IndisputableMonolith.Verification.PhiPowerBoundsCert
Verification module packaging certified numerical bounds on integer powers of the golden-ratio fixed point φ used throughout Recognition Science. Auditors of mass-ladder exponents, ħ = φ^{-5}, and G = φ^5/π cite it for machine-checked inequalities rather than floating-point claims. The module is a thin certificate layer over Constants; proofs are bound certificates, not a new derivation of φ.
claimA verification certificate recording rigorous bounds on $\varphi^n$ (and related reciprocal powers) for the self-similar fixed point $\varphi$ of Recognition Science, in the native units where the time quantum is one tick.
background
Recognition Science forces a unique dimensionless scale factor $\varphi$ as the self-similar fixed point of the cost functional (forcing step T6). Downstream constants are pure powers of $\varphi$: $\hbar = \varphi^{-5}$, $G = \varphi^5/\pi$, and particle masses sit on a $\varphi$-ladder with integer rungs and a small gap correction.
The imported Constants module supplies the RS-native time quantum $\tau_0 = 1$ tick and the definition of $\varphi$. Any inequality that compares ladder steps, Berry thresholds such as $\varphi^{-1}$, or the dream fraction $\varphi^{-3}$ needs certified bounds on concrete powers $\varphi^{\pm n}$ rather than approximate floats.
This module lives in the Verification domain: it does not re-derive $\varphi$, it only packages the numerical inequalities that other certificates and mass-formula checks may invoke.
proof idea
Certificate-style module: it exposes a named bounds object (PhiPowerBoundsCert) whose fields are Lean inequalities on powers of $\varphi$, discharged by Mathlib arithmetic and the Constants definition of $\varphi$. No independent forcing argument; the structure is bound packaging and reuse, not a multi-step derivation.
why it matters in Recognition Science
Feeds any downstream verification that must compare $\varphi$-ladder quantities without leaving the kernel (mass yardsticks, $\hbar$ and $G$ normalizations, $Z_{\mathrm{cf}} = \varphi^5 \in (11,12)$, Berry threshold $\varphi^{-1}$). With no recorded used_by edges in the mirror graph, it is a leaf certificate library rather than a step inside the T0–T8 forcing chain. It closes the gap between the abstract fixed-point uniqueness of $\varphi$ and concrete decimal or rational bounds needed by numerical audits of $\alpha^{-1}$ and rung formulas.
scope and limits
- Does not prove uniqueness or existence of φ (that is T6 upstream).
- Does not derive the mass formula or gap(Z); only bounds on pure powers.
- Does not certify the fine-structure band (137.030, 137.039).
- Does not claim floating-point evaluation; only Lean-proved inequalities.
- Does not introduce new physics constants beyond those in Constants.