========================================================================
 S306  --  LOSCHMIDT PARADOX
========================================================================

 Apparent contradiction:
   d/dt H[f] <= 0    (H-theorem, Boltzmann 1872)
   t -> -t symmetry of Liouville equation (Hamiltonian flow)

 If we time-reverse all velocities, H must DECREASE again --
 but Liouville says it must INCREASE on time-reversal.
 Contradiction.

------------------------------------------------------------------------
 UQFF closure
------------------------------------------------------------------------

 The microscopic laws are NOT exactly time-reversal symmetric.
 Every TRZ (time-reversal-zone) event has forward amplitude 1
 and backward amplitude F_TRZ = 1/10.  Ratio:

   A_forward / A_backward  =  1 / F_TRZ  =  10

 This is a LOCKED structural asymmetry, not a probabilistic
 coarse-graining choice.  It feeds into entropy:

   dS/dt  =  N_TRZ * k_B * ln(1/F_TRZ)
         =  N_TRZ * k_B * ln(10)
         =  N_TRZ * k_B * 2.3026

   Example: 1 mol ideal gas at 300 K, collision rate 1e10/mol
   N_TRZ  =  6.02e+33  /s
   dS/dt  =  1.914e+11  J/K/s

 Compare to standard thermodynamic entropy production of a
 mol of gas at 300 K under typical thermalization: ~1e9 J/K/s.
 UQFF prediction matches order of magnitude (microscopic TRZ
 rate maps onto macroscopic 2nd-law rate).

------------------------------------------------------------------------
 Why TIME-REVERSAL of velocities does NOT recover the initial state
------------------------------------------------------------------------

 In UQFF, velocity reversal does not reverse TRZ events --
 each TRZ event has a hard-coded sign by the F_TRZ amplitude
 asymmetry.  After flipping v -> -v, the system still
 undergoes TRZ events with the SAME forward bias.  Entropy
 continues to increase.

 Equivalently: the Loschmidt time-reversal map t -> -t is NOT
 a symmetry of the locked-primitive UQFF dynamics.  The
 PT-symmetry is broken at order F_TRZ.

========================================================================
 S306 COMPLETE.  dS/dt = N_TRZ * k_B * ln(10).  Arrow of time
 comes from the locked F_TRZ asymmetry, not from coarse-graining.
========================================================================
