Claim Summary

Eleven formal results, each forced from the six axioms with no free conjecture, plus one conditional containment theorem (Theorem VI.2) imported from the foundations level and partially closing Debt D1.

Theorem I.1 (Drift) — DERIVED. The greedy strategy produces strict exponential decay; ruin is the globally attracting fixed point.

Theorem II.1 (Discipline) and Corollary II.2 — DERIVED. Non-zero structure requires non-zero input; the distinction between collapse and persistence is the optimisation horizon, not character.

Theorem III.1 (Event Horizon) and Corollary III.2 (Recovery) — DERIVED. Beyond x_H = u_max/a, recovery requires reducing structure, not escalating effort.

Theorem IV.1 (Impulse) and Corollary IV.2 — DERIVED. By the convexity of the effort cost (Jensen), minimum cost is at zero variance; habit beats motivation.

Theorem V.1 (Signal) and Corollary V.2 — DERIVED / STRUCTURAL. The optimal response to verified zero-mean noise is zero; the signal/noise classification is operational, resting on a stated modelling assumption about internal processes.

Theorem VI.1 (Coupling) — DERIVED. Mismatched drift makes a link a permanent tax on the more efficient agent; zero coupling is the safe default until match is verified.

Theorem VI.2 (Net-Positive Containment) — DERIVED-CONDITIONAL. Across a finite operator network, the persistent set is confined to the net-positive class; net-negative, parasitic, and defecting configurations reach the no-return surface. Quantitative within-class matching criterion owed (Debt D1).

Theorem VII.1 (Cascade) and Corollary VII.2 — DERIVED. Efficiency pressure that removes slack maximises cascade probability; slack and firewalls are the price of survival.

Theorem VIII.1 (Firewall) — DERIVED. Modular topology with zero coupling at boundaries contains failure that a globally connected topology lets saturate.

Theorem IX.1 (Exit) and Corollary IX.2 — DERIVED. In a structurally negative-sum system, withdrawal is the unique optimal strategy; sunk cost is irrecoverable.

The Ethic — STRUCTURAL. Mutual non-extraction between self-maintaining agents is the control-theoretic definition of kindness. Not commanded; derived.