phi_ladder
plain-language theorem explainer
The phi-ladder is the set of all real numbers obtained as integer powers of the golden ratio φ. Nucleosynthesis and stillness proofs cite it to restrict mass-to-light ratios and tier values to discrete scaling steps. The definition is a direct set comprehension over ℤ-exponents supplied by PhiForcing.
Claim. The φ-ladder is the set $S = { x ∈ ℝ | ∃ n ∈ ℤ, x = φ^n }$, where φ is the golden ratio fixed point of the J-cost function.
background
Recognition Science treats existence as the outcome of defect collapse to zero under the J-cost. The phi-ladder supplies the discrete scaling skeleton on which stable configurations sit. The module OntologyPredicates defines RSExists and RSTrue via cost minimization, then introduces the ladder as the concrete set of allowed values.
proof idea
The definition is a direct set comprehension that applies the integer exponentiation from PhiForcing. No lemmas or tactics are required beyond the set-builder syntax.
why it matters
This set is the target of all_ml_on_phi_ladder, which places every population M/L value on the ladder, and of ml_nucleosynthesis, which equates the nucleosynthesis-derived ratio to φ itself. It realizes the self-similar fixed point φ forced at T6 and supplies the discrete steps for the eight-tick octave.
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