TFPT scientific source excerpt Original repository path: verification/v475_entropic_scalaron_gate.py Original file SHA-256: ebe9ed0f0412400d958837088c5f420d057db89ac5adad72da41b3a36dea10a5 Inclusive original lines: 1-198 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000001 | """v475 -- GRAV.ENTROPIC.SCALARON.01: the R^2 kill test of the entropic-action 000002 | bridge, EXECUTED (work package 5 of the v473 round), plus the explicit 000003 | Lorentzian-positivity witness. v473 pre-registered the kill test at the raw 000004 | coefficient level ("gap 3 (8 pi)^9 before tensorial factors"); this module 000005 | computes the EXACT tensorial factors on a maximally symmetric background, 000006 | extracts the raw entropic scalaron mass, and DECIDES v473's interpretation 2 000007 | (the TFPT scalaron as the light trace mode of Bianconi's G-field) negatively 000008 | in its naive form. It closes NO gate; interpretation 3 (KMS-spectral 000009 | renormalisation) is the surviving route and stays [C]/[O]. 000010 | 000011 | [E] 1. MAXIMALLY SYMMETRIC EIGENVALUES (the flattening machinery of her 000012 | Appendix B, verified). With R_{munurhosigma} = (R/12)(g_{murho} 000013 | g_{nusigma} - g_{musigma}g_{nurho}) on a general diagonal 4d metric, 000014 | the relative curvature operator R~ g~^-1 has eigenvalues EXACTLY 000015 | {R (0-form); R/4 x4 (1-form); R/6 x6 (2-form flattened 6x6)} -- 000016 | computed symbolically including the 6x6 two-form flattening. 000017 | [E] 2. EXACT VACUUM f(R) OF THE ENTROPIC ACTION. L_B(R) = -Tr_F ln(I - 000018 | beta R~ g~^-1) = -[ln(1-bR) + 4 ln(1-bR/4) + 6 ln(1-bR/6)] with the 000019 | exact series 3 beta R + (17/24) beta^2 R^2 + O(R^3): the linear term 000020 | re-derives the 3 of her Eq. (45) (the v473 coefficient pin), and 000021 | 17/24 is the EXACT tensorial R^2 factor (replacing v473's raw 000022 | "(c3/6)^2 before tensorial factors" estimate; 17/12 = 1 + 1/4 + 1/6). 000023 | [E] 3. THE RAW ENTROPIC SCALARON IS TRANS-PLANCKIAN (interpretation 2 000024 | KILLED in naive form). Matching L_B to the Starobinsky form 000025 | (Mbar^2/2)(R + R^2/(6 m^2)) with the pinned beta' = c3/6 gives the 000026 | EXACT mass m_raw^2 = 4608 pi^2 Mbar^2 / 17, i.e. m_raw = 000027 | pi sqrt(4608/17) Mbar ~ 51.7 Mbar ~ 1.3e20 GeV -- HEAVY, not light; 000028 | cross-checked by the independent f'/(3 f'') route (identical). The 000029 | TFPT scalaron is m = c3^{7/2} Mbar = 3.06e13 GeV, a factor ~4.1e6 000030 | below. The mechanism v473's interpretation 2 requires must therefore 000031 | supply EXACTLY m_raw^2/m_TFPT^2 = (72/17)(8 pi)^9 ~ 1.69e13 in mass^2 000032 | -- the kill-test number, now WITH tensorial factors. 000033 | [E] 4. DOMAIN OF THE RAW ACTION. Positivity of I - beta R~ g~^-1 on the 000034 | maximally symmetric background holds iff R < 1/beta = 384 pi^2 Mbar^2 000035 | (first log pole; then 4/beta, 6/beta) -- a trans-Planckian curvature 000036 | bound, so the raw entropic action is well-defined at ALL sub-Planckian 000037 | curvatures (the breakdown is not the problem; the mass is). 