phi_bit_more_efficient
plain-language theorem explainer
The inequality establishes that the natural log of phi is strictly less than the natural log of 2, so each phi-bit encodes more states than a binary bit. Recognition theorists comparing CP6 channel capacity to Shannon capacity would cite it when showing phi^12 exceeds 2^12. The proof is a one-line application of the monotonicity of the logarithm to the already-proved bounds 1 < phi < 2.
Claim. $log phi < log 2$
background
The module develops information theory in base phi rather than base 2. A phi-bit is the information unit log_phi, while a Shannon bit is log_2; the recognition channel capacity of CP6 is stated to scale as phi^12. The upstream lemma phi_lt_two supplies the key bound phi < 2, proved by comparing sqrt(5) < 3 and unfolding the definition of phi. The tick definition supplies the eight-tick octave as the fundamental period, and the J-cost core supplies the Recognition Composition Law that underpins the phi-ladder.
proof idea
One-line wrapper that applies Real.log_lt_log to the hypothesis phi_gt_one (which gives phi > 1) and the upstream lemma phi_lt_two (which gives phi < 2). The tactic linarith discharges the positivity side condition inside the application.
why it matters
It supplies the elementary comparison needed to claim that recognition capacity exceeds Shannon capacity for the same state space, directly supporting the module's headline result that CP6 capacity scales as phi^12. The declaration sits inside the eight-tick octave (T7) and the D=3 spatial setting (T8) of the forcing chain; it closes the efficiency gap between phi-bits and binary bits before the recognition_event_12_dof theorem is invoked. No open scaffolding remains on this specific inequality.
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