"""v473 -- GRAV.ENTROPIC.ACTION.01: the entropic-action bridge -- Bianconi's "Gravity from entropy" (Phys. Rev. D 111, 066001 (2025); arXiv:2408.14391) as an EXTERNAL CANDIDATE for the local volume action behind TFPT's already-closed, parameter-free Einstein equation (v358/v359). This module quantifies the bridge; it does NOT adopt the external action as TFPT structure and closes NO gate. TFPT's field equation G_ab + Lambda g_ab = (1/c3) T_ab is derived thermodynamically (entanglement equilibrium, v359) -- honestly typed "equation of state, not a from-action quantisation" [O]. Bianconi's entropic action S_B = -Tr_F ln(G~ g~^-1) (the quantum relative entropy between the spacetime metric and the matter-induced metric, Araki-related via Delta^{1/2}_{G~,g} = G~ g~^-1, her Appendix C) is a local covariant action whose variation yields modified Einstein equations that reduce to Einstein + emergent nonnegative Lambda at low coupling -- exactly the missing variational layer, IF the bridge identities hold. What is checkable today: [E] 1. CARRIER HODGE COUNT. Bianconi's matter fiber is Omega^{0,1,2}; on the five-slot carrier C^5 (NOT on 4d spacetime, where the fiber is 1+4+6=11) the count is 1+5+10 = 16 = 2^(g_car-1) = dim S+_{D5}, and the Hodge star L^1 ~= L^4 folds it onto the even exterior algebra L^even C^5 -- the carrier half-spinor's own Pascal row (v2/v44). Vector-space identity ONLY; Clifford/Dirac/leaf/Calderon compatibility stays open (check 7). [E] 2. COEFFICIENT MATCH (the one quantitative pin). Bianconi's weak-coupling Lagrangian is L = 3 beta R - alpha L_matter + ... (her Eq. (45); the 3 = the three form blocks 0,1,2). Matching 3 beta'/l_P^2 (l_P^2 = G) to the TFPT EH normalisation 1/(16 pi G) = c3/(2G) fixes her free constant EXACTLY: beta'_B = c3/6 = 1/(48 pi). Structural echo: the 6 = 3 blocks x 2 (EH 1/2) = p2 = |R+(A3)| -- recorded as consistency, NOT a new proof. [E] 3. ENTROPY POTENTIAL = QUADRATIC LAMBDA. Her emergent constant is Lambda_G = (1/2 beta) Tr_F(G - I - ln G) with s(x) = x - 1 - ln x >= 0, s(1) = 0, s''(1) = 1 (exact series) => Lambda_G = (1/4 beta) Tr eps^2 + O(eps^3): Lambda >= 0 and QUADRATIC around the entropic fixed point. [E] 4. THE LAMBDA CHANNEL CLOSES ALGEBRAICALLY. With the TFPT displacement eps_* = e^{-1/alpha} Q the quadratic entropy gives Lambda ~ e^{-2/alpha}, the SAME suppression as the closed TFPT branch rho_Lambda/Mbar^4 = (3/4 pi^2) e^{-2/alpha} (v60); matching prefactors with beta' = c3/6 and l_P^2 Mbar^2 = c3 yields the EXACT normalisation target Tr_F Q^2 = 32 c3^4 = 1/(128 pi^4) = (2/3) delta_top -- a closed algebraic target for the two-form block (binom(5,2) = 10 = the pair sector), [C] until an explicit Q is constructed. [E] 5. LOG = ALL-SCALE HEAT KERNEL (Frullani). -ln A = int_0^inf dt/t (e^{-tA} - e^{-t}) -- her log action is an all-scale-integrated heat-kernel action, the correct genre to compare with TFPT's relative spectral action S_rel,chi (KMS cutoff); verified numerically to 1e-12. [E] 6. R^2 KILL TEST QUANTIFIED (the honest wall). The raw log expansion -Tr ln(I+X) = -Tr X + (1/2) Tr X^2 - ... carries a curvature-squared coefficient of order (beta')^2 = (c3/6)^2; TFPT's Starobinsky sector needs Mbar^2/(12 M_scal^2) = 1/(12 c3^7) (M_scal^2 = c3^7 Mbar^2, v36). The EXACT mismatch is 1/(12 c3^7) / (c3/6)^2 = 3 c3^{-9} = 3 (8 pi)^9 ~ 1.20e13 -- THIRTEEN orders of