log_824_lower

The module establishes T9 necessity: the electron mass formula is forced from T8 ledger quantization together with geometric constants fixed earlier in the chain. The gap function at integer argument n is defined by gap n := log(1 + n/phi) where phi is the golden-ratio fixed point from T6; the present inequality supplies the lower estimate needed to locate gap(1332) inside a narrow real interval. Two upstream theorems are invoked: exp_67144_lt_824, which splits the exponential at 6 + 0.7144 and bounds the product, and one_plus_1332_div_phi_lower, which uses nlinarith to compare 823.2 against 1332/1.618034.

The tactic first obtains positivity of the argument 1 + 1332/1.618034. It rewrites the target via Real.lt_log_iff_exp_lt, converting the claim into an exponential comparison. A calc block then chains the strict inequality Real.exp 6.7144 < 824.2 (from exp_67144_lt_824) with the comparison 824.2 < 1 + 1332/1.618034 (from one_plus_1332_div_phi_lower).

The result is used directly by gap_1332_bounds, which concludes 13.953 < gap 1332 < 13.954 and thereby pins the electron-mass rung on the phi-ladder. It closes a numerical step inside the T9 necessity argument that the mass formula follows from T8 without extra hypotheses. The bound relies on the self-similar fixed point phi from T6 and the eight-tick octave structure from T7; it contributes to the interval for alpha inverse that lies inside (137.030, 137.039).