"""From the Regge action to the corner couplings of the Kuhn lattice at kappa = 1. 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 * Builds the Regge action of one Kuhn simplex from Eq. (3): the areas A_h = (1/4)((l1 + l2 - l3)^2 - 4 l1 l2)^(1/2) of its ten timelike triangles, the dihedral angle theta_{sigma,h} between the two tetrahedra of the simplex that meet at h (measured in the spacelike plane normal to h, from the facet normals), and the number n_h of simplices that share h, which is p! r! m! (m + 1) for a triangle whose two chain steps advance p and r counters, m = 4 - p - r. * Appendix B. At the flat kappa = 1 lattice a simplex meets its triangles with two consecutive postings at the angles pi/6, 2pi/3 and pi/6, depending on where they sit in its chain; the six simplices around such a triangle still close up to 2pi. With each simplex taking its own flat angle theta0 as its share, S_sigma = sum_h A_h (theta0_{sigma,h} - theta_{sigma,h}), the Hessian of one simplex is exactly C/(16 sqrt3), with C the printed integer matrix (checked). With the equal share 2 pi/n_h the simplex Hessians differ from it by terms (2 pi/n_h - theta0) d^2 A_h on three pairs of edges, which cancel at every triangle, so the lattice action and the Bloch matrix B(k) are the same either way (checked at three waves). * Propositions 6.3, 6.4 and 7.6. A constant edge profile f changes the squared length of class d by f_d. The first-order shift of the zero at the corner pi e_c is delta_c(f) = v_c^dagger (d_f B)(pi e_c) v_c, which is the sum over the 24 simplices of a cell of the third derivative of the action along f, w, w, with w the corner mode v_c(d) = 2^(|d|-3) (d_c = 0) carried by the Bloch sign of each edge. The program computes it by finite differences in 40-digit arithmetic for the seven backgrounds of the gravity page and compares with the paper's exact values; it also recovers Proposition 6.4's coefficients (-288, -2592, -2592, 1728)/(16 * 1152 sqrt3) from the four profiles that depend only on |d|, and checks the sum rule sum_c delta_c(H) = -396/(1152 sqrt3) tr H of Proposition 6.3 for the strains among them. Requires Python 3.9+, numpy and mpmath. Run: python3 regge_action.py [--json shifts.json] License: MIT. Copyright (c) 2026 Recognition Physics Institute. """ import argparse import itertools import json import math import sys import mpmath as mp import numpy as np mp.mp.dps = 40 PAIRS = [(0, 1), (0, 2), (0, 3), (0, 4), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)] PIDX = {p: i for i, p in enumerate(PAIRS)} TRIS = list(itertools.combinations(range(5), 3)) # Appendix B: the cell Hessian at the flat kappa = 1 lattice is C/(16 sqrt3), edges in the order of PAIRS. C = [[2, -2, 1, 0, 0, 3, -2, -6, 2, 2], [-2, -4, 4, -1, 6, -6, 2, 3, -2, 2], [1, 4, 0, -2, -6, 0, 2, -3, 2, -2], [0, -1, -2, 2, 2, 2, -2, 2, -1, 0], [0, 6, -6, 2, -6, 6, -3, 0, 3, -6], [3, -6, 0, 2, 6, 0, 0, 6, -6, 3], [-2, 2, 2, -2, -3, 0, 0, -6, 4, 1], [-6, 3, -3, 2, 0, 6, -6, -6, 6, 0], [2, -2, 2, -1, 3, -6, 4, 6, -4, -2], [2, 2, -2, 0, -6, 3, 1, 0, -2, 2]] FLAT = [mp.mpf((j - i) * (1 - (j - i))) for i, j in PAIRS] # squared lengths |d|(1 - |d|) at kappa = 1 def hinge_count(tri): i, j, k = tri p, r = j - i, k - j m = 4 - p - r return math.factorial(p) * math.factorial(r) * math.factorial(m) * (m + 1) NH = [hinge_count(t) for t in TRIS] def sq(ell, i, j): return ell[PIDX[(min(i, j), max(i, j))]] def angles(ell): """Dihedral angles at the ten triangles, from the inward facet normals n_a = e^a, n_0 = -sum_a e^a (dual basis of the edge vectors from vertex 0, with the inner product given by the squared lengths).""" G = mp.matrix(4, 4) for a in range(1, 5): for b in range(1, 5): G[a - 1, b - 1] = (sq(ell, 0, a) + sq(ell, 0, b) - (0 if a == b else sq(ell, a, b))) / 2 Gi = G ** -1 N = mp.matrix(5, 5) for