03-ilang-space / 13-numerical-run
13numerical runverified
Summary A precommitted numerical test, in five parts, of earlier results on concrete systems, and of whether local records give a one-dimensional chain.
common.py5 KB · raw
"""Shared helpers for T1-T5 of 13-numerical-run (ensembles, view, W, alpha, points, N_V, ell)."""
import json
from pathlib import Path
import numpy as np
OUT = Path(__file__).parent / "output"
OUT.mkdir(exist_ok=True)
SAME_POINT_TOL = 1e-12 # places with 1 - W < 1e-12 are the same point
def gue(rng, d, s=1.0):
"""GUE(s) of 07: off-diagonal Re/Im variance s^2/2, diagonal real variance s^2.
Draw order: real-part matrix, then imaginary-part matrix."""
a = rng.standard_normal((d, d)) + 1j * rng.standard_normal((d, d))
return s * (a + a.conj().T) / 2.0
def goe(rng, d, s=1.0):
"""GOE(s) of 07: off-diagonal variance s^2, diagonal variance 2 s^2."""
a = rng.standard_normal((d, d))
return s * (a + a.T) / np.sqrt(2.0)
def haar_state(rng, d):
"""Haar-random unit vector: normalized complex Gaussian (real draw, then imaginary draw)."""
z = rng.standard_normal(d) + 1j * rng.standard_normal(d)
return z / np.linalg.norm(z)
def slope(lams, ys):
"""Least-squares slope of log|y| against log(lambda)."""
return float(np.polyfit(np.log(np.asarray(lams, float)), np.log(np.abs(np.asarray(ys, float))), 1)[0])
def view(psis):
"""V_{hh'} = <psi_h'|psi_h> (= (1.7)); psis has rows psi_h."""
psis = np.asarray(psis)
return psis @ psis.conj().T
def wmat(V):
"""(2.11): W(h,h') = |V_hh'|^2 / (p_h p_h')."""
p = np.real(np.diag(V))
return np.abs(V) ** 2 / np.outer(p, p)
def loop_w(V, idx):
"""W(h1,...,hk) = G_{h1h2}...G_{hkh1} = V_{h1h2}...V_{hkh1}/(p_h1...p_hk)."""
p = np.real(np.diag(V))
w = 1.0 + 0j
for j in range(len(idx)):
w *= V[idx[j], idx[(j + 1) % len(idx)]] / p[idx[j]]
return w
def alpha(W):
"""(3.5): alpha = arccos sqrt(W)."""
return np.arccos(np.sqrt(np.clip(W, 0.0, 1.0)))
def points(W):
"""Classes of places with 1 - W < SAME_POINT_TOL (transitive closure)."""
n = W.shape[0]
parent = list(range(n))
def find(i):
while parent[i] != i:
i = parent[i]
return i
for i in range(n):
for j in range(i + 1, n):
if 1.0 - W[i, j] < SAME_POINT_TOL:
parent[find(j)] = find(i)
classes = {}
for i in range(n):
classes.setdefault(find(i), []).append(i)
return sorted(classes.values())
def neighbour_graph(Wp):
"""(4.3) on points: x~x' iff W(x,x')>0 and no y with max(alpha(x,y),alpha(y,x')) < alpha(x,x').
Returns edge list, alpha matrix, and the smallest |alpha(x,x') - min_y max(...)| (diagnostic)."""
A = alpha(Wp)
m = Wp.shape[0]
edges, margins = [], []
for i in range(m):
for j in range(i + 1, m):
if not Wp[i, j] > 0:
continue
others = [y for y in range(m) if y not in (i, j)]
best = min((max(A[i, y], A[y, j]) for y in others), default=np.inf)
if np.isfinite(best):
margins.append(abs(A[i, j] - best))
if not best < A[i, j]:
edges.append((i, j))
return edges, A, (min(margins) if margins else None)
def path_analysis(m, edges, A):
"""Is the graph on m vertices a path graph (connected, acyclic, degrees <= 2)?
