07-ilang-weight / 03-weight-of-mark
03weight of markverified
Determines the cost of a held body with its own mark and its weight in the field of a distant source, for two couplings and three field graphs, and whether the weight follows the cost for every level.
check_weight_of_mark.py16 KB · raw
"""Check for 07-ilang-weight/03-weight-of-mark, derivation v1.
One field object (N_f = 1, n = 1/|X|) on a ring, a square torus and a cubic torus.
Held body: point term g_b P_x, g_b = sigma*gamma*M_k; held source: point term g_s P_xs.
Exact diagonalization is compared with the closed forms of the derivation:
cost (3.2), (3.4), (3.5); weight (3.6)-(3.8); ring form (7.6); weight against cost
(3.12)-(3.17); mirror (3.18). Outputs: output/check_log.txt, output/check_results.json.
"""
import json
from pathlib import Path
import numpy as np
import scipy.sparse as sp
from scipy.optimize import brentq
from scipy.sparse.linalg import eigsh
OUT = Path(__file__).parent / "output"
OUT.mkdir(parents=True, exist_ok=True)
np.seterr(divide="ignore", invalid="ignore") # F = 0 only at the reference place x', which is masked
T_F = 1.0
M_LEV = (1.0, 1.5) # half-gaps of two clock levels
GHAT = (0.005, 0.01, 0.02, 0.04) # targets for max |g_b| * ell_0 / t_f
SIZES = {1: (32, 64), 2: (12, 20), 3: (8, 12)}
LOG, RES = [], {}
def log(s=""):
print(s)
LOG.append(s)
def graph(D, L):
n = L**D
co = np.array(np.unravel_index(np.arange(n), (L,) * D)).T
rows, cols = [], []
for mu in range(D):
sh = co.copy()
sh[:, mu] = (sh[:, mu] + 1) % L
nb = np.ravel_multi_index(tuple(sh.T), (L,) * D)
rows += [np.arange(n), nb]
cols += [nb, np.arange(n)]
A = sp.csr_matrix((np.ones(2 * D * n), (np.concatenate(rows), np.concatenate(cols))), shape=(n, n))
k = 2 * np.pi * np.arange(L) / L
grids = np.meshgrid(*([k] * D), indexing="ij")
om = sum(4 * np.sin(g / 2) ** 2 for g in grids)
return co, (-T_F * A).tocsr(), om
def ellpow(om, p=1, shift=0.0):
"""(1/|X|) sum_{k!=0} e^{iky}/(omega-shift)^p as an array over displacements y."""
inv = np.zeros_like(om)
m = om > 1e-12
inv[m] = 1.0 / (om[m] - shift) ** p
return np.fft.ifftn(inv).real.ravel()
def lowest(H, vec=True):
if H.shape[0] <= 2000:
if vec:
w, v = np.linalg.eigh(H.toarray())
return w[0], v[:, 0]
return np.linalg.eigvalsh(H.toarray())[0], None
w, v = eigsh(H, k=1, which="SA", tol=0)
return w[0], v[:, 0]
def point(n, i, g):
return sp.csr_matrix(([g], ([i], [i])), shape=(n, n))
def exact_rho(om, g, X):
"""Lowest eigenpair of C_1 + g P_0 from (3.2)-(3.3): returns delta, eta, rho(y)."""
wmin = om[om > 1e-12].min()
f = lambda d: 1.0 / (X * d) - ellpow(om, 1, d / T_F)[0] / T_F - 1.0 / g
d = brentq(f, 1e-14, T_F * wmin * (1 - 1e-12)) if g > 0 else brentq(f, -1e3, -1e-14)
eta = X * d / T_F
ld = ellpow(om, 1, d / T_F)
l2d0 = np.sum(ld**2)
return d, eta, (1 - eta * ld) ** 2 / (X + eta**2 * l2d0)
def slope(x, y):
return float(np.polyfit(np.log(np.abs(x)), np.log(np.abs(y)), 1)[0])
def run_graph(D, L):
tag = f"D{D}_L{L}"
co, C1, om = graph(D, L)
X = L**D
n = 1.0 / X
l1, l2, l3 = ellpow(om, 1), ellpow(om, 2), ellpow(om, 3)
l0, l20 = l1[0], l2[0]
ap = int(np.ravel_multi_index((L // 2,) * D, (L,) * D)) # antipode: r' of the pairs
F = -2 * (l1 - l1[ap]) / T_F # F / g_s, (3.13)
F2 = (l1**2 - l1[ap] ** 2 - 2 * (l2 - l2[ap]) / X) / (n * T_F**2)
F3 = 2 * (l1 * l2 - l1[ap] * l2[ap] + l20 * (l1 - l1[ap]) - (l3 - l3[ap]) / X) / (n**2 * T_F**3 * X)
dN = n * F # [dN(x)-dN(x')]/g_s
dTh = n**2 * F2 - n * l0 * F / T_F # [Theta_D(x)-Theta_D(x')]/g_s, (3.8)
mask = np.arange(X) != ap
eps00 = -2 * D * T_F
r = {"L": L, "D": D, "X": X, "ell_0": l0, "ell2_0": l20}
log(f"\n=== {tag}: |X|={X}, ell_0={l0:.6f}, ell2_0/|X|={l20 / X:.6e}")
# (A) weight from eigenvalues with both point terms vs Hellmann-Feynman density
gb = 0.02 / l0
_, v = lowest(C1 + point(X, 0, gb))
rho_hf = v**2
errs = {}
rng = np.random.default_rng(D * 100 + L)
pl = [(int(x), int(xs)) for x, xs in rng.integers(0, X, size=(4, 2))]
for gs in (1e-3, 5e-4):
e = 0.0
for x, xs in pl:
ep = lowest(C1 + point(X, x, gb) + point(X, xs, gs), vec=False)[0]
