ghca

A study of local timescales in Greenberg–Hastings cellular automata, and an exploratory program using GH excitable dynamics as the substrate for a reward-driven learning mechanism on a graph.

Two strands

1. Lattice GH cellular automata (original). Excitable-media dynamics on a 2D lattice with an (active, passive) refractory cycle, plus tools to enumerate which initial configurations self-sustain ("persistence probability" over (active, passive) parameter space).

2. Learning on a GH network (in progress). GH dynamics generalised to an arbitrary weighted graph, with per-node timescales, as the substrate for a learning mechanism. The framing is inside-out (Buzsáki): the medium generates its own repertoire of dynamical patterns, and a strict scalar reward (Sutton) selects and stabilises a subset through action. Memory, attention and executive function are treated as categories read out from one homogeneous substrate, not as built-in modules.

Files

File	Purpose
ghca_main.py	Lattice GH substrate: Population class, run, plot, animate
ghca_core.py	Encode/embed integer configurations, run + animate
ghca_plot.py	Persistence-probability maps over (active, passive) space
ghca_net.py	GH dynamics on a graph: per-node timescales, weighted-threshold excitation, spontaneous firing, homeostatic threshold; topology builders and order-parameter observables
ghca_learn.py	Reward-modulated learner: eligibility-trace conduction (Line A) and timescale (Line B) plasticity, order-parameter critic, layered-graph builder
ghca_plasticity.py	Multi-axis closed-loop engine: tri-axis ($\tau$-adaptation, $\theta$-homeostasis, $W$-routing) substrate adaptation
ghca_causal.py	Causal instrumentation (C-series): partial-observation S_obs, wave variables W=f(S), and do(S) / do(W) / do(θ) intervention operators
experiments/e0_characterization.py	E0 — substrate characterisation (find the self-sustaining band)
experiments/e1_conditioning.py	E1 — stimulus→response conditioning (A-vs-B dissociation)
experiments/e2_delayed_response.py	E2 — delayed response / working memory (τ-controlled memory)
experiments/e2_information.py	E2 addendum — memory as a τ-tuned information-destruction rate
experiments/topology_cycle_capacity.py	E2 addendum — cycle-space (circuit-rank) bound on how many reentrant loops a topology admits
experiments/topology_cycle_packing_exact.py	Correction to the above — its greedy maximises cycle length, not count; exact ILP packing, certified on the ring
experiments/persistent_set_structure.py	Does the persistent set have a tractable description? Not linear; combinatorial when tau_p <= tau_a (companion to SSRN 4047679)
experiments/phase_space_polytope.py	Addendum: exact hulls in direct phase-space coords — small saturating outer polytope, necessary-not-sufficient; (1,1) conservation law
experiments/persistent_set_3x3.py	The gap signature generalises to 3x3 as a plaquette multiset — 40M configs, zero impure at tau_p <= tau_a, 2096x compression
experiments/persistent_set_dec.py	Discrete-calculus ladder: curl (winding) insufficient, ALL relative-phase invariants insufficient at tau_p > tau_a; threshold = + absolute state multiset
experiments/spectrum_mechanism_hunt.py	Theory hunt: 3 mechanism candidates falsified (bridging, lumpability x2), 3 exact laws found (period law T = k*S + d, spectrum constancy on attractors, dwell = f(spectrum))
experiments/spectrum_automaton.py	The spectrum automaton is fate-exact on pure classes (12/12 cells): at tau_a >= tau_p persistence == finite-state z-reachability; mixed classes appear exactly at tau_p > tau_a
experiments/spectrum_automaton_3x3.py	Scaling: fate-exact at 3x3 exhaustively (40M configs, 0 leaks) and 4x4 sampled (0 leaks); soundness FAILS at tau_p > tau_a on 3x3 — 2x2's universal soundness was a smallness accident
experiments/live_subshift_entropy.py	Live-SFT entropy: predicted regime signature FALSIFIED at 3x3 (strict and diagonal cells share h=1.8826, via an identical 8-node scrambled-core SCC); config entropy 0 everywhere — SFT entropy measures the abstraction's fibre ambiguity, not the dynamics
experiments/spectrum_sufficiency_certificate.py	Machine checks for docs/spectrum_sufficiency_proof.md: certificate schema (ING-1/2/3) proves sufficiency at (1,1)/(2,1); death == simultaneous double-dwell (P4); signature moves only via dwell (P5); regime = ING-3 alone (P6)
