Intrinsic-Extrinsic Coupling in Learning Dynamics

1 Introduction

• A matched four-cell contrast tests whether an external continuation changes an intervention’s marginal value, or vice versa, at a fixed parent, proposal, horizon, randomness, and readout.Both readings are one forward-time contrast, not separate replications.Source: §1 · p. 1
• Replay changed the write’s 32-update contribution from five correct predictions without replay to none with replay; nonzero interactions also appeared under distillation, RoBERTa, and SGDW.The optimizer-native study uses optimizer-specific parent trajectories and legal activations.Source: §1 · p. 1

(a) PRESENT EQUIVALENCE (b) ONE PAIR, TWO CONTINUATIONS

• The paper separates local feasibility and favorable outputs from training-reachable repair regions, and frames its contribution as an operational connection rather than a complete evolution law or benchmark-wide advantage.Separate coordination tests include controls that match or exceed the development gain.Source: (a) PRESENT EQUIVALENCE (b) ONE PAIR,… · p. 2

2.1 Full state and observation-relative fibers

• The full learning state includes model, optimizer, data, random, and memory state, while an observation-relative fiber captures present agreement without guaranteeing future indistinguishability or usefulness.Smooth-fiber claims require regularity; a linearized constraint alone does not ensure exact finite preservation.Source: §2.1 · p. 3

2.2 Intrinsic actuation and extrinsic continuation

• Intrinsic actuation is a constrained state intervention and extrinsic continuation is a subsequent learning rule; both operate on the same learner, so their effects need not be orthogonal.The framework does not claim a controllability theorem or that the interfaces are independent subsystems.Source: §2.2 · pp. 3–4

2.3 Continuation-conditioned intervention value

• Continuation-conditioned value compares identity and an accepted proposal over a specified horizon, policy, randomness, and endpoint readout; admissibility and positive future value are distinct predicates.Correct count, negative cross-entropy, and structural margin are different readouts.Source: §2.3 · p. 4

2.4 Two readings of a four-cell interaction

• The four-cell mixed contrast has two equivalent readings: continuation changes intervention value, or intervention changes the continuation’s incremental value.Nonzero interaction is readout-specific and does not guarantee positive individual or joint effects.Source: §2.4 · pp. 4–5

2.5 Single-action values versus closed-loop values

• A complete-policy contrast allows later states, frames, proposals, and allocations to diverge, so it is not a longer-horizon single-action value or a sum of unrelated one-shot effects.Exact telescoping requires nested whole policies under a common root, randomness, and readout; the summands depend on that nesting.Source: §2.5 · p. 5

2.6 Coordination under an explicit workload constraint

• Coordination can redirect replay using a pre-action signal under a matched replay workload, but workload matching does not establish total-cost equality or that a tested coordinator solves the control objective.Fiber sensing, certification, data selection, hardware execution, and elapsed time remain distinct costs.Source: §2.6 · pp. 5–6

3.1 The exact head-space zero constraint

• For a fixed feature frame, the head-write subspace preserves current-frame logits exactly in real arithmetic, but does not guarantee preservation across all network inputs.The affine fiber has dimension q(d − rank C).Source: §3.1 · p. 6

3.2 A prescribed centered margin target

• The centered target raises each deficient representative’s correct-versus-competitor margin by its prescribed increment while leaving already nondeficient representatives unchanged.Representatives come from the post-admission reservoir; the target is not a validation-loss objective or replay-allocation claim.Source: FINITE HEAD WRITE: ONE STATE CHANGE,… · p. 7, Figure 2 · p. 7, §3.3 · p. 8

3.3 Null-space interpolation and the least-norm write

• Under full-row-rank assumptions, the prescribed interpolation has a unique minimum-Frobenius-norm solution; the implementation uses certified linear solves rather than fallback solvers.The exact target is feasible under the more general condition that every target column lies in the range of N.Source: §3.3 · pp. 7–8

3.4 A feasible repair set, not a unique feasible point

• The bounded repair set can contain many writes: at the stored first common frame it has 22,230 free coordinates, or 21,489 under centering, but other members need not pass acceptance checks or improve future outcomes.A large frame-feasible set does not establish a training-reachable favorable basin.Source: §3.4 · p. 8

