Functional compatibility as a determinant of persistent neural learning

• Continual learning methods reduce interference, but they leave unresolved whether a property of incoming learning determines how much can coexist with preserved behaviour.Existing approaches include regularization, replay, constrained gradients, function-space protection, and attention to plasticity loss.(Kirkpatrick et al., 2017; Rolnick et al., 2019; Lopez-Paz & Ranzato, 2017; Farajtabar et al., 2020; Saha et al., 2021; Benjamin et al., 2019)Source: §1 · p. 1
• Functional compatibility measures how much current active learning geometry lies in directions compatible with behaviour declared for retention, rather than measuring forgetting after the update.Adaptive Functional Metaplasticity (AFM) quantifies and exploits this pre-update quantity by comparing protected learning with the same learner’s no-protection endpoint from the identical state.Source: §1 · p. 1
• Changing compatibility from the same neural state should change persistent new learning under the same retention requirement if compatibility is causal rather than merely correlated with interference.Source: §1 · p. 2
• The study’s novelty is a matched-state, multi-level intervention on quantified functional compatibility that separately controls unrestricted learning opportunity and retention requirement.This tests compatibility by intervention rather than post hoc observation, extending prior work on gradient alignment, protected subspaces, and interference geometry.(Riemer et al., 2019; Farajtabar et al., 2020; Wang et al., 2021; Benjamin et al., 2019; Störk, 2026; Shu et al., 2026; Gong et al., 2026; Kitkana & Arora, 2026; Abbes et al., 2026)Source: §1 · p. 2

2.1 Problem formulation and design principle

• AFM distinguishes persistent parameter adaptation from exact finite-output protection and population behavior, warning that finite evidence does not imply preservation on unseen inputs.The framework treats population claims as requiring additional assumptions or certificates.Source: §2.1 · p. 2

2.1.1 Same-state no-protection comparator

• The same-state no-protection comparator measures how much learning from the actual accepted unrestricted transaction can be stored persistently under retention geometry.It matches parameters, state, minibatch, active coordinates, optimization rules, buffers, and declared randomness, while remaining counterfactual.Source: §2.1.1 · pp. 2–3

2.1.2 Persistent compatible assimilation

• Compatible assimilation requests a predeclared fraction of projected comparator charge, while the compatible-gradient fraction quantifies active gradient energy available within protected geometry.The assimilation request and compatibility measure are distinct quantities.Source: §2.1.2 · p. 3

2.1.3 Finite endpoint completion

• AFM separates safe persistent base motion from a finite function-space residual that restores requested current and protected outputs, rejecting transactions when consistency, capacity, support, or certificates fail.The residual is zero away from finite support and is not treated as persistent assimilation.Source: §2.1.3 · p. 3

2.2 Adaptive Functional Metaplasticity

• Adaptive Functional Metaplasticity combines sensitivity sketches, multi-timescale allocation, task-free certification and routing, and function-preserving structural renewal within bounded protected transactions.Protected updates route by observable signatures, update memory and traces, compute counterfactual and compatible references, enforce certified base acceptance, and separately control reopening and renewal.Source: §2.2 · p. 3, §2.2.1 · p. 3, §2.2.2 · p. 4

2.2.5 Protected update

• The protected-update sequence preserves the distinction between private candidates, persistent bases, and finite residuals, and no failed certificate permits installing the unrestricted endpoint.This operationalizes routing, evidence updates, compatible projection, retention checks, endpoint completion, and controlled state changes.Source: §2.2.5 · pp. 4–5

2.3 Theoretical guarantees

• The theoretical guarantees show that normalized compatible assimilation can feasibly use the complete projected comparator, while exact finite completion preserves declared outputs and emulates the no-protection endpoint under stated checks.The compatible factor κt is unavoidable in general, whereas finite residual guarantees apply only when consistency, capacity, support, and numerical conditions hold.Source: §2.3 · p. 5

2.3.1 Retention, stationarity, and local optimality; 2.3.2 Integrated frontier and unavoidable obstructions

• The integrated theory provides pathwise retention, compatible stationarity, and finite-evidence equality, but exposes spectral leakage, information, capacity, routing, renewal, and nonlinear-scope limits.For a behavior Jacobian J, the minimum worst-case first-order leakage of a (d−r)-dimensional plastic subspace is σr+1(J); convex dynamic regret requires a separate affine specialization.Source: §2.3.1 · p. 6, §2.3.2 · p. 6

2.4 AFM framework synthesis

• Across CORe50, CLEAR-10, and CLAD-C, AFM outperformed the strongest tested classical non-oracle family, remained competitive with six modern challengers, and causally linked compatibility to persistent learning.The audit found non-vacuous finite endpoint transactions and theorem-aligned assimilation on accepted nonzero interventions; limitations include finite evidence, observable-routing requirements, retention allowance, curvature, and computational overhead.Source: §2.4 · p. 7

2.5.1 Admissible specification

• AFM specifies all predeclared routing, loss, trust, certification, memory, renewal, numerical, and resource components, with measurable learner-history conditions and typed obstruction outcomes.An optional unprotected initialization may create the initial representation, which is then frozen before protected decisions begin.Source: §2.5 · p. 7, §2.5.1 · p. 7

2.5.2 Losses and retained behaviors

• The learner handles arbitrary ordered minibatches without iid or stationarity assumptions, while certified quadratic bounds determine a trust radius and force abstention when valid certificates are unavailable.Finite computation-graph interval automatic differentiation can construct the required loss and behavior certificates on certified parameter boxes.Source: §2.5.2 · p. 8

