J_bit
plain-language theorem explainer
The minimum non-trivial cost in the Recognition Science ledger model is set to the natural logarithm of the golden ratio. Workers on self-similar forcing cite this base unit when normalizing J-costs across scales. The declaration is a direct noncomputable definition with no proof obligations or reduction steps.
Claim. Define the minimum non-trivial cost by $J = ln φ$, where $φ = (1 + √5)/2$ is the unique positive root of $x^2 = x + 1$.
background
The Phi Forcing module shows that self-similarity in a discrete ledger equipped with J-cost forces the golden ratio. The J-cost of a scale factor x obeys the Recognition Composition Law J(xy) + J(x/y) = 2 J(x) J(y) + 2 J(x) + 2 J(y) and vanishes only at x = 1. Self-similarity therefore requires a non-unit scale whose cost matches the identity, which occurs precisely when the scale satisfies x² = x + 1.
proof idea
This is a direct definition. It assigns the real logarithm of φ to the minimum non-trivial cost without invoking lemmas, tactics, or upstream results.
why it matters
The definition supplies the cost unit required by the self-similar scale and phi-uniqueness arguments summarized in the module documentation. It realizes the T6 step in the unified forcing chain where phi is forced as the self-similar fixed point. Mass formulas and the eight-tick octave later normalize against this unit; the alpha band lies downstream of the same normalization.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.