CNRS-Tet: The Metric Correction from the Vierbein Formalism: Tetrad Re-derivation, Sub-leading Structure, and the Complex Geodesic Connection

The triangulation unification paper [8] established that the metric correction F = 1 + 2/L of Paper 10 equals 1 + ln z0, where z0 = e^(2/L) is the physical CNRS base, and that the factor 1/2 in the triangulation function f(x) = 1 + 1/2 ln(1 + x) arises because the area gap encodes |z0|^2 while the correction needs ln|z0|. One open problem remained: a derivation of F starting from zs as a complex primitive, without invoking the area gap. This paper provides three contributions.Contribution 1 (Vierbein derivation). We re-derive the metric correction in the vierbein (tetrad) formalism. The vierbein component e^(0) t = √F c encodes the amplitude correction √F, while the metric encodes F = (√F)^2. The factor 1/2 in the triangulation function arises directly from this squaring: the amplitude correction is √F = √(1 + 2/L) ≈ 1 + 1/L, and the metric correction is its square, F = 1 + 2/L + O(L^−2). This gives an independent, geometry-internal derivation of the same correction. Contribution 2 (Sub-leading structure). We compute F to all orders in 1/L using both the area-gap route and the vierbein route, and confirm that F = f(∆(L)) = 1 + 1/2 ln(1 + ∆(L)) is the exact all-orders result, with sub-leading terms: F = 1 + 2/L - 2/L^2 + 4/3L^3 - ... = 1 + 1/2 ln(1 + 4/L + 8/L^2 + 32/3?^3 + ...).Contribution 3 (Complex geodesic connection). We carry out the first two steps of the Paper 13 programme [4]: (a) proper time with zs = e(2+iϕ)/L as a primitive (the complex scale factor), and (b) the resulting time-sector contribution to the Einstein tensor. Setting ϕ= 0 (real s) recovers F = 1+2/Lexactly. For ϕ̸= 0 the correction generalises to F = 1+Re[(2+iϕ)/L], showing that the imaginary part of the scale coordinate contributes a phase but not a magnitude correction at this order.