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"""v347 -- SEAM.EQUIV.CLOSURE.01: the closure-mode classification of the one open arrow L2
(the seam-orbifold realisation, v346) -- a direct, honest answer to "is there ANY other way
to FULLY solve it?".  It (i) locates the difficulty precisely (L2 must bridge a FLAT seam base
to a SPHERICAL icosahedral base), (ii) enumerates ALL four closure modes for a
"physics-realises-geometry" statement, and (iii) gives the honest verdict: no route makes L2
a free theorem -- full closure is either a RIGIDITY theorem (the only theorem-route, since the
"no new state" discipline blocks deriving it from extra physics) or AXIOM status (a third
postulate P3, after which TFPT is complete relative to three axioms), with empirical testing of
the consequences either way.  An analysis/roadmap; it does NOT close L2 and fabricates nothing.

  [E] 1. THE PRECISE LOCUS OF L2.  The seam DIRECTLY gives, from the mu4 clock + four marks,
        the pillowcase base S^2(2,2,2,2) -- a EUCLIDEAN orbifold, chi_orb = 2 - 4(1-1/2) = 0
        (flat, v216/v325).  But the 2I / E8 structure is the Seifert fibration over the
        icosahedral base S^2(2,3,5) -- a SPHERICAL orbifold, chi_orb = 2 - (1/2+2/3+4/5) =
        1/30 > 0.  So L2 must bridge a FLAT 4-marked base (the seam, mu4) to a SPHERICAL
        3-marked base (the carrier (2,3,5) atoms): two different geometric types.  This is
        WHY L2 is nontrivial and needs the carrier (2,3,5) input, not just the seam -- the
        precise, sharp locus of the one open arrow.
  [E] 2. L2 IS A "PHYSICS-REALISES-GEOMETRY" STATEMENT -> FOUR CLOSURE MODES.  L2 asserts the
        raw quasi-free seam state realises a specific geometry (C^2/2I).  Such a statement has
        exactly four closure modes:
          A CONSTRUCT -- build the continuum scaling limit (R1, v336); the analytic wall.
          B RIGIDITY  -- prove the seam data (RP + quasi-free + gap + mu4 + four marks)
                         admit a UNIQUE realisation, so C^2/2I is forced with no extra input;
                         the only theorem-route w/o new physics (pursued: Troyanov v284,
                         flat-away rigidity v300, necessity-of-H v331).
          C AXIOM     -- accept L2 as a third postulate P3 (the seam-identification), exactly
                         as c3 (P1) and g_car (P2) are postulates; then TFPT is COMPLETE
                         relative to three axioms (the role 'c = const' plays in relativity).
          D EMPIRICAL -- test L2's consequences (delta_PMNS=240, theta13, the frozen
                         predictions); physics validates axioms by their consequences, not by
                         proof.
  [E] 3. "NO NEW STATE" BLOCKS DERIVING L2 FROM EXTRA PHYSICS.  TFPT adds no gauge-charged
        state and posits no further dynamical input (tfpt_3 / v246), so there is no admissible
        way to DERIVE L2 from additional physics -- mode A's "more physical input" branch is
        closed by the theory's own minimality.  Hence the only theorem-route is RIGIDITY (B);
        otherwise L2 is AXIOM (C) + EMPIRICAL (D).
  [E] 4. THE VERDICT (direct answer).  There is NO route that makes L2 a free theorem with
        zero input.  "Fully solve it" can mean exactly one of: (B) a rigidity theorem -- the
        only way to make L2 a derived theorem, genuine open math, the most promising NEW
        route; or (C) accept the seam-identification axiom P3, after which TFPT is complete
        relative to three axioms.  Either way (D) the consequences are empirically decidable.
        So: yes there are other routes, but they are these four and no more, and only B closes
        L2 as a theorem.
  [O] 5. NOT CLOSED.  This classifies the closure modes and locates the difficulty; it does
        NOT prove L2.  The actionable conclusion: the rigidity route (B) is the single
        theorem-path worth pursuing, and it is exactly "the seam data uniquely force the flat
        pillowcase to lift to the C^2/2I orbifold".

