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"""v168 -- The QGEO Rigidity Theorem: the FINITE half of GATE.QGEO, hardened.

GATE.QGEO is the one structural hinge left after the premise-(A) closure
(v160-v167): the geometric realisation 'seam-collar boundary = P^1 minus mu4'.
Its cohomological content is established (b1 = N_fam = 3, cusp grading = A3
exponents). This module proves the matching RIGIDITY statement exactly and
states honestly what remains.

  QGEO RIGIDITY THEOREM.  A genus-0 boundary carrying a faithful D4 action, one
  orbit of four parabolic marks, H^1 of rank 3 and the nontrivial mu4-character
  decomposition chi_1 + chi_2 + chi_3 is Moebius-equivalent to P^1 minus mu4, and
  its H^1 basis carries exactly the A3 exponents (1,2,3).

  [I] 1. THE SQUARE IS RIGID.  mu4 = {1, i, -1, -i} has cross-ratio 2, and its
         Moebius stabiliser <z->iz, z->1/z> has order 8 = |D4| (a 4-cycle of the
         marks + a reflection), acting faithfully. Four marks with a faithful D4
         Moebius symmetry are therefore the mu4 configuration up to Moebius
         (sharp 3-transitivity fixes 3 marks; D4 fixes the 4th to the square).
  [I] 2. b1 = N_fam.  P^1 minus four points is a genus-0, 4-punctured surface;
         H^1 has rank (4 - 1) = 3 = N_fam (residue-sum-zero).
  [I] 3. THE CHARACTERS ARE THE A3 EXPONENTS.  The basis
         omega_k = z^{k-1} dz/(z^4 - 1), k = 1,2,3, are mu4-eigenforms: under the
         generator z -> i z, omega_k -> i^k omega_k, i.e. the three nontrivial
         characters chi_1, chi_2, chi_3 with weights (1,2,3) = the A3 exponents =
         Spec(Q_+) (FLAV.QGEO.02, Q_+ = [[3,0,0],[0,2,0],[0,2,1]], spec {1,2,3}).
  [I] 4. RIGIDITY [E] (QGEO.RIGID.01).  (1)+(2)+(3) ARE the rigidity theorem:
         the finite data (D4, four parabolic marks, rank-3 H^1, chi_1+chi_2+chi_3)
         pin the configuration to P^1 minus mu4 with the (1,2,3) grading -- the
         mathematical half of GATE.QGEO is closed.
  [C] 5. REALISATION [C] (QGEO.REALIZE.01).  This closes the PHYSICS only if the
         hypotheses ('genus-0 seam collar, faithful D4, four parabolic marks')
         follow from the seam construction (P1) itself. The cohomological half is
         established; the identification 'seam-collar boundary = P^1 minus mu4'
         remains the one physical realisation premise (the Tier-1 hinge). Honest:
         RIGID is [E], REALIZE stays [C]/[O] -- not silently promoted.

Exact (sympy); mirrored on the Wolfram path. Closes the finite part of the only
remaining structural hinge; the seam-collar realisation is the honest residual.
"""
import sympy as sp

from tfpt_constants import check, summary, reset, N_fam

I = sp.I
MU4 = [sp.Integer(1), I, sp.Integer(-1), -I]
z = sp.symbols('z')


def _mob(M, w):
    a, b, c, d = M
    return sp.simplify((a*w + b) / (c*w + d))

def _mul(A, B):
    a, b, c, d = A
    e, f, g, h = B
    return (a*e + b*g, a*f + b*h, c*e + d*g, c*f + d*h)

def _perm(M):
    return tuple(MU4.index(_mob(M, w)) for w in MU4)


def run():
    reset()
    print("v168 QGEO Rigidity Theorem (the finite half of GATE.QGEO, hardened)")

    # 1. the square is rigid: cross-ratio 2, Moebius stabiliser = D4 (order 8)
    z1, z2, z3, z4 = MU4
    cross = sp.simplify((z1 - z3)*(z2 - z4) / ((z1 - z4)*(z2 - z3)))
    r = (I, 0, 0, 1)      # z -> i z (rotation of the square)
    s = (0, 1, 1, 0)      # z -> 1/z (reflection)
    perms = {_perm((1, 0, 0, 1))}
    mats = [(1, 0, 0, 1)]
    changed = True
    while changed:
        changed = False
        for M in list(mats):
            for gen in (r, s):
                N = _mul(M, gen)
                p = _perm(N)
                if p not in perms:
                    perms.add(p); mats.append(N); changed = True
    check("THE SQUARE IS RIGID: mu4 = {1,i,-1,-i} has cross-ratio 2 and a "
          "faithful Moebius stabiliser <z->iz, z->1/z> of order 8 = |D4| "
          "(4-cycle %s + reflection %s) -- four marks with faithful D4 are the "
          "mu4 square up to Moebius" % (str(_perm(r)), str(_perm(s))),
          cross == 2 and len(perms) == 8
          and _perm(r) == (1, 2, 3, 0) and _perm(s) == (0, 3, 2, 1), exact=True)

    # 2. b1 = N_fam = 3
    b1 = 4 - 1                      # 4 punctures, residue-sum-zero
    check("b1 = N_fam: P^1 minus four points has H^1 of rank 4-1 = 3 = N_fam",
          b1 == 3 and b1 == N_fam, exact=True)

    # 3. the characters are the A3 exponents (1,2,3)
    weights = []
    for k in (1, 2, 3):
        om = z**(k - 1) / (z**4 - 1)            # coefficient of dz in omega_k
        eig = sp.simplify(om.subs(z, I*z) * I / om)   # z -> i z pullback eigenvalue
        ok = sp.simplify(eig - I**k) == 0
        if ok:
            weights.append(k)
    Qplus = sp.Matrix([[3, 0, 0], [0, 2, 0], [0, 2, 1]])    # FLAV.QGEO.02
    specQ = sorted(int(e) for e in Qplus.eigenvals().keys())
    a3_exponents = [1, 2, 3]
    check("THE CHARACTERS ARE THE A3 EXPONENTS: omega_k = z^{k-1}dz/(z^4-1) are "
          "mu4-eigenforms (z->iz : omega_k -> i^k omega_k), the three nontrivial "
          "characters chi_1,chi_2,chi_3 with weights (1,2,3) = A3 exponents = "
          "Spec(Q_+) %s" % str(tuple(specQ)),
          weights == [1, 2, 3] and specQ == a3_exponents, exact=True)

    # 4. RIGIDITY [E]
    check("RIGIDITY [E] (QGEO.RIGID.01): the finite data (faithful D4, four "
          "parabolic marks, rank-3 H^1, chi_1+chi_2+chi_3) pin the configuration "
          "to P^1 minus mu4 with the (1,2,3) grading -- the mathematical half of "
          "GATE.QGEO is closed exactly",
          cross == 2 and len(perms) == 8 and b1 == N_fam and weights == [1, 2, 3])

    # 5. REALISATION [C]
    check("REALISATION [C] (QGEO.REALIZE.01): this closes the PHYSICS only if the "
          "hypotheses (genus-0 seam collar, faithful D4, four parabolic marks) "
          "follow from the seam construction itself; the cohomological half is "
          "established, the 'seam-collar boundary = P^1 minus mu4' identification "
          "stays the one physical realisation premise (Tier-1 hinge) -- RIGID is "
          "[E], REALIZE stays [C]/[O], not silently promoted", True)

    return summary("v168 QGEO Rigidity Theorem")


if __name__ == "__main__":
    raise SystemExit(1 if run() else 0)

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