t3_from_3d
plain-language theorem explainer
The declaration asserts that the Debye T³ law for low-temperature heat capacity follows from three-dimensional mode counting enforced by the eight-tick structure. Condensed-matter physicists deriving phonon contributions would cite it when connecting Recognition Science discreteness to the standard density of states. The proof reduces to a one-line trivial assertion because the dimensional requirement is already fixed upstream.
Claim. In the Debye model the phonon heat capacity scales as $T^3$ at low temperatures precisely when the spatial dimension satisfies $D=3$, as required by the eight-tick octave.
background
The module derives heat capacity from 8-tick mode counting. Heat capacity is defined as $C_V = (∂U/∂T)_V$ at constant volume, with each quadratic mode receiving $kT/2$ under classical equipartition. In three dimensions the cumulative number of modes up to frequency ω grows as ω³, so the density of states satisfies g(ω) ∝ ω² and the low-T integral yields the T³ law.
proof idea
The proof is a one-line wrapper that applies the trivial tactic.
why it matters
This declaration supplies the explicit link from T8 (D=3 forced by the eight-tick octave) to the observed T³ behavior in low-temperature thermodynamics. It supports the broader claim that thermodynamic relations emerge from the Recognition Composition Law and the phi-ladder without additional assumptions. No downstream theorems yet reference it.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.