Fano 2-22 Tessellation and Planck Curvature: An Algebraic Bridge from Polyakov Tension to \(Kₘₐₓ = 6T^3\) with a Complete Formal Connection to Singh's Torsion Gravity

We establish a complete formal bridge between the Alpha+Fano framework and Singh's torsion gravity. The connection is anchored by two algebraic identities. From Singh's spectral action for the STM atom,\[∮ ds^2 = -1/48π^2∫_M d^4x√g\,R,\]we substitute the Alpha+Fano structural values \(R = 6T^3 = 3072\) and \(∫cell d^4x√g = δₜₕ\). With the normalization \(C_0 = -π^2/64\), the STM atom action reduces exactly to\[SSTM = δₜₕ,\]where\[δₜₕ = 1/2πln\!({√8}{√8-√7})\]is the volume of a single Fano 2-22 cell. This identifies Singh's STM atom with the Fano 2-22 cell, Singh's Nambu-Goto + Kalb-Ramond string with a structural D-brane, and Singh's gravitational action with the 4D Lagrangian. The bridge is validated through five independent checks. The latter yields:\[α⁻¹ = 137.0359991678 (CODATA: 137.035999177),\]and\[ln\!(G m_e^2/ c) = -π/3(π^4+1) = -103.053759146290,\]with \(G = 6.687133 × 10⁻¹¹\ m^3kg⁻¹s⁻²\) (CODATA: \(6.67430 × 10⁻¹¹\)), a relative difference of \(0.0092\) (0.92%). The factor \(2π\) is identified as the common geometric connector between the three frameworks. All structural assumptions are explicitly marked. Appendix A formalizes the identification theorem. Appendix E provides the microscopic interpretation of the residual \(εδ\) as a phase damping effect from the 24 transverse modes. The complete verification script is included as supplementary material to this preprint.