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#!/usr/bin/env python3
"""v651 -- QGEO.ORBASM.01: the FULL Z3-ORBIFOLD ASSEMBLY of the seam on the
lattice -- the projection sum Z_orb = sum_{sectors} eps * Z[sector] with
every admissible phase wiring, the discrete-torsion question settled by
machine cohomology, the GSO-like spin wirings built explicitly, and the
seam assembly identified against the independent seam data.

THE H^2 QUESTION (settled up front, machine-checked in H2):
H^2(Z3, U(1)) = 0 -- for cyclic groups the Schur multiplier is trivial:
every U(1)-valued 2-cocycle on Z3 is symmetric, so the discrete-torsion
phase eps(g,h) = omega(g,h)/omega(h,g) is IDENTICALLY 1, and the three
mu3-valued cocycle classes all trivialize over mu9 in U(1).  There is
NO discrete torsion for a pure Z3 orbifold.  The residual U(1)-trivial
phase tables omega^{a g h + b h} act as exact CHARGE-CLASS RELABELINGS
of the projection (H2.2) -- the real freedom of the assembly sits in
the SPIN structure: deck^3 = (-1)^F (v623, exact Z6 trace arithmetic in
orbifold_modular_probe.py M4) makes the orbifold a Z6 quotient of the
fermionic seam double, and the physical choice is the GSO-like parity
wiring per twist sector.

CONVENTIONS (inherited read-only from orbifold_modular_probe.py):
seam double = one complex free fermion, N-site chain, eps(k) = -cos k,
half filling, NS base.  The FULL Z6 sector grid: spatial boundary
condition psi(x+N) = -e^{i pi gt/3} psi(x), momenta
k = 2 pi (m + 1/2 + gt/6)/N (gt = 2g reproduces the Z3 twist g; gt = 3
is the Ramond sector with two exact zero modes for 4 | N); temporal
insertion V_ht = exp(i pi ht (Nhat - N/2)/3) (V_2h = U_h of the
modular probe; V_3 = (-1)^F for even N/2).  Exact free product:
Z6[gt,ht] = prod_k 2 cosh((L eps_k - i pi ht/3)/2), real and >= 0 for
all 36 sectors (k -> k+pi pairs eps -> -eps; the two R zero modes give
(2 cos(pi ht/6))^2 >= 0, exactly 0 at (3,3) -- the R,(-1)^F zero).
tau = i L/N.  T is the discrete Dehn twist (translation by N sites):
per mode e^{i k N} = e^{i pi (3+gt)/3} uniformly, so EXACTLY
Z6_T[gt,ht] = Z6[gt, ht+gt+3] for 12 | N (c-number = 1): the Z6
spin-structure shift of the modular probe, now on the full grid.
Continuum target: Z6_cft[gt,ht] = |theta[gt/6, ht/6](0|tau)/eta|^2 with
Casimir C(gt) = (6 nu^2 - 6 nu + 1)/3, nu = frac(1/2 + gt/6) (= 4 x
the v628 per-class coefficient; C = -1/6, -1/9, 1/18, 1/3, 1/18, -1/9
for gt = 0..5).

THE CANDIDATE ASSEMBLIES (all as weight tables w(gt,ht) on the Z6
grid; Z_cand = sum w(gt,ht) Z6[gt,ht]):
  A   naive bosonic Z3:   (1/3) sum_{g,h in Z3} Z[g,h]
                          = sum_g P_{Q = 0 mod 3}[g]   (no R sectors)
  A'  "torsion" omega^{gh}: the U(1)-trivial relabeling, keeps
                          Q = -g mod 3 per twist sector
  B   the Z6 deck assembly: (1/6) sum_{gt,ht in Z6} Z6[gt,ht]
                          = sum_{gt} P_{c=0 mod 6}[gt] -- the
                          deck-invariant projection with the exact
                          deck^3 = (-1)^F structure (all six spatial
                          deck powers, Z6 charge projection of the
                          modular probe M4.3)
  C1  GSO without R:      Z6 charge projection c = 0 but only the
                          three NS-type spatial sectors (gt even)
  C2  anti-GSO:           c = 3 mod 6 on the NS-type sectors
  C3  Arf/GSO twin:       (1/6) sum (-1)^{gt ht} Z6[gt,ht]
                          = P_{c = 3 gt mod 6}[gt] (= B on even gt,
                          opposite parity class on odd gt)
  W1/W2 must-fail weights: i^{gh} (an inadmissible mu4 "torsion") and
                          (1/3)(Z[g,0] - Z[g,1] - Z[g,2]) -- NOT
                          projectors; they exist to prove the
                          integrality test has teeth.

Checks:

  (H2.1) [machine] cohomology enumeration: exactly 27 mu3-valued
         2-cocycles on Z3 = 9 coboundaries x 3 classes; EVERY cocycle
         is symmetric, so the discrete-torsion phase
         omega(g,h)/omega(h,g) == 1 identically; every class becomes a
         coboundary over mu9 subset U(1): H^2(Z3, U(1)) = 0 -- no
         discrete torsion exists for the Z3 orbifold.
  (H2.2) [machine] the U(1)-trivial phase tables omega^{a g h + b h}
         are EXACT charge-class relabelings: their temporal weight is
         the projector onto Q = -(a g + b) mod 3, nothing else.

  (M1.1) [machine] the Z6 sector grid: all 36 Z6[gt,ht] real >= 0,
         exact degeneracies Z6[gt,ht] = Z6[-gt,ht] = Z6[gt,-ht],
         Z6[3,3] = 0 exactly (the R,(-1)^F zero), and the even
         subgrid reproduces the modular-probe Z[g,h] convention
         (independent numpy crosscheck vs mpmath pipeline).
  (M1.2) [E] all 35 nonvanishing sectors match the continuum
         |theta[gt/6, ht/6]/eta|^2 at < 1 % on ln Z after removing the
         exact bulk N L/pi (N = 48; tau = i/2, i, 2i) -- NEW measured
         content beyond the modular probe: the R line and the
         half-twisted (odd deck power) lines; the continuum reference
         is itself exactly S-covariant (machine).
  (M1.3) [E] the assembly table: Z_cand at tau = i/2, i, 2i for
         N = 48, 96 (all six candidates); the O(1) candidates match
         their continuum assemblies at < 1 %; C2 has NO O(1) content:
         its leading exponent is Delta = 41/18 = 1/9 + 13/6 (the
         lightest Q = 3 states live in the TWISTED sectors, below the
         untwisted 5/2), fitted at < 2 %.

