"""v475 -- GRAV.ENTROPIC.SCALARON.01: the R^2 kill test of the entropic-action bridge, EXECUTED (work package 5 of the v473 round), plus the explicit Lorentzian-positivity witness. v473 pre-registered the kill test at the raw coefficient level ("gap 3 (8 pi)^9 before tensorial factors"); this module computes the EXACT tensorial factors on a maximally symmetric background, extracts the raw entropic scalaron mass, and DECIDES v473's interpretation 2 (the TFPT scalaron as the light trace mode of Bianconi's G-field) negatively in its naive form. It closes NO gate; interpretation 3 (KMS-spectral renormalisation) is the surviving route and stays [C]/[O]. [E] 1. MAXIMALLY SYMMETRIC EIGENVALUES (the flattening machinery of her Appendix B, verified). With R_{munurhosigma} = (R/12)(g_{murho} g_{nusigma} - g_{musigma}g_{nurho}) on a general diagonal 4d metric, the relative curvature operator R~ g~^-1 has eigenvalues EXACTLY {R (0-form); R/4 x4 (1-form); R/6 x6 (2-form flattened 6x6)} -- computed symbolically including the 6x6 two-form flattening. [E] 2. EXACT VACUUM f(R) OF THE ENTROPIC ACTION. L_B(R) = -Tr_F ln(I - beta R~ g~^-1) = -[ln(1-bR) + 4 ln(1-bR/4) + 6 ln(1-bR/6)] with the exact series 3 beta R + (17/24) beta^2 R^2 + O(R^3): the linear term re-derives the 3 of her Eq. (45) (the v473 coefficient pin), and 17/24 is the EXACT tensorial R^2 factor (replacing v473's raw "(c3/6)^2 before tensorial factors" estimate; 17/12 = 1 + 1/4 + 1/6). [E] 3. THE RAW ENTROPIC SCALARON IS TRANS-PLANCKIAN (interpretation 2 KILLED in naive form). Matching L_B to the Starobinsky form (Mbar^2/2)(R + R^2/(6 m^2)) with the pinned beta' = c3/6 gives the EXACT mass m_raw^2 = 4608 pi^2 Mbar^2 / 17, i.e. m_raw = pi sqrt(4608/17) Mbar ~ 51.7 Mbar ~ 1.3e20 GeV -- HEAVY, not light; cross-checked by the independent f'/(3 f'') route (identical). The TFPT scalaron is m = c3^{7/2} Mbar = 3.06e13 GeV, a factor ~4.1e6 below. The mechanism v473's interpretation 2 requires must therefore supply EXACTLY m_raw^2/m_TFPT^2 = (72/17)(8 pi)^9 ~ 1.69e13 in mass^2 -- the kill-test number, now WITH tensorial factors. [E] 4. DOMAIN OF THE RAW ACTION. Positivity of I - beta R~ g~^-1 on the maximally symmetric background holds iff R < 1/beta = 384 pi^2 Mbar^2 (first log pole; then 4/beta, 6/beta) -- a trans-Planckian curvature bound, so the raw entropic action is well-defined at ALL sub-Planckian curvatures (the breakdown is not the problem; the mass is). [E] 5. LORENTZIAN-POSITIVITY WITNESS (v473's [C] caveat made explicit). In her warm-up sector G = g + alpha dphi x dphi on Minkowski diag(-1,1,1,1): a TIMELIKE gradient dphi = (v,0,0,0) gives G g^-1 eigenvalues {1 - alpha v^2, 1, 1, 1} -- nonpositive for v^2 >= 1/alpha, so ln(G g^-1) is UNDEFINED on an open field region; the spacelike and Euclidean cases stay positive for all v. Explicit witness that Lorentzian positivity is a genuine restriction; TFPT's OS/reflection-positive route is the sturdier construction (recorded in v473, now exhibited). [C] 6. VERDICT. Of v473's three interpretations, (1) survives (the entropic action covers the Einstein + Lambda level -- beta' pinned, Lambda channel closed by v473/v474), (2) is dead in naive form (the raw G-trace mode is trans-Planckian), (3) KMS-spectral renormalisation remains the ONLY route to the R^2 sector [C]. Nothing closes; no marker moves; v359's typing stays [O]. NET TYPING: [E] the five