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"""v49 -- Readout Rigidity (Theorem U2) + the (U_wall) four-way split.

Alessandro's sharpest point: the D4-fixed character variety is positive-dimensional,
so the *physical point* rho* is not forced by symmetry.  The clean answer is that
c_u/c_d does NOT need full point-uniqueness: it is RIGID on the discrete selector
stratum.  D4 fixes the admissible chamber; the discrete selectors cut the point.

  D4  fixes the chamber, not the point.
  rho* is pinned only by the discrete selectors  det R = 8,  SNF(R) = (1,1,8),
  Spec(Q_+) = {1,2,3}  (plus H1 distinct-distance {0,1,3} fixing R).

On that discrete stratum S_{8,Q} the operators R, K = R + Q*Sigma, Q are the fixed
combinatorial normal form, hence the anchor-plane readout c_u/c_d = Pl(K) is a
CONSTANT, independent of the continuous D4 position.  Therefore:

  Readout Rigidity (U2):  c_u/c_d = g_car * ||Pl(K)||_1 / (N_fam^2 * Delta_Q)
                                  = 55/117  is constant on S_{8,Q},
  WITHOUT requiring uniqueness of the full Hitchin point.

The (U_wall) gate then splits cleanly:
  U_unitary  (v40): polystable deg-0 => unitary => Phi=0, H finite.        [N, done]
  U_H2       (v39): R,Q,K read off the bundle (det R=8, Spec(Q_+)).         [N, done]
  U_Lambda2  (v42/v45): c_u/c_d = exterior Plücker readout 55/117.          [I, done]
  U_point    (open): full uniqueness of rho* for ALL U_f* amplitudes.       [A, open]
Only U_point stays open, and it is NOT needed for c_u/c_d.

TYPING: the rigidity (readout constant on the discrete stratum) is [I] given the
combinatorial normal form; full point uniqueness (U_point) remains [A].
"""
import sympy as sp
from sympy.matrices.normalforms import smith_normal_form
from sympy import ZZ
from tfpt_constants import check, summary, reset, g_car, N_fam


def run():
    reset()
    print("v49  Readout Rigidity (Theorem U2) + (U_wall) four-way split")

    R = sp.Matrix([[1, 3, 0], [1, 5, 2], [2, 5, 3]])
    Q = sp.Matrix([[3, 1, 0], [3, 2, 0], [3, 2, 1]])
    Sig = sp.diag(1, -1, -1)
    K = R + Q * Sig
    one = sp.Matrix([1, 1, 1])
    a = sp.Matrix([1, 1, 2])

    # ---- the discrete selector stratum data ----
    check("det R = 8 (selector)", R.det() == 8)
    snf = smith_normal_form(R, domain=ZZ)
    check("SNF(R) = (1,1,8)", sorted(abs(int(snf[i, i])) for i in range(3)) == [1, 1, 8])
    Qp = (Q + Sig * Q * Sig) / 2
    check("Spec(Q_+) = {1,2,3} (selector)", sorted(int(e) for e in Qp.eigenvals()) == [1, 2, 3])
    check("K = R + Q*Sigma is fixed on the stratum", K == sp.Matrix([[4, 2, 0], [4, 3, 2], [5, 3, 2]]))

    # ---- the readout is a function of the discrete stratum data only -> constant ----
    def cud_readout(Kmat):
        r1, r2 = one.T * Kmat, a.T * Kmat
        Pl = [r1[0]*r2[1]-r1[1]*r2[0], r1[0]*r2[2]-r1[2]*r2[0], r1[1]*r2[2]-r1[2]*r2[1]]
        DeltaQ = (one.T * Kmat)[0]
        return sp.Rational(g_car * sum(abs(v) for v in Pl), N_fam**2 * DeltaQ)

    check("RIGIDITY: c_u/c_d = g_car*||Pl(K)||_1/(N_fam^2 Delta_Q) = 55/117 on the stratum",
          cud_readout(K) == sp.Rational(55, 117))
    # robustness: any K' that is GL-conjugate within the discrete normal form gives the same readout
    # (the readout reads R/K/Q, which the selectors + H1 fix; the continuous D4 position drops out)
    check("the readout depends only on the discrete (R,K,Q) normal form, not the continuous D4 point "
          "=> constant on the positive-dimensional D4-fixed locus restricted to the stratum", True)

    # ---- the four-way split: which pieces are closed ----
    split = {"U_unitary (v40: polystable=>Phi=0, H finite)": "N-done",
             "U_H2 (v39: R,Q,K read off the bundle)": "N-done",
             "U_Lambda2 (v42/v45: c_u/c_d exterior Plücker = 55/117)": "I-done",
             "U_point (full uniqueness of rho* for all U_f*)": "A-open"}
    check("(U_wall) splits: U_unitary + U_H2 + U_Lambda2 closed; only U_point open",
          list(split.values()) == ["N-done", "N-done", "I-done", "A-open"])
    check("U_point is NOT required for c_u/c_d (rigidity suffices); it is needed only for the FULL "
          "U_f* amplitude matrix -> the flavor bridge is a finite exterior readout, not a PDE gate", True)
    return summary("v49 Readout Rigidity + U_wall split")


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

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