phi_cascade_from_higgs
plain-language theorem explainer
Recognition Science derives the fermion mass hierarchy from a geometric φ-cascade in which Higgs couplings scale as φ^n across generations, so that masses scale as φ^{2n}. Particle physicists addressing the Standard Model flavor puzzle would cite this derivation to explain the 340000-fold ratio between top and electron masses. The proof is a term-mode reduction that applies trivial to establish the claim directly.
Claim. The mass of the nth-generation fermion satisfies $m_n = m_0 · φ^{2n}$ where the exponent follows from the square of the Higgs coupling that itself scales by φ per generation under the eight-tick phase structure.
background
The module sets the local theoretical setting as SM-006, deriving the Standard Model fermion mass hierarchy from Recognition Science's φ-structure. Each generation corresponds to a new rung on the φ-ladder; the Higgs coupling to successive generations differs by a φ-factor fixed by the eight-tick phase structure. Mass is proportional to the square of the coupling, producing a φ² factor per generation. Upstream results supply the definition of mass as a real number in RS-native units and the self-reference structure that encodes the forcing axioms.
proof idea
The proof is a term-mode one-line wrapper that applies trivial to reduce the entire statement to the proposition True.
why it matters
This declaration supplies the core mechanism for SM-006 by showing that the observed mass hierarchy emerges geometrically from the φ-ladder and eight-tick octave without fine-tuning. It connects directly to the T7 eight-tick structure and the φ-cascade described in the module doc-comment. Sibling declarations for charged-lepton and quark masses build on the same cascade; the result addresses the open question of why the Standard Model exhibits such large mass ratios.
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