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"""v143 -- The graded hull is a graded-FROBENIUS structure: the Z_4
grading of E8 (v128) carries exact coset duality (-C_k = C_{-k}), the
glue average is the exact carrier projector, each glue sector is a
single Weyl-orbit (multiplicity-one simple current), and the sector
fusion is exactly the C[Z_4] Q-system multiplication (v125).  R2's
identification therefore HOLDS AT THE FINITE LIE LEVEL; what remains
is only the operator-algebraic (conformal-net) step.  [I]/[L] exact.

The R2 statement is 'the seam-Calderon inclusion realises the C[Z_4]
Q-system' (v125).  Its finite shadow -- that the 248-dimensional hull
itself is the Q-system's graded Frobenius module -- is verified here
exactly on the explicit v128 root model:

  [I] 1. COSET DUALITY.  For every one of the 240 roots,
         coset(-r) = -coset(r) mod 4: the negatives of C_1 are
         exactly C_3, while C_0 and C_2 are self-dual.  Since the
         Killing form pairs g_alpha with g_{-alpha} (and the Cartan
         sits in degree 0), the invariant pairing restricts to a
         NONDEGENERATE pairing g_k x g_{-k} -> C: the grading is a
         Frobenius grading.
  [I] 2. THE GLUE AVERAGE IS THE CARRIER PROJECTOR.  The character
         average E = (1/4) sum_k i^{k * deg} equals 1 on C_0 and 0 on
         C_1, C_2, C_3 exactly (orthogonality): the conditional-
         expectation shadow of the v125 Q-system, with quasi-basis
         index |Z_4| = 4 (one unitary weight per glue class).
  [I] 3. GRADED BRACKET (re-verified).  For every pair of roots whose
         sum is again a root, coset(r1 + r2) = coset(r1) + coset(r2)
         mod 4 -- the v128 additivity, re-checked here so this module
         is self-contained.
  [L] 4. SINGLE WEYL ORBITS = MULTIPLICITY ONE.  Each glue sector is
         ONE W(D5) x W(A3) orbit: C_1 = (16, 4bar) (even half-spinors
         x fundamental weights), C_2 = (10, 6), C_3 = (16bar, 4) --
         and the sector dimensions 64, 60, 64 equal the products of
         the irreducible dimensions: every sector is an irreducible
         multiplicity-one g_0-module, i.e. a SIMPLE CURRENT; the
         sector ring is Z_4 = the C[Z_4] fusion (v125), with the
         halfway subgroup {0,2} = the SO(16)_1 step (v92).
  [P] 5. RESIDUE (recorded): the identification now holds at the
         finite Lie level (graded Frobenius + simple-current fusion
         = the Q-system's module structure realised in 248
         dimensions); what remains of R2 is exactly the operator-
         algebraic step -- 'the seam-Calderon inclusion of the
         16-Majorana NET carries this same Q-system' -- one
         conformal-net statement, no remaining finite freedom.
"""
from itertools import combinations

from tfpt_constants import check, summary, reset
from v128_graded_hull import build_roots


def run():
    reset()
    print("v143 graded Frobenius (R2 at the finite Lie level)")

    roots = build_roots()
    neg = lambda r: (tuple(-x for x in r[0]), tuple(-x for x in r[1]))

    # 1. coset duality
    dual_ok = all(roots[neg(r)] == (-c) % 4 for r, c in roots.items())
    sizes = [sum(1 for c in roots.values() if c == k) for k in range(4)]
    pair_c1_c3 = all(roots[neg(r)] == 3 for r, c in roots.items() if c == 1)
    check("COSET DUALITY: coset(-r) = -coset(r) mod 4 for all 240 "
          "roots; -C_1 = C_3 bijectively, C_0 and C_2 self-dual; "
          "sizes (52,64,60,64) -- with the Killing form pairing "
          "g_alpha with g_{-alpha}, the invariant pairing is "
          "nondegenerate on each g_k x g_{-k}: a FROBENIUS grading",
          dual_ok and sizes == [52, 64, 60, 64] and pair_c1_c3)

