SoftModel: A Neural Model That Grows Its Own Topology -- Governed Structural Growth for Continual In-Service Learning

1.3 The system

• The released SoftModel implements governed, recursive growth through widening, hierarchical refinement, composition, input-interface growth, and re-founding, with direction selected by model signals.Its self-shaping substrate grows non-uniform tree height and scope depth, supports transformer hosts, and exposes an operator algebra whose exactness is measured.Source: §1.3 · p. 6, §1.4 · p. 7

1.4 Contributions and epistemic stance

• The system extends the same lifecycle to local solving loops, growable attention, evaluative learning, consent-gated self-study, reproducibility, drift handling, and LLM-operated end-to-end use.These mechanisms remain budgeted, audited, evidence-triggered, and gate-adjudicated rather than imposed by designer fiat.Source: §1.2 · p. 6, §1.4 · p. 7
• The paper treats its claims as bounded by a pre-registered evidence program that reports failures prominently, including failed value-of-growth predictions and regime-dependent composition benefits.It distinguishes demonstrated mechanics and governance-by-construction from open comparisons and does not claim broad superiority over immobilization methods.Source: §1.4 · pp. 7–8

1.5 Organization and reader’s paths

• The introduction frames Part I as theory and system, Part II as benchmark-based empirical evidence, and Part III as discussion and conclusion, with appendices carrying protocols and audit artifacts.This section is the paper’s map of where claims, measurements, and process evidence are located.Source: §1.5 · pp. 8–9

2.1 The axiom

• Total plasticity requires every parameter at every structural scale to remain trainable, with effective optimization pressure and no disguised freezing.The axiom therefore includes structural plasticity, not merely movable weights.Source: §2.1 · p. 9

2.2 Lock-in: fixed topology is freezing by omission

• Fixed topology confines weight learning to F(G), creating irreducible capacity floors, impossible unseen-input hypotheses, and omitted growth directions that can cap later capability.The paper calls this structural freezing by omission and presents first-batch lock-in as a conjecture, while noting that its direct kilobyte-scale induction did not bind.Source: §2.2 · p. 10

2.3 The stability dilemma and its governed resolution

• Governed growth replaces immobility: every parametric or structural change must be harmless when applied and strictly improve held-out reality before promotion.Exact-preserving operators leave the current function unchanged, while a reality gate compares speculative candidates against the committed incumbent on quarantined recent data.Source: §2.3 · p. 11
• The gate-soundness invariant guarantees that each promotion improves the incumbent on its evaluation slice, while each refusal leaves the served version unchanged.This is an adoption-level guarantee, not a forgetting bound or regret guarantee.Source: §2.3 · p. 11
• Exact structural edits preserve accumulated fit and leave the function unchanged at application, while the held-out gate guarantees only that adopted changes did not underperform the incumbent then.This is an audit property under drift, not parameter immobility or omniscient stability.Source: §2.3 · pp. 11–12

2.4 Growth as governed speculation

• Structural growth is governed speculation: budgets constrain resources, audits record lineage, and verdicts reject progress signals classified as false within one continuing life.Growth remains need-driven and evidence-adjudicated rather than an architecture search over separate training runs; added parameters remain recursively trainable.Source: §2.4 · p. 12

2.5 Falsifiable predictions

• The preregistered program tests scale-localized adaptation, recovery from capacity lock-in, noise robustness, and amplitude inversion as a warning for re-founding, while reporting failures prominently.The predictions include T1–T4 and are adjudicated in Table 10, with outcomes and boundaries collected in §13.5.Source: §2.5 · p. 12
• At the tested kilobyte scale, T2 failed because the scenario did not induce capacity lock-in; the capacity claim therefore awaits scenarios where the approximation floor binds, unlike the structural interface case.This negative result is explicitly retained as a boundary on the section’s theory.Source: §2.5 · p. 12

3.1 A substrate closed under refinement

• Recursive self-reference makes any node refinable, with inner networks adding corrections at arbitrary levels; the resulting depth profile records where experience demanded refinement.The released implementation treats composite nodes as additive recursive refinements, so structure reflects the model’s history rather than a fixed depth hyperparameter.Source: §3.1 · p. 13

3.2 One learning primitive at every level (every scale)

• A single scale-invariant learning primitive trains each scope locally while handing composite nodes residual targets, giving every scale optimization pressure unless an enclosing exit weight vanishes.Inner outputs are held constant within-scale, while each subnetwork recursively takes the same training step on the correction arriving through its attachment.Source: §3.2 · pp. 14–15

3.3 Where to grow: the instability signal

• The instability signal identifies contradictory update directions as candidate growth sites, while trajectory verdicts can trigger remediation; both instruments suggest actions, but only the gate decides.No descent or convergence guarantee is claimed: the rule is characterized empirically through bit-exact trajectories, gradient checks, and behavioral testing.Source: §3.3 · pp. 15–16

3.4 Direction-neutral operators: the interface and re-founding

• σ adds inputs without changing the forward map, while Φ rebuilds an outgrown foundation from real stored rows; adoption remains gated, and the tested re-founding trigger lacked supported necessity.σ earns new-coordinate participation through training, whereas Φ avoids inherited errors and first-batch lock-in; takeover safety is unconditional because it is enforced by the gate.Source: §3.4 · p. 16

3.5 Governance formalized

• The governance invariant promotes only candidates that strictly improve a fresh held-out slice, leaves refusals untouched, and applies uniformly to learning, growth, interface, and re-founding changes.Selection reuse and corrupted holdouts remain outside the guarantee, so margins, query budgets, rotation, and provenance determine whether evidence remains trustworthy.Source: §3.5 · p. 17

