IndisputableMonolith.Crystallography.BraggAngleFromPhiLadder
This module defines the Bragg peak spacing ratio as the golden ratio phi drawn from the Recognition Science phi-ladder. It supplies supporting definitions for diffraction cost and angle certificates that connect the J-cost function to observable X-ray peaks. Condensed-matter physicists or RS modelers cite these when deriving crystal geometry from the T6 fixed-point and T8 dimension results. The module is definitional with basic inequalities and no complex proofs.
claimThe Bragg peak spacing ratio equals $r = phi$, where $phi$ is the self-similar fixed point of the J-cost function $J(x) = (x + x^{-1})/2 - 1$. The associated diffraction cost is nonnegative and the angle certificate is inhabited.
background
Recognition Science obtains phi from the J-uniqueness theorem (T5) as the fixed point satisfying the Recognition Composition Law. The imported Constants module fixes the fundamental time quantum at one tick. The Cost module supplies the recognition cost function used to define diffractionCost. This crystallography module applies those primitives to successive Bragg angles, yielding the ratio phi together with nonnegativity and existence statements.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the crystallography interface required by downstream applications of the eight-tick octave and D = 3 results. It translates the phi-ladder mass formula into peak-ratio predictions that can be checked against diffraction data, closing one link between the forcing chain and laboratory observables.
scope and limits
- Does not derive explicit Miller indices or lattice constants for named crystals.
- Does not incorporate temperature, strain, or finite-size corrections.
- Does not prove that phi is the only admissible ratio under all scattering geometries.
- Does not supply numerical error bounds or comparison with measured spectra.