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"""v343 -- FOUR.ROUTES.01: the honest investigation of the four black-hole-birth solution
routes (A finite causal diamond, B Carlip-Cardy near-horizon CFT, C modular/thermal flow,
D self-reproduction attractor).  Direct answer to 'which closes ALL problems and is 100%
provable': NONE individually -- but they CONVERGE on the same two irreducible residuals as
v335 (SEAM.EQUIV.01 + the decoupled QG contour), reached now from the cosmological side.
This is a roadmap/analysis module (like v275/v297); it fabricates nothing and closes
nothing -- it classifies what each route closes and what residual each inherits.

  [E] 1. ROUTE A (finite causal diamond) REFRAMES QG.AMB to a FINITE thermal trace, but
        inherits the GHP-contour residual.  In a horizon-bounded static patch the relevant
        object is a thermal trace Tr e^{-beta H}, FINITE for a gapped H bounded below: the de
        Sitter entropy S_dS = 32 pi^4 e^{2 ainv} is finite & positive (128 c3^4 = 1/(32 pi^4)
        exact, v54/v190), and the gapped transfer has finite susceptibility chi = 729/665
        (v337).  BUT the conformal mode still has the wrong sign c_conf(4) = -3/2 (v332), so
        it STILL needs the GHP contour rotation (v334), whose nonperturbative validity is
        [O].  Verdict A: QG.AMB well-posedness/finiteness [E]; full nonperturbative
        construction [O] (= v334's residual).  Does NOT touch SEAM.EQUIV.01.
  [C] 2. ROUTE B (Carlip-Cardy near-horizon CFT) gives a SECOND route to existence + central
        charge, but NOT the holomorphy bit.  Carlip's near-horizon Virasoro + Cardy
        S = 2 pi sqrt(c L0/6) reproduces the entropy and is CONSISTENT with c = g_car+N_fam =
        8 -- but L0 is free (the known Carlip periodicity ambiguity), so c=8 is consistent,
        NOT forced; and Carlip yields the central charge ONLY, not the modular data /
        holomorphy det K=1 (the actual SEAM.EQUIV.01 discriminator).  Verdict B: existence+c
        [C] (a second path, like v336), does NOT close det K=1 -- inherits SEAM.EQUIV.01.
  [E] 3. ROUTE C (modular/thermal flow) is a REDUCTION, not a closure.  The modular flow
        exists and is KMS at beta=1 (Tomita-Takesaki, a theorem, v239), so 'the dynamics =
        modular evolution of the horizon state' is well-defined ABSTRACTLY.  BUT that this
        flow is the GEOMETRIC boost (= physical time, so it is the real dynamics) is exactly
        QGEO.SYM.01, now a corollary of SEAM.EQUIV.01 (v335).  Verdict C: dynamics = modular
        flow [E abstract]; the geometric content reduces to SEAM.EQUIV.01 -- a reduction.
  [E] 4. ROUTE D (self-reproduction attractor) gives parameter-freeness, not a construction.
        A gapped boundary transport has, by Perron-Frobenius, a UNIQUE attractor (v56) -- so
        the cyclic self-reproduction map has a unique fixed point [E].  BUT 'the full dynamics
        IS that fixed point' is the [P] cyclic interpretation (kept apart from the [E] core),
        and the structure it acts on is still SEAM.EQUIV.01.  Verdict D: uniqueness/parameter-
        freeness [E]; full-dynamics-from-cycle [P] -- not a closure.
  [E] 5. THE VERDICT (direct answer).  NONE of the four routes closes all problems or is 100%
        provable alone.  They CONVERGE: {B, C, D} all bottom out at SEAM.EQUIV.01; {A} bottoms
        out at the QG contour (v334).  So the four routes reduce the entire BH-birth picture
        to the SAME two irreducible residuals as v335 (SEAM.EQUIV.01 + the decoupled QG
        contour) -- NOT zero.  The BH premise makes BOTH more natural (A: a finite thermal
        trace instead of an asymptotic path integral; B: a near-horizon CFT instead of a
        lattice scaling limit) but ELIMINATES neither.  Number of independent closures = 0;
        number of irreducible residuals after all four routes = 2 (unchanged from v335).
        The value of the investigation: the four routes are not four separate gambles -- they
        converge on the ONE keystone, so effort should focus on SEAM.EQUIV.01.

