k_R
plain-language theorem explainer
Recognition Science defines k_R as the natural logarithm of the golden ratio phi, serving as the information cost per recognition event in native units. Black hole thermodynamics researchers cite it when replacing the classical Boltzmann constant in entropy and temperature formulas for ultramassive objects like TON 618. The definition is introduced directly from the ledger structure without additional derivation steps.
Claim. The recognition Boltzmann constant is defined by $k_R := (k_R =) ln(phi)$, where $phi$ denotes the golden ratio fixed point. This quantity equals the fundamental cost per ledger bit and replaces $k_B$ in RS-native thermodynamics.
background
The module UltramassiveBH formalizes the RS treatment of black holes with mass exceeding 10^10 solar masses, emphasizing that J-cost remains finite on (0, infinity) so the interior is a maximal-cost state rather than a curvature singularity. Key supporting definitions include the J-cost function from JcostCore (satisfying the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y)) and the phi-ladder mass formula. The upstream definition in Constants.BoltzmannConstant states: k_R = ln(phi) — the fundamental cost per ledger bit. This replaces k_B in RS-native thermodynamics.
proof idea
one-line definition that directly applies the natural logarithm to the imported golden-ratio constant phi.
why it matters
This definition supplies the numerical prefactor in the RS entropy formula S_BH = k_R * A / (4 l_0^2) listed as result 2 in the module documentation. It feeds the downstream C006_certificate and the suite of bounds and identities (k_R_pos, k_R_lt_half, k_R_eq_J_bit) that certify the Boltzmann analog. The placement ties directly to forcing-chain step T6 (phi as self-similar fixed point) and the eight-tick octave, closing the ledger-to-thermodynamics link for ultramassive black holes.
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