  (M2.1) [central] THE SPECTRAL TEST: the six candidate weight tables
         are exact per-sector charge projectors (temporal weight
         phi in {0,1} machine-exact) => every extracted degeneracy is
         a NONNEGATIVE INTEGER by exact arithmetic on the enumerated
         Fock spectrum (first levels tabulated); the (level, sector,
         charge, count) data at N = 48 and N = 96 are IDENTICAL with
         level residuals shrinking ~ N^{-2}; the must-fail weights
         W1/W2 produce complex / negative "degeneracies" -- the test
         has teeth.
  (M2.2) [machine] trace <-> spectrum consistency: for all 8 weight
         tables the assembled trace at L = 3N equals the enumerated
         spectral sum within the exp(-2 pi L/N * 1.34) tail bound
         (mpmath, 40 digits -- the C2 cancellations are 20 orders
         below double precision).
  (M2.3) [central] modular S MEASURED per candidate (tau = i/2 <-> 2i,
         N = 24/48/96): A, A', B, C3 converge monotonically with the
         stable ~N^{-4} rate (A' converges TOO: charge conjugation of
         the seam double identifies the omega^{gh} and omega^{-gh}
         relabelings -- the H2-trivial phases are S-consistent, an
         honest measured surprise); C1 and C2 are S-BROKEN: their
         deviations do not decrease (C1 lacks the R-type sectors that
         S generates; C2's charge-3 tower has no S-image).
  (M2.4) [central] T EXACT on the lattice (the deck^3 = -1 content):
         B, C1, C3 are machine-exactly T-invariant; A violates T
         EXACTLY by twice its Q = 3 mod 6 content (the operator
         identity Z_T(A) = Z(A) - 2 Z(C2), machine); C2 is exactly
         T-ANTI-invariant (Z_T = -Z); the per-sector Dehn identity
         Z6_T[gt,ht] = Z6[gt,ht+gt+3] holds machine-exactly on odd
         (half-twisted) sectors too.
  (M2.5) [E] vacuum: unique (multiplicity 1) for A, A', B, C1, C3;
         C2 has NO vacuum (dead as a theory).

  (M3.1) [E] the untwisted sector of the surviving assemblies is the
         charge-projected NS seam: neutral degeneracies {1, 4, 9} at
         Delta = 0, 1, 2 -- independent continuum q-series vs lattice
         enumeration (v623: untwisted = base, verbatim).
  (M3.2) [E] the twisted characters: Delta_sigma = 1/9 with assembly
         multiplicity 2 (the v639 sigma/sigma-bar pair, h_sigma =
         1/36 per deck class); the charged {1/6, 5/6} parafermion pair
         at 1/9 + 1/6 is projected OUT exactly as the omega^{k q}
         charge bookkeeping demands; the next neutral twisted level is
         1/9 + 1/3 (x2); independent q-series {1, 1, 2} matches.
  (M3.3) [E] the ground charge class is c = 0 in ALL SIX gt-sectors
         (extends modular-probe M4.3 to the odd deck powers: the
         Delta = 1/36 and Delta = 1/4 grounds are charge-0) -- the
         deck-invariant class carries every sector ground.
  (M3.4) [E/C] THE VERDICT: the survivors of {integer spectrum,
         unique vacuum, S measured clean, T exact} are exactly
         {B, C3}; the seam data discriminate: B keeps every measured
         sector ground (the c = 0 deck-invariant class, M3.3) and the
         trivial Klein phase eta = (1,1) of the RP probe; C3 (the
         Arf/GSO twin) excises ALL odd-sector grounds -- modular-
         consistent as an abstract fermionic orbifold but seam-alien.
         The seam orbifold is B.  HONEST: A and A' die by the EXACT
         lattice T (deck^3 = -1) plus, for A', the v639 twisted
         exponent; C1 dies by S; C2 by vacuum + T-anti; the B-vs-C3
         kill is the measured c = 0 ground bookkeeping plus the Klein
         triviality -- a data-backed [C] reading, not a modularity
         theorem.

  (M4.x) [E] CHIRAL PHASES: the half-band trace Z_chir[g,h] =
         prod_{k in (0,pi)} 2cosh((L eps_k - i lam_h)/2) is genuinely
         COMPLEX in the twisted sectors (the eps -> -eps pairing
         breaks for gt not = 0 mod 3); its phase against the chiral
         theta[g/3,h/3]/eta is measured over tau = i/2, i, 2i at
         N = 48, 96: CONSTANT in tau (drift ~1e-4 at N = 48, ~N^{-2})
         with value e^{-2 pi i a~ b}, a~ = the symmetric-interval
         representative of a = g/3 (spectral-flow-reduced branch; for
         g = 2 the measured phase is +2 pi/9 = -2 pi (a-1) b, not
         -4 pi/9) -- the lattice trace is the theta quotient WITHOUT
         the characteristic phase e^{2 pi i ab}, measured.

  (M5.1) [C] typing: what stands, what is missing.

Verdict enums (frozen): ORB-ASSEMBLY-SEAM (all pass; exactly one
seam assembly), ORB-ASSEMBLY-DEGENERATE (consistency tests pass but
the seam data cannot discriminate), ORB-ASSEMBLY-FAILS (grid or
control level fails), MIXED (anything else).

FIREWALL: GATE.QGEO does not move; no marker changes; the B-vs-C3
kill is data-backed [C] (measured ground classes + Klein triviality),
not a modularity theorem -- v652 hardens it structurally.
Python-only, counted per GATE.WOLFRAM.02.

Conventions inherited read-only from v650_orbifold_modular.py,
v645_klein_rp.py, v622/v623/v628/v639.

PROVENANCE: discovery probe orbifold_assembly_probe.py (2026-08-02,
17/17, verdict ORB-ASSEMBLY-SEAM).
"""

import itertools
from fractions import Fraction

import numpy as np
from mpmath import mp

mp.dps = 40

CHECKS = []


def check(name, ok, detail=""):
    CHECKS.append((name, bool(ok)))
    print("[%s] %s%s" % ("PASS" if ok else "FAIL", name,
                         (" -- " + detail) if detail else ""))


# ------------------------------------------------------------------ constants
OMEGA = np.exp(2j * np.pi / 3.0)
N_MAIN, N_ALT = 48, 96
NS_S = (24, 48, 96)               # S-covariance N series (rotated: N/2)
BETAS = (0.5, 1.0, 2.0)           # tau = i beta
DMAX_ENUM = 1.34                  # Fock enumeration cutoff (total Delta)
DREPORT = 1.26                    # spectra reported up to here
DMAX_UNT = 2.30                   # extended untwisted-only enumeration
TOL_MP = 1e-25                    # mpmath machine identities
TOL_INT = 1e-9                    # integrality / projector arithmetic
TOL_FIT = 0.01                    # 1 % continuum matches
CAND = ("A", "A'", "B", "C1", "C2", "C3")
WFAIL = ("W1", "W2")
REAL_DEV = [0.0]                  # running lnZ6 reality tracker


# ------------------------------------------------------------------ machinery
def modes6(N, gt):
    """Z6-twisted NS momenta k = 2 pi (m + 1/2 + gt/6)/N."""
    return 2.0 * np.pi * (np.arange(N) + 0.5 + (gt % 6) / 6.0) / N


def eps6(N, gt):
    """Dispersion -cos k with exact zero-mode snapping (R sector)."""
    e = -np.cos(modes6(N, gt))
    e[np.abs(e) < 1e-13] = 0.0
    return e


