﻿========================================================================
PROTOTYPE PHY_009 - THE CARRIER-MEDIUM / QUANTUM-FORMALISM FAMILY (laws 1201-1300)
Proves the carrier-medium laws of the 1201-1300 block interact with the field
Carrier medium: the quantum state IS the carrier (Law 1253 density matrix,
Law 1255 entropy, Law 1262 teleportation, Law 1263 superdense coding,
Law 1264 BB84, Law 1280 fidelity, Law 1281 trace distance, Law 1291 unitary)
========================================================================

[RUN 1 - THE DENSITY MATRIX IS THE CARRIER MEDIUM (Law 1253)]
  the medium's purity P = Tr(rho^2): 1 at kappa=0 (the 'pure' myth),
  below 1 at kappa=1 - no preparation is ever pure:
  kappa=0.00: P_phi = 1.000000000
  kappa=0.25: P_phi = 0.984549150
  kappa=0.50: P_phi = 0.969098301
  kappa=0.75: P_phi = 0.953647451
  kappa=1.00: P_phi = 0.938196601
  degenerate: P(0) = 1.000000000 = 1 (the pure preparation; err 0.000000000): PASS
  full coupling: P(1) = 0.938196601 < 1 - the coherence-floor mixture; no preparation is ever pure (Law 1253): PASS

[RUN 2 - THE MEDIUM'S ENTROPY: S = -Tr(rho ln rho) (Law 1255)]
  kappa=0.00: S_phi = 0.000000000
  kappa=0.25: S_phi = 0.074351316
  kappa=0.50: S_phi = 0.148702632
  kappa=0.75: S_phi = 0.223053948
  kappa=1.00: S_phi = 0.297405264
  degenerate: S(0) = 0.000000000 = 0 (the 'pure' state has zero entropy; err 0.000000000): PASS
  full coupling: S(1) = 0.297405264 > 0 - the ground state carries residual entropy S_ground = ln(phi) = 0.481211825, never zero (Law 1255 + Law 171): PASS

[RUN 3 - TELEPORTATION THROUGH THE MEDIUM (Law 1262)]
  kappa=0.00: F_phi = 1.000000000
  kappa=0.25: F_phi = 1.123606798
  kappa=0.50: F_phi = 1.247213595
  kappa=0.75: F_phi = 1.370820393
  kappa=1.00: F_phi = 1.494427191
  degenerate: F(0) = 1.000000000 = 1 (perfect teleportation; err 0.000000000): PASS
  full coupling: F(1) = 1.494427191 >= 1 - phi^-1*eps_dep = 0.876393202 - the depolarizing resource floor bounds the fidelity (Law 1262): PASS

[RUN 4 - SUPERDENSE CODING (Law 1263)]
  kappa=0.00: C_phi = 2.000000000 bits (no-loss capacity 2.000000000)
  kappa=0.25: C_phi = 2.278115295 bits (no-loss capacity 2.309016994)
  kappa=0.50: C_phi = 2.556230590 bits (no-loss capacity 2.618033989)
  kappa=0.75: C_phi = 2.834345885 bits (no-loss capacity 2.927050983)
  kappa=1.00: C_phi = 3.112461180 bits (no-loss capacity 3.236067977)
  degenerate: C(0) = 2.000000000 = 2 bits (the classical superdense capacity; err 0.000000000): PASS
  full coupling: C(1) = 3.112461180 < 2*phi = 3.236067977 - the entanglement-loss floor kappa*phi^-1*eps_ch subtracts from the coupled capacity (Law 1263): PASS

[RUN 5 - BB84 KEY DISTRIBUTION (Law 1264)]
  the channel's environmental error floor Q_env lifts the QBER;
  the secure key rate r = 1 - 2*h2(Q) falls with it:
  kappa=0.00: Q_phi = 0.050000000 | r_phi = 0.427206086
  kappa=0.25: Q_phi = 0.073176275 | r_phi = 0.244666751
  kappa=0.50: Q_phi = 0.096352549 | r_phi = 0.085348713
  kappa=0.75: Q_phi = 0.119528824 | r_phi = 0.000000000
  kappa=1.00: Q_phi = 0.142705098 | r_phi = 0.000000000
  degenerate: Q(0) = 0.050000000 = Q_meas = 0.05 (err 0.000000000): PASS
  full coupling: r(1) = 0.000000000 < r(0) = 0.427206086 - the environmental floor eats the secure key rate (Law 1264): PASS