experiments/viz_gh_vs_symbolic.py	Animation: the medium, its space-time raster, and the symbol automaton side by side for a live and a dead run — the double-dwell death and the L2 class-exchange, visible
experiments/clock_shift_merge.py	The regime law reduces to clock-shift merging: rearrangement-invariance is universal; clock-shift invariance == the regime; at tau_a >= tau_p EVERY orbit merges with its clock-shift (100%)
experiments/clock_shift_healing.py	Lemma R (proven): step == clock-shift iff dwell-free; clock-shift = time-step on live attractors (S+1 >= 4); LIVE healing has the exact closed form tau_a + 2*tau_p + 1 (12 cells), witnesses = one young-wave family; 2S saturators are all dead drains
experiments/damage_relaxation.py	Discrete perturbation theory (damage spreading): the clock-shift is a zero mode; at tau_a >= tau_p every scattered damage relaxes back in exactly tau_a + 2*tau_p + 1; never-relaxing pairs == split-fate pairs, one for one — the regime law as a damage-healing transition
experiments/damage_relaxation_3x3.py	Scaling: the transition survives at 3x3 (split-fate = 0 at every tau_a >= tau_p cell, up to 10M configs) but the 2x2 closed form does NOT lift (relax times 9/12/10/14, topology-dependent) and the right criterion is MERGE, not return-to-uniform ((1,1): 14,052 never-uniform pairs, all merged)
experiments/dwell_debt_confinement.py	The confinement variable: damage == 1 + dwell-debt (exact identity); live debt confined (2x2: ceil(S/2), 3x3: S) at tau_a >= tau_p vs unit-slope drift T-k (k=1 at (1,2), k=2 at (2,3)) at tau_p > tau_a — the regime law as debt confinement vs escape
experiments/debt_streak_lemmas.py	Whiteboard attempt on the confinement lemma: clean 2x2 skeleton (streak = window = debt = ceil(S/2)/S laws) FALSIFIED at 3x3 (streaks 5/6/8/6 vs debt 2/3/4/4) — confinement is episode overlap, not short streaks; fifth 2x2-smallness artifact
experiments/debt_anchor_gradient.py	The open lemma decomposes: gradient bound (regime-independent, G = ceil(S/2) at 2x2) + ANCHOR LAW (the regime: at tau_a >= tau_p some cell always has debt exactly 0, both lattices; anchor escapes to 100+ at tau_p > tau_a) — confinement <= A + diam*G
experiments/anchor_law_blackboard.py	Blackboard ladder B1–B5 on the anchor law: debt floor exactly −1 = merged cells; merge/unmerge events classified (balance exact at dwell-free cells, +16 in-flight at (1,1)); the regime-carrying atomic fact isolated (a merged receptive cell never sees an active v-neighbour — fails at tau_p > tau_a); live shapes rigid (≥2 zeros, ≤1 merged, no (−1,≥1) edge) — ⚠ ≤1 merged is a 2x2 artifact: up to 6 merged at 3x3, see the coherence_invariant_scope.py row below
experiments/anchor_law_certificate.py	"Theorem 4": anchor law + merge CERTIFIED at (2,1) by a non-circular 276-state pair-map certificate — an exhaustive check over that closure, not a hand proof; the name is historical — plus the compression barrier: seven local ledger invariants all leak (3,728…1,216 violations); the wave-coherence structure is necessary for any local proof
experiments/coherence_invariant_scope.py	Scoping the coherence invariant: Theorem 4's certificate METHOD scales to 3x3 (re-certified by full re-enumeration at 3x3 (2,1), 483,446 pair states, converges in 4 steps — no argument transfers the result; it is checked again, not lifted); no clock-shift quotient exists (60/8,100 mixed keys — absolute phase is load-bearing); <=1-merged is the sixth 2x2 artifact (up to 6 merged at 3x3); two-zero rigidity survives
experiments/coherence_formulate_2x2.py	Formulation step 1: orbit co-membership falsified both directions; +1-LOCKSTEP proven necessary (all 276, within 4 steps) but not sufficient (468 lockstep pairs; the 192 extras are archived as the counterexample set for every future candidate)
experiments/coherence_window_S.py	The coherence invariant, found: coherent = within S steps downstream of an exact clock-shift state; window exactly S at all 8 tested 2x2 cells AND at 3x3 (2,1) (483,446 pairs, depth 3) -- every tested cell has tau_a >= tau_p, which is the regime the saturation argument assumes; the claim is not tested at tau_a < tau_p; separates 276/0 vs the 192 impostors; open lattice-free obligation: window saturation F^{S+1}(diag) ⊆ F^{<=S}(diag)
