IndisputableMonolith.Verification.PhiPositivityCert
Certificate module asserting that the Recognition Science self-similarity constant φ is strictly positive. Verification and mass-ladder developments cite it whenever a denominator, log, or rung exponent needs φ > 0. The argument is a short positivity check on the closed-form fixed point of the cost functional, imported from the Constants layer.
claimThe golden-ratio fixed point $\varphi = (1+\sqrt{5})/2$ of the Recognition self-similarity equation satisfies $\varphi > 0$. The module packages this fact as a reusable positivity certificate for downstream verification.
background
Recognition Science forces a unique dimensionless scale $\varphi$ as the self-similar fixed point of the J-cost (forcing step T6). In RS-native units one has $c=1$, $\hbar=\varphi^{-5}$, and masses live on the $\varphi$-ladder; every such expression presupposes $\varphi>0$.
The Constants import supplies the fundamental tick $\tau_0=1$ and the closed form of $\varphi$. This module sits in the Verification domain: it does not redefine $\varphi$, but records a machine-checkable positivity fact so that later certificates (mass formulae, $\alpha$-band bounds, eight-tick combinatorics) can invoke it without re-proving the inequality.
proof idea
Definition-and-certificate module rather than a deep derivation. It exposes a named positivity lemma for $\varphi$, discharged by the standard algebraic identity $\varphi=(1+\sqrt{5})/2$ together with $\sqrt{5}>0$ from Mathlib. No multi-step tactic script; the content is a thin wrapper around the Constants definition of $\varphi$.
why it matters in Recognition Science
Positivity of $\varphi$ is a silent hypothesis under almost every quantitative RS claim: the mass yardstick $\varphi^{r-8+\mathrm{gap}(Z)}$, the Berry threshold $\varphi^{-1}$, $Z_{\mathrm{cf}}=\varphi^5\in(11,12)$, and the dream fraction $\varphi^{-3}$. Without a centralized certificate, each verification file would re-open the same inequality. The module has no recorded downstream edges yet; it is infrastructure for the Verification layer that will feed mass, coupling, and octave certificates once those files import it.
scope and limits
- Does not derive the value of φ; only certifies positivity of the already-defined constant.
- Does not prove uniqueness of φ (that is T6 in the forcing chain).
- Does not establish any mass, coupling, or dimensional claim by itself.
- Does not address φ < 2 or other interval bounds beyond strict positivity.