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"""v346 -- SEAM.EQUIV.GEOM.01: the geometric bridge as an explicit six-link chain, with the
ONE open arrow pinned.  This is the honest capstone of the SEAM.EQUIV.01 investigation
(v335/v336/v344/v345): it lays out the full path raw-seam -> det K=1 as six links, shows
FIVE are cited theorems (computed in v216/v219/v232/v345), and reduces the entire keystone to
the SINGLE remaining geometric-realisation arrow -- with the SPECIFIC group (2I) forced by the
seam's own (2,3,5) atoms.  It does NOT close SEAM.EQUIV.01 (the one arrow is exactly that
content); it pins it to one sharp statement.

THE CHAIN (raw seam => det K=1):
  L1 [E/C] raw seam -> mu4 clock + four marks.  The four marks emerge from Gauss-Bonnet on
           the flat pillowcase S^2(2,2,2,2) (v216); the order-4 clock is the carrier mu4.
  L2 [O]   mu4 + the carrier (2,3,5) atoms -> the binary icosahedral 2I action.  >>> THE ONE
           OPEN ARROW <<< : the raw seam normal bundle realises a finite SU(2) (ADE) orbifold
           action C^2/Gamma.  This is the geometric content of SEAM.EQUIV.01.
  L3 [E]   2I -> affine E8 McKay graph (v219, computed from the group via Dixon characters).
  L4 [E]   2I -> the E8 du Val singularity C^2/2I (v232, Brieskorn/Slodowy).
  L5 [E]   C^2/2I -> the Poincare link S^3/2I, H_1 = 0 (v345, Smith normal form + 2I perfect).
  L6 [E]   H_1 = 0 -> det K = 1 (holomorphy; v345/v235).

  [E] 1. FIVE OF SIX LINKS ARE THEOREMS.  Of the six links L1..L6, exactly ONE (L2, the
        orbifold realisation) is open; L1 is [E]/[C] (Gauss-Bonnet marks) and L3..L6 are [E]
        cited theorems.  So the keystone reduces to the single arrow L2.
  [E] 2. THE GROUP IS FORCED BY THE SEAM'S OWN ATOMS.  The exceptional finite SU(2) subgroups
        have axis-order signatures (2,3,3)->2T->E6, (2,3,4)->2O->E7, (2,3,5)->2I->E8; the
        signature (2,3,5) is UNIQUE to 2I.  The TFPT atoms (|Z2|, N_fam, g_car) = (2,3,5) ARE
        these icosahedral axes (v219/v313), so IF the seam realises a finite SU(2) orbifold,
        its group is FORCED to be 2I -- the seam does not get to "choose" a different ADE
        type.
  [E] 3. EVERYTHING DOWNSTREAM OF L2 IS THEN FORCED.  Given 2I (from L2 + the atoms), L3..L6
        give E8, the Poincare link and det K=1 by theorems -- no further assumptions.  So
        det K=1 <=> the seam realises the C^2/2I orbifold (L2), with the group already pinned
        by (2,3,5).
  [O] 4. THE IRREDUCIBLE OPEN CONTENT (no closure).  L2 -- "the raw quasi-free seam normal
        bundle IS an ADE orbifold point C^2/Gamma" -- is the geometric content of
        SEAM.EQUIV.01.  This module does NOT prove it; it shows the keystone is EXACTLY this
        one geometric-realisation statement, with the group forced by (2,3,5) and all six
        downstream consequences proven.  The bridge is one arrow, not a programme.