experiments/coherence_window_saturation.py	Saturation mechanised: rigid age law (+1 / reset / hold only at ages 0, 1, S); on-orbit recurrence FALSIFIED at all 8 cells (gaps 5–12 vs S 2–8); every hold is a younger preimage merging in (12/12, 40/40 at the ceiling) — the obligation is a covering lemma
experiments/coherence_covering_lemma.py	The covering lemma reduced to atoms: GH non-injectivity is exactly 0<->S swaps (proven); every age-hold is witnessed by a pure swap — u-side at the ceiling (25,998/25,998 at 3x3), v-side at the age-1 holds (38,256/38,256), sides never mix; single-cell witnesses concentrate on low-degree cells per capita (full census: corner 8,736 / edge 4,928 / centre 333 = 2,184 / 1,232 / 333 per cell) -- the centre does witness; an earlier revision reported "never centre" from a 2,000-state enumeration prefix and that was false
docs/coherence_invariant.md	Results doc for the coherence thread — the invariant, the supporting laws sorted by epistemic status (one PROVEN: the swap law; the rest CERTIFIED over a stated domain; "Theorem 4" is a 276-state certificate, not a proof), the substrate/analysis boundary, and an honest-scope section. The window law is a regime law: true at tau_a >= tau_p, false outside it
docs/coherence_larger_lattices_handoff.md	Handoff: four predictions for larger lattices (window universality; boundary concentration; torus needs bigger swaps; quiet-run mechanism), each with falsifier, plus the 4x4 compute routes and the encoding trap
experiments/topology_winding_capacity.py	E2 addendum — the GGH (1980) winding number as the exact sustain criterion; calibrates the length gate above
experiments/scaling_capacities.py	Scaling (Track 3b, size half) — does substrate size buy memory (E2) / attention (E4) / executive control (E5)?
experiments/lattice_capacities.py	Representation + lattice2d port of all three capacity mechanisms; shows E5's hidden layer has 0 recurrent edges
experiments/lattice_animation.py	GIFs for the three lattice mechanisms (reentry, wave annihilation, held option)
experiments/lattice_timescale_demo.py	Ports the input-timing τ rule to a lattice; two results, two negatives (see docs/lattice_timescale_notes.md)
experiments/lattice_afferent_timing.py	Resolves the lock-in: τ learning needs a privileged afferent channel
experiments/lattice_afferent_depth.py	How far exogenous timing penetrates a recurrent medium (answer: it does not)
experiments/lattice_attention_gate.py	A 1-D attention strip of the same cells gating plasticity — a clock, not a filter
experiments/lattice_reward_edges.py	Reward as a fourth edge; τ encodes a stimulus–reward interval
experiments/lattice_attention_value.py	A value chain teaches the attention strip where to gate
experiments/lattice_layers.py	2-D layers instead of 1-D edges; a synchronous burst timed to reward
experiments/lattice_sensorimotor.py	The action is transmission, not emission; bootstrapping is free
experiments/lattice_avoidance.py	Avoidance is not sign-symmetric with approach (transmission is provably monotone)
experiments/lattice_identity.py	Plastic cell identity via homeostatic θ — a structural negative
experiments/lattice_tonic.py	Tonic drive: no window; neither stalled thread unlocks
experiments/e3_timed_response.py	E3 — timed response (identity × latency double dissociation)
experiments/e3_factored_credit.py	E3 composition study — factored credit + curriculum vs shared reward
experiments/e4_attention.py	E4 — selective attention as biased WTA by wave annihilation
experiments/e5_executive.py	E5 — executive control / task switching (a slow-loop option gates fast routing)
experiments/e6_horde.py	E6 — emergent categories (three GVF demons read one frozen substrate)
experiments/c0_instrumentation.py	C0 — instrument the causal variables (W=f(S), partial spikes)
experiments/c1_graph_certificates.py	C1 — validate Theorem-1 epiphenomenality certificate on known SCMs
experiments/c2_fat_handed.py	C2 — do(W) is fat-handed when W=f(S) (achievable-band of behaviour)
experiments/c3_do_theta.py	C3 — do(θ) (timescales/couplings) is the well-posed causal handle
experiments/c4_outcome_relativity.py	C4 — outcome-relativity & degeneracy (causal-emergence cap)
experiments/closed_loop_plasticity.py	Closed-loop plasticity: single-task benchmark evaluating Readout Independence Ratio (RIR)
experiments/sequential_closed_loop.py	Sequential learning: Task A $\to$ Task B $\to$ Task A reversal learning evaluating anti-forgetting
result/	Saved simulation outputs (.npy/.npz) and experiment data