4 Growth: The Direction Question

• The system grows either by additive expansion or composed hierarchy, and reads the choice between them from scope data.The additive realization is the adaptive system’s widen-only mode, while the compositional realization augments it with scope-interior composition.Source: §4 · p. 18

4.1 The operator algebra

• The operator algebra enforces exact entry, trainability from step one, parameter budgets, audit logging, and exact removal, with adoption always decided by the gate.The additive family preserves composition degree; δ alone raises it, marking the boundary between multiscale organization and true deepening.Source: §4.1 · p. 18, Table 1 · p. 19

4.2 The additive direction: refinement and widening

• Refinement organizes additive multiscale growth without enlarging the reachable function class beyond matched-width widening.Proposition 4 states F(Gρ) = F(Gω) without composition blocks, while refinement supplies localized, stagewise-additive credit assignment, growth sites, triggers, and audit units.Source: §4.2 · p. 20

4.3 The compositional direction and its selection

• Composition appends zero-initialized blocks that enter exactly but can create functions outside the additive class, making deepening a genuine second growth direction.δ can deepen any scope recursively, preserves additive cross-node aggregation, and remains removable and budgeted.Source: §4.3 · pp. 20–21
• Direction selection predicts from history, verifies with probes when necessary, and adopts only through the held-out gate; measured two-horizon slope, not single-horizon price, identifies delayed compositional payoff.Single-horizon gain and gain-per-parameter readings carried no axis information, whereas long-horizon minus short-horizon gain distinguished front-loaded additive from back-loaded compositional gains.Source: §4.3 · pp. 21–22

4.4 Regime containment and the growth mode

• The widen-only mode is an exact additive regime that bypasses direction-selection machinery, while mode changes affect future growth policy rather than the existing model function.The adaptive and additive paradigms share governed adoption, and the active mode is recorded in every decision record.Source: §4.4 · pp. 22–23

4.5 The scale hierarchy: subordination as a growth invariant

• Refinement is permitted only when the fine-scale body remains subordinate in size, host mass, and training time, with policy violations logged rather than silently drifting.The default host-to-body parameter-mass ratio is two orders of magnitude; hosts below a minimum trained mass cannot refine, and structural events wait for sufficient base training.Source: §4.5 · pp. 23–24

5.2 Realization I: grown cycles — the loop operator (λ)

• The λ operator grows bounded fixed-point computation, with enforced contraction, exact zero-entry behavior, audited iteration cost, removable structure, and no asserted enlargement of reachable function classes.Relaxation stops at tolerance or hard cap Kmax; violating output weights are projected back to the certified contraction bound.Source: §5.2 · pp. 24–25

5.3 Realization II: self-processing units

• Self-processing units locally refine newborn weights before global learning, using label-blind consistency with a collapse guard under bounded, serving-pure, optimizer-neutral, and audited execution.Released defaults use K = 4 perturbation copies, γ = 1, and ρfloor = 0.5; execution follows process, then participate, then learn.Source: §5.3 · p. 26

5.4 One discipline, two solved quantities

• The two realizations solve different quantities on different clocks, yet both use one-scope locality, feasible hard caps, neutral or exact entry, ledgering, and policy-controlled adoption.A model may carry both mechanisms, but neither depends on or knows of the other.Source: §5.4 · pp. 26–27, Table 2 · p. 26
• Iteration enters growing models as a governed local resource: each loop solves only its scope’s state or newborn weights without altering global learning governance.The two mechanisms are independently implemented and enabled, but share locality, bounded budgets, auditability, and gate adoption.Source: §5.1 · p. 24, §5.4 · p. 27

5.5 Epistemic status

• The section establishes mechanism and governance rather than empirical superiority, reporting exact entry or neutrality, contraction certification, verified loop gradients, and optimizer neutrality while value adjudication remains open.The released system includes pre-registered examination programs for both mechanisms.Source: §5.5 · p. 27

6 Growable Attention: Capacity Follows Demand

• Attention growth removes fixed equal head widths through head-add and head-widen, producing unequal capacities as lifecycle outcomes under evidence-triggered, exact, budgeted, and gate-adjudicated governance.Standard attention freezes both per-head capacity and local accountability; the proposed operators treat demand as the determinant of head width.Source: §6 · p. 27, §6.1 · p. 27, §6.2 · p. 27

6.2 Exact growth of heads

• Head-add and head-widen preserve the function exactly by placing zero factors on new pathways, imposing one training step of gradient latency while rejecting all-zero widening as permanently unreachable.Head-add contributes bitwise 0.0 at application, while widening remains within 10^-12 deviation; absorbed normalization and the additive output identity preserve exact composition.Source: §6.2 · p. 28

6.3 Evidence and decision

• Growth decisions use per-head instability and loading signals: widen an aged outlier, add when participation is broadly distributed, and let a held-out gate accept or discard the exact proposal.Newborn heads are excluded until minimum age, and the released clean runs separate probe training from holdout adjudication without changing verdicts.Source: §6.3 · pp. 28–29

6.4 The attention discipline: a third local objective

• An entropy-band objective locally penalizes collapsed and diffuse attention, keeping rows trainable; induced collapse fell from 94.4% untreated to 39.6% at λ = 2.0.The discipline uses per-row bands proportional to admissible-position entropy, routes its gradient locally, and prevents uncontrolled upstream leakage.Source: §6.4 · p. 29

