coherence_gain
plain-language theorem explainer
The coherence gain for an ensemble of N particles is defined as the ratio of its coherent effective source (linear sum N a) to its incoherent effective source (random-walk magnitude sqrt(N) a). Researchers modeling gravitomagnetic amplification in superconductors cite this to quantify phase-coherence enhancement, with N around 10^22 yielding gains near 10^11. The definition is a direct ratio of the two source functions.
Claim. For an ensemble of $N$ particles each with individual amplitude $a > 0$, the coherence gain is the ratio of the coherent effective source $N a$ to the incoherent effective source $a sqrt(N)$.
background
In the Gravity.CoherenceGain module an Ensemble is a structure holding a positive integer $N$ and positive real individual amplitude $a$, representing $N$ particles each emitting a vector source whose net magnitude depends on relative phase. Coherent alignment sums the vectors linearly to total magnitude $N a$; incoherent random phases produce a random-walk magnitude $a sqrt(N)$ by the central limit theorem. The module imports ledger factorization and phi-forcing structures to embed these sources inside the Recognition Science treatment of gravitational effective mass.
proof idea
The definition is a one-line wrapper that divides coherent_effective_source by incoherent_effective_source on the supplied ensemble.
why it matters
This definition is used by the CoherenceGainCert structure and the theorem coherence_gain_eq_sqrt_N, which proves the ratio equals sqrt(N). It supplies the quantitative link for the coherence gate in superconductors, matching claimed gravitomagnetic enhancements and placing the sqrt(N) factor inside the Recognition Science phi-ladder scaling of gravitational sources.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.