magnitude. A direct identification of the raw entropic action with the TFPT R^2 sector is therefore FALSE without a mechanism (scalaron = light trace mode of the G-field generating c3^{-9}, or KMS-spectral renormalisation) -- the pre-registered kill test. [C] 7. THE COMPRESSION CONJECTURE (recorded, not proven). The bridge proper is P_Sigma (G~ g~^-1) P_Sigma = Delta_Sigma^{1/2} -- the physical Calderon compression of her relative metric operator equals the TFPT seam modular operator; if proven, S_B is a local volume representation of the TFPT relative spectral action. Open with the operator-level Hodge items (Clifford action, Dirac intertwining, leaf involution, Calderon polarisation). Lorentzian positivity of G~ g~^-1 is NOT automatic (timelike gradients); TFPT's OS/reflection-positive route is the sturdier construction -- recorded. [O] 8. NOT CLOSED. The equation-of-state status of v359 stays [O]; adopting S_B would close it ONLY after (7) and the R^2 mechanism (6). No status marker moves; external reference, zero TFPT knobs added. NET TYPING: [E] the five exact identities + the quantified gap; [C] the compression conjecture, the eps_* assignment and the Q-target; [O] the variational-completion gate itself. External anchor: G. Bianconi, "Gravity from entropy", Phys. Rev. D 111, 066001 (2025), arXiv:2408.14391 (DOI 10.1103/PhysRevD.111.066001). Mixed exact (sympy, Wolfram-mirrored) + one numerical quadrature (mpmath). """ import mpmath as mp import sympy as sp from tfpt_constants import check, summary, reset, g_car, N_fam pi = sp.pi c3 = sp.Rational(1, 8) / pi def run(): reset() print("v473 GRAV.ENTROPIC.ACTION.01: the Bianconi entropic-action bridge (PRD 111, 066001 (2025))") # 1. carrier Hodge count: Omega^{<=2}(C^5) has dim 1+5+10 = 16 = dim S+_{D5} dims = [sp.binomial(g_car, k) for k in (0, 1, 2)] dim_splus = 2 ** (g_car - 1) even = [sp.binomial(g_car, k) for k in (0, 2, 4)] fiber_4d = sum(sp.binomial(4, k) for k in (0, 1, 2)) check("CARRIER HODGE COUNT [E]: dim Omega^{0,1,2}(C^5) = %s = 16 = 2^(g_car-1) = " "dim S+_{D5}; Hodge star L^1 ~= L^4 folds onto L^even C^5 (dims %s); " "two-form block binom(5,2) = 10; on 4d spacetime the fiber is only " "1+4+6 = %d -- the 16 REQUIRES the five-slot carrier (vector-space " "identity only)" % ("+".join(str(d) for d in dims), even, fiber_4d), sum(dims) == dim_splus == sum(even) == 16 and dims[2] == 10 and fiber_4d == 11) # 2. coefficient match: 3 beta' / G = 1/(16 pi G) => beta' = c3/6 = 1/(48 pi) beta_p = sp.solve(sp.Eq(3 * sp.Symbol('bp'), sp.Rational(1, 16) / pi), sp.Symbol('bp'))[0] p2_A3 = sp.binomial(4, 2) # |R+(A3)| = 4*3/2 = 6 positive roots check("COEFFICIENT MATCH [E]: 3 beta'_B = 1/(16 pi) = c3/2 => beta'_B = %s = " "c3/6 = 1/(48 pi); the 6 = 3 form blocks x 2 (EH 1/2) = p2 = |R+(A3)| = %d " "(structural echo, not a proof)" % (beta_p, p2_A3), sp.simplify(beta_p - c3 / 6) == 0 and sp.simplify(beta_p - sp.Rational(1, 48) / pi) == 0 and p2_A3 == 6) # 3. entropy potential s(x) = x - 1 - ln x: nonnegative, quadratic minimum at x=1 x, eps = sp.symbols('x eps', positive=True) s = x - 1 - sp.log(x) series = sp.series(s.subs(x, 1 + eps), eps, 0, 4).removeO() smin = sp.solve(sp.diff(s, x), x) check("ENTROPY POTENTIAL [E]: s(x) = x - 1 - ln x has unique stationary point " "x = %s with s = 0, s'' = 1, series s(1+eps) = %s => Lambda_G = " "(1/4 beta) Tr eps^2 + O(eps^3) >= 