2.5.3 Task-free routing and semantic mismatch

• Task-free AFM routes by observable context signatures rather than task labels and explicitly charges mismatch between router-selected and evaluator-defined behaviors.Routing mismatch terms vanish when semantics agree but are not assumed small, and structural replacement cannot be hidden inside parameter-only bounds.Source: §2.5.3 · p. 9

2.5.4 Streamed whole-behavior sensitivity memory

• AFM protects complete frozen commit-block behaviors using streamed Jacobian memory and Frequent Directions sketches, while charging the measured gap between deployed activation state and frozen validated snapshot.Transfer commits only after the observed gap falls below a pre-fixed threshold; otherwise the deployed predictor remains unchanged and a transfer obstruction is returned.Source: §2.5.4 · pp. 9–10

2.5.5 Guarded compact-cardinal shielding of certified candidates

• Certified candidates are frozen at anytime-valid crossings, retained under bounded routing slots and evidence, and served by a predictable least-recently-served policy without copying private parameters into deployment.The deployed transaction accounts jointly for safe-base motion and structural shield replacement, with exact current-minibatch endpoint emulation requiring losses to factor through deployed logits.Source: §2.5.5 · p. 10
• The protected update compares a genuine same-state unrestricted endpoint with a certified compatible safe endpoint, applies retention-charge and trust constraints, and returns an obstruction when the counterfactual endpoint is unavailable.The unrestricted endpoint is never installed; explicit checks cover collinearity, projection idempotence, trust-cap membership, descent, and realized charge.Source: §2.5.5 · pp. 10–11
• The compact-cardinal shield restores exact current and protected logits, safeguards unselected candidates, transfers the selected candidate, remains silent on guards, and has zero residual outside finite support.At most one bump is active at an address, preventing multi-center accumulation; duplicate contradictory requests at one observable address are impossible to satisfy deterministically.Source: §2.5.5 · pp. 14, 16
• AFM rejects and fully rolls back any transaction failing comparator, consistency, capacity, address, endpoint, numerical, or service certificates rather than installing an unrestricted endpoint.Rollback covers parameters, mutable buffers, shield state, module modes, declared private randomness, support radii, guards, and candidate records; failed attempts remain auditable.Source: §2.5.5 · pp. 14–15
• Accepted joint updates preserve the declared base behavior budget, leave active protected behavior exactly unchanged, and achieve zero selected-transfer objective after one service when ξ_t = 1.The construction attains ratio one against the genuine no-protection endpoint while retaining ρ_tr = 0.25 as the measured reference.Source: §2.5.5 · pp. 14–15
• Least-recently-served scheduling selects every continuously eligible candidate within at most C_max available service rounds, and ξ = 1 completes consistent finite transfer after one accepted service.For 0 < ξ < 1, repeated accepted services provide geometric contraction; no compatibility-rate, positive-cosine, or interpolation-conditioning assumption is required.Source: §2.5.5 · p. 15

2.5.6 Conflict, variation, evidence, and renewal richness

• The framework measures conflict as the discarded protected-complement gradient component and diagnoses renewal richness through exact trial-slot motion under zero-gated dormancy.A nonzero selected trial-slot component necessarily includes gate-coordinate motion, while reset-induced Jacobian changes enter the leakage certificate.Source: §2.5.6 · p. 18
• AFM reopening uses raw routed outcomes and a predictable challenger, so semantic changes without declared observable evidence provide zero information.Observable challenger advantage may be positive, negative, zero, or time-varying; such changes are covered by the impossibility term.Source: §2.5.6 · p. 18

2.5.7 Complete AFM algorithm

• The complete algorithm routes observations, certifies candidates on later outcomes, maintains append-only protected records, and allocates protection using finite policies and metaplastic traces.Private fitting parameters are not copied directly into deployment, and re-anchoring creates a new record segment rather than rewriting an active one.Source: §2.5.7 · p. 19
• AFM accepts a persistent base only after path-fraction, retention, descent, projection, trust-region, and numerical checks, then commits the deployed transaction only when all finite endpoint and capacity checks pass.A failed protected transaction restores the entire predictive state; the unrestricted base endpoint is never used as fallback.Source: §2.5.7 · p. 19
• Renewal is permitted only through exactly function-preserving zero-gated resets followed by recomputed protected geometry and accepted nonzero trial-slot motion.If the protected activation checks fail, the reset is rolled back.Source: §2.5.6 · p. 18, §2.5.7 · p. 19

2.6 Task-free routing, consolidation, and reopening

• Under separated observable contexts, nearest-threshold routing with convex EMA centroids identifies each context’s correct slot without task labels.The guarantee follows because same-context signatures remain within threshold of their centroid, while other-context centroids remain beyond it; unseen contexts open empty slots.Source: §2.6 · p. 20

2.6.1 Frozen-representation calibration and observable families

• Frozen-representation calibration makes routing statistics fixed before protected decisions, while clipped thresholds with violated separation premises forfeit positive routing conclusions but preserve pathwise guarantees.Declared prefix references are replayed through the final frozen backbone before centers, radii, scales, temporal state, and thresholds are fixed.Source: §2.6.1 · p. 20
• A finite predeclared signature family can combine valid e-processes with fixed weights, controlling route-identification error while making no finite claim when no observable expert has positive information.The evaluated implementation uses M = 1 with a predeclared hybrid signature, whereas the theorem permits a stronger finite-family construction.Source: §2.6.1 · p. 20

