"""v449 -- SEAM.EQUIV.MMST.UNIFORM.01: uniform-in-N control of the edge scaling limit -- the empirical content of the ONE external MMST hypothesis, strengthened. The single residual of SEAM.EQUIV.01 (after v444/v447/v448) is the cited MMST/OS theorem (v336): the chiral massless scaling limit EXISTS, i.e. ALL n-point functions converge UNIFORMLY in the lattice spacing. v444 grounded the two-point, v448 the four-point, each at a single conformal cross-ratio. This module adds the property the abstract theorem actually asserts -- UNIFORMITY: on the same v367/v368 p+ip collar, TWO logically independent edge four-point cross-ratios both converge to their (distinct) free-fermion conformal values under refinement, the deviation is Cauchy, and |deviation| * N stays bounded by a single N-independent constant across BOTH observables (an empirical 1/N uniform rate). This does NOT prove the theorem -- it pins, on our explicit collar, exactly the convergence the cited theorem supplies abstractly. [E] 1. CROSS-RATIO A converges: Q_A(1,2,4,7)/12 -> sin(5pi/12)/sin(pi/12)=2+sqrt3 (Cauchy under refinement, finest within 1%). [E] 2. CROSS-RATIO B (independent positions) converges: Q_B(1,3,5,9)/12 -> its own conformal value 2 (Cauchy, finest within 1%) -- a SECOND observable with a DIFFERENT limit, so the convergence is not a single tuned point. [E] 3. UNIFORM 1/N RATE: for both cross-ratios and every N in {36,48,60}, |Q(N)-Q_inf| * N <= 4 (one N-independent constant across two independent observables) -- the empirical hallmark of uniform-in-N convergence. [E] 4. BULK NEG CONTROL: the bulk-row cross-ratio is off-scale (gapped, exponential) -- only the EDGE converges, so it is the edge CFT. [C]/[O] 5. VERDICT: the MMST uniform-convergence HYPOTHESIS is empirically satisfied on our collar at two-point (v444), four-point (v448) and now multi-observable uniform-rate level; SEAM.EQUIV.01 stays [O] -- the abstract continuum existence theorem (v336) is strengthened in evidence, not replaced by a proof. Python-only (numpy; free-fermion BdG collar; the conformal targets 2+sqrt3 and 2 are kinematic cross-ratios, no new exact algebraic identity to Wolfram-mirror). """ import numpy as np from tfpt_constants import check, summary, reset, g_car SX = np.array([[0, 1], [1, 0]], complex) SY = np.array([[0, -1j], [1j, 0]], complex) SZ = np.array([[1, 0], [0, -1]], complex) CONF = {(1, 2, 4, 7): 2 + np.sqrt(3), (1, 3, 5, 9): 2.0} def _real2d(Nx, Ny, M): onx = -0.5 * SZ + (1 / (2j)) * SX ony = -0.5 * SZ + (1 / (2j)) * SY ons = M * SZ N = Nx * Ny def idx(x, y): return (x % Nx) * Ny + y H = np.zeros((2 * N, 2 * N), complex) for x in range(Nx): for y in range(Ny): a = idx(x, y) H[2 * a:2 * a + 2, 2 * a:2 * a + 2] += ons b = idx(x + 1, y) H[2 * a:2 * a + 2, 2 * b:2 * b + 2] += onx H[2 * b:2 * b + 2, 2 * a:2 * a + 2] += onx.conj().T if y < Ny - 1: c = idx(x, y + 1) H[2 * a:2 * a + 2, 2 * c:2 * c + 2] += ony H[2 * c:2 * c + 2, 2 * a:2 * a + 2] += ony.conj().T return H, idx def _gcorr(Nx, Ny, M): H, idx = _real2d(Nx, Ny, M) w, v = np.linalg.eigh(H) occ = v[:, w < -1e-9] return occ @ occ.conj().T, idx def _cmag(G, idx, y0, x0, x1): a, b = idx(x0, y0), idx(x1, y0) return np.linalg.norm(G[2 * a:2 * a + 2, 2 * b:2 * b + 2]) def _chord(r, Nx): return (Nx / np.pi) * np.sin(np.pi * r / Nx) def _Qmeas(G, idx, Nx, frac, y0): u = Nx // 12 xs = [f * u for f in frac] g = lambda a, b: _cmag(G, idx, y0, xs[a], xs[b]) return (g(0, 1) * g(2, 3)) / (g(0, 2) * g(1, 3)) def run(): reset() print("v449 SEAM.EQUIV.MMST.UNIFORM: uniform-in-N control of the edge scaling " "limit -- two independent cross-ratios converge at a single 1/N rate") sizes = [36, 48, 60] fracs = [(1, 2, 4, 7), (1, 3, 5, 9)] # measure both cross-ratios on the SAME ground state per size (edge row y=0) Q = {f: [] for f in fracs} Gfine = idxfine = None for Nx in sizes: Ny = Nx // 3 G, idx = _gcorr(Nx, Ny, 1.0) if Nx == sizes[-1]: Gfine, idxfine = G, idx for f in fracs: Q[f].append(_Qmeas(G, idx, Nx, f, 0)) # ---- 1/2. each cross-ratio Cauchy + converges to its conformal value ---- conv_ok = {} for f in fracs: qs = np.array(Q[f]) target = CONF[f] dev = np.abs(qs - target) monotone = bool(np.all(np.diff(dev) <= 1e-9)) # deviation shrinks under refinement finest = dev[-1] ok = monotone and finest < 0.01 * abs(target) conv_ok[f] = ok label = "A" if f == (1, 2, 4, 7) else "B" tname = "sin(5pi/12)/sin(pi/12)=2+sqrt(3)" if f == (1, 2, 4, 7) else "2" check("CROSS-RATIO %s CONVERGES [E]: Q%s%s -> %s=%.4f, measured " "%.4f,%.4f,%.4f (Nx=36,48,60); |Q-Q_inf| shrinks monotonically " "%.4f->%.4f->%.4f, finest <1%%" % (label, label, str(f).replace(" ", ""), tname, target, qs[0], qs[1], qs[2], dev[0], dev[1], dev[2]), ok) # ---- 3. uniform 1/N rate across BOTH observables ---- prods = [] for f in fracs: qs = np.array(Q[f]) for k, Nx in enumerate(sizes): prods.append(abs(qs[k] - CONF[f]) * Nx) uniform = max(prods) <= 4.0 check("UNIFORM 1/N RATE [E]: |Q(N)-Q_inf|*N <= %.3f for BOTH cross-ratios and " "every Nx in {36,48,60} (max=%.3f) -- a single N-independent constant " "bounds the deviation across two independent observables, the empirical " "hallmark of uniform-in-N convergence" % (4.0, max(prods)), uniform) # ---- 4. bulk neg control (off-scale) ---- Qb = _Qmeas(Gfine, idxfine, sizes[-1], (1, 2, 4, 7), (sizes[-1] // 3) // 2) bulk_off = abs(Qb - CONF[(1, 2, 4, 7)]) > 10.0 check("BULK NEG CONTROL [E]: the bulk-row cross-ratio Q_bulk=%.2f is off-scale " "vs the conformal %.4f (gapped, exponential) -- only the EDGE converges, " "so the uniform convergence is the edge CFT" % (Qb, CONF[(1, 2, 4, 7)]), bulk_off) # ---- 5. verdict [C]/[O] ---- grounded = all(conv_ok.values()) and uniform and bulk_off check("VERDICT [C]/[O]: the MMST uniform-convergence hypothesis is empirically " "satisfied on our collar -- two independent edge cross-ratios converge to " "distinct conformal values at one 1/N rate, extending v444/v448 from " "single points to a uniform multi-observable statement; SEAM.EQUIV.01 " "stays [O], the abstract existence theorem (v336) is strengthened in " "evidence, not proved here", grounded and g_car == 5) return summary("v449 SEAM.EQUIV.MMST.UNIFORM: two independent edge four-point " "cross-ratios converge to 2+sqrt(3) and 2 (Cauchy), with " "|Q-Q_inf|*N bounded by one N-independent constant across both " "(uniform 1/N rate) vs an off-scale bulk control -- the empirical " "content of the cited MMST uniform-convergence theorem; " "SEAM.EQUIV.01 stays [O]") if __name__ == "__main__": raise SystemExit(1 if run() else 0)