_LNZ6 = {}


def lnZ6(N, L, gt, ht):
    """mp real ln Z6[gt,ht]; None if the sector vanishes exactly."""
    key = (N, L, gt % 6, ht % 6)
    if key in _LNZ6:
        return _LNZ6[key]
    e = eps6(N, gt)
    lam = mp.pi * (ht % 6) / 3
    nz = int((e == 0.0).sum())
    if nz:
        assert nz == 2 and (gt % 6) == 3
        zf = 2 * mp.cos(lam / 2)          # per zero mode; squared >= 0
        if abs(zf) < mp.mpf(10) ** -30:   # ht = 3: 2 cos(pi/2) = 0
            _LNZ6[key] = None
            return None
    tot = mp.mpc(0)
    Lm = mp.mpf(L)
    for ee in e:
        if ee == 0.0:
            continue
        tot += mp.log(2 * mp.cosh((Lm * mp.mpf(float(ee)) - 1j * lam) / 2))
    if nz:
        tot += 2 * mp.log(abs(zf))
    dev = abs(mp.e ** (1j * mp.im(tot)) - 1)
    REAL_DEV[0] = max(REAL_DEV[0], float(dev))
    val = mp.re(tot)
    _LNZ6[key] = val
    return val


def weight_tables():
    """The candidate + must-fail weight tables on the Z6 grid."""
    W = {}
    A = np.zeros((6, 6), complex)
    Ap = np.zeros((6, 6), complex)
    W1 = np.zeros((6, 6), complex)
    W2 = np.zeros((6, 6), complex)
    for g in range(3):
        for h in range(3):
            A[2 * g, 2 * h] = 1.0 / 3.0
            Ap[2 * g, 2 * h] = OMEGA ** (g * h) / 3.0
            W1[2 * g, 2 * h] = (1j) ** (g * h) / 3.0
            W2[2 * g, 2 * h] = (1.0 if h == 0 else -1.0) / 3.0
    W["A"], W["A'"], W["W1"], W["W2"] = A, Ap, W1, W2
    W["B"] = np.full((6, 6), 1.0 / 6.0, complex)
    C1 = np.zeros((6, 6), complex)
    C1[0::2, :] = 1.0 / 6.0
    W["C1"] = C1
    C2 = np.zeros((6, 6), complex)
    for ht in range(6):
        C2[0::2, ht] = (-1.0) ** ht / 6.0
    W["C2"] = C2
    C3 = np.zeros((6, 6), complex)
    for gt in range(6):
        for ht in range(6):
            C3[gt, ht] = (-1.0) ** (gt * ht) / 6.0
    W["C3"] = C3
    return W


WT = weight_tables()


def Zcand(N, L, name, tmap=False):
    """mp assembly value sum w(gt,ht) Z6[gt, ht (+gt+3 if tmap)]."""
    tot = mp.mpc(0)
    for gt in range(6):
        for ht in range(6):
            w = WT[name][gt, ht]
            if w == 0:
                continue
            hh = (ht + gt + 3) % 6 if tmap else ht
            lz = lnZ6(N, L, gt, hh)
            if lz is None:
                continue
            tot += mp.mpc(w) * mp.e ** lz
    return tot


def Zreal(N, L, name, tmap=False):
    """Real assembly value; the imaginary part cancels through the
    gt <-> -gt / ht <-> -ht degeneracies, which hold at the double-
    precision mode level (the eps_k are doubles)."""
    z = Zcand(N, L, name, tmap)
    assert abs(mp.im(z)) <= mp.mpf(10) ** -10 * (abs(mp.re(z)) + 1)
    return mp.re(z)


def phi_table(name):
    """Temporal weight phi(gt, Q mod 6) = sum_ht w e^{i pi ht Q/3}."""
    P = np.zeros((6, 6), complex)
    for gt in range(6):
        for q in range(6):
            P[gt, q] = sum(WT[name][gt, ht] * np.exp(1j * np.pi * ht * q / 3.0)
                           for ht in range(6))
    return P


PHI = {name: phi_table(name) for name in CAND + WFAIL}


# probe-convention numpy crosscheck (verbatim machinery style)
def ln2cosh_np(z):
    zs = np.where(z.real >= 0, z, -z)
    return zs + np.log(1.0 + np.exp(-2.0 * zs))


def lnZ_probe(N, L, g, h):
    """Modular-probe convention: real ln Z[g,h], Z3 grid, numpy."""
    lam = 2.0 * np.pi * (h % 3) / 3.0
    k = 2.0 * np.pi * (np.arange(N) + 0.5 + (g % 3) / 3.0) / N
    val = ln2cosh_np(0.5 * (L * (-np.cos(k)) - 1j * lam)).sum()
    assert abs(np.exp(1j * val.imag) - 1.0) < 1e-9
    return val.real


def lnZ6cft(gt, ht, beta, M=3000):
    """Continuum ln |theta[gt/6, ht/6](0|i beta)/eta|^2 (product form)."""
    lnq = -2.0 * np.pi * beta
    lam = np.pi * (ht % 6) / 3.0
    a0 = (0.5 + (gt % 6) / 6.0) % 1.0
    C = (6.0 * a0 * a0 - 6.0 * a0 + 1.0) / 3.0
    m = np.arange(M)
    tot = 0.5 * C * lnq
    for a in (m + a0, m + 1.0 - a0):
        qa = np.exp(lnq * a)
        tot += np.sum(np.log(1.0 + 2.0 * np.cos(lam) * qa + qa * qa))
    return tot


def theta_ab(a, b, beta, M=80):
    """theta[a,b](0|i beta) = sum_n e^{-pi beta (n+a)^2 + 2 pi i (n+a) b}."""
    n = np.arange(-M, M + 1)
    return np.sum(np.exp(-np.pi * beta * (n + a) ** 2
                         + 2j * np.pi * (n + a) * b))


def ln_eta(beta, M=4000):
    n = np.arange(1, M)
    return -np.pi * beta / 12.0 + np.sum(np.log1p(-np.exp(-2.0 * np.pi
                                                          * beta * n)))


# ================================================================== H2
print("=" * 72)
print("H2: discrete torsion for Z3 -- the cohomology, machine-enumerated")
print("=" * 72)

cocycles = []
for f in itertools.product(range(3), repeat=9):
    ok = True
    for a in range(3):
        for b in range(3):
            for c in range(3):
                if (f[3 * a + b] + f[3 * ((a + b) % 3) + c]
                        - f[3 * b + c] - f[3 * a + (b + c) % 3]) % 3:
                    ok = False
                    break
            if not ok:
                break
        if not ok:
            break
    if ok:
        cocycles.append(f)
cob = set()
for beta in itertools.product(range(3), repeat=3):
    cob.add(tuple((beta[a] + beta[b] - beta[(a + b) % 3]) % 3
                  for a in range(3) for b in range(3)))
classes = set()
for f in cocycles:
    classes.add(min(tuple((f[i] - c[i]) % 3 for i in range(9))
                    for c in cob))
sym = all(f[3 * a + b] == f[3 * b + a]
          for f in cocycles for a in range(3) for b in range(3))
triv = True
for rep in classes:
    found = False
    for gam in itertools.product(range(9), repeat=3):
        if all((gam[a] + gam[b] - gam[(a + b) % 3]
                - 3 * rep[3 * a + b]) % 9 == 0
               for a in range(3) for b in range(3)):
            found = True
            break
    triv &= found
check("H2.1 [machine] cohomology of Z3: exactly %d mu3-valued 2-cocycles"
      " = %d coboundaries x %d classes; EVERY cocycle is symmetric, so "
      "the discrete-torsion phase omega(g,h)/omega(h,g) == 1 identically"
      " (no torsion phase exists), and every class trivializes over mu9 "
      "in U(1): H^2(Z3, U(1)) = 0 -- there is NO discrete torsion for "
      "the pure Z3 orbifold" % (len(cocycles), len(cob), len(classes)),
      len(cocycles) == 27 and len(cob) == 9 and len(classes) == 3
      and sym and triv)

ok22 = True
for a in range(3):
    for b in range(3):
        for g in range(3):
            tgt = (-(a * g + b)) % 3
            for q in range(3):
                phi = sum(OMEGA ** (a * g * h + b * h) * OMEGA ** (h * q)
                          for h in range(3)) / 3.0
                ok22 &= abs(phi - (1.0 if q == tgt else 0.0)) < 1e-12
check("H2.2 [machine] the U(1)-trivial phase tables omega^{a g h + b h} "
      "are EXACT charge-class relabelings: the temporal weight is the "
      "projector onto Q = -(a g + b) mod 3 for all nine (a,b) tables -- "
      "the only Z3 'torsion' freedom is which charge class is kept; the "
      "physical freedom sits in the deck^3 = (-1)^F spin wiring", ok22)


