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"""v574 -- QGEO.WINDING.SELF.01: Stage B of the No-Carrier-Reconstruction
audit, first narrowing -- the involution TYPE is killed by a rank theorem
and the winding normalisation N_fam = 3 is FORCED by the double
self-consistency (ladder + saturation): two of the five declared
geometric inputs relocate to consistency-forced.

PROVENANCE.  v567 (Stage A) reconstructed Q from the carrier-free inputs
(S)(W)(E)(M)(L) and relocated the audit to GATE.QGEO.01: are those inputs
themselves forced?  This module ablates two of them.  Audited by the
discovery probe (stageb_winding_probe.py, 12/12, verdict FORCED).

[E] 1. THE TYPE IS FORCED (rank theorem, symbolic): for ANY involution
    with a two-dimensional positive eigenspace the sheet direction
    V' = Q P_- has rank <= 1 (char poly x^2 (x - t) identically in Q),
    so the binary trace code Spec V = {0,1,2} (v566) is unreachable --
    only the corpus type (one positive direction) can carry the
    self-code.
[E] 2. (E) BOUNDS N: the winding demand forces the first column to
    (N,N,N), so q11 = N must itself be an A3 exponent -- N in {4,5,6}
    give EMPTY classes; N <= 3.
[E] 3. the ladder self-consistency (v568) is ROBUST: for every
    N in {1,2,3} it selects EXACTLY ONE member of the parabolic class
    (28/16/14 members -> 1/1/1).
[E] 4. THE CENTRAL RESULT -- N IS FORCED: the saturation
    self-consistency sat = N^4 (v567's (X) read as a self-consistency)
    holds ONLY at N = 3 (N = 1: sat 32 != 1; N = 2: sat 512 != 16;
    N = 3: sat 81 = 3^4) -- the DOUBLE self-consistency pins
    (N, Q) = (3, compiler Q) uniquely across the whole N-ablated
    family: N_fam = 3 relocates from declared input to
    consistency-forced within the reconstruction.  Equivalently by the
    V-spectra: {0,2,3} / {0,1,3} / {0,1,2} -- only N = 3 carries the
    binary trace code p_n = 2 + 2^n.

[E] 5. THE HONEST DECOMPOSITION (spectrum ablation): freeing the
    exponent spectrum itself, the double self-consistency does NOT pin
    N absolutely -- the one surviving family is spectrum = {1, 2, N}
    with the shape Q(N) = [[N,1,0],[N,2,0],[N,2,1]] (verified in-module
    for N = 4, 5, 6, each ladder-selfconsistent with sat = N^4; the
    full 35-spectrum scan lives in the discovery probe and finds no
    other double-selfconsistent point).  The forcing decomposes
    precisely: the self-consistency forces the SHEET spectrum {1,2}
    (the binary trace code) and the compiler SHAPE; the A3 exponent
    demand (E) contributes exactly ONE datum, N = 3.

[E] 6. REDUNDANCY -- (M) AND (L) FALL TOO: relative to the self-code
    demands (parabolic + ladder + sat = 81) the monodromy budget (M)
    and the seam-line demand (L) are REDUNDANT at the declared bound:
    dropping (L), and even dropping BOTH (M) and (L) (class widens to
    1274 members), the self-code still picks EXACTLY the compiler Q --
    the declared geometric input list of the reconstruction shrinks to
    (S)-existence, (W)-form and (E).

[E] 7. THE MAXIMAL ABLATION -- (E) FALLS TOO: dropping the exponent
    demand entirely, the demands (W: first column 3(1,1,1)) +
    integrality + the self-code package STILL pin the compiler Q
    uniquely (29 ladder candidates, one hit).  (E), (M), (L) are ALL
    redundant at the declared bound: the reconstruction needs only the
    sheet involution (type forced), the winding column with N_fam = 3
    (the E8-glue count, v1) and the self-code demands.

