"""MC-286 validation: Einstein-Compatible Continuity-Field Action.

Validates the canonical scalar-field stress-energy/conservation identity in a
flat test chart. This is a symbolic E1 witness for the algebraic reduction,
not a curved-spacetime numerical simulation or empirical physics certificate.
"""
import sympy as sp
import json, hashlib, platform, sys

t,x,y,z=sp.symbols("t x y z", real=True)
coords=(t,x,y,z)
C=sp.Function("C")(*coords)
V=sp.Function("V"); Vp=sp.Function("Vp")
eta=sp.diag(-1,1,1,1)
dC=sp.Matrix([sp.diff(C,c) for c in coords])
kinetic=sp.simplify((dC.T*eta*dC)[0])
T=sp.Matrix(4,4, lambda mu,nu: sp.simplify(dC[mu]*dC[nu]-eta[mu,nu]*(sp.Rational(1,2)*kinetic+V(C))))
boxC=sp.simplify(sum(eta[mu,nu]*sp.diff(C,coords[mu],coords[nu]) for mu in range(4) for nu in range(4)))
res=[]
for nu in range(4):
    div=sum(eta[mu,mu]*sp.diff(T[mu,nu],coords[mu]) for mu in range(4))
    div=sp.expand(div).replace(lambda e:isinstance(e,sp.Derivative) and e.expr==V(C), lambda e: Vp(C)*sp.diff(C,e.variables[0]))
    target=sp.simplify((boxC-Vp(C))*sp.diff(C,coords[nu]))
    res.append(sp.simplify(div-target))
validation={
    "model":"Einstein-compatible continuity-field action with canonical scalar C",
    "signature":"(-,+,+,+)",
    "lagrangian_C":"-1/2 ∂_μ C ∂^μ C - V(C)",
    "stress_energy_tensor":"T_{μν} = ∂_μ C ∂_ν C - η_{μν}[1/2 ∂_α C ∂^α C + V(C)]",
    "euler_lagrange":"□C - V'(C) = 0",
    "divergence_identity":"∂^μ T_{μν} = (□C - V'(C)) ∂_ν C",
    "box_C":str(boxC),
    "divergence_residuals":[str(r) for r in res],
    "all_symbolic_checks_pass":all(r==0 for r in res),
    "boundary":"This machine check validates the scalar-field algebra in a flat test chart and the conservation implication. It does not numerically simulate curved spacetime, prove physical existence of C, remove singularities, or establish GR-QFT unification.",
    "python":sys.version.split()[0], "sympy":sp.__version__, "platform":platform.platform(),
}
payload=json.dumps(validation, sort_keys=True, separators=(",",":")).encode()
validation["certificate_sha256"]=hashlib.sha256(payload).hexdigest()
print(json.dumps(validation, indent=2))
assert validation["all_symbolic_checks_pass"]