# ================================================================== M1
print("=" * 72)
print("M1: the Z6 sector grid and the candidate assemblies")
print("=" * 72)

# M1.1 grid structure
dev_deg, dev_sub = 0.0, 0.0
zero_ok = True
for (beta, N) in ((0.5, 48), (1.0, 48), (2.0, 48)):
    L = int(round(beta * N))
    for gt in range(6):
        for ht in range(6):
            z = lnZ6(N, L, gt, ht)
            zc = lnZ6(N, L, (-gt) % 6, ht)
            zh = lnZ6(N, L, gt, (-ht) % 6)
            if z is None:
                zero_ok &= (gt % 6, ht % 6) == (3, 3) and zc is None \
                    and zh is None
                continue
            dev_deg = max(dev_deg, abs(float(z - zc)), abs(float(z - zh)))
    for g in range(3):
        for h in range(3):
            dev_sub = max(dev_sub, abs(float(lnZ6(N, L, 2 * g, 2 * h))
                                       - lnZ_probe(N, L, g, h)))
check("M1.1 [machine] the Z6 grid: all 36 Z6[gt,ht] are real >= 0 (max "
      "reality dev %.1e), the degeneracies Z6[gt,ht] = Z6[-gt,ht]"
      " = Z6[gt,-ht] hold at the double-precision mode level (the "
      "eps_k are doubles; the identities are exact in exact "
      "arithmetic), Z6[3,3] = 0 exactly (the R,(-1)^F zero, both"
      " zero modes annihilate), and the even subgrid reproduces the "
      "modular-probe Z[g,h] convention (independent numpy crosscheck)"
      % REAL_DEV[0],
      REAL_DEV[0] < 1e-12 and dev_deg < 1e-11 and dev_sub < 1e-9
      and zero_ok,
      "max degeneracy dev = %.3e, subgrid dev = %.3e"
      % (dev_deg, dev_sub))

# M1.2 continuum reference: exact S-covariance + the 35-sector match
dev_scov = 0.0
for beta in (0.5, 1.0, 2.0):
    for gt in range(6):
        for ht in range(6):
            if (gt % 6, ht % 6) == (3, 3) or (ht % 6, (-gt) % 6) == (3, 3):
                continue
            dev_scov = max(dev_scov, abs(lnZ6cft(gt, ht, 1.0 / beta)
                                         - lnZ6cft(ht, (-gt) % 6, beta)))
dev_th = 0.0
for beta in (0.5, 1.0):
    for gt in range(6):
        for ht in range(6):
            if (gt, ht) == (3, 3):
                continue
            t = theta_ab(gt / 6.0, ht / 6.0, beta)
            dev_th = max(dev_th, abs(lnZ6cft(gt, ht, beta)
                                     - 2.0 * (np.log(abs(t))
                                              - ln_eta(beta))))
dev_tab = 0.0
print("  lnZhat = lnZ - N L/pi vs continuum, N = 48 (new sectors shown):")
print("  %-5s %-7s %14s %14s %11s" % ("tau", "(gt,ht)", "lattice",
                                      "continuum", "|dev|"))
for beta in BETAS:
    L = int(round(beta * N_MAIN))
    for gt in range(6):
        for ht in range(6):
            if (gt, ht) == (3, 3):
                continue
            zl = float(lnZ6(N_MAIN, L, gt, ht)) - N_MAIN * L / np.pi
            zc = lnZ6cft(gt, ht, beta)
            d = abs(zl - zc)
            dev_tab = max(dev_tab, d)
            if beta == 1.0 and (gt, ht) in ((1, 0), (1, 1), (3, 0),
                                            (3, 1), (5, 2)):
                print("  %-5s (%d,%d)  %14.8f %14.8f %11.3e"
                      % ("i", gt, ht, zl, zc, d))
check("M1.2 [E] all 35 nonvanishing Z6 sectors match the continuum "
      "|theta[gt/6,ht/6]/eta|^2 at < 1 %% on lnZhat (N = 48; tau = i/2,"
      " i, 2i) -- NEW beyond the modular probe: the R line (gt or ht = "
      "3) and the half-twisted deck lines (gt, ht odd); the continuum "
      "reference is itself exactly S-covariant (dev %.1e) and equals "
      "the theta series (dev %.1e)" % (dev_scov, dev_th),
      dev_tab < TOL_FIT and dev_scov < 1e-9 and dev_th < 1e-9,
      "max lattice-continuum |dev| = %.3e" % dev_tab)

# M1.3 the assembly table + continuum assemblies + C2 exponent
print("  assembly table lnZhat_cand = ln Z_cand - N L/pi:")
print("  %-4s %-3s" % ("tau", "N")
      + "".join("%12s" % c for c in CAND))
dev_asm = 0.0
for beta in BETAS:
    for N in (N_MAIN, N_ALT):
        L = int(round(beta * N))
        row = []
        for cd in CAND:
            zl = float(mp.log(Zreal(N, L, cd))) - N * L / np.pi
            row.append(zl)
            if cd != "C2" and N == N_MAIN:
                zc = np.log(sum((WT[cd][gt, ht]
                                 * np.exp(lnZ6cft(gt, ht, beta))
                                 for gt in range(6) for ht in range(6)
                                 if (gt, ht) != (3, 3))).real)
                dev_asm = max(dev_asm, abs(zl - zc))
        print("  %-4s %-3d" % (("i/2" if beta == 0.5 else
                                ("i" if beta == 1.0 else "2i")), N)
              + "".join("%12.6f" % v for v in row))
lc1 = mp.log(Zreal(N_MAIN, 96, "C2"))
lc2 = mp.log(Zreal(N_MAIN, 144, "C2"))
E0_NS48 = eps6(N_MAIN, 0)[eps6(N_MAIN, 0) < 0].sum()
d_c2 = (-float(lc2 - lc1) / 48.0 - E0_NS48) * N_MAIN / (2.0 * np.pi)
D_C2 = 41.0 / 18.0        # 1/9 + 13/6: the lightest TWISTED Q = 3 state
check("M1.3 [E] the assembly table stands: the five O(1) candidates "
      "match their continuum assemblies at < 1 %% (max dev %.3e); C2 "
      "has NO O(1) content -- its leading exponent is Delta = %.4f vs "
      "41/18 = 1/9 + 13/6 (the lightest Q = 3 mod 6 states live in the"
      " TWISTED sectors, below the untwisted 5/2; rel dev %.2e < 2 %%)"
      % (dev_asm, d_c2, abs(d_c2 - D_C2) / D_C2),
      dev_asm < TOL_FIT and abs(d_c2 - D_C2) / D_C2 < 0.02)


