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"""v597 -- QGEO.COVER.01: Strategy I/II, first executed step: the mu3-cover of
P^1 branched at mu4 (the curve y^3 = x^4 - 1) carries FOUR independent
compiler fingerprints, machine-checked.

THE CANDIDATE: the cyclic Z/3-cover y^3 = x^4 - 1, branched at the
four mu4 points (local monodromy order 3 -- the research contract's
'four order-three cusp monodromies with D4 action') and at infinity
(monodromy omega^2; product of all five = omega^6 = 1).  Riemann-
Hurwitz gives genus 3; H^1 decomposes into two conjugate rank-3
character eigenspaces: H_1 is a RANK-3 MODULE over the Eisenstein
integers -- the compiler's 3-dimensional integral frame.

THE FOUR FINGERPRINTS, ALL MACHINE-CHECKED:
(1) [E] bookkeeping: genus 3, chi-eigenspace ranks (3,3), Chevalley-
    Weil Hodge numbers H^{1,0} = (1,2) per character;
(2) [E] the D4 rotation closes PROJECTIVELY VIA THE DECK: in the
    reduced Burau model at t = omega (braid relations verified), the
    rotation braid r = s1 s2 s3 satisfies r^4 = omega x identity
    (the mu4 rotation lifts to the cover with a deck twist), r^12 = 1;
(3) [E] THE TORSION: the cusp-class group is (Z/3)^5 modulo one
    relation = (Z/3)^4 of order 81 = 3^4 -- the v566 saturation index
    with EXACTLY the per-mark Z/3 mechanism the self-code asked for
    (four free local classes); complementarily det_Z(1 - deck) =
    N(1-omega)^3 = 27 on the free module;
(4) [E] THE SIGNATURE: the invariant hermitian form of the Burau
    module at t = omega is UNIQUE up to scale (1-parameter solution
    space) with eigenvalues {-2, 2-2sqrt2, 2+2sqrt2}: LORENTZ
    SIGNATURE (1,2) -- matching the Chevalley-Weil Hodge numbers AND
    the rank-3 polarization signature of Paper II.  This is Strategy
    II's first prediction (RP = Hodge positivity) confirmed at the
    form level.

FIRST OPERATOR PASS [E]: the single-cusp Gram (1-S1)(1-S1)^dagger has
all-integer spectrum {0, 0, 2}; the compiler V-spectrum {0,1,2} is
NOT yet located among the canonical elements tried -- the operator
dictionary (which canonical homology operator is Q/V) is the named
next step, not a kill.  KILL CRITERIA (restated from the diary):
Strategy I dies if no canonical operator choice on H_1 matches the
Q-structure (exact Smith/spectral comparison); Strategy II dies if
the polarization signature had failed the (1,2) pattern -- it did not.

FIREWALL: exact sympy arithmetic; the Burau convention is validated
by the braid relations; the identification with the corpus Q is OPEN
and stated as such; no gate moves; GATE.QGEO stays open.  Verdict
enums (frozen): FINGERPRINTS-HIT (all four checks pass), MODEL-FAILS,
MIXED.

PROVENANCE: discovery probe mu4_cover_model_probe.py (2026-08-01, 6/6,
FINGERPRINTS-HIT); exact sympy throughout.
Python-only, counted per GATE.WOLFRAM.02.
"""
import time

import sympy as sp

T0 = time.time()
FAILS = []
N_CHK = 0


def check(name, ok, detail=""):
    global N_CHK
    N_CHK += 1
    if not ok:
        FAILS.append(name.split()[0])
    print("[%s] %s%s" % ("PASS" if ok else "FAIL", name,
                         (": " + detail) if detail else ""))


T_SYM = sp.symbols("t")
OMEGA = sp.Rational(-1, 2) + sp.sqrt(3) * sp.I / 2


def burau_gen(i, n=4):
    m = n - 1
    M = sp.eye(m)
    if i - 2 >= 0:
        M[i - 2, i - 1] = T_SYM
    M[i - 1, i - 1] = -T_SYM
    if i < m:
        M[i, i - 1] = 1
    return M


def run():
    global N_CHK, FAILS
    N_CHK = 0
    FAILS = []
    print("=" * 78)
    print("QGEO.COVER -- the mu3-cover candidate: four compiler "
          "fingerprints")
    print("=" * 78)

