J_bit_pos
plain-language theorem explainer
J_bit_pos asserts that the minimum non-trivial cost J_bit, defined as the natural logarithm of the golden ratio, is strictly positive. Researchers deriving Planck-scale constants or stellar assembly models cite this to anchor the coherence energy scale in the Recognition ledger. The proof is a one-line wrapper applying the standard logarithm positivity fact to the already-established inequality phi greater than one.
Claim. $0 < J_{bit}$ where $J_{bit} := {ln} φ$ and $φ = (1 + √5)/2$ is the golden ratio.
background
The PhiForcing module establishes that self-similarity in a discrete ledger equipped with J-cost forces the scale ratio to satisfy the golden-ratio equation. J_bit is introduced as the minimum non-trivial cost via the definition J_bit := Real.log φ. The local setting is the forcing argument that begins from a discrete ledger, imposes self-similarity, and concludes that the only non-trivial solution is φ itself.
proof idea
This is a one-line wrapper that applies the Mathlib lemma Real.log_pos directly to the sibling fact phi_gt_one.
why it matters
The result supplies the positivity step required by PlanckScaleMatching (for the extremum condition and planck_scale_matching_cert) and by StellarAssembly and ConstantDerivations. It anchors the coherence quantum E_coh = φ^{-5} as the minimum energy for coherent recognition and supports the T5 J-uniqueness step in the forcing chain. Downstream certificates treat J_bit_pos as an input that guarantees the mass ladder and energy scales remain positive.
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