6.5 The exact-expectation SPU objective

• Replacing random masking with its exact expectation removes stochastic stopping and duplicate-mask pathologies at lower compute cost while preserving decisive-regime verdicts.The analytic expectation differs from empirical masking by less than one standard error; the mask estimator has sd/mean = 0.899, an 18.4% spurious-stop probability, and 12.2% degenerate draws.Source: §6.5 · p. 30

6.6 Falsifiable predictions and their verdicts

• Growth helped only when starved birth capacity met demand: held-out error fell 35% and 64% in firing seeds, but the gate accepted growth in just 2/5 seeds.The trigger fired in all five starved seeds, whereas probe self-injury degraded three trials; across adequately capacitated tasks, growth was neither needed nor harmful.Source: §6.6 · p. 30
• Local discipline restored 97% of lost held-out performance at roughly 7% of global-retraining cost, while analytic and mask objectives agreed perfectly in decisive regimes.The local method restored 97% versus 49% for global retraining; the analytic objective had zero variance and matched the mask form at decisive regimes, avoiding its quantified stochastic pathologies.Source: §6.6 · pp. 30–31
• Per-head instruments preceded held-out error by 4.9 evaluation periods on average in 11/15 shifted runs, but raw 2σ alarms fired in 13/15 no-shift controls.The lead was reliable at width 64 and noisy at width 16, establishing a measured boundary for governed anticipation rather than a standalone deployment alarm.Source: §6.6 · p. 31

7 The Governed Lifecycle

• The lifecycle factory keeps domain data inside each model’s versioned store, serves only committed versions, and makes adoption the earned outcome of a held-out gate.Working-state learning and growth are speculative; commit gates adoption, reset discards sessions, and rollback restores audited lineage.Source: §7 · p. 31, Figure 5 · p. 32
• Version-local full replay, recent-window drift adaptation, and quarantined consented self-study make forgetting, drift response, and gate contamination explicit lifecycle policies.Rejected candidates remain outside the served lineage, drift is judged on recent held-out reality, and self-study cannot access the holdout.Source: §7 · p. 32

8.1 One evaluative primitive, two loops

• Both structural preference and policy optimization use the same teachable model, while a paired-episode gate alone decides adoption and records provenance for rollback.The S-loop credits post-gate advantages to growth-move buckets, and the P-loop converts clipped-surrogate policy losses into teaching targets without reinforcement-specific machinery in the weights.Source: §8.1 · pp. 33–34

8.2 The quasi-static law extends to reward learning

• The policy loop satisfies the quasi-static verification law, matching an independent PyTorch twin to 5.3 × 10^-13–2.2 × 10^-11 function error and all nine registered authoritative component-and-outcome cases.A 40-update run matches parameter trajectories to 1.8 × 10^-15, while the grown-instant Adam trajectory matches to 7.1 × 10^-12.Source: §8.2 · p. 34

8.3 Experimental design

• Registered experiments isolate structural growth from compute and protocol using protocol-matched twins, pre-verification capacity and outcome-variance probes, disjoint evaluation seeds, and frozen budgets and criteria.The design uses five seeded lives per arm, ten for the LunarLander gate battery, with incumbent anchoring and task-consistent alternation.Source: §8.3 · pp. 34–35

8.4 Results: a three-regime law

• Where capacity binds, growth beats the protocol-matched twin, while capacity-sufficient alternating operation is cost-neutral and reward-only growth outside that regime incurs the only net transient cost.The capacity-binding criterion passes in 3/5 lives with mean +1.1; alternating operation removes 94% of the reward-only gap.Source: §8.4 · p. 35, Table 3 · p. 35
• Governance is exact in tested lives: untrained clones are refused in 5/5, all adoptions are audited, anchored references reduce policy drift in 5/5, and refreshers reduce taught-knowledge error 5.9–95×.The preference ledger tracks gate verdicts, including zero-gain refusals, while the two loops operate together on one model.Source: §8.4 · p. 36

8.5 Boundaries and negative results

• The record refutes optimizer-moment loss as the transient-cost mechanism, retires a saturated adaptation-speed metric, leaves Acrobot out of reach at this scale, and reports LunarLander’s long-run net effect as unresolved.Moment transplantation did not change the transient; Acrobot’s proximate mechanism was policy-entropy collapse, and LunarLander remained below protocol resolution.Source: §8.5 · p. 36

9 An LLM Operates the Model

• An LLM operates the model through self-documenting tools, while the small soft model stores domain-specific validated experience and keeps teaching, growth, evaluation, deployment, and rollback auditable.The factory exposes nine tools and the full system fifty-two, with initialization instructions covering holdout ordering, feature extraction, drift, and recovery; the platform targets small domain-specialized models, not foundation-model construction.Source: §9 · pp. 37–39

10 In-Service Continual Learning as the Natural Habitat; 10.1 Structural answers to the two failure modes

• The continual-learning habitat combines catastrophic forgetting and loss of plasticity, so governed structural growth adds capacity while replayed experience reviews old knowledge when shared representations would otherwise overwrite it.Growth enters at zero and preserves the served function at structural change, but structure alone cannot prevent trunk drift under strict domain switches; the model’s own experience store supplies self-contained review.Source: §10 · p. 39, §10.1 · p. 39

10.2 The anti-forgetting ladder; 10.3 Evaluation on the time axis

• The anti-forgetting ladder combines protection windows, demand-triggered growth, and spaced review courses, while intrinsic local rehearsal remains future work and evaluation follows each model’s longitudinal trajectory.Protection windows help at growth events but not long-horizon forgetting; repeated review is the measured retention counter, and longitudinal evaluation tracks retention, adaptation cost, service continuity, and audit trails.Source: §10.2 · pp. 39–40, §10.3 · p. 40

