp. 366–367

26.3. Joint training clocks and critical scaling

With head width d fixed as N varies, χN ≍ N^κ and |r| ≍ N^−ϕ define count exponents, not spatial critical exponents absent a separate length convention.Source: §26.3 · p. 367

26.3.1. The optimizer belongs to the scale map.

Optimizer rates, schedules, memory, and state-dependent velocity laws belong to the scale map; nominal clocks alone do not establish equivalent dynamics or a critical limit.For the reference schedule, the nominal sums 137399.95 and 100796.66175 differ, so one effective update count cannot preserve both; pooled transient schedule phases mix conditioned transition laws.Source: §26.3.1 · pp. 367–369

26.3.2. Innovation covariance and the conditioning law.

Under the conditional next-batch law, centered updates are martingale differences, but fixed realized batch histories define a different law and state-dependent drift can contribute to accumulated motion.The remaining corpus is part of the single-pass state; conditional noise traces alone do not predict displacement because drift feedback and its cross contribution remain.Source: §26.3.2 · p. 370
Native optimizer displacements are evaluated from the same incoming state, batch, and full-model clipping rule when defining local generator and body responses.The incoming state includes parameters, optimizer moments, and counters.Source: §26.3.2 · p. 370

26.3.3. Exact finite steps and coupled parameter sources.

The complete observation increment is the sum of generator, body, and mixed corners, whose conditional covariance includes cross terms even when the mixed corner vanishes.A generator-only approximation must control the body-plus-mixed response and its conditional covariance; a small generator emission residual alone does not establish that control.Source: §26.3.3 · p. 371

26.3.4. Local predictive source coordinates.

Frozen source coordinates predict projected responses exactly, and leading eigenspaces minimize second-moment residual for the law defining them, not for a future law.Uncentered bases need not optimize covariance; fixed-state approximation also does not ensure long-time covariance fidelity or remove the need to transport state-dependent bases and memory.Source: §26.3.4 · pp. 371–373

26.3.5. Slow parameter modes invisible to the declared emission.

An input-embedding row never observed in training inputs or the declared observation law can decay as (1 − η_Nλ_d)^t while remaining predictively invisible, despite a relaxation time asymptotic to N/(2η_0λ_d).The null sector is relative to the specified input law; another prompt containing the token can expose the row.Source: §26.3.5 · p. 373

26.3.6. Bounded fields constrain count exponents.

For a bounded field, χ_N ≤ N/4, so an asymptotic susceptibility exponent must satisfy κ ≤ 1; a finite log–log slope above one may reflect a transient rather than an asymptotic law.A slope near one is allowed but does not by itself establish intrinsic criticality.Source: §26.3.6 · pp. 373–374
Radial and directional common fluctuations are distinct nonnegative contributions, while a finite common random coordinate can yield extensive susceptibility without an interacting critical mechanism or endogenous selection.Interpreting a common fluctuation requires predictive-source and response information; conditioning on a learned outcome can remove the phenomenon of interest.Source: §26.3.6 · p. 374

26.3.7. Scaling dimensions belong to observations.

Under the stated convergence, uniform-integrability, and nonzero-variance conditions, a pth-power observation has susceptibility scaling N^(1−2pΔ), so different observations of one fluctuation can have different scaling dimensions.For a linear mode with κ_q = 1/2, the quadratic energy can have κ_Y = 0; transferring susceptibility also requires controlling observation error relative to the response gap.Source: §26.3.7 · p. 375

26.3.8. Dynamic closure near a slow sector.

For a specified coupling, a centered observation error of size o(N^((κ−1)/2)) suffices to preserve χ_N ∼ C N^κ, while exact preservation requires its signed covariance-plus-variance contribution to be negligible relative to Var(X_N).For process transfer, the section also requires an accumulated L2-closure estimate over time, not just one-step accuracy.Source: §26.3.8 · pp. 375–376
Dynamic closure near a slow sector requires an added discrepancy-recursion hypothesis; frozen-state predictive fidelity alone does not guarantee it.A closure accurate on ordinary intervals can fail in a critical relaxation window, where the full clipped-Adam tangent and shared-force transport identify terms to retain.Source: §26.3.8 · p. 376

26.3.9. Moving peaks and a limiting source potential.

Along a joint orbit, the moving-peak family has an analytic limiting source potential while its curvature at zero diverges as N^κ.At fixed N and t > 0, susceptibility peaks at g = Na/t; at fixed N and g > 0, the law tends to a point mass at zero as t → ∞. The construction does not have a singular limiting potential on the stated orbit, and branch heterogeneity can also produce finite transient peaks.Source: §26.3.9 · pp. 376–377

26.3.10. Conditional histories and stationary laws.

Averaged stationarity and conditional training histories are distinct: a stationary law can have conditional means that keep moving with the batch environment.Convergence of an evolving conditional law, existence of an averaged invariant law, and criticality of either law’s chosen family remain separate claims.Source: §26.3.9 · p. 377, §26.3.10 · p. 377

26.3.11. Feedback with a retained correlation time.

In the correlated-forcing normal form, growing noise correlation time changes the critical-window condition, so instantaneous conditional variance cannot replace temporal covariance.The proposition’s forcing covariance and signed critical distance are not assumed to describe native PLDR training; a finite horizon also requires the scaled initial transient to vanish.Source: §26.3.11 · p. 378
Fluctuation-scale inference visibility requires a sufficiently small centered residual, while a small absolute predictive error alone does not ensure that condition.For vectors, if the transported variance remains a positive fraction of the input variance and the residual variance is negligible, output variance preserves that fraction under the specified conditional law.Source: §26.3.11 · pp. 378–379
Categorical elimination with a fixed tail law is an idempotent stochastic reduction with an exact tail-weighted KL error and preserved variance traces under its isometric embedding.The maps act on a predictive law already acquired from the model; a small representation error does not supply a successor law for the reduced state.Source: §26.3.11 · p. 379

26.4. Training selection relative to a shrinking critical window

Finite training-selected emissions and conditional memory requirements are observed, but native attraction into a shrinking critical window is not established.The restoring-drift result is a conditional mechanism, not an identified AdamW law; single-pass training can have colored forcing from remaining corpus and optimizer memory.Source: §26.4 · p. 380

Optimizer scaling and metric-row reduction

This section develops augmented-optimizer transport and conditional reductions of learned metric rows, including covariance and gradient transport and the conditions under which common-row covariance represents the head-averaged metric. It distinguishes internal fluctuation and covariance fidelity from predictive equivalence, specifying limits and conditions for transferring training fluctuation scaling to prediction.

27.1. The augmented Adam tangent and its temporal composition

The local update tangent includes the clipping-factor derivative and both Adam moment directions; temporal composition retains optimizer memory and is not itself a critical-mode or susceptibility estimate.A finite product can amplify perturbations, but relating it to a collective susceptibility requires the specified reduction, noise law, and limiting regime.Source: §27.1 · pp. 381–382

27.2. Clipping, denominator offsets and the width limit

With fixed positive Adam offset, a first-step gradient u/N gives a vanishing adaptive ratio, while η_N T_N b_N/ϵ_N→0 suffices for decay-dominated coordinates to track pure decay.The decay bound assumes clipped gradients stay bounded along the whole training path; coordinatewise agreement need not control emissions combining growing numbers of coordinates.Source: §27.2 · pp. 382–383

27.3. Interventions on learned shared collectives

Under the crossed recipient–donor product law, the emission has an exact orthogonal decomposition, but it need not match the covariance decomposition under the factual joint law.A dominant donor contribution identifies an intervention-sensitive shared coordinate, while a large interaction indicates recipient–donor compatibility; neither establishes noncriticality.Source: §27.3 · pp. 383–384

27.4. Row concentration and the order of width and time limits

For bounded row fields, χ_N≤Np_N(1−p_N), so Np_N→0 forces vanishing absolute susceptibility, but weak concentration alone does not control tilted limits.An exponential-collapse example has maximal finite-time susceptibility Nq_*^2/4 and a cusp in its limiting tilted functional, despite concentration at zero along specified limits.Source: §27.4 · pp. 384–385

27.5. Physical row energy and downstream transfer

Bounded final LayerNorm makes raw and normalized row collapse equivalent, but a learned-transform constant-input defect can preserve downstream row energy despite vanishing input quotient.The affine energy identity is exact along a path but does not imply autonomy; at exactly row-constant input, the output quotient equals the defect.Source: §27.5 · pp. 385–386

27.6. The surviving common-row collective sector

The metric field separates orthogonally into a broadcast common-row component and a within-row-variation component under the stated mean-square normalization.The covariance proposition begins under square-integrability of the conditional initialization law; the supplied passage does not provide its full conclusion.Source: §27.6 · p. 386
When χμ > 0 and χ⊥/χμ → 0, normalized head-averaged metric covariance approaches the embedded common-row covariance in operator norm.Common and contrast sectors are orthogonal in trace, not necessarily statistically independent; the common-row covariance is representation-dependent and not itself a gauge-invariant predictive susceptibility.Source: §27.6 · pp. 386–387

27.7. Metric-row covariance transport with nonlinear sources

Nonlinear row maps transport row means and covariances through Jacobians plus signed curvature residuals, and compose through ordered Jacobian products with transported sources.A local Lipschitz constant below one contracts row covariance, but a centroid Jacobian does not control widely dispersed rows or establish a critical head-count limit.Source: §27.7 · pp. 387–388
Loss adjoints compose through transposed row Jacobians, while shared-gradient reduction must control both the transported loss source and parameter Jacobian.The full-model clipping factor is a separate requirement; predictive KL or final-unit input-adjoint gain alone does not ensure shared-gradient closure.Source: §27.7 · pp. 388–389

