phi_11_hierarchy_lower
The inequality establishes phi^11 > 180 as a conservative lower bound supporting the phi-ladder for fermion mass hierarchies. Physicists modeling particle mass ratios via Recognition Science predictions would cite this result. The proof substitutes the Fibonacci closed form for phi^11 and discharges the comparison with linear arithmetic on the bound phi > 1.61.
claim$180 < phi^{11}$ where $phi = (1 + sqrt(5))/2$ is the golden ratio.
background
This theorem sits in the module of calculated proofs for registry predictions, covering the fermion mass hierarchy prediction P-002 with explicit phi^6 and phi^11 structure. The local setting uses the phi-ladder mass formula in which masses scale as yardstick times phi to a power involving rung and gap(Z), together with the Recognition Composition Law and the eight-tick octave. Upstream results supply the key identity phi^11 = 89 phi + 55 by Fibonacci recurrence and the tighter bound phi > 1.61.
proof idea
The term-mode proof first rewrites the goal via the lemma phi^11 = 89 * phi + 55, then invokes linear arithmetic on the resulting inequality using the supplied lower bound phi > 1.61.
why it matters in Recognition Science
The result supplies the phi_11 lower bound inside the registry predictions certificate that confirms the mass hierarchy prediction P-002. It aligns with the Recognition Science phi-ladder and the mass formula yardstick * phi^(rung - 8 + gap(Z)), closing one of the calculated proofs listed in the module for the COMPLETE_PROBLEM_REGISTRY.
scope and limits
- Does not establish any upper bound on phi^11.
- Does not map specific particles to particular rungs on the phi-ladder.
- Does not derive the golden ratio itself from the forcing chain T0-T8.
- Does not address the cosmological constant bounds proved in the same module.
formal statement (Lean)
150theorem phi_11_hierarchy_lower : (180 : ℝ) < (phi : ℝ)^11 := by
proof body
Term-mode proof.
151 rw [phi_eleventh_eq]
152 linarith [phi_gt_onePointSixOne]
153
154/-- **HIERARCHY STRUCTURE**: Mass ratios are φ-powers of integer differences. -/