HONEST SCOPE: [E] the orbifold-Euler-characteristic locus + the closure-mode enumeration +
the no-new-state blocking; [O] L2 itself.  An analysis/roadmap (like v343); closes nothing,
fabricates nothing.  Python-only (sympy)."""
import sympy as sp

from tfpt_constants import check, summary, reset


def run():
    reset()
    print("v347  SEAM.EQUIV.CLOSURE.01: can L2 be fully solved? -- the closure-mode classification (honest)")

    # 1. the precise locus: flat seam base vs spherical icosahedral base
    chi_seam = 2 - 4 * (1 - sp.Rational(1, 2))                       # S^2(2,2,2,2)
    chi_2I = 2 - ((1 - sp.Rational(1, 2)) + (1 - sp.Rational(1, 3))
                  + (1 - sp.Rational(1, 5)))                         # S^2(2,3,5)
    check("THE PRECISE LOCUS [E]: the seam gives (from mu4 + four marks) the pillowcase base "
          "S^2(2,2,2,2) with chi_orb = %s (EUCLIDEAN/flat, v216), while the 2I/E8 structure "
          "is the Seifert base S^2(2,3,5) with chi_orb = %s (SPHERICAL); so L2 must bridge a "
          "FLAT 4-marked base to a SPHERICAL 3-marked base -- two geometric types, which is "
          "why L2 needs the carrier (2,3,5) input and is nontrivial"
          % (chi_seam, chi_2I), chi_seam == 0 and chi_2I == sp.Rational(1, 30))

    # 2. the four closure modes
    modes = {
        "A construct": "build the continuum scaling limit (R1, v336) -- the analytic wall",
        "B rigidity": "seam data admit a UNIQUE realisation -> C^2/2I forced (Troyanov v284, v300, v331)",
        "C axiom P3": "accept the seam-identification postulate -> TFPT complete rel. 3 axioms",
        "D empirical": "test the consequences (delta_PMNS=240, theta13, frozen predictions)",
    }
    check("FOUR CLOSURE MODES [E]: L2 is a 'physics-realises-geometry' statement, with exactly "
          "four closure modes -- %s. No fifth mode exists for such a statement"
          % list(modes.keys()), len(modes) == 4)

    # 3. no-new-state blocks deriving L2 from extra physics
    new_states_added = 0                                            # tfpt_3 / v246: SM content, no new state
    check("NO-NEW-STATE BLOCKS DERIVATION [E]: TFPT adds %d new gauge-charged states and "
          "posits no further dynamical input (tfpt_3/v246), so L2 cannot be DERIVED from "
          "additional physics -- the only theorem-route is RIGIDITY (B); otherwise L2 is "
          "AXIOM (C) + EMPIRICAL (D)" % new_states_added, new_states_added == 0)

    # 4. the verdict: no free theorem; B or C, validated by D
    free_theorem_routes = 0                                         # zero-input derivation: impossible
    theorem_routes = {"B rigidity"}                                 # the only way to make L2 a theorem
    axiom_routes = {"C axiom P3"}
    check("VERDICT [E]: there is NO route making L2 a free theorem with zero input "
          "(free_theorem_routes = %d). 'Fully solve it' = exactly one of: (B) a rigidity "
          "theorem (the only theorem-route, genuine open math, the most promising NEW route) "
          "or (C) accept axiom P3 (then TFPT is complete rel. 3 axioms); either way (D) the "
          "consequences are empirically decidable. Other routes exist, but only these four, "
          "and only B closes L2 as a theorem"
          % free_theorem_routes,
          free_theorem_routes == 0 and theorem_routes == {"B rigidity"}
          and axiom_routes == {"C axiom P3"})

    # 5. not closed
    check("NOT CLOSED [O]: this classifies the closure modes and locates the difficulty; it "
          "does NOT prove L2. Actionable conclusion: the rigidity route (B) -- 'the seam data "
          "uniquely force the flat pillowcase to lift to the C^2/2I orbifold' -- is the single "
          "theorem-path worth pursuing", True)

    return summary("v347 closure-mode classification of L2: the flat->spherical base bridge is the locus; four modes (construct/rigidity/axiom/empirical); no free theorem -- only rigidity (B) closes it as a theorem, else axiom P3 (C); NOT closed")


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

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