exact computations; [C] the verdict / surviving route; [O] unchanged (AP2 compression + the R^2 gate itself). Extends GRAV.ENTROPIC.ACTION.01 (v473) / GRAV.ENTROPIC.HODGE.01 (v474). Exact (sympy; Wolfram-mirrored). """ import itertools import sympy as sp from tfpt_constants import check, summary, reset pi = sp.pi c3 = sp.Rational(1, 8) / pi Mbar = sp.Symbol('Mbar', positive=True) def run(): reset() print("v475 GRAV.ENTROPIC.SCALARON.01: the R^2 kill test executed + the Lorentzian-positivity witness") # 1. maximally symmetric eigenvalues of R~ g~^-1, incl. the 6x6 flattening R = sp.Symbol('R', positive=True) gd = sp.symbols('g0:4', positive=True) g = sp.diag(*gd) ginv = g.inv() def riem(m, n, r, s): return (R / 12) * (g[m, r] * g[n, s] - g[m, s] * g[n, r]) # 0-form block: eigenvalue R (trivial); 1-form block: Ricci_mu^nu ricci = sp.Matrix(4, 4, lambda m, n: sum(riem(m, a, n, b) * ginv[a, b] for a in range(4) for b in range(4))) N1 = sp.simplify(ricci * ginv) pairs = list(itertools.combinations(range(4), 2)) g2f = sp.Matrix(6, 6, lambda A, B: sp.simplify( g[pairs[A][0], pairs[B][0]] * g[pairs[A][1], pairs[B][1]] - g[pairs[A][0], pairs[B][1]] * g[pairs[A][1], pairs[B][0]])) R2f = sp.Matrix(6, 6, lambda A, B: sp.simplify( 2 * riem(pairs[A][0], pairs[A][1], pairs[B][0], pairs[B][1]))) N2 = sp.simplify(R2f * g2f.inv()) check("MAX-SYMMETRIC EIGENVALUES [E] (her Appendix-B flattening verified): on a " "general diagonal 4d metric with R_munurhosigma = (R/12)(gg - gg), the " "relative curvature operator R~ g~^-1 has eigenvalues EXACTLY {R (scalar); " "R/4 x4 (Ricci block); R/6 x6 (Riemann on flattened 2-forms)} -- symbolic, " "including the 6x6 two-form flattening", N1 == sp.Rational(1, 4) * R * sp.eye(4) and N2 == sp.Rational(1, 6) * R * sp.eye(6)) # 2. exact vacuum f(R): series 3 beta R + (17/24) beta^2 R^2 b = sp.Symbol('beta', positive=True) fB = -(sp.log(1 - b * R) + 4 * sp.log(1 - b * R / 4) + 6 * sp.log(1 - b * R / 6)) ser = sp.series(fB, R, 0, 3).removeO() lin = ser.coeff(R, 1) quad = ser.coeff(R, 2) check("EXACT VACUUM f(R) [E]: L_B(R) = -[ln(1-bR) + 4 ln(1-bR/4) + 6 ln(1-bR/6)] " "= %s R + %s R^2 + O(R^3) -- the linear 3 beta re-derives her Eq. (45) " "(= the v473 pin), and 17/24 = (1/2)(1 + 1/4 + 1/6) is the EXACT tensorial " "R^2 factor (v473 quoted the raw (c3/6)^2 'before tensorial factors')" % (lin, quad), lin == 3 * b and quad == sp.Rational(17, 24) * b ** 2) # 3. the raw entropic scalaron mass, two independent routes beta_val = (c3 / 6) / (8 * pi * Mbar ** 2) # beta = beta' l_P^2, l_P^2 = 1/(8 pi Mbar^2) # route A: match l_P^-4 [3 b R + (17/24) b^2 R^2] to (Mbar^2/2)[R + R^2/(6 m^2)] m2 = sp.Symbol('m2', positive=True) lp4 = (8 * pi * Mbar ** 2) ** 2 # 1/l_P^4 eh_match = sp.simplify(lp4 * 3 * beta_val - Mbar ** 2 / 2) m2_A = sp.solve(sp.Eq(lp4 * sp.Rational(17, 24) * beta_val ** 2, Mbar ** 2 / (12 * m2)), m2)[0] # route B: f(R) scalaron m^2 = f'(0)/(3 f''(0)) in the same units m2_B = sp.simplify((3 * beta_val) / (3 * 2 * sp.Rational(17, 24) * beta_val ** 2)) m2_raw = sp.simplify(m2_A) m_raw_num = float(sp.sqrt(m2_raw / Mbar ** 2)) m2_tfpt = c3 ** 7 * Mbar ** 2 enhance = sp.simplify(m2_raw / m2_tfpt) check("RAW ENTROPIC SCALARON [E] (interpretation 2 KILLED in naive form): with " "the pinned beta' = c3/6 the EH normalisation matches identically (residual " "%s) and BOTH routes give m_raw^2 = %s = 4608 pi^2/17 Mbar^2, i.e. m_raw = " "%.1f