If so: vertex order, step lengths alpha, and ell (4.6) between the two ends."""
adj = {v: set() for v in range(m)}
for i, j in edges:
adj[i].add(j)
adj[j].add(i)
deg = [len(adj[v]) for v in range(m)]
seen, n_comp = set(), 0
for s in range(m):
if s in seen:
continue
n_comp += 1
seen.add(s)
stack = [s]
while stack:
v = stack.pop()
for w in adj[v]:
if w not in seen:
seen.add(w)
stack.append(w)
connected = n_comp == 1
acyclic = len(edges) == m - n_comp # forest iff |E| = |V| - #components
is_path = bool(connected and acyclic and max(deg) <= 2)
res = {"degrees": deg, "n_components": n_comp, "connected": connected, "n_edges": len(edges),
"acyclic": acyclic, "max_degree": max(deg), "is_path": is_path}
if is_path:
ends = [v for v in range(m) if deg[v] <= 1]
order, prev = [ends[0]], None
while len(order) < m:
nxt = [w for w in adj[order[-1]] if w != prev][0]
prev = order[-1]
order.append(nxt)
steps = [float(A[order[k], order[k + 1]]) for k in range(m - 1)]
# ell by Floyd-Warshall on N_V with alpha weights
D = np.full((m, m), np.inf)
np.fill_diagonal(D, 0.0)
for i, j in edges:
D[i, j] = D[j, i] = A[i, j]
for k in range(m):
D = np.minimum(D, D[:, [k]] + D[[k], :])
res.update({"order": order, "steps_alpha": steps,
"ell_ends": float(D[order[0], order[-1]]),
"sum_steps": float(sum(steps))})
return res
def cjson(a):
a = np.asarray(a)
if np.iscomplexobj(a):
return {"re": a.real.tolist(), "im": a.imag.tolist()}
return a.tolist()
def save(name, data):
with open(OUT / name, "w") as f:
json.dump(data, f, indent=1)
t1_leading_order.py2 KB · raw
"""T1 (computation): leading order alpha/lambda -> d_0, test of (5.10)."""
import itertools
import numpy as np
from scipy.linalg import expm
from common import gue, view, wmat, alpha, points, slope, cjson, save
n, dc = 8, 6
grid = [1e-1, 3e-2, 1e-2, 3e-3, 1e-3]
rng = np.random.default_rng(1301)
K = [gue(rng, dc) for _ in range(n)] # records K_h, h = 1..8 in draw order
chi = np.zeros(dc, complex); chi[0] = 1.0 # |chi> = e_1
phi = np.full(n, 1 / np.sqrt(n))
u = np.array([Kh @ chi - (chi.conj() @ Kh @ chi) * chi for Kh in K])
d0 = np.array([[np.linalg.norm(u[h] - u[k]) for k in range(n)] for h in range(n)])
pairs = list(itertools.combinations(range(n), 2))
rows, E = [], []
for lam in grid:
psis = np.array([phi[h] * (expm(-1j * lam * K[h]) @ chi) for h in range(n)]) # (12.8)
V = view(psis)
W = wmat(V)
A = alpha(W)
rel = {f"{h+1},{k+1}": float(abs(A[h, k] / lam - d0[h, k]) / d0[h, k]) for h, k in pairs}
E.append(max(rel.values()))
rows.append({"lambda": lam, "W": W.tolist(), "alpha": A.tolist(), "rel_err_per_pair": rel,
"E": E[-1], "n_points": len(points(W))})
print(f"lambda={lam:.0e} E={E[-1]:.6e} points={len(points(W))}")
s = slope(grid, E)
passed = bool(E[-1] < 1e-2 and s >= 0.9)
print(f"slope={s:.6f} decision={'pass' if passed else 'fail'}")
save("t1_leading_order.json", {
"part": "T1", "n": n, "d_c": dc, "seed": 1301, "grid": grid,
"records_K": [cjson(Kh) for Kh in K], "u": cjson(u), "d0": d0.tolist(),
"per_lambda": rows, "E": E, "fitted_slope_E": s,
"rule": "pass iff E(1e-3) < 1e-2 and slope >= 0.9", "decision": "pass" if passed else "fail"})
t2_dimension_distances.py3 KB · raw
"""T2 (computation): (a) affine rank of {u_h}, tests (7.7), (7.11); (b) d_0^2 statistics, tests (7.6), (7.10)."""
