J_bit
J_bit defines the fundamental bit cost in Recognition Science as the natural logarithm of the golden ratio φ. Researchers deriving physical constants from the RS foundation cite it when scaling coherence energies or time units in the phi-ladder. The declaration is a direct noncomputable assignment with no lemmas or reduction steps.
claim$J_{bit} = ln(φ)$ where $φ = (1 + √5)/2$ is the unique positive fixed point of the J-cost function satisfying $J(φ) = 1$.
background
Recognition Science starts from the Composition Law and derives the J-cost $J(x) = ½(x + x^{-1}) - 1$ as the unique cost function (T5). Its self-similar fixed point is the golden ratio φ (T6), which generates the eight-tick octave and forces D = 3. The ConstantDerivations module converts these algebraic objects into physical constants by expressing c, ħ, G, and α as ratios of RS-native quantities, all algebraic in φ.
proof idea
Direct definition: J_bit is assigned the value Real.log φ_val, where φ_val is the golden-ratio constant imported from PhiForcing. No tactics or lemmas are applied.
why it matters in Recognition Science
J_bit supplies the base logarithmic scale for bit costs that later definitions (E_coh, c_rs, G_rs) multiply by powers of φ to obtain the derived constants. It occupies the Level 1–2 step in the module’s derivation chain, converting the abstract J-cost into a concrete numerical unit that makes ħ = φ^{-5} and G = φ^5/π hold in RS units.
scope and limits
- Does not prove J_bit > 0 (handled by sibling J_bit_pos).
- Does not derive numerical values of c, ħ, or G.
- Does not connect J_bit to experimental measurements or units outside RS-native scaling.
formal statement (Lean)
69noncomputable def J_bit : ℝ := Real.log φ_val
proof body
Definition body.
70
71/-- J_bit > 0 since φ > 1. -/