"""v360 -- GRAV.GAPCORR.01 (Direction 2): the gap-induced higher-curvature correction to GR -- the disciplined result. The question: does the seam transfer gap {1,(2/3)^6,(1/3)^6} yield a NEW, calculable, FORCED deviation from GR (a near-term kill test) beyond the known R^2 scalaron? Run with the v354/v355 discriminator (a coefficient counts only if it is atom-forced, never a fit), the honest answer is: NO new near-term forced prediction -- the calculable GR deviation IS the existing R^2 scalaron, and a distinct gap-induced term sits at the seam/Planck scale with a coefficient that is not atom-forced. This module records that outcome (positive content + an explicit decline), it does not manufacture a coefficient. [E] 1. THE LEADING HIGHER-CURVATURE TERM IS THE R^2 SCALARON (already in TFPT). The spectral-action a4 = R^2/72 (v36) gives the R^2 term; TFPT fixes its scale, M_scal^2/Mbar^2 = c3^7 (exponent 7 = g_car + N_fam - 1), M_scal = c3^(7/2) Mbar ~ 3.06e13 GeV. This IS the calculable beyond-Einstein correction. [C] 2. IT IS THE ENTANGLEMENT-EQUILIBRIUM NEXT ORDER (consistency). the higher-curvature (Wald) entropy correction to the first law gives the f(R) = R + R^2 equation (v28), so the scalaron is the next-order term of the SAME entanglement-equilibrium route that gave the Einstein equation (v359) -- a structural confirmation, not a new term. [E] 3. IT IS ALREADY FALSIFIABLE NOW. the R^2/Starobinsky attractor predicts n_s = 1 - 2/N_star and r = 12/N_star^2 on the frozen band N_star in [50,60] -- a live CMB-S4 kill test (already on the scorecard). So Direction 2's "calculable GR deviation" already EXISTS and is already testable; no new prediction is needed. [E] 4. THE GAP IS A DIMENSIONLESS RATE -> SEAM/PLANCK-SCALE CORRECTIONS. Delta = 6 ln(3/2) is dimensionless; a distinct gap-induced curvature^2 term enters at the physical scale xi * v_geo with xi = 1/Delta ~ 0.41 (in seam units) and v_geo ~ M_Pl, i.e. at the seam/Planck scale -- NOT near-term observable. [O] 5. DECLINE (the discriminator). a NEW dimensionless deviation coefficient distinct from the scalaron would need the subleading heat-kernel / Wald data, which is NOT a forced atom combination (the scalaron exponent 7 = g+N-1 and the gap exponent 6 = 2 N_fam are different structures; c3^7 and (2/3)^6 are unrelated). Per v354/v355 we DECLINE to manufacture an atom-unforced coefficient. HONEST CONCLUSION: Direction 2 yields no new near-term forced prediction; the answer is the existing scalaron + Planck-scale corrections. HONEST SCOPE: [E] the scalaron-as-leading-correction + its existing n_s/r kill test + the gap-is-dimensionless / Planck-scale reasoning; [C] the Wald-first-law consistency; [O] the explicit decline of a manufactured new coefficient. A disciplined-search result (like v355): the positive content is that the calculable GR deviation is the scalaron; the decline is the anti-numerology outcome. Python-only (sympy/mpmath).""" import mpmath as mp import sympy as sp from tfpt_constants import check, summary, reset, g_car, N_fam mp.mp.dps = 25 pi = sp.pi c3 = sp.Rational(1, 8) / pi def run(): reset() print("v360 GRAV.GAPCORR.01 (Direction 2): the gap-correction is the R^2 scalaron; a new coefficient is DECLINED") # 1. leading higher-curvature term = the R^2 scalaron scal_exp = g_car + N_fam - 1 # 7 M_ratio = c3 ** 7 # M_scal^2/Mbar^2 M_scal = float(mp.mpf(1) / (8 * mp.pi)) ** 3.5 * 2.435e18 check("LEADING HIGHER-CURVATURE TERM = R^2 SCALARON [E]: spectral-action a4=R^2/72 (v36); " "M_scal^2/Mbar^2 = c3^7 (exponent 7 = g_car+N_fam-1 = %d), M_scal = c3^(7/2) Mbar ~ " "%.2e GeV -- the calculable beyond-Einstein correction, already in TFPT" % (scal_exp, M_scal), scal_exp == 7 and M_ratio == c3 ** 7 and 2e13 < M_scal < 4e13) # 2. it is the entanglement-equilibrium next order (consistency, citable) check("ENTANGLEMENT-EQUILIBRIUM NEXT ORDER [C]: the higher-curvature (Wald) entropy " "correction to the first law gives the f(R)=R+R^2 equation (v28), so the scalaron is " "the next-order term of the SAME route that gave the Einstein equation (v359) -- a " "structural confirmation, not a new term", True) # 3. already falsifiable now (n_s, r) N_lo, N_hi = 50, 60 ns_lo, ns_hi = 1 - 2 / N_lo, 1 - 2 / N_hi r_lo, r_hi = 12 / N_hi ** 2, 12 / N_lo ** 2 check("ALREADY FALSIFIABLE NOW [E]: the R^2/Starobinsky attractor predicts n_s=1-2/N* in " "[%.3f,%.3f] and r=12/N*^2 in [%.4f,%.4f] on N* in [50,60] -- a live CMB-S4 kill " "test; Direction 2's calculable GR deviation already EXISTS and is testable" % (ns_lo, ns_hi, r_lo, r_hi), 0.95 < ns_lo < ns_hi < 0.97 and 0.003 < r_lo < r_hi < 0.005) # 4. the gap is a dimensionless rate -> seam/Planck-scale corrections gap = 6 * mp.log(mp.mpf(3) / 2) xi = 1 / gap check("GAP IS DIMENSIONLESS -> SEAM/PLANCK SCALE [E]: Delta=6 ln(3/2)=%.4f is a dimensionless " "rate; a distinct gap-induced curvature^2 term enters at xi*v_geo with xi=1/Delta=%.4f " "(seam units) and v_geo ~ M_Pl -- i.e. the seam/Planck scale, NOT near-term observable" % (float(gap), float(xi)), 2.4 < float(gap) < 2.5 and 0.4 < float(xi) < 0.42) # 5. decline (the discriminator) lam2 = (sp.Rational(2, 3)) ** 6 gap_exp = 2 * N_fam # 6 check("DECLINE [O]: a NEW dimensionless deviation coefficient distinct from the scalaron " "needs the subleading heat-kernel/Wald data, NOT a forced atom combination -- the " "scalaron exponent 7=g+N-1 and the gap exponent 6=2 N_fam=%d are different structures, " "and c3^7 vs (2/3)^6=%s are unrelated. Per v354/v355 we DECLINE to manufacture an " "atom-unforced coefficient: Direction 2 yields NO new near-term forced prediction; " "the answer is the existing scalaron + Planck-scale corrections" % (gap_exp, lam2), gap_exp == 6 and scal_exp == 7 and gap_exp != scal_exp) return summary("v360 GRAV.GAPCORR.01 (Direction 2): the calculable GR deviation IS the R^2 scalaron (c3^7, already falsifiable via n_s/r and the next-order entanglement-equilibrium term); a distinct gap-induced correction sits at the seam/Planck scale with a coefficient that is NOT atom-forced -> DECLINED (the disciplined anti-numerology outcome). No new near-term forced prediction.") if __name__ == "__main__": raise SystemExit(1 if run() else 0)