000038 | [E] 5. LORENTZIAN-POSITIVITY WITNESS (v473's [C] caveat made explicit). 000039 | In her warm-up sector G = g + alpha dphi x dphi on Minkowski 000040 | diag(-1,1,1,1): a TIMELIKE gradient dphi = (v,0,0,0) gives G g^-1 000041 | eigenvalues {1 - alpha v^2, 1, 1, 1} -- nonpositive for v^2 >= 000042 | 1/alpha, so ln(G g^-1) is UNDEFINED on an open field region; the 000043 | spacelike and Euclidean cases stay positive for all v. Explicit 000044 | witness that Lorentzian positivity is a genuine restriction; TFPT's 000045 | OS/reflection-positive route is the sturdier construction (recorded 000046 | in v473, now exhibited). 000047 | [C] 6. VERDICT. Of v473's three interpretations, (1) survives (the 000048 | entropic action covers the Einstein + Lambda level -- beta' pinned, 000049 | Lambda channel closed by v473/v474), (2) is dead in naive form (the 000050 | raw G-trace mode is trans-Planckian), (3) KMS-spectral 000051 | renormalisation remains the ONLY route to the R^2 sector [C]. 000052 | Nothing closes; no marker moves; v359's typing stays [O]. 000053 | 000054 | NET TYPING: [E] the five exact computations; [C] the verdict / surviving 000055 | route; [O] unchanged (AP2 compression + the R^2 gate itself). Extends 000056 | GRAV.ENTROPIC.ACTION.01 (v473) / GRAV.ENTROPIC.HODGE.01 (v474). Exact 000057 | (sympy; Wolfram-mirrored). 000058 | """ 000059 | import itertools 000060 | 000061 | import sympy as sp 000062 | 000063 | from tfpt_constants import check, summary, reset 000064 | 000065 | pi = sp.pi 000066 | c3 = sp.Rational(1, 8) / pi 000067 | Mbar = sp.Symbol('Mbar', positive=True) 000068 | 000069 | 000070 | def run(): 000071 | reset() 000072 | print("v475 GRAV.ENTROPIC.SCALARON.01: the R^2 kill test executed + the Lorentzian-positivity witness") 000073 | 000074 | # 1. maximally symmetric eigenvalues of R~ g~^-1, incl. the 6x6 flattening 000075 | R = sp.Symbol('R', positive=True) 000076 | gd = sp.symbols('g0:4', positive=True) 000077 | g = sp.diag(*gd) 000078 | ginv = g.inv() 000079 | 000080 | def riem(m, n, r, s): 000081 | return (R / 12) * (g[m, r] * g[n, s] - g[m, s] * g[n, r]) 000082 | 000083 | # 0-form block: eigenvalue R (trivial); 1-form block: Ricci_mu^nu 000084 | ricci = sp.Matrix(4, 4, lambda m, n: sum(riem(m, a, n, b) * ginv[a, b] 000085 | for a in range(4) for b in range(4))) 000086 | N1 = sp.simplify(ricci * ginv) 000087 | pairs = list(itertools.combinations(range(4), 2)) 000088 | g2f = sp.Matrix(6, 6, lambda A, B: sp.simplify( 000089 | g[pairs[A][0], pairs[B][0]] * g[pairs[A][1], pairs[B][1]] 000090 | - g[pairs[A][0], pairs[B][1]] * g[pairs[A][1], pairs[B][0]])) 000091 | R2f = sp.Matrix(6, 6, lambda A, B: sp.simplify( 000092 | 2 * riem(pairs[A][0], pairs[A][1], pairs[B][0], pairs[B][1]))) 000093 | N2 = sp.simplify(R2f * g2f.inv()) 000094 | check("MAX-SYMMETRIC EIGENVALUES [E] (her Appendix-B flattening verified): on a " 000095 | "general diagonal 4d metric with R_munurhosigma = (R/12)(gg - gg), the " 000096 | "relative curvature operator R~ g~^-1 has eigenvalues EXACTLY {R (scalar); " 000097 | "R/4 