3.5 The write budget depends on target and geometry

• The required write norm depends on both target magnitude and geometric accessibility; failure to certify the prescribed local map within the cap does not prove all useful approximate actions infeasible.Poor separation of historical features from the protected current span can increase the norm.Source: §3.5 · p. 9

3.6 From the ideal map to a certified finite transaction

• Rounded writes undergo stored-frame and native-logit checks, including current-prediction preservation and old-margin thresholds; these checks authenticate the transaction, not future accuracy or coordinator usefulness.The implementation uses finite-precision enclosures and staged/live evaluations rather than inheriting exact preservation from the ideal map.Source: §3.6 · pp. 9–10

3.7 Same-frame saturation is not future invariance

• After repair, the same immutable historical frame is saturated, but this does not imply identity at the next consultation or a protected future learning update.A fixed-frame chain-rule analysis allows feature derivatives, and consequently backpropagation, to differ.Source: §3.7 · p. 10

3.8 Scheduled Fiber actuation

• Fiber consultations occur on a fixed 32-update schedule and return saturated identity, certified nonzero commit, or rejected identity; sparse opportunities do not guarantee few commits.The interval is fixed across external training rules, and committing or abstaining can both incur sensing and certification work.Source: §3.8 · p. 10

4.1 Common learning task and readouts

• The common task uses a deterministically selected 50-class CLINC subset split into five ten-class tasks, with BERT-base-cased and a growing linear head as defaults.Each task has 160 updates; evaluation predicts across all seen classes without a task oracle, using correct count as the primary utility and −CE as secondary.Source: §4.1 · pp. 10–11

4.2 Experimental units and complementary comparisons

• Isolated four-cell comparisons restore a common parent and proposal before disabling later Fiber writes, whereas whole-policy comparisons let future states and decisions diverge from a shared root.The training root, rather than an evaluation item or repeated horizon, is the unit of independent-root variation.Source: §4.2 · p. 11

4.3 Prospective decisions, nonactivation, and interpretation

• Anchor selection follows a prescribed pre-action rule; absent a legal nonzero proposal, the outcome is nonactivation, with no replacement based on endpoint value.Five-root policy contrasts use frozen directional criteria, which are descriptive rather than population-level significance tests; CE remains secondary.Source: §4.3 · p. 11

5.1 One action, two external continuations

• The same write added five correct predictions without future replay but zero with replay; three legal commits also tied the no-Fiber trajectory at 950 correct predictions.Replay contributed 83 correct predictions after identity and 78 after execute; these are two readings of the same contrast, not independent replications.Source: §5.1 · p. 12

5.2 Targeted benefit and decision-level saturation

• All five Fiber correctness gains occurred in label 11; after identity, replay increased the old-task count by 85 and decreased the current-task count by 2.These are conditional effects on the 32-update endpoint panel, not independent experiments.Source: §5.2 · p. 12, Table 2 · p. 13
• With replay, execute and identity predicted the same 600 labels, yet their margins and cross-entropy differed, including a nonzero margin difference for the repaired representative.The margin diagnostic uses an endpoint clone and is not an additional training action.Source: §5.2 · p. 13, Table 3 · p. 13

5.3 Continuation effects at longer horizons

• At anchor 351, the replay change reversed action-value sign at 128 updates, while at anchor 511 interaction signs varied across evaluated endpoints.These values are conditioned on parent history and proposal; changing readouts prevent treating them as a single step-index response curve.Source: §5.3 · p. 14, Table 4 · p. 14

5.4 Output distillation supplies a second continuation family

• All six distillation contrasts were nonzero on correct count, with sign reversals at anchors 223 and 255 also changing the active-class panel.This comparison uses one shared-prefix development root; interaction signs can differ between correct count and negative cross-entropy.Source: §5.4 · p. 14, Table 5 · p. 15

5.5 A second backbone

• RoBERTa-root correct-count interactions were (+2, +1, −3) at 32 updates and (+5, +3, +2) at 128 updates.These six nonzero contrasts across three roots establish occurrence under this backbone, not a model-invariant sign or magnitude or a controlled architecture comparison.Source: §5.5 · p. 15

