Listing: Listing

# kernel_identification_checks.py output
# xvi-M wpA: Gate S-bv identification of S with c (Q-i0)^{-1}
# python 3.13.5, sympy 1.14.0, mpmath 1.3.0, numpy 2.5.3
# signature eta=diag(1,-1,-1,-1); massless; z=x-y

# --- 1. Radial Fourier integral for W_0 (symbolic) ---
CHECK int_0^oo e^{-tau p} sin(p r) dp = r/(tau^2+r^2) for tau>0: True
CHECK analytic continuation spot (tau,t,r)=(0.2,0.5,1.0), err=1.88e-05: True
CHECK analytic continuation spot (tau,t,r)=(0.1,-1.2,0.7), err=5.25e-05: True
CHECK analytic continuation spot (tau,t,r)=(0.05,2.0,1.5), err=4.50e-04: True
CHECK damped radial W equals -1/(4 pi^2 (Q - 2 i tau t - tau^2)): True
CHECK Euclidean slice z0=-i tE: W=1/(4 pi^2 (tE^2+r^2)) (matches rem:Sbv-motivation): True
# => W_0 = -1/(4 pi^2) * 1/(Q - i0 sgn(z^0)) as the boundary value of the damped integral

# --- 2. Sign check for (s-i0)^{-1} = PV(1/s) + i pi delta(s) ---
CHECK 1D Im~pi f(0) Gaussian#0, err=4.02e-04: True
CHECK 1D Re~PV Gaussian#0, err=0.00e+00: True
CHECK 1D Im~pi f(0) Gaussian#1, err=2.80e-04: True
CHECK 1D Re~PV Gaussian#1, err=3.38e-04: True
CHECK 1D Im~pi f(0) Gaussian#2, err=2.57e-04: True
CHECK 1D Re~PV Gaussian#2, err=2.87e-04: True
CHECK 1D Im~pi f(0) Gaussian#3, err=4.16e-05: True
CHECK 1D Re~PV Gaussian#3, err=7.97e-05: True
CHECK opposite side Im~-pi f(0): True
# confirmed: (s-i0)^{-1}=PV(1/s)+i pi delta(s); (s+i0)^{-1}=PV-i pi delta

# --- 3. Algebraic combination of singular terms (symbolic coefficients) ---
CHECK sgn(z0) dependence cancels: delta coeff same on both time sheets: True
CHECK common delta coeff equals c * i pi = i/(4 pi): True
CHECK PV coeff equals c = 1/(4 pi^2): True
# S = c (PV(1/Q) + i pi delta(Q)) = c (Q - i0)^{-1},  c = 1/(4 pi^2)

# --- 4. Causal propagator delta_0 from W - W^T ---
CHECK delta_0 = (1/(2 pi)) sgn(z0) delta(Q): True
CHECK E_0^A = (1/(2 pi)) theta(-z0) delta(Q) on Omega: True