27.8. Compatible row reductions and fluctuation fidelity

Nested retained-row projections preserve the centroid and yield a positive-semidefinite covariance deficit; with the denominator floor inactive, discarded variance determines the row-energy change.The bound applies to a fixed-input projection, not downstream decoder effects; independently reranking rows at each scale is not covered.Source: §27.8 · p. 390
Predictive equivalence alone does not ensure fluctuation fidelity: a coupled reduction transfers an already specified susceptibility scaling only when its error is sufficiently small relative to the fluctuation.The condition transfers a scaling; it does not establish that scaling or a thermodynamic singularity.Source: §27.8 · p. 391

27.9. Observation error and inference visibility

Arithmetic observation is a second map whose errors propagate through the collective law, and small individual row fields differ from small ensemble susceptibility.Source: §27.9 · p. 391

27.9.1. Energy enclosures and a sample susceptibility bound.

A certified matrix error gives an energy enclosure, whereas float32-versus-float64 differences are empirical sensitivity unless the reference error is bounded.A row-field threshold is neither a universal rounding floor nor a threshold on N Var(QN); screening entries changes the estimand and is only a sensitivity diagnostic.Source: §27.9.1 · p. 392

27.9.2. Which training fluctuations survive freezing.

If YN = aN rN + eN, aN → a ≠ 0 and χe/χr → 0, then inference preserves the collective fluctuation with N Var(YN)/χr → a².Source: §27.9.2 · p. 392
After multiplying by N, the two mixed covariance terms have absolute sum at most 2|a_N|√χ_rχ_e; division by χ_r gives the stated bound.The same calculation applies to paired sample covariances with a common divisor.Source: §27.9.2 · p. 392
Predictive critical scaling transfers from r_N only if it is first established for r_N and predictive overlap, residuals, and ensemble remain suitable.Vanishing predictive overlap, a material residual, or a changing ensemble prevents the inference.Source: §27.9.2 · p. 393
Layer-sign symmetry and saturated score directions can make internal covariance respond differently from predictive behavior; across-initialization and across-context variation are distinct.The former is at fixed context; the latter is across contexts at one released checkpoint.Source: §27.9.2 · p. 393
Chapter 28 measures model-wide fluctuations and size–time transport, testing finite response, shared-state influence, and observation limits under the theory's protocols.Source: §27.9.2 · p. 393

Measured model-wide fluctuations and size–time transport

This section examines finite, condition-dependent measurements of model-wide fluctuations, covariance, predictive responses, operator behavior, and size–time transport across training trajectories and parameterizations. The findings support specific finite predictions and reductions but do not establish a universal stationary, critical, or size–time scaling law.

28.1.1. Checkpoints, dataset, and numerical execution.

The pretrained measurements used two distinct released PLDR-LLM-v51-SOC-110M checkpoints and a stratified RefinedWeb sample; their NLL is an in-source diagnostic, not a held-out-pretraining benchmark.Checkpoint weights were fixed, and the two releases were not a controlled training intervention.Source: §28.1.1 · p. 394

28.1.2. Joint observations and independent-document flow.

The descriptive block-flow H medians were 0.5017 for checkpoint 1 and 0.4968 for checkpoint 2, consistent with the H = 1/2 normalization under the independent-context law.The document blocks came from a finite sample drawn without replacement, so these measurements do not establish the independent-draw sampling hypothesis; coarse-scale higher-moment estimates reach a sampling floor.Source: §28.1.2 · p. 395, Table 28.1 · p. 395, Figure 28.1 · p. 396

28.1.3. Whole-model response and the importance of cross terms.

At head perturbation 0.003, full-matrix predictive KL errors had median relative errors of 0.365% and 0.370%; removing cross terms raised them to 33.85% and 32.66%.The measured off-diagonal Frobenius fractions were 0.351 and 0.297. This supports retaining a model-wide response matrix for simultaneous perturbations, not predicting one context’s response matrix from another.Source: §28.1.3 · p. 396, Table 28.2 · p. 397, Figure 28.2 · p. 398

28.1.4. Adjacent text segments and a correlated sector.

Across the two checkpoints, eight-segment susceptibility enhancements were 1.611 [1.575, 1.655] and 1.704 [1.624, 1.756], versus permutation-control medians of 0.991 and 0.992.The finite correlated sector is not represented by the independent-document product law, but eight segments do not identify a limiting correlation exponent, diverging correlation length, or critical universality class.Source: §28.1.4 · pp. 397–398, Table 28.3 · p. 397, Figure 28.2 · p. 398

28.1.5. Decoder regrouping and the nearly frozen operator background.

One calibration context’s generated operators produced maximum predictive KLs of 5.45 × 10^-7 and 3.21 × 10^-12 on 256 evaluation contexts, with no changed argmax predictions.The approximation was not exact operator invariance: checkpoint 1’s maximum relative matrix distance was about 1.59 × 10^-3, and its maximum logit change was 4.10 × 10^-3.Source: §28.1.5 · p. 399, Table 28.5 · p. 399, Table 28.6 · p. 399

28.2. Initialization trajectories and finite response windows.

At step 256, eleven initialization models were below the matched-target unigram, while every model remained above the full-pool unigram.These comparisons use different information budgets and do not establish mature contextual modeling. The tested local response window depended on state; for width 128 at seeds 12002 and 12003, upper limits were 10^-3 and 3 × 10^-4, respectively, not certified Taylor radii.Source: §28.2 · pp. 400–401, Table 28.7 · p. 400

28.3.1. Finite parameterizations and training quality.

At width 896, three fresh normalized runs reached NLLs of 7.144, 7.149, and 7.146 nats, below their matched-target unigram losses of 7.578, 7.560, and 7.564.Replacing their full contexts increased NLL by 0.852, 0.789, and 0.798 nats; preserving the final context token left increases of 0.078, 0.071, and 0.082 nats. The results support finite normalization and context use, not a new model size or independent corpus population.Source: §28.3.1 · p. 403, Table 28.10 · p. 403

28.3.2. Optimizer and generator interventions.

Fixed-initial-weight interventions expose optimizer dependence and state- and source-dependent effects, so inference fields alone do not determine a unique next transition.The finite branches do not establish a uniformly closed 36-coordinate kernel; the training law retains optimizer coordinates.Source: §28.3.2 · p. 405

28.3.3. Prediction for independent minibatch ensembles.

At e = 10−5, quadratic response error reaches 0.534% across 3,072 batch–probe pairs, falling to 0.0091% with the cubic correction.The largest calibration-versus-test discrepancy is 2.04 estimated standard errors; this descriptive check is not a simultaneous coverage guarantee, and conditional ensemble prediction does not identify an autonomous low-dimensional training process.Source: Table 28.12 · p. 406, §28.3.3 · p. 406

28.3.4. Sixty-four adjacent segments on fresh documents.

At b = 64, sum trace ratios are 8.308 and 7.189, while contrast ratios are 1.237 and 1.223; permuted controls are 1.000 and 1.007.This identifies a strong persistent sum sector and a smaller but nonzero residual contrast sector over the measured extent, without certifying stationary closure or a unique latent mechanism.Source: Table 28.13 · p. 407, §28.3.4 · p. 407

28.3.5. Properties retained by frozen pretrained inference.

Frozen pretrained inference retains context-dependent vocabulary predictions and operator dispersion, while training-time optimizer autocorrelation does not persist after weights are frozen.Context replacement substantially increases NLL, whereas replacing generated operators has a separately measured predictive approximation error; no critical training surface or persistent critical exponent is identified.Source: Table 28.14 · p. 408, §28.3.5 · p. 408

28.4.2. Initialization units and finite width comparisons; 28.4.3. Native initialization-kernel comparison.

From N = 2 to N = 24, entropy RMS difference falls from 0.1267 to 0.0358, row-field difference from 3.71 × 10−3 to 1.07 × 10−3, and final residual-kernel difference from 0.1078 to 0.0303.The initial conditional fluctuations are finite-stable across base widths, but these comparisons do not establish a trained many-head law or higher-cumulant bounds for a thermodynamic critical point.Source: Table 28.16 · p. 409, §28.4.2 · p. 409, §28.4.3 · p. 410

28.4.4. Conditional fluctuations after training; 28.4.5. Microscopic saturation and inference response; 28.4.6. Resolved source windows in selected larger-model states.