[E] 8. WHAT THE WINDING FORM REALLY DOES: dropping the winding demand
    entirely (arbitrary first column), the self-code package leaves a
    FOUR-element column family {(0,1,1), (0,3,3), (3,3,3), (5,1,5)}
    (full scan in the discovery probe; the four members re-verified
    in-module) -- and exactly ONE is uniform: (3,3,3), the compiler.
    The winding demand's load-bearing content is precisely the
    UNIFORMITY of the column (1 of 4).

NAMED LIMITS AS LOAD-BEARING CONTENT: what stays declared is the
involution EXISTENCE, the winding FORM and N_fam = 3 (independently
anchored in the E8 glue); the remaining
GATE.QGEO.01 content is the derivation of (S)-existence, (W)-form and
(E) from mu4/D4/H^1 geometry (narrowed by FOUR items now, NOT closed);
enumeration bound B = 6 declared; family data N_fam appears here as the
UNKNOWN of the ablation, not as an input; NO P2 reduction -- AX.P2.01 /
ANCHOR.GEN.01 markers untouched.  Fences: Smith 1861 / Hermite 1851
CLASSICAL; verdict enums frozen before the run.  Python-only, counted
per GATE.WOLFRAM.02.  Discovery:
experiments/tfpt-discovery/stageb_winding_probe.py (2026-07-31, 12/12,
FORCED).
"""
import time
from itertools import product

import sympy as sp
from sympy.matrices.normalforms import smith_normal_form

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

B_ENTRY = 6


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 ""))


def info(name, detail=""):
    print("[INFO] %s%s" % (name, (": " + detail) if detail else ""))


I3 = sp.eye(3)
IDX7 = ((0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1), (2, 2))
SIG = sp.diag(1, -1, -1)
Q_CORPUS = sp.Matrix([[3, 1, 0], [3, 2, 0], [3, 2, 1]])

def run():
    global N_CHK, FAILS
    N_CHK = 0
    FAILS = []
    print("=" * 78)
    print("QGEO.WINDING.SELF -- Stage B, first narrowing")
    print("=" * 78)

    # --- S1: the involution TYPE is killed by a rank theorem --------------------
    q = sp.MatrixSymbol("q", 3, 3)
    Pminus_21 = sp.diag(0, 0, 1)      # opposite type: 2-dim POSITIVE eigenspace
    Vgen = sp.Matrix(q) * Pminus_21
    x = sp.symbols("x")
    cp = sp.factor(Vgen.charpoly(x).as_expr())
    check("S1.1 [E, theorem -- THE TYPE IS FORCED] for ANY involution with a "
          "TWO-dimensional positive eigenspace the sheet direction V' = Q "
          "P_- has rank <= 1: its characteristic polynomial is x^2 (x - t) "
          "identically in Q (symbolic: %s), so Spec V' has at most ONE "
          "nonzero value -- the binary trace code Spec V = {0, 1, 2} "
          "(v566, p_n = 2 + 2^n) is UNREACHABLE: only the corpus type "
          "(one positive direction) can carry the self-code" % cp,
          sp.simplify(cp / x**2 * 0 + (Vgen.charpoly(x).as_expr()
                                       - x**3 + sp.trace(Vgen) * x**2))
          == 0)


    def admissible_class(N):
        """(S)(W)(E)(M)(Z) enumeration at winding normalisation N."""
        out = []
        if N > 3:
            return out                     # (E): q11 = N must be an exponent
        rest = {1, 2, 3} - {N}
        lo, hi = min(rest), max(rest)
        for q12, q13, q22, q23, q32, q33 in product(range(B_ENTRY + 1),
                                                    repeat=6):
            # (E) integer prefilter on the lower block
            if q22 + q33 != lo + hi or q22 * q33 - q23 * q32 != lo * hi:
                continue
            if (q12, q13) == (0, 0):
                continue
            Q = sp.Matrix([[N, q12, q13], [N, q22, q23], [N, q32, q33]])
            Qm = (Q - SIG * Q * SIG) / 2
            Q2 = Qm * Qm
            if Q2 == sp.zeros(3, 3):
                continue
            P = Q2 / N
            if sp.expand(P * P - P) != sp.zeros(3, 3):
                continue
            out.append(Q)
        return out


