IndisputableMonolith.Algebra.PhiRing
Algebra.PhiRing supplies algebraic identities and positivity facts for the golden ratio phi derived from sqrt(5). Researchers building the phi-ladder or J-uniqueness arguments would cite it. The module collects elementary lemmas on the quadratic equation, conjugate relations, and basic inequalities.
claimLet $\phi = \frac{1 + \sqrt{5}}{2}$. Then $\phi > 1$, $\phi^2 = \phi + 1$, $\psi = \phi^{-1}$ satisfies the conjugate equation, and the product and sum identities $\phi \psi = 1$, $\phi + \psi = \sqrt{5}$ hold.
background
The module belongs to the Algebra domain and imports Mathlib, Cost, and CostAlgebra. It introduces phi via its minimal polynomial and positivity statements such as sqrt5_pos and phi_pos. The setting is the self-similar fixed point required by the unified forcing chain after J-uniqueness.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
This module feeds the RecognitionCategory module by supplying the algebraic primitives for phi. It fills the T6 step that forces phi as the fixed point and supports later constants such as G = phi^5 / pi.
scope and limits
- Does not prove the full forcing chain from T0 to T8.
- Does not define the J-cost function or Recognition Composition Law.
- Does not address mass formulas or the alpha band.