Condensed Derivations

The formal skeleton on which all voices rest The Dictionary of Variables x(t) — State (structure, health, capital, coherence, survivable order) u(t) — Control (effort, work, attention, applied energy) −ax — Drift (entropy, decay, friction, default loss) ξ(t) — Observation noise (zero conditional mean) u_max — Capacity (yield strength, liquidity, biological limit)

ℒ(x, u) — Instantaneous cost

J — Action (total accumulated cost over trajectory) λ — Effort weight (price of control input)

E₁, E₂ — Stored structured energy (coupled agents) a₁, a₂ — Drift constants; u₁, u₂ — internal control inputs

Derivation I — Drift. Dynamics: dx/dt = −ax + u. Greedy strategy: min u² ⇒ u = 0. Result: x(t) = x₀ e^(−at). Drift is the default solution.

Derivation II — Discipline. Objective: min ∫[(x−1)² + λu²] dt. Steady-state minimisation yields x = 1/(1+λa²), u = a/(1+λa²). Effort is strictly positive. Discipline is the entropy tax.

Derivation III — Event Horizon. Constraint: u ≤ u_max. Horizon: x_H = u_max/a. For x > x_H: dx/dt < 0 at maximum effort. Recovery requires reduction of x, not escalation of u.

Derivation IV — Impulse. For square-integrable controls with equal mean input, cost u² is strictly convex. By Jensen’s inequality, 𝔼[u²] ≥ (𝔼[u])². Minimum cost at constant u (zero variance).

Derivation V — Signal. Stochastic dynamics: dx/dt = −ax + u + ξ(t), 𝔼[ξ] = 0.

Reactive control: 𝔼[u²] = a²x² + 𝔼[ξ²]. Operator: u_response = 0 for verified noise.

Derivation VI — Link. Coupled dynamics: Ė₁ = −a₁E₁ + u₁ − k(E₁−E₂). Mismatched drift ⇒ persistent gradient ⇒ permanent tax. Default: k = 0 until match verified.

Derivation VII — Cascade. Propagation inequality: |ΔL_A| · k_AB > S_B ⇒ shock propagates. Efficiency (S→0, k→max) maximises fragility. Slack is damping.

Derivation VIII — Firewall. Diffusion: ∂ρ/∂t = D∇²ρ, D ∝ k. Global topology saturates. Modular topology (k=0 at boundaries) contains failure.

Derivation IX — Exit. Game value: J = ∫(R−C) dt. Structurally negative-sum: sup_u 𝔼[J_future] < 0 for all T ≥ T_min. Stopping rule: 𝔼[J_future] < 0 ⇒ EXIT.