The Holographic Circlette: Part 15 — The Rosetta Stone: How 3D Geometry Emerges from a Discrete Lattice

Audit note. The degree-3 vertex analysis and resulting 1/3 transport-distribution penalty derived here are anchored at the vertex-figure level of the 4.8.8 Archimedean tiling, which is the local two-dimensional reduction of the canonical three-dimensional Truncated Cubic Honeycomb substrate. The 3D-bulk extension to the macroscopic gauge web (where the coordination number at relevant vertices is z = 5, not z = 3) is anchored as an open structural problem in ANCHOR §15 item 52(c) / item 63 (pending an explicit Wilsonian coarse-graining theorem or analytical Ollivier--Ricci linearity proof). Accordingly, the Einstein-tensor coefficient derivation presented in this paper is rigorous at the vertex-figure level and computationally validated on the 4.8.8 lattice (ANCHOR §15 item 52 partial closure 2026-05-19); the macroscopic-3D-bulk extension is the canonical open target. Readers should treat the form-de We derive how pi factors governing coupling constants across force sectors emerge from dimensional projections of the discrete lattice: 1D to pi/2 (Dashen), 2D to 4*pi (pion decay), 3D to 8*pi*G (Einstein tensor). This provides the bridge from the 2D tensor network to 3+1D General Relativity. 2026-06-20 legacy canon revision: This is a canon-reconciled legacy version. Emergent-geometry/gravity framing superseded The paper retains its historical derivation trail but carries a 2026-06-20 canon revision note identifying current status and superseded claims. 2026-06-21 canon refresh: This version incorporates the 2026-06-21 ANCHOR/DRIFT/PTMS canon refresh and rebuilt local PDF.