import numpy as np
from common import gue, goe, save
def u_vec(K):
chi = np.zeros(K.shape[0], complex); chi[0] = 1.0 # e_1 = |chi>
return K @ chi - (chi.conj() @ K @ chi) * chi
def affine_rank(us):
D = us[1:] - us[0] # u_h - u_{h_1}
R = np.hstack([D.real, D.imag]) # real coordinates
sv = np.linalg.svd(R, compute_uv=False)
return int(np.sum(sv > 1e-9 * sv[0])), sv.tolist()
# (a) ranks
rank_rows, all_ok = [], True
for dc in [2, 3, 5, 9]:
for n in [3, 6, 12, 20]:
for ens, base, draw, quoted in [("GUE", 1302, gue, min(n - 1, 2 * (dc - 1))),
("GOE", 1402, goe, min(n - 1, dc - 1))]:
seed = base + 100 * dc + n
rng = np.random.default_rng(seed)
us = np.array([u_vec(draw(rng, dc).astype(complex)) for _ in range(n)])
r, sv = affine_rank(us)
ok = r == quoted
all_ok &= ok
sva = np.array(sv) / sv[0]
rank_rows.append({"ensemble": ens, "d_c": dc, "n": n, "seed": seed, "rank": r,
"quoted": quoted, "equal": ok, "singular_values": sv,
"diag_min_kept_ratio": float(sva[:r].min()),
"diag_max_dropped_ratio": float(sva[r:].max()) if r < len(sva) else None})
print(f"{ens} d_c={dc} n={n:2d} seed={seed} rank={r} quoted={quoted} {'ok' if ok else 'MISMATCH'}")
dec_a = "pass" if all_ok else "fail"
print("T2(a):", dec_a)
# (b) distances, d_c = 9, 4000 pairs
dc, npairs = 9, 4000
stats, ok_b = {}, True
for ens, seed, draw, q_rel in [("GUE", 1303, gue, 1 / np.sqrt(dc - 1)), ("GOE", 1304, goe, np.sqrt(2 / (dc - 1)))]:
rng = np.random.default_rng(seed)
d2 = np.empty(npairs)
for t in range(npairs):
uh = u_vec(draw(rng, dc).astype(complex))
uk = u_vec(draw(rng, dc).astype(complex))
d2[t] = np.linalg.norm(uh - uk) ** 2
mean, rel = float(d2.mean()), float(d2.std(ddof=1) / d2.mean())
q_mean = 2 * (dc - 1)
dev_mean, dev_rel = abs(mean - q_mean) / q_mean, abs(rel - q_rel) / q_rel
ok = bool(dev_mean <= 0.03 and dev_rel <= 0.10)
ok_b &= ok
stats[ens] = {"seed": seed, "sample_mean": mean, "quoted_mean": q_mean, "rel_dev_mean": dev_mean,
"sample_rel_std": rel, "quoted_rel_std": float(q_rel), "rel_dev_rel_std": dev_rel,
"within": ok, "d0_squared": d2.tolist()}
print(f"{ens}: mean={mean:.5f} (quoted {q_mean}, dev {dev_mean:.4%}), rel std={rel:.5f} "
f"(quoted {q_rel:.5f}, dev {dev_rel:.4%})")
dec_b = "pass" if ok_b else "fail"
print("T2(b):", dec_b)
save("t2_dimension_distances.json", {
"part": "T2", "a_ranks": rank_rows, "a_rule": "pass iff every rank equals (7.7) resp. (7.11)",
"a_decision": dec_a,
"a_diag_global_min_kept_ratio": min(r["diag_min_kept_ratio"] for r in rank_rows),
"a_diag_global_max_dropped_ratio": max(r["diag_max_dropped_ratio"] for r in rank_rows
if r["diag_max_dropped_ratio"] is not None),
"b_stats": stats, "b_sample_std": "ddof=1",
"b_rule": "pass iff, for both ensembles, |mean-quoted|/quoted <= 3% and |relstd-quoted|/quoted <= 10%",
"b_decision": dec_b})
t3_loop_phases.py5 KB · raw
"""T3 (computation): loop phases, tests (10.10), (10.15) and Step 6(a) of 10."""
