=== Beam Analysis Results ===

Beam Centroids:
<x> = -0.539 mm
<px> = 0.323 MeV/c
<y> = 0.134 mm
<py> = -0.606 MeV/c
<pz> = 3089.634 MeV/c

Correlation Matrix:
Units: [mm, MeV/c]
 3.148e+02 -2.811e+02 -1.946e-01  6.147e-01
-2.811e+02  2.523e+02  2.167e-01 -7.438e-01
-1.946e-01  2.167e-01  2.114e+01 -5.741e+01
 6.147e-01 -7.438e-01 -5.741e+01  2.118e+02

Normalized Twiss Parameters (G4beamline style):
X plane:
εt = 2.512e-01 mm-rad
βx = 36649.593 mm/rad
αx = 10.592
γx = 0.003 rad/mm

Y plane:
εt = 2.512e-01 mm-rad
βy = 2460.517 mm/rad
αy = 2.163
γy = 0.002 rad/mm

Geometric Twiss Parameters (position-angle):
X plane:
εx = 6.636e-03 mm-rad
βx = 47445.351 mm
αx = 13.712
γx = 0.004 1/mm

Y plane:
εy = 1.112e-02 mm-rad
βy = 1900.503 mm
αy = 1.671
γy = 0.002 1/mm

Courant-Snyder Invariant Check:
βγ - α² (X) normalized = 1.000000 (should be 1.0)
βγ - α² (Y) normalized = 1.000000 (should be 1.0)
βγ - α² (X) geometric = 1.000000 (should be 1.0)
βγ - α² (Y) geometric = 1.000000 (should be 1.0)

Dispersion:
Dx = -139.535 mm
Dx' = 0.028118 rad
Dy = -18.381 mm
Dy' = 0.039135 rad