    def parabolic_members(cls):
        out = []
        for Q in cls:
            U, V = Q * sp.diag(1, 0, 0), Q * sp.diag(0, 1, 1)
            ws = [I3]
            fr = [I3]
            for _ in range(4):
                fr = [w * G for w in fr for G in (U, V)]
                ws += fr
            Mc = sp.Matrix([[w[i, j] for i in range(3) for j in range(3)]
                            for w in ws])
            if Mc.rank() == 7:
                out.append(Q)
        return out


    def ladder_members(para):
        out = []
        for Q in para:
            V = Q * sp.diag(0, 1, 1)
            ev = [e for e in V.eigenvals() if e != 0]
            dom = max(ev, key=lambda e: abs(e))
            vlad = sp.Matrix([1, dom, dom**2])
            if sp.simplify(V * vlad - dom * vlad) == sp.zeros(3, 1):
                out.append((Q, dom))
        return out


    def saturation(Q):
        U, V = Q * sp.diag(1, 0, 0), Q * sp.diag(0, 1, 1)
        ws = [I3]
        fr = [I3]
        for _ in range(5):
            fr = [w * G for w in fr for G in (U, V)]
            ws += fr
        Mc = sp.Matrix([[w[i, j] for (i, j) in IDX7] for w in ws])
        if Mc.rank() != 7:
            return None
        s_ = smith_normal_form(Mc.T)
        sat = 1
        for i in range(7):
            if s_[i, i] != 0:
                sat *= abs(s_[i, i])
        return sat


    # --- S2: (E) bounds N --------------------------------------------------------
    kills = all(len(admissible_class(N)) == 0 for N in (4, 5, 6))
    check("S2.1 [E] the exponent demand (E) BOUNDS the winding "
          "normalisation: (W) forces the first column to (N, N, N), so "
          "q11 = N must itself be an A3 exponent -- N in {4, 5, 6} give "
          "EMPTY admissible classes (enumeration at B = %d), N <= 3 by "
          "theorem" % B_ENTRY, kills)

    # --- S3: the three surviving families ---------------------------------------
    results = {}
    for N in (1, 2, 3):
        cls = admissible_class(N)
        para = parabolic_members(cls)
        lads = ladder_members(para)
        sats = [(Q, dom, saturation(Q)) for (Q, dom) in lads]
        results[N] = dict(n_adm=len(cls), n_para=len(para), lads=sats)
        info("N = %d" % N, "admissible %d, parabolic %d, ladder-selected %d"
             % (len(cls), len(para), len(lads)))

    check("S3.1 [E] the ladder self-consistency (v568) is ROBUST across the "
          "family: for EVERY N in {1, 2, 3} it selects EXACTLY ONE member "
          "of the parabolic class (%d/%d/%d members -> 1/1/1)"
          % (results[1]["n_para"], results[2]["n_para"],
             results[3]["n_para"]),
          all(len(results[N]["lads"]) == 1 for N in (1, 2, 3)))

    sel = {N: results[N]["lads"][0] for N in (1, 2, 3)}
    check("S3.2 [E] the selected members and their data: N = 1 -> dom 3, "
          "sat %s; N = 2 -> dom 3, sat %s; N = 3 -> dom 2, sat %s = 3^4 -- "
          "only N = 3 selects the COMPILER Q and the binary ladder (1,2,4)"
          % (sel[1][2], sel[2][2], sel[3][2]),
          sel[3][0] == Q_CORPUS and sel[3][1] == 2
          and sel[1][1] == 3 and sel[2][1] == 3)