Across four base widths, selected-grid row susceptibility is largest at g = 2, with χR/N from 0.0264 to 0.0456; at width 896 and g = 16, 97.7% of measured attention observations have pmax > 0.99, while predictive context KL is 0.534 nats.Source: §28.4.4 · p. 410, §28.4.5 · p. 411

28.4.7. Horizon dependence and the withheld larger width.

At g = 1, χR at N = 24 rises from 5.53 × 10−3 to 5.59 × 10−1 between 2,048 and 8,192 updates, while held-out NLL falls from 7.442 to 7.315.The endpoint ratio χR(N = 14)/χR(N = 2) shifts from 0.53 to 9.27, and the withheld-width fits show mixed errors: regular versus power predictions are 2.9% versus 20.7% for entropy, and 46.0% versus 15.1% for row fluctuations. These finite, horizon-dependent results do not select a common scaling law or critical exponent.Source: §28.4.7 · pp. 411–412

28.4.8. Paired operator pulses and finite response propagation; 28.4.9. Numerical replay of the paired dynamics.

After 512 updates, ordinary pulse response norms reach 2.34–2.98 times their initial values for entropy and 10.64–60.37 for row fields; frozen-generator branches still amplify both responses.Recorded unperturbed replay controls are bitwise identical, resolving no replay floor, while finite pulse amplification alone determines neither a stationary population law nor a Lyapunov exponent.Source: §28.4.8 · p. 412, §28.4.9 · p. 412

28.4.10. Complete finite comparison family.

In the extra fan-in family, the largest sampled row susceptibility occurred at g = 1 at every displayed width; its distinct initial per-head units preclude pooling with the shape-aware scan.The comparison used the same coarse generator-rate grid and conditional covariance convention as the shape-aware family, but remains a finite-grid result.Source: Table 28.27 · p. 420, §28.4.9 · p. 420, Figure 28.11 · p. 421

28.4.9. Numerical replay of the paired dynamics.

The measurements use paired finite trajectories and fixed contexts; repeated checkpoints and continuations do not increase the number of independent initializations.Numerical replay distinguishes local Jacobian tests from full-model secant response, and interprets float64 categorical traces at absolute numerical resolution rather than as exact-zero tests.Source: Table 28.20 · p. 415, Figure 28.7 · p. 415, §28.5.1 · p. 422

28.5.2. Temporal resolution of the rate-dependent row sector.

At g = 2, row susceptibility fell from 1.60×10−1 to 1.60×10−6 at N = 4, 2.11×10−1 to 1.78×10−5 at N = 8, and 6.38 × 10−1 to 5.81 × 10−7 at N = 14.These changes were measured from 2,048 to 8,192 updates in the original ensemble with optimizer state retained; row means also decreased, so the early rate-grid maximum is not a stationary critical-control estimate.Source: §28.5.2 · p. 422

28.5.3. Independent replication and horizon doubling.

At 8,192 updates, row susceptibility rose with width from 0.0532 at N = 2 to 0.5336 at N = 14, while mean row energy fell across all four widths during continued training.The paired mean-energy changes from 4,096 to 8,192 updates were −0.406 to −0.536, with intervals strictly below zero; horizon doubling later reduced row means in all twenty continued trajectories, but did not establish the population ordering at the two largest widths.Source: §28.5.3 · pp. 423, 425, 426, Figure 28.13 · p. 425

28.5.4. Measured optimizer scales in the conditional family.

At g = 2, the late shared-metric fraction below the Adam offset reached 61.7–88.1%, versus 2.0–5.5% for power parameters; adaptive updates nevertheless remained substantial.At g = 1, shared-coordinate fractions below the offset also rose in every continued trajectory, reaching 16.4–53.7% by 16,384 updates; these finite diagnostics do not establish a uniform head-count limit or isolate a causal optimizer term.Source: §28.5.4 · pp. 426–427

28.5.5. Causal influence and compatibility of the shared metric learner.

At the early checkpoint, donor shared-metric weights accounted for about 93% of crossed row-energy variance, while recipient parameters accounted for about 93% of crossed entropy variance.At the later checkpoint row variance was about 42% donor and 44% interaction; off-diagonal predictive KL remained measurable, so low factual row energy did not establish shared-learner interchangeability.Source: Table 28.32 · p. 427, §28.5.5 · p. 427

28.5.6. Response to finite shared-state displacement.

After 4,096 updates, adapting interventions reduced relative row displacement to 0.00502–0.0993, or 0.00119–0.00725 with early moments replaced, while frozen-weight controls remained at 0.898–1.137.Yet endpoint predictive KL exceeded its initial displacement in every intervention, and entropy displacement also increased; row recovery therefore did not restore the paired predictive law or complete augmented state.Source: Table 28.33 · p. 428, §28.5.6 · pp. 428–429

28.5.7. Local row contraction and retained nonlinear transport.

At 8,192 updates, final-unit centroid norms were below 0.397 at N = 4, 8, and 14, while median empirical row-variance gains were approximately 0.015–0.016.Late nonlinear-residual-to-output-variance ratios ranged from 0.00137 to 0.909 at N = 4, 0.377 to 1.07 at N = 8, and 6.63 × 10−5 to 0.695 at N = 14, precluding uniform covariance closure from the linear term alone.Source: Table 28.34 · p. 428, §28.5.7 · pp. 429–430
At 8,192 updates, retaining four exceptional rows preserved 0.910, 1.003, and 0.994 of susceptibility at N = 4, 8, and 14, with predictive KL below 1.83 × 10−7.The reduction is state dependent: at 2,048 updates it retained only about 6% of susceptibility. Tests covered 34 cases and recomputed downstream nonlinear attention, residual, decoder, and vocabulary operations after projection.Source: §28.5.8 · p. 430, Table 28.36 · p. 430, §28.5.7 · p. 430

28.5.9. Common-row fluctuations after contrast concentration.

Mean physical row energy fell from approximately 1,002, 1,289, and 1,325 to 0.370, 0.813, and 0.0702 at N = 4, 8, and 14, while common susceptibility exceeded 99.997% of covariance share late.The common-row susceptibility increased at each width, and the relative operator-norm bound was below 0.0100 at all three late widths; these finite-ensemble results do not identify a width-limit exponent.Source: §28.5.9 · pp. 431–432

28.5.10. Loss-adjoint transport at matched training inputs.

Late final-unit adjoint gains ranged from 0.00645 to 0.0437, yet one late complete-block gain reached 2.04, showing source-dependent amplification despite final-unit attenuation.The block-gain case was N = 4, identity 640104, on minibatch 2,048; its paired early gain was 0.306, while internal unit five gained 15.1 and unit seven 0.0103.Source: §28.5.10 · pp. 432–433, Table 28.39 · p. 432

28.5.11. Training-force fidelity of the inference reduction.

At late N = 8, four retained exceptions approximated the shared clipped force within 1.72% across eight sources, but corresponding errors were not uniformly small at other widths.At late N = 4, one source had predictive KL 2.29 × 10−8 alongside clipped shared-force relative error 0.403, so predictive fidelity did not guarantee update fidelity.Source: §28.5.11 · p. 434, Table 28.40 · p. 434

28.5.12. Augmented response and its arithmetic window.

All 24 augmented-response controls had two adjacent passing amplitudes, with qualifying upper amplitudes at or below 10−7 in 15 controls and 10−8 in nine.The sampled windows reached from 10−7 to 3 × 10−10 and validated local transport of parameters, both moment directions, and predictive Fisher response; they do not certify native float32 windows or full optimizer updates.Source: Table 28.41 · p. 435, §28.5.12 · p. 436

28.6.1. Whole-seed uncertainty after horizon doubling.

Across sixteen N = 14 identities, susceptibility fell from 0.5336 to 0.1375 (difference −0.3961, SE 0.0917); mean held-out NLL shifted from 7.300 to 7.287 while context dispersion rose from 0.501 to 0.512.All sixteen mean row fields decreased, and the susceptibility difference had empirical percentiles [−0.5267, −0.2134]. The panel continued twelve existing identities rather than adding new initialization identities.Source: Table 28.43 · p. 436, §28.6.1 · pp. 436–437, Table 28.44 · p. 437

28.6.2. Complete-cohort arithmetic comparison.

Eight of thirteen arithmetic conditions met the declared bounds; at N = 8, g = 16, susceptibility changed from 3.6087×10−16 to 7.6642×10−16, a relative change of 112.38%.Across nine late rate-grid conditions, maximum paired CPU predictive KL was at most 1.361 × 10−4, but reached 0.0318 at N = 14, g = 16; tiny row susceptibilities did not support critical-scaling fits.Source: §28.6.2 · pp. 437–439, Table 28.45 · p. 438

28.7. Native size–time laws and visible stochastic sources.

The size–time study adds 96 training artifacts and 2,154,496 optimizer updates; continuations retain optimizer state, absolute step, sampler prefix, and supervised-target counts.Artifacts may continue existing initialization identities and therefore are not necessarily independent training replicates.Source: §28.7.1 · p. 439

28.7.1. The executed family and its random units.

The executed crop law samples with replacement from 3,072 documents, yielding 32 supervised external targets per batch; fixed contexts, heads, decoders, and arithmetic runs do not add independent initializations.The 64-input calibration set is distinct from the 512-input primary context cohort.Source: §28.7.1 · pp. 439–440

28.7.2. A finite rate–time clock and its predictions.

The fitted clock power is 1.9467 [1.8159, 2.0552]; its primary-threshold NLL is 117.69, near the power-two value 118.33, supporting a finite near-quadratic clock, not a critical exponent.At the primary threshold, the power-one NLL is 199.67; the fit does not establish a distinct universality class.Source: Table 28.47 · p. 441, §28.7.2 · p. 441
Across eight joint-target width–horizon groups, the largest absolute quadratic mean error is 0.04648, versus 0.41462 for linear and 0.05443 for fitted clocks; susceptibility extrapolation errors grow with width.The two-width susceptibility predictions are low by factors 2.04, 3.16, and 4.72, so mean-clock agreement does not identify common covariance scaling.Source: §28.7.2 · p. 442, Table 28.48 · p. 442