# ================================================================== M2
print("=" * 72)
print("M2: consistency tests -- spectra, traces, S, T, vacuum")
print("=" * 72)


def enum_sector(N, gt, dmax):
    """Exact Fock enumeration: all states with total Delta <= dmax.
    Returns (E0_ns-relative energy dE, Delta, Q) triples."""
    e = eps6(N, gt)
    E0 = e[e < 0].sum()
    E0_ns = eps6(N, 0)[eps6(N, 0) < 0].sum()
    Qsea = int((e < 0).sum()) - N // 2
    dg = (E0 - E0_ns) * N / (2.0 * np.pi)
    cut = (dmax - dg) * 2.0 * np.pi / N
    if cut < 0:
        return []
    sel = np.abs(e) <= cut + 1e-12
    costs = np.abs(e[sel])
    dq = np.where(e[sel] > 0, 1, np.where(e[sel] < 0, -1, 1))
    order = np.argsort(costs)
    costs, dq = costs[order], dq[order]
    out = []

    def dfs(i, E, q):
        out.append((E0 - E0_ns + E, dg + E * N / (2.0 * np.pi), Qsea + q))
        for j in range(i, len(costs)):
            if E + costs[j] > cut + 1e-12:
                break
            dfs(j + 1, E + costs[j], q + dq[j])

    dfs(0, 0.0, 0)
    return out


def spectrum(N, dmax):
    """Cluster the six-sector enumeration into levels; map each level
    to the nearest p/36 fraction.  Returns (clusters, max residual):
    cluster = (Fraction, {(gt, Q): count}, {(gt, Q): [dE...]})."""
    allst = []
    for gt in range(6):
        for (dE, d, Q) in enum_sector(N, gt, dmax):
            allst.append((d, gt, Q, dE))
    allst.sort()
    clusters, cur = [], [allst[0]]
    for st in allst[1:]:
        if st[0] - cur[-1][0] > 0.012:
            clusters.append(cur)
            cur = [st]
        else:
            cur.append(st)
    clusters.append(cur)
    out, resid = [], 0.0
    for cl in clusters:
        mean = np.mean([st[0] for st in cl])
        p = int(round(36.0 * mean))
        resid = max(resid, max(abs(st[0] - p / 36.0) for st in cl))
        cnt, des = {}, {}
        for (d, gt, Q, dE) in cl:
            cnt[(gt, Q)] = cnt.get((gt, Q), 0) + 1
            des.setdefault((gt, Q), []).append(dE)
        out.append((Fraction(p, 36), cnt, des))
    return out, resid


SPEC48, RES48 = spectrum(N_MAIN, DMAX_ENUM)
SPEC96, RES96 = spectrum(N_ALT, DMAX_ENUM)


def degeneracy(name, cnt):
    return sum(n * PHI[name][gt % 6, Q % 6] for (gt, Q), n in cnt.items())


# M2.1 the spectral test
proj_ok = True
for name in CAND:
    for gt in range(6):
        for q in range(6):
            p = PHI[name][gt, q]
            proj_ok &= min(abs(p), abs(p - 1.0)) < TOL_INT
int_ok = {name: True for name in CAND}
print("  candidate spectra (N = 48; exact projector arithmetic):")
print("  %-7s" % "level" + "".join("%7s" % c for c in CAND)
      + "   %-16s %s" % ("W1", "W2"))
w1_bad, w2_bad = False, False
for (fr, cnt, _) in SPEC48:
    if fr > Fraction(int(DREPORT * 36), 36):
        continue
    row = []
    for name in CAND:
        d = degeneracy(name, cnt)
        ok = abs(d.imag) < TOL_INT and abs(d.real - round(d.real)) \
            < TOL_INT and round(d.real) >= 0
        int_ok[name] &= ok
        row.append(int(round(d.real)))
    d1 = degeneracy("W1", cnt)
    d2 = degeneracy("W2", cnt)
    w1_bad |= (abs(d1.imag) > 0.1 or abs(d1.real - round(d1.real)) > 0.1)
    w2_bad |= (d2.real < -0.1)
    print("  %-7s" % str(fr) + "".join("%7d" % v for v in row)
          + "   %-16s %.3f" % ("%.3f%+.3fi" % (d1.real, d1.imag),
                               d2.real))
same = True
key48 = {(str(fr), gt, Q): n for (fr, cnt, _) in SPEC48
         for (gt, Q), n in cnt.items()}
key96 = {(str(fr), gt, Q): n for (fr, cnt, _) in SPEC96
         for (gt, Q), n in cnt.items()}
same = (key48 == key96)
check("M2.1 [central] THE SPECTRAL TEST: all six candidate weight "
      "tables are exact per-sector Z6-charge projectors (phi in {0,1} "
      "machine-exact) => every degeneracy is a NONNEGATIVE INTEGER by "
      "exact arithmetic on the enumerated Fock spectrum; the (level, "
      "sector, charge, count) data at N = 48 and N = 96 are IDENTICAL "
      "(level residuals %.2e -> %.2e, the ~N^-2 shrink); the must-fail "
      "weights have teeth: W1 (i^{gh}) gives complex non-integer, W2 "
      "gives negative 'degeneracies'" % (RES48, RES96),
      proj_ok and all(int_ok.values()) and same and w1_bad and w2_bad
      and RES96 < RES48 / 2.5 and RES48 < 0.012)

# M2.2 trace <-> spectrum consistency at L = 3N (mp)
L3 = 3 * N_MAIN
E0_ns = eps6(N_MAIN, 0)[eps6(N_MAIN, 0) < 0].sum()
tail = 400.0 * float(mp.e ** (-2 * mp.pi * 3 * (DMAX_ENUM - 0.02)))
dev_tr = 0.0
for name in CAND + WFAIL:
    tr = Zcand(N_MAIN, L3, name) * mp.e ** (mp.mpf(L3) * mp.mpf(E0_ns))
    en = mp.mpc(0)
    for (fr, cnt, des) in SPEC48:
        for (gt, Q), dEs in des.items():
            w = mp.mpc(complex(PHI[name][gt % 6, Q % 6]))
            for dE in dEs:
                en += w * mp.e ** (-mp.mpf(L3) * mp.mpf(dE))
    dev_tr = max(dev_tr, float(abs(tr - en)))
check("M2.2 [machine] trace <-> spectrum consistency: for all 8 weight "
      "tables the assembled lattice trace at L = 3N equals the "
      "enumerated spectral sum within the tail bound %.1e (mpmath, 40 "
      "digits; C2's content sits ~20 orders below double precision)"
      % tail, dev_tr < tail, "max |trace - enum| = %.3e" % dev_tr)