2.6.2 Operational consolidation and evidence-based reopening

• Anytime-valid consolidation certifies a frozen candidate when its evidence-bound average conditional validation risk is at most τ, without stationarity or iid assumptions.After certification, the candidate’s count, losses, evidence, outputs, sketch, and transfer objective are frozen; later staleness testing can cancel it under separately controlled error.Source: §2.6.2 · pp. 21–22

2.6.3 Separately controlled reopening and observable route refinement

• Outcome contradiction and observable displacement receive independent lifetime controls: route refinement preserves protected source state, while semantic-only change permits reopening but not a new route under stable signatures.False-split control includes any separately allocated calibration failure probability, and observational indistinguishability remains an explicit routing-mismatch obstruction.Source: §2.6.3 · pp. 22–23
• Permitted route splitting requires positive observable signature drift; with zero signature drift, no finite route-splitting delay is claimed, although ordinary reopening or release remains available.The joint delay waits for the slower outcome or signature crossing and deterministic gates, while continuous-mixture e-processes adapt to unknown positive drift without an expanding grid.Source: §2.6.3 · pp. 23–24

2.7 Whole-behavior memory and spectral protection

• The deterministic streamed-memory certificate bounds leakage for every round and every unit vector in the protected range, while the routed empirical behavior has t = 0 and the result is deterministic.The results quantify leakage represented by bounded sensitivity sketches and characterize the first-order rank-plasticity frontier.Source: §2.7 · p. 24, §2.7.1 · p. 25

2.8.1 Retention and compatible adaptation

• Every accepted protected update restores the declared finite protected-evidence behavior exactly, while broader deployed maps incur only bounded joint-state and structural replacement drift.The active-interval result follows by telescoping one-step bounds, with the persistent safe-base leg, structural replacement, and exact finite-map restoration charged separately.Source: §2.8.1 · pp. 26–27
• Activation and release are decomposed into snapshot-to-activation transfer, exact active-interval retention, and uncontrolled post-release drift rather than conflated with protected forgetting.Delayed activation and capacity release are therefore reported separately from protected forgetting.Source: §2.8.1 · p. 27
• The safe-base projected-stationarity identity holds pathwise for smooth non-convex losses, while shield-emulated deployed progress remains a separate exact ledger with current-batch ratio one.The stationarity proof uses the pre-step-shield loss variation; deployed progress is not substituted into the base stationarity argument.Source: §2.8.1 · p. 27
• Indefinitely available compatible movement, exact finite protection, and ratio-one current endpoint emulation coexist on accepted tails with finite cumulative certified behavior charge.This holds when the safe-base loss-budget condition is finite; broader joint behavior additionally requires summable structural and routing charges.Source: §2.8.1 · p. 27
• Persistent safe-base movement is certified by observable lower bounds on accepted updates and divergent retained square-root budget mass, not inferred solely from first-order compatibility or shield updates.It is sufficient that omitted rounds have finite square-root budget mass; the certificate depends on the realized safe radius, projected gradient, and endpoint backtracking factor.Source: §2.8.1 · p. 28

Logarithmic trace bank

• K = O(log H) exponential moving-average traces approximate a power-law relevance profile over ages 1, . . . , H with constant multiplicative distortion.Theorem 2.32 establishes this approximation for every 0 < α ≤1 and K ≥2.Source: §2.9 · p. 28
• A single exponential approximation to f(τ) = τ −α over horizon H has polynomial worst-case distortion, with C4 ≥ Hα/4α.This lower bound follows for odd H ≥3 and shows why a logarithmic bank is needed to represent scale-free relevance efficiently.Source: §2.9 · p. 29

Rank/timescale allocation

• The frontier cost penalizes both retention leakage and plastic gradient energy blocked by protection, so increasing rank is not free and cannot dominate by freezing more coordinates.The controller is therefore evaluated against simultaneous retention and adaptation costs.Source: §2.9 · pp. 28–29

Allocation transfer

• Accurate prediction of realized retention weights yields near-oracle functional leakage, and exact prediction makes the rank-r allocation oracle optimal for that round.The guarantee compares the policy’s predicted allocation with a top-r oracle projector.Source: §2.9 · p. 29

Policy selection

• Exponential weights over declared policies provides a high-probability metaplastic allocation regret bound whose selected costs jointly represent spectral retention residual and blocked gradient fraction.The controller is consequently coupled to retention and compatible adaptation rather than analyzed separately.Source: §2.9 · pp. 29–30

Atomization invariance

• The trace, policy, spectral-loss, allocation-transfer, and Hedge results remain valid under metaplastic atomization invariance.The proposition extends the stated results to each record contribution.Source: §2.9 · p. 30

Atomization and controller guarantees

• Finite PSD atomization preserves the controller’s trace, spectral-cost, allocation-transfer, and Hedge arguments by replacing each PSD term with a certified nonnegative weighted sum.The proof relies only on nonnegative PSD sums, trace residuals under I − Q_t, operator-norm perturbations, and bounded expert losses.Source: Give atom a the explicit realized signal · p. 30
• Each atom retains its own K traces and policy weight, which determine predicted and realized covariances through corresponding nonnegative weighted sums.Source: Give atom a the explicit realized signal · p. 30

Memory-certificate condition

• An atom-specific memory certificate is valid only when weighted certified PSD sensitivity/error decompositions upper-bound the same whole-behavior derivative energy, Frequent-Directions error, and anchor-transport terms.Without such certificates, algebraic splitting does not strengthen the derivative guarantee, and the original record-level certificate is retained; certified atoms may be sketch eigen-directions, parameter groups, or rank-one synaptic sensitivities.Source: Give atom a the explicit realized signal · p. 30

