IndisputableMonolith.Verification.Necessity.PhiNecessity
Module packaging the minimal self-similarity axioms that force the golden ratio as the unique preferred scale. A verification or foundations reader cites it for the T6 step: self-similarity plus a positive fixed scale yields φ. The argument chains discrete self-similarity into a fixed-point equation whose only positive solution is φ.
claimUnder a minimal self-similarity structure on a positive preferred scale $s$, one has $s = 1 + 1/s$, hence $s = \varphi := (1+\sqrt{5})/2$, the unique positive root of $x^2 = x + 1$.
background
Recognition Science forces the dimensionless scale factor $\varphi$ as the unique self-similar fixed point (forcing chain T6). The cost functional $J$ and the Recognition Composition Law already single out multiplicative structure; self-similarity then pins the numerical value of the rung ratio.
This module sits in the Verification/Necessity layer. It imports RS constants (including the native tick $\tau_0$) and the elementary golden-ratio support lemmas: $\varphi^2 = \varphi + 1$, the fixed-point identity $\varphi = 1 + 1/\varphi$, and uniqueness of the positive root of $x^2 = x + 1$. Those facts depend only on real algebra and Mathlib's goldenRatio API.
The local notion is a bare self-similarity interface (a positive preferred scale closed under the discrete doubling/halving that the eight-tick octave later uses). No continuum limit or full J-calculus is required at this stage.
proof idea
The module is theorem-bearing, not a pure definition file. It introduces a minimal self-similarity predicate, derives positivity of the preferred scale, obtains the fixed-point relation $s = 1 + 1/s$ from discrete self-similarity, and invokes the uniqueness lemma for the positive root of $x^2 = x + 1$ to conclude $s = \varphi$. Downstream named results package the same chain as "self-similarity forces $\varphi$" and "$\varphi$ is mathematically necessary."
why it matters in Recognition Science
Closes the T6 landmark: $\varphi$ is forced as the self-similar fixed point once a positive preferred scale exists. That value propagates into every RS-native constant ($\hbar = \varphi^{-5}$, $G = \varphi^5/\pi$, the mass ladder yardstick $\varphi^{\mathrm{rung}-8+\mathrm{gap}(Z)}$, Berry threshold $\varphi^{-1}$, $Z_{\mathrm{cf}} = \varphi^5$). The module supplies the necessity certificates that later verification and inevitability pipelines quote when they claim $\varphi$ is not an empirical fit. No external used-by edges are recorded on the page; the sibling theorems themselves are the export surface for the rest of the monolith.
scope and limits
- Does not derive self-similarity from the J-cost or RCL; that is assumed as interface.
- Does not prove the eight-tick octave or D = 3 (T7–T8).
- Does not fix dimensionful units beyond the pure scale ratio φ.
- Does not address mass gaps, α, or continuum field equations.