IndisputableMonolith.Physics.DebyeFrequencyFromPhiLadder
Defines Debye frequency as a phi-ladder threshold quantity in RS-native units, via a non-negative domain cost that vanishes at equilibrium and a positive canonical threshold. Supplies a DebyeFreqCert bundle and an inhabited certificate. Condensed-matter or lattice-dynamics work in the RS stack would cite it when tying phonon cutoffs to the same J-cost and phi scaffolding used for masses and constants.
claimOn the $\varphi$-ladder, a domain cost $C$ is non-negative and vanishes at equilibrium. A canonical threshold $\theta>0$ is fixed from that cost. Debye frequency is packaged as a certificate $\mathrm{DebyeFreqCert}$ asserting that the phonon cutoff is the ladder quantity determined by $\theta$ in RS units ($c=1$, tick $\tau_0$).
background
Recognition Science places all dimensionful scales on a discrete $\varphi$-ladder generated by the unique self-similar fixed point of the J-cost (T5–T6 in the forcing chain). The Cost import supplies the recognition cost $J$, with $J(x)=(x+x^{-1})/2-1$, obeying the Recognition Composition Law. Constants supplies the RS time quantum $\tau_0=1$ tick and the native unit conventions ($c=1$, $\hbar=\varphi^{-5}$).
In ordinary solid-state physics the Debye frequency $\omega_D$ is the cutoff of the linear phonon spectrum. Here the module treats that cutoff as a ladder-derived threshold rather than an empirical input: a domain cost measures departure from equilibrium on the recognition ledger, and the canonical threshold is the positive scale at which that cost licenses a collective mode cutoff.
Sibling definitions in the module are the cost functional, its equilibrium vanishing and non-negativity, the positive canonical threshold, and the certificate type that packages the Debye claim.
proof idea
Definition-and-certificate module, not a single deep theorem. It introduces domainCost, proves non-negativity and vanishing at equilibrium, defines canonicalThreshold with a positivity lemma, then bundles these into DebyeFreqCert with an inhabited cert witness. Argument structure is: cost axioms from the Cost import, threshold extraction, certificate assembly. No long tactic scripts; the load-bearing content is the identification of $\omega_D$ with the ladder threshold in RS units.
why it matters in Recognition Science
Closes a condensed-matter gap in the RS physics layer: phonon cutoffs sit on the same $\varphi$-ladder and J-cost scaffolding as particle masses (yardstick $\cdot\varphi^{\mathrm{rung}-8+\mathrm{gap}(Z)}$) and the forced constants. Downstream use is not yet wired in this graph (used_by empty), so the module is a leaf certificate ready for solid-state or thermal derivations that need a first-principles $\omega_D$ rather than a fit parameter. It sits downstream of Constants and Cost only, keeping the Debye claim auditable against T5 J-uniqueness and the native tick $\tau_0$.
scope and limits
- Does not derive numerical Debye temperatures for named materials.
- Does not prove uniqueness of the domain cost beyond the stated equilibrium and non-negativity properties.
- Does not connect Debye frequency to measured specific heats or phonon DOS data.
- Does not force spatial dimension or the eight-tick octave; those live in the T7–T8 chain.
- Does not supply a used_by parent theorem in the current graph.