"""v478 -- GRAV.ENTROPIC.LEGS.01: first computable steps on the TWO remaining research legs of the entropic-action bridge (v473-v477 round): (A) the AP2 CONTINUUM leg -- the state-side modular Hamiltonian of the compressed critical state flows to the CHM/Bisognano-Wichmann geometric form, i.e. exactly the object TFPT's Einstein derivation consumes (v323/v358); and (B) the MEASURE leg -- the v477 one-moment condition reduced to a single-scale corollary with an exact closed form for the KMS time t0, and the nearby integer explicitly DECLINED per the anti-numerology discipline (v354/v355). Both legs stay [O]; nothing closes. Leg A (AP2 continuum). v476 forced the state-side reading: Delta_Sigma is built FROM the compressed relative metric/covariance. The continuum question is whether that object is the GEOMETRIC modular operator (CHM/BW boost). On the exactly solvable critical chain (infinite-volume sinc kernel, high precision) the answer is exhibited: [N] 1. CALABRESE-CARDY c = 1. The compressed interval state has S(L) = (c/3) ln L + k with c_est = 1.00089, 1.00038, 1.00021 for the pairs (8,16), (12,24), (16,32) -- monotone convergence to the Dirac value c = 1 (= 2 Majoranas x 1/2, the v450 reading); a gapped control chain gives c_est ~ 2e-4 -> 6e-8 (area law). The compressed state carries the conformal/BW structure, quantitatively. [N] 2. CHM PARABOLA. The nearest-neighbour couplings of the Peschel modular Hamiltonian K = ln((1-C_A)/C_A) organise on the CHM weight beta(x) = x(L-x)/L with correlation 0.866 -> 0.967 -> 0.986 (L = 8, 16, 24, monotone), and the rescaled mid-plateau drifts by shrinking steps -- the state-side Delta_Sigma flows to the geometric CHM/BW form in the continuum limit (constants deliberately NOT asserted: the lattice EH is only asymptotically local). [N] 3. QUASI-LOCALITY STRUCTURE. At half filling the EVEN-range bands of K vanish identically (particle-hole structure; r = 2 band = 0 to precision) and the odd tails are small (mean |K_{i,i+3}| / mean |K_{i,i+1}| ~ 0.018) -- the modular Hamiltonian is quasi-local with a parabolic envelope, the lattice shadow of the local boost. Leg-A verdict [C]: the state-side reading (v476) meets TFPT's geometric modular input (CHM ball, v323/v358) in the continuum limit; the continuum AP2 statement itself stays [O]. Leg B (the measure). v477 reduced the R^2 route to mu2/mu1^2 = (72/17)(8 pi)^9. The sharpest special case: [E] 4. SINGLE-SCALE COROLLARY. For a single-scale measure dmu = delta(t - t0) dt the moments give mu2/mu1^2 = e^{t0} EXACTLY (symbolic), so the condition forces t0 = ln(72/17) + 9 ln(8 pi) = 30.461...: ONE dimensionless KMS time with an exact closed form. DECLINED explicitly: t0 is NOT h(E8) = 30 (difference 0.461 -- the near-miss is recorded and rejected per the v354/v355 no-free-pattern rule); no TFPT atom equals t0; deriving the measure from the entropic structure stays OPEN. [C/O] 5. Scope: leg A is a lattice exhibit (not a continuum proof); leg B is a parametrisation (not a derivation). No status marker moves. NET TYPING: [N] the three lattice exhibits (high-precision mpmath, 50 digits); [E] the single-scale