""" co_stabilization.py Mutual-dependence knockout toy: a first co-stabilization candidate. (Benchmark v1.9) THE QUESTION. v1.8 measured composition — two coupled generators whose joint trace empties the single-generator equivalence class. But composition is not yet life. The ladder note's synthesis names the missing threshold: "a self-maintaining composition of processes whose functions make differences for one another" — CO-STABILIZATION, each function holding the others in a viable regime. This file tests a first candidate operationalization: designed mutual dependence on a fixed ring. The pre-run proposal treated a super-additive knockout cascade as the signature of co-stabilization. The run measured that cascade, but its failed noise prediction shows that the model establishes dependency fragility, not yet ecological self-maintenance. The original predictions remain below; the post-run calibration states what survived. THE SETUP. N nodes ("functions"), each carrying a health h_i in [0, 1]; viable if h_i >= v_thresh. A ring topology (each node coupled to two neighbors). Dynamics: h_i(t+1) = clip( s + c * mean_{j~i} h_j(t) , 0, 1 ) with s = 1 - c chosen so that the all-healthy state h_i = 1 is a fixed point at EVERY coupling c. This is the crux: at rest, every regime looks identical — all nodes viable, health 1. The coupling c is the mutual-dependence dial: c = 0 each node is self-sufficient (s = 1). Pure coexistence. c -> 1 each node's health is almost entirely its neighbors' (s -> 0). A standing web that holds itself up — and can fall together. THREE PROBES: REST — run undisturbed; record viability. (Predicted: blind — every c looks perfectly healthy.) KNOCKOUT — freeze one node at h = 0, relax to steady state, count how many of the OTHER N-1 nodes fall below v_thresh. The cascade. Co-stabilization score = mean over knockout targets of (survivors lost beyond the 1 removed) — additive vs super- additive. NOISE — inject i.i.d. per-step health noise, measure the fraction of node-steps that stay viable. The other face of coupling: mutual support averages distributed shocks away. The pair of probes asks whether the same coupling that cascades under targeted knockout also buffers distributed noise — a robustness/fragility trade. (It does not, in this model, and that is the finding; see RESULT.) PREDICTIONS (stated before the first run; original language retained, with interpretation recalibrated in RESULT and POST-RUN CALIBRATION): P1 REST is blind: viability = 1.0 for every c. Watching a healthy ecology cannot distinguish coexistence from co-stabilization — the coverage/Bateson principle's fourth appearance (exp6 bindings, exp7 lures, v1.8 symbionts, here dependencies). P2 KNOCKOUT separates them: at c = 0 a knockout costs exactly 1 node (additive; cascade ratio ~ 1); as c rises the survivors' health is dragged down and extra nodes fall — super-additive collapse — with a threshold in c, above which one knockout takes a finite fraction (at c -> 1, on a ring pinned to 0, the whole line decays: total collapse). P3 NOISE robustness rises with c: averaging over neighbors buffers i.i.d. shocks, so the viable fraction under noise increases with coupling — the same c that P2 makes fragile. P4 THE RISKY ONE — the knockout threshold is SHARP (a phase transition in c), not gradual: below it a knockout is local, above it it cascades system-wide. If instead the cascade grows smoothly with c, there is no critical coupling and co-stabilization is a matter of degree, not a threshold — also worth knowing, and the docstring will say which. RESULT (run 1, N=16 ring, 200 noise trials — console numbers): P1 CONFIRMED WITHIN THE CONSTRUCTION: rest viability = 1.00 at every c because s = 1-c makes h=1 a fixed point by design. The homogeneous rest state cannot identify c; this does not establish that ecological co-stabilization generally lacks passive signatures. P2 CONFIRMED AS DEPENDENCY: the knockout cascade is 0 for c <= 0.7, then +2 extra nodes at