log_phi_gt_0481
plain-language theorem explainer
The theorem establishes a rigorous lower bound log(φ) > 0.481 using the monotonicity of the exponential function and Taylor series approximations. Numerics researchers in the Recognition Science framework would cite it when bounding the golden ratio logarithm for gap calculations. The proof proceeds by rewriting the inequality as φ > exp(0.481), then bounding the exponential via a 10-term Taylor sum plus error term and comparing to a known lower bound on φ.
Claim. $0.481 < log φ$ where $φ$ is the golden ratio.
background
The Numerics.Interval.Log module supplies rigorous interval bounds for the natural logarithm via Mathlib's monotonicity results and Taylor error estimates. For arguments in (0, ∞) the logarithm is increasing, so interval images are contained in the image of the endpoints; for log(1 + x) with |x| < 1 the Taylor series with remainder |x|^{n+1}/((n+1)(1 - |x|)) is used. The present declaration works indirectly by converting the target log inequality into an exponential comparison, relying on the precomputed Taylor sum and error at 0.481 together with the lower bound φ > 1.61803395.
proof idea
The proof rewrites the goal via Real.lt_log_iff_exp_lt, records that 0.481 lies in [0, 1], invokes Real.exp_bound' to obtain the 10-term Taylor sum plus remainder, equates both pieces to the precomputed constants exp_taylor_10_at_0481 and exp_error_10_at_0481 by simp and norm_num, then chains the resulting upper bound on exp(0.481) through the lemma exp_0481_lt_phi and the lower bound phi_gt_161803395.
why it matters
The bound is invoked by f_gap_gt_strong to strengthen the gap-term estimate inside the alpha-bounds development. It supplies a concrete numerical anchor for the golden-ratio logarithm that appears throughout the Recognition Science forcing chain (T5–T6) where φ is the self-similar fixed point; the same constant enters mass-ladder and alpha-band calculations. No open scaffolding remains at this leaf.
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