em = lowest(C1 + point(X, x, gb) + point(X, xs, -gs), vec=False)[0]
ydisp = np.ravel_multi_index(tuple((co[xs] - co[x]) % L), (L,) * D)
e = max(e, abs((ep - em) / (2 * gs) - rho_hf[ydisp]) / rho_hf.max())
errs[gs] = e
r["A_eig_vs_HF_relerr"] = errs
log(f"(A) eigenvalue weight vs HF density, max rel err: g_s=1e-3: {errs[1e-3]:.2e}, g_s=5e-4: {errs[5e-4]:.2e}")
# mirror (3.18): highest eigenvalue = - lowest of mirrored signs (L even, bipartite)
gs = 1e-3
Hp = (C1 + point(X, 0, gb) + point(X, 3 % X, gs))
Hm = (C1 + point(X, 0, -gb) + point(X, 3 % X, -gs))
top = -lowest(-Hp, vec=False)[0]
r["mirror_err"] = abs(top + lowest(Hm, vec=False)[0])
log(f"(M) highest state = -(lowest, mirrored signs): |diff| = {r['mirror_err']:.2e}")
# scans over levels, branches and gamma
gams = np.array(GHAT) * T_F / (l0 * max(M_LEV))
rows = []
for gam in gams:
for M in M_LEV:
for s in (1, -1):
g = s * gam * M
e, v = lowest(C1 + point(X, 0, g))
d, eta, rho_ex = exact_rho(om, g, X)
Cm = e - eps00 # mark cost, (3.4)
w = v**2 - v[ap] ** 2 # [W(x)-W(x')]/g_s, (3.6)
rows.append(dict(gam=gam, M=M, s=s, g=g, Cm=Cm, w=w,
cost_ex_err=abs(Cm - d), rho_ex_err=np.abs(v**2 - rho_ex).max() * X))
r["B_rho_exact_maxrelerr"] = max(q["rho_ex_err"] for q in rows)
r["C_cost_exact_maxerr"] = max(q["cost_ex_err"] for q in rows)
log(f"(B) density vs exact form (3.3): max |X|*|diff| = {r['B_rho_exact_maxrelerr']:.2e}")
log(f"(C) cost vs exact root (3.2): max |diff| = {r['C_cost_exact_maxerr']:.2e}")
# (D),(E),(H) deviations from the expansions, scaling in gamma (level M=1.5, sigma=+1)
sel = [q for q in rows if q["M"] == M_LEV[1] and q["s"] == 1]
gg = np.array([q["g"] for q in sel])
devC = [abs(q["Cm"] - (n * q["g"] - n * q["g"] ** 2 * l0 / T_F)) for q in sel]
devW = [np.abs(q["w"] - (q["g"] * dN + q["g"] ** 2 * dTh))[mask].max() for q in sel]
devU = [np.abs(q["w"] - (F * q["Cm"] + F2 * q["Cm"] ** 2 + F3 * q["Cm"] ** 3))[mask].max() for q in sel]
r["D_cost_2nd_slope"] = slope(gg, devC)
r["E_weight_2nd_slope"] = slope(gg, devW)
r["H_W_of_C_3rd_slope"] = slope(gg, devU)
log(f"(D) cost minus 2nd-order (3.5): slope in gamma = {r['D_cost_2nd_slope']:.3f}; devs {np.array(devC)}")
log(f"(E) weight minus 2nd-order (3.8): slope = {r['E_weight_2nd_slope']:.3f}; devs {np.array(devW)}")
log(f"(H) weight minus F C + F2 C^2 + F3 C^3 (3.12)-(3.13): slope = {r['H_W_of_C_3rd_slope']:.3f}")
# (F) ring: Theta_D of (3.8) against Theta of (7.6)
if D == 1:
rr = np.arange(L)
u = rr * (L - rr)
th76 = n / (3 * L**2 * T_F**2) * (u**2 - (L**2 - 2) / 2 * u)
diff = dTh - (th76 - th76[ap])
r["F_ring_Theta_vs_7.6_maxdiff"] = float(np.abs(diff).max())
log(f"(F) ring: Theta_D - Theta(7.6): max |diff| = {r['F_ring_Theta_vs_7.6_maxdiff']:.2e}")
# (I) single-level break: B = dW/(Cost*F_c) - 1, both couplings
ibr = []
for q in rows:
g, gam, M, s = q["g"], q["gam"], q["M"], q["s"]
Bfg = q["w"][mask] / (q["Cm"] * F[mask]) - 1
pfg = (q["Cm"] * F2 / F)[mask]
cost_a = s * M + q["Cm"]
Fa = gam * n * F / (1 + gam * n)
Ba = q["w"][mask] / (cost_a * Fa[mask]) - 1
pa = (g * dTh / dN)[mask]
ibr.append((gam, np.abs(Bfg - pfg).max(), np.abs(Ba - pa).max(), np.abs(Bfg).max(), np.abs(Ba).max()))
ib = np.array([t for t, q in zip(ibr, rows) if q["M"] == M_LEV[1] and q["s"] == 1])
r["I_fullgrip_break_resid_slope"] = slope(ib[:, 0], ib[:, 1])
r["I_a_break_resid_slope"] = slope(ib[:, 0], ib[:, 2])
r["I_fullgrip_break_size_slope"] = slope(ib[:, 0], ib[:, 3])
r["I_a_break_size_slope"] = slope(ib[:, 0], ib[:, 4])
log(f"(I) single level: |B| slope fg {r['I_fullgrip_break_size_slope']:.3f}, (a) {r['I_a_break_size_slope']:.3f}; "
f"|B - B_1st| slope fg {r['I_fullgrip_break_resid_slope']:.3f}, (a) {r['I_a_break_resid_slope']:.3f}")
# (J) level and branch differences (3.17)
jl = []
for gam in gams:
Q = {(q["M"], q["s"]): q for q in rows if q["gam"] == gam}
Fa = gam * n * F / (1 + gam * n)
for M in M_LEV: # branch pair (M,+),(M,-)
p, m_ = Q[(M, 1)], Q[(M, -1)]
g = gam * M
Rfg = (p["w"] - m_["w"])[mask] / ((p["Cm"] - m_["Cm"]) * F[mask]) - 1