import numpy as np
from scipy.linalg import expm
from common import gue, goe, view, loop_w, slope, cjson, save
dc, ntri = 4, 50
grid = [1e-2, 3e-3, 1e-3]
chi = np.zeros(dc, complex); chi[0] = 1.0 # |chi> = e_1 (real)
phi = np.full(3, 1 / np.sqrt(3)) # W is independent of phi_h != 0
def u_vec(K):
return K @ chi - (chi.conj() @ K @ chi) * chi
def omega(v, w):
return float(np.vdot(v, w).imag) # Im <v|w>
def loop_phase(Ks, lam):
psis = np.array([phi[j] * (expm(-1j * lam * Ks[j]) @ chi) for j in range(3)]) # (12.8)
return float(np.angle(loop_w(view(psis), [0, 1, 2])))
# (a) generic GUE records
rng = np.random.default_rng(1305)
tris = [[gue(rng, dc) for _ in range(3)] for _ in range(ntri)]
om = [omega(u_vec(K[1]) - u_vec(K[0]), u_vec(K[2]) - u_vec(K[0])) for K in tris]
sel = [t for t in range(ntri) if abs(om[t]) > 1e-2]
a_rows, maxrel = [], []
for lam in grid:
per = {}
for t in sel:
Phi = loop_phase(tris[t], lam)
Phi2 = -lam ** 2 * om[t]
per[t] = {"Phi": Phi, "Phi2": Phi2, "rel_err": abs(Phi - Phi2) / abs(Phi2)}
maxrel.append(max(v["rel_err"] for v in per.values()))
a_rows.append({"lambda": lam, "per_triangle": per, "max_rel_err": maxrel[-1]})
print(f"(a) lambda={lam:.0e} max rel err={maxrel[-1]:.6e}")
s_a = slope(grid, maxrel)
dec_a = "pass" if (maxrel[-1] < 1e-2 and s_a >= 0.9) else "fail"
print(f"(a) selected {len(sel)}/{ntri}, slope={s_a:.6f}, decision={dec_a}")
# diagnostics (not decisions): ranking of triangles at lambda = 1e-3, and an mpmath (40 digits)
# recomputation of Phi for the triangle attaining the maximum, to exclude rounding as the cause
import mpmath as mp
last = a_rows[-1]["per_triangle"]
ranking = sorted(last, key=lambda t: -last[t]["rel_err"])
diag_a = {"triangles_by_rel_err_at_1e-3": [{"triangle": t, "omega": om[t], "rel_err": last[t]["rel_err"]}
for t in ranking[:5]],
"median_rel_err_per_lambda": [float(np.median([v["rel_err"] for v in r["per_triangle"].values()]))
for r in a_rows]}
mp.mp.dps = 40
t0, lam = ranking[0], mp.mpf("1e-3")
E = [mp.expm(-1j * lam * mp.matrix(tris[t0][j].tolist())) * mp.matrix(chi.tolist()) for j in range(3)]
ov = lambda a, b: sum(mp.conj(a[i]) * b[i] for i in range(dc)) # <a|b>
Wl = ov(E[1], E[0]) * ov(E[2], E[1]) * ov(E[0], E[2]) # G_12 G_23 G_31
diag_a["mpmath_check"] = {"triangle": t0, "Phi_mpmath": float(mp.arg(Wl)), "Phi_double": last[t0]["Phi"],
"rel_dev": float(abs(mp.arg(Wl) - last[t0]["Phi"]) / abs(mp.arg(Wl)))}
print("(a) diagnostics:", diag_a["triangles_by_rel_err_at_1e-3"][:2], diag_a["mpmath_check"])
# (b)(i) commuting records K_h = x_h X
X = gue(np.random.default_rng(1306), dc)
x = np.random.default_rng(1307).standard_normal((ntri, 3))
tris_i = [[x[t, j] * X for j in range(3)] for t in range(ntri)]
# (b)(ii) real records, GOE
rng = np.random.default_rng(1308)
tris_ii = [[goe(rng, dc).astype(complex) for _ in range(3)] for _ in range(ntri)]
b_out, dec_b_ok = {}, True
for name, T in [("i_commuting", tris_i), ("ii_real_GOE", tris_ii)]:
rows, mx = [], []
for lam in grid:
ph = [loop_phase(K, lam) for K in T]
mx.append(max(abs(p) for p in ph))