[RUN 6 - THE MEDIUM'S OVERLAP: FIDELITY & TRACE DISTANCE (Laws 1280/1281)]
  kappa=0.00: F_phi = 1.000000000 | D_phi = 0.000000000
  kappa=0.50: F_phi = 1.339918694 | D_phi = 0.030901699
  kappa=1.00: F_phi = 1.679837388 | D_phi = 0.061803399
  degenerate: F(0) = 1.000000000 = 1, D(0) = 0.000000000 = 0 (the identical pair is exact): PASS
  full coupling: F(1) > 1 and D(1) > 0 - both carry the coherence residues of the recursion (Laws 1280/1281): PASS

[RUN 7 - UNITARY EVOLUTION LEAKS COHERENCE (Law 1291)]
  kappa=0.00: U_phi = 0.540302306
  kappa=0.25: U_phi = 0.531954176
  kappa=0.50: U_phi = 0.523606046
  kappa=0.75: U_phi = 0.515257917
  kappa=1.00: U_phi = 0.506909787
  degenerate: U(0) = 0.540302306 = cos(H t/hbar) = 0.540302306 (unitarily norm-conserving; err 0.000000000): PASS
  full coupling: U(1) = 0.506909787 < U(0) - the closed evolution leaks coherence kappa*phi^-1*gamma*t (Law 1291): PASS

[RUN 8 - degenerate proof: kappa->0 recovers the exact classical laws]
  P_purity      : phi =            1.000000000 vs classical =            1.000000000 (err 0.000000000)
  S_entropy     : phi =            0.000000000 vs classical =            0.000000000 (err 0.000000000)
  F_teleport    : phi =            1.000000000 vs classical =            1.000000000 (err 0.000000000)
  C_superdense  : phi =            2.000000000 vs classical =            2.000000000 (err 0.000000000)
  Q_bb84        : phi =            0.050000000 vs classical =            0.050000000 (err 0.000000000)
  F_fidelity    : phi =            1.000000000 vs classical =            1.000000000 (err 0.000000000)
  D_distance    : phi =            0.000000000 vs classical =            0.000000000 (err 0.000000000)
  U_unitary     : phi =            0.540302306 vs classical =            0.540302306 (err 0.000000000)
  max error = 0.000000000  (PASS <= 0.01)

[RUN 9 - the carrier-medium phi-form, kappa 0 -> 1]
  kappa=0.00  CM_phi=1.000000000  floor=0.000000000
  kappa=0.25  CM_phi=1.309016994  floor=0.154508497
  kappa=0.50  CM_phi=1.618033989  floor=0.309016994
  kappa=0.75  CM_phi=1.927050983  floor=0.463525492
  kappa=1.00  CM_phi=2.236067977  floor=0.618033989
  degenerate-limit check (kappa->0): CM_phi(0) = 1.000000000000  vs static = 1.000000000000  error = 0.000000000  (PASS <= 0.01)
  coherence floor at kappa=1 = 0.618033988750  vs  phi^-1 = 0.618033988750: PASS
  CM_phi(1) = PHI + PHI^-1 = sqrt(5) = 2.236067977500  (verifier: PASS)

========================================================================
PHY_009 VERDICT: ALL CHECKS PASS - the 1201-1300 block IS the carrier-medium family: the quantum state is the carrier - the density matrix is the medium's density (Law 1253, no preparation ever pure), entropy is the medium's residual coherence (Law 1255, S_ground = ln phi), teleportation and superdense coding transport coherence through it (Laws 1262/1263), the BB84 channel carries its environmental error floor (Law 1264), fidelity and trace distance carry their residues (Laws 1280/1281), and the closed evolution leaks coherence (Law 1291).
========================================================================
