thermal_energy_at_unit_T
plain-language theorem explainer
The theorem shows that thermal energy at unit temperature in RS units equals ln(φ). Researchers deriving thermodynamic relations from the Recognition Science ledger would cite this when connecting temperature scales to bit cost. The proof reduces directly via substitution of T=1 and the definition of k_R followed by ring simplification.
Claim. For real $T$ satisfying $T=1$, the thermal energy satisfies $k_R T = ln φ$, where $k_R$ is the Recognition Science Boltzmann analog defined as the ledger bit cost $ln φ$.
background
The module derives the Boltzmann constant analog k_R from the fundamental ledger bit cost. Upstream, k_R is defined as the natural logarithm of the golden ratio φ forced by the self-similar fixed point. Temperature is measured in units where average cost per degree of freedom matches this bit cost, so the linear relation E = k_R T follows without extra parameters. Upstream results include the cost function induced by multiplicative recognizers and the J-cost of recognition events from observer forcing, both grounding energy in ledger structure.
proof idea
The proof is a one-line wrapper that rewrites with the hypothesis T=1, unfolds the definition of k_R, and simplifies the product via ring tactics.
why it matters
This fills the C-006.6 slot in the Boltzmann derivation chain, connecting thermal energy at unit temperature to ln(φ) from the ledger structure forced at T6. It supports the claim that k_R emerges with zero free parameters as the exchange rate between temperature and energy, consistent with the Recognition Composition Law and the eight-tick octave. The result anchors the temperature-energy relation in the framework; no direct downstream uses are recorded yet.
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