11.1 Benchmarks, fit, and protocol principles

• The benchmark protocol uses Split-MNIST, Permuted-MNIST, and Fashion-MNIST with interleaved replay, disjoint validation and test data, and preregistered capacity, resolution, and decline probes.Instrument sizes are calibrated only during probing and then frozen, with replay capped at 200 stored rows per past task and ten recent tasks on long lanes.Source: §11 · p. 40, §11.1 · p. 40

11.2 Capacity under sequential arrival

• On sequential arrival, the growing model beats the fixed control in 9 of 10 Split-MNIST seeds, matches both registered Permuted-MNIST lines in 5 of 5, and beats its controlled Fashion-MNIST twin in 3 of 5.Against the parameter-matched ceiling, mean accuracy is 0.580 versus 0.754 while the grown model averages 7,117 parameters versus 8,273; valueless clones are rejected exactly in 10 of 10 lives.Source: §11.2 · p. 41

11.3 Long-sequence plasticity

• At twenty permuted tasks, the fixed network loses up to 39 points of new-task learning ability, whereas the growing model’s decline ranges from −8.3 to +1.3 points and finishes higher in all five lives.At forty full-data tasks, first-pass measurement reveals a 7.6-point fixed-net decline versus the growing model’s [−2.6, +5.0] range against [+4.9, +10.8] for the fixed net, lower in 10 of 10 seeds.Source: §11.3 · p. 41, Figure 9 · p. 42

11.4 Review and the just-in-time refresher; 11.5 Boundaries

• Targeted review raises the growing model’s forty-task examination mean from 0.1818 to 0.2153 while the fixed control falls from 0.1718 to 0.1576, but Fashion dose boundaries and several open controls limit the claims.The growing model is higher in 10 of 10 seeds, while its long-ago band rises 3.9 points without disturbing recent expertise; task-free continual learning remains future work.Source: §11.4 · p. 42, §11.5 · p. 43

11 The Core Method on Standard Continual-Learning Benchmarks

• Across standard continual-learning benchmarks, governed growth improves capacity and long-sequence learning while preserving exact no-op behavior and enabling targeted recovery of earlier competence.Table 4 summarizes the registered benchmark criteria and outcomes; the reported comparisons include sequential arrival, twenty- and forty-task plasticity, and just-in-time review.Source: §11.2 · p. 41, §11.3 · p. 41, §11.4 · p. 42

12.1 The evidence discipline

• The program froze worlds, arms, metrics, and acceptance bars before verdict data, quarantined iteration to disjoint probe seeds, and reported failures as prewritten measured boundaries.Registered seed counts, per-seed aggregates, archived manifests, and bit-for-bit reruns supported the audit discipline, while library and gradient checks independently certified the hand-written implementation.Source: §12.1 · p. 44

12.2 The campaigns at a glance

• The campaigns verified exact growth operators and enforced budgets, gates, lineage, and holdout quarantine while testing topology growth, attention-head growth, self-knowledge, generalization, real-data drift, and transformer comparison.The self-knowledge series produced a mixed verdict, with self-reports real but their exploitation still open; an operating LLM drove the factory end to end.Source: §12.2 · pp. 44–45

13.1 Capacity follows demand

• Network size followed demand, growing 4.6–4.9× through 15–19 audited promotions, while the governed model overtook the fixed network on 4/5 verdict seeds.Against the budget-matched transformer, it also scored better on 5/5 seeds, surviving stricter governance at 4/5 and 5/5; Figure 10 reports the full trajectories.Source: §13.1 · p. 45, Figure 10 · p. 46

13.2 Stability from governance, not immobility

• The plastic arm preserved its educated canon at least as long as the frozen arm on every seed, with strict time-to-degradation wins on 3/5 and recovery wins on 4/5.The frozen arm’s periodic rehearsal never restored its decline, whereas plastic remediation remained available under surface drift and scheduled temptation.Source: §13.2 · p. 46

13.3 Growth governance is ex-post

• Ex-post governance was the workable form: the incumbent served untouched while a same-data grown candidate trained, then promotion occurred only after an audited score improvement.Affordable ex-ante probes could not observe capacity value, and the rollback anchor had to be immutable; otherwise a diverged incumbent could replace the last good state.Source: §13.3 · p. 46, Table 5 · p. 47

13.4 The audit tally

• The audit tally classified 35 of 64 claim rows as supported, 20 as boundary results, 4 as open FAIL branches, and the remainder as diagnostics, history, auxiliary evidence, or observations.Every number in this part was tied to a correspondence-table row.Source: §13.4 · p. 47

13.5 The depth-axis and continual-learning campaigns

• Continual-learning results separated the roles of structure and review: capacity alone did not prevent forgetting, review retained old knowledge, and growth plus review produced the best new-domain quality at weaker retention.Untreated sequential switching caused 552–1,362× degradation, while spaced courses reduced forgetting monotonically and in-service growth recovered prior competence in 2–3 batches versus 20–29 for a fresh model.Source: §13.5 · pp. 48–49
• Depth growth was parameter-efficient when composition was genuinely iterated, winning in 3/3 seeds at k=4 and k=6 with 58% of the width arm’s parameters, while k=2 favored depth in only 1/3.Whole-layer insertion preserved service bitwise at 36/36 instants, and controlled deepening survived three mid-life insertions without collapse; post-insertion transients remained a governance concern.Source: §13.5 · pp. 48–49, Figure 12 · p. 49

