First-Principles Derivation of the MOND Interpolating Function and Scale from Horizon Thermodynamics

Paper V of the Entropy-Response Gravity (Bipartite KMS Gravity) programme. Derives the MOND interpolating function mu(x)=x/sqrt(1+x^2) (Standard form) and the relation a_0=cH_0/sqrt(4*pi^2-1) from a Bunch-Davies vacuum threshold condition (Rindler causal horizon = vacuum coherence length), with an independent H_0 implication and testable redshift evolution. Version 4 note: wording corrected per the framework's separation of dimensionless content from magnitude -- the framework derives the dimensionless ratio a_0/(cH_0)=1/sqrt(4*pi^2-1); the scale is set by the input H_0 (not derived). Interpolating-function label corrected to the Standard form (n=2; Simple = x/(1+x), n=1) throughout. Results unchanged.