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

The triangulation unification paper 6 established that the metric correction F = 1 + 2/L of Paper 10 2 equals 1 + ln z0, where z0 = e^ (2/L) is the physical CNRS base (conditional on the static diagonal ansatz and the identification z0 = e^ (2/L) ), 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 = √Fc 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 = sqrt (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 via asymptotic structure rather than numerical coincidence. Contribution 2 (Exactness within the static ansatz). F = 1 + 2/L is the exact, all-orders result within the static diagonal ansatz — not a leading-order approximation. Since 1 + ∆ (L) = e^ (4/L) exactly (where ∆ (L) = e^ (4/L) − 1 is the area gap), taking the logarithm gives 1/2 ln (1 + ∆) = 2/L exactly. Sub-leading terms appear only in the expansion of ∆ (L) for large L; they do notpropagate into F. The triangulation is closed within this ansatz. Contribution 3 (Complex geodesic: partial derivation and target specification). We carry out the first step of the Paper 13 programme 4: proper time with zs = e^ ( (2+iϕ) /L) as a primitive (the complex scale factor). Setting ϕ = 0 (real s) recovers F = 1 + 2/L exactly. For ϕ ̸= 0 the imaginary part enters only at order ϕ2/L2, not at linear order, so the metric correction is stable under Born-rule projection. The full derivation — obtaining F from zs as a primitive without imposing ϕ = 0 — remains open and is precisely specified as Open Problem 1.