String Tension and Confinement

The v16 S_a auxiliary fields diffuse from coloured soliton cores and deepen χ along the line between them — a flux-tube-like channel emerging from GOV-01 + GOV-02 alone, with all parameters derived from χ₀ = 19.

Full script

"""17 - Confinement & Flux Tubes: v16 S_a Auxiliary Fields

Two coloured solitons are placed on a COLOR-level grid with the v16
S_a auxiliary fields enabled (kappa_tube > 0).  The S_a fields diffuse
from each soliton core and their smoothed colour variance (SCV) sources
an extra term in GOV-02 that deepens chi *between* the sources,
forming a flux-tube-like channel.

This is NOT full QCD confinement (no linear V = sigma*r potential),
but demonstrates: S_a diffusion, SCV-driven chi deepening, and a
string-tension proxy from chi midpoint vs separation.

All S_a parameters are DERIVED from chi0 = 19:
  gamma = epsilon_W = 0.1     (weak helicity coupling)
  L     = beta0     = 7       (chi0 - 12, QCD beta function)
  D     = gamma*L^2 = 4.9     (diffusion coefficient)
  kappa_tube = (z2 - rank_G)*kappa = 30*kappa  (30 = 32-2, geometry)
"""

import numpy as np
import lfm
from lfm.constants import KAPPA, KAPPA_C, KAPPA_STRING, LAMBDA_H

N          = 64
SEP        = 20         # half-separation between solitons (cells)
AMP        = 8.0
STEPS      = 15_000
KAPPA_TUBE = 10.0 * KAPPA  # 10x default — stronger signal on 64^3 grid

cfg = lfm.SimulationConfig(
    grid_size=N,
    field_level=lfm.FieldLevel.COLOR,
    kappa_c=KAPPA_C,
    kappa_tube=KAPPA_TUBE,
    kappa_string=KAPPA_STRING,
    lambda_self=LAMBDA_H,
    dt=0.02,
)
sim = lfm.Simulation(cfg)

print("17 - Confinement & Flux Tubes via v16 S_a Fields")
print("=" * 62)

cx = cy = N // 2
z1, z2 = N // 2 - SEP // 2, N // 2 + SEP // 2
sim.place_soliton((cx, cy, z1), amplitude=AMP)
sim.place_soliton((cx, cy, z2), amplitude=AMP)
sim.equilibrate()

sa = sim.sa_fields
print(f"S_a allocated:   {'YES' if sa is not None else 'NO'}")
print(f"S_a shape:       {sa.shape}")
print(f"S_a max (init):  {sa.max():.4e}")

chi_mid_before = lfm.measure_chi_midpoint(sim.chi, (cx, cy, z1), (cx, cy, z2))
print(f"\nBEFORE run:\n  chi midpoint: {chi_mid_before:.4f}")

sim.run(steps=STEPS, record_metrics=False)

chi_mid_after = lfm.measure_chi_midpoint(sim.chi, (cx, cy, z1), (cx, cy, z2))
sa_after = sim.sa_fields
print("\nAFTER run:")
print(f"  chi midpoint:        {chi_mid_after:.4f}")
print(f"  chi midpoint change: {chi_mid_after - chi_mid_before:+.4f}")
print(f"  S_a max (equil):     {sa_after.max():.4e}")

profile   = lfm.flux_tube_profile(sim.chi, sa_after, (cx, cy, z1), (cx, cy, z2))
chi_line  = profile["chi_profile"]
print(f"\nFlux-tube chi profile: min={chi_line.min():.3f}  max={chi_line.max():.3f}")

tension = lfm.string_tension(sim)
print(f"String-tension proxy:  {tension:.6f}")

scv = lfm.smoothed_color_variance(sa_after)
print(f"\nSCV max (soliton cores):  {scv.max():.4e}")
print(f"SCV at midpoint (z={N//2}):   {scv[cx, cy, N//2]:.4e}")

print("\n" + "=" * 62)
deepened = chi_mid_after < chi_mid_before
print(f"Flux tube formed:        {'YES' if deepened else 'NO'}")
print(f"Chi deepened at midpoint: {chi_mid_before:.3f} -> {chi_mid_after:.3f}")
print(f"SA diffusion active:     {'YES' if sa_after.max() > 1e-8 else 'NO'}")

Expected output

17 - Confinement & Flux Tubes via v16 S_a Fields
==============================================================
S_a allocated:   YES
S_a shape:       (3, 64, 64, 64)
S_a max (init):  6.4000e+01

BEFORE run:
  chi midpoint: 18.9846

AFTER run:
  chi midpoint:        18.7422
  chi midpoint change: -0.2424
  S_a max (equil):     3.8741e+01

Flux-tube chi profile: min=12.184  max=18.995
String-tension proxy:  0.010127

SCV max (soliton cores):  4.1024e+01
SCV at midpoint (z=32):   1.3891e+00

==============================================================
Flux tube formed:        YES
Chi deepened at midpoint: 18.985 -> 18.742
SA diffusion active:     YES