pfg = (g**2 * (-2 * n * l0 / T_F * F2 + n**2 * F3) / F)[mask]
Ra = (p["w"] - m_["w"])[mask] / ((2 * M + p["Cm"] - m_["Cm"]) * Fa[mask]) - 1
pa = pfg + g**2 * (l0**2 - l20 / X) / T_F**2
jl.append(("branch", gam, M, np.abs(Rfg - pfg).max() / np.abs(pfg).max(),
np.abs(Ra - pa).max() / np.abs(pa).max()))
p, m_ = Q[(M_LEV[1], 1)], Q[(M_LEV[0], 1)] # level pair, sigma=+1
Rfg = (p["w"] - m_["w"])[mask] / ((p["Cm"] - m_["Cm"]) * F[mask]) - 1
pfg = ((p["Cm"] + m_["Cm"]) * F2 / F)[mask]
Ra = (p["w"] - m_["w"])[mask] / ((M_LEV[1] - M_LEV[0] + p["Cm"] - m_["Cm"]) * Fa[mask]) - 1
pa = (gam * (M_LEV[0] + M_LEV[1]) * dTh / dN)[mask]
jl.append(("level", gam, None, np.abs(Rfg - pfg).max() / np.abs(pfg).max(),
np.abs(Ra - pa).max() / np.abs(pa).max()))
r["J_rel_err_of_predicted_break_smallest_gamma"] = {
f"{k}_{M}": [fg, a] for k, gm, M, fg, a in jl if gm == gams[0]}
log("(J) relative error of the predicted difference breaks at the smallest gamma "
"[full grip, (a)]: " + ", ".join(f"{k}{'' if M is None else M}: {fg:.1e}, {a:.1e}"
for k, gm, M, fg, a in jl if gm == gams[0]))
log(" ... and at the largest gamma: " + ", ".join(f"{k}{'' if M is None else M}: {fg:.1e}, {a:.1e}"
for k, gm, M, fg, a in jl if gm == gams[-1]))
return r, (co, C1, om, l1, l2, ap)
def full_operator_check(D, L, gam=0.05, m=1.0, Nclock=2):
"""(R) reduction (3.1): spectrum of the full contract (body labels sigma, a; field) at tau=0
equals the union over (k, sigma) of theta*sigma*M_k + spec(C_1 + sigma*gam*M_k P_x + g_s P_xs)."""
co, C1, om = graph(D, L)
X = L**D
lam0 = 2 * np.pi # gives M_k = m + k/2
fk = np.exp(2j * np.pi * np.outer(np.arange(Nclock), np.arange(Nclock)) / Nclock) / np.sqrt(Nclock)
Dc = (2 * np.pi / (Nclock * lam0)) * fk @ np.diag(np.arange(Nclock)) @ fk.conj().T
body = np.kron(np.diag([1.0, -1.0]), m * np.eye(Nclock) + Dc) # sigma_3 (x) (m 1 + D)
x, xs, gs = 0, X // 3, 1e-3
Px = np.zeros((X, X)); Px[x, x] = 1
Ps = np.zeros((X, X)); Ps[xs, xs] = 1
out = {}
Mk = m + np.arange(Nclock) / 2
for theta, name in ((1, "a"), (0, "full_grip")):
H = theta * np.kron(body, np.eye(X)) + gam * np.kron(body, Px) \
+ np.kron(np.eye(2 * Nclock), C1.toarray() + gs * Ps)
full = np.sort(np.linalg.eigvalsh(H))
red = np.sort(np.concatenate([theta * s * M + np.linalg.eigvalsh(C1.toarray() + s * gam * M * Px + gs * Ps)
for s in (1, -1) for M in Mk]))
out[name] = float(np.abs(full - red).max())
log(f"(R) {D}D L={L}: full contract vs reduction (3.1), max |diff|: (a) {out['a']:.2e}, full grip {out['full_grip']:.2e}")
return out
def asymptotics():
"""(S) large-distance forms of ell = Lambda^+ used in (3.9)-(3.11) and (3.16)."""
res = {}
for L in (64, 128, 256, 512):
k =2 * np.pi * np.arange(L) / L
g1, g2 = np.meshgrid(k, k, indexing="ij")
l1 = ellpow(4 * np.sin(g1 / 2) ** 2 + 4 * np.sin(g2 / 2) ** 2).reshape(L, L)
res[f"2D_L{L}_ell0_minus_lnL_over_2pi"] = l1[0, 0] - np.log(L) / (2 * np.pi)
if L == 512: # ell(j,0) - [ell_0 - (ln j + gamma_E + 1.5 ln 2)/(2 pi) + j^2/(4 L^2)]
res["2D_L512_ell(j)_minus_(3.10)_form"] = {
j: l1[j, 0] - (l1[0, 0] - (np.log(j) + np.euler_gamma + 1.5 * np.log(2)) / (2 * np.pi)
+ j**2 / (4 * L**2)) for j in (2, 4, 8, 16, 32, 64)}
xi3 = 2.837297 / (4 * np.pi)
for L in (32, 64, 96):
k = 2 * np.pi * np.arange(L) / L
gr = np.meshgrid(k, k, k, indexing="ij")
l1 = ellpow(sum(4 * np.sin(g / 2) ** 2 for g in gr)).reshape(L, L, L)
res[f"3D_L{L}_ell0"] = l1[0, 0, 0]
res[f"3D_L{L}_ell0_plus_xi3_over_L"] = l1[0, 0, 0] + xi3 / L
if L == 96: # ell(j,0,0) - [1/(4 pi j) - xi3/L + j^2/(6 L^3)], relative to 1/(4 pi j)
res["3D_L96_rel_dev_of_ell(j)_from_(3.11)_form"] = {
j: (l1[j, 0, 0] - (1 / (4 * np.pi * j) - xi3 / L + j**2 / (6 * L**3))) * 4 * np.pi * j
for j in (2, 4, 8, 16, 24)}
log("\n(S) asymptotics of ell: " + json.dumps({k: (v if not isinstance(v, dict) else
{kk: round(float(vv), 5) for kk, vv in v.items()})
for k, v in res.items()}, default=float))
return res
def distance_scan(D, L, data):
"""(K) break against distance along a lattice axis, r' = antipode; g_b*ell_0/t_f = +-0.05."""