x4 (Ricci block); R/6 x6 (Riemann on flattened 2-forms)} -- symbolic, " 000098 | "including the 6x6 two-form flattening", 000099 | N1 == sp.Rational(1, 4) * R * sp.eye(4) and 000100 | N2 == sp.Rational(1, 6) * R * sp.eye(6)) 000101 | 000102 | # 2. exact vacuum f(R): series 3 beta R + (17/24) beta^2 R^2 000103 | b = sp.Symbol('beta', positive=True) 000104 | fB = -(sp.log(1 - b * R) + 4 * sp.log(1 - b * R / 4) + 6 * sp.log(1 - b * R / 6)) 000105 | ser = sp.series(fB, R, 0, 3).removeO() 000106 | lin = ser.coeff(R, 1) 000107 | quad = ser.coeff(R, 2) 000108 | check("EXACT VACUUM f(R) [E]: L_B(R) = -[ln(1-bR) + 4 ln(1-bR/4) + 6 ln(1-bR/6)] " 000109 | "= %s R + %s R^2 + O(R^3) -- the linear 3 beta re-derives her Eq. (45) " 000110 | "(= the v473 pin), and 17/24 = (1/2)(1 + 1/4 + 1/6) is the EXACT tensorial " 000111 | "R^2 factor (v473 quoted the raw (c3/6)^2 'before tensorial factors')" 000112 | % (lin, quad), 000113 | lin == 3 * b and quad == sp.Rational(17, 24) * b ** 2) 000114 | 000115 | # 3. the raw entropic scalaron mass, two independent routes 000116 | beta_val = (c3 / 6) / (8 * pi * Mbar ** 2) # beta = beta' l_P^2, l_P^2 = 1/(8 pi Mbar^2) 000117 | # 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)] 000118 | m2 = sp.Symbol('m2', positive=True) 000119 | lp4 = (8 * pi * Mbar ** 2) ** 2 # 1/l_P^4 000120 | eh_match = sp.simplify(lp4 * 3 * beta_val - Mbar ** 2 / 2) 000121 | m2_A = sp.solve(sp.Eq(lp4 * sp.Rational(17, 24) * beta_val ** 2, 000122 | Mbar ** 2 / (12 * m2)), m2)[0] 000123 | # route B: f(R) scalaron m^2 = f'(0)/(3 f''(0)) in the same units 000124 | m2_B = sp.simplify((3 * beta_val) / (3 * 2 * sp.Rational(17, 24) * beta_val ** 2)) 000125 | m2_raw = sp.simplify(m2_A) 000126 | m_raw_num = float(sp.sqrt(m2_raw / Mbar ** 2)) 000127 | m2_tfpt = c3 ** 7 * Mbar ** 2 000128 | enhance = sp.simplify(m2_raw / m2_tfpt) 000129 | check("RAW ENTROPIC SCALARON [E] (interpretation 2 KILLED in naive form): with " 000130 | "the pinned beta' = c3/6 the EH normalisation matches identically (residual " 000131 | "%s) and BOTH routes give m_raw^2 = %s = 4608 pi^2/17 Mbar^2, i.e. m_raw = " 000132 | "%.1f Mbar ~ 1.3e20 GeV -- TRANS-PLANCKIAN, not light; the TFPT scalaron is " 000133 | "c3^{7/2} Mbar = 3.06e13 GeV; required enhancement EXACTLY m_raw^2/m_TFPT^2 " 000134 | "= %s = (72/17)(8 pi)^9 ~ %.3e in mass^2 (the kill-test number WITH " 000135 | "tensorial factors)" % (eh_match, m2_raw / Mbar ** 2, m_raw_num, 000136 | enhance, float(enhance)), 000137 | eh_match == 0 and 000138 | sp.simplify(m2_A - m2_B) == 0 and 000139 | sp.simplify(m2_raw - sp.Rational(4608, 17) * pi ** 2 * Mbar ** 2) == 0 and 000140 | sp.simplify(enhance - sp.Rational(72, 17) * (8 * pi) ** 9) == 0 and 000141 | abs(float(enhance) - 1.694e13) / 1.694e13 < 1e-3) 000142 | 000143 | # 4. domain: first log pole at R = 1/beta = 384 pi^2 Mbar^2 (trans-Planckian) 000144 | pole = sp.simplify(1 / beta_val) 000145 | check("DOMAIN [E]: positivity of I - beta R~ g~^-1 holds iff R < 1/beta = %s = " 000146 | "384 pi^2 Mbar^2 ~ %.0f Mbar^2 (then 4/beta, 6/beta) -- a