"""v144 -- The det-ratio cancellation, exact within the SdS family: the product of the two horizon radii obeys r_b r_c = 1 - Delta^2/3 EXACTLY (the traceless cubic's e_2-rigidity), so with the v132 anomaly zeta(0) = -2/3 per sphere the non-zero-mode determinant ratio is det'_n / det'_N = (r_b r_c)^{4/3} = (1 - Delta^2/3)^{4/3}: NO first-order term in the horizon split -- v131's '1 + O(deficit)' holds exactly in the within-family (near-Nariai) sense, with deficit coefficient |mu_4| (|Z_2|/N_fam)^3 = 32/27 (audit: the v103 canonical curvature times the glue order). [I] exact arithmetic; the finite-weight absorption across levels stays [P], with its obstruction now stated sharply. v131 left R1's last computation as '[P] 4. show det'_n/det'_dS = 1 + O(deficit)'. Within the SdS family this is exact arithmetic: [I] 1. e_2-RIGIDITY. The SdS horizon cubic is traceless (t^3 - 3t + 2c, v101/v103): with s = r_b + r_c and p = r_b r_c the two relations p - s^2 = -3 (e_2) and r_o = -s (e_1 = 0) give, for the split Delta = r_b - r_c, Delta^2 = s^2 - 4p = 12 - 3 s^2 and hence EXACTLY p = r_b r_c = 1 - Delta^2 / 3. [I] 2. THE DET-RATIO IN CLOSED FORM. The GP operator on S^2(r) is L_r = r^{-2} (-Delta_unit - 2); with the v132 anomaly zeta(0)|det' = -2/3 per sphere, det'(L_r) = r^{4/3} det'(L_1), so the two-sphere product ratio against Nariai (r_b = r_c = 1) is det'_n / det'_N = (r_b r_c)^{4/3} = (1 - Delta^2/3)^{4/3}. [I] 3. NO FIRST-ORDER TERM. d(det'-ratio)/d(Delta) = 0 at Delta = 0 exactly: the leading correction is quadratic in the split -- the v131 cancellation, derived (not assumed) from e_2-rigidity + the established anomaly. [I] 4. DEFICIT FORM (audit). With the Nariai mass deficit eps = 1 - c: eps = (3/8) Delta^2 + O(Delta^4), so det'_n/det'_N = 1 - (32/27) eps + O(eps^2), coefficient 32/27 = |mu_4| (|Z_2|/N_fam)^3 = 4 (2/3)^3 (the v103 canonical curvature times the glue order; recorded as an audit identity, not promoted). [P] 5. HONEST SCOPE (recorded). This closes v131's step in the WITHIN-FAMILY sense. The finite-weight comparison (S_n/S_dS in {1, 2/3, 1/3}) still runs through the v132/v133 anomaly budget: under a naive GLOBAL rescaling reading, det'^{-1/2} squared would multiply the Born square by (S_n/S_dS)^{-4/3}, shifting the exponent 6 -> 14/3 (not an atom) -- so the [P] absorption MUST proceed via the Hubble-unit conversion (v132's reading), not via global rescaling. The frozen transfer spectrum is untouched; the obstruction the remaining [P] has to resolve is now stated sharply. """ import sympy as sp from tfpt_constants import check, summary, reset DELTA, T = sp.symbols('Delta t', positive=True) def run(): reset() print("v144 det-ratio family cancellation (R1's v131 step, within-family)") s = 2 * sp.sqrt(1 - DELTA ** 2 / 12) p = 1 - DELTA ** 2 / 3 rb, rc, ro = (s + DELTA) / 2, (s - DELTA) / 2, -s # 1. e2-rigidity: the three roots satisfy a traceless cubic with e2 = -3 poly = sp.expand((T - rb) * (T - rc) * (T - ro)) e1 = sp.simplify(rb + rc + ro) e2 = sp.simplify(rb * rc + rb * ro + rc * ro) prod_exact = sp.simplify(rb * rc - p) check("e_2-RIGIDITY: with s = r_b + r_c, the traceless cubic " "(e_1 = 0, e_2 = -3) forces Delta^2 = 12 - 3 s^2 and " "EXACTLY r_b r_c = 1 - Delta^2/3 (verified symbolically " "on the explicit root triple)", e1 == 0 and e2 == -3 and prod_exact == 0 and sp.simplify(poly.coeff(T, 2)) == 0 and sp.simplify(poly.coeff(T, 1)) == -3) # 2. det-ratio in closed form (zeta(0) = -2/3 per sphere, v132) zeta0 = sp.Rational(-2, 3) # det'(c L) = c^{zeta(0)} det'(L); L_r = r^{-2} L_1 => det'(L_r) = r^{-2 zeta0} det'(L_1) scaling_exponent = sp.simplify(-2 * zeta0) ratio = (rb * rc) ** sp.Rational(4, 3) check("DET-RATIO CLOSED FORM: det'(L_r) = r^{4/3} det'(L_1) " "(v132 anomaly zeta(0) = -2/3, scaling exponent -2*zeta(0) " "= 4/3 per sphere); two-sphere product ratio against " "Nariai = (r_b r_c)^{4/3} = (1 - Delta^2/3)^{4/3}", scaling_exponent == sp.Rational(4, 3) and sp.simplify(ratio - (1 - DELTA ** 2 / 3) ** sp.Rational(4, 3)) == 0) # 3. no first-order term in the split series = sp.series(ratio, DELTA, 0, 4).removeO() d1 = sp.diff(ratio, DELTA).subs(DELTA, 0) d2_coeff = series.coeff(DELTA, 2) check("NO FIRST-ORDER TERM: d(ratio)/d(Delta)|_0 = 0 exactly; " "leading correction quadratic, ratio = 1 - (4/9) Delta^2 + " "O(Delta^4) -- the v131 cancellation DERIVED from " "e_2-rigidity + the established anomaly", d1 == 0 and d2_coeff == sp.Rational(-4, 9) and series.coeff(DELTA, 1) == 0) # 4. deficit form (audit) c_par = sp.simplify(p * s / 2) # cubic constant: 2c = e3*(-1) => c = p*s/2 eps_series = sp.series(1 - c_par, DELTA, 0, 4).removeO() eps_coeff = eps_series.coeff(DELTA, 2) ratio_per_eps = sp.Rational(-4, 9) / eps_coeff atom = -sp.Integer(4) * sp.Rational(2, 3) ** 3 check("DEFICIT FORM (audit): eps = 1 - c = (3/8) Delta^2 + " "O(Delta^4), so det-ratio = 1 - (32/27) eps + O(eps^2); " "coefficient 32/27 = |mu_4| (|Z_2|/N_fam)^3 = 4 (2/3)^3 " "(the v103 canonical curvature x glue order; audit-typed)", eps_coeff == sp.Rational(3, 8) and ratio_per_eps == sp.Rational(-32, 27) == atom) # 5. honest scope shifted = sp.Integer(6) - sp.Rational(4, 3) check("HONEST SCOPE [P] (recorded): within-family cancellation " "closed; the finite-weight absorption stays open -- a " "naive global-rescaling reading would shift the Born " "exponent 6 -> 14/3 (not an atom), so the absorption must " "proceed via the Hubble-unit conversion (v132); the frozen " "transfer spectrum is untouched", shifted == sp.Rational(14, 3)) return summary("v144 det-ratio family cancellation") if __name__ == "__main__": raise SystemExit(1 if run() else 0)