IndisputableMonolith.PhiSupport
This module supplies the algebraic lemmas for the golden ratio φ satisfying φ² = φ + 1. All mass and mixing derivations in the Recognition Science framework import it to justify phi-ladder scaling. The proofs are direct algebraic expansions from the quadratic definition.
claim$\phi = \frac{1 + \sqrt{5}}{2}$ satisfies $\phi^2 = \phi + 1$ with $\phi > 1$.
background
Recognition Science forces φ as the self-similar fixed point (T6) of the J-cost function. This module supplies the supporting algebraic facts. It imports the base time quantum τ₀ = 1 tick from Constants. The three siblings are phi_squared (the equation φ² = φ + 1), phi_fixed_point' (an equivalent fixed-point form), and one_lt_phi (the inequality φ > 1).
proof idea
This is a definition module whose siblings contain direct algebraic verifications. Each lemma expands the quadratic definition of φ and checks the identity or inequality by elementary arithmetic.
why it matters in Recognition Science
The module feeds the core derivations in T9 (ElectronMass.Defs), T10 (LeptonGenerations), T11 (CKMGeometry), and T12 (QuarkMasses). It closes the algebraic support for the phi-ladder required by the forcing chain after T6.
scope and limits
- Does not derive φ from the J-cost fixed-point equation.
- Does not connect φ to physical units or constants.
- Does not address the eight-tick octave or spatial dimension D = 3.
used by (9)
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IndisputableMonolith.Physics.CKMGeometry -
IndisputableMonolith.Physics.ElectronMass.Defs -
IndisputableMonolith.Physics.LeptonGenerations.Defs -
IndisputableMonolith.Physics.LeptonGenerations.Necessity -
IndisputableMonolith.Physics.QuarkMasses -
IndisputableMonolith.RRF.Physics.ElectronMass.Defs -
IndisputableMonolith.RRF.Physics.LeptonGenerations.Defs -
IndisputableMonolith.RRF.Physics.LeptonGenerations.Necessity -
IndisputableMonolith.RRF.Physics.QuarkMasses