    # 2. glue average = carrier projector (character orthogonality)
    II = complex(0, 1)
    avg = lambda deg: sum(II ** (k * deg) for k in range(4)) / 4
    check("GLUE AVERAGE = CARRIER PROJECTOR: (1/4) sum_k i^{k deg} "
          "= 1 on C_0 and 0 on C_1, C_2, C_3 exactly -- the "
          "conditional-expectation shadow of the v125 Q-system with "
          "quasi-basis index |Z_4| = 4",
          abs(avg(0) - 1) < 1e-15
          and all(abs(avg(d)) < 1e-15 for d in (1, 2, 3)))

    # 3. graded bracket (self-contained re-check)
    root_set = set(roots)
    add = lambda r, s: (tuple(a + b for a, b in zip(r[0], s[0])),
                        tuple(a + b for a, b in zip(r[1], s[1])))
    pairs = 0
    grade_ok = True
    items = list(roots.items())
    for i, (r1, c1) in enumerate(items):
        for r2, c2 in items[i + 1:]:
            s = add(r1, r2)
            if s in root_set:
                pairs += 1
                if roots[s] != (c1 + c2) % 4:
                    grade_ok = False
    check(f"GRADED BRACKET: coset(r1+r2) = coset(r1)+coset(r2) mod 4 "
          f"for every root pair whose sum is a root ({pairs} pairs) "
          "-- the v128 additivity re-verified, this module is "
          "self-contained", grade_ok and pairs > 0)

    # 4. single Weyl orbits = multiplicity one
    c1_roots = [r for r, c in roots.items() if c == 1]
    c2_roots = [r for r, c in roots.items() if c == 2]
    # W(D5) = signed permutations with an even number of sign flips;
    # orbit invariants on the D5 part: sorted |entries| + sign parity.
    # The parity is an invariant ONLY when all entries are nonzero
    # (flipping a zero coordinate is invisible, so vectors with a zero
    # entry realise both parities inside one orbit).
    def d5_orbit_tag(v):
        flips = sum(1 for x in v if x < 0)
        parity = flips % 2 if all(x != 0 for x in v) else 0
        return (tuple(sorted(abs(x) for x in v)), parity)
    # W(A3) = S4 on the 4 coordinates: orbit invariant = sorted entries
    a3_orbit_tag = lambda w: tuple(sorted(w))
    c1_d5 = {d5_orbit_tag(r[0]) for r in c1_roots}
    c1_a3 = {a3_orbit_tag(r[1]) for r in c1_roots}
    c2_d5 = {d5_orbit_tag(r[0]) for r in c2_roots}
    c2_a3 = {a3_orbit_tag(r[1]) for r in c2_roots}
    dims_ok = (len(c1_roots) == 16 * 4 and len(c2_roots) == 10 * 6)
    check("SINGLE WEYL ORBITS: C_1 is one W(D5)xW(A3) orbit (one "
          "half-spinor sign class x one A3 weight class) with "
          "64 = 16*4 = dim(16,4bar); C_2 likewise with 60 = 10*6 = "
          "dim(10,6) -- each glue sector is an irreducible "
          "multiplicity-one g_0-module (a SIMPLE CURRENT); the "
          "sector ring is Z_4 = the C[Z_4] fusion (v125), halfway "
          "subgroup {0,2} = the SO(16)_1 step (v92)",
          len(c1_d5) == 1 and len(c1_a3) == 1
          and len(c2_d5) == 1 and len(c2_a3) == 1 and dims_ok)

    check("RESIDUE [P] (recorded): R2's identification holds at the "
          "finite Lie level (graded Frobenius + simple-current "
          "fusion = the Q-system's 248-dim module); what remains is "
          "ONE conformal-net statement -- the seam-Calderon "
          "inclusion of the 16-Majorana net carries this Q-system "
          "-- with no remaining finite freedom", True)

    return summary("v143 graded Frobenius")


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

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