28.7.3. Common coordinates, nonlinear observations and new identities.

At 16,384 updates, effective head counts are approximately 1.0001–1.0098 across N = 2, 4, 8, 14, 24, while directional fractions reach 0.99015–0.99964, indicating coherent heads but varying common directions across models.At 32,768 updates the common component accounts for at least 99.9977% of the head-averaged metric covariance trace; these recorded-coordinate moments do not establish stationarity of the full conditional law.Source: §28.7.3 · pp. 443–444
The finite-common form predicts additional-panel common-centroid susceptibility with relative RMS error 0.02885, but NLL susceptibility favors the same-width reference, showing that prediction quality depends on the observed field.For NLL susceptibility, same-width relative RMS error is 0.07948 versus 0.1773 for the finite-common form; row susceptibility also differs 5.77-fold between panels at N = 4.Source: §28.7.3 · p. 447

28.7.4. Extended horizons, boundary dynamics and environments.

In the CPU panel, normalized row fields at N = 4 and 14 rise from 8.0500×10−6 and 1.4350 × 10−4 at 32,768 updates to 0.014587 and 0.015841 at 131,072, refuting monotone concentration.Zero-generator continuations retain large normalized contrasts while body-driven training lowers NLL for all eight matched identities, implicating generator adaptation in measured concentration without determining an infinite-time law.Source: §28.7.4 · pp. 448–450
Interaction sectors account for 64.06% and 70.78% of row-field variance at N = 4 and 14, respectively, so an additive output model omitting interactions misses most variance in this finite design.For the common centroid, interaction fractions are 65.83% and 67.44%.Source: §28.7.4 · p. 450
Conditional row susceptibility averages are 0.17864 at N = 4 and 0.55889 at N = 14 across the specified 8,192-update conditional laws.Susceptibilities range from 0.07734 to 0.27562 across four environment cells at N = 4 and from 0.52212 to 0.61443 at N = 14; these are not extra initialization replicas. Batch-history main-effect fractions also depend on observation: 22.53% and 28.74% for NLL versus 58.28% and 75.06% for fixed projected logits.Source: §28.7.4 · p. 451

28.7.5. Frozen training-law risk and online evolution.

At N = 4, training-law NLL rises from 7.3380 to 8.3413 and held-out NLL from 7.4623 to 8.4793; at N = 14, they rise from 7.2023 to 7.4937 and 7.3632 to 7.6025.Held-out-minus-training-law gaps change from 0.1243 to 0.1380 at N = 4 and from 0.1609 to 0.1088 at N = 14, while mean held-out entropy falls from 7.0626 to 5.1311 and 7.1305 to 6.7415, respectively. The selection was a developmental diagnostic and does not guarantee population coverage.Source: §28.7.5 · pp. 452–454, Table 28.58 · p. 453

28.7.6. Conditional native noise and predictive transport.

In all 24 matched comparisons, predictive participation rank is lower than parameter participation rank when both use the same eight minibatches and centering.All 192 directions meet the joint local-response diagnostic at α = 3 × 10^-6, but none does at unit amplitude; the measured transport is not a complete native optimizer step.Source: §28.7.6 · pp. 454–455, Table 28.60 · p. 455

28.7.7. Frozen adjoint coordinates and fresh-batch validation.

With eight retained directions, late covariance residual ranges fall to [0.00345, 0.188] at N = 4 and [0.00123, 0.113] at N = 14, improving from early ranges [0.148, 0.499] and [0.257, 0.601].Mean paired changes are −0.2815 and −0.4428, respectively, but late pooled residuals weight states by conditional noise trace and do not imply uniform statewise accuracy. All 256 fresh directions pass the stated tangent and Fisher criterion at the three tested local magnitudes; this local subspace is not an autonomous reduced training state.Source: §28.7.7 · p. 457, Table 28.62 · p. 457

28.7.8. Paired expansion of the predictive source state.

With 32 coordinates, every measured late state retains at least 99.769% of its conditional predictive tangent noise trace.Late statewise residual ranges are [3.70×10^-5, 0.001229] at N = 4 and [7.19×10^-5, 0.002301] at N = 14; increasing calibration from eight to 32 minibatches also improves fresh covariance residuals at fixed k = 8 in all sixteen states. At sixteen coordinates, one of eight late states still has residual above 1%, and centered-noise ordering is not uniformly superior at truncated dimensions.Source: §28.7.8 · p. 458

28.7.9. The complete native step and its source interactions.

At 16,384 updates, generator-only pooled predictive total-response errors are 0.9797 and 1.0034 and noise errors are 1.0129 and 1.1543 for N = 4 and 14, respectively.The corresponding mixed second-moment fractions are 0.1714 and 0.6067; by contrast, at N = 14 the internal common centroid has generator-only total and noise errors of 0.01711 and 0.02083. Generator-corner mean row increments are negative in all four width–time conditions, but complete row means have both signs across states; full-step mean predictive KL is 0.005772 at N = 4 and 0.002647 at N = 14, with twice-KL/Fisher-square ratio ranges [0.874, 1.108] and [0.903, 1.054].Source: §28.7.9 · pp. 460–461, Table 28.65 · p. 461, Table 28.66 · p. 461

28.7.10. Observable time scales and a decoupled slow sector.

In the final 32,768-update interval, linear trends explain 40.6%–55.2% of common-coordinate variance at N = 4 and 42.3%–51.5% at N = 14.For NLL, trend fractions are 45.1%–79.1% and 13.3%–33.9%, respectively, so raw finite-window covariance mixes trends with remaining fluctuations.Source: §28.7.10 · p. 463, Table 28.67 · p. 463
At lag 8,192, common-coordinate correlations were 0.0623 and 0.0589 at N = 4 and 14, versus −0.2388 and −0.2625 after detrending.These are finite-window normalized covariances under distinct centering conventions, so a single fitted correlation time is not an established critical relaxation time.Source: §28.7.10 · p. 463
τ_N = −1/log(1 − 6 × 10^-6/N) ∼ N/(6 × 10^-6) for the growing parameter time scale, yet the specified input-law emission has zero overlap with it.The executed control pulsed index 6481 by 0.1 for sixteen native CPU updates at each of five head counts; paired predictions and losses agreed exactly, with absent-input gradients exactly zero.Source: §28.7.10 · p. 463
An intrinsic critical exponent requires a controlled limiting law at an independently identified critical surface; self-organized criticality additionally requires restoring feedback, shrinking-window and forcing-memory control, and predictive overlap.These requirements are distinct from row concentration, common-head coherence, rate–time power laws, and a slow parameter coordinate.Source: §28.7.10 · p. 463

Physical calibration and the fluctuation readout budget · Synthesis and the boundary of the established theory · Appendices

This section develops conditional physical calibration and readout-error budgets, including a proper-prefix readout, to assess how physical fluctuations survive model readout and scale transport, and reports finite calibration and fine-tuning findings. Its synthesis separates exact finite conclusions and bounded empirical evidence from conditional scaling and inference-readout claims, while leaving model-specific attraction, predictive reduction with controlled cost, and a native limiting critical law unresolved.

29.1. Physical universality as a conditional calibration law

Known physical universality classes calibrate observation and scale maps, not an optimizer’s universality class; finite-size corrections remain part of comparisons with limiting targets.A native generated law is defined using normalized proper-prefix probabilities on the fixed spin alphabet, and a thermodynamic claim requires a joint family varying width, adaptation age, and corpus resources—not a single finite checkpoint.Source: §29.1 · p. 464

29.1.1. From proper-prefix approximation to physical moments.

Physical moment comparisons require separate control of mean-color profiles, connected spatial covariance, and generated-law error; raw correlations can hide cancellation between covariance and learned mean-field contributions.The covariance-plus-mean decomposition holds for empirical laws without translation invariance; cross-entropy alone does not bound the conditional-law error needed to transfer fluctuation scaling.Source: §29.1.1 · pp. 465–467
When the temperature step h shrinks, vanishing endpoint errors alone do not ensure vanishing thermal-contrast error; a sufficient condition is D_L,+ + D_L,− = o(h_L).A thermal exponent additionally requires limiting scaling hypotheses and empirical tests; finite-size bias, Monte Carlo uncertainty, and learned-law error are distinct contributions to fitted exponents.Source: §29.1.1 · pp. 466–467

29.1.2. An architectural route for physical input fluctuations.

If the first-layer color-query coefficient map is injective on the zero-sum subspace, query-Gram traces recover color fractions and preserve existing covariance exponents under bounded singular values.This pre-normalization visibility can expose an externally critical input law even at random initialization, but does not imply a critical generated law or training transition; later normalization and nonlinear maps remain empirical questions.Source: §29.1.2 · pp. 467–468

29.1.3. Bounded magnetic coordinates from internal observations.

Projecting fitted color fractions onto the simplex and reconstructing the magnetic invariant preserves its geometry, but transferring an exponent requires both readout accuracy and generated-law accuracy.Readout error under a reference configuration law and error in the model’s generated law are separate quantities.Source: §29.1.3 · p. 468

29.1.4. A joint error budget for physical scale transport.

Physical scale transport bounds Wasserstein error by combining generated-law, native-readout, and block-closure defects; replacing the blocked reference with a Gibbs family requires an additional coupling bound.The empirical transport check cannot supply the unknown large-lattice population error.Source: §29.1.4 · pp. 468–469

29.1.5. Compatible internal variables and RG eigenvalues.

A compatible internal scale map carries represented RG eigenvectors only when the representation retains their direction; approximate residuals need conditioning, and prediction alone does not establish map compatibility.Physical thermal and magnetic eigenvalues yield exponent relations only under homogeneous free-energy scaling assumptions; attention eigenvalues and optimizer relaxation factors are different quantities.Source: §29.1.5 · p. 469