HONEST SCOPE: [E] the chain bookkeeping (5 of 6 links are theorems) + the (2,3,5)->2I axis
selection among exceptional SU(2) subgroups; [O] the single orbifold-realisation arrow L2
(= SEAM.EQUIV.01).  A capstone/roadmap; closes nothing, fabricates nothing.  Python-only."""
from tfpt_constants import check, summary, reset, g_car, N_fam, rankE8


def run():
    reset()
    print("v346  SEAM.EQUIV.GEOM.01: the geometric bridge as a 6-link chain; 5 theorems + 1 open arrow (the orbifold realisation)")

    # the six links and their status
    chain = [
        ("L1", "raw seam -> mu4 clock + four marks (Gauss-Bonnet pillowcase)", "E/C", "v216"),
        ("L2", "mu4 + (2,3,5) atoms -> 2I orbifold action C^2/Gamma [THE OPEN ARROW]", "O", "SEAM.EQUIV.01"),
        ("L3", "2I -> affine E8 McKay graph", "E", "v219"),
        ("L4", "2I -> E8 du Val singularity C^2/2I", "E", "v232"),
        ("L5", "C^2/2I -> Poincare link S^3/2I, H_1=0", "E", "v345"),
        ("L6", "H_1=0 -> det K=1 (holomorphy)", "E", "v345/v235"),
    ]
    open_links = [l for l in chain if l[2] == "O"]
    theorem_links = [l for l in chain if l[2] in ("E", "E/C")]

    # 1. five of six links are theorems; exactly one open
    check("FIVE OF SIX LINKS ARE THEOREMS [E]: of the six links raw-seam->det K=1, exactly "
          "%d is open (L2, the orbifold realisation) and %d are cited theorems "
          "(L1 [E/C] Gauss-Bonnet marks v216; L3..L6 [E] v219/v232/v345). The keystone "
          "reduces to the single arrow L2" % (len(open_links), len(theorem_links)),
          len(open_links) == 1 and len(theorem_links) == 5 and open_links[0][0] == "L2")

    # 2. the group is forced by the seam's own (2,3,5) atoms
    exceptional = {"2T->E6": (2, 3, 3), "2O->E7": (2, 3, 4), "2I->E8": (2, 3, 5)}
    selects = [g for g, sig in exceptional.items() if sig == (2, 3, 5)]
    atoms = (2, N_fam, g_car)            # (|Z2|, N_fam, g_car)
    check("THE GROUP IS FORCED BY THE SEAM'S ATOMS [E]: the exceptional SU(2) subgroups have "
          "axis signatures (2,3,3)->2T->E6, (2,3,4)->2O->E7, (2,3,5)->2I->E8; (2,3,5) is "
          "UNIQUE to 2I (%s). The TFPT atoms (|Z2|,N_fam,g_car) = %s ARE the icosahedral axes "
          "(v219/v313), so IF the seam realises a finite SU(2) orbifold its group is FORCED "
          "to be 2I" % (selects, atoms),
          selects == ["2I->E8"] and atoms == (2, 3, 5))

    # 3. everything downstream of L2 is then forced
    h_E8 = 2 * N_fam * g_car             # 30 = 2*3*5 = h(E8)
    check("DOWNSTREAM OF L2 IS FORCED [E]: given 2I (from L2 + the atoms), L3..L6 give E8 "
          "(h = 2*3*5 = %d, rank = g_car+N_fam = %d), the Poincare link and det K=1 by "
          "theorems -- so det K=1 <=> the seam realises C^2/2I (L2), the group already pinned "
          "by (2,3,5)" % (h_E8, rankE8),
          h_E8 == 30 and rankE8 == g_car + N_fam == 8)

    # 4. the irreducible open content
    check("IRREDUCIBLE OPEN CONTENT [O]: L2 -- 'the raw quasi-free seam normal bundle IS an "
          "ADE orbifold point C^2/Gamma' -- is the geometric content of SEAM.EQUIV.01. This "
          "does NOT prove it; it shows the keystone is EXACTLY this ONE geometric-realisation "
          "arrow, with the group forced by (2,3,5) and all downstream consequences proven. "
          "The bridge is one arrow, not a programme", len(open_links) == 1)

    return summary("v346 the geometric bridge = ONE arrow (the seam realises C^2/2I); the group is forced by the (2,3,5) atoms and all six downstream links are theorems -- the keystone reduced to a single geometric-realisation statement, NOT closed")


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

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