corollary (symbolic); [C] the meeting-point reading; [O] both legs. Extends GRAV.ENTROPIC.COMPRESS.01 (v476) / .SCALEFLOW.01 (v477). Numerical + one exact corollary; Python-only by nature (mpmath eigensolver), the exact corollary flagged in the Wolfram README. """ import mpmath as mp import sympy as sp from tfpt_constants import check, summary, reset mp.mp.dps = 50 def sinc_cov(L): """Infinite-volume half-filled critical covariance, compressed to L sites.""" M = mp.matrix(L, L) for i in range(L): for j in range(L): if i == j: M[i, j] = mp.mpf('0.5') else: d = i - j M[i, j] = mp.sin(mp.pi * d / 2) / (mp.pi * d) return M def entropy(C): w, _ = mp.eigsy(C) S = mp.mpf(0) for k in range(C.rows): lam = w[k] if 0 < lam < 1: S -= lam * mp.log(lam) + (1 - lam) * mp.log(1 - lam) return S def modular_h(C): w, V = mp.eigsy(C) return V * mp.diag([mp.log((1 - w[k]) / w[k]) for k in range(C.rows)]) * V.T def corr(xs, ys): n = len(xs) mx, my = sum(xs) / n, sum(ys) / n cov = sum((xs[i] - mx) * (ys[i] - my) for i in range(n)) vx = mp.sqrt(sum((x - mx) ** 2 for x in xs)) vy = mp.sqrt(sum((y - my) ** 2 for y in ys)) return cov / (vx * vy) def run(): reset() print("v478 GRAV.ENTROPIC.LEGS.01: the two remaining legs -- AP2 continuum exhibit + the measure corollary") # 1. Calabrese-Cardy c = 1 (critical) vs c = 0 (gapped control) S = {L: entropy(sinc_cov(L)) for L in (8, 12, 16, 24, 32)} c_est = [float(3 * (S[b] - S[a]) / mp.log(mp.mpf(b) / a)) for a, b in ((8, 16), (12, 24), (16, 32))] import numpy as np Ng = 240 h = np.zeros((Ng, Ng)) for i in range(Ng - 1): h[i, i + 1] = h[i + 1, i] = -1.0 for i in range(Ng): h[i, i] = (-1.0) ** i wg, vg = np.linalg.eigh(h) Cg = (vg[:, wg < 0] @ vg[:, wg < 0].T) def s_gap(L): off = (Ng - L) // 2 lam = np.linalg.eigvalsh(Cg[off:off + L, off:off + L]) lam = lam[(lam > 1e-12) & (lam < 1 - 1e-12)] return float(-(lam * np.log(lam) + (1 - lam) * np.log(1 - lam)).sum()) c_gap = 3 * (s_gap(32) - s_gap(16)) / float(mp.log(2)) check("CALABRESE-CARDY c = 1 [N] (leg A): the compressed critical interval has " "S(L) = (c/3) ln L + k with c_est = %s (pairs (8,16),(12,24),(16,32)) -- " "monotone convergence to the Dirac c = 1 (= 2 Majoranas x 1/2, the v450 " "reading); gapped control c_est = %.1e (area law) -- the compressed state " "carries the conformal/BW structure quantitatively" % (["%.5f" % c for c in c_est], c_gap), all(abs(c - 1) < 2e-3 for c in c_est) and abs(c_est[2] - 1) < abs(c_est[1] - 1) < abs(c_est[0] - 1) and abs(c_gap) < 1e-4) # 2. CHM parabola: NN couplings vs beta(x) = x(L-x) corrs, plateaus = [], [] for L in (8, 16, 24): K = modular_h(sinc_cov(L)) nn = [-K[i, i + 1] for i in range(L - 1)] xs = [i + mp.mpf('0.5') for i in range(L - 1)] par = [x * (L - x) for x in xs] corrs.append(float(corr(nn, par))) mid = (L - 1) // 2 plateaus.append(float(nn[mid] * L / par[mid])) drift = [abs(plateaus[i + 1] - plateaus[i]) for i in range(2)] check("CHM PARABOLA [N] (leg A): the NN couplings of K = ln((1-C_A)/C_A) " "organise on the CHM weight beta(x) = x(L-x)/L with correlation %s " "(L = 8, 16, 24 -- monotone toward 1), mid-plateau %s with shrinking " "drift %s -- the state-side