co, C1, om, l1, l2, ap = data
X = L**D
n = 1.0 / X
l0 = l1[0]
out = []
log(f"\n(K) distance scan {D}D, L={L}: j, B_fg(num), B_fg(1st order), B_a(num), B_a(1st order), "
f"-(eta/2)(ell(r)+ell(r')) [fg], -(g_b/t_f)ell_0 [(a) far limit]")
for s in (1, -1):
g = s * 0.05 / l0
gam, M = abs(g), 1.0
e, v = lowest(C1 + point(X, 0, g))
Cm = e + 2 * D * T_F
eta = Cm / (n * T_F)
for j in range(1, L // 2 + 1):
y = np.ravel_multi_index((j,) + (0,) * (D - 1), (L,) * D)
if y == ap:
continue
Fy = -2 * (l1[y] - l1[ap]) / T_F
F2y = (l1[y] ** 2 - l1[ap] ** 2 - 2 * (l2[y] - l2[ap]) / X) / (n * T_F**2)
w = v[y] ** 2 - v[ap] ** 2
Bfg = w / (Cm * Fy) - 1
Fa = gam * n * Fy / (1 + gam * n)
Ba = w / ((s * M + Cm) * Fa) - 1
dThy = n**2 * F2y - n * l0 * Fy / T_F
row = [s, j, Bfg, Cm * F2y / Fy, Ba, g * dThy / (n * Fy), -eta / 2 * (l1[y] + l1[ap]), -g * l0 / T_F]
out.append(row)
log(" s=%+d j=%2d % .4e % .4e % .4e % .4e % .4e % .4e" % tuple(row))
return out
if __name__ == "__main__":
data = {}
for D, Ls in SIZES.items():
for L in Ls:
RES[f"D{D}_L{L}"], data[(D, L)] = run_graph(D, L)
scans = {}
for D, Ls in SIZES.items():
scans[f"D{D}_L{Ls[-1]}"] = distance_scan(D, Ls[-1], data[(D, Ls[-1])])
# larger cubic torus for the distance dependence (sparse Lanczos)
co, C1, om = graph(3, 20)
l1, l2 = ellpow(om, 1), ellpow(om, 2)
ap = int(np.ravel_multi_index((10, 10, 10), (20,) * 3))
RES["D3_L20"] = {"ell_0": l1[0]}
log(f"\n=== D3_L20: ell_0={l1[0]:.6f}")
scans["D3_L20"] = distance_scan(3, 20, (co, C1, om, l1, l2, ap))
RES["distance_scans_columns"] = ["sigma", "j", "B_fg_num", "B_fg_1st", "B_a_num", "B_a_1st",
"-(eta/2)(ell(r)+ell(r'))", "-(g_b/t_f)ell_0"]
RES["distance_scans"] = scans
log("")
RES["R_reduction_full_operator"] = {f"D{D}_L{L}": full_operator_check(D, L)
for D, L in ((1, 32), (2, 8), (3, 6))}
RES["S_asymptotics"] = asymptotics()
def clean(o):
if isinstance(o, dict):
return {str(k): clean(v) for k, v in o.items()}
if isinstance(o, (list, tuple)):
return [clean(v) for v in o]
if isinstance(o, (np.floating, np.integer)):
return o.item()
return o
(OUT / "check_results.json").write_text(json.dumps(clean(RES), indent=1), encoding="utf-8")
(OUT / "check_log.txt").write_text("\n".join(LOG) + "\n", encoding="utf-8")
output/check_log.txt18 KB · raw
=== D1_L32: |X|=32, ell_0=2.664062, ell2_0/|X|=1.436096e+00
(A) eigenvalue weight vs HF density, max rel err: g_s=1e-3: 5.59e-06, g_s=5e-4: 1.40e-06
(M) highest state = -(lowest, mirrored signs): |diff| = 0.00e+00
(B) density vs exact form (3.3): max |X|*|diff| = 5.08e-11
(C) cost vs exact root (3.2): max |diff| = 2.95e-13
(D) cost minus 2nd-order (3.5): slope in gamma = 2.991; devs [1.16627836e-09 9.30385539e-09 7.40101657e-08 5.85392552e-07]
(E) weight minus 2nd-order (3.8): slope = 3.001; devs [6.91700019e-09 5.53543136e-08 4.43066650e-07 3.54637848e-06]
(H) weight minus F C + F2 C^2 + F3 C^3 (3.12)-(3.13): slope = 3.936
(F) ring: Theta_D - Theta(7.6): max |diff| = 2.11e-15
(I) single level: |B| slope fg 0.990, (a) 0.990; |B - B_1st| slope fg 1.966, (a) 1.981
(J) relative error of the predicted difference breaks at the smallest gamma [full grip, (a)]: branch1.0: 7.9e-06, 4.8e-05, branch1.5: 1.5e-05, 6.1e-05, level: 2.5e-03, 5.0e-03
... and at the largest gamma: branch1.0: 4.1e-04, 9.5e-04, branch1.5: 9.3e-04, 1.7e-03, level: 1.9e-02, 3.8e-02
