log_phi_ne_zero
plain-language theorem explainer
Physicists deriving discrete gauge symmetries from the forcing chain cite this lemma to ensure that integer multiples of ln φ generate well-defined phase shifts under 8-tick compactification. It appears in proofs that normalize log-ratios by ln φ without division by zero. The argument is a direct one-line application of ne_of_gt to the upstream positivity statement for log φ.
Claim. $0 < 1 + 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 < 1/2 <
background
The GCIC module closes an acknowledged gap in the Response paper by deriving the discrete gauge r ~ r + n·ln φ from T6 (φ-self-similarity) and T7 (8-tick neutrality) rather than treating it as an input. The upstream theorem log_phi_pos states that 0 < Real.log phi and is obtained via Real.log_pos applied to one_lt_phi. This lemma supplies the non-vanishing denominator required when normalizing log-ratios to obtain the compact phase Θ ∈ ℝ/ℤ.
proof idea
The proof is a one-line wrapper that applies ne_of_gt directly to log_phi_pos.
why it matters
It is invoked by compactPhase_gauge_invariant to establish that compactPhase(r + n·ln φ) = compactPhase(r), thereby confirming invariance under the discrete gauge. The result also feeds the phi-ladder lattice constructions. It supplies the missing non-zero condition that lets the 8-tick compactification follow from the forcing-chain steps T6 and T7 without additional hypotheses.
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