IndisputableMonolith.Foundation.Phi_Square_Identity
Foundation module packaging the golden-ratio square identity and related cost-domain facts. It defines a domain cost, proves nonnegativity, introduces a positive canonical threshold, and bundles them in an inhabited certificate. Researchers tracing how φ enters RS cost and ladder constructions would cite it. Content is definitional plus short real-arithmetic certificates, not a deep existence proof.
claimModule recording $\varphi^2=\varphi+1$ for the golden ratio $\varphi=(1+\sqrt{5})/2$, a domain cost functional with nonnegativity, a strictly positive canonical threshold, and an inhabited certificate type that packages these algebraic and inequality facts for reuse.
background
Recognition Science forces $\varphi$ as the self-similar fixed point (T6) of the cost structure built on $J(x)=(x+x^{-1})/2-1$. The classical relation $\varphi^2=\varphi+1$ is the algebraic skeleton of that fixed point and reappears in the phi-ladder mass formula and octave bookkeeping.
The module lives in Foundation and imports Constants ($\tau_0=1$ tick) and Cost. Sibling definitions introduce a domain cost (evaluation of cost on a chosen domain), equality at evaluation points, nonnegativity of that cost, and a positive canonical threshold. A certificate structure then packages the square identity with those inequalities so downstream proofs can assume one inhabited object rather than re-prove elementary $\varphi$-algebra.
proof idea
Definition-and-certificate module, not a long derivation. Domain cost and canonical threshold are definitions; nonnegativity and positivity are short real-arithmetic arguments. The certificate is a structure bundling the identity with the threshold facts; an inhabited instance supplies a canonical witness. No deep tactic scripts or appeals to the full forcing chain: the work is packaging known golden-ratio algebra for later foundation lemmas.
why it matters in Recognition Science
Anchors the elementary $\varphi^2=\varphi+1$ fact inside the Foundation layer that supports T6 (phi forced as self-similar fixed point) and later cost or threshold arguments. No direct used-by edges are recorded on this page, but the certificate and threshold siblings are the reusable handles any proof needs when quoting the square identity together with a positive cost cutoff. Upstream of phi-ladder and octave constructions that treat the same identity as given.
scope and limits
- Does not derive φ from the J-cost fixed-point equation; takes the constant from Constants.
- Does not prove uniqueness of φ among positives from the square identity alone.
- Does not link the identity to the Recognition Composition Law or eight-tick period.
- Does not supply numerical bounds on α, masses, or rung gaps.
- Does not discharge broader forcing-chain goals (T0–T8).