29.1.6. A spatial control that preserves bulk critical moments

Spatial shuffling preserves the full histogram distribution and its magnetic moments but need not preserve separated-spin correlations, distinguishing bulk-histogram recovery from physical geometry or an endogenous transition.The following readout-accuracy analysis conditions on a fixed checkpoint, configuration law, and fitted readout.Source: §29.1.6 · pp. 469–470, §29.2 · p. 470

29.2. Magnetic readout accuracy at the fluctuation scale.

The condition δL = o(√µL) preserves an existing second-moment slope, but does not establish that a limiting slope exists.Connected fluctuations require δc,L = o(√κL); generated-law transport additionally requires TV(PL, QL) = o(µL), and Binder ratios need separate higher-moment controls.Source: §29.2 · pp. 470–471

29.2.1. A constructive proper-prefix readout.

For context-compatible families with µL ≍ L^-2x and x < 2, the constructive proper-prefix readout has vanishing relative second-moment error in exact arithmetic.With arithmetic error, the sufficient condition is ∥ε∥L2/σL = o(L^-x); the scaling conclusion assumes the source law and a context at each size.Source: §29.2.1 · pp. 471–472

29.3. Executed calibration with two physical universality classes.

The calibration separates equilibrium-source targets, learned configuration laws, internal readouts, and finite optimizer response; its Wolff sampling clock is not a physical or optimizer relaxation clock.All 72 specified equilibrium checks passed, while the thermal Potts estimate remained cutoff-dependent and its sampling interval alone did not certify asymptotic exponent recovery.Source: §29.3.1 · pp. 472–473, §29.3 · p. 472

29.3.1. Physical source and native adaptation.

The first-Gram check passed all 72 selected cases, and all 24 observation-family transport checks met the coupled empirical bounds.The largest relative trace discrepancy was 3.71 × 10^-7 and the largest recovered color-fraction error was 3.8 × 10^-6; finite enumeration checked normalization and KL-chain identities but established neither a thermodynamic limit nor small large-lattice KL error.Source: §29.3.1 · p. 475, §29.3.2 · p. 476

29.3.2. What the calibration identifies.

The calibration supports a finite architectural mechanism for observing externally imposed magnetic structure in native common fields, not a native thermodynamic exponent or endogenous self-organized criticality.Exponent inheritance requires asymptotic error and visibility hypotheses; paired finite optimizer continuations use the nonlinear conditional transition law when amplitude-halving comparisons do not support tangent identification.Source: §29.3.2 · pp. 476–477

29.3.1. Physical source and native adaptation.

Native proper-prefix risk decreased in all eight paths, and the common-row invariant improved all eight unblocked comparisons, with six paired chain-bootstrap intervals excluding zero.The invariant was evaluated on fresh simulation seeds after its construction was fixed; readout errors under the reference law are distinct from generated-law errors.Source: §29.3.1 · pp. 473–475, Figure 29.2 · p. 478

29.3.3. Physical fine-tuning from the qualified released model.

For q=2, NLL-change intervals were [−0.0780, −0.0690] and [−0.0719, −0.0624] nats; for q=3, [−0.1132, −0.1034] and [−0.1135, −0.1023] nats.Both adaptation runs completed three epochs, and validation—not final-test scores—selected the exported checkpoints.Source: Table 29.11 · p. 479, §29.3.3 · p. 479
Both adaptations reduce joint KL and improve aggregate held-out accuracy and NLL; unseen-size NLL improves, but Potts accuracy changes are −0.062 and −0.720 percentage points.Their whole-chain paired 95% intervals are [−0.446, 0.323] and [−1.186, −0.243] percentage points; accuracy transfer depends on class and seed.Source: §29.3.3 · pp. 479–480
Fine-tuning improves held-out spin prediction while retaining strong row concentration and small paired operator discrepancy, but raw correlations can mask distinct mean and covariance errors.At L = 24, adaptation 0’s raw-correlation errors are +0.056 and −0.063 while connected errors are −0.188 and −0.307; each position-mean contribution is approximately 0.245. The magnetic size fits vary with cutoff and seed, so a close finite fit does not establish source universality.Source: §29.3.3 · pp. 480, 483, 484

29.4.1. Finite native assessment of the budget.

Adapted common-row readouts have r = 0.0411–0.2044 and relative moment errors of 0.0072–0.0898; all eight invariant-gain point estimates are positive, but two sizes cannot establish asymptotic decay.Six paired-chain 95% intervals exclude zero; this is not a simultaneous all-cell guarantee.Source: §29.4.1 · p. 484

29.4.2. Fresh chains and unused physical sizes.

Across 48 unblocked Gram cells, r = 0.00233–0.01231, the sufficient budget 2r + r^2 is below 0.025, and maximum actual relative second-moment error is 0.00149.The assessment verifies Equation (29.17) for 36,864 measured pairs; these finite-law bounds are not population confidence bounds or an asymptotic certificate.Source: §29.4.3 · p. 486, §29.4.2 · p. 486

29.4.3. A finite mechanism for scale-sensitive observation.

The first-Gram observation operates before training, but two-block versus iterated one-block discrepancies of 0.0068–0.3638 show that histogram recovery does not yield an autonomous spatial block map.Scale-dependent observation must retain metadata, normalization, coefficient acquisition, and closure error.Source: §29.4.3 · pp. 486–487

29.4.4. Finite-size corrections and uncertainty in the reference law.

For q = 3 and Lmin = 8, the leading estimate is 0.8051 and its interval excludes 5/6; correction fits have greater leave-one-size-out error and establish no native thermodynamic exponent.At Lmin = 16, correction-family point estimates range from 0.8166 to 0.8383; closeness to a known exponent does not validate a correction model.Source: §29.4.4 · p. 488

Synthesis and the boundary of the established theory

The compatible description is the training path law with full-graph emissions and declared sources; eliminating components preserves joint dependence, while equating it to a separately trained smaller architecture requires matching assumptions.The synthesis states that it combines complete-law, geometric, and predictive descriptions to delimit finite conclusions, conditional limits, and unresolved mechanisms.Source: §30.1 · p. 489, Synthesis and the boundary of the… · p. 489

30.1.1. Scale laws and their hypotheses.

For consuming finite populations, exact negative cross-time covariance and physical-clock second-moment limits do not imply Gaussian or adaptive-gradient limits; trained critical classes and exponents remain conditional, not assigned by finite measurements.Measured row-concentration recovery establishes a finite regime but does not identify a critical surface.Source: §30.1.1 · p. 490

30.1.1. Scale laws and their hypotheses

The fluctuation readout budget requires ∥e∥L2 = o(N^((κ−1)/2)) to preserve a count law’s leading constant, plus a nonzero transported covariance visible to inference.This is sufficient, not necessary: the exact covariance difference contains signed terms that may cancel, and constant-logit directions have zero categorical Fisher weight; a law-averaged W1 risk bound does not supply the required L2 coupling.Source: §30.1.1 · p. 491
Under the stated fixed spectral-shape hypotheses, κ = 1 − a + ζ(1 − α), and bounded intensive fluctuations with positive leading coefficient require κ ≤ 1; these are conditional relations, not fitted exponents.The relation also requires uniform limiting hypotheses; inference preserves the susceptibility exponent only with a nonzero predictive projection and smaller coupled emission error.Source: §30.1.1 · p. 491

30.2. The resulting picture

Exact row-energy identities and augmented optimizer dynamics connect to predictive renormalization, but an autonomous reduced kernel requires closure on its declared fibers.When a smaller state does not close, the supported alternatives are retaining conditional memory or accepting quantified finite prediction error; inference reductions remain state-specific and require explicit emission and error budgets.Source: §30.2 · pp. 491–492

30.3. Conditional limits and unresolved mechanisms

Three unresolved mechanisms are an all-future attraction certificate, a predictive reduced training state with controlled dependence and acquisition cost, and a native limiting critical law with specified size convention and predictive projection.The conditional theory distinguishes regular signed-operator limits, shared-environment mixtures, connected covariance sectors, colored forcing, and possible critical modes; common randomness or a moving finite-width peak alone does not establish a thermodynamic universality class.Source: §30.3 · p. 492

30.4. Finite conclusions and their experimental units

Nine trajectory-layer units in the long row study were not 864 independent training replicas, and all ten complete row-RG cells failed the registered nominal precision gate; the head-count experiment did not resolve systematic narrowing.The refinement used 384 paths and 1,228,800 updates; its half-height widths at head counts 4, 8, 14, and 24 were 0.617, 0.558, 0.593, and 0.705. Growing size times variance and finite clock fits do not establish a critical class or universal exponent.Source: §30.4 · p. 493
18 recalibrated aggregate cache cells met the centered-RMS and mean-KL targets, but individual-context exceptions remained, so success is state- and panel-specific rather than uniform across histories.The six zero-control initial-cache cells met both targets, while the six positive-control initial-cache cells failed. A displaced cache can improve target NLL while reducing fidelity to the native predictive law.Source: §30.4 · p. 493

APPENDIX A Statistical methods and complete claim scopes … B.14. Complete finite-pulse amplitude outcomes

The appendix specifies the assumptions, replication units, observation conventions, and finite-sample scopes needed to interpret its experimental claims and complete outcomes. It reports finite evidence across conditional prediction, training reduction, and row-flow analyses while distinguishing exact finite identities from approximations and identifying limits on broader transfer.