State and scope

• Persistent state grows with the fixed number of retained atoms rather than elapsed lifetime, while atomization otherwise serves only the allocation controller.Source: Give atom a the explicit realized signal · p. 30

Function-preserving structural renewal

• Exact reset leaves the realized predictor, current loss, and every active protected behavior unchanged, while accepted compatible updates certify nonvacuous gate activation.If activation is not accepted, transactional rollback restores the exact pretrial state.Source: §2.10 · p. 31
• Renewal failure probability is at most δe once cumulative conditional richness reaches log(1/δe), and zero richness makes success impossible for that renewal family.The guarantee uses actual predictable richness and requires no known lower bound.Source: §2.10 · p. 31
• Under isotropic Gaussian gradients and an orthogonal compatible projector of rank h, h > 0 and ω > 0 imply positive richness, but isotropy is unnecessary for the universal theorem.The general result does not assume commutation of the structural and compatibility projections.Source: §2.10 · p. 32
• Fixed structural pools preserve lifetime resource use: failed or unactivated trials reuse dormant slots, while exhausted capacity triggers a structural-capacity obstruction instead of allocating parameters.Successful activations consume slots until separately certified structural moves restore exact zero-gated dormancy.Source: §2.10 · p. 32

2.11.1 Convex affine-behavior dynamic regret

• Under compact convex retention sets, convex differentiable losses, bounded gradients, and an exact projection oracle, AFM satisfies global dynamic regret with an unavoidable compatibility price.The first compatibility-price term cannot be removed by any retention-feasible learner.Source: §2.11.1 · p. 32, §2.11.2 · p. 35

2.11.2 Integrated frontier theorem

• The integrated frontier theorem guarantees task-free routing, safe transfer, retention, non-convex stationarity, timescale optimality, reopening control, renewal, bounded resources, and convex performance under declared certificates.Its probabilistic conclusions hold jointly with probability at least 1 −δ, while pathwise conclusions hold on every realized trajectory; the guarantees do not require mismatch, curvature, conflict, variation, delay, or drift terms to be small.Source: §2.11.2 · pp. 33–35
• AFM preserves accepted protected behavior, supports arbitrary non-convex adaptation through projected stationarity, and converts temporal prediction error into excess leakage while competing on a composite frontier cost.Compatible base plasticity can coexist with finite certified safe-base behavior charge and exact finite deployed protection; broader joint maps incur an explicit structural-shield charge.Source: §2.11.2 · pp. 33–34
• The theorem uses bounded persistent state independent of elapsed lifetime, explicit numerical-precision policies, and obstruction reporting when structural capacity, richness, oracle availability, or feasibility is insufficient.The declared memory expression is W = O(d) + O(Lmaxd) + O(Cmaxℓcandd) + O((Jmax + Cmax)d) + O(CmaxdZ) plus further fixed-size terms, while oracle failure is not replaced by uncertified approximation.Source: §2.11.2 · pp. 34–35

2.12.1 Matching obstructions and maximality

• The constructions establish unavoidable limits from indistinguishable contexts, finite histories, nonlinear curvature, anchor drift, incompatibility, zero information or richness, finite memory, and finite-state lifetime testing.These results force explicit terms, abstention, information release, or assumptions that make the obstructions small.Source: §2.12.1 · pp. 35–36
• Componentwise maximality shows that deleting any listed obstruction term falsifies the corresponding universal guarantee, although the result does not establish sharp constants or minimax optimality.The obstructions cover routing, consolidation, leakage, curvature, anchor drift, incompatibility, reopening, renewal richness, finite memory, and infinite-horizon testing.Source: §2.12.1 · p. 37

2.12.2 Guarantee and obstruction summary

• The integrated theorem links eight mechanism families to guarantees and their dependencies, including routing separation, validated consolidation, derivative-leakage control, compatible adaptation, information-dependent reopening, function-preserving renewal, finite precision, and compatibility-costly dynamic regret.The summary explicitly retains obstruction terms for contradictory requests, unavailable service, finite rank, direct conflict, insufficient compatible richness, and fixed structural resources.Source: §2.12.2 · p. 37

3.1 Compatibility as a causal variable

• Compatibility changed persistent learning under matched conditions, whereas unrestricted, distillation, and EWC converted it less efficiently and replay and DER++ showed no seed-resolved slope.Thus, compatibility describes available retention-constrained geometry, while learning rules determine how much of that geometry is exploited.Source: §3.1 · p. 38, §3.7.2 · p. 45

3.4.1 Compatibility construction

• The intervention varied learning-signal direction within identical parent states, protected evidence, and current inputs using a generalized function-space eigenproblem and analyzed realized compatibility.Parent checkpoints preserved model, optimizer, stream, replay, and random-number-generator states; 750 causal states spanned three systems, six compatibility conditions, and seven method branches.Source: §3.3 · pp. 40–41, §3.4.1 · p. 41

3.4.3 Common retention budget

• Every compatibility condition and method shared the same absolute retention budget, so β controlled retention strictness independently of requested compatibility.The implementation searched 33 endpoint scales and selected the best feasible persistent endpoint without assuming monotone protected drift.Source: §3.4.3 · pp. 41–42

3.5 Natural-state validation

• Natural-state validation followed ordinary continual-learning streams from the same three architectures, five seeds, and parent checkpoints without selecting states by compatibility.Measurements used discarded validation branches and deterministic probes, preserving the subsequent ordinary parent trajectory.Source: §3.5 · pp. 42–43

