IndisputableMonolith.Foundation.Phi_Square_Identity
Foundation module packaging the golden-ratio square identity φ² = φ + 1 against the RS cost J, together with a domain cost, a canonical positive threshold, and an inhabitable certificate bundle. Analysts of the T5–T6 forcing chain cite it when they need a checked algebraic identity linking φ-powers to J-values. The module is mostly definitions plus short nonnegativity and evaluation lemmas.
claimPackage the identity $\varphi^2 = \varphi + 1$ with the RS cost $J(x) = (x + x^{-1})/2 - 1$, a domain cost functional, a strictly positive canonical threshold, and a certificate record asserting the square identity holds in the cost calculus.
background
Recognition Science forces the unique cost $J(x) = \cosh(\log x) - 1$ (equivalently $(x + x^{-1})/2 - 1$) at T5 and the golden ratio $\varphi$ as the self-similar fixed point at T6. The elementary relation $\varphi^2 = \varphi + 1$ therefore sits inside every subsequent $\varphi$-ladder computation (mass rungs, eight-tick periods, coupling bands).
This module lives in Foundation and imports Constants (RS-native units, including the tick $\tau_0$) and Cost (the $J$-calculus). It introduces a domain cost, its pointwise evaluation and nonnegativity, a canonical positive threshold, and a certificate type PhiSqIdentCert whose inhabitation records that the square identity is available to the cost layer.
No external paper proposition is attached in the source; the module is infrastructure for later forcing and mass-ladder arguments that rewrite $\varphi$-powers via the square relation.
proof idea
Definition-heavy module. Domain cost and canonical threshold are introduced as defs; nonnegativity and positivity are short tactic or term proofs from the Cost import and the positivity of $\varphi$. The certificate record is a structure; inhabitation is a one-line constructor application once the algebraic identity and threshold facts are in hand. No deep inductive or analytic argument appears.
why it matters in Recognition Science
Supplies a reusable, type-checked handle on $\varphi^2 = \varphi + 1$ inside the $J$-cost calculus, which every later $\varphi$-ladder identity (mass formula yardstick $\cdot \varphi^{\mathrm{rung}-8+\mathrm{gap}(Z)}$, eight-tick octave, $\alpha$ band) ultimately rewrites through. Downstream use edges are not yet recorded on this page, so the module presently functions as Foundation scaffolding rather than a cited leaf of a named parent theorem. It closes the algebraic gap between T6 ($\varphi$ forced) and concrete power reductions used in Constants and Cost clients.
scope and limits
- Does not derive uniqueness of J or force φ; those are T5–T6 upstream.
- Does not prove mass-ladder or alpha-band numerics.
- Does not establish eight-tick or D = 3 results.
- Does not claim a paper proposition number beyond Foundation infrastructure.
- Certificate inhabitation is local; it does not discharge global forcing hypotheses.