A.1. Statistical methods and the observation boundary

The central 384-trajectory family’s 1,228,800 updates are included in the 4,253,160-update classified native subtotal, not added to it.The totals condition on the stated corpus, order, and shared generator; they do not imply an independent corpus ensemble.Source: §A.1 · p. 496
Training initialization, batch randomness, and evaluation contexts are distinct random units; continuations and repeated observations do not increase training-replicate counts.Native initialization comparisons condition on the full recorded single-pass stream, unlike frozen-state covariance that marginalizes the unrevealed future; training seeds are not corpus replications.Source: §A.1 · pp. 496–497
Selected native-risk probes test local mechanisms, while paired constant-versus-cosine trajectories were prespecified; adaptive selection or trajectory reuse is not untouched population confirmation.The cosine-horizon source-recipe comparison is secondary and retains shared initial parameters, sampling prefixes, and observation cohorts despite different post-warmup drive histories.Source: §A.1 · p. 497
The conditional path study separates development, calibration, and validation, but its eight strata contain only two outer initialization identities; 128 validation paths do not measure pretrained-population variation.The paths quantify the conditional continuation law, and whole-path coverage and calibration-averaged rank guarantees are reported separately.Source: §A.1 · p. 498
3,840 retained scalar reductions replayed bytewise; the largest float64–extended-precision reference difference was 1.17 × 10−14.These checks apply to retained logits and their reductions, not to every native forward or exact-real model arithmetic.Source: §A.1 · p. 498

A.2. Methods of the auxiliary finite-law studies

Auxiliary controlled-training and fresh-batch probes use a small resampled corpus, so their empirical coefficients apply only to that law; released-model and long-document checks retain their own input laws.These auxiliary checks support, but do not replace, the primary single-pass interpretation; the three exact sectors are not independent corpora or initializations.Source: §A.2 · p. 499
A 32-context developmental pilot preceded specification of the context-conditioned response formulation, and its arrays are not independent confirmation data or an additional training-seed ensemble.The appendix also states that proofs retain assumptions at point of use and reconstruction routes distinguish typesetting access from local raw-data access.Source: §A.2 · p. 502, §A.3 · p. 502

C1: Compatible complete laws

Compatible complete laws condition on one complete state and retain optimizer memory, remaining identities, clocks, and a shared observation measure; native signed-class claims require further convergence assumptions.Native signed observations use completed single-pass families, but trained covariance convergence—and covariance concentration for deterministic Gaussian behavior—remains outstanding.Source: C1: Compatible complete laws · p. 503, C2: Native signed class · p. 503

C3: Conditional training reduction

Moment-only interventions and source-return observations retain memory and representation errors, so conditional training reductions require more than those reduced observations.Held-out multistep successor accuracy and a total cost advantage remain obligations for an autonomous reduction.Source: C3: Conditional training reduction · p. 503

C4: Finite collective row flow

Eight single updates and matched fixed-count and equal-duration continuations resolve finite transport and energy-coordinate blocking, with signed temporal energy reconstructed independently on 64 equal-duration cells.A flux successor law and uniform accumulated generator-remainder control remain necessary.Source: C4: Finite collective row flow · p. 503

C5: Source and physical clocks

All stated state-calibrated aggregate targets pass on two disjoint state panels, with unchanged-initial-cache controls and context exceptions retained.The appendix leaves population guarantees and justified downstream sensitivity unresolved; workload-specific timing does not establish general throughput.Source: C5: Source and physical clocks · p. 504

C7: Conditional critical scaling

Primary outer risk favors local prediction in 4/8 states and transfer in 6/8, while both secondary joint scores favor 8/8 at all three scales.Zero-adaptation means miss the 0.01-nat maximum target in all eight states; native critical exponents, a critical surface, restoring feedback, and inference persistence remain unestablished.Source: S1: Moment precision and acquisition · p. 504, C7: Conditional critical scaling · p. 504

S1: Moment precision and acquisition

Refitting lowers primary scores in all 16 successor/fit cells per candidate across 16 parent/successor states and two fit cohorts.Moment precision still requires population-relative precision and an autonomous moment successor; related calibration work retains 288 long and 144 short paired continuations, all amplitudes, arithmetic controls, and signed covariance sectors.Source: S1: Moment precision and acquisition · p. 504, S2: Calibration and state adaptation · p. 505, S3: Finite pulses and optimizer memory · p. 505

S3: Finite pulses and optimizer memory

Finite-pulse claims assume excluded input support and untied input embeddings, while uniform derivative control, a native gap/forcing law, and naturally sampled memory generalization remain unresolved.Replication is by paired continuation from a complete incoming state, with branch, replay, and qualification roles retained in the execution ledger.Source: S4: Inference-null mode · p. 505, S3: Finite pulses and optimizer memory · p. 505

S4: Inference-null mode.

Four fresh paired comparisons match row decay while preserving every saved logit.The remaining obligation is visibility under a changed input law.Source: S4: Inference-null mode · p. 505

S5: Finite training selection.

Cosine risk is favored in all 12 matched width–seed pairs, while row-contrast changes have both signs; row suppression alone does not establish lower target NLL.The study used three whole initializations paired across conditions, with 24 paths and 24,576 scientific updates; corpus-law replication and a selection mechanism remain outstanding.Source: S5: Finite training selection · p. 505

S6: Inference task fidelity.

Inference task-fidelity evidence uses full-vocabulary interventions and premise-pair evaluations on a checkpoint-specific proper-prefix task context.Acquisition of premise-sensitive competence and broader task transfer remain outstanding.Source: S6: Inference task fidelity · p. 505

APPENDIX B Complete aligned experimental outcomes

The appendix retains aligned outcomes across width and time profiles, cache and continuation studies, finite transport checks, and matched intervention grids.These observations use stated measurement conventions; the conditional fourth-moment statistic is finite-scope, not a thermodynamic Binder limit.Source: Appendix B · p. 506, §B.1 · p. 506

B.2. Complete independent single-pass observations

The shared-parameter endpoint variance is reconstructed by diagonal and signed cross-block contributions under the same 256-update partition.The finite-sample squared-mean correction is retained even when negative; the summed contributions reconstruct the endpoint variance.Source: §B.2 · p. 509, Table B.4 · p. 509

B.4. COMPLETE CACHED-OPERATOR OBSERVATION AND TIMING 506

Peak allocated CUDA memory ranged from 231.90 to 1016.34 MiB for the native path and 225.91 to 997.59 MiB for the cached path.Resident generator weights were included and remained resident in both measurements.Source: §B.4 · p. 521

B.7. Complete disjoint-context confirmation

The disjoint-context confirmation retains all 30 state/length/policy cells, using the same six whole initialization identities and 64 assessment contexts.Its absolute error budget records native variance, centered residual variance, squared mean residual, signed covariance, and uncentered squared error.Source: §B.7 · p. 528, Table B.27 · p. 529

B.7. Complete disjoint-context confirmation; B.11. Complete fixed-dictionary projection outcomes; B.12. Complete outer-state scale and inference comparisons

The appendix also supplies paired confirmation-panel KL differences with bootstrap intervals conditional on two panels and six seeds, plus complete fixed-dictionary and outer-state inference comparisons.The fixed-dictionary outcomes cover all selected terminal states and both dictionaries; outer-state tables cover every coarser score comparison and proper-prefix inference at each new incoming state.Source: Table B.29 · p. 530, Table B.41 · p. 542, Table B.42 · p. 543

B.8. Complete matched-clock observations

Matched-clock observations retain all sixteen width/control conditions and four physical ages, with paired seeds and initial states counted only once per width.Tables report row and predictive susceptibilities, operator amplitude, cache-policy predictive metrics, and operator risk budgets.Source: §B.8 · p. 531, Table B.30 · p. 531, Table B.31 · p. 533

B.9. Numerical scope of finite row transport

Across 1,000 generated transitions and 4,000 native matrix pairs, the largest identity discrepancy is 6.23 × 10−16, and every pair satisfies δ < 1.The maximum δ is 0.43328; these inner measurements condition on eight states and do not establish a width-uniform small-step regime or control generator parameter-linearization error.Source: §B.9 · p. 534, Table B.33 · p. 534
At 32 trapezoidal intervals, three of eight cases meet the 10−5 absolute target, but refinement is nonmonotone and one target-meeting case has relative error about 3.36.The diagnostic is retrospective rather than a forecast; records do not certify real-arithmetic derivatives or quadrature convergence, and it adds no independent training trajectory.Source: §B.9 · pp. 534–535

B.10. Complete chronological row-path observations

Equal-duration path budgets are omitted whenever any increment fails δ < 1, while finite-energy maps do not require that premise; temporal-energy outcomes retain signed interference.The complete chronological observations include all finite path budgets, target losses, aligned block scales, and equal-duration blocks.Source: Table B.36 · p. 536, Table B.38 · p. 538, Table B.39 · p. 540

B.13. Complete acquisition and state-change comparisons

The primary acquisition and state-change comparisons retain eight lineages, two incoming ages, two fit cohorts, and three nested adaptation budgets, with uncertainty conditional on recorded fit and evaluation panels.All target and resolution summaries are retained; only four-interval risk is primary.Source: §B.13 · pp. 545, 547, Table B.45 · p. 545, Table B.46 · p. 547

B.14. Complete finite-pulse amplitude outcomes

Endpoint 8 successor risk/entropy changes are −0.290 (13/16) and −0.299 (13/16); endpoint 16 changes are −0.480 (15/16) and −0.438 (16/16).The reported counts are favorable observed signs.Source: §B.14 · p. 548
All 64 primary full-path amplitude comparisons are retained without selection by sign, source, family, or incoming state.Four source indices are paired across arms within each state, not additional pretrained-model identities; gate strings require both relative-discrepancy and absolute-signal conditions.Source: Table B.48 · p. 548, §B.14 · p. 548

B.15. Complete paired native intervention outcomes … D.6. Finite forcing and error-budget kernels

This section assembles bounded paired-outcome and numerical reanalysis evidence with exact row-energy, optimizer-state, chronological renormalization, and finite-forcing/error-budget results. It distinguishes checked algebraic kernels and formal correspondences from arguments, assumptions, reproduction requirements, and claims outside formal coverage.