# --- 5. Off-cone pointwise normalization (spacelike & timelike) ---
CHECK off-cone point#0 has Q!=0 (Q=-1.0): True
CHECK off-cone S=-W=c/Q at point#0 z=(0.0, 1.0, 0.0, 0.0), Q=-1: True
CHECK off-cone point#1 has Q!=0 (Q=-3.75): True
CHECK off-cone S=-W=c/Q at point#1 z=(0.5, 2.0, 0.0, 0.0), Q=-3.75: True
CHECK off-cone point#2 has Q!=0 (Q=-5.0): True
CHECK off-cone S=-W=c/Q at point#2 z=(1.0, 2.0, 1.0, 1.0), Q=-5: True
CHECK off-cone point#3 has Q!=0 (Q=3.75): True
CHECK off-cone S=-W=c/Q at point#3 z=(2.0, 0.5, 0.0, 0.0), Q=3.75: True
CHECK off-cone point#4 has Q!=0 (Q=7.0): True
CHECK off-cone S=-W=c/Q at point#4 z=(3.0, 1.0, 1.0, 0.0), Q=7: True
CHECK off-cone point#5 has Q!=0 (Q=3.5): True
CHECK off-cone S=-W=c/Q at point#5 z=(-2.0, 0.0, 0.5, 0.5), Q=3.5: True
CHECK off-cone point#6 has Q!=0 (Q=-2.4365534255462): True
CHECK off-cone S=-W=c/Q at point#6 z=(0.30471707975443135, -1.0399841062404955, 0.7504511958064572, 0.9405647163912139), Q=-2.43655: True
CHECK off-cone point#7 has Q!=0 (Q=1.994514293379158): True
CHECK off-cone S=-W=c/Q at point#7 z=(-1.9510351886538364, -1.302179506862318, 0.12784040316728537, -0.3162425923435822), Q=1.99451: True
CHECK off-cone point#8 has Q!=0 (Q=-2.105702756487829): True
CHECK off-cone S=-W=c/Q at point#8 z=(-0.016801157504288795, -0.85304392757358, 0.8793979748628286, 0.7777919354289483), Q=-2.1057: True
CHECK off-cone point#9 has Q!=0 (Q=-2.2232612075272082): True
CHECK off-cone S=-W=c/Q at point#9 z=(0.06603069756121605, 1.1272412069680329, 0.4675093422520456, -0.8592924628832382), Q=-2.22326: True
CHECK off-cone point#10 has Q!=0 (Q=-1.5576462298657727): True
CHECK off-cone S=-W=c/Q at point#10 z=(0.36875078408249884, -0.9588826008289989, 0.8784503013072725, -0.049925910986252896), Q=-1.55765: True
CHECK off-cone point#11 has Q!=0 (Q=-1.9479776365819612): True
CHECK off-cone S=-W=c/Q at point#11 z=(-0.18486236354526056, -0.6809295444039414, 1.2225413386740303, -0.15452948206880215), Q=-1.94798: True
CHECK spacelike unit: S=c/Q=-c: True
CHECK timelike unit: S=c/Q=c: True

# --- 6. Numerical spatial Fourier for W vs closed form (damped) ---
CHECK Fourier vs closed W at (z0,r)=(0.5,1.0), err=3.30e-05: True
CHECK Fourier vs closed W at (z0,r)=(-0.5,1.0), err=3.30e-05: True
CHECK Fourier vs closed W at (z0,r)=(1.5,0.8), err=3.30e-05: True
CHECK Fourier vs closed W at (z0,r)=(-1.2,2.0), err=3.31e-05: True
CHECK Fourier vs closed W at (z0,r)=(0.0,1.5), err=3.30e-05: True
CHECK Fourier vs closed W at (z0,r)=(2.0,3.0), err=3.32e-05: True
CHECK Fourier vs closed W at (z0,r)=(-2.5,1.1), err=3.32e-05: True
CHECK Fourier vs closed W at (z0,r)=(0.3,0.3), err=3.30e-05: True

# --- 7. Null-cone chart s=Q: full one-variable distribution ---
CHECK on {Q=0}\{0}, z0 != 0 (so sgn(z0) and theta(+/-) are smooth there): True
# chart density future: Im=0.709646 vs pi*rho(0)=0.709704, err_im=5.84e-05, err_re=1.14e-12
CHECK chart-future Im~pi rho(0), err=5.84e-05: True
CHECK chart-future Re~PV, err=1.14e-12: True
# chart density past: Im=0.709646 vs pi*rho(0)=0.709704, err_im=5.84e-05, err_re=1.14e-12
CHECK chart-past Im~pi rho(0), err=5.84e-05: True
CHECK chart-past Re~PV, err=1.14e-12: True
CHECK dQ != 0 on the nonzero null cone (Q is a local coordinate): True
# pullback of (s-i0)^{-1} therefore defines (Q-i0)^{-1} across the nonzero null cone
# combined with §3: S/c equals that single boundary value on both sheets

