"""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)