J_bit
Stellar assembly models employ the elementary ledger bit cost defined as the natural logarithm of the golden ratio. This base unit converts integer multiples of recognition cost differentials into mass-to-light ratios on the phi ladder during collapse. Researchers deriving M/L from photon emission versus mass storage costs would cite the definition. It is supplied by direct assignment with no lemmas or reductions.
claimThe elementary ledger bit cost is $J_{bit} = ln φ$, where $φ$ denotes the golden ratio.
background
The stellar assembly module derives mass-to-light ratios from recognition cost differentials between photon emission and bound mass storage. The convex cost function is $J(x) = ½(x + 1/x) - 1$, which satisfies the Recognition Composition Law and equals $cosh(ln x) - 1$. Cost differences are expressed as integer multiples of the bit cost, so that $Δδ = n · J_{bit}$ yields $M/L = φ^n$ with $n$ fixed by the eight-tick structure and restricted to the interval [0, 3].
proof idea
The definition is introduced by direct assignment of the natural logarithm of the golden ratio. No upstream lemmas are invoked; the assignment follows immediately from the identification of the base ledger unit.
why it matters in Recognition Science
This definition supplies the scaling unit for the cost differential in the stellar M/L derivation. It supports the module result that observed ratios fall on the phi ladder with $M/L ≈ φ^1 ≈ 1.618$ in solar units. The construction connects to the self-similar fixed point for phi and to the eight-tick octave that determines admissible integers n. No downstream theorems are recorded.
scope and limits
- Does not compute n for specific stellar populations.
- Does not derive the golden ratio from the forcing chain.
- Does not calibrate against observed M/L data.
- Does not extend the bit cost to non-stellar systems.
formal statement (Lean)
52noncomputable def J_bit : ℝ := Real.log φ
proof body
Definition body.
53
54/-- J_bit is positive (φ > 1 → ln φ > 0) -/