The Scaffolding Principle and the Fermion Mass Hierarchy: Ten Numerical Tests of CRM as a Generative Methodology

Paper 18: Fermion Mass Hierarchy Title: The Scaffolding Principle and the Fermion Mass Hierarchy: Ten Numerical Tests of CRM as a Generative Methodology Description:Paper 18 in the Constructive Reverse Mathematics (CRM) of Mathematical Physics series. This paper systematically tests whether removing the LPO "scaffolding" (continuous limits, exact symmetries) from Standard Model Yukawa renormalization-group flow reveals BISH-level mechanisms that explain the fermion mass hierarchy. Ten numerical investigations across two phases yield uniformly negative results: the mass hierarchy is not determined by SM infrared dynamics at any loop order, discretization scale, or parameterization tested. The entire computation lies in BISH with no LPO boundary — the first such domain in the CRM series. Introduces the "scaffolding principle" hypothesis (negative test) and demonstrates CRM as a powerful diagnostic with null generative capacity for the flavor problem. The Lean 4 formalization (Mathlib, v4.28.0-rc1) builds with 0 errors, 0 warnings, 0 sorry. Keywords: constructive reverse mathematics, fermion mass hierarchy, Yukawa coupling, renormalization group, scaffolding principle, BISH, Standard Model, Lean 4, Mathlib Complete Paper List Other papers in the Constructive Reverse Mathematics Series (see Zenodo for current files): Paper 1: WithdrawnPaper 2: Bidual gap and WLPO — DOI: 10.5281/zenodo.17107493Paper 3: WithdrawnPaper 4: Quantum spectra axiom calibration — DOI: 10.5281/zenodo.17059483Paper 5: Schwarzschild curvature verification — DOI: 10.5281/zenodo.18489703Paper 6: Heisenberg uncertainty (v2, CRM over Mathlib) — DOI: 10.5281/zenodo.18519836Paper 7: Physical bidual gap, trace-class operators — DOI: 10.5281/zenodo.18527559Paper 8: 1D Ising model and LPO — DOI: 10.5281/zenodo.18516813Paper 9: Ising formulation-invariance — DOI: 10.5281/zenodo.18517570Paper 10: Logical geography of mathematical physics — DOI: 10.5281/zenodo.18580342 v 2.0 Technical Synthesis of Papers 1–16Paper 11: Entanglement, CHSH, Tsirelson bound — DOI: 10.5281/zenodo.18527676Paper 12: Constructive history of mathematical physics — DOI: 10.5281/zenodo.18581707 v 2.0 Historical Synthesis of Papers 1–16Paper 13: Event horizon as logical boundary — DOI: 10.5281/zenodo.18529007Paper 14: Quantum decoherence — DOI: 10.5281/zenodo.18569068Paper 15: Noether theorem — DOI: 10.5281/zenodo.18572494Paper 16: Technical note: Born rule — DOI: 10.5281/zenodo.18575377Paper 17: Bekenstein-Hawking formula — DOI: 10.5281/zenodo.18597306Paper 18: Fermion mass hierarchy and scaffolding principle — DOI: 10.5281/zenodo.18600243Paper 19: WKB tunneling and LLPO — DOI: 10.5281/zenodo.18602596Paper 20: Observable-dependent logical cost, 1D Ising magnetization and WLPO — DOI: 10.5281/zenodo.18603079Paper 21: Bell nonlocality and LLPO — DOI: 10.5281/zenodo.18603251Paper 22: Markov's Principle and radioactive decay — DOI: 10.5281/zenodo.18603503 Note: GitHub repository is not maintained; please see Zenodo for current files.