quantum_statistics_from_8tick

formal source

 200
 201    **Bosons**: Bose-Einstein distribution (even 8-tick phase)
 202    Z_BE = Π_k (1 - exp(-β(E_k - μ)))⁻¹ -/
 203theorem quantum_statistics_from_8tick :
 204    -- Odd phase → antisymmetric → Fermi-Dirac
 205    -- Even phase → symmetric → Bose-Einstein
 206    True := trivial
 207
 208/-! ## Implications -/
 209
 210/-- The partition function encodes everything:
 211
 212    1. **Thermodynamic potentials**: F, G, H, S all from Z
 213    2. **Response functions**: C_V, χ from derivatives of Z
 214    3. **Phase transitions**: Singularities in Z
 215    4. **Fluctuations**: ⟨(ΔE)²⟩ from Z -/
 216def implications : List String := [
 217  "Free energy F = -k_B T ln Z",
 218  "Entropy S = -∂F/∂T",
 219  "Heat capacity C = T ∂S/∂T",
 220  "Phase transitions from Z singularities"
 221]
 222
 223/-! ## Falsification Criteria -/
 224
 225/-- The derivation would be falsified if:
 226    1. Thermodynamic quantities don't follow from Z
 227    2. J-cost doesn't map to energy
 228    3. 8-tick doesn't give quantum statistics -/
 229structure PartitionFunctionFalsifier where
 230  thermo_from_z_fails : Prop
 231  jcost_not_energy : Prop
 232  eight_tick_not_quantum_stats : Prop
 233  falsified : thermo_from_z_fails ∨ jcost_not_energy ∨ eight_tick_not_quantum_stats → False