"""v115 -- The anchor pins the residue: an EXACT normal form for the (U_wall) flat-bundle residue, and the answer to the v114 branch question -- the anchor splitting forces the d1 = 0 slice. [I] symbolic lemmas + exact matrix + [N] Riemann-Hilbert verification and uniqueness scan. The v33 (U_wall) flat bundle was a NUMERICAL Riemann-Hilbert solution. This module shows its residue is (gauge-equivalent to) an exact matrix whose every invariant is a compiler atom, and uses it to answer the open v114 question. [I] 1. MU4-AVERAGE LEMMA (symbolic, generic X). Conjugation-averaging over mu_4 is the diagonal projection: sum_k U^k X U^{-k} = |mu_4| * diag(X). Hence the exponents at infinity are 4*diag(A_0), and the anchor splitting O(-2)+O(-1)^2 holds IFF diag A_0 = (2,1,1)/4 = a / |mu_4| -- the residue diagonal IS the anchor over mu_4 (v33's diagonal was forced, not chosen). [I] 2. THE (8,0,5)/144 LEMMA. For a Hermitian A_0 with the anchor diagonal and the cusp spectrum {0, 1/3, 2/3} (weights {0,1,2}/ N_fam), the off-diagonal weight is pinned: |a12|^2 + |a13|^2 + |a23|^2 = 13/144, and with a13 = 0 the determinant condition solves UNIQUELY to (|a12|^2, |a13|^2, |a23|^2) = (8, 0, 5) / 144 -- numerators (rank E8, 0, g_car), denominator (|mu_4| N_fam)^2, total 13 = Delta_Q = 2 p_2(a) + 1 (the quark denominator 117 = 9*13). Exact linear algebra, no fit. [I] 3. EXACT RESIDUE NORMAL FORM. The real representative A_0* = [[1/2, sqrt2/6, 0], [sqrt2/6, 1/4, sqrt5/12], [0, sqrt5/12, 1/4]] has characteristic polynomial lam(lam-1/3)(lam-2/3) EXACTLY. [N] 4. A_0* IS THE RIEMANN-HILBERT SOLUTION. Numerically (DOP853, rtol 1e-10): the big-circle monodromy of the Fuchsian system built on A_0* is trivial to ~1e-10, the four loop monodromies satisfy prod M_k = 1, and the unitarised holonomy has |diag M~_0| = (0, 1/2, 1/2) with tr(M~_0 U) = 1. The v33/v40 numerical point is gauge-equivalent to A_0* (the phases of a12, a23 are diagonal gauge). [N] 5. THE ANCHOR FORCES THE d1 = 0 SLICE (v114 residue answered). A multi-seed Riemann-Hilbert scan over the full anchor-pinned spectral manifold lands, at every flat solution found, on the SAME gauge orbit (off-diagonal moduli (sqrt8, 0, sqrt5)/12) with |diag M~_0| = (0, 1/2, 1/2). Combined with v114: the whole harmonic diagonal is anchor + torsion forced; delta = 1/2 needs no selection anymore. [P] 6. RESIDUE (recorded, not claimed): a13 = 0 and the gauge-orbit uniqueness are numerical findings (single component across all seeds); promoting them to [I]/[L] (finite algebraic geometry / Groebner on the RH constraints) is the remaining step. """ import numpy as np import scipy.linalg as sl import sympy as sp from scipy.integrate import solve_ivp from scipy.optimize import minimize from tfpt_constants import check, summary, reset 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]) I3 = np.eye(3, dtype=complex) ANCHOR_DIAG = np.array([0.5, 0.25, 0.25]) def fuchs_rhs(ak): def rhs(t, y, path): z, dz = path(t) amat = sum(ak[j] / (z - PUNCT[j]) for j in range(4)) return ((amat @ y.reshape(3, 3)) * dz).reshape(-1) return rhs def m_circle(a0, rtol=1e-10): ak = [UIS[k] @ a0 @ UIC[k] for k in range(4)] rhs = fuchs_rhs(ak) def path(t): return 3 * np.exp(1j * t), 3j * np.exp(1j * t) sol = solve_ivp(lambda t, y: rhs(t, y, path), [0, 2 * np.pi], I3.reshape(-1), rtol=rtol, atol=1e-12, method='DOP853') return sol.y[:, -1].reshape(3, 3) def loop_mono(a0, k, eps=0.25, rtol=1e-10): ak = [UIS[j] @ a0 @ UIC[j] for j in range(4)] rhs = fuchs_rhs(ak) 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 sol = solve_ivp(lambda t, y: rhs(t, y, path), [0, 1], I3.reshape(-1), rtol=rtol, atol=1e-12, method='DOP853') return sol.y[:, -1].reshape(3, 3) def harmonic_diag(a0): """Unitarise via the unique invariant Hermitian form (v40 method).""" ms = [loop_mono(a0, 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 if np.linalg.eigvalsh(h).min() < 1e-8: return None w = np.asarray(sl.sqrtm(h), dtype=complex) m0t = w @ ms[0] @ np.linalg.inv(w) prodres = np.linalg.norm(ms[0] @ ms[1] @ ms[2] @ ms[3] - I3) return np.diag(m0t), np.trace(m0t @ UN), prodres def a0_from(x): c12 = x[0] + 1j * x[1] c13 = x[2] + 1j * x[3] c23 = x[4] + 1j * x[5] return np.array([[ANCHOR_DIAG[0], c12, c13], [np.conj(c12), ANCHOR_DIAG[1], c23], [np.conj(c13), np.conj(c23), ANCHOR_DIAG[2]]], complex) def spec_pen(a): ev = np.sort(np.linalg.eigvalsh(a)) return float(np.sum((ev - np.array([0, 1 / 3, 2 / 3])) ** 2)) def run(): reset() print("v115 anchor residue (exact normal form; anchor forces d1 = 0)") # 1. mu4-average lemma xsym = sp.Matrix(3, 3, lambda i, j: sp.Symbol(f'x{i}{j}', complex=True)) usym = sp.diag(1, sp.I, -sp.I) ssum = sp.zeros(3) for k in range(4): uk = usym ** k ssum += uk * xsym * uk.inv() check("MU4-AVERAGE LEMMA [symbolic]: sum_k U^k X U^{-k} = 4 diag(X) " "-- mu_4 conjugation-averaging IS the diagonal projection; " "exponents at infinity = 4 diag(A0), so the anchor splitting " "O(-2)+O(-1)^2 <=> diag A0 = (2,1,1)/4 = a/|mu_4|", sp.simplify(sp.expand(ssum) - 4 * sp.diag(xsym[0, 0], xsym[1, 1], xsym[2, 2])) == sp.zeros(3)) # 2. the (8,0,5)/144 lemma x, y = sp.symbols('x y', nonnegative=True) sol = sp.solve([sp.Eq(x + y, sp.Rational(13, 144)), sp.Eq(sp.Rational(1, 32) - y / 2 - x / 4, 0)], [x, y]) check("THE (8,0,5)/144 LEMMA: anchor diagonal + cusp spectrum pin " "the off-diagonal weight sum to 13/144; with a13 = 0 the " "det = 0 condition solves UNIQUELY to (|a12|^2, |a23|^2) = " "(8, 5)/144 -- numerators (rank E8, g_car), denominator " "(|mu_4| N_fam)^2 = 144, total 13 = Delta_Q (quark " "denominator 117 = 9 x 13)", sol == {x: sp.Rational(1, 18), y: sp.Rational(5, 144)} and sp.Rational(1, 18) == sp.Rational(8, 144) and 8 + 5 == 13 and (4 * 3) ** 2 == 144 and 9 * 13 == 117) # 3. exact residue normal form astar = sp.Matrix([[sp.Rational(1, 2), sp.sqrt(2) / 6, 0], [sp.sqrt(2) / 6, sp.Rational(1, 4), sp.sqrt(5) / 12], [0, sp.sqrt(5) / 12, sp.Rational(1, 4)]]) lam = sp.Symbol('lam') check("EXACT RESIDUE NORMAL FORM: A0* = [[1/2, sqrt2/6, 0], " "[sqrt2/6, 1/4, sqrt5/12], [0, sqrt5/12, 1/4]] has char poly " "lam(lam - 1/3)(lam - 2/3) EXACTLY -- eigenvalues = cusp " "weights {0,1,2}/N_fam automatic", sp.simplify(astar.charpoly(lam).as_expr() - lam * (lam - sp.Rational(1, 3)) * (lam - sp.Rational(2, 3))) == 0) # 4. A0* is the Riemann-Hilbert solution a0_num = np.array(astar.evalf(20), dtype=complex) mc = m_circle(a0_num) res = harmonic_diag(a0_num) dg, trt, prodres = res check("A0* IS THE RH SOLUTION [N]: big-circle monodromy trivial " f"(||M_inf - 1|| = {np.linalg.norm(mc - I3):.1e} < 1e-7), " f"prod M_k = 1 ({prodres:.1e}), unitarised |diag M~_0| = " "(0, 1/2, 1/2), tr(M~_0 U) = 1 -- the v33/v40 numerical point " "is gauge-equivalent to this exact matrix", np.linalg.norm(mc - I3) < 1e-7 and prodres < 1e-6 and abs(dg[0]) < 1e-6 and abs(abs(dg[1]) - 0.5) < 1e-6 and abs(abs(dg[2]) - 0.5) < 1e-6 and abs(trt - 1) < 1e-5) # 5. multi-seed uniqueness scan (the anchor forces the d1 = 0 slice) rng = np.random.default_rng(3) hits = [] for _ in range(8): x0 = rng.normal(0, 0.25, 6) r1 = minimize(lambda v: spec_pen(a0_from(v)), x0, method='Nelder-Mead', options={'maxiter': 3000, 'xatol': 1e-13, 'fatol': 1e-15}) if r1.fun > 1e-12: continue def objective(v): a0 = a0_from(v) pen = spec_pen(a0) if pen > 1e-4: return 100 * pen + 10.0 return 100 * pen + float( np.linalg.norm(m_circle(a0, rtol=1e-8) - I3) ** 2) r2 = minimize(objective, r1.x, method='Nelder-Mead', options={'maxiter': 2500, 'xatol': 1e-11, 'fatol': 1e-13}) if r2.fun < 1e-7: a0 = a0_from(r2.x) out = harmonic_diag(a0) if out is None: continue dgi, trti, _ = out hits.append((abs(a0[0, 1]), abs(a0[0, 2]), abs(a0[1, 2]), abs(dgi[0]), abs(dgi[1]))) hits = np.array(hits) ref = np.array([np.sqrt(8) / 12, 0.0, np.sqrt(5) / 12]) check(f"ANCHOR FORCES THE d1 = 0 SLICE [N]: {len(hits)} independent " "RH solutions found; ALL on the same gauge orbit (off-diag " "moduli (sqrt8, 0, sqrt5)/12) and ALL with |diag M~_0| = " "(0, 1/2, 1/2) -- the v114 branch question is answered: the " "harmonic diagonal is anchor + torsion forced", len(hits) >= 3 and np.abs(hits[:, :3] - ref).max() < 1e-4 and np.abs(hits[:, 3]).max() < 1e-5 and np.abs(hits[:, 4] - 0.5).max() < 1e-5) # 6. residue check("RESIDUE [P] (recorded, not claimed): a13 = 0 and the " "gauge-orbit uniqueness are numerical (one component across " "all seeds); promoting them to [I]/[L] via finite algebraic " "geometry (Groebner on the RH constraints) is the remaining " "step of the U_wall point selection", True) return summary("v115 anchor residue") if __name__ == "__main__": raise SystemExit(1 if run() else 0)