log_phi_ne_zero
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 in Recognition Science
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.
scope and limits
- Does not prove irrationality of log phi.
- Does not give a numerical bound on |log phi|.
- Does not address lattices in dimensions other than one.
Lean usage
have h : (r + n * Real.log phi) / Real.log phi = r / Real.log phi + n := by rw [add_div, mul_div_cancel_of_imp (fun h => absurd h log_phi_ne_zero)]formal statement (Lean)
116theorem log_phi_ne_zero : Real.log phi ≠ 0 := ne_of_gt log_phi_pos
proof body
Term-mode proof.
117
118/-- `log (φ⁻¹) = -log φ`. -/