"""v117 -- The monodromy is the Weyl group of A3: the (U_wall) holonomy is an EXACT 24-element representation (S4 = W(A3), standard x sign), and the harmonic diagonal (0, 1/2, 1/2) is a theorem of that representation. [I] exact matrix + exact group enumeration; [N] identification with the analytic monodromy of the exact A0* system. v116 made the residue side exact; the holonomy values were the last [N] item. This module closes them: the unitarised monodromy of the exact A0* system is itself EXACT -- entries in (1/2)Z[i] -- and generates, together with the mu_4 deck twist U, a group of order 24. [I] 1. THE EXACT MONODROMY. M0 = [[0, -(1+i)/2, (1-i)/2], [-(1+i)/2, -i/2, -1/2], [(1-i)/2, -1/2, i/2]] is unitary with det 1, tr 0, char poly lam^3 - 1 (the cusp class {1, w, w^2} EXACTLY), M0^3 = 1, and diag M0 = (0, -i/2, +i/2) -- d1 = 0 and delta = 1/2 are now exact matrix entries, not numerical observations. The v20 hand value, the v40/v41 observation, the v114 torsion value and the v115 anchor slice all meet in this one exact matrix. [I] 2. TORSION AND FLATNESS. (M0 U)^4 = 1 with tr(M0 U) = 1 (the {1, i, -i} branch, d1 = 0 slice -- exactly where v114/v115 located the physical point), and the flatness identity prod_k U^k M0 U^{-k} = 1 holds exactly (v114 normal form). [I] 3. THE GROUP IS W(A3) = S4. Exact enumeration of gives ORDER 24 with order statistics (1, 9, 8, 6) for element orders (1, 2, 3, 4) and character values (3, -1, 0, 1) -- uniquely the symmetric group S4 in its standard x sign representation (= the reflection representation of the WEYL GROUP OF A3, twisted by sign). The flavor-wall monodromy group IS the Weyl group of the family lattice A3 of the glue D5 + A3 + mu4: U = image of a 4-cycle (the mu_4 deck), M0 = a 3-cycle (the cusp / family rotation), 24 = |W(A3)| = 4!. [N] 4. IDENTIFICATION. The ODE monodromy of the exact A0* system, unitarised in the harmonic frame (v40 method), equals the exact M0 to ~1e-7, and the invariant form H is U-invariant to ~1e-10 (so the harmonic frame preserves the equivariance). [P] 5. STATUS (recorded): holonomy VALUES are now [I] (properties of the exact W(A3) representation); the identification of that representation with the analytic monodromy of A0* stays [N] (transcendental ODE matching, sharp to 1e-7). """ import numpy as np import scipy.linalg as sl import sympy as sp from scipy.integrate import solve_ivp from tfpt_constants import check, summary, reset II = sp.I M0_EXACT = sp.Matrix([[0, -(1 + II) / 2, (1 - II) / 2], [-(1 + II) / 2, -II / 2, sp.Rational(-1, 2)], [(1 - II) / 2, sp.Rational(-1, 2), II / 2]]) U_EXACT = sp.diag(1, II, -II) def enumerate_group(gens): def freeze(m): return tuple(sp.simplify(x) for x in m) elems = {freeze(sp.eye(3)): sp.eye(3)} frontier = [sp.eye(3)] while frontier: new = [] for e in frontier: for g in gens: x = sp.expand(g * e) kx = freeze(x) if kx not in elems: elems[kx] = x new.append(x) frontier = new return list(elems.values()) def ode_monodromy(): """Unitarised monodromy of the exact A0* system (v40 method).""" un = np.diag([1, 1j, -1j]) uis = [np.linalg.matrix_power(un, k) for k in range(4)] uic = [np.linalg.inv(u) for u in uis] punct = np.array([1, 1j, -1, -1j]) eye3 = np.eye(3, dtype=complex) a0 = np.array([[0.5, np.sqrt(8) / 12, 0], [np.sqrt(8) / 12, 0.25, np.sqrt(5) / 12], [0, np.sqrt(5) / 12, 0.25]], complex) ak = [uis[k] @ a0 @ uic[k] for k in range(4)] def loop(k, eps=0.25): pk = punct[k] c = pk - eps * pk / abs(pk) def path(t): if t < 0.3: return c * (t / 0.3), c / 0.3 elif t < 0.7: s = (t - 0.3) / 0.4 ang = 2 * np.pi * s return (pk + (c - pk) * np.exp(1j * ang), (c - pk) * 1j * 2 * np.pi * np.exp(1j * ang) / 0.4) else: s = (t - 0.7) / 0.3 return c + (0 - c) * s, (0 - c) / 0.3 def rhs(t, y): z, dz = path(t) am = sum(ak[j] / (z - punct[j]) for j in range(4)) return ((am @ y.reshape(3, 3)) * dz).reshape(-1) sol = solve_ivp(rhs, [0, 1], eye3.reshape(-1), rtol=1e-11, atol=1e-13, method='DOP853') return sol.y[:, -1].reshape(3, 3) ms = [loop(k) for k in range(4)] aop = np.vstack([np.kron(ms[k].T, ms[k].conj().T) - np.eye(9) for k in range(4)]) vh = np.linalg.svd(aop)[2] h = vh.conj().T[:, 8].reshape(3, 3) h = (h + h.conj().T) / 2 if np.trace(h).real < 0: h = -h w = np.asarray(sl.sqrtm(h), dtype=complex) m0 = w @ ms[0] @ np.linalg.inv(w) h_uinv = np.linalg.norm(un @ h @ un.conj().T - h) / np.linalg.norm(h) return m0, h_uinv def run(): reset() print("v117 monodromy = W(A3) (exact holonomy; delta = 1/2 a theorem)") lam = sp.Symbol('lam') # 1. the exact monodromy check("THE EXACT MONODROMY [I]: M0 (entries in (1/2)Z[i]) is unitary " "with det 1, tr 0, char poly lam^3 - 1 (cusp class EXACT), " "M0^3 = 1, and diag M0 = (0, -i/2, +i/2) -- d1 = 0 and " "delta = 1/2 are exact matrix entries now, not observations", sp.simplify(M0_EXACT.H * M0_EXACT) == sp.eye(3) and sp.simplify(M0_EXACT.det()) == 1 and sp.simplify(M0_EXACT.trace()) == 0 and sp.simplify(M0_EXACT.charpoly(lam).as_expr() - (lam ** 3 - 1)) == 0 and sp.simplify(M0_EXACT ** 3) == sp.eye(3) and [sp.simplify(M0_EXACT[i, i]) for i in range(3)] == [0, -II / 2, II / 2]) # 2. torsion and flatness tmat = M0_EXACT * U_EXACT prod = sp.eye(3) for k in range(4): uk = U_EXACT ** k prod = prod * (uk * M0_EXACT * uk.inv()) check("TORSION AND FLATNESS [I]: (M0 U)^4 = 1 with tr(M0 U) = 1 " "(the {1,i,-i} branch, d1 = 0 slice -- exactly where v114/" "v115 located the point) and prod_k U^k M0 U^{-k} = 1 exactly", sp.simplify(tmat ** 4) == sp.eye(3) and sp.simplify(tmat.trace()) == 1 and sp.simplify(prod) == sp.eye(3)) # 3. the group is W(A3) = S4 elems = enumerate_group([U_EXACT, M0_EXACT]) orders, chars = {}, {} for x in elems: p, o = x, 1 while sp.simplify(p) != sp.eye(3): p = sp.expand(p * x) o += 1 orders[o] = orders.get(o, 0) + 1 tr = sp.nsimplify(sp.simplify(x.trace())) chars[(o, tr)] = chars.get((o, tr), 0) + 1 check("THE GROUP IS W(A3) = S4 [I]: exact enumeration of " "gives ORDER 24, order statistics (1, 9, 8, 6) and character " "values (3, -1, 0, 1) -- uniquely S4 in standard x sign = the " "(twisted) reflection representation of the WEYL GROUP OF A3; " "U = a 4-cycle (the mu_4 deck), M0 = a 3-cycle (family " "rotation), 24 = |W(A3)| = 4!", len(elems) == 24 and orders == {1: 1, 2: 9, 3: 8, 4: 6} and chars == {(1, sp.Integer(3)): 1, (2, sp.Integer(-1)): 9, (3, sp.Integer(0)): 8, (4, sp.Integer(1)): 6} and sp.factorial(4) == 24) # 4. identification with the analytic monodromy m0_num, h_uinv = ode_monodromy() m0_ref = np.array(M0_EXACT.evalf(20), dtype=complex) check("IDENTIFICATION [N]: the unitarised ODE monodromy of the " "exact A0* system equals the exact M0 " f"(||diff|| = {np.linalg.norm(m0_num - m0_ref):.1e} < 1e-6) " f"and H is U-invariant ({h_uinv:.1e} < 1e-8) -- the harmonic " "frame preserves the equivariance", np.linalg.norm(m0_num - m0_ref) < 1e-6 and h_uinv < 1e-8) # 5. status check("STATUS [P] (recorded): holonomy VALUES now [I] (exact " "properties of the W(A3) representation: d1 = 0, delta = 1/2, " "cusp class, torsion branch); the identification with the " "analytic monodromy of A0* stays [N] (ODE matching, 1e-7). " "The delta thread v20 -> v40/v41 -> v114 -> v115 -> v117 is " "closed end to end", True) return summary("v117 monodromy = W(A3)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)