rows.append({"lambda": lam, "Phi_per_triangle": ph, "max_abs_Phi": mx[-1]})
s = slope(grid, mx)
dec_b_ok &= s >= 2.8
# diagnostic (10.15): W(-lambda) = W(lambda)^* at lambda = 1e-2, max over triangles
diag = None
if name == "ii_real_GOE":
dev = []
for K in T:
pp = np.array([phi[j] * (expm(-1j * 1e-2 * K[j]) @ chi) for j in range(3)])
pm = np.array([phi[j] * (expm(+1j * 1e-2 * K[j]) @ chi) for j in range(3)])
dev.append(abs(loop_w(view(pm), [0, 1, 2]) - np.conj(loop_w(view(pp), [0, 1, 2]))))
diag = float(max(dev))
b_out[name] = {"per_lambda": rows, "max_abs_Phi": mx, "fitted_slope": s,
"diag_max_dev_W(-lam)_vs_conj_W(lam)_at_1e-2": diag}
print(f"(b) {name}: max|Phi|={mx}, slope={s:.6f}, diag={diag}")
dec_b = "pass" if dec_b_ok else "fail"
print("(b) decision:", dec_b)
save("t3_loop_phases.json", {
"part": "T3", "d_c": dc, "grid": grid,
"a": {"seed": 1305, "omega_per_triangle": om, "selected_triangles": sel, "n_selected": len(sel),
"per_lambda": a_rows, "max_rel_err": maxrel, "fitted_slope": s_a, "diagnostics": diag_a,
"records": [[cjson(K) for K in tri] for tri in tris],
"rule": "pass iff max rel err < 1e-2 at lambda=1e-3 and slope >= 0.9", "decision": dec_a},
"b": {"seeds": {"X": 1306, "x_h": 1307, "GOE": 1308}, "X": cjson(X), "x_h": x.tolist(), **b_out,
"rule": "pass iff fitted slope >= 2.8 in both controls", "decision": dec_b}})
t4_light_cone.py4 KB · raw
"""T4 (computation): light cone, test of (8.16), with negative control.
Diagnostic (labelled): mpmath (40 digits) recomputation of Delta W to bound double-precision rounding."""
import itertools
import mpmath as mp
import numpy as np
from scipy.linalg import expm
from common import gue, haar_state, view, wmat, slope, cjson, save
L = 5
grid = [0.1, 0.05, 0.025, 0.0125]
rng = np.random.default_rng(1309)
cells = [haar_state(rng, 2) for _ in range(L)] # c_1..c_5, in this order
omega = cells[0]
for c in cells[1:]:
omega = np.kron(omega, c) # c_1 is the leftmost tensor factor
Jk = {k: gue(np.random.default_rng(1310 + k), 4) for k in range(1, L)}
rng = np.random.default_rng(1320); Kh = [gue(rng, 2) for _ in range(3)]
rng = np.random.default_rng(1330); Bb = [gue(rng, 2) for _ in range(2)] # B_beta1, B_beta2
phi = np.full(3, 1 / np.sqrt(3))
pairs = list(itertools.combinations(range(3), 2))
def on(op, first):
"""Embed an operator acting on cells first, first+1, ... (1-indexed) into the 2^L medium."""
nc = int(round(np.log2(op.shape[0])))
return np.kron(np.kron(np.eye(2 ** (first - 1)), op), np.eye(2 ** (L - first - nc + 1)))
Jsum = sum(on(Jk[k], k) for k in range(1, L))
def W_of(B_cell_op, lam):
psis = np.array([phi[h] * (expm(-1j * lam * (on(Kh[h], 1) + B_cell_op + Jsum)) @ omega) for h in range(3)])
return wmat(view(psis)) # (12.8), (2.11)
def mp_W(B_cell_op, lam):
mp.mp.dps = 40
lam = mp.mpf(lam)
out = []
for h in range(3):
M = on(Kh[h], 1) + B_cell_op + Jsum
Mm = [[mp.mpc(complex(M[i, j])) for j in range(32)] for i in range(32)]
v = [mp.mpc(complex(z)) for z in omega]
acc, term, n = list(v), list(v), 0
while True: # Taylor series of exp(-i lam M) v
n += 1
