The ERB Framework: Deriving the Standard Model Flavor Hierarchy from a Single Mechanical Postulate - Renormalized Structure of Quantum Field Theory

We present version 26 of the Emergent Resonant Brane (ERB) framework, a reconstruction program in which quantum field theory emerges as the long-wavelength effective description of an underlying elastic resonance substrate. The microscopic model rests on a single mechanical postulate: two coupled scalar fields with mass-density ratio ξ = mB/mA = 3/2, the smallest ratio of two distinct primes.

From this seed, without free parameters in the mixing sector, we derive: (i) the fine-structure constant α_EM = 1/137.071 via Berry-phase holonomy (0.025% error, all inputs fixed prior); (ii) the complete CKM quark-mixing matrix with mean absolute error MAE = 6.94×10⁻⁴ and Jarlskog invariant J = 2.95×10⁻⁵ (1.69% error, correct sign); (iii) the geometric origin of CP violation, which is a structural necessity of the asymmetric brane interface rather than a postulated phase; and (iv) exactly three fermion generations as the stationary nodes of an SO(3) resonance cavity.

Version 26 introduces a Wilsonian renormalization group framework connecting the sine-Gordon substrate to the infrared scalar effective theory, and two new contributions: a candidate dimensionless formula for the pion mass as the Goldstone mode of interface torsion, and a hypothesis that ξ = 3/2 is a stable infrared fixed point of the density-ratio RG flow.

All results are labeled with explicit status markers: PROVEN, DERIVED, CANDIDATE, HYPOTHESIS, or OPEN. Six open problems are stated with the specific derivations required. The Mathieu amplitude φ_a = 0.00345 is numerically determined; its analytic form is OPEN (P3). The energy scale Λ = 464 GeV is anchored to the W-boson mass rather than derived from the substrate stress-energy tensor (P1). The non-Abelian Lie algebra closure for SU(3) is not yet formally demonstrated (P2).

The program makes one primary falsifiable prediction: δ_CP ≈ 352° for neutrino CP violation, testable by the DUNE experiment approximately 2030. Falsification criterion: |δ_CP(measured) − 352°| > 20°.

An executable audit suite (ERB_Audit_Suite_v26.py) reproduces all results and is included as supplementary material.

P.S.: Detailed derivations of the coupling constant and the scale, previously established in older versions, are included in this release as technical supplements for completeness. Final Self-Contained is in progress.