The Coherence Derivation: From Ξ Lagrangian to the MOND Interpolation Function

We derive the MOND interpolation function from coherence-field microphysics within the General Coherence Field Theory (GCFT) framework. The derivation proceeds in four steps. First, the McGaugh interpolation function ν(x) = 1/(1 − e^{−√x}) is identified as algebraically identical to 1 + ⟨n_BE⟩(√x), the Bose-Einstein occupation number (CDN-272). Second, the √x argument is shown to follow uniquely from harmonic oscillation of Ξ modes in the gravitational potential, with the proportionality constant cancelling exactly in the energy-to-temperature ratio (CDN-273). Third, the velocity-dependent coupling sector is derived as a first-order Doppler shift of the same modes, unifying the static (MOND) and dynamic (flyby) channels of the theory (CDN-274). Fourth, the complete derivation chain — from Lagrangian through mode structure, thermal occupation, and Doppler perturbation to phenomenology — resolves all twelve theory debts accumulated across the programme (D1–D12). The result is parameter-free: ν(x) is fixed entirely by Bose-Einstein statistics applied to harmonic gravitational modes, with a₀ entering as the coherence temperature. Covers the derivation arc CDN-270–274.