    # --- S4: the double self-consistency pins N ---------------------------------
    sat_ok = {N: (sel[N][2] == N**4) for N in (1, 2, 3)}
    check("S4.1 [E, THE CENTRAL RESULT -- the winding normalisation is "
          "FORCED] the saturation self-consistency sat = N^4 (v567's (X), "
          "read as a self-consistency in the family) holds ONLY at N = 3: "
          "N = 1 gives sat 32 != 1, N = 2 gives 512 != 16, N = 3 gives "
          "81 = 3^4 -- the DOUBLE self-consistency (ladder + saturation) "
          "pins (N, Q) = (3, compiler Q) uniquely across the whole "
          "N-ablated family: N_fam = 3 relocates from declared input to "
          "consistency-forced within the reconstruction",
          sat_ok == {1: False, 2: False, 3: True}
          and sel[1][2] == 32 and sel[2][2] == 512 and sel[3][2] == 81)

    specs = {}
    for N in (1, 2, 3):
        V = sel[N][0] * sp.diag(0, 1, 1)
        specs[N] = tuple(sorted(V.eigenvals().keys(), key=lambda e: sp.N(e)))
    check("S4.2 [E] the V-spectra of the selected members: N = 1 -> "
          "{0, 2, 3}, N = 2 -> {0, 1, 3}, N = 3 -> {0, 1, 2}: only N = 3 "
          "carries the binary trace code p_n = 2 + 2^n (the v566 "
          "Spec V = {0, 1, 2}); the winding normalisation and the trace "
          "code select the SAME point",
          specs[1] == (0, 2, 3) and specs[2] == (0, 1, 3)
          and specs[3] == (0, 1, 2))

    # --- S4b: the spectrum ablation -- the honest decomposition -----------------
    shape_ok = True
    for N in (4, 5, 6):
        QN = sp.Matrix([[N, 1, 0], [N, 2, 0], [N, 2, 1]])
        Qp = (QN + SIG * QN * SIG) / 2
        if Qp.eigenvals() != {sp.Integer(1): 1, sp.Integer(2): 1,
                              sp.Integer(N): 1}:
            shape_ok = False
        VN = QN * sp.diag(0, 1, 1)
        dom = max([e for e in VN.eigenvals() if e != 0], key=lambda e: abs(e))
        vlad = sp.Matrix([1, dom, dom**2])
        if sp.simplify(VN * vlad - dom * vlad) != sp.zeros(3, 1):
            shape_ok = False
        if saturation(QN) != N**4:
            shape_ok = False
    check("S4.3 [E, THE HONEST DECOMPOSITION -- spectrum ablation] freeing "
          "the exponent spectrum itself, the double self-consistency does "
          "NOT pin N absolutely: the ONE surviving family is spectrum = "
          "{1, 2, N} with the shape Q(N) = [[N,1,0],[N,2,0],[N,2,1]] -- "
          "verified here for N = 4, 5, 6 (each ladder-selfconsistent with "
          "sat = N^4; full 35-spectrum scan in the discovery probe finds NO "
          "other double-selfconsistent point).  So the forcing decomposes "
          "PRECISELY: the self-consistency forces the SHEET spectrum {1,2} "
          "(the binary trace code) and the compiler SHAPE; the exponent "
          "demand (E) -- the A3 glue geometry -- contributes exactly ONE "
          "datum, N = 3.  The family number is pinned by (self-consistency) "
          "x (A3 exponents), each load-bearing", shape_ok)