HONEST SCOPE: [E] the finiteness facts (A), the modular/attractor theorems (C/D), and the
convergence count; [C] the Cardy consistency (B); [O]/[P] the inherited residuals.  An
analysis module; closes nothing, fabricates nothing.  Python-only (sympy/mpmath)."""
import sympy as sp
import mpmath as mp

from tfpt_constants import check, summary, reset

mp.mp.dps = 25
pi = sp.pi


def run():
    reset()
    print("v343  FOUR.ROUTES.01: which BH-birth route closes all & is 100% provable? (honest: none; they converge)")

    c3 = sp.Rational(1, 8) / pi

    # 1. Route A: finite thermal trace, but inherits the GHP contour
    dS_identity = sp.simplify(128 * c3 ** 4 - 1 / (32 * pi ** 4)) == 0      # v54
    ainv = mp.mpf("137.0359992168407")
    SdS = 32 * mp.pi ** 4 * mp.e ** (2 * ainv)
    chi = 1 / (1 - sp.Rational(2, 3) ** 6)                                  # 729/665, v337
    c_conf = -(sp.Integer(4) - 1) * (sp.Integer(4) - 2) / sp.Integer(4)     # -3/2, v332
    routeA_closes = "QG.AMB well-posedness/finiteness [E]; full construction [O] (v334 contour)"
    check("ROUTE A [E]/[O]: finite causal diamond REFRAMES QG.AMB to a FINITE thermal trace "
          "-- S_dS = 32 pi^4 e^{2 ainv} finite>0 (128 c3^4 = 1/(32 pi^4) exact, v54), gapped "
          "susceptibility chi = %s finite (v337) -- BUT c_conf(4) = %s < 0 still needs the "
          "GHP contour (v334, [O]); so A gives %s. Does NOT touch SEAM.EQUIV.01"
          % (chi, c_conf, routeA_closes),
          dS_identity and mp.isfinite(SdS) and SdS > 0 and chi == sp.Rational(729, 665)
          and c_conf == sp.Rational(-3, 2))

    # 2. Route B: Cardy c=8 consistent but ambiguous; no holomorphy
    c_seam = 8
    S = mp.mpf("100")                                                      # any horizon entropy
    L0 = S ** 2 / (4 * mp.pi ** 2) * 6 / c_seam                            # solve Cardy for L0
    cardy_consistent = mp.almosteq(2 * mp.pi * mp.sqrt(c_seam * L0 / 6), S)
    check("ROUTE B [C]: Carlip near-horizon Virasoro + Cardy S = 2 pi sqrt(c L0/6) is "
          "CONSISTENT with c = g_car+N_fam = 8 (solvable for L0=%.3f) -- but L0 is FREE (the "
          "Carlip periodicity ambiguity), so c=8 is consistent NOT forced, and Carlip yields "
          "the central charge ONLY, not the holomorphy det K=1 discriminator. A second route "
          "to existence+c (like v336); does NOT close det K=1 -- inherits SEAM.EQUIV.01"
          % float(L0), cardy_consistent and c_seam == 8)

    # 3. Route C: modular flow is a theorem, geometric content = SEAM.EQUIV.01
    kms_beta = 1                                                           # Tomita-Takesaki, v239
    check("ROUTE C [E]->reduction: the modular flow exists & is KMS at beta=%d (Tomita-"
          "Takesaki, v239), so 'dynamics = modular evolution of the horizon state' is "
          "well-defined ABSTRACTLY -- BUT that it is the GEOMETRIC boost (physical time) is "
          "QGEO.SYM.01 = a corollary of SEAM.EQUIV.01 (v335). A reduction to SEAM.EQUIV.01, "
          "not an independent closure" % kms_beta, kms_beta == 1)

    # 4. Route D: unique attractor [E], self-reproduction [P]
    transfer_spec = [sp.Integer(1), sp.Rational(2, 3) ** 6, sp.Rational(1, 3) ** 6]
    gapped = transfer_spec[1] < 1                                          # Perron-Frobenius gap, v56
    check("ROUTE D [E]/[P]: a gapped transport (spec {1,(2/3)^6,(1/3)^6}, gap, v56) has by "
          "Perron-Frobenius a UNIQUE attractor [E] -- so the cyclic self-reproduction map "
          "has a unique fixed point (parameter-freeness). BUT 'the full dynamics IS that "
          "fixed point' is the [P] cyclic interpretation, and the structure is still "
          "SEAM.EQUIV.01 -- not a closure", gapped)

    # 5. the verdict: none is 100% provable; convergence on the same 2 residuals as v335
    independent_closures = 0
    irreducible_residuals = {"SEAM.EQUIV.01", "QG-contour (v334)"}          # = v335 status
    routes_reducing_to_seam = {"B", "C", "D"}
    routes_improving_qg = {"A"}
    check("VERDICT [E] (direct answer): NONE of the four routes closes all problems or is "
          "100%% provable alone (independent closures = %d). They CONVERGE: {B,C,D} bottom "
          "out at SEAM.EQUIV.01, {A} at the QG contour -- the SAME %d irreducible residuals "
          "as v335 (%s), NOT zero. The BH premise makes both MORE natural (A finite thermal "
          "trace; B near-horizon CFT) but eliminates neither. Focus = SEAM.EQUIV.01"
          % (independent_closures, len(irreducible_residuals), sorted(irreducible_residuals)),
          independent_closures == 0 and len(irreducible_residuals) == 2
          and routes_reducing_to_seam == {"B", "C", "D"} and routes_improving_qg == {"A"})

    return summary("v343 four BH-birth routes: none closes all / is 100% provable; all four converge on SEAM.EQUIV.01 + the decoupled QG contour (= v335)")


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

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