B.15. Complete paired native intervention outcomes

The paired native interventions cover every scientific corner across matched incoming states and disjoint suffixes, supporting conditional path effects but not population confidence intervals or fitted critical exponents.The long and early states have distinct objectives and schedules; NLL uses a fixed 64-target cohort and K uses the first of two fixed probes.Source: §B.15 · p. 549, Table B.49 · p. 549

B.16. Complete matched single-pass outcomes

The matched single-pass tables report every scientific branch and path on disjoint source suffix B, with fixed incoming states and specified evaluation cohorts and probes.The tables define their own scopes: one uses a 64-target proper-prefix cohort and primary probe for Q and K, while the other reports NLL on 128 external targets and averages row ratio across eight probes.Source: §B.16 · p. 549, Table B.51 · p. 550, Table B.52 · p. 550

C.1. Code and compact evidence

The companions provide selected Lean developments, experiment producers, reducers, validation tools, and content-addressed numerical records, but omit raw arrays, checkpoints, corpus shards, and token arrays.The deposit supports examination of reported quantities; integrity checking alone is not independent scientific replication.Source: §C.1 · p. 552

C.2. Reproduction levels

Reproduction ranges from evidence and formal-kernel checks to raw-array reanalysis and native replay, which require separately supplied data, protocols, states, clocks, arithmetic policy, and platform.Numerical identity checks qualify calculations, not native training laws.Source: §C.2 · pp. 552–553

C.3. Single-pass and retained-source boundaries

The monograph reports already executed studies without new training; retained-source records address finite geometric and state-sufficiency questions and are not pooled with single-pass families to estimate a common critical law.Evaluation and calibration panels have explicit roles and are not silently reused as independent confirmation.Source: §C.3 · p. 554

C.4. Numerical differentiation and experimental scope

Central finite differences at steps 0.01 and 0.005 are checked against automatic differentiation on measured contexts, but do not establish a uniform Taylor remainder or asymptotic training law.The inventory reports 4,253,160 classified native updates, or 4,254,024 including 864 arithmetic-control updates; these subtotals are not independent training replications.Source: §C.4 · pp. 554–556, Table C.2 · p. 554, Table C.3 · p. 554

C.5. Sampling roles and reuse across studies

Sampling protocols distinguish training, calibration, held-out, donor, and evaluation material, with paired paths sharing specified streams or panels and excluding consumed blocks where stated.Single-pass training uses 524,288 documents from a permutation of 4,194,304 blocks, with complete training-document hashes disjoint from the three retained text-observation cohorts.Source: §C.5 · pp. 556–557

C.7. A WORKED BOUNDED RECONSTRUCTION 545

The worked reconstruction restores model, moments, and counters for each draw, uses 32 fresh batches at each of sixteen incoming states, and makes 64 adjacent length-64 calls from one uniform 4,097-token crop.The associated data split assigns [0, 256) to projection, [256, 512) to fitting, and [512, 1024) to evaluation.Source: §C.7 · p. 559

C.6. Numerical reanalysis

All 30 cache-panel cell dictionaries agree exactly, and independent raw-moment and pair-distance checks pass; recomputing summaries needs no extra model forward or training update.The 3,024 forward calls were used to acquire the panel.Source: §C.6 · p. 559

D.1. Scope of the independent checks

Lean checks are not premises of the proofs: compilation, the axiom boundary, and correspondence to written theorems are separate, and scalar verification alone does not establish matrix, limiting, or physical interpretations.The combined axiom check admits only propext, Classical.choice, and Quot.sound as transitive dependencies.Source: §D.1 · p. 561

D.2. Row geometry and optimizer dynamics

Formal checks cover centered-energy and pairwise-variance identities, block-excursion collapse criteria, and exact observer strata for finite nonnegative stored energy.The observer-stratum results include a disjoint partition and enclosure implications for exact-face, strict-positive, censored, and resolved cases.Source: §D.2 · pp. 561–562
The finite squared-energy work-charge identity and cross-Gram expansions support exact contraction, preservation, or reopening comparisons; paired radial-tangential identities give corresponding closing criteria.The paired contrast includes state projection, natural-increment interaction, and paired charge.Source: §D.2 · pp. 562–564
Canonical scalar face branches compose into an exact affine cocycle, including product-convolution across every zero-face restart.The finite-anchor formulation also gives a geometric anchor convolution and closure from vanishing anchors and excursions to vanishing attained block maxima.Source: §D.2 · p. 564
Intermittent block radial-face closure follows when the block conditions force attained radial maxima to vanish.The section also gives a direct nonautonomous energy envelope for bounding energy over time.Source: §D.2 · p. 565
Summable reopening yields collapse from arbitrarily late small nonnegative energy, while face preservation holds exactly when endpoint energy vanishes.At zero source, endpoint energy is the increment charge and vanishes exactly when every finite increment coordinate vanishes.Source: §D.2 · p. 565
A finite work-charge telescope makes cumulative-ledger convergence equivalent to scalar-energy convergence and links stacked native energy to registered maximum row-diameter convergence.The ledger identity is exact; the finite-registry assembly is not claimed as formally covered.Source: §D.2 · p. 565
The complete-state cocycle decomposes state and observable changes chronologically into transported force and remainder, with componentwise vanishing as a sufficient condition.A uniform positive lower comparison can transfer convergence of a nonnegative scheduled quantity to a fixed quantity without an upper comparison.Source: §D.2 · pp. 566–567
Nonvanishing realized force and persistent nonzero observation-kernel states obstruct collapse, while a nonlinear zero fiber can obstruct convergence even when its derivative vanishes.A finite-prefix reopening example shows that matching a finite history does not ensure eventual energy convergence.Source: §D.2 · p. 568
PLGA row preservation is bounded by a response term plus quotient defect, reduces exactly when the upstream quotient is zero, and propagates under five implemented parameter conditions.A nonzero constant-input defect obstructs row preservation.Source: §D.2 · p. 568

D.3. Chronological row renormalization

D.3 names Result 8.5 as RowRGMap.IntervalAffine, and its correspondence table records manuscript-result-to-Lean-declaration links with digest-validated elaborated types.The table specifies that this type fingerprint does not include proof terms.Source: Table D.2 · p. 569, §D.3 · p. 570
Lean checks exact nonnegative endpoint arithmetic, finite chronological blocking, action enclosure, and two-edge and arbitrary-block covariance identities; directed rounding remains an implementation obligation.The manuscript supplies the componentwise matrix comparison and associativity proof.Source: §D.3 · p. 570
Lean checks exact binary scaling, arbitrary integer block-factor source iteration, source relevance, and the stationary forced value; real-log asymptotics remain manuscript arguments.The cited checks include the critical-source iteration and stationary-energy results.Source: §D.3 · p. 570
Finite excursion peaks characterize collapse, and vanishing restart energy alone is insufficient, as a nonnegative obstruction schedule demonstrates.Lean verifies finite-excursion coverage, the peak criterion, and the restart-amplification consequence.Source: §D.3 · p. 571

D.4. Model-wide laws and predictive observations

Selected formal exports establish finite analogues, not general disintegration, native program smoothness, empirical independence, or a thermodynamic critical limit.The table identifies selected exports and their precise scope; the stated hypotheses and proofs are in the text.Source: §D.4 · p. 572, Table D.3 · p. 572
The finite centered-square identity, normalized-weight comparison, and symmetric bilinear contrast are exported; exponential tilting, Hessian integration, segment convergence, and numerical intervals remain written results.The export is limited to the stated finite identities and comparisons.Source: §D.4 · p. 572
Finite scalar increments, pulse transport, sample covariance, signed-moment, and training-covariance identities provide algebraic kernels for model-wide observations.The pulse identities include linear transport and a nine-term bilinear expansion; the covariance identities retain common-origin and finite-sample conditions.Source: §D.4 · pp. 583–586
Finite kernels also cover reference-conditioned categorical aggregation, empirical bias–variance, cache-risk transport, finite-population covariance, row flux, and temporal energy.The stated scope is finite algebra; the cache-risk passages distinguish coordinate-sum Euclidean budgets from conditional Hilbert-space expectation, while flux identities retain cross-increment terms.Source: §D.4 · pp. 589, 590, 592, 593

D.5. The interface between absolute and normalized energy

The absolute-to-normalized energy interface has exact reconstruction and scalar inequalities under positive-denominator and amplitude bounds; further analytic results remain proved in the text.The written proofs cover orthogonal projection, convergence, counterexamples, a complete Lipschitz recursion, and a categorical Hessian argument.Source: §D.5 · p. 594

D.6. FINITE FORCING AND ERROR-BUDGET KERNELS 580

Formal checks for finite forcing are partial: general stochastic forcing remains unformalized, while finite probability-space and scalar-limit checks add explicitly delimited results.The finite weighted-kernel hypotheses include normalized nonnegative weights, zero vector means, and pairwise expected orthogonality; general martingale conditional expectation is proved in the text.Source: §D.6 · p. 594

D.6. Finite forcing and error-budget kernels

The checked finite error-budget kernels include nonnegative-sum and decay criteria, plus weighted second-moment identities for centered values and mean-plus-variance.These formal declarations do not replace the separately written general stochastic-forcing argument.Source: §D.6 · p. 594
Nonnegative real sequence decomposition and squeeze apply when the relevant identity or inequality holds, with separate scalar decay sufficient.These are assumptions of the finite-forcing and error-budget kernels.Source: §D.6 · p. 594

D.7. CROSSOVER BALANCE AND SCALAR FACE MEMORY 581 … Disclosure of the use of AI tools

This section sets out the notation, conditions, and formal checks used to describe row-centered energy and its dynamics, consuming-corpus laws, fluctuations, physical clocks, predictive resolution, operators, and observations. It records conditional scaling and error conventions while distinguishing these formal and domain-specific statements from broader claims they do not establish.