# M2.3 modular S, measured per candidate
print("  S at tau = i/2 <-> 2i per candidate, |ln Z(N,N/2) - ln Z(N/2,N)|:")
print("  %-4s %12s %12s %12s %9s %9s" % ("cand", "N=24", "N=48", "N=96",
                                         "rate1", "rate2"))
SDEV, SRATE = {}, {}
for name in CAND:
    ds = []
    for N in NS_S:
        X = mp.log(Zreal(N, N // 2, name))
        Y = mp.log(Zreal(N // 2, N, name))
        ds.append(float(abs(X - Y)))
    SDEV[name] = ds
    r = [np.log(ds[0] / ds[1]) / np.log(2.0),
         np.log(ds[1] / ds[2]) / np.log(2.0)]
    SRATE[name] = r
    print("  %-4s %12.4e %12.4e %12.4e %9.3f %9.3f"
          % (name, ds[0], ds[1], ds[2], r[0], r[1]))
s_clean = {}
for name in ("A", "A'", "B", "C3"):
    ds, r = SDEV[name], SRATE[name]
    s_clean[name] = (ds[0] > ds[1] > ds[2] and abs(r[0] - r[1]) < 0.8
                     and ds[2] < 1e-4)
s_broken = {name: (SDEV[name][2] > 1e-3
                   and SDEV[name][2] > 0.5 * SDEV[name][0])
            for name in ("C1", "C2")}
check("M2.3 [central] modular S MEASURED: A, A', B, C3 converge "
      "monotonically over N = 24/48/96 with stable rates ~ N^-4 (B: "
      "%.2f/%.2f -- the assemblies inherit the probe's S-covariant-"
      "artifact cancellation; A' converges TOO: charge conjugation of "
      "the seam double identifies the omega^{gh} and omega^{-gh} "
      "relabelings, so the H2-trivial phase is S-consistent -- an "
      "honest surprise vs the naive weight-symmetry count), while C1 "
      "and C2 are S-BROKEN: their deviations do NOT decrease (C1 "
      "stalls at %.2e -- it lacks the R-type sectors S generates; C2 "
      "sits at %.1f -- its charge-3 tower has no S-image)"
      % (SRATE["B"][0], SRATE["B"][1], SDEV["C1"][2], SDEV["C2"][2]),
      all(s_clean.values()) and all(s_broken.values()),
      "clean %s" % {k: "%.1e" % SDEV[k][2] for k in s_clean})

# M2.4 T exact (the deck^3 = -1 content); A''s T-image is genuinely
# complex (the torsion-like relabeling has no real T-closure), so the
# deviations are computed in complex arithmetic throughout.
TDEV = {}
for name in CAND:
    Z0 = Zcand(N_MAIN, 24, name)
    ZT = Zcand(N_MAIN, 24, name, tmap=True)
    TDEV[name] = float(abs(ZT - Z0) / abs(Z0))
idA = float(abs((Zcand(N_MAIN, 24, "A") - Zcand(N_MAIN, 24, "A",
                                                tmap=True))
                - 2 * Zcand(N_MAIN, 24, "C2"))
            / abs(Zcand(N_MAIN, 24, "A")))
idC2 = float(abs(Zcand(N_MAIN, 24, "C2", tmap=True)
                 + Zcand(N_MAIN, 24, "C2"))
             / abs(Zcand(N_MAIN, 24, "C2")))
t96 = {name: float(abs(Zcand(N_ALT, 48, name, tmap=True)
                       - Zcand(N_ALT, 48, name))
                   / abs(Zcand(N_ALT, 48, name)))
       for name in ("B", "C3")}


def lnZraw_mp(N, L, gt, lam_extra):
    """mp complex ln Tr[prod e^{i theta_k n_k} e^{-LH}] with per-mode
    theta_k = lam_extra + N k_k (the Dehn insertion)."""
    k = modes6(N, gt)
    e = -np.cos(k)
    tot = mp.mpc(0)
    for (kk, ee) in zip(k, e):
        th = mp.mpf(float(lam_extra)) + mp.mpf(N) * mp.mpf(float(kk))
        a = mp.mpf(L) * mp.mpf(float(ee)) / 2
        tot += mp.log(mp.e ** a + mp.e ** (1j * th - a))
    return tot


dev_dehn = 0.0
for (gt, ht) in ((1, 0), (5, 2), (2, 1)):
    lam = np.pi * ht / 3.0
    lhs = lnZraw_mp(N_MAIN, 24, gt, lam) - 1j * mp.mpf(float(lam)) \
        * mp.mpf(N_MAIN) / 2
    hh = (ht + gt + 3) % 6
    rhs = lnZ6(N_MAIN, 24, gt, hh)
    dev_dehn = max(dev_dehn, float(abs(mp.e ** (lhs - rhs) - 1)))
check("M2.4 [central] T EXACT on the lattice (the deck^3 = -1 "
      "content): B, C1, C3 are machine-exactly T-invariant (devs %.1e,"
      " %.1e, %.1e at N=48; B, C3 at N=96: %.1e, %.1e); A violates T "
      "EXACTLY by twice its Q = 3 mod 6 content -- the operator "
      "identity Z_T(A) = Z(A) - 2 Z(C2) holds at %.1e, dev_T(A) = "
      "%.3e; C2 is exactly T-ANTI-invariant (|Z_T + Z|/Z = %.1e, dev = "
      "%.3f); A''s T-image is genuinely complex, dev = %.3e; the per-"
      "sector Dehn identity Z6_T[gt,ht] = Z6[gt,ht+gt+3] holds on odd "
      "sectors at the double-limited mode-phase level (dev %.1e)"
      % (TDEV["B"], TDEV["C1"], TDEV["C3"], t96["B"], t96["C3"],
         idA, TDEV["A"], idC2, TDEV["C2"], TDEV["A'"], dev_dehn),
      TDEV["B"] < TOL_MP and TDEV["C1"] < TOL_MP and TDEV["C3"] < TOL_MP
      and max(t96.values()) < TOL_MP and idA < TOL_MP
      and TDEV["A"] > 1e-4 and idC2 < TOL_MP
      and abs(TDEV["C2"] - 2.0) < 1e-12 and dev_dehn < 1e-11)

# M2.5 vacuum multiplicity
vac = {}
for name in CAND:
    fr0 = [cl for cl in SPEC48 if cl[0] == 0]
    vac[name] = int(round(degeneracy(name, fr0[0][1]).real)) \
        if fr0 else 0
check("M2.5 [E] the vacuum: unique (multiplicity 1) for A, A', B, C1, "
      "C3; C2 has NO vacuum at all (its first state sits at Delta = "
      "41/18, the twisted charge-3 level): dead as a theory",
      all(vac[n] == 1 for n in ("A", "A'", "B", "C1", "C3"))
      and vac["C2"] == 0, "vac = %s" % vac)


# ================================================================== M3
print("=" * 72)
print("M3: which assembly is the seam")
print("=" * 72)


def poly_mul(P, Q, tmax):
    R = {}
    for (t1, z1), c1 in P.items():
        for (t2, z2), c2 in Q.items():
            if t1 + t2 <= tmax:
                R[(t1 + t2, z1 + z2)] = R.get((t1 + t2, z1 + z2), 0) \
                    + c1 * c2
    return R


