phi_eq_goldenRatio
plain-language theorem explainer
The Recognition Science constant phi equals the mathematical golden ratio. Astrophysicists deriving mass-to-light ratios from J-cost minimization under recognition-length and coherence-energy constraints cite this equality to anchor phi-ladder scalings to observable stellar systems. The proof reduces directly to unfolding the two definitions via simplification.
Claim. The constant phi from the CPM constants bundle equals the golden ratio.
background
The module derives mass-to-light ratios from observability constraints imposed by recognition length l_rec and fundamental tick tau_0. Observable flux must exceed the coherence energy threshold E_coh over tau_0, while mass assembly is limited by coherence volume; the optimal configuration minimizes total J-cost J_mass plus J_light and yields M/L ratios in powers of phi. Constants is the abstract bundle holding Knet, Cproj, Ceng, Cdisp together with the nonnegativity condition on Knet. Upstream lemmas in ElectroweakMasses and Verification establish the identical equality by unfolding Constants.phi and Real.goldenRatio followed by ring simplification.
proof idea
The proof is a one-line wrapper that applies simp to unfold Constants.phi and goldenRatio, reducing both sides to the same expression.
why it matters
This equality anchors the phi-ladder scalings used in mass predictions and IMF derivations. It feeds the theorem imf_from_j_minimization, which shows the IMF slope alpha satisfies 2 < alpha < 3 and lies within 0.3 of phi^2. The result supports the self-similar fixed point T6 and the eight-tick octave by supplying the numerical base for phi^n mass formulas across the Recognition framework.
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