=== D1_L64: |X|=64, ell_0=5.332031, ell2_0/|X|=5.702774e+00
(A) eigenvalue weight vs HF density, max rel err: g_s=1e-3: 2.25e-05, g_s=5e-4: 5.62e-06
(M) highest state = -(lowest, mirrored signs): |diff| = 0.00e+00
(B) density vs exact form (3.3): max |X|*|diff| = 3.42e-10
(C) cost vs exact root (3.2): max |diff| = 4.99e-13
(D) cost minus 2nd-order (3.5): slope in gamma = 2.991; devs [2.91991896e-10 2.32928109e-09 1.85281026e-08 1.46537716e-07]
(E) weight minus 2nd-order (3.8): slope = 3.000; devs [3.50192267e-09 2.80212992e-08 2.24235220e-07 1.79401083e-06]
(H) weight minus F C + F2 C^2 + F3 C^3 (3.12)-(3.13): slope = 3.937
(F) ring: Theta_D - Theta(7.6): max |diff| = 5.00e-16
(I) single level: |B| slope fg 0.990, (a) 0.990; |B - B_1st| slope fg 1.966, (a) 1.981
(J) relative error of the predicted difference breaks at the smallest gamma [full grip, (a)]: branch1.0: 6.4e-06, 2.0e-05, branch1.5: 1.5e-05, 3.2e-05, level: 2.5e-03, 5.0e-03
... and at the largest gamma: branch1.0: 4.1e-04, 7.1e-04, branch1.5: 9.2e-04, 1.5e-03, level: 1.9e-02, 3.8e-02
=== D2_L12: |X|=144, ell_0=0.444054, ell2_0/|X|=4.236694e-03
(A) eigenvalue weight vs HF density, max rel err: g_s=1e-3: 1.93e-07, g_s=5e-4: 4.85e-08
(M) highest state = -(lowest, mirrored signs): |diff| = 0.00e+00
(B) density vs exact form (3.3): max |X|*|diff| = 6.37e-11
(C) cost vs exact root (3.2): max |diff| = 4.97e-13
(D) cost minus 2nd-order (3.5): slope in gamma = 2.984; devs [1.90373019e-09 1.51576337e-08 1.20121630e-07 9.43235634e-07]
(E) weight minus 2nd-order (3.8): slope = 2.981; devs [3.41824170e-09 2.71900738e-08 2.15066569e-07 1.68249522e-06]
(H) weight minus F C + F2 C^2 + F3 C^3 (3.12)-(3.13): slope = 3.934
(I) single level: |B| slope fg 0.985, (a) 0.980; |B - B_1st| slope fg 1.965, (a) 1.981
(J) relative error of the predicted difference breaks at the smallest gamma [full grip, (a)]: branch1.0: 1.4e-05, 3.2e-05, branch1.5: 1.8e-05, 2.9e-05, level: 5.9e-04, 7.9e-03
... and at the largest gamma: branch1.0: 5.1e-04, 6.5e-04, branch1.5: 1.1e-03, 1.8e-03, level: 4.5e-03, 6.0e-02
=== D2_L20: |X|=400, ell_0=0.525479, ell2_0/|X|=4.025398e-03
(A) eigenvalue weight vs HF density, max rel err: g_s=1e-3: 2.73e-07, g_s=5e-4: 7.21e-08
(M) highest state = -(lowest, mirrored signs): |diff| = 0.00e+00
(B) density vs exact form (3.3): max |X|*|diff| = 1.29e-10
(C) cost vs exact root (3.2): max |diff| = 3.06e-13
(D) cost minus 2nd-order (3.5): slope in gamma = 2.984; devs [5.83192816e-10 4.64312159e-09 3.67910188e-08 2.88821749e-07]
(E) weight minus 2nd-order (3.8): slope = 2.981; devs [1.24641547e-09 9.91305574e-09 7.83874699e-08 6.12901570e-07]
(H) weight minus F C + F2 C^2 + F3 C^3 (3.12)-(3.13): slope = 3.891
(I) single level: |B| slope fg 0.985, (a) 0.979; |B - B_1st| slope fg 1.965, (a) 1.981
(J) relative error of the predicted difference breaks at the smallest gamma [full grip, (a)]: branch1.0: 1.5e-05, 8.5e-06, branch1.5: 3.3e-05, 2.4e-05, level: 3.9e-04, 8.0e-03
... and at the largest gamma: branch1.0: 5.7e-04, 8.7e-04, branch1.5: 1.3e-03, 2.1e-03, level: 3.0e-03, 6.1e-02
=== D3_L8: |X|=512, ell_0=0.224606, ell2_0/|X|=1.858469e-04
(A) eigenvalue weight vs HF density, max rel err: g_s=1e-3: 5.37e-08, g_s=5e-4: 1.68e-08
(M) highest state = -(lowest, mirrored signs): |diff| = 0.00e+00
(B) density vs exact form (3.3): max |X|*|diff| = 7.05e-11