Documentation

• docs/learning_experiments.md — the full design: substrate spec, strict-reward learning framework, the two parallel plasticity lines (conduction weights vs local timescales), input/cue/feedback formats, hyperparameters, and the staged experiment series E0–E6.
• docs/e0_results.md — findings from E0 (substrate characterisation): range-1 fixates, the live threshold band widens with range (threshold-range scaling), an organised spiral band at r=2/a=6/θ≈4, and the dominant loop period tracking τ (period = 1.00·τ + 0.95, r = 0.9992).
• docs/e1_results.md — findings from E1 (conditioning): a strict scalar reward carves the stimulus→action mapping; the predicted dissociation holds (Line A = 0.91, Line B = 0.35 ≤ chance, A+B = 0.86 final accuracy over 6 seeds).
• docs/e2_results.md — findings from E2 (working memory): memory is a τ-controlled reentrant loop; the dissociation inverts — Line A retains only at zero delay, Line B learns τ below the loop transit time and holds memory to D=200. Needs a shared regional timescale (per-node τ hits a weakest-link problem).
• docs/topology_cycle_capacity.md — E2 addendum (topology analysis, no learning): the circuit rank β₁ = m − N + c bounds how many independent reentrant loops a substrate admits, and the E2 sustain gate (τ < L) collapses that ceiling by 20–70× to the usable count — which falls with τ on lattice/small-world/RGG (the ring is a flat exception: its packed cycles all exceed the τ range). A capacity/duration tradeoff on any Line-B τ policy. Capacity is a bound, not a measured count. ⚠ Superseded in part — see topology_cycle_packing_exact.md: the greedy packing maximises cycle length rather than count, so every K_dyn there is low by 1.4–7.5×, the 20–70× collapse is really 3.9–20×, and the ring is not flat.
• docs/spectrum_sufficiency_proof.md — theory. A machine-checked proof that the gradient spectrum decides persistence at (1,1) and (2,1) via a three-ingredient certificate schema (closure + activity + complement-death), each ingredient a finite check the companion experiment asserts. Two exact laws reframe the open problem: death ⟺ two cells ever dwell simultaneously (P4), and the signature moves only via dwell, one adjacent-gap exchange at a time (P5). The regime boundary is relocated to a single question: why is the dead state's basin boundary signature-measurable exactly when τa ≥ τp?
• docs/topology_cycle_packing_exact.md — correction to the above. pack_long_cycles picks max(longer, key=len); maximising the count wants min. One word costs 1.0–4.7×, an exact set-packing ILP up to 7.5×. On ring(60,k=3) the optimum is certified at all four τ tested (45/36/30/25, meeting min(β₁, ⌊m/(τ+1)⌋)) versus 6 in the merged doc — so ring capacity does fall with τ. Best-known K is still monotone decreasing in τ on all four topologies, so the capacity/duration tradeoff survives; only the numbers and the "flat ring" claim change.
• docs/persistent_set_structure.md — new, external-facing (structural companion to Moitra & Sen, SSRN 4047679). Does the persistent set admit a tractable description? Geometrically no — it is not linearly separable from the dying set in one-hot coordinates in any of 22 cells with P > 0, so no polytope in those coordinates describes it. Combinatorially yes whenever τ_p ≤ τ_a: persistence is a function of the sorted cyclic phase-gap multiset, compressing the space 16–161× into homogeneous classes (14641 configs → 91 classes at (5,5)), with compression growing in S. One-way implication only — (2,5) is invariant with τ_p > τ_a. Dynamics validated against ghca_net.Network: 7461 transitions, 0 mismatches.
• docs/topology_winding_capacity.md — E2 addendum (calibration of the above): the winding number of a continuous cycle, the invariant from Greenberg, Greene & Hastings (1980), is an exact sustain criterion (45/45) where the length gate scores 40/45 — and every length-gate miss is τ = L, the marginal death boundary. Winding is a dynamical quantity, so it calibrates the topological bound rather than replacing it; read too early it underperforms the gate.
• docs/scaling_capacities.md — scaling (Track 3b, size half): size helps only where size is the binding constraint. E2 memory is size-limited and matches the τ < L law 6/6; E4 attention is scale-invariant (0.08 sensitivity spread over an 8× arena range — bias-to-noise sets the collision locus, not room); E5 executive control shows no established size effect (16× N_H, Spearman p=0.51, overlapping CIs, per-seed bimodality dominating). A null on the size axis, next to 3c/P4's positive result for interference — different capacity, different constraint.
• docs/lattice_capacities.md — representation + 2-D port (with animations): each capacity uses a different substrate primitive (reentrant loop / colliding waves / held option gating a feedforward conjunction), and E5's context ring is E2's loop reused. E5's hidden layer has 0 H→H edges — purely feedforward AND-gates, which is why the N_H null holds. E2's τ < L law ports to the 2-D torus 28/28; E4's psychometric curve ports unchanged and stays size-invariant. E5 shows a threshold at the option's sustain boundary and no graded size dependence above it, on either knob.