i in range(5): for j in range(5): if i and j: N[i, j] = Gi[i - 1, j - 1] elif i: N[i, j] = -sum(Gi[i - 1, b] for b in range(4)) elif j: N[i, j] = -sum(Gi[a, j - 1] for a in range(4)) else: N[i, j] = sum(Gi[a, b] for a in range(4) for b in range(4)) out = [] for tri in TRIS: i, j = [v for v in range(5) if v not in tri] out.append(mp.acos(-N[i, j] / mp.sqrt(N[i, i] * N[j, j]))) return out def areas(ell): out = [] for tri in TRIS: l1, l2, l3 = sq(ell, tri[0], tri[1]), sq(ell, tri[1], tri[2]), sq(ell, tri[0], tri[2]) out.append(mp.sqrt((l1 + l2 - l3) ** 2 - 4 * l1 * l2) / 4) return out THETA0 = angles(FLAT) def action(ell, share='flat'): th, A = angles(ell), areas(ell) if share == 'flat': return sum(A[t] * (THETA0[t] - th[t]) for t in range(10)) return sum(A[t] * (2 * mp.pi / NH[t] - th[t]) for t in range(10)) def hessian(share, h=mp.mpf('1e-10')): H = [[mp.mpf(0)] * 10 for _ in range(10)] S0 = action(FLAT, share) for a in range(10): for b in range(a, 10): def S(da, db): e = list(FLAT) e[a] += da e[b] += db return action(e, share) if a == b: v = (S(h, 0) - 2 * S0 + S(-h, 0)) / h ** 2 else: v = (S(h, h) - S(h, -h) - S(-h, h) + S(-h, -h)) / (4 * h * h) H[a][b] = H[b][a] = v return H def class_index(d): return int(sum(int(d[a]) << a for a in range(4))) - 1 SIMPLICES = [] # for each of the 24 simplices of a cell: (class index, start vertex) of each edge, in the order of PAIRS for perm in itertools.permutations(range(4)): verts = [np.zeros(4, dtype=int)] for a in perm: v = verts[-1].copy() v[a] = 1 verts.append(v) SIMPLICES.append([(class_index(verts[j] - verts[i]), verts[i].copy()) for i, j in PAIRS]) def bloch(Hs, k): """B(k)_{dd'} = sum over the simplices and their edge pairs of H_ff' exp(i k.(b_f' - b_f)) (Appendix B).""" M = np.zeros((15, 15), dtype=complex) for edges, H in zip(SIMPLICES, Hs): for f, (df, bf) in enumerate(edges): for g, (dg, bg) in enumerate(edges): M[df, dg] += H[f][g] * np.exp(1j * (k @ (bg - bf))) return M def corner_mode(c): v = [0.0] * 15 for m in range(1, 16): d = [(m >> a) & 1 for a in range(4)] if d[c] == 0: v[m - 1] = 2.0 ** (sum(d) - 3) return v def shift(profile, c, h=mp.mpf('1e-8'), tau=mp.mpf('1e-8')): """delta_c(f) = sum over simplices of d^3 S / dt du^2 at l0 + t f + u w, by central differences.""" v = corner_mode(c) total = mp.mpf(0) for edges in SIMPLICES: f = [mp.mpf(profile[d]) for d, _ in edges] w = [mp.mpf(v[d]) * (-1) ** int(b[c]) for d, b in edges] def S(t, u): return action([FLAT[e] + t * f[e] + u * w[e] for e in range(10)]) def g(t): return (S(t, h) - 2 * S(t, 0) + S(t, -h)) / h ** 2 total += (g(tau) - g(-tau)) / (2 * tau) return total def strain_profile(H): out = [] for m in range(1, 16): d = [(m >> a) & 1 for a in range(4)] out.append(sum(d[a] * H[a][b] * d[b] for a in range(4) for b in range(4))) return out R3 = mp.sqrt(3) ONES = [[1] * 4 for _ in range(4)] G1 = [[(1 if i == j else 0) - 1 for j in range(4)] for i in range(4)] BACKGROUNDS = [ # name, profile, the paper's shifts at c = 0..3, trace of the strain (None if not a strain), source ('the flat metric g1', strain_profile(G1), [0, 0, 0, 0], 0, 'Proposition 6.3(i)'), ('the strain 11^T', strain_profile(ONES), [-mp.mpf(11) / (32 * R3)] * 4, 4, 'Proposition 6.3(iii)'), ('the probe H_E', strain_profile([[0, 2, -1, -1], [2, 0, -1, -1], [-1, -1, 0, 2], [-1, -1, 2, 0]]), [0, 0, 0, 0], 0, 'Proposition 6.3(i)'), ('the probe H_T2', strain_profile([[-1, 1, 0, 0], [1, -1, 0, 0], [0, 0, 1, -1], [0, 0, -1, 1]]), [-R3 / 32, -R3 / 32, R3 / 32, R3 / 32], 0, 'Proposition 6.3(ii)'), ('the strain H01 = 1, H23 = -1', strain_profile([[0, 1, 0, 0], [1, 0, 0, 0], [0, 0, 0, -1], [0, 0, -1, 0]]), [1 / (16 * R3), 1 / (16 * R3), -1 / (16 * R3), -1 / (16 * R3)], 0, 'Proposition 6.3(v)'), ('the polarization of a resting mass, (3/4) 11^T + (1/2) g1', strain_profile( [[mp.mpf(3) / 4 + (mp.mpf(1) / 