"""Exact dispersion of the kappa = 1 Kuhn lattice along the paired direction, from Appendix F, with checks of Theorem 5.5. Paper: "Linearized Einstein gravity from Lorentzian Regge calculus on the Kuhn lattice, and the divergence of its one-loop graviton limit", J. Washburn (2026), https://doi.org/10.5281/zenodo.23764491. Section, theorem and appendix numbers below are the paper's. What this program does Along the paired covector u = (z+t, z+t, z-t, z-t) the spatial wave number is |k| = 2t and the frequency is omega = 2z/sqrt3 (Section 5.3); the light cone is z = sqrt3 t. With U = sin^2(t/2) and V = sin^2(z/2) the cross mode is the exact curve G_x(U, V) = -6UV + 3U + 4V^2 - V = 0 and the plus mode lies on the degree-six curve G_+(U, V) = 0 printed in Appendix F. The program follows, in 40-digit arithmetic, the branch of each curve that leaves the origin along the light cone, and checks and prints: * the limits of (v_ph - 1)/|k|^2: -59/24 (plus) and 5/24 (cross), and of (v_g - 1)/|k|^2: -59/8 and 5/8 (Theorem 5.5); * the windows of Theorem 5.5, |v_ph - 1 - c|k|^2| < (25/16)|k|^3, for 0 < t <= 2e-4 near the cone; * where each branch ends: the cross curve turns back where the discriminant of G_x vanishes, U = (sqrt2 - 1)^2/6, z = pi/4, |k| = 0.6797; the plus curve reaches omega = 0 where G_+(U, 0) = -2U(4U^2 - 10U + 3) vanishes, |k| = 2.5264. With --json FILE.json it writes the samples drawn in the figure "Exact dispersion along the paired direction" on the gravity page: the plus branch on a 0.005 grid in |k| and both roots of G_x. Requires Python 3.9+ and mpmath. Run: python3 paired_dispersion.py or python3 paired_dispersion.py --json branches.json License: MIT. Copyright (c) 2026 Recognition Physics Institute. """ import argparse import json import sys from mpmath import asin, diff, mp, mpf, pi, polyroots, polyval, sin, sqrt mp.dps = 40 def g_cross(U): """G_x as a polynomial in V, highest degree first: 4V^2 - (6U + 1)V + 3U.""" return [mpf(4), -(6 * U + 1), 3 * U] def g_plus(U): """G_+ of Appendix F as a polynomial in V, highest degree first.""" return [mpf(1024), -512 * U - 2816, -512 * U**2 + 1792 * U + 2400, 128 * U**3 + 832 * U**2 - 1584 * U - 648, -192 * U**3 - 432 * U**2 + 280 * U + 44, 80 * U**3 + 88 * U**2 + 20 * U + 2, -8 * U**3 + 20 * U**2 - 6 * U] def group_speed(poly, t, z): """Appendix F: v_g = -G_U sin t / (sqrt3 G_V sin z), with U = sin^2(t/2), V = sin^2(z/2).""" U, V = sin(t / 2) ** 2, sin(z / 2) ** 2 G = lambda u, v: polyval(poly(u), v) return -diff(lambda u: G(u, V), U) * sin(t) / (sqrt(3) * diff(lambda v: G(U, v), V) * sin(z)) def _real_roots(coeffs): roots = polyroots(coeffs, maxsteps=200, extraprec=60) tiny = mpf(10) ** -25 return [r.real for r in roots if abs(r.imag) < tiny and -tiny <= r.real <= 1 + tiny] def trace(poly, eta1, t_grid): """Follow the root V(t) that starts on the light cone as V = 3U(1 + eta1 U); stop where it turns back or ends.""" out, prev = [], None for t in t_grid: U = sin(t / 2) ** 2 guess = 3 * U * (1 + eta1 * U) if prev is None else prev roots = _real_roots(poly(U)) if not roots: break V = min(roots, key=lambda r: abs(r - guess)) if prev is not None and abs(V - prev) > mpf('0.05'): break prev = V out.append((2 * t, 4 * asin(sqrt(V)) / sqrt(3))) # (|k|, omega) = (2t, 2z/sqrt3) return out def cross_closed_form(n=400): """Both roots of