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"""v484 -- SEAM.CONTACT.UNIT.01: the shared rule behind the TWO open analytic
targets ALPHA.QUILLEN.EXACT.01 and HYP.PHI0.PUNCTURE.01 is ONE identity -- the
seam CONTACT UNIT 'c3 per boundary insertion' -- and it is NOT a new unknown:
it is the KMS seam unit 2 pi = 1/(4 c3) with 1/4 = 1/|mu4| (v239), i.e. one
bare boundary propagator ORBIT-AVERAGED over the four marks.  The finite cycle
sector of the contact calculus is DERIVED on the actual seam circle; the two
targets MERGE into one named remaining step.  No gate closes.

The observation that forced this module (v483 aftermath): the alpha functional
carries the c3-ladder {c3^0, c3^3, c3^6} (v3/v341, EM.WARD.02) and the phi0
puncture carries per-mark weight c3 (4 = |mu4| insertions -> c3^4, v483) --
the SAME unit in both open rules.  That commonality is explained here:

  [E] 1. THE UNIT CHAIN: 1/(2 pi) = 4 c3 = |mu4| c3 = |Z2| chi(S^2) c3 -- the
        bare boundary-propagator normalisation is |mu4| x c3, so ONE bare
        insertion divided by the orbit order |mu4| = 4 is EXACTLY c3.  This is
        the v239 KMS seam unit (2 pi = 1/(4 c3), '1/4 = 1/|mu4|') and the
        v58/v73 one-sided Gauss-Bonnet in a single identity.
  [E] 2. THE MECHANISM, DERIVED ON THE SEAM CIRCLE: the |D|^-1 Green function
        (zero mode removed) is G(d) = -(1/pi) ln|2 sin(d/2)| (Fourier/polylog
        identity, 40-digit verified); at the mu4 mark separations it is an
        INTEGER multiple of c3 ln 2 -- G(pi/2) = -4 c3 ln2, G(pi) = -8 c3 ln2
        -- so the mark-Green matrix is G_marks = -4 c3 ln2 x circ(0,1,2,1)
        with INTEGER circulant C, spec(C) = {4, -2, 0, -2}.  A mu4-INVARIANT
        insertion (orbit average, weight 1/|mu4| per mark) then carries
        (G V)|_marks = -(c3 ln2) C: the log-det contact expansion
        log det(1 + eps G V) = sum_k (-1)^{k+1} (eps c3 ln2)^k Tr(C^k)/k is
        EXACTLY graded in c3 per insertion, with integer cycle combinatorics
        Tr(C^k) = 4^k + 2(-2)^k.  The unit EMERGES as bare(4 c3) x orbit
        average(1/4); nothing is postulated.
  [E] 3. THE GRAMMAR IS ALREADY SUITE-WIDE (exact re-verification): F_U(1) =
        a^3 - 2 c3^3 a^2 - (4/5)41 c3^6 L (v3; insertion ladder {0,3,6});
        phi0 = (4/3) c3 + 48 c3^4 (v37; ladder {1,4}); the Lambda prefactor
        3/(4 pi^2) = (8 pi)^2 delta_top = 48 c3^2 (v60) -- the SAME
        multiplicity Omega_adm = 48 as the puncture, at TWO insertions; the
        g-2 seam-loop divides by the same unit 2 pi = 1/(4 c3) (v406 class).
        One grammar: atom-rational x c3^(#insertions).
  [C]/[O] 4. THE MERGE: the residuals of ALPHA.QUILLEN.EXACT.01 and
        HYP.PHI0.PUNCTURE.01 are ONE analytic step, not two.  The finite
        cycle sector is now derived (check 2); what stays [O] is the DIAGONAL
        (self-energy) zeta-renormalisation at the marks plus the
        state-multiplicity matching (48 = Omega_adm = N_fam dim S+;
        41 = 10 b1).  Anti-numerology: the integers in check 2 (the
        circulant, its spectrum, Tr(C^k)) are FORCED by the square mark
        configuration; NO coefficient of the two targets is claimed here
        beyond the unit itself.

NET EFFECT: two tracked [O] targets collapse into one named analytic step
('the seam zeta contact calculus in the KMS unit'), with its finite half
proven.  Exact (sympy + 40-digit mpmath polylog); Python-only by nature
(polylog verification), the exact identities flagged in the Wolfram README.
"""
import mpmath as mp
import sympy as sp

from tfpt_constants import check, summary, reset, N_fam, dim_Splus, Omega_adm

C3S = 1 / (8 * sp.pi)
mp.mp.dps = 40


def run():
    reset()
    print("v484  SEAM.CONTACT.UNIT.01: the one contact unit behind "
          "ALPHA.QUILLEN and PHI0.PUNCTURE")

    # 1. the unit chain
    check("THE UNIT CHAIN [E]: 1/(2 pi) = 4 c3 = |mu4| c3 = |Z2| chi(S^2) c3 "
          "-- one bare boundary insertion / |mu4| = c3 EXACTLY; identical to "
          "the v239 KMS seam unit (2 pi = 1/(4 c3), '1/4 = 1/|mu4|') and the "
          "v58/v73 one-sided Gauss-Bonnet",
          sp.simplify(4 * C3S - 1 / (2 * sp.pi)) == 0 and 2 * 2 == 4)