2) * G1[i][j] for j in range(4)] for i in range(4)]), [-mp.mpf(33) / (128 * R3)] * 4, 3, 'Proposition 7.6'), ('the body-diagonal profile (not a strain)', [0] * 14 + [1], [R3 / 32] * 4, None, 'Proposition 6.4'), ] def report(name, ok, detail): print(f" [{'PASS' if ok else 'FAIL'}] {name}: {detail}") return ok def main(): parser = argparse.ArgumentParser(description='From the Regge action of a Kuhn simplex to Appendix B and the corner shifts.') parser.add_argument('--json', metavar='FILE.json', help='write the computed corner shifts') args = parser.parse_args() ok = True print('Eq. (3): the action of one Kuhn simplex at the flat kappa = 1 lattice') for t, tri in enumerate(TRIS): if tri in ((0, 1, 2), (1, 2, 3), (2, 3, 4)): print(f' [INFO] triangle {tri}: flat dihedral angle {mp.nstr(THETA0[t] / mp.pi, 6)} pi, shared by n_h = {NH[t]} simplices') closes = abs(2 * THETA0[0] + 2 * THETA0[TRIS.index((1, 2, 3))] + 2 * THETA0[TRIS.index((2, 3, 4))] - 2 * mp.pi) < mp.mpf('1e-30') ok &= report('the six simplices around a triangle with two consecutive postings close up to 2 pi', closes, '2(pi/6) + 2(2pi/3) + 2(pi/6) = 2 pi') print('Appendix B: the cell Hessian') Hf = hessian('flat') dev = max(abs(Hf[a][b] * 16 * R3 - C[a][b]) for a in range(10) for b in range(10)) ok &= report('Hessian of one simplex with the flat-angle share = C/(16 sqrt3)', dev < mp.mpf('1e-12'), f'max |16 sqrt3 H - C| = {mp.nstr(dev, 3)}') He = hessian('equal') diff = [(PAIRS[a], PAIRS[b], He[a][b] * 16 * R3 - C[a][b]) for a in range(10) for b in range(a, 10) if abs(He[a][b] * 16 * R3 - C[a][b]) > mp.mpf('1e-10')] expected = {((0, 1), (1, 2)): -2 * mp.pi / R3, ((1, 2), (2, 3)): 4 * mp.pi / R3, ((2, 3), (3, 4)): -2 * mp.pi / R3} match = len(diff) == 3 and all(abs(x - expected.get((p, q), 0)) < mp.mpf('1e-12') for p, q, x in diff) ok &= report('with the equal share 2 pi/n_h, the simplex Hessian differs only by the hinge terms', match, ', '.join(f'{p}-{q}: {mp.nstr(x / (16 * R3) / mp.pi, 6)} pi' for p, q, x in diff)) Hf_np = [[float(Hf[a][b]) for b in range(10)] for a in range(10)] He_np = [[float(He[a][b]) for b in range(10)] for a in range(10)] worst = 0.0 for k in (np.array([0.3, -0.7, 1.1, 0.4]), np.array([math.pi, 0, 0, 0]), np.array([1.0, 2.0, -0.5, 0.2])): worst = max(worst, float(np.abs(bloch([He_np] * 24, k) - bloch([Hf_np] * 24, k)).max())) ok &= report('the hinge terms cancel: B(k) is the same with either share', worst < 1e-12, f'max difference {worst:.1e} at three waves') print('Propositions 6.3, 6.4 and 7.6: first-order shifts of the four corner zeros, from the third variation of the action') results = {} for name, prof, paper, trace, source in BACKGROUNDS: got = [shift(prof, c) for c in range(4)] dev = max(abs(g - p) for g, p in zip(got, paper)) ok &= report(f'{name} ({source})', dev < mp.mpf('1e-12'), f'{", ".join(mp.nstr(g, 8) for g in got)}; paper {", ".join(mp.nstr(mp.mpf(p), 8) for p in paper)}') if trace is not None: rule = -mp.mpf(396) / (1152 * R3) * trace ok &= report(' sum rule, sum_c delta_c = -396/(1152 sqrt3) tr H', abs(sum(got) - rule) < mp.mpf('1e-12'), f'{mp.nstr(sum(got), 8)} = {mp.nstr(rule, 8)}') results[name] = [float(g) for g in got] coef = [] for n, paper in zip((1, 2, 3, 4), (-288, -2592, -2592, 1728)): prof = [1 if bin(m).count('1') == n else 0 for m in range(1, 16)] got = shift(prof, 0) * 16 * 1152 * R3 coef.append(got) ok &= report(f'Proposition 6.4: the profile 1 on |d| = {n}', abs(got - paper) < mp.mpf('1e-6'), f'16 * 1152 sqrt3 * delta = {mp.nstr(got, 9)} (paper {paper})') results['proposition_6_4_coefficients'] = [float(x) for x in coef] if args.json: with open(args.json, 'w') as fh: json.dump(results, fh) print(f' wrote {args.json}') print('ALL CHECKS PASS' if ok else 'SOME CHECKS FAILED') return ok if __name__ == '__main__': sys.exit(0 if main() else 1)