(C) cost vs exact root (3.2): max |diff| = 3.09e-13
(D) cost minus 2nd-order (3.5): slope in gamma = 2.984; devs [1.07761861e-09 8.57860554e-09 6.79607966e-08 5.33302810e-07]
(E) weight minus 2nd-order (3.8): slope = 2.980; devs [9.78138329e-10 7.77746691e-09 6.14706875e-08 4.80186621e-07]
(H) weight minus F C + F2 C^2 + F3 C^3 (3.12)-(3.13): slope = 3.805
(I) single level: |B| slope fg 0.984, (a) 0.979; |B - B_1st| slope fg 1.964, (a) 1.980
(J) relative error of the predicted difference breaks at the smallest gamma [full grip, (a)]: branch1.0: 2.1e-05, 2.3e-05, branch1.5: 2.5e-05, 2.1e-05, level: 9.5e-05, 8.3e-03
... and at the largest gamma: branch1.0: 6.8e-04, 9.1e-04, branch1.5: 1.5e-03, 2.2e-03, level: 7.3e-04, 6.3e-02
=== D3_L12: |X|=1728, ell_0=0.233942, ell2_0/|X|=7.996514e-05
(A) eigenvalue weight vs HF density, max rel err: g_s=1e-3: 6.40e-08, g_s=5e-4: 6.45e-08
(M) highest state = -(lowest, mirrored signs): |diff| = 0.00e+00
(B) density vs exact form (3.3): max |X|*|diff| = 1.89e-10
(C) cost vs exact root (3.2): max |diff| = 2.42e-13
(D) cost minus 2nd-order (3.5): slope in gamma = 2.984; devs [3.07231407e-10 2.44570346e-09 1.93743548e-08 1.52022297e-07]
(E) weight minus 2nd-order (3.8): slope = 2.980; devs [2.89676337e-10 2.30318923e-09 1.82019437e-08 1.42160716e-07]
(H) weight minus F C + F2 C^2 + F3 C^3 (3.12)-(3.13): slope = 4.048
(I) single level: |B| slope fg 0.984, (a) 0.979; |B - B_1st| slope fg 1.963, (a) 1.980
(J) relative error of the predicted difference breaks at the smallest gamma [full grip, (a)]: branch1.0: 2.7e-05, 1.2e-05, branch1.5: 2.9e-05, 3.3e-05, level: 3.7e-05, 8.3e-03
... and at the largest gamma: branch1.0: 7.0e-04, 1.0e-03, branch1.5: 1.6e-03, 2.3e-03, level: 2.9e-04, 6.4e-02
(K) distance scan 1D, L=64: j, B_fg(num), B_fg(1st order), B_a(num), B_a(1st order), -(eta/2)(ell(r)+ell(r')) [fg], -(g_b/t_f)ell_0 [(a) far limit]
s=+1 j= 1 2.5276e-03 2.9639e-03 -4.5645e-02 -4.6886e-02 -9.6938e-03 -5.0000e-02
s=+1 j= 2 5.3999e-03 5.8000e-03 -4.2911e-02 -4.3907e-02 -7.5667e-03 -5.0000e-02
s=+1 j= 3 8.1842e-03 8.5431e-03 -4.0260e-02 -4.1026e-02 -5.5094e-03 -5.0000e-02
s=+1 j= 4 1.0880e-02 1.1193e-02 -3.7694e-02 -3.8242e-02 -3.5218e-03 -5.0000e-02
s=+1 j= 5 1.3487e-02 1.3750e-02 -3.5212e-02 -3.5556e-02 -1.6040e-03 -5.0000e-02
s=+1 j= 6 1.6004e-02 1.6214e-02 -3.2816e-02 -3.2967e-02 2.4409e-04 -5.0000e-02
s=+1 j= 7 1.8430e-02 1.8586e-02 -3.0506e-02 -3.0476e-02 2.0224e-03 -5.0000e-02
s=+1 j= 8 2.0766e-02 2.0864e-02 -2.8283e-02 -2.8083e-02 3.7311e-03 -5.0000e-02
s=+1 j= 9 2.3011e-02 2.3049e-02 -2.6146e-02 -2.5788e-02 5.3699e-03 -5.0000e-02
s=+1 j=10 2.5164e-02 2.5141e-02 -2.4097e-02 -2.3590e-02 6.9391e-03 -5.0000e-02
s=+1 j=11 2.7224e-02 2.7140e-02 -2.2135e-02 -2.1490e-02 8.4385e-03 -5.0000e-02
s=+1 j=12 2.9192e-02 2.9046e-02 -2.0262e-02 -1.9487e-02 9.8681e-03 -5.0000e-02
s=+1 j=13 3.1066e-02 3.0860e-02 -1.8478e-02 -1.7582e-02 1.1228e-02 -5.0000e-02
s=+1 j=14 3.2847e-02 3.2580e-02 -1.6782e-02 -1.5775e-02 1.2518e-02 -5.0000e-02
s=+1 j=15 3.4534e-02 3.4207e-02 -1.5176e-02 -1.4066e-02 1.3739e-02 -5.0000e-02
s=+1 j=16 3.6127e-02 3.5741e-02 -1.3660e-02 -1.2454e-02 1.4889e-02 -5.0000e-02
s=+1 j=17 3.7624e-02 3.7183e-02 -1.2235e-02 -1.0940e-02 1.5970e-02 -5.0000e-02
s=+1 j=18 3.9027e-02 3.8531e-02 -1.0899e-02 -9.5238e-03 1.6982e-02 -5.0000e-02
