k_R_pos
plain-language theorem explainer
The positivity of the recognition Boltzmann constant follows directly from the golden ratio exceeding unity. Researchers modeling black-hole entropy or holographic information capacity in Recognition Science cite this to ensure all derived thermodynamic quantities remain positive. The proof reduces to a single application of the real logarithm positivity lemma.
Claim. $0 < k_R$ where $k_R = ln phi$ and $phi$ is the golden ratio satisfying $phi > 1$.
background
In the Recognition Science treatment of ultramassive black holes the entropy takes the form $S_{BH} = k_R · A/(4 ℓ₀²)$ with $k_R = ln phi$ serving as the fundamental cost per ledger bit. The module imports the definition $k_R := Real.log phi$ from Constants.BoltzmannConstant together with the upstream fact $1 < phi$ from Constants.one_lt_phi. Local setting stresses that J-cost stays finite everywhere on (0,∞) so the black-hole interior is a maximal J-cost state rather than a curvature singularity.
proof idea
The proof is a one-line term wrapper that applies Real.log_pos to the upstream lemma one_lt_phi.
why it matters
This positivity result underpins the RS entropy formula and feeds directly into rs_entropy_pos, horizonBandwidth_pos and entropy_is_bandwidth_capacity. It anchors thermodynamic positivity required by the phi-forcing chain and the eight-tick octave, confirming that entropy equals holographic bandwidth capacity in the Recognition Science framework.
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