phi_critical_numeric
plain-language theorem explainer
The critical energy scale at the golden ratio satisfies 0.09 < J(φ) < 0.12. Condensed matter modelers using Recognition Science for lattice phase transitions would cite this interval to bound coherence energies in superconducting predictions. The proof reduces the expression via the closed-form identity for J(φ), substitutes the inverse relation from the quadratic equation, and resolves the bounds with linear arithmetic on the supplied numerical estimates for φ.
Claim. Let $J(x) = (x + x^{-1})/2 - 1$. Then $0.09 < J(ϕ) < 0.12$, where $ϕ$ is the golden ratio.
background
The JCostPhaseTransition module defines the critical energy scale as the J-cost function evaluated at the golden ratio, where J-cost encodes the recognition energy for phase transitions in condensed-matter lattices. The upstream phi_critical_value theorem supplies the explicit reduction $J(ϕ) = (ϕ + ϕ^{-1})/2 - 1$. The module draws on Constants for the golden-ratio definition, its quadratic identity $ϕ^2 = ϕ + 1$, and the tight numerical bounds 1.61 < ϕ < 1.62.
proof idea
The tactic proof first rewrites via the phi_critical_value theorem. It derives the inverse identity ϕ^{-1} = ϕ - 1 from phi_sq_eq together with field simplification and positivity. It then applies the two bounding lemmas on ϕ and closes both sides of the conjunction with linear arithmetic.
why it matters
This supplies the concrete numerical interval required for the falsifiable prediction that phi-structured superconducting lattices yield critical temperatures 80-120 K when coherence energy sits near 0.09 eV. It quantifies the J-uniqueness step (T5) inside the condensed-matter application of the forcing chain and supports downstream doping estimates at carrier density 1/ϕ^2.
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