    # --- S4c: the redundancy ablation -- (M) and (L) fall too -------------------
    def selfcode_hits(require_M):
        hits = []
        n_adm = 0
        for q12, q13, q22, q23, q32, q33 in product(range(B_ENTRY + 1),
                                                    repeat=6):
            if q22 + q33 != 3 or q22 * q33 - q23 * q32 != 2:
                continue
            Q = sp.Matrix([[3, q12, q13], [3, q22, q23], [3, q32, q33]])
            Qp = (Q + SIG * Q * SIG) / 2
            if not all(x.is_integer for x in Qp):
                continue
            if Qp.eigenvals() != {sp.Integer(1): 1, sp.Integer(2): 1,
                                  sp.Integer(3): 1}:
                continue
            if require_M:
                Qm = (Q - SIG * Q * SIG) / 2
                Q2 = Qm * Qm
                if Qm == sp.zeros(3, 3) or Q2 == sp.zeros(3, 3):
                    continue
                P = Q2 / 3
                if sp.expand(P * P - P) != sp.zeros(3, 3):
                    continue
            n_adm += 1
            V = Q * sp.diag(0, 1, 1)
            ev = [e for e in V.eigenvals() if e != 0]
            if not ev:
                continue
            dom = max(ev, key=lambda e: abs(e))
            vlad = sp.Matrix([1, dom, dom**2])
            if sp.simplify(V * vlad - dom * vlad) != sp.zeros(3, 1):
                continue
            U = Q * sp.diag(1, 0, 0)
            ws = [I3]
            fr = [I3]
            for _ in range(4):
                fr = [w * G for w in fr for G in (U, V)]
                ws += fr
            if sp.Matrix([[w[i, j] for i in range(3) for j in range(3)]
                          for w in ws]).rank() != 7:
                continue
            if saturation(Q) == 81:
                hits.append(Q)
        return n_adm, hits


    n_ml, hits_ml = selfcode_hits(require_M=True)     # (L) dropped
    n_none, hits_none = selfcode_hits(require_M=False)  # (M) and (L) dropped
    check("S4.4 [E, REDUNDANCY -- (M) and (L) fall too] relative to the "
          "self-code demands (parabolic dim 7 + ladder + sat = 81), the "
          "monodromy budget (M) and the seam-line demand (L) are REDUNDANT "
          "at the declared bound: dropping (L) leaves %d admissible members "
          "and the self-code picks EXACTLY the compiler Q; dropping BOTH "
          "(M) and (L) widens the class to %d members and the self-code "
          "STILL picks exactly the compiler Q -- the declared geometric "
          "input list of the reconstruction shrinks to (S)-existence, "
          "(W)-form and (E), with the type and N already forced"
          % (n_ml, n_none),
          len(hits_ml) == 1 and hits_ml[0] == Q_CORPUS
          and len(hits_none) == 1 and hits_none[0] == Q_CORPUS
          and n_none > 1000)

    # --- S4d: the maximal ablation -- (E) falls too ------------------------------
    hits_noE = []
    cands_noE = 0
    for lam_ in range(1, 20):
        for q13 in range(B_ENTRY + 1):
            q12 = 1 - q13 * lam_
            if not (0 <= q12 <= B_ENTRY):
                continue
            for q23 in range(B_ENTRY + 1):
                q22 = lam_ - q23 * lam_
                if not (0 <= q22 <= B_ENTRY):
                    continue
                for q33 in range(B_ENTRY + 1):
                    q32 = lam_ * lam_ - q33 * lam_
                    if not (0 <= q32 <= B_ENTRY):
                        continue
                    Q = sp.Matrix([[3, q12, q13], [3, q22, q23],
                                   [3, q32, q33]])
                    V = Q * sp.diag(0, 1, 1)
                    ev = [e for e in V.eigenvals() if e != 0]
                    if not ev:
                        continue
                    dom = max(ev, key=lambda e: abs(sp.N(e)))
                    if dom != lam_:
                        continue
                    cands_noE += 1
                    U = Q * sp.diag(1, 0, 0)
                    ws = [I3]
                    fr = [I3]
                    for _ in range(4):
                        fr = [w * G for w in fr for G in (U, V)]
                        ws += fr
                    if sp.Matrix([[w[i, j] for i in range(3)
                                   for j in range(3)]
                                  for w in ws]).rank() != 7:
                        continue
                    if saturation(Q) == 81:
                        hits_noE.append(Q)
    check("S4.5 [E, THE MAXIMAL ABLATION -- (E) falls too] dropping the "
          "exponent demand ENTIRELY (no spectrum condition on Q_+ at all), "
          "the demands (W: first column 3(1,1,1)) + integrality (B = %d) + "
          "the self-code package (parabolic + ladder + sat = 81) STILL pin "
          "the compiler Q uniquely: %d ladder candidates, exactly 1 "
          "self-code hit = the compiler.  Together with S4.4: (E), (M) and "
          "(L) are ALL redundant at the declared bound -- the "
          "reconstruction needs only the sheet involution (type forced), "
          "the winding column with N_fam = 3 (the E8-glue count, v1: "
          "240 = 16 x 5 x 3) and the self-code demands"
          % (B_ENTRY, cands_noE),
          len(hits_noE) == 1 and hits_noE[0] == Q_CORPUS
          and cands_noE >= 20)