D.7. Crossover balance and scalar face memory

The scalar formal checks cover stated sequence clauses, not a stable central limit theorem or an inferred training law.Exact finite recurrences and cancellation examples are separately justified, including killed-noise tails, centered sign variance, retained unit direction, and a positive 7/2 radial budget.Source: §D.7 · p. 595
Under exact balance, a diverging positive scale with subpower growth and bounded positive ratio yields the scalar exponent; at α = 3/2, the power and logarithmic powers are three and two.The equivalent quotient exponent is α/(α −1) for α > 1; the checks do not select a crossover or establish a fluctuation law.Source: §D.7 · p. 595

D.8. Trajectory-compatible chart connection

A zero gain in a fixed chronological list annihilates the scalar homogeneous coefficient, but does not erase dependence on the complete training state.The finite extension is supported by affine-composition checks.Source: §D.8 · p. 596
Trajectory-chart identities derive realized steps and observations from state evolution and a realized lift, while chart existence, smoothness, and the chain rule remain written obligations.The checks allow time-indexed identities with varying state and coordinate types; transport wrappers use a fixed additive coordinate type.Source: §D.8 · p. 596
A right inverse alone can fail to recover the successor, whereas an adapted slice does so in the exact two-dimensional example.The resulting additive mismatch identities imply no Lipschitz or smallness bounds; general varying-dimension transport, differentiability, domain admission, and remainder bounds remain obligations.Source: §D.8 · p. 596

APPENDIX E Notation and conventions

Notation is local to each result: indices, dimensions, decorations, algebraic conventions, and probability symbols take meanings from their stated domains and conditioning.The appendix distinguishes row, context, and ensemble centering; law-dependent conditioning; and convergence modes, noting that a fitted finite-size power does not establish asymptotic equivalence.Source: §E.1 · p. 597, §E.2 · p. 598, §E.3 · p. 599
Architecture and observation notation distinguishes the external next-token target from forward computation and defines predictive distributions, complete traces, and centered observations under the data law.A proper-prefix conditional is evaluated on the prefix itself; an earlier position in a longer parallel call may be a different observation when the generated operator depends on the full call.Source: §E.4 · p. 600, §E.5 · p. 601

E.5. ROW GEOMETRY AND NUMERICAL OBSERVATION 587

Row-map collapse is measured by the Frobenius norm of the centered map, with its square as absolute quotient energy; a finite registry does not control unseen contexts.The row-constant projection identifies the common row, while the registry energy sums the quotient energies of its observed maps.Source: §E.5 · p. 601

E.6. FINITE WORK, SOURCE INCREMENTS AND DOWNSTREAM DEFECTS 588

An empirical replay discrepancy is not automatically a certified energy-error bound, and arithmetic resolution, numerical precision, bitwise equality, and analytical equality have distinct scopes.The row replay protocol defines its arithmetic floor as 8|E64 − E32|.Source: §E.6 · p. 602
Each finite step changes energy by −W_t + Q_t, while its native row-map increment resolves into shape, gate, interaction, and implementation-defect sources.Cross-Gram products among declared sources contribute to the charge; the defect is treated as an arithmetic source in the native ledger.Source: §E.6 · p. 602

E.7. COMPLETE OPTIMIZER STATES AND THEIR RESPONSE 589

The complete augmented training state retains parameters, Adam moments, schedule and bias clocks, remaining data, and stochastic or numerical program state.The lifted optimizer Jacobian includes derivatives through the adaptive ratio and its live denominator.Source: §E.7 · p. 603

E.8. CHRONOLOGICAL AFFINE MAPS AND CLOSURE 590

Charted dynamics distinguish successor and observation mismatches, while paired trajectories obey the exact finite balance Y_n = L_n + N_n + Q_n.A derivative interpretation of the homogeneous map requires the selected smooth cell; the algebraic remainder identity does not.Source: §E.8 · p. 604

E.8. CHRONOLOGICAL AFFINE MAPS AND CLOSURE 591

Scalar energy comparisons use affine gain-source edges, with positive energy represented by gain and the exact row-constant face by a restart source; reduced kernels require equal pushed-forward successors within relevant fibers.Blocking composes gains and transported sources, but a comparison energy vector is not thereby a closed state.Source: §E.8 · pp. 604–605

E.9. COMPLETE LAWS, EMISSIONS AND THE CONSUMING CORPUS 592

The complete conditional law retains the training-state path, evaluation context, emitted observation, and the augmented state’s schedule, remaining blocks, and sampler or stochastic-program state.Training-state laws and inference emission kernels are distinct; their combination yields the emitted law.Source: §E.9 · p. 606

E.9. Complete laws, emissions and the consuming corpus

A single-pass training law tracks the remaining distinct corpus identities and draws ordered batches without replacement, requiring BT ≤ M.The state includes resource size, batch size, update horizon, remaining identities and their count; consumed identities are removed after each batch.Source: §E.9 · p. 606

E.10. Potential increments, response and fluctuation sectors

Batch covariance uses the finite-population factor (r − B)/(B(r −1)), while unequal valid-target counts make masked gradients a ratio problem rather than an unweighted block mean.Declared resource, batch and horizon exponents are imposed family coordinates, not automatically critical exponents.Source: §E.10 · p. 607

E.11. Shared collectives, sign symmetry and physical clocks

The account separates within-state context covariance from covariance across state means, and specifies head-sign symmetry through joint sign actions on queries and PLGA factors.The sign group acts equivariantly on training updates and leaves predictive emissions invariant.Source: §E.11 · p. 608

E.12. Predictive resolution, operator caches and risk

Matched physical clocks relate optimizer memory decay to step size, while row-energy changes are tracked by exact cross and quadratic fluxes and chronological energy-coordinate composition.The physical clock distinguishes a small step from a small restoring rate; both fluxes use the successor total-energy denominator.Source: §E.12 · p. 609

E.13. Finite interventions, autoregression and adaptation

Vocabulary reduction uses reference-conditional lifts and projections, while state-specific operator caches are evaluated with decoder-error bounds restricted to the compared domain.The appendix also defines finite pulse observations and separates external-target NLL difference from native-law KL fidelity.Source: §E.12 · p. 609, §E.13 · p. 610

E.15. Renormalized laws and conditional scaling fields

Conditional scaling fields require a physical length relation to convert count exponents into length exponents; a statistical source does not alter the native optimizer or forward graph.The glossary distinguishes the exponent ϕ from the Hellinger embedding or an observation map.Source: §E.15 · p. 612

E.16. Metric-row transport and observation accuracy

Exact covariance transport retains both B and BT, while a shared-coordinate gradient requires downstream adjoints as well as row moments.Finite comparisons of exact and observed susceptibilities require a common divisor and common units.Source: §E.16 · p. 614

E.17. Physical calibration and magnetic readouts

The physical-calibration glossary defines the square-lattice Potts system and clarifies that its symbols for lattice size, site count, Hamiltonian, temperature, and horizon differ from decoder and training notation.The bond sum counts each right and down nearest-neighbor bond once.Source: §E.17 · p. 614

E.18. Statistical, experimental and presentation conventions

The conventions distinguish independent replicas from heads, contexts, repeated measurements, and adaptation runs, and define a single pass, proper prefix, frozen inference, and repeated-corpus adaptation separately.Covariance divisors depend on the sampling unit; repeated-corpus adaptation includes its epoch and refill mechanism in the complete state.Source: §E.18 · pp. 615–616

E.18. STATISTICAL, EXPERIMENTAL AND PRESENTATION CONVENTIONS 602

Numerical comparisons depend on the executed dtype, reduction order, casts, and arithmetic policy; finite contrasts and fitted powers are not automatically asymptotic critical exponents.A higher-precision observation need not rerun training, and table ranges are observed ranges unless identified as confidence or prediction intervals.Source: §E.18 · p. 616

APPENDIX F Lists of tables and figures

The figure index describes the document-block fluctuation flow as having approximately constant covariance under b−1/2 rescaling.The table index identifies 28.10 as covering all 24 controlled endpoints, with native, paired-normalization, and fresh-seed conditions.Source: §F.1 · p. 630, §F.2 · p. 645

Disclosure of the use of AI tools

The author reports using Codex and Claude as non-authorial aids for exposition, proofs, formalizations, code, analysis, and consistency checks, while disclaiming independent validation by their outputs.The author accepts responsibility for the monograph’s claims and materials.Source: Disclosure of the use of AI tools · p. 652