3.6.1 Inferential unit

• Seed-level slopes, rather than method rows, were the independent causal replicates, while normalized ratios were interpreted cautiously because tiny ∆0 can produce extreme values.The analysis averaged 50 state slopes within each seed and used five seed-level slopes with two-sided 95% Student-t intervals; natural CNN ∆0 reached 3.89 × 10^-4. The AFM diagnostic was empirical rather than a proof of theorem assumptions.Source: §3.6.1 · p. 43, §3.6.2 · p. 43, §3.6.3 · p. 43

3.7.1 Intervention fidelity

• Finite unrestricted progress was matched across compatibility conditions, with within-state relative spread averaging 0.27%–0.28% and never exceeding 0.40%.Matching used a one-dimensional bisection along each original direction; the reported validation directly verifies finite-∆0 matching, not gradient-norm matching.Source: §3.4.2 · p. 41, Table 4 · p. 44, §3.7.1 · p. 44

3.7.2 Causal effect of compatibility under an intermediate retention budget

• At β = 0.5, projected compatibility slopes were 1.014 for CNN, 1.001 for ViT, and 0.965 for text, with all seed-level slopes positive.The cross-system descriptive mean was 0.993 with 95% t interval [0.980, 1.006], and the result replicated across all three fully trainable systems.Source: Table 5 · p. 44, §3.7.2 · p. 44

3.7.3 Retention strength changes the observable compatibility slope

• Across retention budgets, the projected compatibility slope was zero without retention allowance, positive for every positive β, and approached one sooner in vision than text.The two CIFAR-10 systems reached approximately unit slope by β = 0.05–0.10, whereas the text transformer required a more permissive budget.Source: Table 6 · p. 45, §3.7.3 · p. 45
• Retention strength controls how much compatible learning is accepted, distinguishing the compatibility factor κ from the accepted path fraction bλ.Thus, compatibility is not the only determinant of persistent learning.Source: §3.7.3 · p. 45, §3.8 · p. 48

3.7.4 Native AFM mechanism audit

• Native AFM passed its retention and theorem-aligned checks across controlled nonzero interventions, completing 3,745/3,750 finite cases and achieving 0.0 recorded endpoint errors with deployed progress ratio 1.0.These finite completions verify the declared endpoint mechanism, not population retention away from finite support; requested-zero negative margins lacked certified theorem conditions.Source: §3.7.4 · p. 45, §3.9 · p. 48

3.7.5 Natural-state validation

• Native AFM satisfied the D ≤0.005 retention criterion and had positive theorem-aligned margins on all 750 natural states, but high compatibility did not ensure high realized persistent progress.In text, compatibility averaged 0.950, mean native AFM path fraction was 0.107, and median persistent ratio was approximately 0.119.Source: §3.7.5 · p. 46, §3.10 · p. 48

3.7.6 Local causal scale and ordinary natural scale

• The causal intervention isolates local geometry by matching the finite unrestricted decrease, whereas natural-state validation examines much larger ordinary stream-generated decreases, especially in vision.Therefore, the causal slope should not be assumed to describe arbitrary large updates.Source: §3.7.6 · p. 47, §3.9 · p. 48

3.10 Controlled causal experiment synthesis

• Functional compatibility causally changes persistent progress under matched retention constraints, with projected slopes large, reproducible, and close to one across CNN, ViT, and text-transformer systems.At β = 0.5, the slope is approximately one in all three fully unfrozen systems; the effect depends on retention-feasible expression of compatible directions.Source: §3.8 · pp. 47–48, §3.10 · p. 48

3.9 Scope and limitations of the causal analysis

• The evidence supports a broad geometric result rather than universal method dominance: natural validation is observational, strict retention thresholds are branch-specific, and compatibility alone does not determine persistence.The requested-zero condition reflects minimum rather than exact zero compatibility, with realized means 0.0107 for CNN and 0.0413 for ViT; ratios can also become unstable when ∆0 is small.Source: §3.7.5 · p. 47, §3.9 · p. 48, §3.10 · p. 49

3.11 Complete cross-system slope summary

• Seed-level effects provide the primary inferential analysis, while aggregate slopes summarize cross-system results and medians clarify skewed natural-state ratios.The unrestricted branch has ratio exactly one by construction; other CNN means can be distorted by states with small ∆0.Source: §3.11 · p. 49, §3.12 · p. 50

4.1 Robust across directions and models

• At β = 0.5, ten-seed compatibility slopes were 1.015, 1.001, 0.999, and 0.960 for the convolutional, baseline vision-transformer, stronger vision-transformer, and text-transformer systems.The descriptive equal-system pooled slope was 0.994, and vision slopes remained near one across the expanded local multiscale regime.Source: §4.1 · p. 51, Figure 2 · p. 52

4.2 Compatibility in ordinary learning

• In 750 natural states, mean κ was 0.228, 0.422, and 0.950 for the convolutional, vision-transformer, and text-transformer systems, respectively, but the text model’s mean persistent path fraction was only 0.107.Natural-state behavior supports compatibility’s causal relevance while showing that retention-constrained path capacity and the learning rule also determine persistent progress.Source: §4.2 · p. 51, Figure 3 · p. 53
• AFM achieved the highest dataset-mean primary score on CLEAR-10 and CLAD-C, while FGH exceeded it on CORe50 by 2.8%; AFM led all six challengers in the equal-weighted summary.AFM exceeded CCL-DC, MKD, LPR, FGH, aL-SAR, and OCAR by 7.0%, 11.4%, 16.3%, 21.7%, 31.7%, and 36.8%, respectively; comparisons were not equal-compute or equal-storage.Source: §4.2 · p. 53, Table 15 · p. 54, §4.3 · p. 54