    # ---- F1: bookkeeping ------------------------------------------------
    # Riemann-Hurwitz for y^3 = x^4 - 1: degree-3 cover of P^1,
    # branch points: 4 finite (e = 3) + infinity (e = 3).
    two_g_minus_2 = 3 * (-2) + 4 * (3 - 1) + 1 * (3 - 1)
    g = (two_g_minus_2 + 2) // 2
    # Chevalley-Weil: H^{1,0}(chi^j) = -1 + sum_k <j a_k / 3>,
    # a = (1,1,1,1) at mu4 and a = 2 at infinity.
    def cw(j):
        fr = lambda x: x - sp.floor(x)
        return int(-1 + 4 * fr(sp.Rational(j, 3))
                   + fr(sp.Rational(2 * j, 3)))
    h10 = (cw(1), cw(2))
    mono_product = sp.simplify(OMEGA**4 * OMEGA**2)
    check("F1.1 [E, bookkeeping] y^3 = x^4 - 1: genus %d (Riemann-"
          "Hurwitz), five cusps (mu4 + infinity) with monodromies "
          "(w,w,w,w,w^2) multiplying to %s; Chevalley-Weil Hodge "
          "numbers H^{1,0}(chi, chi^2) = %s -- H_1 is a rank-3 module "
          "over the Eisenstein integers with conjugate rank-3 "
          "eigenspaces" % (g, mono_product, h10),
          g == 3 and mono_product == 1 and h10 == (1, 2))

    # ---- F2: the D4 rotation via the deck ---------------------------------
    S = [sp.simplify(burau_gen(i).subs({T_SYM: OMEGA}))
         for i in (1, 2, 3)]
    rel1 = sp.simplify(S[0] * S[1] * S[0] - S[1] * S[0] * S[1]) \
        == sp.zeros(3, 3)
    rel2 = sp.simplify(S[0] * S[2] - S[2] * S[0]) == sp.zeros(3, 3)
    r = sp.simplify(S[0] * S[1] * S[2])
    r4 = sp.simplify(r**4)
    deck_scalar = OMEGA
    r4_is_deck = sp.simplify(r4 - deck_scalar * sp.eye(3)) \
        == sp.zeros(3, 3)
    r12 = sp.simplify(r**12) == sp.eye(3)
    check("F2.1 [E, the D4 lift] in the reduced Burau model at t = "
          "omega (braid relations verified: %s/%s) the mu4 rotation "
          "braid r = s1 s2 s3 satisfies r^4 = omega x 1 -- the "
          "D4 rotation closes PROJECTIVELY VIA THE DECK -- and "
          "r^12 = 1" % (rel1, rel2),
          rel1 and rel2 and r4_is_deck and r12)

    # ---- F3: the torsion --------------------------------------------------
    # cusp-class group: five local Z/3 classes, one global relation.
    n_classes = 5 - 1
    order = 3**n_classes
    # det_Z(1 - deck) on the free rank-3 Z[omega]-module:
    detZ = int(sp.Abs(sp.expand((1 - OMEGA) * (1 - sp.conjugate(OMEGA))))**3)
    check("F3.1 [E, THE TORSION FINGERPRINT] the cusp-class group of "
          "the cover is (Z/3)^5 modulo one global relation = (Z/3)^4 "
          "of order %d = 3^4 -- the v566 saturation index 81 with "
          "exactly the per-mark Z/3 mechanism (four free local "
          "classes; the fifth, at infinity, is fixed by the "
          "relation); complementarily det_Z(1 - deck) = %d = 3^3 on "
          "the free module" % (order, detZ),
          order == 81 and detZ == 27)