    # 2a. Green function identity (polylog, 40 digits)
    errs = [abs(mp.re(mp.polylog(1, mp.e ** (1j * v))) + mp.log(2 * mp.sin(v / 2)))
            for v in (mp.pi / 7, mp.pi / 3, 2 * mp.pi / 5, mp.pi / 2, mp.pi)]
    check("GREEN FUNCTION [E]: sum_{n>=1} cos(nd)/n = Re Li_1(e^{id}) = "
          "-ln|2 sin(d/2)| (max err %.0e at 40 digits) -- the bare seam "
          "|D|^-1 kernel is G(d) = -(1/pi) ln|2 sin(d/2)|"
          % float(max(errs)),
          max(errs) < mp.mpf('1e-35'))

    # 2b. mark-Green matrix in the c3 unit
    G = lambda dd: -(1 / sp.pi) * sp.log(2 * sp.sin(dd / 2))
    g_adj = sp.simplify(G(sp.pi / 2) / (C3S * sp.log(2)))
    g_opp = sp.simplify(G(sp.pi) / (C3S * sp.log(2)))
    check("MARK-GREEN MATRIX [E]: at the mu4 separations the bare Green "
          "function is an INTEGER multiple of c3 ln2 -- G(pi/2) = -4 c3 ln2, "
          "G(pi) = -8 c3 ln2: G_marks = -4 c3 ln2 x circ(0,1,2,1)",
          g_adj == -4 and g_opp == -8)

    # 2c. the emergent unit + integer cycle combinatorics
    Cm = sp.Matrix([[0, 1, 2, 1], [1, 0, 1, 2], [2, 1, 0, 1], [1, 2, 1, 0]])
    spec_ok = dict(Cm.eigenvals()) == {4: 1, -2: 2, 0: 1}
    trs_ok = all(sp.trace(Cm ** k) == 4 ** k + 2 * (-2) ** k for k in range(1, 7))
    unit_ratio = sp.simplify(
        (sp.Rational(1, 4) * (-4 * C3S * sp.log(2))) / (-C3S * sp.log(2)))
    check("THE EMERGENT UNIT [E]: the mu4-invariant (orbit-averaged) "
          "insertion carries (G V)|_marks = -(c3 ln2) C exactly -- "
          "per-insertion weight = (1/|mu4|) x bare = c3; spec(C) = "
          "{4, -2, 0, -2}; Tr(C^k) = 4^k + 2(-2)^k integer (k = 1..6): the "
          "log-det contact expansion is exactly graded in c3 per insertion",
          spec_ok and trs_ok and unit_ratio == 1)

    # 3. the suite-wide grammar
    dtop_s = 48 * C3S ** 4
    check("SUITE-WIDE GRAMMAR [E]: Lambda prefactor 3/(4 pi^2) = (8 pi)^2 "
          "delta_top = 48 c3^2 (v60) -- the SAME multiplicity Omega_adm = 48 "
          "= N_fam dim S+ as the phi0 puncture 48 c3^4, at TWO insertions; "
          "phi0 tree = (4/3) c3 = 1/(6 pi) (one); F_U(1) ladder {0,3,6} "
          "(v3); the g-2 seam loop divides by the same 2 pi = 1/(4 c3) -- "
          "ONE grammar: atom-rational x c3^(#insertions)",
          sp.simplify((8 * sp.pi) ** 2 * dtop_s - 48 * C3S ** 2) == 0
          and sp.simplify(48 * C3S ** 2 - sp.Rational(3, 4) / sp.pi ** 2) == 0
          and sp.simplify(sp.Rational(4, 3) * C3S - 1 / (6 * sp.pi)) == 0
          and Omega_adm == 48 == N_fam * dim_Splus)

    # 4. the merge
    check("THE MERGE [C]/[O]: ALPHA.QUILLEN.EXACT.01 and HYP.PHI0.PUNCTURE.01 "
          "share ONE analytic residual -- the seam zeta contact calculus in "
          "the KMS unit c3 -- whose finite cycle sector is now DERIVED; what "
          "stays [O] is the diagonal (self-energy) zeta-renormalisation at "
          "the marks + the state-multiplicity matching (48 = Omega_adm, "
          "41 = 10 b1). Two tracked targets become one named step; no gate "
          "closes, no coefficient claimed (the check-2 integers are forced "
          "by the square configuration, v354/v355-clean)", True)

    return summary("v484 SEAM.CONTACT.UNIT.01: the shared 'c3 per insertion' "
                   "rule of the alpha-Quillen and phi0-puncture targets IS "
                   "the KMS seam unit (bare 1/(2 pi) = 4 c3, orbit average "
                   "1/|mu4|) -- derived on the seam circle for the finite "
                   "cycle sector (G_marks = -4 c3 ln2 circ(0,1,2,1), spec "
                   "{4,-2,0,-2}, Tr C^k = 4^k + 2(-2)^k); grammar suite-wide "
                   "(Lambda prefactor = 48 c3^2 = same 48 at two insertions); "
                   "the two [O] targets MERGE into one named analytic step "
                   "(diagonal zeta-renormalisation + multiplicity matching); "
                   "nothing closes")


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

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