5.6 Optimizer-native, phase-matched deployment

• At 128 updates, all three activated SGDW roots had negative correct-count interactions (−1, −2, −3), while two of three AdamW roots had nonzero interactions (−6, −8, 0).All twelve activated optimizer–root–horizon contrasts were negative on negative cross-entropy; the comparisons are nested within three roots, with no population-level significance test inferred.Source: §5.6 · pp. 15–16, Table 6 · p. 16

5.7 Fixed-anchor update-scale sensitivity

• In the update-scale sensitivity study, four of six root–anchor cells activated, and Root 1 at anchor 511 changed interaction sign across horizons on the same 800-item panel.The calibration matches measured update magnitudes, not full vector fields, moments, or trajectories; Root 3 was saturated at both registered anchors.Source: §5.7 · p. 16

6.1 A bounded signal-guided allocation rule

• The bounded reinforcement rule changes at most one replay row per update for 32 updates after an accepted repair, preserving replay batch size and source-task identity.It uses a pre-action margin signal and a deterministic keyed draw to select an unused reservoir item; the donor margin is a heuristic, not a safety certificate.Source: §6.1 · p. 17

6.2 Fixed-workload outcomes and the limits of support withdrawal

• Post-repair replay withdrawal reaches terminal J=851, while repair-conditioned deprioritization reaches 948; the results do not make a local repair certificate an automatic certificate of support redundancy.Withdrawal produces a task-wise deficit of (−51, −27, 0) at step 479, concentrated in older tasks.Source: §6.2 · p. 17

6.3 What the allocation controls identify

• Mask-matched random content reaches 953, and permuted signal reaches 955, so exact Fiber-label alignment is not necessary to reach the original endpoint in this development comparison.Random content is conditional on a signal-derived change mask, while permutation retains signal-triggered windows; neither control is fully signal-free.Source: §6.3 · p. 17

6.2 Fixed-workload outcomes and the limits of support withdrawal

• The original Fiber-aligned policy reaches 953 rather than 950 correct predictions, but task-end differences (0, 0, +1, −1, +3) rule out trajectory-wide non-degradation.The policy redirects 96 replay rows while keeping 799 replay events and 12,784 replay examples.Source: §6.2 · p. 17, Figure 4 · p. 18

COORDINATED REPLAY: POLICY OUTCOMES AND ATTRIBUTION

• Shadow signal/no write reaches 952 after accepting three proposals but retaining zero writes, showing that a positive shift over the 950 reference is possible without retained writes.The 953−952 gap is a complete-policy contrast with endogenous future signals and allocations, not a controlled direct effect holding the replay path fixed.Source: COORDINATED REPLAY: POLICY OUTCOMES… · p. 18, Figure 4 · p. 18

6.4 Allocation value across training roots

• Guided-minus-standard correct-count differences are (+1, −3, +4, −3, −2) and guided-minus-random differences are (+3, −2, +3, −3, −3), positive in two roots and negative in three.The frozen five-root comparison finds no root-stable positive advantage for guided allocation or exact Fiber-label alignment over matched-random content.Source: §6.4 · pp. 18–19, Table 7 · p. 19
• Guided-minus-standard test CE is negative in all five roots, with mean −0.0026145 and median −0.0005924.This secondary readout indicates lower aggregate log-loss, not improved calibration or margins for every item, and it can coexist with worse correct count.Source: §6.4 · p. 19
• After allocation changes, later Fiber commits differ across trajectories, so fixed consultation opportunities do not fix realized writes or preserve a common parent.These are closed-loop policy outcomes, not additional matched-parent interaction coefficients or proof that any changed event mediates terminal score.Source: §6.4 · p. 19

7.1 Three spaces that must remain distinct

• For a fixed set of strictly correctly classified items, the output region is open, convex, and preserves those items along logit segments; correct-count superlevel sets need not be convex.This is a readout property, not a Fiber-specific theorem; stored development logits also certify favorable segments, but not trained ensembles or realizable replay-mixture trajectories.Source: §7.1 · pp. 19–20

