Constants Boltzmann Constant K R Pos

The Boltzmann analog

The Boltzmann constant k_B is the exchange rate between temperature and energy in statistical mechanics: the equation E = k_B · T says that a degree of freedom at temperature T carries energy k_B · T. In the Recognition Science framework, this constant is not a free parameter. The framework derives its own analog, called k_R (a ledger-based constant), from the golden ratio φ = (1+√5)/2, which the framework's structure forces as a fundamental scale. The definition is k_R = ln(φ), the natural logarithm of the golden ratio, which is approximately 0.481 in the framework's natural units.

The specific declaration k_R_pos is a theorem in the framework's machine-checked library of formal theorems. It proves that k_R is strictly greater than zero. The proof is direct: since φ is greater than 1, its natural logarithm is positive. This is not a numerical approximation or a physical assumption; it is a formal consequence of the definition. The same library also proves that k_R is nonzero, which is required for any calculation that divides by k_R, and that k_R is less than 0.5, placing it in the narrow interval (0.47, 0.49).

The positivity of k_R matters because it guarantees that the framework's temperature scale is physically sensible: a positive constant means that higher temperature always corresponds to higher thermal energy, and that the energy at unit temperature is positive. The framework identifies k_R with the cost of a single bit in its recognition ledger, a discrete record of events. This identification is a theorem in the library: k_R equals the ledger bit cost J_bit. In this account, temperature is the average cost per degree of freedom, and the Boltzmann analog is the exchange rate set by the ledger's self-similarity scale.

What k_R_pos does not claim is important. It does not claim that the framework's k_R equals the measured SI value of the Boltzmann constant. The SI connection is a separate calibration step: k_B^SI = k_R · (E_coh_SI / T_coh_SI), where E_coh and T_coh are coherence energy and temperature scales. The theorem also does not claim that temperature itself is fundamental; it only establishes a property of a derived constant within the framework. Finally, k_R_pos does not prove that the framework's derivation of φ is correct; that is a separate theorem about the ledger structure, not about the Boltzmann analog.

/-- **DEFINITION C-006**: The RS Boltzmann analog k_R.

    k_R = ln(φ) — the fundamental cost per ledger bit.
    This replaces k_B in RS-native thermodynamics. -/
noncomputable def k_R : ℝ := Real.log Constants.phi

/-- **THEOREM C-006.1**: k_R is positive.

    Proof: φ > 1, so ln(φ) > 0. -/
theorem k_R_pos : k_R > 0 := by
  unfold k_R
  apply Real.log_pos
  exact Constants.one_lt_phi

/-- **THEOREM C-006.2**: k_R is nonzero.

    This is required for thermodynamic calculations (division by k_R). -/
theorem k_R_ne_zero : k_R ≠ 0 := by
  exact ne_of_gt k_R_pos

/-- **THEOREM C-006.5**: k_R = J_bit (the ledger bit cost).

    This is the fundamental identity: the Boltzmann analog equals
the cost of a single bit in the recognition ledger. -/
theorem k_R_eq_J_bit : k_R = Constants.J_bit := rfl

What this page does not claim

k_R_pos does not claim that k_R equals the measured SI value of the Boltzmann constant. The theorem does not claim that temperature is fundamental; it only establishes a property of a derived constant. The positivity proof does not establish the correctness of the framework's derivation of φ.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Constants/BoltzmannConstant.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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