term = [(-1j * lam / n) * mp.fsum(Mm[i][j] * term[j] for j in range(32)) for i in range(32)]
acc = [acc[i] + term[i] for i in range(32)]
if max(abs(t) for t in term) < mp.mpf("1e-45"):
break
out.append(acc)
G = lambda a, b: mp.fsum(mp.conj(b[i]) * a[i] for i in range(32)) # <b|a>
nrm = [G(out[h], out[h]).real for h in range(3)]
return {(h, k): abs(G(out[h], out[k])) ** 2 / (nrm[h] * nrm[k]) for h, k in pairs}
res, ok_all, control_ok = {}, True, True
W0 = {lam: W_of(0.0, lam) for lam in grid}
mpW0 = {lam: mp_W(0.0, lam) for lam in grid}
for r in range(4):
Bop = on(Bb[0], r + 1)
rows, mx, mx_mp, ctrl = [], [], [], []
for lam in grid:
dW = W_of(Bop, lam) - W0[lam]
per = {f"{h+1},{k+1}": float(dW[h, k]) for h, k in pairs}
mx.append(max(abs(v) for v in per.values()))
mW = mp_W(Bop, lam)
dmp = {p: mW[p] - mpW0[lam][p] for p in pairs}
mx_mp.append(float(max(abs(v) for v in dmp.values())))
dWc = W_of(on(0.7 * np.eye(2), r + 1), lam) - W0[lam]
ctrl.append(float(np.max(np.abs(dWc[np.triu_indices(3, 1)]))))
rows.append({"lambda": lam, "Delta_W_per_pair": per, "max_abs_Delta_W": mx[-1],
"diag_mpmath_max_abs_Delta_W": mx_mp[-1],
"diag_rel_dev_double_vs_mpmath": abs(mx[-1] - mx_mp[-1]) / mx_mp[-1],
"control_max_abs_Delta_W": ctrl[-1]})
s, s_mp = slope(grid, mx), slope(grid, mx_mp)
ok = s >= 3 + r - 0.2
ok_all &= ok
c_ok = all(c <= 1e-12 for c in ctrl)
control_ok &= c_ok
res[f"r={r}"] = {"per_lambda": rows, "fitted_slope": s, "threshold": 3 + r - 0.2, "slope_ok": ok,
"exploratory_within_0.2_of_3+r": abs(s - (3 + r)) <= 0.2,
"diag_mpmath_fitted_slope": s_mp, "control_ok": c_ok}
print(f"r={r}: max|dW|={['%.3e' % v for v in mx]} slope={s:.4f} (thr {3+r-0.2}) mp-slope={s_mp:.4f} "
f"control max={max(ctrl):.2e}")
dec, dec_c = ("pass" if ok_all else "fail"), ("pass" if control_ok else "fail")
print("decision:", dec, " control:", dec_c)
save("t4_light_cone.json", {
"part": "T4", "L": L, "grid": grid, "seeds": {"cells": 1309, "J_k": "1310+k", "K_h": 1320, "B_beta": 1330},
"cell_states": [cjson(c) for c in cells], "J": {k: cjson(Jk[k]) for k in Jk},
"K": [cjson(K) for K in Kh], "B": [cjson(B) for B in Bb], "results": res,
"rule": "pass iff slope >= 3+r-0.2 for every r", "decision": dec,
"control_rule": "B_beta1 = 0.7*1: pass iff max|Delta W| <= 1e-12 at every lambda", "control_decision": dec_c})
t5_chain.py5 KB · raw
"""T5 (computation and check): local records and a chain at finite lambda, 16 qubit cells (dim 65536).
Computation: W, points, N_V, path analysis for local g=0, local g=0.5 and the nonlocal control.
Check: closed form W = prod_k cos^2(lambda (F_hk - F_h'k)) for g = 0 (local records and control)."""
import itertools
import numpy as np
import scipy.sparse as sp
from scipy.sparse.linalg import expm_multiply
from common import view, wmat, points, neighbour_graph, path_analysis, save
L, n = 16, 10
D = 2 ** L
grid = [0.05, 0.3, 1.0]
sx = sp.csr_matrix(np.array([[0, 1], [1, 0]], complex))
sy = sp.csr_matrix(np.array([[0, -1j], [1j, 0]], complex))
def emb(op, k, ncell):
"""Operator on cells k..k+ncell-1 (1-indexed), cell 1 = leftmost tensor factor."""
