Derivation of κ=Rsurf/Rtot\kappa = R_{\mathrm{surf}}/R_{\mathrm{tot}} κ=Rsurf/Rtot from the PDL Axioms: the Indifference Lemma and Partial Resolution of Open Problem OP1 (D36)

Open Problem OP1 of D36 asks for a derivation of κ=Rsurf/Rtot\kappa = R_{\mathrm{surf}}/R_{\mathrm{tot}} κ=Rsurf/Rtot from the four PDL axioms C1–C4, without invoking D25 as a definition. The present paper provides a systematic investigation and reaches three conclusions. First, a two-factor decomposition is established: κ=(φ/3)⋅(rval/Rtot)\kappa = (\varphi/3)\cdot(r_{\mathrm{val}}/R_{\mathrm{tot}}) κ=(φ/3)⋅(rval/Rtot), where the first factor P1=φ/3=Rsurf/rvalP_1 = \varphi/3 = R_{\mathrm{surf}}/r_{\mathrm{val}} P1=φ/3=Rsurf/rval is the conditional engagement probability of a valence relation (established in D05 via the golden-ratio redistribution under C4), and the second factor P2=rval/RtotP_2 = r_{\mathrm{val}}/R_{\mathrm{tot}} P2=rval/Rtot is the probability that a relation drawn uniformly from the full relational budget is a valence relation. The identity κ=P1⋅P2\kappa = P_1\cdot P_2 κ=P1⋅P2 is verified to machine precision (error