IndisputableMonolith.Materials.PhiLadderPhononResonance
This module defines the phonon resonance frequency on the Recognition Science phi-ladder as ω_p(k) = ω_0 φ^k. Materials researchers modeling high-Tc hydrides cite it for discrete rung selection in optimization landscapes. The module supplies supporting definitions and elementary properties of the rung map with no complex derivations.
claimThe phonon resonance frequency at rung \(k\) is given by \(\omega_p(k) = \omega_0 \cdot \phi^k\).
background
The module imports Constants, where the fundamental RS time quantum satisfies τ₀ = 1 tick, and Cost. It operates inside the phi-ladder construction already present in the Recognition framework, where discrete rungs are indexed by integers and scaled by successive factors of the self-similar fixed point φ.
Sibling declarations introduce phonon_rung as the integer index, together with lemmas establishing that the map is positive, strictly increasing, and satisfies a constant ratio between adjacent rungs. These objects sit in the Materials domain and prepare the ground for resonance calculations that later feed superconductor models.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the phonon resonance model that is deepened in HydrideSCOptimization for the single-parameter φ-rung search in high-Tc hydrides (H₃S, LaH₁₀, YH₆). It fills Track E6 of Plan v5 and supplies the resonance frequency that the downstream module uses to optimize the hydrogen-dominant landscape.
scope and limits
- Does not derive the numerical value of the base frequency ω₀.
- Does not prove any resonance condition for superconductivity.
- Does not incorporate temperature or damping effects.
- Does not address multi-phonon or anharmonic interactions.