log_phi_ne_zero
plain-language theorem explainer
The theorem establishes that the natural logarithm of the golden ratio is nonzero. Researchers formalizing gauge invariance on the phi-ladder or discrete phase compactification cite it to justify division by log phi without division-by-zero. The proof is a one-line term application of ne_of_gt to the already-proved positivity of log phi.
Claim. $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$ is false, hence $0 < 0$
background
The phi-ladder lattice module formalizes the geometric sequence of powers of the golden ratio phi = (1 + sqrt(5))/2 on the positive reals, which becomes an additive lattice r * log phi on the log scale. This structure is forced by self-similarity (T6) and supports Poisson summation on the log line. The sibling theorem log_phi_pos asserts 0 < Real.log phi because phi > 1, with the same statement appearing in upstream modules on biodiversity scaling and black-hole horizon states.
proof idea
The proof is a one-line term wrapper that applies ne_of_gt directly to the upstream theorem log_phi_pos.
why it matters
It supplies the non-vanishing needed for the gauge-invariance theorem compactPhase_gauge_invariant, which shows that compactPhase(r + n * log phi) equals compactPhase(r) for integer n. This step closes the lattice property required for reciprocal symmetry on the phi-ladder and for the sub-conjecture on phi-Poisson summation. It directly supports the claim that n * log phi generates valid integer displacements without collapse.
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