return sp.kron(sp.kron(sp.identity(2 ** (k - 1), format="csr"), op),
sp.identity(2 ** (L - k - ncell + 1), format="csr"), format="csr")
SX = [emb(sx, k, 1) for k in range(1, L + 1)]
XXYY = sum(emb(sp.kron(sx, sx) + sp.kron(sy, sy), k, 2) for k in range(1, L)) # k = 1..15
omega = np.zeros(D, complex); omega[0] = 1.0 # |0>^{(x)16}
phi = np.full(n, 1 / np.sqrt(n))
f = np.array([[np.exp(-(i + 3 - k) ** 2 / (2 * 1.5 ** 2)) for k in range(1, L + 1)] for i in range(1, n + 1)])
R = np.random.default_rng(1340).standard_normal((n, L))
R = R * np.sqrt((f ** 2).sum(1) / (R ** 2).sum(1))[:, None] # row-wise rescaling
def records(F):
return [sum(F[i, k] * SX[k] for k in range(L)) for i in range(n)]
def closed_form_W(F, lam):
return np.array([[np.prod(np.cos(lam * (F[i] - F[j])) ** 2) for j in range(n)] for i in range(n)])
cases = {"local_g0": (f, 0.0), "local_g0.5": (f, 0.5), "control_g0": (R, 0.0)}
out = {"part": "T5", "L": L, "n_places": n, "grid": grid, "f": f.tolist(), "R_rescaled": R.tolist(),
"seed_R": 1340, "cases": {}}
for name, (F, g) in cases.items():
Ks = records(F)
Ms = [K + g * XXYY for K in Ks]
rows = []
for lam in grid:
vecs = [expm_multiply(-1j * lam * M, omega) for M in Ms] # (12.8)
psis = np.array([phi[i] * vecs[i] for i in range(n)])
W = wmat(view(psis)) # (2.11)
pts = points(W)
rep = [c[0] for c in pts]
Wp = W[np.ix_(rep, rep)]
edges, A, margin = neighbour_graph(Wp) # (4.3)
pa = path_analysis(len(pts), edges, A) # (4.6) for ell
row = {"lambda": lam, "W": W.tolist(), "points": [[h + 1 for h in c] for c in pts],
"N_V_edges": [[[h + 1 for h in pts[i]], [h + 1 for h in pts[j]]] for i, j in edges],
"alpha_points": A.tolist(), "min_W": float(W.min()), **pa,
"diag_min_margin_in_(4.3)": margin,
"diag_max_norm_dev": float(max(abs(np.linalg.norm(v) - 1) for v in vecs))}
if pa["is_path"]:
row["order_places"] = [[h + 1 for h in pts[v]] for v in pa["order"]]
if g == 0.0:
row["check_max_abs_dev_from_closed_form"] = float(np.max(np.abs(W - closed_form_W(F, lam))))
if g == 0.5 and lam == 1.0: # diagnostic: independent Taylor propagation (40 substeps)
tay = []
for M in Ms:
v = omega.copy()
for _ in range(40):
acc, term, m = v.copy(), v.copy(), 0
while np.linalg.norm(term) > 1e-18:
m += 1
term = (-1j * lam / 40 / m) * (M @ term)
acc = acc + term
v = acc
tay.append(v)
Wt = wmat(view(np.array([phi[i] * tay[i] for i in range(n)])))
row["diag_taylor_max_abs_dev_vectors"] = float(max(np.max(np.abs(a - b)) for a, b in zip(vecs, tay)))
row["diag_taylor_max_abs_dev_W"] = float(np.max(np.abs(W - Wt)))
rows.append(row)
print(f"{name} lambda={lam}: points={len(pts)} edges={len(edges)} degrees={pa['degrees']} "
f"path={pa['is_path']} order={row.get('order_places')} ell={pa.get('ell_ends')} "
f"check={row.get('check_max_abs_dev_from_closed_form')} margin={margin:.3e} minW={W.min():.3e}")
out["cases"][name] = {"g": g, "per_lambda": rows}
c1 = all(r["is_path"] for r in out["cases"]["local_g0"]["per_lambda"])
c2 = not [r for r in out["cases"]["control_g0"]["per_lambda"] if r["lambda"] == 0.3][0]["is_path"]
out.update({"condition_local_g0_path_at_all_lambda": c1, "condition_control_not_path_at_0.3": c2,
"rule": "supported iff local g=0 path at all three lambda and control not path at lambda=0.3",
"decision": "pass" if (c1 and c2) else "fail"})
print("conditions:", c1, c2, "decision:", out["decision"])
save("t5_chain.json", out)
output/t1_leading_order.json42 KB · raw
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"d_c": 6,
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}output/t3_loop_phases.json195 KB · raw
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