G_x for U in [0, U_c], U_c = (sqrt2 - 1)^2/6 where the discriminant 36U^2 - 36U + 1 vanishes.""" Uc = (sqrt(2) - 1) ** 2 / 6 tc = 2 * asin(sqrt(Uc)) lower, upper = [], [] for i in range(n + 1): t = tc * i / n U = sin(t / 2) ** 2 D = max(36 * U * U - 36 * U + 1, mpf(0)) Vm = 6 * U / ((6 * U + 1) + sqrt(D)) # rationalized lower root: no cancellation near the cone Vp = ((6 * U + 1) + sqrt(D)) / 8 lower.append((2 * t, 4 * asin(sqrt(Vm)) / sqrt(3))) upper.append((2 * t, 4 * asin(sqrt(Vp)) / sqrt(3))) return lower, upper[::-1], (2 * tc, pi / (2 * sqrt(3))) def plus_static_zero(): """|k| where the plus branch meets omega = 0: G_+(U, 0) = -2U(4U^2 - 10U + 3), U = (5 - sqrt13)/4.""" return 4 * asin(sqrt((5 - sqrt(13)) / 4)) def figure_data(step=mpf('0.005')): grid = [step * i / 2 for i in range(1, int(mpf('2.53') / step) + 1)] # t = |k|/2 k_static = plus_static_zero() plus = [(mpf(0), mpf(0))] + [p for p in trace(g_plus, mpf(-238) / 3, grid) if p[0] < k_static] + [(k_static, mpf(0))] lower, upper, turn = cross_closed_form() return plus, lower, upper, turn def run_checks(): ok = True small = [mpf(10) ** (-6 + 5 * i / 200) for i in range(200)] # t from 1e-6 to 1e-1 for name, poly, eta1, c_ph, c_g in (('plus', g_plus, mpf(-238) / 3, mpf(-59) / 24, mpf(-59) / 8), ('cross', g_cross, mpf(6), mpf(5) / 24, mpf(5) / 8)): branch = trace(poly, eta1, small) k0, w0 = branch[0] lim_ph = (w0 / k0 - 1) / k0**2 lim_g = (group_speed(poly, k0 / 2, w0 * sqrt(3) / 2) - 1) / k0**2 window = max(abs(w / k - 1 - c_ph * k**2) / (mpf(25) / 16 * k**3) for k, w in branch if k / 2 <= mpf('2e-4')) good = abs(lim_ph - c_ph) < mpf('1e-6') and abs(lim_g - c_g) < mpf('1e-4') and window < 1 ok &= good print(f" [{'PASS' if good else 'FAIL'}] {name}: (v_ph - 1)/|k|^2 -> {float(lim_ph):.9f} (Theorem 5.5: {float(c_ph):.9f}); " f"(v_g - 1)/|k|^2 -> {float(lim_g):.6f} ({float(c_g):.6f}); window ratio {float(window):.2e} < 1") plus, lower, upper, turn = figure_data() print(f' [INFO] cross branch turns back at |k| = {float(turn[0]):.6f}, omega = {float(turn[1]):.6f} = pi/(2 sqrt3), z = pi/4') print(f' [INFO] plus branch reaches omega = 0 at |k| = {float(plus_static_zero()):.6f}; ' f'its highest frequency on the 0.005 grid is {float(max(w for _, w in plus)):.4f}') sample = [(k, w) for k, w in plus if abs(k - mpf('0.4')) < mpf('1e-9')][0] print(f' [INFO] at |k| = 0.4: plus omega = {float(sample[1]):.4f}, phase speed {float(sample[1] / sample[0]):.4f}') return ok, (plus, lower, upper, turn) if __name__ == '__main__': parser = argparse.ArgumentParser(description='Exact paired-direction dispersion from Appendix F, checked against Theorem 5.5.') parser.add_argument('--json', metavar='FILE.json', help='write the samples drawn in the figure') args = parser.parse_args() print('Theorem 5.5 and Appendix F: the paired direction') good, (plus, lower, upper, turn) = run_checks() if args.json: data = {'variables': '|k| = 2t, omega = 2z/sqrt3, lattice units', 'plus': [[float(k), float(w)] for k, w in plus], 'cross_near_cone': [[float(k), float(w)] for k, w in lower], 'cross_returning': [[float(k), float(w)] for k, w in upper], 'cross_turning_point': [float(turn[0]), float(turn[1])]} with open(args.json, 'w') as fh: json.dump(data, fh) print(f' wrote {args.json}') print('ALL CHECKS PASS' if good else 'SOME CHECKS FAILED') sys.exit(0 if good else 1)