log_lower_numerical
plain-language theorem explainer
This theorem supplies a concrete numerical lower bound for the natural logarithm of the golden-ratio approximation 1.618033. Researchers deriving the electron mass in Recognition Science cite it when establishing interval bounds on log phi that feed the phi-ladder mass formula. The proof rewrites the claim via the exponential-logarithm equivalence and invokes a Taylor remainder estimate for exp at 0.481211, reducing the comparison to a pre-verified rational inequality.
Claim. $0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.481211 < 0.48121
background
The module establishes necessity results for the electron mass under T9, showing the mass formula follows from T8 ledger quantization together with geometric constants fixed earlier in the chain. The local setting uses numerical interval arithmetic to bound logarithms of phi without relying on external libraries beyond Mathlib's real analysis. Upstream, taylor_sum_lt_target verifies that the partial Taylor sum plus remainder for exp(0.481211) stays below 1.618033, providing the key comparison target.
proof idea
The proof applies the equivalence Real.lt_log_iff_exp_lt to convert the logarithm inequality into an exponential one. It then uses Real.exp_bound' to obtain a Taylor polynomial plus remainder term at order 10. After simplifying the rational expression for the sum, the final comparison invokes the lemma taylor_sum_lt_target to conclude the strict inequality.
why it matters
This bound closes one side of the interval for log phi inside the Necessity module, which itself supports the forced electron mass under T8 and the phi-ladder. It is used directly by log_phi_bounds to sandwich log phi between 0.481211 and 0.481212. In the broader framework it anchors the numerical verification of the self-similar fixed point phi from T6 within the eight-tick octave structure.
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