4.3 Nonlinear geometry sets the boundary

• At 1% natural norm, only one CNN state versus 231 vision-transformer states supported all four same-norm compatibility levels; at 10% and above, neither architecture supported the continuum.The matched intervention fails because finite nonlinear geometry makes one or more same-norm directions non-descending, not because compatibility’s causal effect is necessarily absent.Source: §4.3 · pp. 54, 56
• At 1% natural update norm, compatibility increased absolute persistent progress in 231 feasible vision-transformer states, with slope 3.064 × 10−4, while the normalized-ratio slope was −0.572 and crossed zero.At 10% and above, neither architecture supported the jointly feasible four-level intervention, so infeasibility does not imply a zero or negative compatibility effect.Source: §4.3 · p. 54, Figure 4 · p. 55

4.4 Purpose and scope

• The tested causal proposition is conditional: at a fixed neural state and retention criterion, changing compatible incoming learning changes persistent storage, while retention budgets, learning rules, and finite geometry limit its extension.The follow-up studies therefore report positive, null, and infeasible outcomes as distinct parts of the claim’s empirical scope.Source: §4.4 · p. 56, §4.5 · p. 57

4.7.2 Key result

• At β = 0.01 in the CNN, multiscale matched slopes rose from 0.10272 at s = 0.05 to 0.96935 at s = 0.9, whereas at β = 0.05 they stayed near one.Thus compatibility is not the sole determinant of attainable persistent progress; the retention budget also matters.Source: §4.7.2 · p. 58

4.8.2 Results

• At β = 0.5, independent-direction matched slopes had 95% seed-level intervals [1.03046, 1.06024], [1.02516, 1.03442], and [0.97849, 0.99454] for CNN, ViT, and text transformer.Within-κ directional variation was much smaller than between-κ variation, with all systems having mean state fractions of 1.0 below that threshold.Source: §4.8.2 · p. 60, §4.8.3 · p. 60, Table 21 · p. 60

4.9.4 Interpretation

• The generality result was reproduced in a stronger ViT configuration without changing the intervention, but it is not evidence from foundation-model-scale systems.At β = 0.5, all four AFM/projection matched slopes were strongly positive and near one.Source: §4.9.4 · p. 61, §4.9.2 · p. 61

4.10.2 Assessment result

• All 65 negative theorem-aligned margins near requested κ = 0 lacked certification of the relevant finite curvature or step condition, so they cannot be counted as certified theorem violations.Numerical verification matched reported κ values within 1.5883 × 10−9 and reported ρ values exactly across all 65 cases.Source: §4.10.2 · p. 62, §4.10.3 · p. 62

4.11.3 Feasibility

• At 1% natural update norm, the ViT supported 231/500 feasible states across ten seeds, whereas the CNN supported only one and neither architecture supported any at 10%, 50%, or 100%.Feasibility required positive same-norm finite progress for all four target-compatibility directions; 232 of 4,000 attempted state-scale conditions passed overall.Source: §4.11.3 · p. 63, §4.11.2 · p. 63

4.12.2 ∆0-heterogeneity stratification

• Absolute persistent-progress slopes were nearly unchanged across low-, medium-, and high-∆0 heterogeneity, but ratio-slope intervals changed from positive to strongly negative, reaching [−13.001, −3.271].At β = 0.5, the absolute-slope intervals were approximately [3.03618, 3.09315]×10−4, [3.02131, 3.12192]×10−4, and [2.99120, 3.12784]×10−4 across the three strata.Source: §4.12.2 · p. 65, §4.12.3 · p. 65

4.12.3 Interpretation

• The fixed-norm bridge shows that compatibility remains causally associated with absolute persistent learning where jointly feasible, but realized unrestricted progress ∆0 becomes direction-dependent, making ρ unstable as a standalone causal summary.Adjusting for realized ∆0 yielded a projection coefficient of about 2.694, but this is a sensitivity analysis because ∆0 is post-intervention.Source: §4.12.1 · p. 64, §4.12.3 · p. 65, §4.13 · p. 65

4.13 Finite-step feasibility boundary

• Above the feasible local regime, increasing fixed update norm destroys the common four-level positive-descent continuum before 10% of median natural update norm in both tested vision systems.Where joint feasibility disappears, no matched four-level causal slope exists; the boundary reflects finite curvature and nonlinear loss-surface structure.Source: §4.13 · p. 65

4.14 Integrated interpretation of the follow-up experiments

• Across ten seeds, CNNs, ViTs, and a text transformer, compatibility produced a positive dose response with pooled slope 0.9936 at β = 0.5.The effect also remained near one across independently generated directions and a designed local multiscale range.Source: §4.14 · p. 66

4.12.1 Primary decomposition

• In 231 feasible ViT states, absolute persistent progress increased with compatibility with slope 3.06428 × 10−4 and CI95% = [3.03053, 3.09315] × 10−4, while the normalized ratio could fail or reverse.The mean persistent-update acceptance fraction was 1.0 and the mean finite-completion fraction was 0.994118.Source: §4.12.1 · p. 64, Table 26 · p. 64, §4.14 · p. 66

5.1 Study design and inferential unit; 5.2 Causal systems and reconstruction

• The study links chronological benchmarks, controlled compatibility interventions, and follow-up stress tests, treating repeated within-state measurements as clustered by seed rather than independent rows.Experiments used convolutional and transformer systems for vision and text, reconstructed from identical parent checkpoints containing model, optimizer, stream, replay, and random-state information.Source: §5.1 · p. 67, §5.2 · p. 67