0, QUADRATIC around the fixed point" % (smin, series), smin == [1] and s.subs(x, 1) == 0 and sp.diff(s, x, 2).subs(x, 1) == 1 and sp.simplify(series - (eps ** 2 / 2 - eps ** 3 / 3)) == 0) # 4. Lambda channel: (1/4 beta) e^{-2/a} TrQ^2 = (3/4 pi^2) e^{-2/a} Mbar^2 # with beta = beta' l_P^2, l_P^2 Mbar^2 = G Mbar^2 = 1/(8 pi) = c3 TrQ2 = sp.Symbol('TrQ2') sol = sp.solve(sp.Eq(TrQ2 / (4 * (c3 / 6) * c3), sp.Rational(3, 4) / pi ** 2), TrQ2)[0] delta_top = 48 * c3 ** 4 check("LAMBDA CHANNEL [E->C]: eps_* = e^{-1/alpha} Q makes Lambda_G ~ e^{-2/alpha} " "= the closed v60 branch (3/4 pi^2) e^{-2/alpha}; prefactor match " "(beta' = c3/6, l_P^2 Mbar^2 = c3) => Tr_F Q^2 = %s = 32 c3^4 = " "1/(128 pi^4) = (2/3) delta_top -- exact algebraic target for the " "two-form (pair) block, Q itself unconstructed [C]" % sol, sp.simplify(sol - 32 * c3 ** 4) == 0 and sp.simplify(sol - sp.Rational(1, 128) / pi ** 4) == 0 and sp.simplify(sol - sp.Rational(2, 3) * delta_top) == 0) # 5. Frullani: -ln a = int_0^inf dt/t (e^{-t a} - e^{-t}) (all-scale heat kernel) mp.mp.dps = 30 frul = lambda a: mp.quad(lambda t: (mp.e ** (-t * a) - mp.e ** (-t)) / t, [0, mp.inf]) err = max(abs(frul(a) + mp.log(a)) for a in (mp.mpf('0.4'), mp.mpf('2.5'))) check("LOG = ALL-SCALE HEAT KERNEL [E] (Frullani): -ln A = int dt/t (e^{-tA} - " "e^{-t}), max residual %.1e -- the entropic action is an all-scale " "heat-kernel action, the right genre against S_rel,chi (KMS cutoff)" % err, err < mp.mpf('1e-12')) # 6. R^2 kill test: raw (c3/6)^2 vs TFPT 1/(12 c3^7); exact gap 3 c3^-9 = 3 (8 pi)^9 raw = (c3 / 6) ** 2 staro = 1 / (12 * c3 ** 7) gap = sp.simplify(staro / raw) check("R^2 KILL TEST [E/X-guard]: raw log-expansion coefficient (c3/6)^2 vs " "TFPT Starobinsky Mbar^2/(12 M_scal^2) = 1/(12 c3^7) (M_scal^2 = c3^7 " "Mbar^2, v36) -- EXACT gap %s = 3 (8 pi)^9 ~ %.3e (13 orders): a direct " "identification of the raw entropic action with the TFPT R^2 sector is " "FALSE without a mechanism (light G-trace mode generating c3^{-9}, or " "KMS-spectral renormalisation)" % (gap, float(gap)), sp.simplify(gap - 3 * (8 * pi) ** 9) == 0 and abs(float(gap) - 1.2002728758784e13) / 1.2e13 < 1e-3) # 7. the compression conjecture (recorded) check("COMPRESSION CONJECTURE [C] (recorded, NOT proven): P_Sigma (G~ g~^-1) " "P_Sigma = Delta_Sigma^{1/2} -- the Calderon compression of the relative " "metric operator = the TFPT seam modular operator; plus the operator-level " "Hodge items (Clifford, Dirac, leaf involution, Calderon polarisation); " "Lorentzian positivity NOT automatic -- TFPT's OS route is the sturdier " "construction", True) # 8. nothing closes check("NOT CLOSED [O]: v359's equation-of-state status stays [O]; adopting S_B " "would close it only after (7) + the R^2 mechanism (6); no marker moves, " "zero TFPT knobs added (external anchor: Bianconi, PRD 111, 066001 (2025), " "arXiv:2408.14391)", True) return summary("v473 GRAV.ENTROPIC.ACTION.01: Bianconi entropic-action bridge -- [E] " "1+5+10=16=dim S+ (carrier-only; 4d fiber is 11), beta'_B = c3/6 = 1/(48 pi) " "pinned, Lambda_G quadratic-nonnegative with exact target Tr Q^2 = 32 c3^4, " "Frullani all-scale heat kernel, R^2 gap EXACTLY 3 (8 pi)^9 ~ 1.2e13; [C] " "compression conjecture P (G g^-1) P = Delta^1/2; [O] variational completion " "open, no gate closed") if __name__ == "__main__": raise SystemExit(1 if run() else 0)