01 — Theorem 4.2: Routing–MDL Equivalence & Telescoping Identity

1. Statement of Theorem 4.2 (Route Redirection via Coding Geometry)

Let $G = (V, E)$ be a finite directed graph and let $\Gamma(s, t)$ denote the finite family of admissible simple pathways from source $s \in V$ to target $t \in V$. Under a prefix-free conditional coding assignment $\ell_c(e_i \mid p_i)$, the clue-updated route weight functional satisfies:

$$W_c(\gamma) = \Phi_c(t) - \Phi_c(s) + L_c(\gamma)$$

where:

$\Phi_c: V \to \mathbb{R}$ is the node potential anchored on the graph Laplacian.

$L_c(\gamma) = \sum_{e_i \in \gamma} d_c(e_i \mid p_i)$ is the non-negative path coding description length.

Because $\Delta\Phi_c(s,t) = \Phi_c(t) - \Phi_c(s)$ depends only on the endpoints $(s,t)$ and is constant across every $\gamma \in \Gamma(s,t)$, minimizing total route weight $W_c(\gamma)$ is strictly equivalent to minimizing description length $L_c(\gamma)$:

$$\mathcal{D}(Cw) = \arg\min_{\gamma \in \Gamma(s,t)} W_c(\gamma) = \arg\min_{\gamma \in \Gamma(s,t)} L_c(\gamma)$$

Whenever a clue endomorphism $C: \mathcal{W} \to \mathcal{W}$ compresses the description length of an alternative pathway $\gamma_{\text{PASS}} \in \Gamma(s,t)$ below the unredirected minimizer $\gamma_{\text{BW}} \in \mathcal{D}(w)$, route redirection occurs without altering $\Gamma(s,t)$:

$$\boxed{\mathcal{D}(Cw) \neq \mathcal{D}(w)}$$

---

2. Live StrategyOS Operational Witness (Section 7.2–7.3)

Fixed Admissible Route Family: $\Gamma(s,t) = \{\gamma_{\text{BW}}, \gamma_{\text{EXP}}, \gamma_{\text{OPT}}, \gamma_{\text{PASS}}\}$

Before Clue: $\mathcal{D}(w) = \{\gamma_{\text{BW}}\}$ with $L_{\text{MDL}}(w) = 26.4950\text{ bits}$

After Clue: $\mathcal{D}(Cw) = \{\gamma_{\text{PASS}}\}$ with $L_{\text{MDL}}(Cw) = 15.7106\text{ bits}$

Telescoping Potential Verification:

* $\Phi_c(s) = 722.0000\text{ bits}$, $\Phi_c(t) = 49.8557\text{ bits}$, $\Delta\Phi = -672.1443\text{ bits}$

* Measured telescoping residual error: 0.00000000 bits