The companion triangulation verification note [9] identified, by algebraic inversion, the exact function f(x) = 1 + 1/2 ln(1 + x) satisfying f(∆(L)) = 1+2/L exactly, where ∆(L) = e^(4/L) − 1 is the CNRS area gap for the physical base z0 = e^(2/L). The present paper provides what that note could not: a conditional geometric derivation of f from the CNRS tiling geometry, and a complete unification of the metric correction, clock-rate correction, and Einstein tensor deficit under a single geometric quantity. All results are conditional on two framework assumptions: (i) the static diagonal ansatz of Papers 9–11; (ii) the identification z0 = e^(2/L) as the physical CNRS base. Assumption (ii) is the critical hinge of the programme (Section 1.2). Result 1 (conditional derivation of f ). Given z0 = e^(2/L), the metric correction F = 1+2/L equals 1+ln z0. The area gap ∆(L) = |z0|^2 −1 encodes this exponent in squared form; the function f recovers it via 1/2 ln(1 +∆(L)) = ln z0 = 2/L. Within the class of functions {1 + c · ln(1 + x) : c ∈ R}, f is the unique member satisfying f(|z0|^2 − 1) = 1 + ln z0 for all real z0 > 1. Result 2 (Einstein tensor unification). Within the static diagonal ansatz, Gtt = −6c^2F/L^3 is linear in F . The configurational deficit ratio κ5T^corr tt /κ5T^bd tt = F = f(∆(L)) exactly. The fractional deficit increase F −1 = ln z0 = 2/L equals the CNRS exponent. Unified statement (conditional). Within the static diagonal ansatz and given z0 = e^(2/L): the metric correction 2/L, the clock-rate correction 2/L, and the fractional deficit increase 2/L all equal ln z0, the exponent of the physical CNRS base. This is the first conditional quantitative bridge from a single geometric CNRS quantity to three distinct physical observables. What remains open. The derivation of the correction structure from the full complex treatment of zs as a primitive is substantially advanced by Papers 16–17 [5, 6] but not complete. The identification z0 = e^(2/L) requires a dynamical or geometric uniqueness argument.

The mathematical development in this paper was produced in dialogue with Claude.ai (Anthropic) in Spring 2026, directed by the author. Use of AI assistance is acknowledged in accordance with standard scholarly practice.