Mbar ~ 1.3e20 GeV -- TRANS-PLANCKIAN, not light; the TFPT scalaron is " "c3^{7/2} Mbar = 3.06e13 GeV; required enhancement EXACTLY m_raw^2/m_TFPT^2 " "= %s = (72/17)(8 pi)^9 ~ %.3e in mass^2 (the kill-test number WITH " "tensorial factors)" % (eh_match, m2_raw / Mbar ** 2, m_raw_num, enhance, float(enhance)), eh_match == 0 and sp.simplify(m2_A - m2_B) == 0 and sp.simplify(m2_raw - sp.Rational(4608, 17) * pi ** 2 * Mbar ** 2) == 0 and sp.simplify(enhance - sp.Rational(72, 17) * (8 * pi) ** 9) == 0 and abs(float(enhance) - 1.694e13) / 1.694e13 < 1e-3) # 4. domain: first log pole at R = 1/beta = 384 pi^2 Mbar^2 (trans-Planckian) pole = sp.simplify(1 / beta_val) check("DOMAIN [E]: positivity of I - beta R~ g~^-1 holds iff R < 1/beta = %s = " "384 pi^2 Mbar^2 ~ %.0f Mbar^2 (then 4/beta, 6/beta) -- a TRANS-PLANCKIAN " "curvature bound: the raw entropic action is well-defined at all " "sub-Planckian curvatures; the obstruction is the MASS, not the domain" % (pole, float(pole / Mbar ** 2)), sp.simplify(pole - 384 * pi ** 2 * Mbar ** 2) == 0) # 5. Lorentzian-positivity witness (warm-up sector G = g + alpha dphi dphi) al, v = sp.symbols('alpha v', positive=True) gL = sp.diag(-1, 1, 1, 1) for name, dphi, expect in ( ("timelike", sp.Matrix([v, 0, 0, 0]), 1 - al * v ** 2), ("spacelike", sp.Matrix([0, v, 0, 0]), 1 + al * v ** 2)): M = dphi * dphi.T ev = sorted(((gL + al * M) * gL.inv()).eigenvals().items(), key=lambda kv: sp.default_sort_key(kv[0])) if name == "timelike": ev_time = dict(ev) else: ev_space = dict(ev) gE = sp.eye(4) ME = sp.Matrix([v, 0, 0, 0]) * sp.Matrix([v, 0, 0, 0]).T ev_eucl = ((gE + al * ME) * gE.inv()).eigenvals() check("LORENTZIAN-POSITIVITY WITNESS [E]: on Minkowski a TIMELIKE gradient gives " "G g^-1 eigenvalues {1 - alpha v^2, 1} -- NONPOSITIVE for v^2 >= 1/alpha, " "so ln(G g^-1) is undefined on an open field region; spacelike gives " "{1 + alpha v^2, 1} and Euclidean {1 + alpha v^2, 1}, positive for ALL v " "-- the v473 caveat exhibited: Lorentzian positivity is a genuine " "restriction, the OS/reflection-positive route is sturdier", ev_time.get(1 - al * v ** 2, 0) == 1 and ev_time.get(1, 0) == 3 and ev_space.get(1 + al * v ** 2, 0) == 1 and ev_space.get(1, 0) == 3 and ev_eucl.get(1 + al * v ** 2, 0) == 1 and ev_eucl.get(1, 0) == 3) # 6. verdict check("VERDICT [C]: of v473's three interpretations, (1) SURVIVES (entropic " "action = Einstein + Lambda level: beta' pinned v473, Lambda channel " "closed v473/v474), (2) is DEAD in naive form (raw G-trace mode " "trans-Planckian, this module), (3) KMS-spectral renormalisation is the " "ONLY surviving route to the R^2 sector [C]. Nothing closes; no marker " "moves; v359's equation-of-state typing stays [O]", True) return summary("v475 GRAV.ENTROPIC.SCALARON.01: the R^2 kill test EXECUTED -- [E] " "max-symmetric eigenvalues {R, R/4 x4, R/6 x6} (Appendix-B flattening " "verified), exact vacuum f(R) with series 3bR + (17/24)b^2R^2 (the 3 " "re-derived, 17/24 the exact tensorial factor), raw scalaron m^2 = " "4608 pi^2/17 Mbar^2 (m ~ 51.7 Mbar, TRANS-PLANCKIAN -- interpretation 2 " "killed naive; required enhancement exactly (72/17)(8pi)^9 ~ 1.69e13), " "domain bound 384 pi^2 Mbar^2 (not the problem), Lorentzian timelike " "witness 1 - alpha v^2 <= 0; [C] KMS renormalisation the only surviving " "R^2 route; [O] unchanged") if __name__ == "__main__": raise SystemExit(1 if run() else 0)