A Minimal Relational Sketch for the Emergence of the Golden Ratio in PDL (Version 2 — corrected)

Abstract

Version 2 of PDL document D05. This version corrects four errors present in version 1, two of which produced a conclusion numerically incompatible with the formula this note is cited to justify. Taken at face value, version 1 yields R_surf = 574.772 and hence α⁻¹ = 179.921, against a measured 137.036. Errata: (E1) the derivative of the trade-off functional was computed with the coefficients interchanged, and the functional was described as a quantity to be maximised although it admits only a minimum; (E2) the opening decomposition R_tot = R_core + R_surf + R_sea is incompatible with R_tot = r_val + R_sea; (E3) the orientation of the golden section was inverted; (E4) R_core was identified with r_valence = 930 instead of the mean coherence core r_val/3 = 310. Corrections E3 and E4 are jointly required; neither suffices alone. The corrected self-similarity condition reproduces R_surf = 310φ exactly. The central claim of this version is negative and is stated explicitly: R_surf = φ·r_val/3 is NOT derived from axioms C1–C4. It is designated a named hypothesis, Hφ. Consequently κ, α, σ(N), G_eff, the Hubble prediction and Λ are conditional upon Hφ, and the statement in the Global Mapping (v33) that Gate 3 holds without any named hypothesis requires correction. This version supersedes the epistemic labels attributed to D05 in D39 and D43, which recorded its content as a theorem. No adjustable parameter is introduced: φ is a fixed algebraic number and the factor 1/3 counts the three valence cores. Results not passing through R_surf — the quintuplets, Δn = 4, E_bord, ε_geom, and the gauge chain — are unaffected.