J_bit
J_bit supplies the logarithmic unit for recognition cost as the natural logarithm of the golden ratio. Researchers deriving stellar mass-to-light ratios from ledger principles cite this to anchor phi-ladder scalings without external calibration. The declaration is a direct constant assignment in the mass-to-light module.
claimThe recognition cost bit is defined by $J_{bit} := ln φ$, where $φ$ is the golden ratio.
background
Recognition Science assigns the cost of any recognition event to its J-cost via the multiplicative recognizer on positive ratios. J_bit sets the base unit so that cost differentials produce exact powers of phi. The module derives the stellar mass-to-light ratio from three strategies: J-cost weighting in stellar assembly, phi-tier structure of nuclear and photon fluxes, and geometric constraints from wavelength, time, and coherence scales. Upstream results include the non-negative J-cost assignment for events and the structure of J-cost itself.
proof idea
The declaration is a direct definition that assigns the natural logarithm of the golden ratio to J_bit. No lemmas or tactics are invoked beyond the imported constant phi.
why it matters in Recognition Science
This definition anchors the three-strategy derivation of the mass-to-light ratio that completes zero-parameter status for Recognition Science. It supplies the scaling unit required for the claim that all astrophysical calibrations follow from the ledger. The step aligns with J-uniqueness and the phi fixed-point in the forcing chain.
scope and limits
- Does not compute numerical values or approximations of J_bit.
- Does not prove relations to other cost functions outside this module.
- Does not address specific observational datasets or error bars.
formal statement (Lean)
57noncomputable def J_bit : ℝ := Real.log φ
proof body
Definition body.
58
59/-! ## The Unified M/L Value -/
60
61/-- **DEFINITION**: The derived mass-to-light ratio in solar units.
62
63 The characteristic M/L equals the golden ratio φ ≈ 1.618 solar units.
64 This value emerges from three independent derivation strategies:
65 1. StellarAssembly: J-cost weighting of photon emission vs mass storage
66 2. NucleosynthesisTiers: φ-tier structure of nuclear/photon fluxes
67 3. ObservabilityLimits: Geometric constraints from λ_rec, τ₀, E_coh
68
69 See: LaTeX Manuscript, Chapter "Astrophysical Tests", Section "M/L Derivation".
70
71 FALSIFIER: If observed M/L systematically deviates from the φ-ladder
72 {φ^n : n ∈ ℤ} by more than measurement uncertainty. -/