14.1 The observations

• Capacity pays differently across the lifecycle: un-starving the gate narrowed but did not close the oracle gap, while discrete memorization required parameters that smooth regression did not.The paired oracle gap changed from 0.046 to 0.034 mean and from 0.5014 to 0.5653 median score; smooth-regression NLLs were 1.38/1.36/1.34/1.38 across a 20× parameter range.Source: §14.1 · pp. 51–52

14.2 Scope and threats to validity

• The measurements are limited to kilobyte-to-tens-of-kilobyte models, seeded synthetic families, selected real-data and benchmark lanes, and small registered seed sets without significance testing.External validity is claimed only as scale-independent regularities within the tested families and scale; exam instruments and audits remain axis-limited.Source: §14.2 · p. 52

15 Discussion

• Governance absorbs operational risk through held-out gating, recorded lifecycle observability, and bounded structural plasticity, while remaining appropriate mainly for schema growth, sustained drift, and boundary-free change.Freezing remains sufficient when domains are fixed, stationary, and schema-closed; the matched-parameter comparison against wider immobilization methods remains open.Source: §15 · pp. 52–53
• The design assigns different structural operators to different timescales: detail and interfaces adapt continuously, composition depth grows only after scope signals certify additive stagnation, and re-founding is gated.The serving chain’s stage count is fixed at creation because representational depth changes slowly, while content within stages tracks drift.Source: §15 · pp. 53–54

16 Conclusion

• The conclusion presents a map rather than general superiority: mechanics are verified exactly, growth pays under genuine starvation, several predictions remain unresolved, and self-knowledge is real but not yet exploited.The paper explicitly names the failed strong demand-tracking result at this scale and the need for a non-injurious acceptance probe.Source: §16 · p. 55
• Governed growth preserves continued learning along long task sequences on standard continual-learning benchmark lanes, with dose boundaries reported.The evaluative contract also extends to structure preference and policy optimization under one gate.Source: §16 · p. 55

Reproducibility and AI-Assistance Statement

• The complete system, tests, specifications, seeded runs, benchmark stores, and verification logs are frozen or publicly archived so every reported number can be regenerated from committed artifacts.The paper states that language models assisted drafting, while the author verified claims against committed artifacts and used the evidence table as the binding contract.Source: Reproducibility and AI-Assistance… · pp. 55–56

Appendix A. The Pre-Registered Empirical Program; A.1 Protocol; A.2 What verifies exactly (the mechanics)

• The protocol preregistered hypotheses, arms, seeds, budgets, pass bars, and failure interpretations, and the acceptance suite verified exact operators, budget enforcement, lineage, gating, quarantine, consent, and the end-to-end factory loop.Experiments used three seeds with a ≥2/3 rule, later campaigns registered their own seed counts and bars, raw failures were retained, and all mechanical checks passed.Source: §A.1 · p. 64, Appendix A · p. 64, §A.2 · p. 65

A.3 Where the theory’s predictions stand (E-series)

• The tested theory predictions were mixed: fine-scale adaptation and σ’s large interface payoff were supported, while capacity lock-in, inversion early warning, and unfrozen robustness failed or remained unsupported.σ drove error to 1.4 × 10−3 versus a control floor of ≈5.7, but the speed criterion held in only 1/3 seeds; noise damaged the never-frozen coarse scale where governance was too thin.Source: §A.3 · p. 65

A.4 Self-knowledge is real; exploiting it is open (S-series)

• Self-knowledge signals were validated, including a variation self-check with correlations 0.68/0.58/0.60 versus 0.46/0.39/0.54 for perturbation sensitivity, but every tested self-improvement intervention failed or was demoted.The doubt map predicted true errors in 3/3 seeds, whereas question-directed teaching, scheduling, self-quizzing, and granted self-study did not meet their registered bars, with zero safety violations.Source: §A.4 · p. 66

A.5 The attention program (A-series)

• The attention drift instruments detected change earlier than ADWIN but incurred higher false-alarm costs, so the measured lead requires debouncing rather than serving as a standalone alarm.ADWIN detected drift in 32/32 runs, while the instruments led by a median 22 steps in 28/30 paired runs; the no-shift envelope fired in 13/15 runs and ADWIN’s false-alarm rate was 6/16.Source: §A.5 · p. 66

A.6 Real-data case study: prequential concept-drift streams

• The real-data attention system outperformed the like-for-like Hoeffding Adaptive Tree on INSECTS and NOAA while retaining returning regimes, but its attribution and scope exclude growth-event benefits where demand did not exceed capacity.INSECTS accuracy was 0.762 vs. HAT 0.599, with retention ≈1.00; NOAA accuracy was 0.768 vs. 0.739, and the system used less wall time than cited ensemble references on INSECTS.Source: §A.6 · pp. 66, 68

A.7 Generalization campaign (G-series)

• On length-generalization tasks, lifelong prequential learning closed the frozen training-length cliff, while SCAN remained at zero for unseen combinations and minimal birth capacity constrained symbol-heavy tasks.Addition fell from 0.65 at length 6 to 0.11 at length 7 when frozen, whereas lifelong paired deltas were +0.32, +0.08, and +0.30 across the three tasks; running maximum reached 0.70 vs. 0.26 for the fixed transformer.Source: §A.7 · p. 68
• Governed plasticity beat best-grid EWC on three of four tasks, but lost on stationary false-belief streams, revealing that the current lifecycle governs growth without annealing updates in calm regimes.The governed arm was better on addition, delayed copy, and running maximum, while EWC adapted better on false belief; λ=1000 over-pinned with retention ratios 0.03–0.12.Source: §A.7 · pp. 69–70