• docs/e3_results.md — findings from E3 (timed response): double dissociation confirmed — Line A learns identity (wrong timing), Line B learns timing (not identity). New open problem: naive A+B interferes (both worse than either alone) under a single shared reward — later decomposed: factored credit removes the below-chance collapse (to ≈chance), a slow-first curriculum adds a marginal bimodal lift (joint composition on 1/5 seeds) — direction supported, magnitude not established at n=5.
• docs/e4_results.md — findings from E4 (attention): selective attention as biased winner-take-all by wave annihilation — a textbook psychometric (accuracy 0.96 at modest bias), the annihilation locus linear in the bias, achieved with zero inhibitory nodes.
• docs/e5_results.md — findings from E5 (executive control): a persistent reentrant loop (the E2 mechanism) acts as an option that gates fast routing — switching 0.89 vs 0.20 when the loop is ablated, post-switch accuracy consolidating 0.57→0.92, single-rule routing spared by the ablation (0.87 vs 0.86). The discriminator localises the loop's role to holding the rule across a block.
• docs/e6_results.md — findings from E6 (emergent categories): three GVF demons on one frozen substrate, reading the same feature vector, predict distinct questions well above baseline (memory R²=0.62, attention forecast 0.84, executive R²=0.98); their readouts are near-orthogonal and a generic probe matches an own-region oracle — memory/attention/executive are questions asked of one machine, not modules.
• docs/lattice_results.md — findings from the lattice arc (learned timescales on an excitable sheet, 11 experiments, headlines at n=20): the input-timing τ rule works on a 2-D medium where the old self-referential rule ratchets to its ceiling, but exogenous timing does not penetrate a recurrent medium from a localised sensory strip (|τ−P| 1.54 at the strip, ~2.0 immediately beyond it, 2.92 at the far wall and non-monotone in between — locking nowhere). A 1-D attention chain of the same cells carries a timing reference to any depth (0.00 at depth 92) — a clock, not a filter. Reward as a fourth edge makes τ encode each cell's own stimulus–reward interval (|Δ| 0.16–0.19, 97–98% within ±2, against an unpaired control receiving identical reward events that fails completely), and a backward value chain removes the last hand-set constant. Layered 2-D sheets produce the arc's one emergent output — a travelling wave becomes a synchronous burst timed to reward (91% within ±3 steps at D=70) — and show that value must arrive diffusely and couple modulatorily or synchrony collapses. The action primitive turns out to be transmission, not emission: the transmission edge sits at the learned interval (31.5 / 51.5 / 71.5 for D=30/50/70) and graded credit exists from trial 1, so contingent reward needs no shaping. Two structural negatives: selective avoidance is impossible (transmission is provably monotone in probe time; max per-seed violation +0.0000), and plastic cell identity costs 49–76% of the propagation reach at any rate. Across four independent failures — amplitude, phase, direction, activity level — the arc's transferable law is that a raw signal magnitude or geometry cannot serve as a label on this substrate; only a structural or predictive signal can. Not reinforcement learning: no action changes the world, and the anatomy is still designed.
• docs/causal_experiments.md — C-series plan: using the substrate (where W = f(S) is explicit) as a synthetic-SCM testbed for the spike-wave causal question (arXiv:2511.06602) — validate the paper's certificates on ground truth, then show do(W) is fat-handed under real constitution and do(θ) is the well-posed handle.
• docs/c0_results.md — findings from C0: W=f(S) verified; the wave carries info beyond partial spikes for a collective code (growing as observation gets sparser) but not for a labeled-line code — informativeness is structure-dependent.
• docs/c1_results.md — findings from C1: on six canonical graphs the Theorem-1 certificate matches ground-truth do(W) — including the confounded case (association without causation) and front-door (causal despite an observed mediator).
• docs/c2_results.md — findings from C2 (headline): when W=f(S) is constituted, one do(W=w) admits a huge behavioural band (33 σ) for a micro-reading behaviour vs ~0 for a collective one — do(W) is fat-handed and its causal verdict depends on the realization.
• docs/c3_results.md — findings from C3: do(θ) (timescales/couplings) is the well-posed handle — single-valued reproducible response, intervention ambiguity 0.014 σ vs do(W)'s 33 σ; θ is exactly what plasticity acts on.
• docs/c4_results.md — findings from C4: the causal role is (handle, outcome)-relative (do(θ) matrix is diagonal); the wave is the natural causal variable only where behaviour is collective (macro- sufficiency 1.03 vs 0.11 — causal emergence).
• docs/synthesis.md — tying note: the E-series and C-series are one argument — θ (timescales, couplings) is both the variable the learner adapts and the only well-posed causal handle; spikes and waves are two readouts of one parameterised dynamics.