5.3 Functional compatibility intervention

• Compatibility was manipulated in function space across six requested levels, with realized κ recomputed for analysis and nonzero-request errors at most 1.1 × 10−5.The requested-zero condition was a minimum-compatibility boundary, with mean realized values of 0.010681, 0.041266, and 8.3 × 10−5 in the convolutional, vision-transformer, and text systems.Source: §5.3 · p. 67

5.4 Matching unrestricted learning; 5.5 Retention budgets and methods

• Unrestricted learning opportunity was matched by bisection along each original direction, yielding within-state relative ∆0 spreads of 0.283%, 0.282%, and 0.268% across the three systems.This makes ρ = ∆persistent/∆0 interpretable as the persistent fraction of approximately equal unrestricted learning opportunity; retention methods were evaluated under common drift budgets and endpoint-selection rules.Source: §5.4 · p. 68, §5.5 · p. 68

5.6 AFM framework and protected transaction

• AFM uses same-state no-protection comparators, functional sensitivity sketches, compatibility projection, certified assimilation, finite residual completion, atomic checks, and function-preserving dormant-module renewal.Its protected transaction rejects failed endpoints rather than installing unrestricted updates, while routing relies on declared observables and evidence-based reopening.Source: §5.6 · pp. 68–69

5.7 Natural-state validation; 5.8 Independent-direction experiment; 5.9 Multiscale and generality tests; 5.11 Fixed-norm bridge

• Validation covered natural states, independent directions, multiscale interventions, ten-seed and stronger-architecture generality, and fixed-norm bridges with prespecified feasibility and error checks.At β = 0.5, directional variation was below between-compatibility variation in 1.0 of states across all three systems; the fixed-norm bridge found 232 feasible conditions among 4,000 attempts.Source: §5.7 · p. 69, §5.8 · p. 70, §5.11 · p. 70

5.10 Near-zero boundary assessment; 5.12 Chronological AFM benchmarks; 5.13 Statistics

• Chronological benchmarks used retention, plasticity, and origin-referenced hypervolume across classical and modern controls, while statistics aggregated state slopes to seed-level effects with bootstrap or t-based intervals.The near-zero assessment identified 65 negative empirical margins without certified finite assumptions, and benchmark hypervolume was treated as protocol-specific rather than directly comparable across datasets.Source: §5.10 · p. 70, §5.12 · p. 71, §5.13 · p. 71

6.1.1 Datasets, architecture, and protocol

• The benchmark protocols fix the chronological streams, common compact convolutional base, frozen representation boundary, learner-visible information, and five paired seeds without task or boundary identifiers.Modern challengers retain their method-native auxiliary state, so the comparison controls architecture and information interface but not total storage or compute.Source: §6.1.1 · p. 71, Table 28 · p. 72

6.1.3 Classical comparison

• AFM exceeds the strongest eligible classical non-oracle family on every paired seed, with mean hypervolumes of 0.248, 0.381, and 0.513 on CORe50, CLEAR-10, and CLAD-C.Paired mean advantages are approximately 0.0090, 0.0068, and 0.0110, with every confidence interval above zero.Source: Table 29 · p. 73, §6.1.3 · p. 73

6.2.2 Classical seedwise results and operating-point trade-offs

• AFM’s operating points form a retention–acquisition frontier: higher η increases acquisition, while conservative settings reduce forgetting across the benchmarks.On CLEAR-10, all coordinates reduce forgetting relative to no protection with similar plasticity; on CLAD-C, full assimilation has more peak-to-final forgetting.Source: §6.1.3 · p. 73, §6.2.2 · p. 79

6.1.4 Comparison with modern continual-learning methods

• AFM has the highest mean primary score among modern challengers on CLEAR-10 and CLAD-C, while FGH exceeds it on CORe50 by about 2.8%.The comparison uses AFM’s declared frontier versus challengers’ frozen method-native configurations, not arbitrary challenger hyperparameter sweeps.Source: §6.1.4 · p. 74, Table 32 · p. 75

6.1.5 Ablation and theorem-aligned execution evidence

• The joint AFM transaction improves CORe50 R × P by about 18.6% over both base-only variants, whereas collapsing the multiscale trace bank has essentially no aggregate effect at η = 0.50.Positive paired intervals support the full-AFM improvement; the single-timescale result does not invalidate the multiscale theorem.Source: Table 35 · p. 77, §6.1.5 · p. 77
• Accepted protected updates keep maximum endpoint error below 10^-6, and realized persistent-base progress exceeds the analytic lower bound on every accepted protected update examined.The near-one deployed ratio verifies finite endpoint completion rather than population retention away from finite shield support.Source: Table 36 · p. 77, §6.1.5 · p. 77
• AFM’s host-observed overhead is concentrated in protection geometry, finite counterfactual completion, and endpoint verification, which together consume 94.82%, 91.70%, and 95.78% across CORe50, CLEAR-10, and CLAD-C.Median runtime is approximately 35×, 15×, and 184× matched SGD, respectively; ordinary learning accounts for only 0.36%, 0.44%, and 0.24%.Source: Table 37 · p. 78, §6.1.5 · p. 78

6.1.6 Empirical scope and limitations

• The empirical claim is restricted to continual adaptation over a shared learned representation, not unrestricted end-to-end representation learning or universal dominance.CLAD-C supports assimilation, restoration, transfer, and finite-protection mechanisms, but insufficient signature calibration prevents a positive routing claim.Source: §6.1.6 · p. 78