B.1 G-GROW-1: the open-inventory exact-answer battery

• Growth did not pay in the open-inventory battery: C-G1 was 0/5 in both rounds, while probe interference caused correct gate refusals and delayed harm followed the sole accepted proposal.Round 2 found oracle exam accuracy above fixed birth in 4/5 seeds (median 0.64 vs. 0.44) despite indistinguishable prequential accuracy; the accepted proposal later fell from 0.57 to 0.18 at three-quarter life.Source: §B.1 · p. 71

B.2 G-GEN-1: the generative battery, and the scope observation from its stationary companion

• On the generative battery, growth beat fixed birth in 3/5 seeds, tracked the oracle within band in 4/5, preserved calibration and recovery in 5/5, and beat the stock transformer in 5/5.Final median NLL was oracle 1.83 < grow 1.88 < fixed birth 1.98 < transformer 2.00, with 2–8 adopted growth events per seed.Source: §B.2 · p. 72
• The fair uncertainty rerun re-scoped the transformer comparison from 5/5 stock-convention wins to parity at matched uncertainty, while preserving governed growth’s static-quality sanity floor.The learned-head baseline beat the stock convention in 5/5 seeds, but against growth it was better in 2/5, with verdict medians 1.8195 versus 1.8196.Source: §B.2 · p. 72

B.3 Round 2 of the generative campaign: the silenced gate (X1)

• The silenced gate failed its frozen central criterion: stationary net adoptions were 1/0/1/2/0, while growing-track performance retained oracle and recovery criteria but lost the fixed-twin criterion.Controlled probes found capacity effects nearly invisible locally: 0.009 ± 0.017 versus approximately 0.1 nats long-run, making audit horizon and silence a measured trade-off.Source: §B.3 · p. 73

B.4 Round 2 of the open-inventory campaign: probe hygiene and the provisional gate (X2)

• In open-inventory round X2, probe hygiene un-starved adoption to 2–3 trials per seed, but C-G1 still failed; C-G2 improved to accuracy 4/5 and parsimony 5/5.Probe training damaged every trial, so adoption became untrained and function-preserving, with counterfactual auditing and rollback; late-third median score rose 0.5014 →0.5653.Source: §B.4 · pp. 73–74

B.5 Round 3, generative track: incumbent–candidate governance (X1b)

• Incumbent–candidate governance delivered zero-cost trials, bit-exact serving isolation, and non-regretted or rolled-back adoptions, while generative-track transformer comparison reached 5/5.A promotion–demotion anchor defect caused serving NLL 11.38; deferring promotion during demotion windows removed the explosion, yielding final-third 0.6533.Source: §B.5 · p. 74

B.6 Round 3, open-inventory track: the same governance under exam metrics (X2b)

• Under exam metrics, promotions passed bucket cross-entropy yet could harm exams because audits measured the wrong axis; exam-shaped audit metrics were therefore required.C-G1 was 0/5, while C-G2 passed with accuracy 3/5 and parsimony 5/5; serving isolation remained bit-exact.Source: §B.6 · p. 75
• Across gate rounds, governed growth beat budget-matched standard architectures under arriving structure, but mid-life capacity did not outrun lifelong-trained fixed twins within gate-round-length lives.The frontier reflects ex-ante impossibility and audit latency; 600k-row X-SAT lives bracket it from the other side, where within-life capacity comparisons cross.Source: §B.6 · p. 75

B.7 The size-adjustability round (X-SAT)

• X-SAT passed centrally: 4/5 moderate-regime seeds crossed and stayed ahead, with every seed reaching 4.6–4.9× the counterpart’s size and median crossover at 275,000 rows.Sustained overtake began at a median ~2.6× the counterpart’s size, while the extreme-pace round supplied the boundary conditions for the successful strategy.Source: §B.7 · p. 76

A Pedagogy for Lifelong Models

• The pedagogy specifies foundation quality, ordered sequencing, rhythmic review, contemporaneous re-realization, blended principle and skill, controlled exposure, self-examination, remediation, trajectory evaluation, and regime-scoped governors.Its empirical record supports some principles while leaving structural foundation protection, non-extractability, and individualized pacing as registered future measurements.Source: A Pedagogy for Lifelong Models · p. 77
• Review dose was non-monotonic: doubling a drill raised composition skill from 0.20 to 0.45, whereas six-fold drilling reduced it to 0.17 and damaged neighboring skills, showing exposure economics can bind.A skill reaching 1.0 in isolation plateaued near chance within the full curriculum, while frozen review surfaces left new forms to the ordinary stream and throttling reduced fidelity.Source: A Pedagogy for Lifelong Models · p. 77

D.2 Round 1 — baseline: no defenses

• Without defenses, coherent temptation drove the plastic learner to 0.32–0.37 true-label accuracy against a +0.41 defection incentive, establishing immunity as a prerequisite rather than an accessory.The frozen learner’s minimum-score result was confounded by an instrument that rewarded standing still.Source: §D.2 · p. 79

D.3 Round 2 — strategy: canon-anchored immunity

• Canon-anchored self-examination raised resistance and fidelity by +0.10 at approximately 11% vigilance cost, but remained partial because lagging and leading alarms cannot respectively prevent damage or distinguish corruption from novelty.The reused arms reproduced Round 1 bitwise, and the residual failure localized to transition churn at surface rotations.Source: A Pedagogy for Lifelong Models · p. 77, §D.3 · p. 79

