# SCRIPT: BH-FOLD-PARALLEL-01
# Claim under test: the GR static base near a Schwarzschild horizon and the Bures shape metric near the
# coplanar wall have the SAME square-root structure: in the even observable X (Killing norm N, or Gram w3)
# the metric coefficient is const^2/(4X); in the signed coordinate sqrt(X) (lapse lambda, or sigma3) it is a
# finite constant and the metric is even under the sign flip (smooth double / orientation cover).
# Kill condition: if h_NN*N does not tend to a finite nonzero constant, or h_lambda_lambda is not finite and
# even in lambda, or the 4D case fails the same test, the parallel is false.
import sympy as sp
m, N, lam, w = sp.symbols('m N lam w', positive=True)
# 3D Schwarzschild static base: h = dr^2/(1-2m/r) + r^2 dOmega^2, N = 1-2m/r
r = 2*m/(1-N)
drdN = sp.diff(r, N)
hNN = sp.simplify(drdN**2/N)            # radial coefficient in N
print('3D: h_NN * N  ->', sp.limit(hNN*N, N, 0), ' (expect 4 m^2 = 1/(4 kappa^2), kappa=1/(4m))')
hll = sp.simplify(hNN*(2*lam)**2).subs(N, lam**2)   # dN = 2 lam dlam
hll = sp.simplify(hll)
print('3D: h_lam_lam =', hll, '| at lam=0:', sp.limit(hll, lam, 0), ' (expect kappa^-2 = 16 m^2)')
print('3D: even in lam:', sp.simplify(hll - hll.subs(lam, -lam)) == 0)
# proper distance from horizon to small lam
print('3D: proper distance ~ ', sp.series(sp.integrate(sp.sqrt(hll), (lam, 0, lam)), lam, 0, 2))
# 4D Tangherlini: N = 1 - 2m/r^2 (paper (33.28)), kappa = 1/sqrt(2m)
r4 = sp.sqrt(2*m/(1-N))
h4 = sp.simplify(sp.diff(r4, N)**2/N)
print('4D: h_NN * N ->', sp.simplify(sp.limit(h4*N, N, 0)), ' (expect 1/(4 kappa^2) = m/2)')
h4l = sp.simplify((h4*(2*lam)**2).subs(N, lam**2))
print('4D: h_lam_lam at 0 =', sp.simplify(sp.limit(h4l, lam, 0)), ' even:', sp.simplify(h4l-h4l.subs(lam,-lam))==0)
# Bures shape metric along the wall direction: g = dw^2/(4w), w = sigma^2
s = sp.symbols('s')
gww = 1/(4*w)
print('Bures: g_ww * w ->', sp.simplify(gww*w), ' ; in sigma: g_ss =', sp.simplify(gww*sp.diff(s**2, s)**2).subs(w, s**2))
# Kruskal constants quoted in the manuscript
print('3D Kruskal Q(0) = sqrt(e):', sp.simplify(sp.exp(sp.Rational(1,2))/1 - sp.sqrt(sp.E)) == 0)
r0 = sp.symbols('r0', positive=True)
Dstar = sp.exp(r0/r0)*r0/(r0+r0)
print('4D D*(0) = e/2:', sp.simplify(Dstar - sp.E/2) == 0)