c=0.8-0.9, +4 at c=0.95, and +15 at c=1.0. Knockout exposes the dependency designed into the rule. A super- additive cascade does not by itself demonstrate self-maintenance. P3 FALSIFIED — and the miss is the real finding. The noise viability does NOT rise with coupling; it FALLS monotonically (0.977 -> 0.521). Mechanism: in this model mutual dependence SUBSTITUTES for self- sufficiency (s = 1 - c), so as c rises the restoring force to the viable setpoint vanishes and distributed noise accumulates instead of being reset. There is no robust/fragile trade here: pure mutual dependence is fragile BOTH ways — to targeted knockout AND to distributed noise — while looking perfectly healthy at rest. The cautionary reading: co-stabilization-as-substitution is a monoculture, not resilience. Genuine ecological robustness would require coupling that ADDS a redundant pathway to self-sufficiency rather than replacing it — the redundancy model, named as the follow-up below. What survives is narrower: the homogeneous fixed point does not identify c, while knockout separates dependencies among the three probes run here. P4 NOT ESTABLISHED AS A PHASE TRANSITION: the first threshold crossing lies between sampled c=0.7 and c=0.8, then the finite cascade climbs in steps. Its location depends on viability cutoff, ring size, topology, and the coarse c grid. Total collapse at c=1.0 is the boundary where self-sufficiency is exactly zero. Scaling is required before "critical coupling" would be justified. POST-RUN CALIBRATION. The experiment operationalizes mutual dependence, not co-stabilization itself. Super-additive loss shows that other nodes depend on the knocked-out node under the chosen update rule. Self-maintenance would require the coupled organization to contribute robustness or continued viability that the parts lack alone; this substitution model does the opposite under noise. The failed P3 therefore rejects this first candidate as an ecology in the stronger sense. WHAT THIS DOES NOT SHOW. A fixed ring with a linear averaging rule is not an ecology that composes and dissolves on its own. Nothing is selected or reproduces; health is a scalar threshold, not metabolism; and the all-healthy rest state is identical by construction. Knockout is the separating probe among the three probes run here, not the only conceivable identifying intervention. The apparent onset is a threshold crossing in a finite, thresholded graph, not an established phase transition. Two follow-ups remain: (a) a REDUNDANCY model in which coupling adds a compensating pathway while self-sufficiency is retained, tested across topology, N, viability cutoff, and coupling resolution; and (b) a population model where coupling is built and broken by the processes. Until one of those shows mutual maintenance rather than designed dependency, co-stabilization remains [HYPOTHESIZED]. No consciousness claim follows. Usage:: python co_stabilization.py # console summary (~5 s) python co_stabilization.py --save # + figure to lab/tools/ Related: - theory/computation/static-information-and-living-process.md (the open threshold this candidate failed to establish) - composition.py (v1.8: composition, the pair this generalizes) - theory/core/measurement-as-weak-intervention.md (watching < perturbing, here rest < knockout) - papers/viable-corridor.md (capability loading: robustness bought on one axis, paid on another) """ from __future__ import annotations import argparse from pathlib import Path import numpy as np N = 16 V_THRESH = 0.5 CS = [0.0, 0.2, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 1.0] RELAX_STEPS = 400 NOISE_TRIALS = 200 NOISE_STEPS = 200 NOISE_SIGMA = 0.25 def _ring_neighbors(n: int) -> list[tuple[int, int]]: return [((i - 1) % n, (i + 1) % n) for i in range(n)] def relax(c: float, frozen: int | None = None, steps: int = RELAX_STEPS) -> np.ndarray: """Relax the ring to steady state; optionally pin one node at 0.""" s = 1.0 - c nb = _ring_neighbors(N) h = np.ones(N) if frozen is not None: h[frozen] = 0.0 for _ in range(steps): new = np.array([s + c * 0.5 * (h[a] + h[b]) for (a, b) in nb]) new = np.clip(new, 0.0, 1.0) if frozen is not None: new[frozen] = 0.0 h = new return h def rest_viability(c: float) -> float: h = relax(c) return float((h >= V_THRESH).mean()) def knockout_cascade(c: float) -> float: """Mean over knockout targets of (extra survivors lost beyond the 1 removed). 