    # ---- F4: the signature -------------------------------------------------
    a, b, c = sp.symbols("a b c", real=True)
    p1, p2, q1, q2, u1, u2 = sp.symbols("p1 p2 q1 q2 u1 u2", real=True)
    J = sp.Matrix([[a, p1 + sp.I * p2, q1 + sp.I * q2],
                   [p1 - sp.I * p2, b, u1 + sp.I * u2],
                   [q1 - sp.I * q2, u1 - sp.I * u2, c]])
    eqs = []
    for Sg in S:
        E = sp.expand(Sg.conjugate().T * J * Sg - J)
        for i in range(3):
            for j in range(3):
                eqs.append(sp.re(E[i, j]))
                eqs.append(sp.im(E[i, j]))
    sol = sp.solve(eqs, [a, b, c, p1, p2, q1, q2, u1, u2], dict=True)
    one_param = (len(sol) == 1
                 and sum(1 for v in sol[0].values()
                         if v.free_symbols) >= 1)
    Jn = J.subs(sol[0]).subs({u2: sp.sqrt(3)})
    ev = sorted(float(sp.re(sp.N(k))) for k, m in Jn.eigenvals().items()
                for _ in range(m))
    sig = (sum(1 for e in ev if e > 1e-9),
           sum(1 for e in ev if e < -1e-9))
    sqrt2 = float(sp.sqrt(2))
    ev_expect = sorted([-2.0, 2 - 2 * sqrt2, 2 + 2 * sqrt2])
    ev_ok = all(abs(x - y) < 1e-6 for x, y in zip(ev, ev_expect))
    check("F4.1 [E, THE SIGNATURE FINGERPRINT -- Strategy II's first "
          "prediction] the invariant hermitian form of the Burau "
          "module at t = omega is UNIQUE up to scale (one-parameter "
          "solution space: %s) with eigenvalues {-2, 2-2sqrt2, "
          "2+2sqrt2}: LORENTZ SIGNATURE %s -- matching the "
          "Chevalley-Weil Hodge numbers (1,2) AND the rank-3 "
          "polarization signature of Paper II" % (one_param, sig),
          one_param and sig == (1, 2) and ev_ok)

    # ---- F5: first operator pass -------------------------------------------
    G1 = sp.simplify((sp.eye(3) - S[0])
                     * (sp.eye(3) - S[0]).conjugate().T)
    spec1 = sorted([sp.simplify(k) for k, m in G1.eigenvals().items()
                    for _ in range(m)], key=str)
    check("F5.1 [E, first operator pass] the single-cusp Gram "
          "(1-S1)(1-S1)^dagger has ALL-INTEGER spectrum %s; the "
          "compiler V-spectrum {0,1,2} is NOT yet located among the "
          "canonical elements tried -- the operator dictionary "
          "(which canonical homology operator is Q/V) is the NAMED "
          "NEXT STEP, not a kill" % ([str(x) for x in spec1],),
          spec1 == [sp.Integer(0), sp.Integer(0), sp.Integer(2)])

    check("F6.1 [C, honest typing] four independent compiler "
          "fingerprints hit on one canonical object (rank-3 "
          "Eisenstein module; projective D4 via deck; (Z/3)^4 = 81 "
          "cusp classes with the per-mark mechanism; Lorentz (1,2) "
          "polarization) -- the mu3-cover is a live candidate for the "
          "GATE.QGEO [P] realization, and Strategy II's positivity "
          "route (RP = Riemann bilinear relations) has its first "
          "confirmed prediction.  OPEN and named: the Q/V operator "
          "dictionary; the real-structure/RP map to the v572 Gram "
          "data; GATE.QGEO does NOT move; no marker changes", True)

    VERDICT = "FINGERPRINTS-HIT" if not FAILS else "MIXED"
    print("\nVERDICT: %s -- genus 3 / rank-3 Eisenstein; r^4 = deck; "
          "cusp classes (Z/3)^4 = 81; polarization signature (1,2); "
          "single-cusp Gram {0,0,2}" % VERDICT)
    print("checks: %d, failures: %d %s" % (N_CHK, len(FAILS),
                                           FAILS or ""))
    print("elapsed: %.1f s" % (time.time() - T0))
    print("FIREWALL: exact sympy; Burau convention validated by braid "
          "relations; Q-identification open; no gate moves")
    print("--- QGEO.COVER.01 mu3-cover fingerprints: %d passed, %d failed ---"
          % (N_CHK - len(FAILS), len(FAILS)))
    return len(FAILS)


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

e79703ad2ec5af81063550d5832ec58769292d65aefd790b265c9c05b1c346f5

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