IndisputableMonolith.Materials.MetamaterialBandGapFromPhiLadder
The module defines the reference band-gap center frequency as the RS-native dimensionless value 1 together with gap frequencies on the phi-ladder. Materials physicists deriving photonic band structures from the self-similar fixed point would cite these constructions. The module is organized as a chain of definitions followed by lemmas on positivity, monotonicity and ratio properties.
claimThe reference band-gap center frequency equals the RS-native dimensionless constant one. Gap frequencies are defined on the natural numbers and satisfy positivity, strict increase, and adjacent ratio relations derived from the phi fixed point.
background
The module sits in the Materials domain and imports the fundamental RS time quantum (RS-native). τ₀ = 1 tick from IndisputableMonolith.Constants. It introduces the reference band-gap center frequency as the RS-native dimensionless 1 along with gap frequency definitions built on the phi-ladder. The local theoretical setting applies the Recognition Composition Law and the eight-tick octave to metamaterial frequency ladders.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
This module supplies the phi-ladder band-gap constructions that realize the MetamaterialBandGapCert object. It connects the forcing chain steps T5-T8 and the phi fixed point to materials applications in RS-native units. No downstream uses appear in the current dependency graph.
scope and limits
- Does not compute numerical values for laboratory metamaterials.
- Does not address three-dimensional lattice geometries.
- Does not incorporate electromagnetic constants outside RS-native units.
- Does not treat fabrication tolerances or experimental verification.