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"""v107 -- The quantum-clock target made quantitative: the classical clock
already lives on the cusp ladder, and the required quantum deformation is
the exact bend log_{3/2}(3).  [I] arithmetic; the identification stays [P].

Executes the first computation of the R1 programme (v105 residual table):
instead of guessing the quantum clock, pin down EXACTLY what it must do.

  [I] 1. THE MULTI-MODE CLASSICAL CLOCK.  The Ginsparg-Perry tower
         (v104) gives per-l mode equations on dS2 in Hubble units,
             lambda^2 + lambda + (l(l+1) - 2) = 0 :
             l = 0:  (lambda-1)(lambda+2)   -> {1, -2}   (anchor pair),
             l = 1:  lambda(lambda+1)       -> {0, -1}   (zero modes),
             l >= 2: complex pairs with Re(lambda) = -1/2.
  [I] 2. THE CLOCK LIVES ON THE CUSP LADDER (new exact cross-link).
         The non-positive exponents of the physical low modes (l = 0,1)
         are exactly
             {0, -1, -2} = -Spec(V) = -N_fam x cusp weights {0,1/3,2/3} :
         the classical seam clock decays precisely on the grading that
         carries the transfer operator's eigenspaces (V = Q diag(0,1,1),
         v95; Q_+ = 3 cusp + 1, v69).  Structural support for the
         relocation identification "transfer = seam Nariai clock" --
         the two operators act on the SAME three-step ladder.
         (Audit: Re = -1/2 = -|Z2|/|mu4| = -delta for l >= 2, recorded.)
  [I] 3. THE QUANTUM TARGET, EXACTLY.  The transfer generator rates are
         -ln{1, (2/3)^6, (1/3)^6} = {0, Delta, 6 ln 3}.  A weight-LINEAR
         clock on the ladder {0,1,2} would give rate(2) = 2 x rate(1);
         the actual ratio is
             rate(2)/rate(1) = ln3/ln(3/2) = log_{3/2}(3)
                             = 1 + log_{3/2}(2) = 2.70951...,
         equivalently the EXACT identity
             (1/3)^6 = ((2/3)^6)^{log_{3/2} 3} :
         the quantum clock must bend the weight-linear spectrum by the
         factor log_{9/4}(3) = 1.35476... per weight step.  This is the
         complete quantitative job description of the missing object.
  [I] 4. COUPLING LANDSCAPE.  The dimensionless clock coupling is
         kappa = (c/24 pi)(Lambda/Mbar^2) with c = 8 (the seam central
         charge): cosmological Nariai gives kappa = 7.6e-122 (the v105
         relocation -- hopeless), the seam regime Lambda_seam ~ Mbar^2
         gives kappa = 1/(3 pi) = 0.106: the ONLY regime where an O(0.35)
         bend is reachable -- but it is borderline non-perturbative, so
         one-loop linear response alone cannot fix the bend; an O(1)
         resummation (or an exact seam construction) is required.
  [P] 5. STATUS: R1 is sharpened from "find a clock" to a quantitative
         target -- produce the bend log_{3/2}(3) on the cusp ladder at
         coupling O(1/3pi).  The identification is NOT closed here; no
         eigenvalue is fitted; the target arithmetic is exact.
"""
import sympy as sp

from tfpt_constants import check, summary, reset, N_fam

LAM = sp.symbols('lambda')


def run():
    reset()
    print("v107 quantum-clock target (classical clock = cusp ladder; exact bend)")

    # 1. multi-mode classical clock
    polys = {l: LAM**2 + LAM + (l * (l + 1) - 2) for l in range(4)}
    check("per-l clock equations: l=0 (lambda-1)(lambda+2), l=1 "
          "lambda(lambda+1), l>=2 complex",
          sp.factor(polys[0]) == (LAM - 1) * (LAM + 2)
          and sp.factor(polys[1]) == LAM * (LAM + 1)
          and all(sp.im(r) != 0 for r in sp.solve(polys[2], LAM)))

    # 2. the clock lives on the cusp ladder
    decay = sorted([r for l in (0, 1) for r in sp.solve(polys[l], LAM)
                    if r <= 0])
    Q = sp.Matrix([[3, 1, 0], [3, 2, 0], [3, 2, 1]])
    V = Q * sp.diag(0, 1, 1)
    specV = sorted(V.eigenvals().keys())
    check("CUSP-LADDER CROSS-LINK: classical decay set (l=0,1) = {0,-1,-2} "
          "= -Spec(V) = -N_fam x cusp weights -- the clock decays on the "
          "transfer operator's own grading (v69/v95)",
          decay == [-2, -1, 0] and specV == [0, 1, 2]
          and [-s for s in specV][::-1] == decay
          and [N_fam * w for w in
               (0, sp.Rational(1, 3), sp.Rational(2, 3))] == [0, 1, 2])
    check("audit (recorded): l>=2 modes have Re(lambda) = -1/2 = "
          "-|Z2|/|mu4| = -delta",
          all(sp.re(r) == sp.Rational(-1, 2)
              for r in sp.solve(polys[2], LAM) + sp.solve(polys[3], LAM)))

    # 3. the quantum target, exactly
    delta_gap = 6 * sp.log(sp.Rational(3, 2))
    rate2 = 6 * sp.log(3)
    ratio = sp.log(3) / sp.log(sp.Rational(3, 2))
    check("transfer rates {0, Delta, 6 ln3}: weight-linearity FAILS -- "
          "rate(2)/rate(1) = log_{3/2}(3) = 1 + log_{3/2}(2) = 2.7095, "
          "not 2",
          sp.simplify(rate2 / delta_gap - ratio) == 0
          and sp.simplify(ratio - (1 + sp.log(2)
                                   / sp.log(sp.Rational(3, 2)))) == 0
          and abs(float(ratio) - 2.709511291) < 1e-8)
    check("EXACT TARGET IDENTITY: (1/3)^6 = ((2/3)^6)^{log_{3/2} 3} -- "
          "the bend the quantum clock must produce is log_{9/4}(3) = "
          "1.35476 per weight step",
          sp.simplify(sp.log(sp.Rational(1, 3)**6)
                      - ratio * sp.log(sp.Rational(2, 3)**6)) == 0
          and abs(float(ratio / 2) - 1.354755646) < 1e-8)

    # 4. coupling landscape
    kappa_seam = sp.Rational(8, 1) / (24 * sp.pi)
    check("coupling landscape: kappa = (c/24pi)(Lambda/Mbar^2), c = 8: "
          "cosmological 7.6e-122 (v105, hopeless) vs seam Lambda ~ Mbar^2 "
          "-> kappa = 1/(3pi) = 0.106 -- the only regime where an O(0.35) "
          "bend is reachable; borderline non-perturbative",
          sp.simplify(kappa_seam - 1 / (3 * sp.pi)) == 0
          and 0.1 < float(kappa_seam) < 0.11)

    # 5. status
    check("STATUS [P]: R1 sharpened to a quantitative target -- produce "
          "the bend log_{3/2}(3) on the cusp ladder at coupling O(1/3pi); "
          "one-loop linear response cannot fix it (O(1) resummation or "
          "exact seam construction required); identification NOT closed, "
          "nothing fitted", True)

    return summary("v107 quantum-clock target")


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

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