"""Energy-preserving dephasing example, not a black-hole model.
Python 3.10+, standard library. Qubit order S,E (and R,S,E for reference).
H0=Delta*|1><1|_S tensor I_E, with a degenerate H_E=0.
U=CNOT(S->E) is unitary and [U,H0]=0. Initial environment is |0>.
This does not account for apparatus/switching work or derive a gravity coupling.
"""
import cmath
import json
import math
import random

def cnot(v):
    w=list(v);w[2],w[3]=w[3],w[2];return w

def partial_environment(v):
    d=len(v)//2
    return [[sum(v[2*i+k]*v[2*j+k].conjugate() for k in range(2)) for j in range(d)] for i in range(d)]

def difference(a,b):return max(abs(x-y) for ra,rb in zip(a,b) for x,y in zip(ra,rb))

def bell_overlap(r):return ((r[0][0]+r[0][3]+r[3][0]+r[3][3])/2).real

def reference_cnot(v):
    return cnot(v[:4])+cnot(v[4:])

def run():
    gap=2.3;energies=[0,0,gap,gap]
    columns=[cnot([complex(i==j) for i in range(4)]) for j in range(4)]
    unitarity=max(abs(sum(columns[i][k].conjugate()*columns[j][k] for k in range(4))-(i==j)) for i in range(4) for j in range(4))
    commutator=max(abs((energies[i]-energies[j])*columns[j][i]) for i in range(4) for j in range(4))
    rng=random.Random(1729);energy_error=0.0;channel_error=0.0;recovery_error=0.0
    for _ in range(128):
        a,b=[complex(rng.gauss(0,1),rng.gauss(0,1)) for _ in range(2)]
        norm=math.sqrt(abs(a)**2+abs(b)**2);a,b=a/norm,b/norm
        v=[a,0j,b,0j];w=cnot(v)
        before=sum(energies[i]*abs(v[i])**2 for i in range(4))
        after=sum(energies[i]*abs(w[i])**2 for i in range(4))
        energy_error=max(energy_error,abs(before-after))
        channel_error=max(channel_error,difference(partial_environment(w),[[abs(a)**2,0j],[0j,abs(b)**2]]))
        recovery_error=max(recovery_error,max(abs(x-y) for x,y in zip(v,cnot(w))))
    h=1/math.sqrt(2)
    plus=cnot([h+0j,0j,h+0j,0j]);minus=cnot([h+0j,0j,-h+0j,0j])
    local_difference=difference(partial_environment(plus),partial_environment(minus))
    joint_inner=abs(sum(a.conjugate()*b for a,b in zip(plus,minus)))
    reference=[h+0j,0j,0j,0j,0j,0j,h+0j,0j]
    entangled=reference_cnot(reference)
    output_bell=bell_overlap(partial_environment(entangled))
    recovered_bell=bell_overlap(partial_environment(reference_cnot(entangled)))
    assert max(unitarity,commutator,energy_error,channel_error,recovery_error,local_difference,joint_inner)<1e-12
    assert abs(output_bell-.5)<1e-12 and abs(recovered_bell-1)<1e-12
    return {'scope':'Standard finite-dimensional counterexample, not a new theory or gravitational simulation','random_cases':128,'energy_gap_input':gap,'environment_hamiltonian':'H_E=0, two-dimensional degenerate memory','unitarity_error':unitarity,'hamiltonian_commutator_error':commutator,'energy_expectation_error':energy_error,'dephasing_channel_error':channel_error,'global_inverse_error':recovery_error,'plus_minus_local_difference':local_difference,'plus_minus_joint_overlap_magnitude':joint_inner,'reference_bell_overlap_after_discarding_environment':output_bell,'reference_bell_overlap_after_global_inverse':recovered_bell,'limitations':['No independent physical existence claim for the degenerate memory','No quantum-gravity geometry, causal horizon or endpoint dynamics','No hardware execution or apparatus energy-cost budget']}

if __name__=='__main__':print(json.dumps(run(),indent=2))