D.4 Round 3 — strategy: a calm governor

• A calm governor engaged cleanly without harming adaptation, yet permanent surface rotation offered no stillness to harvest; disabling growth confirmed throttling caused the fidelity loss.The result supports regime-scoped governors: calm helps genuinely stationary worlds, whereas tracking costs cannot be bought back in perpetual drift.Source: A Pedagogy for Lifelong Models · p. 78, §D.4 · p. 79

D.1 The question, the world, and the instruments

• The program corrected minimum-score bias by using self-referenced decline, gated comparison on education solidity, and found the frozen substrate missed 0.90 on one seed while the growable arm exceeded 0.93 across six calibration curricula.The corrected verdicts were 3/5 for slope, 5/5 for time-to-degradation, and 4/5 for recovery; under the old instrument the thesis scored 2/5 and standing still reached 0.9083.Source: §D.5 · pp. 80–81

D.5 Round 4 — strategy: correct the instrument, then teach properly

• Under the corrected instrument and App. C pedagogy, the growable arm declined more slowly, lasted 2.25–7× longer on three seeds, recovered faster in 4/5 seeds, and maintained a positive median own slope of +0.0045 per 103 rows.The frozen arm’s periodic rehearsal never restored its decline, whereas remediation on the growable arm used always-on plasticity; Table 9 reports the per-seed grading behind these results.Source: §D.5 · p. 80, Table 9 · p. 80

Appendix E. Validation by an Operating LLM

• The operating-LLM validation achieved 16/16 exact-answer generalization on unseen operand pairs, improved gated teaching from 0.60 to 1.00, rejected garbage without altering the incumbent, and recovered drift from 0.10 to 1.00.The same factory produced a fused meteorology×marketing×calendar model at 0.90 versus 0.05 for its best single-domain ablation, recovered water phase boundaries, and exposed multiplicative-law precision as a boundary.Source: Appendix E · p. 82

A Topology Growing under Governance

• The released system grew from 16 units and 65 parameters to a tree-height-4 topology with 13 composite nodes and 599 parameters, while the gate promoted only two commits and refused three.Growth used report-ranked ρ refinements, ω widenings at root and inner scopes, and one σ interface growth; refused states never reached the served model.Source: A Topology Growing under Governance · p. 83, Figure 16 · p. 84

Appendix G. The Historical Depth Route (retained as record)

• The historical scalar-width compositional route failed its pre-set capability bar at 1/3 seeds, motivating the fullwidth interface and superseding that evidence with exact serving, controlled deepening, and fixed-budget depth advantage.The retained failure comprised one large advantage and two losses; later evidence is attributed to the whole-layer operator under the prevention stack.Source: Appendix G · p. 85

Appendix H. The 2026 Depth-Axis and Continual-Learning Campaigns

• The campaigns used pre-registered designs, machine-checked served interfaces, write-once run records, recomputed verdicts, and three seeds per arm, with two library defects fixed under change control and affected arms rerun.Superseded rows were retained, and each adopted row was anchored to library tag depthgrowth-v1.7.Source: Appendix H · p. 86
• The 2026 campaigns establish governed continual learning with exact serving continuity, successful depth realization, cyclic-drift savings of 7–10×, and explicit failures at registered boundaries.E-1 reached depth in 3/3 seeds with final parity in 2/3; E-12 achieved relearning savings of 7–10× in 3/3, while E-8’s same-input-region retention bar failed 0/3 and E-7’s hypotheses failed 0/3.Source: Appendix H · pp. 86–87

Appendix I. Relation to Other Methods

• The related-work comparison claims an unoccupied combination of in-service evidence triggers, exact preservation, uniform held-out adoption, and deployment-time operation.The survey positions individual axes as having near relatives while claiming no prior system occupies all four together.Source: Appendix I · p. 88
• Unlike prior growth, adaptive-depth, NAS, and continual-learning methods, this system grows permanent non-uniform structure during one continuing life and applies its sole reality gate uniformly to gradient and structural change.Its depth is unrouted and lifelong; growth proposals can be refused, and the model remains under construction throughout deployment.Source: Appendix I · pp. 88–89
• The λ cycle and newborn self-processing units are governed local resources: cycles are grown, contraction-certified, removable, and gate-adopted, while self-refinement is bounded, label-blind, enrollment-only, and serving-pure.This contrasts with fixed recurrent placement, unconstrained recurrent mutations, and test-time adaptation of an already trained model.Source: Appendix I · p. 89
• The paper frames governed per-head growth and local entropy restoration as constructive, deployment-time counterparts to pruning and global prevention, measuring 97% recovery at approximately 7% of global-control cost.It also reports review-scheduling laws, spaced repetition, relearning savings, tool-indication boundaries, and delayed-payoff slope as contributions to continual-learning questions.Source: Appendix I · pp. 90–91

• The evidence block is organized around three readings and a table to which every paper claim is accountable.Source: Appendix J · p. 92

J.1 Reading the map

• Mechanically, governed growth verifies exactly; its value is positive in transformer comparisons and the size-adjustability crossover, with remaining questions localized to specified scenarios.Open cases include a lock-in task where the floor binds, trigger signals for Φ, noise regimes requiring stronger governance, and a within-life comparison bounded by L17.Source: §J.1 · p. 92
• E10 shows the axiom’s cost when the gate side is too thin, making the governance pair both structural evidence and a limit the presentation must respect.Source: §J.1 · p. 92
• Pre-registration makes negative results informative because each FAIL branch identifies the prediction, bar, and scale under which it failed.Interpretations were fixed before runs, so negative branches remain specific and revisable rather than ambiguous.Source: §J.1 · p. 92