6.2.3 Modern challenger compatibility

• AFM has the higher primary score in 84 of 90 matched comparisons and 17 of 18 dataset means, with FGH the only dataset-level reversal.Relative to challengers, AFM’s dataset-level mean gains range from 0.9% to 72.6% where it leads; FGH is 2.8% higher on CORe50, while AFM leads on CLEAR-10 and CLAD-C.Source: §6.2.3 · p. 80, Table 41 · p. 81
• Several challengers exceed AFM on individual stability or final-accuracy measures even though AFM has the stronger joint primary profile.The conventional metrics therefore qualify, rather than overturn, the primary comparison.Source: §6.2.3 · p. 82, Table 43 · p. 82

6.2.4 CLAD-C Pedestrian analysis

• CLAD-C Pedestrian forgetting reflects a higher AFM peak before a collapse shared with no protection, while finite protection preserves declared evidence rather than the unseen class distribution.Under inverse-frequency weighting, excess forgetting increased from about 0.0269 to 0.0837, with mean attenuation −0.0568 and interval [−0.1113, −0.0023], rejecting a global-count-only explanation without proving temporal absence as the unique cause.Source: §6.2.4 · p. 83, Table 44 · p. 83

6.2.5 Resource accounting

• Modern comparisons retain method-native auxiliary state, so runtime and memory are interpreted alongside predictive performance; the current AFM implementation has substantial runtime overhead, especially on CLAD-C.CCL-DC and MKD carry complete learners and replay, whereas AFM carries candidate, projector, metaplastic, event, and finite-shield state.Source: §6.2.5 · p. 83

6.3.1 Transaction ablation

• At CORe50 η = 0.50, full AFM improves plasticity, online accuracy, and held-out final-test accuracy over normalized base-only protection by 0.0551, 0.0708, and 0.0354.Base-only variants achieve near-perfect retention by suppressing acquisition; single-timescale AFM is nearly identical to full AFM at this coordinate.Source: §6.3.1 · p. 84, Table 45 · p. 84

6.3.2 Execution-level audit

• Across 45 protected runs, AFM completed 41,999 of 44,303 finite-restoration attempts, kept accepted endpoint error below 10−6, and satisfied the persistent-base progress floor.CLAD-C exceeded a tighter diagnostic in 11 of 22,932 accepted events without violating the analytic bound, and routing audits withheld the positive route-identification conclusion there.Source: §6.3.2 · p. 84

6.3.3 Runtime bottleneck analysis

• Protected-projector construction is the largest CLAD-C host-time bottleneck at 37.89%, while candidate fitting is slowest per call but contributes only 1.86% because it runs 37 times.Compact-cardinal shield construction contributes 20.58%; profiling uses low-perturbation host timing and characterizes the frozen algorithm rather than one deterministic CLAD-C trajectory.Source: §6.3.3 · pp. 84–85

6.3.4 Controlled mechanism tests

• Controlled tests produced route splits for observable within-route shifts, reopening under semantic conflict without unsupported splits, and no resolution under identical observations.The transfer-conflict diagnostic was non-informative because no transfer attempt occurred on any of its three seeds.Source: §6.3.4 · pp. 85–86, Table 47 · p. 86

Conceptual distinctions

• AFM measures persistent plasticity against the actual same-state endpoint accepted without protection, separating requested assimilation η_t from the compatibility fraction κ_t.A large η_t does not imply that an incompatible gradient can be stored persistently.Source: §7 · p. 86

Scope of guarantees

• Exact finite restoration and comparator-endpoint reproduction do not establish storage of the unrestricted trajectory or population-level retention.Outside the finite residual support, future behavior is governed by the protected base; finite Pedestrian observations can remain exact while unseen Pedestrian behavior is overtaken by classifier interference.Source: §7 · p. 86

Information limits

• Task-free routing is exact only under observable separation, because no learner can identify a semantic distinction that induces the same observation law.AFM controls false route refinement with separately allocated sequential evidence and does not use evaluator boundaries as learner-visible information.Source: §7 · p. 86

Empirical validation

• At η = 0.50 on n = 5 frozen CORe50 trajectories, full AFM increased plasticity, online accuracy, and held-out final-test accuracy over base-only variants that suppressed acquisition.The single-timescale variant was nearly identical on this finite stream, while failed finite-restoration attempts were rejected rather than committed.Source: §7 · p. 87, Table 48 · p. 87

Limitations and future work

• AFM’s main practical limitation is computational overhead concentrated in protection geometry, finite counterfactual completion, endpoint verification, and repeated protected-projector and shield construction.The bounded-resource theorem concerns structural state and explicit precision, not low wall-clock cost; optimization must preserve the same-state comparator and certified transaction.Source: §7 · p. 87

Interpretation

• Functional compatibility controls the local persistent-learning frontier, while retention constraints determine its use and nonlinear geometry limits how far matched interventions extend.Compatibility is not a universal determinant of persistence independently of retention allowance, method, or finite curvature; the study complements replay, regularization, gradient projection, and plasticity-preservation approaches (Schwarz et al., 2018; Störk, 2026; Abbes et al., 2026).Source: §7 · p. 88

• No specific funding was received for this work.Source: Funding · p. 88

Data availability

• All datasets came from publicly available original sources, and no new primary dataset was generated or redistributed with the submission.Numerical summaries underlying the figures and tables are provided throughout the paper.Source: Data availability · p. 88
• Saved checkpoints and row-level derived outputs are available from the corresponding author on reasonable request, while released code provides reproduction procedures and configurations.Source: Data availability · p. 88