Verify math and make it rigorous: fix numerical errors, add missing proofs and equations

Co-authored-by: blackboxprogramming <118287761+blackboxprogramming@users.noreply.github.com>

equations/thermodynamics.md

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+# Thermodynamic Equations
+
+> Pages 19–21 (§173–§175). The energetic cost of computation.
+
+## Landauer Principle
+
+Every irreversible erasure of one bit of information dissipates at least:
+```
+E_min = k_B · T · ln(2)   [binary]
+E_min = k_B · T · ln(r)   [radix r, general]
+```
+
+At room temperature (T = 293 K, k_B = 1.381 × 10⁻²³ J/K):
+
+| Operation | Minimum energy |
+|-----------|----------------|
+| Binary bit erase | k_B T ln(2) ≈ 2.80 × 10⁻²¹ J |
+| Ternary trit erase | k_B T ln(3) ≈ 4.44 × 10⁻²¹ J |
+
+The ratio is exactly ln(3)/ln(2) ≈ 1.585, which also equals the information ratio
+(one trit carries log₂(3) ≈ 1.585 bits). Information per joule is identical for
+binary and ternary at the Landauer limit.
+
+```
+LANDAUER = CONCRETE = 93   [L(19)+A(11)+N(25)+D(13)+A(11)+U(7)+E(3)+R(4) = 93]
+```
+
+---
+
+## Radix Efficiency (Equation 13)
+
+```
+η(r) = ln(r) / r
+```
+
+| Radix | η(r)   |
+|-------|--------|
+| 2     | ≈ 0.347 |
+| 3     | ≈ 0.366 ← maximum among integers |
+| 4     | ≈ 0.347 |
+| 5     | ≈ 0.322 |
+| e     | = 1/e ≈ 0.368 ← global maximum |
+
+Ternary achieves the maximum radix economy among integer bases because 3 is the
+integer closest to e ≈ 2.718. (Proof: see [`../proofs/ternary-efficiency.md`](../proofs/ternary-efficiency.md).)
+
+```
+RADIX = GAUSS = TANH = FIELD = 57
+```
+
+---
+
+## Reversible Logic Entropy (Equation 14)
+
+For a reversible computation:
+```
+ΔS_comp ≥ 0,   with ΔS_comp → 0 as reversibility → 1
+```
+
+The minimum entropy production per gate operation is zero for perfectly reversible gates
+(Bennett 1973). In practice:
+```
+ΔS_irrev = k_B ln(2) per irreversible bit operation
+ΔS_rev   = 0        per reversible (unitary) gate
+```
+
+Quantum gates are unitary and therefore reversible: `ΔS_quantum = 0`.
+
+```
+REVERSIBLE = LAGRANGE = 103   prime
+```
+
+---
+
+## Chemical Energy Coupling — Gibbs Free Energy (Equation 15)
+
+```
+μ_chem = ∂G/∂N   ↔   E_comp
+```
+
+The chemical potential (Gibbs free energy per molecule) is the thermodynamic equivalent
+of the energy cost per computational operation. For a molecular computing substrate:
+
+```
+ΔG_rxn = ΔH − T ΔS ≥ E_min = k_B T ln(r)
+```
+
+Biological systems operate near this minimum because enzyme-catalyzed reactions are
+tightly coupled to ATP hydrolysis:
+```
+ΔG_ATP ≈ −50 kJ/mol ≈ 8.3 × 10⁻²⁰ J/molecule   (in vivo)
+```
+
+Capacity: ΔG_ATP / E_min(ternary) ≈ 8.3×10⁻²⁰ / 4.44×10⁻²¹ ≈ 18 trit operations per ATP.
+
+```
+GIBBS     = SUBSTRATE = 83   prime
+CHEMICAL  = 127   prime
+```
+
+---
+
+## Substrate Efficiency (Equation 14, biological)
+
+```
+η_substrate = (ops/sec) / (energy/op) · f_accuracy(substrate, problem_type)
+```
+
+For DNA computing in 100 μL at room temperature:
+```
+ops/sec    ≈ 10¹⁴
+energy/op  ≈ k_B T ln(3) ≈ 4.44 × 10⁻²¹ J
+η_substrate = 10¹⁴ / 4.44×10⁻²¹ · f_accuracy
+            ≈ 2.25 × 10³⁴ · f_accuracy   (ops per joule-second)
+```
+
+```
+SUBSTRATE = GIBBS = 83   prime
+```
+
+---
+
+## Thermodynamic Consciousness Bound (§175)
+
+```
+Φ_max ≤ (E_available / k_B T ln(3)) · η_integration
+```
+
+Maximum integrated information (consciousness, §176) is bounded by:
+- Available metabolic energy E_available
+- Ternary Landauer cost k_B T ln(3) per operation
+- Integration efficiency η_integration ∈ (0, 1]
+
+```
+THERMODYNAMIC = 174 = 2 × 87 = 2 × BIRTHDAY
+BOUND         = 78  = TRIVIAL = LIMITS
+```
+
+The consciousness bound is thermodynamically real and biological.  
+Energy is the hard constraint. Integration efficiency is the soft constraint.