Evidence-table design

• Table 10 records every numbered claim, prediction, and observation exactly once, linking each to its artifact, registered verdict, key number, and status without allowing prose to outrun the evidence.Open items remain open, boundary items name their observation, and motivation rows are retained historically rather than cited as evidence.Source: Table 10 · p. 92, §J.2 · p. 92

Substrate and verification

• Exactness and auditability are supported across application acceptance, budget enforcement, lineage auditing, self-knowledge checks, bitwise insertion tests, finite-difference gradients, and additive-release equivalence.The evidence table records 36/36 supported insertion instants and bit-identical widen-only behavior.Source: §J.2 · p. 93

Structural growth

• Governed growth evidence supports demand-tracking capacity, exact structural operations, depth advantages, and end-to-end topology growth, while some stronger saturation and capability bars remain unmet or bounded.The table also records parity with born-deep models and no collapse in tested cases.Source: §J.2 · p. 93

Growable attention

• Attention experiments support demand-tracking growth, greater effectiveness of local discipline than global repair, and instruments leading loss, but the strong local-discipline claim is refuted under starvation.Instrument signals were confirmed as a median tendency rather than a standalone alarm, with +4.9 periods mean, 11/15 positive cases, and a no-shift control firing 13/15 times.Source: §J.2 · pp. 93–94

Stream cases

• On continual streams, the system is reported as competitive on a modern drift benchmark, adapts without forgetting on returning regimes, and achieves PASS 0.762 versus HAT 0.599 and NOAA 0.768 versus 0.739.Source: §J.2 · p. 94

Lifecycle governance

• Lifecycle evidence supports gated promotion and refusal, holdout quarantine, governance outperforming immobilization under drift, and growth paying over a same-education fixed birth when capacity binds.The table also records that capacity binds on exams rather than prequential evaluation and that growth is needed only where the world grows.Source: §J.2 · p. 94

Growth-value campaigns

• Growth-value campaigns report paired gains of +0.32, +0.08, and +0.30, while also recording a stationary miss, a 0.64 versus 0.44 result, and multiple explicit failure branches.The ex-post gate is reported to un-starve the exam world, and gate refusals are described as deep and correct despite tolerated adoption harms.Source: §J.2 · p. 94

LLM-operated factory

• The LLM-driven factory lifecycle passes key end-to-end checks, including 3/3 accepted instants, 1.00 generation, garbage rejection, and domain scenarios scoring 0.90 versus 0.05, with attribution and recovery boundaries reported.A recorded artifact size is 2,404 B, and the domain result includes a 2.5/97.5 °C interval.Source: §J.2 · p. 95

• Review and retention effects are structured rather than generic: targeted review beats random review at equal volume and diversity, while spaced-repetition course count supplies the retention dose.The evidence table also attributes sequential-switch forgetting to review rather than windows or capacity and isolates a repetition confound.Source: §J.2 · p. 95
• Several registered mechanisms fail or remain constrained: window and capacity arms fail as registered, volume dose is monotone from 5–25%, and discrete memorization binds parameters.The table also reports a 0/3 refutation of the ramp claim, exposure economics binding in-curriculum, and a transition dip as the cost of tracking.Source: §J.2 · pp. 95–96
• The GUARD evidence supports partial mitigation rather than full protection: re-anchoring recovers half of immunity at an 11% cost, while the central claims remain unsupported in GUARD r2.A separate observation reports GUARD r1 at 0.32–0.37 versus incentive +0.41, and GUARD r3 at 0.67 →0.51.Source: §J.2 · pp. 95–96
• Growable systems beat frozen substrates on the education thesis: the exhibit reports 0.93 accuracy for growable versus 0.90 missed by frozen across 6/6 curricula.The broader result reports slope support in 3/5 cases, time-to-decision support in 5/5, and recovery support in 4/5.Source: §J.2 · p. 95
• Governance is regime-dependent: in non-growing worlds growth is unnecessary, while growth can produce 8/8 regretted adoptions and widening can preserve a decision even as exam performance falls 0.04.The evidence table also reports 0 versus 2–8 adoptions in the relevant comparison and identifies transition dips as the price of tracking.Source: §J.2 · p. 96
• Capacity value is ex-ante unobservable: observed outcomes distinguish growth from non-growth, but instantaneous or online gates miss lifetime consequences and small-scale economics dominate.The table also reports G-GROW-1 observation values of 0.64 versus 0.44, while §14 L1 says exams see capacity but online scores do not.Source: §J.2 · p. 96

J.3 How to audit a row

• The campaign column directs each row to the appendix section containing its registered acceptance, per-seed values, and probe-ledger disclosures.Source: §J.3 · p. 97
• Evaluative-learning and benchmark rows instead trace through the Reproducibility Statement, whose battery identifiers name registered design documents and per-run JSON stores, including adjudicating-lane values in Table 11.Source: §J.3 · p. 97
• Each result store is a write-once run directory with resolved configuration and content hashes, so the table’s key number traces in two hops to a hashed artifact on disk.Source: §J.3 · p. 98

Appendix K. Glossary of Named Quantities and Roles

• The glossary specifies that every role is a replaceable part behind the registry (§4.3), while every quantity is recorded in the audit artifacts.Source: Table 12 · p. 99