Process & reviews

• docs/process.md — how the project runs its planning and review passes: decoupled in process, linked by a one-directional review→plan hand-off, and why. Read before doing either pass. (AGENTS.md points agents/humans here.)
• docs/core_review.md — independent integrity/overreach audit of the core series (E0–E6, C0–C4): no fabrication; surfaced the E3 framing overreach and the perturb_tau reproducibility bug.
• docs/extensions_review.md — self-audit of the extensions (E7, C5–C7, E8.x): reproducible and honest; the residual risk is the substrate-vs-analysis boundary (afforded vs learned).
• docs/next_steps.md — roadmap / planning: candidate directions scored by which review-surfaced tension they retire.

Progress

• E0 — substrate characterisation and operating point (see results)
• E1 — stimulus→response conditioning (A-vs-B dissociation confirmed)
• E2 — delayed response / working memory (dissociation inverts: B critical)
• E3 — timed response (double dissociation confirmed; A+B interference decomposed: factored credit removes the below-chance reward-conflation collapse 0.11→0.48, a slow-first curriculum adds a marginal, bimodal lift to 0.56 with genuine joint composition on 1/5 seeds — direction supported, magnitude not established at n=5; substrate-resonance capped)
• E4 — selective attention as biased WTA by wave annihilation (psychometric accuracy 0.96 at modest bias; zero inhibitory nodes)
• E5 — executive control / task switching: a persistent loop (E2 mechanism) as an option gating routing (switching 0.89 vs ablated 0.20; switch cost consolidates 0.57→0.92; single-rule spared 0.87 vs 0.86)
• E6 — emergent categories (Horde/GVF readout): three demons on one frozen substrate predict distinct questions (memory R²=0.62, attention 0.84, executive R²=0.98), near-orthogonal, no dedicated wiring — E-series complete

C-series (constitution & causality of spike–wave duality — see docs/causal_experiments.md):

• C0 — instrument the causal variables (W=f(S); wave informative beyond partial spikes for a collective code only)
• C1 — certificate validated on ground truth (all 6 canonical graphs agree; confounded & front-door as key cases)
• C2 — do(W) is fat-handed for a constituted W=f(S) (achievable band 33σ vs ~0)
• C3 — do(θ) is the well-posed causal handle (ambiguity 0.014σ vs 33σ; θ→W→B)
• C4 — outcome-relativity (diagonal do(θ) matrix) & degeneracy (macro-sufficiency 1.03 vs 0.11) — C-series complete

Closed-Loop Substrate Plasticity (see docs/closed_loop_plasticity_results.md):

• Multi-Axis Plasticity — tri-axis closed-loop engine ($\tau$-adaptation, $\theta$-homeostasis, $W$-routing) achieves RIR = $0.851 \pm 0.111$ on E1
• Substrate Credit Assignment & Anti-Forgetting — Task A $\to$ Task B $\to$ Task A sequential reversal learning yields 70.0% ± 13.8% retention (vs 29.6% ± 31.2% weight-only) via topological loop protection

Both E/C series and closed-loop substrate plasticity tracks are complete. See docs/synthesis.md for how E-series and C-series tie together.

Reproduce

uv sync
python3 reproduce_all.py                      # runs 8-step automated test & verification harness

Each experiments/*.py is self-contained and writes its figures to docs/figures/ and data to result/.