# untwisted continuum series: prod (1+z t^{2n+1})^2 (1+ t^{2n+1}/z)^2,
# t = q^{1/2}; neutral coefficients at t^{0,2,4} (Delta = 0, 1, 2)
P = {(0, 0): 1}
for n in range(3):
    for _ in range(2):
        P = poly_mul(P, {(0, 0): 1, (2 * n + 1, 1): 1}, 4)
        P = poly_mul(P, {(0, 0): 1, (2 * n + 1, -1): 1}, 4)
ser_unt = [P.get((2 * d, 0), 0) for d in range(3)]
SPEC_U, _ = spectrum(N_MAIN, DMAX_UNT)
lat_unt = []
for d in range(3):
    fr = Fraction(36 * d, 36)
    cl = [c for c in SPEC_U if c[0] == fr]
    lat_unt.append(int(round(degeneracy("B", {k: v for k, v in
                                              cl[0][1].items()
                                              if k[0] == 0}).real))
                   if cl else 0)
check("M3.1 [E] the untwisted sector of the assembly is the charge-"
      "projected NS seam: neutral degeneracies {%d, %d, %d} at Delta = "
      "0, 1, 2 from the lattice enumeration equal the independent "
      "continuum q-series {%d, %d, %d} (v623: untwisted = base, "
      "verbatim; charge |Q| >= 6 states sit far above)"
      % tuple(lat_unt + ser_unt),
      lat_unt == ser_unt == [1, 4, 9])

# twisted continuum series per sector: u = q^{1/6},
# prod (1+z u^{6n+5})(1+u^{6n+1}/z)(1+z u^{6n+1})(1+u^{6n+5}/z)
P = {(0, 0): 1}
for n in range(2):
    P = poly_mul(P, {(0, 0): 1, (6 * n + 5, 1): 1}, 12)
    P = poly_mul(P, {(0, 0): 1, (6 * n + 1, -1): 1}, 12)
    P = poly_mul(P, {(0, 0): 1, (6 * n + 1, 1): 1}, 12)
    P = poly_mul(P, {(0, 0): 1, (6 * n + 5, -1): 1}, 12)
ser_tw = [P.get((u, 0), 0) for u in (0, 2, 6)]
ser_chg = [P.get((1, -1), 0), P.get((1, 1), 0)]


def sector_count(fr, gt):
    cl = [c for c in SPEC48 if c[0] == fr]
    if not cl:
        return {}
    return {Q: n for (g, Q), n in cl[0][1].items() if g == gt}


lat_tw = [sum(n for Q, n in sector_count(fr, 2).items() if Q % 6 == 0)
          for fr in (Fraction(4, 36), Fraction(16, 36), Fraction(40, 36))]
lat_chg = sector_count(Fraction(10, 36), 2)
d19 = int(round(degeneracy("B", [c for c in SPEC48
                                 if c[0] == Fraction(4, 36)][0][1]).real))
d19_pair = int(round(degeneracy("B", [c for c in SPEC48
                                      if c[0] == Fraction(10, 36)]
                                [0][1]).real))
check("M3.2 [E] the twisted characters in the assembly: Delta = 1/9 "
      "with multiplicity %d (the v639 sigma/sigma-bar pair, h_sigma = "
      "1/36 per deck class); the charged parafermion pair at 1/9 + 1/6 "
      "(lattice counts Q = -1: %d, Q = +1: %d per sector) is projected "
      "OUT (assembly multiplicity %d) exactly as the omega^{k q} charge"
      " bookkeeping demands; neutral twisted tower {1, 1, 2} at "
      "1/9 + {0, 1/3, 1} matches the independent series {%d, %d, %d}"
      % (d19, lat_chg.get(-1, 0), lat_chg.get(1, 0), d19_pair,
         ser_tw[0], ser_tw[1], ser_tw[2]),
      d19 == 2 and d19_pair == 0 and lat_tw == ser_tw == [1, 1, 2]
      and lat_chg.get(-1, 0) == lat_chg.get(1, 0) == 1
      and ser_chg == [1, 1])

# M3.3 ground charge class per sector
grounds = {}
for gt in range(6):
    best, qs = None, None
    for cl in SPEC48:
        sc = sector_count(cl[0], gt)
        if sc:
            best, qs = cl[0], sorted(sc.items())
            break
    grounds[gt] = (best, qs)
print("  sector grounds: %s"
      % {gt: (str(fr), qs) for gt, (fr, qs) in grounds.items()})
g_ok = all(any(Q % 6 == 0 for Q, n in qs) for (fr, qs) in
           grounds.values())
odd_expect = (grounds[1][0] == Fraction(1, 36)
              and grounds[5][0] == Fraction(1, 36)
              and grounds[3][0] == Fraction(9, 36))
b_keeps = all(int(round(degeneracy("B", {(gt, Q): n for Q, n in
                                         grounds[gt][1]}).real)) > 0
              for gt in range(6))
c3_kills = all(int(round(degeneracy("C3", {(gt, Q): n for Q, n in
                                           grounds[gt][1]}).real)) == 0
               for gt in (1, 3, 5))
check("M3.3 [E] the ground charge class is c = 0 in ALL SIX gt-sectors"
      " (extends modular-probe M4.3 to the odd deck powers: the "
      "Delta = 1/36 grounds of gt = 1, 5 and the Delta = 1/4 R-pair "
      "are charge-0): the deck-invariant class carries every sector "
      "ground -- B keeps all of them, C3 excises every odd-sector "
      "ground", g_ok and odd_expect and b_keeps and c3_kills)

# M3.4 the verdict
survivors = [n for n in CAND
             if int_ok[n] and vac[n] == 1 and TDEV[n] < TOL_MP
             and (n in s_clean and s_clean[n])]
check("M3.4 [E/C] THE VERDICT: the survivors of {integer spectrum, "
      "unique vacuum, S measured clean, T exact} are exactly {B, C3} "
      "(A dies by exact T violation %.1e = its charge-3 content; C1 by"
      " S stalling at %.2e; C2 by no-vacuum + T-anti; A' by exact T "
      "violation %.2e AND by the v639 twisted exponent -- its first "
      "twisted state sits at 5/18, not 1/9); the seam data "
      "discriminate B from C3: "
      "B keeps every measured c = 0 sector ground (M3.3) and matches "
      "the trivial Klein phase eta = (1,1) of the RP probe, C3 "
      "(the Arf/GSO twin) excises all odd-sector grounds.  THE SEAM "
      "ASSEMBLY IS B -- the Z6 deck quotient with uniform charge-0 "
      "projection.  HONEST: the B-vs-C3 kill is the measured ground-"
      "class datum plus the Klein triviality (data-backed [C]), not a "
      "modularity theorem" % (TDEV["A"], SDEV["C1"][2], TDEV["A'"]),
      sorted(survivors) == ["B", "C3"] and b_keeps and c3_kills)


# ================================================================== M4
print("=" * 72)
print("M4: chiral phases (E) -- the half-band trace vs theta/eta")
print("=" * 72)

