boltzmann_analog_formula
plain-language theorem explainer
The Recognition Science framework sets its native Boltzmann constant to the natural logarithm of the golden ratio. Researchers deriving information-theoretic entropy or ledger-bit costs in RS-native thermodynamics cite this when replacing k_B. The proof is a one-line term that directly supplies Real.log phi as the witness for the existential.
Claim. There exists a real number $k_R$ such that $k_R = 0.48121182505960347$ (approximately), where this value is exactly the natural logarithm of the golden ratio $phi = (1 + sqrt(5))/2$.
background
The module supplies calculated proofs for registry items in the COMPLETE_PROBLEM_REGISTRY. Item C-006 concerns the Boltzmann analog k_R, predicted to satisfy 0 < k_R < 0.5. Upstream definitions in BoltzmannConstant and UltramassiveBH both introduce k_R as the RS-native replacement for k_B, with the explicit form k_R = ln(phi) and the observation that positivity follows immediately from phi > 1. This places k_R as the fundamental cost per ledger bit in the phi-ladder thermodynamics.
proof idea
The proof is a one-line term wrapper that applies the witness Real.log phi to discharge the existential quantifier.
why it matters
This declaration supplies the explicit calculated value for C-006. It feeds the sibling bounds boltzmann_analog_positive, boltzmann_analog_lt_half and boltzmann_analog_bounds inside the same module. The result sits inside the forcing chain after T6 (phi as self-similar fixed point) and supplies the entropy scale once c = 1 and hbar = phi^{-5} are fixed.
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