IndisputableMonolith.Materials.MetamaterialBandGapFromPhiLadder
This module defines the reference band-gap center frequency and associated gap frequencies derived from the phi-ladder in Recognition Science. Materials physicists modeling photonic or acoustic structures in RS-native units would cite these definitions for frequency scaling. The module consists entirely of successive definitions establishing the reference value and monotonicity properties of the gap sequence.
claimThe reference band-gap center frequency equals the dimensionless value 1 in RS-native units. Gap frequencies form a sequence on the phi-ladder satisfying positivity, strict increase, and adjacent ratios fixed by the golden ratio phi.
background
The module imports Mathlib and IndisputableMonolith.Constants. The upstream Constants module defines the fundamental RS time quantum as τ₀ = 1 tick, supplying the base unit for all frequencies. This module introduces the reference frequency as the band-gap center at dimensionless value 1, then defines gapFreq together with auxiliary statements for positivity and ordering on the phi-ladder.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
This module supplies the frequency definitions that support MetamaterialBandGapCert within the same module. It extends the phi-ladder fixed point (T6) and eight-tick octave (T7) to materials applications, providing the frequency yardstick parallel to the mass formula. It touches the step connecting the forcing chain to concrete band-gap centers.
scope and limits
- Does not derive band gaps from Maxwell or wave equations.
- Does not specify metamaterial geometry or lattice parameters.
- Does not compute numerical gap widths or transmission spectra.
- Does not connect to experimental measurements or material constants.