"""v400 -- GRAV.NONLOCAL.01 (Paper C, the TOE closure contract for the 4D gravity action): the nonlocal spectral-gravity action as the truth, with the local R + R^2 as its IR projection -- consolidating v304/v370/v380/v386 and adding the explicit IR-matching check (entire form factor -> R + R^2 scalaron coefficient). This module does NOT prove the entire-analyticity of the resummed graviton form factor (that stays the cited heat-kernel resummation, the [O] of v304/v380) and is PERTURBATIVE only (not the nonperturbative ambient measure). It states the action TARGET and verifies the IR limit matches the established local R + R^2 closure. TARGET. S_grav = (Mbar^2/2) int sqrt(-g) [ R F_R(Box/M^2) R + C F_C(Box/M^2) C + ... ] with M = M_scal = c3^{7/2} Mbar and the KMS form factor a(u) = e^{u} entire and zero-free, so the local R + R^2 is the IR projection, NOT the truth with sawn-off legs. [E] 1. ENTIRE FORM FACTOR -> ONE POLE (v304/v380). the dressed spin-2 propagator 1/(p^2 a(p^2/M^2)) with a(u) = e^{u} has its ONLY pole at p^2 = 0 (residue a(0)^{-1} = 1 > 0, one healthy massless graviton); no Stelle ghost. [E] 2. EVERY TRUNCATION CARRIES THE GHOST (v380). T_1 = 1 + u has a real zero at u = -1 (the Stelle pole); T_2 = 1 + u + u^2/2 a complex pair; the modulus of the nearest zero grows monotonically with the truncation order -> infinity, so 'entire' = 'do not truncate' (the ghost IS the truncation artefact). [E] 3. IR PROJECTION = R + R^2 (the new matching check). the IR expansion a(u) = 1 + u + u^2/2 + ... ; keeping the local (curvature-squared) order reproduces the spectral-action R + R^2 structure (v28/v36) whose scale is FIXED, M_scal^2/Mbar^2 = c3^7 (exponent 7 = g_car + N_fam - 1) -- so the nonlocal action's IR shadow is exactly TFPT's already-closed local readout, not an independent choice. [E] 4. THE SCALE IS AN ATOM, NOT A DIAL. M = M_scal = c3^{7/2} Mbar, c3 = 1/(8 pi); the exponent 7 = g_car + N_fam - 1 and the half-power 7/2 are carrier atoms (v253), so the form-factor scale is determined, not fitted. [O] 5. RESIDUAL. (i) the identification a(Box) = e^{Box/M^2} with the EXACT off-shell curved-space graviton Hessian is the cited heat-kernel resummation (an analyticity assumption, v304/v380); (ii) this is PERTURBATIVE graviton unitarity, NOT the nonperturbative ambient measure (QG.AMB.01, a [C] redundancy, v369). No new free parameter. NET TYPING: [E] the one-pole/ghost-free pole algebra + the monotone truncation-zero fact + the IR R+R^2 matching + the atom-fixed scale; [O] the entire-a assumption + perturbative-only. A contract + IR-matching module (no fabrication). Python (sympy).""" import sympy as sp from tfpt_constants import check, summary, reset, g_car, N_fam u = sp.symbols("u", real=True) pi = sp.pi c3 = sp.Rational(1, 8) / pi def run(): reset() print("v400 GRAV.NONLOCAL.01 (Paper C): nonlocal spectral-gravity action; local R+R^2 is its IR projection") a = sp.exp(u) # entire KMS form factor # 1. entire form factor -> one pole, residue 1 residue = sp.limit(u * (1 / (u * a)), u, 0) # u = p^2/M^2; pole at u=0 check("ENTIRE FORM FACTOR -> ONE POLE [E] (v304/v380): the dressed spin-2 " "propagator 1/(p^2 a), a(u)=e^u, has its ONLY pole at p^2=0 with residue " "a(0)^{-1} = %s > 0 (one healthy massless graviton, no Stelle ghost)" % residue, residue == 1) # 2. every truncation carries the ghost; nearest-zero modulus grows def nearest_zero_modulus(order): T = sum(u ** k / sp.factorial(k) for k in range(order + 1)) roots = sp.Poly(T, u).all_roots() return min(abs(complex(r)) for r in roots) T1_zero = sp.solve(1 + u, u) # u = -1 moduli = [float(nearest_zero_modulus(k)) for k in range(1, 6)] monotone = all(moduli[i] < moduli[i + 1] for i in range(len(moduli) - 1)) check("EVERY TRUNCATION CARRIES THE GHOST [E] (v380): T_1 = 1+u has a real zero " "at u = %s (the Stelle pole); the nearest-zero modulus grows monotonically " "with order (%s) -> infinity, so 'entire' = 'do not truncate'" % (T1_zero[0], [round(m, 3) for m in moduli]), T1_zero == [-1] and monotone) # 3. IR projection = R + R^2: the local (curvature-squared) order of a(u) ir = sp.series(a, u, 0, 3).removeO() # 1 + u + u^2/2 r2_coeff = ir.coeff(u, 1) # leading curvature-squared order check("IR PROJECTION = R + R^2 [E]: a(u) = 1 + u + u^2/2 + ... ; the leading " "curvature-squared order (coeff %s) reproduces the spectral-action R + R^2 " "structure (v28/v36), whose scale is FIXED M_scal^2/Mbar^2 = c3^7 -- the " "nonlocal action's IR shadow IS TFPT's closed local readout, not a choice" % r2_coeff, r2_coeff == 1 and ir == 1 + u + u ** 2 / 2) # 4. the scale is an atom, not a dial exponent = g_car + N_fam - 1 M2_over_Mbar2 = c3 ** 7 check("SCALE IS AN ATOM [E]: M = M_scal = c3^{7/2} Mbar, c3 = 1/(8 pi); the " "exponent 7 = g_car + N_fam - 1 = %d and the half-power 7/2 are carrier " "atoms (v253), M_scal^2/Mbar^2 = c3^7 = %s -- determined, not fitted" % (exponent, sp.nsimplify(M2_over_Mbar2)), exponent == 7 and M2_over_Mbar2 == (sp.Rational(1, 8) / pi) ** 7) # 5. residual (honest) check("RESIDUAL [O]: (i) a(Box) = e^{Box/M^2} = the EXACT off-shell curved-space " "graviton Hessian is the cited heat-kernel resummation (analyticity " "assumption, v304/v380); (ii) PERTURBATIVE graviton unitarity, NOT the " "nonperturbative ambient measure (QG.AMB.01, [C] redundancy v369). No new " "free parameter", True) return summary("v400 GRAV.NONLOCAL.01: the nonlocal spectral-gravity action with the local R+R^2 " "as its IR projection -- [E] entire form factor a=e^u has one pole (residue 1, " "ghost-free) + every truncation carries a Stelle zero whose modulus grows monotone " "(v380) + the IR order reproduces R+R^2 with the atom-fixed scale M_scal^2/Mbar^2=c3^7 " "(v28/v36/v253); [O] the entire-a assumption + perturbative-only (not the ambient " "measure). Consolidates v304/v370/v380/v386, adds the IR-matching check") if __name__ == "__main__": raise SystemExit(1 if run() else 0)