massHierarchyPattern
The definition states that fermion masses follow exponential scaling on the φ-ladder with generation-dependent exponents, matching observed ratios such as top/charm near φ^10 and muon/electron near φ^11. Particle physicists examining Standard Model hierarchies would cite this when linking Recognition Science to data. The declaration is a direct string assignment with no lemmas or reductions.
claimFermion masses satisfy $m ∝ φ^n$ where the exponent $n$ is generation-dependent, producing ratios such as $m_t/m_c ≈ φ^{10}$, $m_c/m_u ≈ φ^{13}$, $m_τ/m_μ ≈ φ^6$, and $m_μ/m_e ≈ φ^{11}$.
background
The module derives exactly three fermion generations from the 8-tick cycle (period 2^3) distributed across three spatial dimensions, with each generation indexed by parity combinations of three bits. The φ-ladder assigns discrete tiers to physical quantities via integer rungs, as in upstream scale(k) := φ^k and rung functions that label classes by level. Upstream structures such as PhiForcingDerived.of and LedgerFactorization.of calibrate the underlying J-cost that enforces these exponential scalings.
proof idea
This is a one-line definition that directly assigns the descriptive string literal summarizing the φ^n scaling pattern for masses across generations.
why it matters in Recognition Science
This definition supplies the mass hierarchy pattern supporting the three-generations derivation from the 8-tick octave and 3D structure. It connects to framework landmarks T7 (eight-tick octave) and T8 (D=3) plus the mass formula on the φ-ladder. It addresses the open question of why exactly three generations exist, as flagged in the module's PRL paper proposition.
scope and limits
- Does not derive the specific exponents from the forcing chain.
- Does not compute numerical mass values beyond the listed approximations.
- Does not incorporate the yardstick or gap(Z) terms from the general mass formula.
- Does not reference particular upstream lemmas beyond the φ-ladder convention.
formal statement (Lean)
120def massHierarchyPattern : String :=
proof body
Definition body.
121 "Masses scale as φ^n for generation-dependent n"
122
123/-- **THEOREM (Hierarchy from φ-ladder)**: Each generation sits on a different
124 rung of the φ-ladder, giving exponential mass ratios. -/