0 = purely additive (coexistence); large = super-additive.""" extra = [] for k in range(N): h = relax(c, frozen=k) others = np.delete(h, k) dead_others = int((others < V_THRESH).sum()) extra.append(dead_others) # nodes lost beyond the removed one return float(np.mean(extra)) def noise_viability(c: float, seed: int = 0) -> float: """Fraction of node-steps staying viable under i.i.d. health noise.""" rng = np.random.default_rng(seed) s = 1.0 - c nb = _ring_neighbors(N) h = np.ones(N) viable = 0 total = 0 for _ in range(NOISE_STEPS): new = np.array([s + c * 0.5 * (h[a] + h[b]) for (a, b) in nb]) new = np.clip(new + rng.normal(0, NOISE_SIGMA, N), 0.0, 1.0) h = new viable += int((h >= V_THRESH).sum()) total += N return viable / total def run_suite() -> dict: rest, cascade, noise = [], [], [] for c in CS: rest.append(rest_viability(c)) cascade.append(knockout_cascade(c)) nv = [noise_viability(c, seed=s) for s in range(NOISE_TRIALS)] noise.append(float(np.mean(nv))) return {"cs": CS, "rest": rest, "cascade": cascade, "noise": noise} def print_summary(res: dict) -> None: print("=" * 72) print(" BENCHMARK v1.9 — mutual dependence under knockout (candidate model)") print(f" (N={N} ring, knockout cascade + noise robustness vs coupling c)") print("=" * 72) print(f"\n {'c':>6s} {'rest viability':>15s} {'knockout cascade':>18s} " f"{'noise viability':>16s}") for i, c in enumerate(res["cs"]): print(f" {c:6.2f} {res['rest'][i]:15.2f} {res['cascade'][i]:18.2f} " f"{res['noise'][i]:16.3f}") print("\n cascade = mean extra nodes lost per knockout, beyond the 1 removed") print(" (0 = additive; large = super-additive dependency under this rule)") print("\n Reading: the homogeneous rest state is blind to c by design;") print(" knockout exposes super-additive dependency. The predicted noise") print(" robustness is FALSIFIED: substitution coupling is fragile both") print(" ways. This rejects the candidate as ecological co-stabilization;") print(" it does not close the self-maintenance question. See CALIBRATION.") def figure(res: dict, outdir: Path) -> Path: import matplotlib matplotlib.use("Agg") import matplotlib.pyplot as plt cs = res["cs"] fig, axes = plt.subplots(1, 2, figsize=(12, 4.6)) ax = axes[0] ax.plot(cs, res["cascade"], "o-", color="#d62728") ax.set_xlabel("coupling c (mutual dependence)") ax.set_ylabel("extra nodes lost per knockout") ax.set_title("(a) Knockout cascade\n(homogeneous rest is blind by construction)") ax.grid(alpha=0.3) ax2 = ax.twinx() ax2.plot(cs, res["rest"], "s--", color="#999999", alpha=0.6) ax2.set_ylabel("rest viability (watching)", color="#999999") ax2.set_ylim(0, 1.1) ax = axes[1] ax.plot(cs, res["noise"], "^-", color="#1f77b4", label="noise viability") ax.plot(cs, [x / max(res["cascade"]) if max(res["cascade"]) else 0 for x in res["cascade"]], "o--", color="#d62728", alpha=0.6, label="knockout cascade (norm.)") ax.set_xlabel("coupling c (mutual dependence)") ax.set_ylabel("fraction / normalized") ax.set_title("(b) Candidate fails resilience\n(substitution coupling is fragile both ways)") ax.legend(fontsize=8) ax.grid(alpha=0.3) fig.suptitle("Benchmark v1.9 — mutual dependence: knockout reveals fragility", fontsize=11) fig.tight_layout() out = outdir / "inverse_benchmark_co_stabilization.png" fig.savefig(out, dpi=110) return out def main() -> None: ap = argparse.ArgumentParser() ap.add_argument("--save", action="store_true") args = ap.parse_args() res = run_suite() print_summary(res) if args.save: outdir = Path(__file__).resolve().parents[2] / "tools" print(f"\n figure -> {figure(res, outdir)}") if __name__ == "__main__": main()