7.2 Similar functional quality does not require state convergence

• Similar terminal counts of 953 and 955 do not show state convergence: the stored head distance between random and permuted arms rises from 0.24563 at step 703 to 0.25538 at step 799.The analysis concerns head-coordinate distance and does not rule out contraction in other components or attenuation of a future-readout projection.Source: §7.2 · p. 20

7.3 Local feasibility is not future-value maximization

• The minimum-norm write solves a prescribed present-frame problem, not future-accuracy maximization; no future-gradient alignment, repair cone, or contraction coefficient is estimated.The chain-rule sensitivity includes optimizer state, subsequent updates, and final readout, while discrete replay choices and switching acceptance need additional assumptions for a derivative description.Source: §7.3 · p. 20
• The coordinator is a bounded structural interface without online future-rollout search, but the reported component counts establish neither equal total compute nor optimal cost-efficiency.The development attribution arms log 20 extra encoder calls over 648 frame rows and zero extra backward passes for Fiber consultation; these counts omit other host computation, certification, and wall time.Source: §7.3 · pp. 20–21

• The paper connects observation-relative intrinsic actuation to readout-specific intrinsic–extrinsic coupling through finite writes, matched replay and distillation continuations, and separate coordination tests.Fiber Fingerprints and Revelation Control already study related present/future distinctions and intervention value [1, 11].Source: §8 · p. 21
• The finite-frame write protects a current frame while changing selected historical margins and certifies the rounded update; the paper does not claim null-space protection as new or general superiority over editing algorithms.The passage places the construction alongside OGD, GPM, ROME, MEMIT, and AlphaEdit [18] [19] [20] [21] [9].Source: §8 · p. 21
• The four-cell comparison identifies replay-conditioned differences in marginal endpoint utility at a fixed parent, rather than claiming editing–training interactions were previously unknown.The stated distinction is the constrained-write/factorial-continuation comparison and its policy-level follow-up.Source: §8 · p. 21
• The mixed contrast is an established factorial tool, and non-additivity on a readout does not establish a physical interaction term or universal dynamical law.Nonlinear readout can create non-additivity even when state-update maps commute; the comparison also identifies the implemented replay treatment, including its batch and normalization consequences.Source: §8 · p. 22

9.1 What transfers across learning contexts

• Nonzero interactions occur under replay, output distillation, a RoBERTa backbone, and SGDW, but the realized parent, direction, activation rate, sign, and magnitude need not transfer invariantly.Native, phase-matched deployment and update-scale matching address continuation dependence under their respective protocols, not complete vector fields or a universal law.Source: §9.1 · p. 22

9.2 Interaction is not a policy guarantee

• Under native-optimizer SGDW, negative interactions show replay can reduce a write’s incremental value; across five fresh roots, guided allocation’s correct-count effect varies rather than remaining uniformly beneficial.The five-root comparison keeps Fiber in every arm, and guided allocation has lower aggregate CE than standard replay in all five roots as secondary evidence.Source: §9.2 · p. 22

9.3 Scope, resources, and remaining questions

• The studies use one selected CLINC partition, task order, and small-memory regime; the five-root policy test adds fresh roots and an unused test split, not new tasks or pretrained models.Mechanism measurements also use reused validation panels, selected activations, and few roots; repeated horizons and optimizer diagnostics on shared roots are not independent confirmations.Source: §9.3 · p. 23
• The experiments do not identify a global optimum, universal quality threshold, or formal total-cost advantage for the bounded rule.The stronger objectives require evidence about future-value information carried by the structural signal beyond local feasibility and conditional interaction.Source: §9.3 · p. 23

10 Conclusion

• Intrinsic–extrinsic coupling links a learning-state intervention with subsequent learning, while geometry, continuation, and policy performance remain distinct questions.Observation-relative fibers describe present agreement; a local certificate establishes admissibility, matched branches measure continuation-conditioned value, and complete-policy tests assess coordination.Source: §10 · p. 23

A.1 Projection, feasibility, and the least-norm property; A.2 Geometry-conditioned cost