# M4.0 the continuum phase bookkeeping: theta[a,b]/eta product form
dev_pf = 0.0
for (a, b) in ((1.0 / 3.0, 1.0 / 3.0), (2.0 / 3.0, 1.0 / 3.0),
               (1.0 / 3.0, 2.0 / 3.0)):
    for beta in BETAS:
        q = np.exp(-2.0 * np.pi * beta)
        n = np.arange(1, 400)
        prod = np.prod(1.0 + q ** (n - 0.5 + a) * np.exp(2j * np.pi * b)) \
            * np.prod(1.0 + q ** (n - 0.5 - a) * np.exp(-2j * np.pi * b))
        pf = np.exp(2j * np.pi * a * b) * q ** (0.5 * a * a - 1.0 / 24.0) \
            * prod
        ser = theta_ab(a, b, beta) / np.exp(ln_eta(beta))
        dev_pf = max(dev_pf, abs(pf - ser) / abs(ser))
check("M4.0 [machine] the continuum phase bookkeeping: theta[a,b]/eta "
      "= e^{2 pi i a b} q^{a^2/2 - 1/24} prod (1 + q^{n-1/2+a} e^{2 pi "
      "i b})(1 + q^{n-1/2-a} e^{-2 pi i b}) -- the characteristic "
      "phase e^{2 pi i a b} is pinned before measuring",
      dev_pf < 1e-10, "max dev = %.3e" % dev_pf)


def lnZchir(N, L, g, h):
    """Complex half-band (right-moving) trace: k in (0, pi) mod 2 pi."""
    gt = (2 * g) % 6
    lam = 2.0 * np.pi * (h % 3) / 3.0
    k = np.mod(modes6(N, gt), 2.0 * np.pi)
    sel = (k > 0) & (k < np.pi)
    e = -np.cos(k[sel])
    assert sel.sum() == N // 2 and np.all(np.abs(e) > 1e-12)
    return ln2cosh_np(0.5 * (L * e - 1j * lam)).sum()


print("  phase of Z_chir * eta / theta[g/3,h/3] "
      "(target e^{-2 pi i a~ b}, a~ = symmetric rep of g/3):")
print("  %-6s %-3s %10s %10s %10s %10s %10s"
      % ("(g,h)", "N", "arg(i/2)", "arg(i)", "arg(2i)", "drift",
         "|ph-pred|"))
m4_rows = {}
for (g, h) in ((1, 1), (2, 1), (1, 2)):
    a, b = g / 3.0, h / 3.0
    ared = ((a + 0.5) % 1.0) - 0.5     # spectral-flow-reduced |a~|<=1/2
    pred = np.exp(-2j * np.pi * ared * b)
    for N in (N_MAIN, N_ALT):
        phs = []
        for beta in BETAS:
            L = int(round(beta * N))
            t = theta_ab(a, b, beta) / np.exp(ln_eta(beta))
            ph = np.exp(1j * lnZchir(N, L, g, h).imag) \
                * np.conj(t / abs(t))
            phs.append(ph)
        drift = max(abs(p1 - p2) for p1 in phs for p2 in phs)
        mdev = max(abs(p - pred) for p in phs)
        m4_rows[(g, h, N)] = (phs, drift, mdev)
        print("  (%d,%d)  %-3d %10.5f %10.5f %10.5f %10.2e %10.2e"
              % (g, h, N, np.angle(phs[0]), np.angle(phs[1]),
                 np.angle(phs[2]), drift, mdev))
m4_ok = all(m4_rows[(g, h, N_ALT)][1] < 0.02
            and m4_rows[(g, h, N_ALT)][2] < 0.03
            and m4_rows[(g, h, N_ALT)][1] < m4_rows[(g, h, N_MAIN)][1]
            for (g, h) in ((1, 1), (2, 1), (1, 2)))
check("M4.1 [E] THE CHIRAL PHASE IS MEASURED: the half-band trace "
      "Z_chir[g,h] is genuinely complex in the twisted sectors, its "
      "phase against theta[g/3,h/3]/eta is CONSTANT in tau (drift "
      "~1e-4 at N=48 shrinking ~N^-2 to ~2e-5 at N=96) and equals "
      "e^{-2 pi i a~ b} with a~ the SYMMETRIC representative of the "
      "characteristic (|a~| <= 1/2): for (1,1) and (1,2) this is "
      "-2 pi g h/9, for (2,1) it is +2 pi/9 = -2 pi (a-1) b -- the "
      "lattice half-band trace realizes the theta quotient WITHOUT "
      "the characteristic phase e^{2 pi i a b}, on the spectral-flow-"
      "reduced branch (particles at offset 1/6, not 7/6): the chiral "
      "phase bookkeeping of the trace convention is exactly the "
      "reduced theta characteristic, measured", m4_ok)


# ================================================================== M5
print("=" * 72)
print("M5: the typing")
print("=" * 72)

check("M5.1 [C, honest typing] WHAT STANDS on the lattice: "
      "H^2(Z3,U(1)) = 0 by machine enumeration (no discrete torsion; "
      "phases = charge relabelings); the full 36-sector Z6 grid with "
      "continuum match < 1 % including the NEW R and half-twisted "
      "lines; six assemblies built and tested -- exact integer spectra"
      " (projector arithmetic with must-fail teeth), trace-spectrum "
      "consistency, S measured with rates, T exact as Z6 Dehn "
      "arithmetic (deck^3 = -1 closes: the naive Z3 orbifold violates "
      "T by exactly its charge-3 content), vacuum census; the seam "
      "assembly identified as the Z6 deck quotient B with uniform "
      "c = 0 projection (v623 untwisted basis, v639 twisted "
      "characters, M4.3 ground classes, Klein eta = (1,1)); the "
      "chiral phase e^{2 pi i a b} bookkeeping measured.  STILL OPEN: "
      "the continuum OS statement for the assembled orbifold, "
      "non-abelian extensions, the full eta/theta phase theory beyond "
      "the measured constants (T-phases per character), and the "
      "B-vs-C3 discrimination remains a data-backed [C] reading (Arf "
      "stacking is modular-consistent in the abstract).  GATE.QGEO "
      "does not move", True)


# ================================================================== summary
print("=" * 72)
n_pass = sum(1 for _, ok in CHECKS if ok)
print("%d/%d checks passed" % (n_pass, len(CHECKS)))
grid_ok = all(ok for n, ok in CHECKS if n.startswith(("H2", "M1")))
m3_ok = all(ok for n, ok in CHECKS if n.startswith("M3"))
if n_pass == len(CHECKS):
    print("ALL CHECKS PASSED")
    print("VERDICT: ORB-ASSEMBLY-SEAM -- the Z3 orbifold of the seam")
    print("assembles: no discrete torsion exists (H^2 = 0, machine), the")
    print("freedom is the Z6 spin wiring, and exactly ONE assembly -- the")
    print("Z6 deck quotient with uniform charge-0 projection -- passes")
    print("integer spectrum + unique vacuum + measured S + exact T AND")
    print("matches the seam data (untwisted basis, 1/9 twisted pair,")
    print("c = 0 grounds, Klein eta = (1,1)).")
elif grid_ok and not m3_ok:
    print("SOME CHECKS FAILED")
    print("VERDICT: ORB-ASSEMBLY-DEGENERATE -- the consistency tests")
    print("stand but the seam data do not single out one assembly.")
elif not grid_ok:
    print("SOME CHECKS FAILED")
    print("VERDICT: ORB-ASSEMBLY-FAILS -- the grid or control level")
    print("fails: honest negative.")
else:
    print("SOME CHECKS FAILED")
    print("VERDICT: MIXED")


def run():
    """run_all entry point: the checks above already ran at import."""
    return len([1 for _, ok in CHECKS if not ok])


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

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