    # --- S4e-verify (module form: the four survivors, probe-typed scan) ---------
    four = [((0, 1, 1), sp.Matrix([[0, 1, 0], [1, 4, 0], [1, 4, 3]])),
            ((0, 3, 3), sp.Matrix([[0, 1, 0], [3, 2, 0], [3, 2, 1]])),
            ((3, 3, 3), sp.Matrix([[3, 1, 0], [3, 2, 0], [3, 2, 1]])),
            ((5, 1, 5), sp.Matrix([[5, 1, 0], [1, 2, 0], [5, 2, 1]]))]
    ok4 = True
    for col, Q4 in four:
        V4 = Q4 * sp.diag(0, 1, 1)
        ev4 = [e for e in V4.eigenvals() if e != 0]
        dom4 = max(ev4, key=lambda e: abs(sp.N(e)))
        vl4 = sp.Matrix([1, dom4, dom4**2])
        if sp.simplify(V4 * vl4 - dom4 * vl4) != sp.zeros(3, 1):
            ok4 = False
        if saturation(Q4) != 81:
            ok4 = False
    uniform4 = [col for col, _ in four if col[0] == col[1] == col[2]]
    check("S4.6 [E, what the winding FORM really does] dropping the winding "
          "demand entirely, the self-code package leaves the FOUR-element "
          "column family {(0,1,1), (0,3,3), (3,3,3), (5,1,5)} (exclusivity "
          "by the full scan in the discovery probe; the four members "
          "re-verified here: each ladder-selfconsistent with sat = 81) -- "
          "and exactly ONE is uniform: (3,3,3), the compiler.  The winding "
          "demand's load-bearing content is precisely the UNIFORMITY of "
          "the column (1 of 4); it stays a genuine declared input with "
          "sharply located content",
          ok4 and uniform4 == [(3, 3, 3)])

    # --- S5: honesty --------------------------------------------------------------
    check("S5.1 [C, named limits] what remains of Stage B: the demands (E) "
          "(the A3 exponent spectrum), (M) (the monodromy budget) and (L) "
          "(the seam line) are still DECLARED geometric inputs -- deriving "
          "them from mu4/D4/H^1 geometry is exactly the remaining "
          "GATE.QGEO.01; this probe narrows the gate by two items ((S)'s "
          "type by theorem, (W)'s normalisation by double self-consistency) "
          "and does NOT close it; enumeration bound B = %d declared; "
          "NO P2 reduction, markers untouched" % B_ENTRY, True)

    VERDICT = ("FORCED" if not FAILS else "MIXED")
    print("\nVERDICT: %s -- type killed by rank theorem; N <= 3 by (E); "
          "double self-consistency (ladder + sat = N^4) pins "
          "(N, Q) = (3, compiler Q) uniquely" % VERDICT)
    print("checks: %d, failures: %d %s" % (N_CHK, len(FAILS), FAILS or ""))
    print("elapsed: %.1f s" % (time.time() - T0))

    print("--- QGEO.WINDING.SELF.01 Stage-B winding narrowing: %d passed, %d failed ---"
          % (N_CHK - len(FAILS), len(FAILS)))
    return len(FAILS)


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

9f74fab83fc0355ca591e2bb71078f8af4dd4d95b0df5830d01840462d28e4d7

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