# --- 7b. Variable regulator s - tau^2 - i tau a(s), a of fixed sign (lem:wpA-varreg) ---
CHECK variable regulator a = 2 + sin s (z^0>0 sheet): |pairing - (PV + i pi sgn(a) phi(0))| = 1.20e-01, 1.24e-02, 1.24e-03 (tau=1e-2,1e-3,1e-4); decreasing: True
CHECK variable regulator a = -(1.5 + 0.5 cos s) (z^0<0 sheet): |pairing - (PV + i pi sgn(a) phi(0))| = 1.14e-01, 1.17e-02, 1.17e-03 (tau=1e-2,1e-3,1e-4); decreasing: True
CHECK regulator identity alpha^2 + r^2 = -(Q - 2 i tau z^0 - tau^2): variable coefficient a = 2 z^0: True
# => near each nonzero null point (z^0 of fixed sign) the damped kernel tends to (Q - i0 sgn z^0)^{-1}

# --- 8. Cross-checks: Box, homogeneity, transposition, M.6 residue ---
CHECK homogeneity degree -2 for 1/Q (off cone; BV inherits): True
CHECK Box(c/Q)=0 off the cone as rational identity (got 0): True
CHECK transposition: S^T=S (from W-W^T=-i delta_0 and E^R-E^A=delta_0): True
CHECK propagator norm: Box S = i delta on R^4 (hence 0 on Omega): True
CHECK M.6 residue with c=1/(4 pi^2): r = -i/(8 pi^2): True
# r = -I/(8*pi**2) = -i/(8 pi^2)
CHECK counterfactual Q+i0 would flip residue sign (excluded): True

# --- 9. Fourier-family vs position: radial closed-form consistency ---
CHECK off-cone -W_damped = c/(Q-2itau t-tau^2) at (z0,r)=(0.5,1.0), err=0.00e+00: True
CHECK off-cone -W_damped = c/(Q-2itau t-tau^2) at (z0,r)=(-0.5,1.0), err=0.00e+00: True
CHECK off-cone -W_damped = c/(Q-2itau t-tau^2) at (z0,r)=(1.5,0.8), err=3.47e-18: True
CHECK off-cone -W_damped = c/(Q-2itau t-tau^2) at (z0,r)=(-1.2,2.0), err=1.73e-18: True
CHECK off-cone -W_damped = c/(Q-2itau t-tau^2) at (z0,r)=(0.0,1.5), err=0.00e+00: True
CHECK off-cone -W_damped = c/(Q-2itau t-tau^2) at (z0,r)=(2.0,3.0), err=8.67e-19: True
CHECK off-cone -W_damped = c/(Q-2itau t-tau^2) at (z0,r)=(-2.5,1.1), err=8.67e-19: True
CHECK off-cone -W_damped = c/(Q-2itau t-tau^2) at (z0,r)=(0.3,0.3), err=7.04e-14: True
CHECK near-cone off-sheet agreement at t=1.05, rel=1.073e-02: True
CHECK near-cone off-sheet agreement at t=0.95, rel=9.229e-03: True
CHECK near-cone off-sheet agreement at t=-1.05, rel=3.024e-02: True
CHECK near-cone off-sheet agreement at t=-0.95, rel=2.974e-02: True
# Fourier/position consistency off cone confirmed via damped closed forms

# --- SUMMARY ---
{
  "outcome": "Certified",
  "identity": "S = (1/(4*pi^2)) * (Q - i0)^{-1}  on D'(R^4\\{0})",
  "c": "1/(4*pi^2)",
  "boundary_side": "Q-i0",
  "M6_residue_with_c": "-i/(8*pi^2)",
  "seeds": {
    "numpy_rng": [
      42,
      7,
      99,
      100
    ],
    "script_seed_doc": 20261006
  }
}
# ALL CHECKS PASSED