Delta_Sigma flows to the geometric CHM/BW " "form that TFPT's Einstein derivation consumes (v323/v358); constants " "deliberately not asserted (the lattice EH is only asymptotically local)" % (["%.4f" % c for c in corrs], ["%.4f" % p for p in plateaus], ["%.1e" % d for d in drift]), corrs[0] < corrs[1] < corrs[2] and corrs[2] > 0.98 and drift[1] < drift[0]) # 3. quasi-locality: even bands vanish, odd tails small + parabolic L = 24 K = modular_h(sinc_cov(L)) band = {r: [K[i, i + r] for i in range(L - r)] for r in (1, 2, 3)} even_max = max(abs(v) for v in band[2]) tail_ratio = (sum(abs(v) for v in band[3]) / len(band[3])) / \ (sum(abs(v) for v in band[1]) / len(band[1])) xs3 = [i + mp.mpf('1.5') for i in range(L - 3)] corr3 = float(corr([-v for v in band[3]], [x * (L - x) for x in xs3])) check("QUASI-LOCALITY [N] (leg A): at half filling the EVEN-range bands of K " "vanish identically (max |K_{i,i+2}| = %.1e, particle-hole structure), " "the odd tail is small (mean ratio r=3/r=1 = %.4f) and itself " "parabola-enveloped (corr %.3f) -- the modular Hamiltonian is quasi-local " "with the boost envelope, the lattice shadow of the local BW form" % (float(even_max), float(tail_ratio), corr3), float(even_max) < 1e-30 and float(tail_ratio) < 0.05 and corr3 > 0.9) # 4. single-scale corollary (leg B, exact) t0s, pi_s = sp.symbols('t0 pi_s', positive=True) mu1 = t0s * sp.exp(-t0s) mu2 = t0s ** 2 * sp.exp(-t0s) ratio = sp.simplify(mu2 / mu1 ** 2) target = sp.Rational(72, 17) * (8 * sp.pi) ** 9 t0_val = sp.log(target) t0_num = float(t0_val) check("SINGLE-SCALE COROLLARY [E] (leg B): a single-scale measure dmu = " "delta(t - t0) dt has mu2/mu1^2 = %s = e^{t0} exactly, so the v477 " "moment condition forces t0 = ln(72/17) + 9 ln(8 pi) = %.6f -- ONE " "dimensionless KMS time, exact closed form; DECLINED: t0 is NOT " "h(E8) = 30 (difference %.3f; near-miss rejected per the v354/v355 " "no-free-pattern rule); no TFPT atom equals t0 -- deriving the measure " "from the entropic structure stays OPEN" % (ratio, t0_num, t0_num - 30), ratio == sp.exp(t0s) and sp.simplify(t0_val - sp.log(sp.Rational(72, 17)) - 9 * sp.log(8 * sp.pi)) == 0 and abs(t0_num - 30.4610) < 1e-3 and abs(t0_num - 30) > 0.4) # 5. scope check("SCOPE [C/O]: leg A is a lattice exhibit (the state-side modular data " "flows to the CHM/BW geometric form), NOT a continuum proof; leg B is a " "parametrisation (one exact KMS time), NOT a derivation. Both legs stay " "[O]; no status marker moves", True) return summary("v478 GRAV.ENTROPIC.LEGS.01: the two remaining legs, first computable " "steps -- leg A (AP2 continuum): compressed critical state has c_est -> 1 " "(1.0009/1.0004/1.0002; gapped control ~0), EH NN couplings on the CHM " "parabola (corr 0.87 -> 0.99), even bands exactly zero, odd tails 2% -- the " "state-side Delta_Sigma flows to the CHM/BW form TFPT's Einstein derivation " "consumes; leg B (measure): single-scale corollary mu2/mu1^2 = e^{t0} forces " "t0 = ln(72/17) + 9 ln(8pi) = 30.461 exactly, near-miss to 30 DECLINED; [C] " "meeting point; [O] both legs open, no marker moves") if __name__ == "__main__": raise SystemExit(1 if run() else 0)