phi_bit_more_efficient
The inequality log phi < log 2 shows that each phi-bit encodes more states than a binary bit, so fewer units suffice for the same information volume. Information theorists comparing recognition channels to Shannon limits cite this when bounding phi-based encoding efficiency. The proof is a one-line wrapper applying the monotonicity of the logarithm to the established bounds phi greater than 1 and phi less than 2.
claimThe natural logarithm satisfies $ln phi < ln 2$, where phi denotes the golden ratio.
background
The RecognitionEntropy module quantifies information in phi-bits rather than Shannon bits. Phi is the golden ratio fixed point of the Recognition Composition Law, and the module shows that CP6 recognition capacity scales as phi to the 12, exceeding base-2 capacity. The result rests on the upstream lemma phi_lt_two, which proves phi less than 2 by comparing square roots of 5 and 9, together with the implicit bound phi greater than 1.
proof idea
The proof is a one-line wrapper that applies the lemma Real.log_lt_log. It supplies the hypothesis phi greater than 1 via linarith on phi_gt_one and the hypothesis phi less than 2 directly from the upstream lemma phi_lt_two.
why it matters in Recognition Science
This result supports the module claim that recognition exceeds Shannon capacity for the same state space and feeds into phi_bits_exceed_shannon together with recognition_capacity_phi_12. It aligns with the forcing chain where phi emerges as the self-similar fixed point at T6 and the eight-tick octave at T7. No open scaffolding remains for this inequality.
scope and limits
- Does not compute the numerical efficiency ratio log base 2 of phi.
- Does not extend the comparison to bases other than 2.
- Does not incorporate relativistic or quantum corrections to the information measure.
formal statement (Lean)
61theorem phi_bit_more_efficient :
62 Real.log phi < Real.log 2 := by
proof body
Term-mode proof.
63 apply Real.log_lt_log (by linarith [phi_gt_one]) phi_lt_two
64
65/-- The meaning capacity of one recognition event (one 8-tick cycle)
66 is exactly 12 real DOF (the dimension of CP6). -/