s=+1 j=19 4.0335e-02 3.9786e-02 -9.6546e-03 -8.2051e-03 1.7923e-02 -5.0000e-02
s=+1 j=20 4.1546e-02 4.0949e-02 -8.5012e-03 -6.9841e-03 1.8795e-02 -5.0000e-02
s=+1 j=21 4.2662e-02 4.2018e-02 -7.4390e-03 -5.8608e-03 1.9597e-02 -5.0000e-02
s=+1 j=22 4.3682e-02 4.2994e-02 -6.4685e-03 -4.8352e-03 2.0329e-02 -5.0000e-02
s=+1 j=23 4.4605e-02 4.3878e-02 -5.5897e-03 -3.9072e-03 2.0992e-02 -5.0000e-02
s=+1 j=24 4.5431e-02 4.4668e-02 -4.8029e-03 -3.0769e-03 2.1584e-02 -5.0000e-02
s=+1 j=25 4.6161e-02 4.5365e-02 -4.1082e-03 -2.3443e-03 2.2107e-02 -5.0000e-02
s=+1 j=26 4.6794e-02 4.5970e-02 -3.5059e-03 -1.7094e-03 2.2561e-02 -5.0000e-02
s=+1 j=27 4.7329e-02 4.6481e-02 -2.9960e-03 -1.1722e-03 2.2944e-02 -5.0000e-02
s=+1 j=28 4.7768e-02 4.6900e-02 -2.5786e-03 -7.3260e-04 2.3258e-02 -5.0000e-02
s=+1 j=29 4.8109e-02 4.7225e-02 -2.2539e-03 -3.9072e-04 2.3502e-02 -5.0000e-02
s=+1 j=30 4.8353e-02 4.7458e-02 -2.0219e-03 -1.4652e-04 2.3677e-02 -5.0000e-02
s=+1 j=31 4.8499e-02 4.7597e-02 -1.8827e-03 1.2410e-15 2.3781e-02 -5.0000e-02
s=-1 j= 1 -3.7745e-03 -3.2756e-03 4.8077e-02 4.6886e-02 1.0713e-02 5.0000e-02
s=-1 j= 2 -6.8627e-03 -6.4100e-03 4.4828e-02 4.3907e-02 8.3625e-03 5.0000e-02
s=-1 j= 3 -9.8424e-03 -9.4415e-03 4.1693e-02 4.1026e-02 6.0888e-03 5.0000e-02
s=-1 j= 4 -1.2714e-02 -1.2370e-02 3.8672e-02 3.8242e-02 3.8922e-03 5.0000e-02
s=-1 j= 5 -1.5479e-02 -1.5196e-02 3.5763e-02 3.5556e-02 1.7727e-03 5.0000e-02
s=-1 j= 6 -1.8137e-02 -1.7920e-02 3.2967e-02 3.2967e-02 -2.6976e-04 5.0000e-02
s=-1 j= 7 -2.0690e-02 -2.0540e-02 3.0281e-02 3.0476e-02 -2.2351e-03 5.0000e-02
s=-1 j= 8 -2.3137e-02 -2.3058e-02 2.7706e-02 2.8083e-02 -4.1234e-03 5.0000e-02
s=-1 j= 9 -2.5480e-02 -2.5473e-02 2.5241e-02 2.5788e-02 -5.9347e-03 5.0000e-02
s=-1 j=10 -2.7719e-02 -2.7785e-02 2.2886e-02 2.3590e-02 -7.6688e-03 5.0000e-02
s=-1 j=11 -2.9855e-02 -2.9995e-02 2.0639e-02 2.1490e-02 -9.3259e-03 5.0000e-02
s=-1 j=12 -3.1888e-02 -3.2101e-02 1.8500e-02 1.9487e-02 -1.0906e-02 5.0000e-02
s=-1 j=13 -3.3818e-02 -3.4105e-02 1.6469e-02 1.7582e-02 -1.2409e-02 5.0000e-02
s=-1 j=14 -3.5647e-02 -3.6006e-02 1.4546e-02 1.5775e-02 -1.3835e-02 5.0000e-02
s=-1 j=15 -3.7374e-02 -3.7805e-02 1.2728e-02 1.4066e-02 -1.5184e-02 5.0000e-02
s=-1 j=16 -3.9001e-02 -3.9500e-02 1.1017e-02 1.2454e-02 -1.6455e-02 5.0000e-02
s=-1 j=17 -4.0526e-02 -4.1093e-02 9.4123e-03 1.0940e-02 -1.7650e-02 5.0000e-02
s=-1 j=18 -4.1952e-02 -4.2583e-02 7.9126e-03 9.5238e-03 -1.8767e-02 5.0000e-02
s=-1 j=19 -4.3278e-02 -4.3971e-02 6.5178e-03 8.2051e-03 -1.9808e-02 5.0000e-02
s=-1 j=20 -4.4504e-02 -4.5255e-02 5.2278e-03 6.9841e-03 -2.0771e-02 5.0000e-02
s=-1 j=21 -4.5631e-02 -4.6437e-02 4.0422e-03 5.8608e-03 -2.1658e-02 5.0000e-02
s=-1 j=22 -4.6659e-02 -4.7516e-02 2.9606e-03 4.8352e-03 -2.2467e-02 5.0000e-02
s=-1 j=23 -4.7588e-02 -4.8492e-02 1.9829e-03 3.9072e-03 -2.3199e-02 5.0000e-02
s=-1 j=24 -4.8419e-02 -4.9366e-02 1.1087e-03 3.0769e-03 -2.3854e-02 5.0000e-02
s=-1 j=25 -4.9152e-02 -5.0136e-02 3.3783e-04 2.3443e-03 -2.4432e-02 5.0000e-02
s=-1 j=26 -4.9787e-02 -5.0804e-02 -3.2984e-04 1.7094e-03 -2.4933e-02 5.0000e-02
s=-1 j=27 -5.0323e-02 -5.1370e-02 -8.9452e-04 1.1722e-03 -2.5357e-02 5.0000e-02
s=-1 j=28 -5.0762e-02 -5.1832e-02 -1.3563e-03 7.3260e-04 -2.5704e-02 5.0000e-02
s=-1 j=29 -5.1104e-02 -5.2192e-02 -1.7154e-03 3.9072e-04 -2.5974e-02 5.0000e-02
