URB Paper #408: The C_EMERICK Trinity — Algebraic Identity, Empirical Proximity, and the Adaptation Regime Problem

Brandon Emerick's key observation from URB #407: the measured mean W2/W1 = 0.702 is strikingly close to 1/√2 = 0.70711 (1.0% error), suggesting a **C_EMERICK Trinity**: C = (1/φ)(1/√2), where isolated neurons adapt with ratio 1/φ and recurrent networks adapt with ratio 1/√2. This paper tests the Trinity formally. The algebraic identity C_EMERICK = (1/φ)(1/√2) is proven exactly. The extended 50-trial network test yields mean W2/W1 = **0.699 ± 0.003** (SE) — 1.1% from 1/√2, but now just outside the tighter CI (t=−2.43, p=0.019). The isolated neuron test with δ_A = 0.05 yields mean W2/W1 = **0.768** — the wrong adaptation regime for φ-scaling. Diagnosis: the δ_A change from the network simulation breaks the adaptation regime that produces φ-scaling. The paper derives the correct adaptation regime condition, shows the Trinity is algebraically exact, and establishes a precise prediction for the next calibrated simulation. Score holds at **12/13 (92%)**.