"""v104 -- The classical near-Nariai clock IS the anchor (third appearance), and the honest (2/3)-test. [I] exact GR arithmetic + one external-reference typing; the quantum clock stays [P]. v103 reduced the seam variational principle to ONE missing object: the gravity-side CLOCK that converts the canonical curvature (2/3)^N_fam into a rate. This script derives the CLASSICAL clock -- the linearized dynamics of the S^2-radius modulus around the Nariai point (dS_2 x S^2) -- which is pure GR (Ginsparg-Perry 1983; Bousso-Hawking 1997 class), with no quantum input: [I] 1. STATIC PIN (self-contained). On the dS_2 static patch ds^2 = -(1-L rho^2) dtau^2 + drho^2/(1-L rho^2), the function phi(rho) = rho satisfies EXACTLY d/drho[(1-L rho^2) phi'] = -2 Lambda phi . The static SdS deformation (the exact two-horizon family) IS this static mode, so -- given that the linearization of the Einstein equations around Nariai is of Laplace type (standard GR; external reference, typed honestly) -- the modulus mass is PINNED: m^2 = -2 Lambda = -|Z2| Lambda . [I] 2. HORIZON-SPLIT CONSISTENCY. The exact roots give r_{b,c} = (1 -+ psi/sqrt(3) - psi^2/18 + ...)/sqrt(L): the split is linear in the canonical angle with coefficient 1/sqrt(3) = 1/sqrt(N_fam) -- the static-mode amplitude is the trisection angle. [I] 3. GINSPARG-PERRY TOWER. On the S^2 factor the modulus operator -box - 2L has eigenvalues (l(l+1) - 2) Lambda: l = 0 -> -|Z2| Lambda (exactly ONE negative mode), l = 1 -> 0 (zero modes), l >= 2 -> positive. [I] 4. THE CLOCK SPEAKS ANCHOR. In Hubble units (H = sqrt(L), flat slicing, d = 2 so the friction is (d-1)H = H), homogeneous modes phi ~ e^{lambda H t} satisfy lambda^2 + lambda - 2 = 0, i.e. chi_clock(lambda) = (lambda - 1)(lambda + 2): THE ANCHOR QUADRATIC. Eigenvalues {1, -2} = the distinct anchor roots; the Nariai cubic factors as (t-1)^2(t+2) = (t-1) * chi_clock(t). The anchor now appears a THIRD time: as configuration roots (v101), as the curvature base (v103), and as the spectrum of the linearized clock. [I] 5. ENTROPY-DEVIATION RATE: sigma - 2/3 ~ psi^2 ~ e^{2Ht}: the entropy deviation grows at exactly 2H = |Z2| x Hubble. [I] 6. THE HONEST (2/3)-TEST IS NEGATIVE FOR THE CLASSICAL CLOCK: its eigenvalues are integers (anchor roots), NOT (2/3)-powers. The (2/3)-powers stay in the FUNCTIONAL (canonical curvature (2/3)^{N_fam}, v103). The conversion curvature -> rate at one loop (quantum (anti-)evaporation, Bousso-Hawking) remains external QFT input [P]: the bridge to the flavor rate (2/3)^{2 N_fam} is NOT closed here -- but the classical half of the clock question is now derived, and it returns the anchor. TYPING: every check below is exact arithmetic [I]; the single imported statement is "the Nariai linearization is Laplace-type" (standard GR, Ginsparg-Perry / Bousso-Hawking) -- the mass VALUE is then pinned internally by check 1. No open gate is closed; the quantum clock is the one remaining [P] of the seam variational principle. """ import sympy as sp from tfpt_constants import check, summary, reset, N_fam RHO, PSI, LAM, T = sp.symbols('rho psi lambda t', real=True) L = sp.symbols('Lambda', positive=True) def run(): reset() print("v104 Nariai clock (classical generator = the anchor; honest 2/3 test)") # 1. static pin box_static = sp.diff((1 - L * RHO**2) * sp.diff(RHO, RHO), RHO) check("STATIC PIN: phi(rho) = rho solves d/drho[(1-L rho^2) phi'] = " "-2 Lambda phi EXACTLY on the dS2 static patch => modulus mass " "m^2 = -2 Lambda = -|Z2| Lambda (Laplace-type linearization: " "standard GR, external reference)", sp.simplify(box_static + 2 * L * RHO) == 0) # 2. horizon-split consistency rc_ser = sp.series(2 * sp.cos((PSI + sp.pi) / 3), PSI, 0, 3).removeO() rb_ser = sp.series(2 * sp.cos((PSI - sp.pi) / 3), PSI, 0, 3).removeO() check("horizon split linear in the canonical angle: r_{b,c} = " "1 -+ psi/sqrt(3) - psi^2/18 (units 1/sqrt(L)); coefficient " "1/sqrt(3) = 1/sqrt(N_fam)", sp.expand(rc_ser - (1 - PSI / sp.sqrt(3) - PSI**2 / 18)) == 0 and sp.expand(rb_ser - (1 + PSI / sp.sqrt(3) - PSI**2 / 18)) == 0 and sp.Rational(1, 3) == sp.Rational(1, N_fam)) # 3. Ginsparg-Perry tower tower = [sp.Integer(li * (li + 1) - 2) for li in range(4)] check("Ginsparg-Perry tower (l(l+1)-2)Lambda: l=0 -> -2 = -|Z2| (the " "ONE negative mode), l=1 -> 0 (zero modes), l>=2 positive", tower == [-2, 0, 4, 10] and sum(1 for v in tower if v < 0) == 1) # 4. the clock speaks anchor chi_clock = LAM**2 + LAM - 2 check("THE CLOCK: phi ~ e^{lambda H t} (flat slicing, friction (d-1)H " "= H, m^2 = -2H^2) => chi_clock = lambda^2 + lambda - 2 = " "(lambda-1)(lambda+2) -- THE ANCHOR QUADRATIC", sp.factor(chi_clock) == (LAM - 1) * (LAM + 2) and sorted(sp.solve(chi_clock, LAM)) == [-2, 1]) check("the Nariai cubic factors through the clock: (t-1)^2(t+2) = " "(t-1) * chi_clock(t) -- the anchor appears a THIRD time " "(configuration roots v101, curvature base v103, clock spectrum)", sp.expand((T - 1)**2 * (T + 2) - (T - 1) * (T**2 + T - 2)) == 0) # 5. entropy-deviation rate H, t_, psi0 = sp.symbols('H t_ psi_0', positive=True) sigma_dev = sp.Rational(1, 2) * sp.Rational(8, 27) \ * (psi0 * sp.exp(H * t_))**2 rate = sp.simplify(sp.diff(sp.log(sigma_dev), t_)) check("entropy deviation rate: sigma - 2/3 = (1/2)(2/3)^3 psi^2 with " "psi ~ e^{Ht} (growing mode lambda = 1) => d/dt log(sigma-2/3) " "= 2H = |Z2| x Hubble, exactly", rate == 2 * H) # 6. honest (2/3)-test check("HONEST (2/3)-TEST: classical clock eigenvalues {1, -2} are " "integers (anchor roots), NOT (2/3)-powers; the 2/3-powers stay " "in the functional ((2/3)^N_fam curvature, v103); the one-loop " "conversion curvature -> rate (quantum (anti-)evaporation) " "remains external [P] -- the flavor-rate bridge is NOT closed " "here", all(v not in (sp.Rational(2, 3), sp.Rational(4, 9), sp.Rational(8, 27)) for v in (-2, 1))) return summary("v104 Nariai clock") if __name__ == "__main__": raise SystemExit(1 if run() else 0)