no_heat_death
plain-language theorem explainer
The theorem proves that the 8-tick and 45-tick cadences cannot align at every natural-number tick, blocking the synchronized zero-defect state required for heat death. Cosmologists citing the perpetual complexity result would invoke it to establish that local complexity persists at every epoch. The proof is a one-line term-mode negation that supplies the explicit counterexample t=1.
Claim. It is not the case that every natural number tick $t$ satisfies $t ≡ 0 mod 8$ and $t ≡ 0 mod 45$ simultaneously.
background
The PerpetualComplexity module combines two results: positive vacuum energy carried by passive modes and the fact that gcd(8,45)=1. The 8-tick period is the octave forced by the self-similar fixed point in the forcing chain; the 45-tick cycle arises from synchronization minimization. The module states that this pair “guarantees that the universe generates local complexity at every epoch. Thermal death is impossible.”
proof idea
The term proof first applies push_neg to the negated universal quantifier, then supplies the witness pair ⟨1, Or.inl (by norm_num)⟩. Norm_num verifies that 1 mod 8 ≠ 0, discharging the disjunct and completing the existential.
why it matters
This corollary populates the no_heat_death field of the PerpetualComplexityCert structure, which in turn certifies the perpetualComplexity theorem. It directly discharges Theorem 10.1 of Dark_Energy_Mode_Counting.tex §10 by showing that the coprimeness of the eight-tick octave and the 45-tick cycle prevents global synchronization. In the Recognition Science framework the result closes the argument that perpetual misalignment forces ongoing complexity generation.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.