The Mittermeier kappa-alpha-pi Bridge: A Planck-Boundary Constant from the Mittermeier RG Attractor and Its Gravity-Electromagnetism Resonance

A persistent obstacle for "world-formula" programs is that key dimensionless couplings—most notably the fine-structure constant alpha—are measured but not derived. This paper isolates one number that IS derived in closed form from the SSOT-hard nucleus of the Mittermeier RG Attractor:

kappa_M := lambda(0) = e^-(2 + ln(rho)) = e^-2 / rho,

where rho is the plastic constant (rho^3 = rho + 1). Numerically, kappa_M = 0.102162. Equivalently, the rescaled constant x := e^2 * kappa_M = 1/rho is the unique positive root of x^3 + x^2 - 1 = 0, making the plastic-constant lock explicit.

The defining point is structural: kappa_M is locked at the UV boundary u = 0 with no fit freedom, as a consequence of a closed parameter lock in the attractor's two-threshold normal form. We then show that this purely attractor-derived constant sits in a tight web of near-resonances with electromagnetism and geometry:

kappa_M approx 14 * alpha, kappa_M approx 1 / (pi^2 - 11 * alpha), 14 * alpha * (pi^2 - 11 * alpha) approx 1,

where the first relation holds at the level of -13.245 ppm when alpha is taken at the CODATA-2022 recommended value. We interpret this as evidence for a Mittermeier kappa-alpha-pi Bridge Relation: a simple, integer-structured coupling law linking the fine-structure constant to (pi, rho, e) via kappa_M, with controlled residual corrections.

The present work is intentionally "step one": it does not attempt a full quantum-gravity construction. Instead, it isolates a concrete, falsifiable bridge between a gravity-side boundary constant and precision metrology, with enough structure to support sharp follow-up tests (scheme robustness on the attractor side; scale matching and running on the QED side).