J_bit
J_bit names the base unit of non-trivial J-cost as the natural logarithm of the golden ratio. Ledger modelers and phi-ladder mass calculations reference this constant when normalizing costs under self-similarity. The declaration is a direct one-line assignment to Real.log φ together with a positivity remark.
claim$J_{bit} := ln φ$, where $φ = (1 + √5)/2$ is the golden ratio. This quantity is the minimum non-trivial cost in the J-cost structure of a discrete ledger.
background
The PhiForcing module shows that self-similarity on a discrete ledger equipped with J-cost forces the scale factor to satisfy the golden-ratio equation. J-cost is the functional obeying the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). Upstream results supply the ledger axioms: collision-free structures from OptionAEmpiricalProgram, edge-length relations from SimplicialLedger, and mechanism-design constraints from GameTheory.
proof idea
The declaration is a direct definition that assigns J_bit to Real.log φ. No lemmas or tactics are invoked; the attached comment simply records positivity.
why it matters in Recognition Science
J_bit supplies the base cost unit required by the phi_forced theorem in the same module, which concludes that any self-similar discrete ledger must adopt ratio φ. It anchors the T6 step in the forcing chain where phi emerges as the unique self-similar fixed point. The definition closes the gap between the abstract J-cost axioms and the concrete eight-tick octave scaling.
scope and limits
- Does not prove positivity of J_bit.
- Does not derive φ from ledger axioms.
- Does not link the cost unit to measured constants such as α or G.
formal statement (Lean)
220noncomputable def J_bit : ℝ := Real.log φ
proof body
Definition body.
221
222/-- J_bit is positive. -/