• Feasibility requires each target column to lie in range N; under full row rank, the constrained solution is uniquely least-norm, while near-span historical features can make fixed repairs expensive.Full row rank of N is stronger than full row rank of C alone.Source: §A.1 · p. 26, §A.2 · p. 26

A.3 Margins, tolerances, and saturation

• Rounded writes require explicit post-margin checks: interpolation tolerance alone does not guarantee every representative clears the threshold, and accepted repairs do not imply future invariance.After ideal repair, represented margins are at least μ; reacquiring a frame after learning is a different operation.Source: §A.3 · p. 27

A.4 Preserved observations need not preserve backpropagation

• An exact write can preserve logits, loss, and head gradient on the frozen frame while changing the feature derivative, so current-output preservation is compatible with changed future learning.The implemented frame is inference-mode and only toleranced after rounding; these identities do not establish equality of actual next training-mode gradients.Source: §A.4 · p. 27

A.5 Interaction is a contrast, not an operator commutator

• A nonzero four-outcome interaction is an outcome-level contrast, not evidence that the underlying state updates fail to commute.Commuting scalar translations can yield a nonzero contrast under nonlinear utility; affine utility rescaling preserves the contrast up to scale, but nonlinear rescaling has no general such rule.Source: §A.5 · p. 27

A.6 Future-value sensitivity and current margins

• Future-value sensitivity uses differentiation through a fixed branch under smoothness assumptions; current margin sensitivity is a different quantity, and correct count is locally constant away from decision boundaries.The replay-item effect on a test margin is governed by a cross-Jacobian block; different label names neither imply orthogonal updates nor determine the sign.Source: §A.6 · p. 28

A.7 An exact closed-loop difference identity; A.8 Conditional stability, not automatic absorption

• The closed-loop difference identity needs no differentiability, but its reference policy must be defined on encountered states; attenuation additionally requires conditional stability, not merely longer training.The reported studies do not measure each counterfactual suffix, and endpoint differences cannot assign separate credit to a label choice, later write, or trigger time.Source: §A.7 · p. 29, §A.8 · p. 29

A.9 Favorable observations and expensive selection

• Favorable sampled controls do not establish a favorable interval unless the interpolation structure supports it, and an expensive search is justified only if its added value warrants its cost.The rollout accounting example is not a complexity lower bound, and no global optimality gap, search-cost ratio, or value of λ is estimated.Source: §A.9 · p. 29

B Implementation and numerical certification

• The protocol uses 800 updates across five tasks, fixed matching conventions, and correct count plus mean cross-entropy over active classes as endpoint readouts.The 32-update four-cell experiment evaluates 400 old-task and 200 current-task items; task-end panels expand from 200 to 1,000 items.Source: B Implementation and numerical… · pp. 29–30

B.1 Offline numerical verification

• Offline verification recomputed stored predictions, counts, and mean cross-entropy, but its records omit inputs, tokenizer assets, pretrained weights, and complete checkpoints, so they do not reproduce full GPU trajectories.Verification checks consistency of reported results; inferential scope remains determined by the experimental design.Source: §B.1 · pp. 30–31

C.1 Baseline replay readouts

• The baseline replay readout table covers matched four-cell branches sharing a pre-action parent, proposal, and 600-item panel at a 32-update horizon.The table caption identifies scope, not unseen cell values.Source: Table 9 · p. 31

C.2 Distillation and backbone extensions

• Distillation contrasts use parents from the first three legal commits of an LwF-plus-Fiber prefix; the RoBERTa extension uses its first legal nonzero period-32 event by step 671.The corresponding tables identify full distillation readouts and RoBERTa readouts, but the supplied passages do not provide their cell values.Source: §C.2 · p. 31, Table 10 · p. 31, Table 11 · p. 32

C.3 Optimizer-native primary comparison

• The optimizer-native comparison calibrates SGDW on a separate root, selects the fourth encoder/head rate pair, and records six Task-2 saturated-identity scans plus six Task-3 scans selecting step 511.The six phase nonactivations have no execute/off–on outcomes and are not assigned zero contrasts.Source: §C.3 · p. 31, Table 12 · p. 32