TRANS-PLANCKIAN " 000147 | "curvature bound: the raw entropic action is well-defined at all " 000148 | "sub-Planckian curvatures; the obstruction is the MASS, not the domain" 000149 | % (pole, float(pole / Mbar ** 2)), 000150 | sp.simplify(pole - 384 * pi ** 2 * Mbar ** 2) == 0) 000151 | 000152 | # 5. Lorentzian-positivity witness (warm-up sector G = g + alpha dphi dphi) 000153 | al, v = sp.symbols('alpha v', positive=True) 000154 | gL = sp.diag(-1, 1, 1, 1) 000155 | for name, dphi, expect in ( 000156 | ("timelike", sp.Matrix([v, 0, 0, 0]), 1 - al * v ** 2), 000157 | ("spacelike", sp.Matrix([0, v, 0, 0]), 1 + al * v ** 2)): 000158 | M = dphi * dphi.T 000159 | ev = sorted(((gL + al * M) * gL.inv()).eigenvals().items(), 000160 | key=lambda kv: sp.default_sort_key(kv[0])) 000161 | if name == "timelike": 000162 | ev_time = dict(ev) 000163 | else: 000164 | ev_space = dict(ev) 000165 | gE = sp.eye(4) 000166 | ME = sp.Matrix([v, 0, 0, 0]) * sp.Matrix([v, 0, 0, 0]).T 000167 | ev_eucl = ((gE + al * ME) * gE.inv()).eigenvals() 000168 | check("LORENTZIAN-POSITIVITY WITNESS [E]: on Minkowski a TIMELIKE gradient gives " 000169 | "G g^-1 eigenvalues {1 - alpha v^2, 1} -- NONPOSITIVE for v^2 >= 1/alpha, " 000170 | "so ln(G g^-1) is undefined on an open field region; spacelike gives " 000171 | "{1 + alpha v^2, 1} and Euclidean {1 + alpha v^2, 1}, positive for ALL v " 000172 | "-- the v473 caveat exhibited: Lorentzian positivity is a genuine " 000173 | "restriction, the OS/reflection-positive route is sturdier", 000174 | ev_time.get(1 - al * v ** 2, 0) == 1 and ev_time.get(1, 0) == 3 and 000175 | ev_space.get(1 + al * v ** 2, 0) == 1 and ev_space.get(1, 0) == 3 and 000176 | ev_eucl.get(1 + al * v ** 2, 0) == 1 and ev_eucl.get(1, 0) == 3) 000177 | 000178 | # 6. verdict 000179 | check("VERDICT [C]: of v473's three interpretations, (1) SURVIVES (entropic " 000180 | "action = Einstein + Lambda level: beta' pinned v473, Lambda channel " 000181 | "closed v473/v474), (2) is DEAD in naive form (raw G-trace mode " 000182 | "trans-Planckian, this module), (3) KMS-spectral renormalisation is the " 000183 | "ONLY surviving route to the R^2 sector [C]. Nothing closes; no marker " 000184 | "moves; v359's equation-of-state typing stays [O]", True) 000185 | 000186 | return summary("v475 GRAV.ENTROPIC.SCALARON.01: the R^2 kill test EXECUTED -- [E] " 000187 | "max-symmetric eigenvalues {R, R/4 x4, R/6 x6} (Appendix-B flattening " 000188 | "verified), exact vacuum f(R) with series 3bR + (17/24)b^2R^2 (the 3 " 000189 | "re-derived, 17/24 the exact tensorial factor), raw scalaron m^2 = " 000190 | "4608 pi^2/17 Mbar^2 (m ~ 51.7 Mbar, TRANS-PLANCKIAN -- interpretation 2 " 000191 | "killed naive; required enhancement exactly (72/17)(8pi)^9 ~ 1.69e13), " 000192 | "domain bound 384 pi^2 Mbar^2 (not the problem), Lorentzian timelike " 000193 | "witness 1 - alpha v^2 <= 0; [C] KMS renormalisation the only surviving " 000194 | "R^2 route; [O] unchanged") 000195 | 000196 | 000197 | if __name__ == "__main__": 000198 | raise SystemExit(1 if run() else 0)