IndisputableMonolith.Foundation.GoldenRatio_Uniqueness_v3
Foundation module packaging a uniqueness certificate for the golden ratio as the self-similar fixed point of the Recognition cost. Anyone citing T6 (phi forced) or the forcing chain would land here. The argument is certificate-shaped: a domain cost, its nonnegativity and equality properties, a positive canonical threshold, and an inhabited cert record.
claimThe golden ratio $\varphi=(1+\sqrt{5})/2$ is uniquely characterized as the positive self-similar fixed point under the Recognition $J$-cost. The module supplies a domain cost functional, a canonical positive threshold, and a certificate record asserting nonnegativity of that cost together with the threshold bound that pins $\varphi$.
background
Recognition Science forces constants from a single cost functional $J$. Landmark T5 identifies $J(x)=(x+x^{-1})/2-1$ (equivalently $\cosh(\log x)-1$) as the unique symmetric cost obeying the Recognition Composition Law. Landmark T6 then forces $\varphi$ as the self-similar fixed point of that cost structure: the unique scale at which the ladder closes under the same $J$.
This module sits in the Foundation layer and imports the RS constants (including the native time quantum) and the Cost API. Sibling objects introduce a domain-restricted cost, its evaluation identity and nonnegativity, and a canonical threshold with a positivity lemma. Those pieces are bundled into a certificate type GoldenRatio_v3Cert with an inhabited instance, so downstream forcing steps can consume a single package rather than a scatter of lemmas.
The local setting is therefore not a fresh derivation of $J$, but a uniqueness-and-threshold packaging of $\varphi$ once $J$ and the cost interface are already available.
proof idea
Definition-and-certificate module rather than a single deep proof. It defines a domain cost and records elementary facts (evaluation identity, nonnegativity). It defines a canonical threshold and proves that threshold is positive. Those facts are assembled into a certificate structure with an inhabited instance, so the uniqueness claim for $\varphi$ is carried as a packaged cert rather than a bare equality theorem. No long tactic script is required at module scope; the work is the interface and the positivity/nonnegativity lemmas that fill the cert fields.
why it matters in Recognition Science
T6 in the forcing chain (T0–T8) states that $\varphi$ is forced as the self-similar fixed point once $J$ is unique. This module is the v3 packaging of that uniqueness story: domain cost, canonical threshold, and an inhabited certificate that later Foundation and Constants consumers can import without re-proving the elementary inequalities.
Used_by is empty at the graph snapshot, so the module is presently a leaf supplier rather than a mid-chain hinge. Its value is still structural: the eight-tick octave (T7), $D=3$ (T8), and the mass ladder all assume a single rigid $\varphi$. A clean cert for golden-ratio uniqueness is what keeps those later steps from smuggling an unforced scale. The alpha band and RS-native unit choices ($\hbar=\varphi^{-5}$, $G=\varphi^5/\pi$) likewise presuppose that $\varphi$ is not an arbitrary fitting parameter.
scope and limits
- Does not re-derive J-uniqueness (T5); assumes the Cost API and RCL background.
- Does not prove the full T0–T8 forcing chain; only the phi-uniqueness certificate package.
- Does not derive physical constants (c, hbar, G, alpha) or the mass ladder.
- Does not assert dimensional results (T8) or the eight-tick period (T7).
- Does not expose a single top-level equality theorem; the claim is certificate-shaped.