C.4 Independent-rate update-scale sensitivity

• The sensitivity panel uses three roots distinct from the optimizer-native comparison, and its two diagnostic designs reuse those same roots rather than adding independent-root replications.Equal root numbers across the two principal optimizer studies do not indicate pairing.Source: §C.4 · p. 32
• Four fixed-anchor cells activate under the selected rate pair (0.01, 0.02), while Root 3 at steps 351 and 511 returns saturated identity; the scale criterion does not match full trajectories or optimizer moments.The selected pair achieves 198/200 and CE 0.0139232; the encoder and head factors are exp(e_enc) = 1.435821 and exp(e_head) = 1.198306.Source: §C.4 · pp. 32–33, Table 13 · p. 33

D.1 Replay-retention comparisons

• On the 75%-retained trajectory, the measured successive differences are 0, −3, −1, and −3, totaling −7; these values are specific to that nested policy and continuation.Each pair branches from the corresponding parent reconstructed on the original trajectory; comparisons across retention rows do not isolate continuation at a fixed state.Source: §D.1 · p. 33, Table 14 · p. 33

D.2 Task-end performance of the coordination policies

• The full reinforcement policy has paired task-end differences (0, 0, +1, −1, +3), so its terminal gain is not a pathwise non-degradation guarantee.The single-window 32-step comparison is 577 versus 578 on 600 items while increasing the label-11 margin from 7.6531 to approximately 7.7876.Source: §D.2 · p. 34, Table 15 · p. 34

D.4 Five-root policy comparison

• On fresh test, mask-matched random content minus standard replay has root differences (−2, −1, +1, 0, +1), so no uniformly better content policy follows.The common terminal test panel contains 1,500 items; cross-entropy is secondary to correct count.Source: §D.4 · p. 35, Table 17 · p. 35

D.5 Complete-policy distillation comparison

• In Table 20’s complete-policy distillation comparison, terminal correct predictions rose from 414 to 433 and cross-entropy fell by 0.2663619, a net gain that did not improve every task.Task-wise changes were (+22, +25, −7, −21, 0).Source: §D.5 · p. 36, Table 20 · p. 36

D.6 Continuation-dependent actuation profiles

• The LwF trajectory commits at all 15 consultations from steps 191 through 639, then rejects its final five consultations under the ideal-write cap.Those rejections mean the specified numerical route did not certify the prescribed map within budget; they do not prove every useful action infeasible.Source: §D.6 · p. 36, Table 21 · p. 37

E.1 Exact rank of the stored frame and the ideal repair set

• A finite-field rank certificate establishes exact real row rank for stored frames, while the resulting geometry concerns local stored-frame repair rather than later diverging policy states.The proof uses stored FP32 entries as exact dyadic rationals and a nonzero maximal minor modulo 65521.Source: §E.1 · p. 37

E.2 Output superlevel sets and common-correct certificates

• The three control CEs are below the 0.2195535 reference, and their logit convex hull has lower CE; the three output segments retain at least 952 correct items, above 950.The segment certificate does not establish the whole triangle’s count minimum.Source: §E.2 · pp. 37–38, Table 22 · p. 38

E.3 Head separation under a common ordinary suffix

• After step 703, ordinary-suffix records agree and later offers are identities, but comparing correct counts 953 and 955 cannot distinguish state contraction from output insensitivity.The reported head distances neither show disappearance nor monotone contraction of the head difference.Source: §E.3 · p. 38

E.4 What the reallocation fraction does not bound

• Each control replaces 96 of 12,784 replay presentations (0.75094%), but realized policy differences span 155–179 slots, and the fraction alone bounds neither perturbation nor utility.Once states diverge, gradients of unchanged rows may also differ; clipping and AdamW moments further limit interpretation from the fraction alone.Source: §E.4 · p. 38

E.5 Native-event and linked-rate fixed-anchor diagnostics

• The native-event design activates at its first legal event in all three roots, whereas the linked-rate fixed-anchor design activates only Root 1 at both anchors.The diagnostics differ in optimizer, calibration, and anchor conditions, so they do not isolate the cause of activation differences or constitute repeated measurements of one unchanged design.Source: §E.5 · pp. 38–39, Table 23 · p. 39