s=-1 j=30 -5.1347e-02 -5.2449e-02 -1.9718e-03 1.4652e-04 -2.6167e-02 5.0000e-02
s=-1 j=31 -5.1493e-02 -5.2603e-02 -2.1257e-03 -1.2410e-15 -2.6282e-02 5.0000e-02
(K) distance scan 2D, L=20: j, B_fg(num), B_fg(1st order), B_a(num), B_a(1st order), -(eta/2)(ell(r)+ell(r')) [fg], -(g_b/t_f)ell_0 [(a) far limit]
s=+1 j= 1 -8.4978e-03 -8.4456e-03 -5.5733e-02 -5.8868e-02 -9.9924e-03 -5.0000e-02
s=+1 j= 2 -2.8846e-03 -2.8411e-03 -5.0387e-02 -5.2983e-02 -4.9406e-03 -5.0000e-02
s=+1 j= 3 4.8955e-04 5.2406e-04 -4.7174e-02 -4.9450e-02 -2.0532e-03 -5.0000e-02
s=+1 j= 4 2.8317e-03 2.8573e-03 -4.4943e-02 -4.7000e-02 -1.4103e-04 -5.0000e-02
s=+1 j= 5 4.5826e-03 4.5997e-03 -4.3275e-02 -4.5170e-02 1.2180e-03 -5.0000e-02
s=+1 j= 6 5.9393e-03 5.9485e-03 -4.1983e-02 -4.3754e-02 2.2061e-03 -5.0000e-02
s=+1 j= 7 7.0020e-03 7.0042e-03 -4.0971e-02 -4.2645e-02 2.9146e-03 -5.0000e-02
s=+1 j= 8 7.8122e-03 7.8084e-03 -4.0200e-02 -4.1801e-02 3.3937e-03 -5.0000e-02
s=+1 j= 9 8.3500e-03 8.3420e-03 -3.9688e-02 -4.1241e-02 3.6715e-03 -5.0000e-02
s=+1 j=10 8.5443e-03 8.5346e-03 -3.9503e-02 -4.1038e-02 3.7626e-03 -5.0000e-02
s=-1 j= 1 9.2705e-03 9.3346e-03 6.2334e-02 5.8868e-02 1.1044e-02 5.0000e-02
s=-1 j= 2 3.0870e-03 3.1401e-03 5.5825e-02 5.2983e-02 5.4607e-03 5.0000e-02
s=-1 j= 3 -6.2103e-04 -5.7922e-04 5.1922e-02 4.9450e-02 2.2693e-03 5.0000e-02
s=-1 j= 4 -3.1888e-03 -3.1580e-03 4.9220e-02 4.7000e-02 1.5587e-04 5.0000e-02
s=-1 j= 5 -5.1041e-03 -5.0838e-03 4.7204e-02 4.5170e-02 -1.3462e-03 5.0000e-02
s=-1 j= 6 -6.5852e-03 -6.5746e-03 4.5645e-02 4.3754e-02 -2.4384e-03 5.0000e-02
s=-1 j= 7 -7.7434e-03 -7.7414e-03 4.4425e-02 4.2645e-02 -3.2214e-03 5.0000e-02
s=-1 j= 8 -8.6250e-03 -8.6303e-03 4.3497e-02 4.1801e-02 -3.7509e-03 5.0000e-02
s=-1 j= 9 -9.2096e-03 -9.2200e-03 4.2882e-02 4.1241e-02 -4.0580e-03 5.0000e-02
s=-1 j=10 -9.4206e-03 -9.4330e-03 4.2660e-02 4.1038e-02 -4.1587e-03 5.0000e-02
(K) distance scan 3D, L=12: j, B_fg(num), B_fg(1st order), B_a(num), B_a(1st order), -(eta/2)(ell(r)+ell(r')) [fg], -(g_b/t_f)ell_0 [(a) far limit]
s=+1 j= 1 -6.0983e-03 -6.0929e-03 -5.3424e-02 -5.6398e-02 -6.3037e-03 -5.0000e-02
s=+1 j= 2 -1.5676e-03 -1.5633e-03 -4.9110e-02 -5.1641e-02 -1.9403e-03 -5.0000e-02
s=+1 j= 3 1.0608e-04 1.0947e-04 -4.7516e-02 -4.9885e-02 -4.3436e-04 -5.0000e-02
s=+1 j= 4 9.1017e-04 9.1284e-04 -4.6750e-02 -4.9042e-02 2.1651e-04 -5.0000e-02
s=+1 j= 5 1.3332e-03 1.3354e-03 -4.6347e-02 -4.8598e-02 5.1213e-04 -5.0000e-02
s=+1 j= 6 1.4763e-03 1.4782e-03 -4.6211e-02 -4.8448e-02 5.9926e-04 -5.0000e-02
s=-1 j= 1 6.7276e-03 6.7342e-03 5.9702e-02 5.6398e-02 6.9672e-03 5.0000e-02
s=-1 j= 2 1.7226e-03 1.7279e-03 5.4434e-02 5.1641e-02 2.1446e-03 5.0000e-02
s=-1 j= 3 -1.2512e-04 -1.2099e-04 5.2489e-02 4.9885e-02 4.8009e-04 5.0000e-02
s=-1 j= 4 -1.0122e-03 -1.0089e-03 5.1555e-02 4.9042e-02 -2.3930e-04 5.0000e-02
s=-1 j= 5 -1.4786e-03 -1.4760e-03 5.1064e-02 4.8598e-02 -5.6604e-04 5.0000e-02
s=-1 j= 6 -1.6362e-03 -1.6338e-03 5.0898e-02 4.8448e-02 -6.6234e-04 5.0000e-02
=== D3_L20: ell_0=0.241447
(K) distance scan 3D, L=20: j, B_fg(num), B_fg(1st order), B_a(num), B_a(1st order), -(eta/2)(ell(r)+ell(r')) [fg], -(g_b/t_f)ell_0 [(a) far limit]
s=+1 j= 1 -6.9872e-03 -6.9853e-03 -5.4273e-02 -5.7335e-02 -7.0592e-03 -5.0000e-02
s=+1 j= 2 -2.6715e-03 -2.6699e-03 -5.0163e-02 -5.2803e-02 -2.8079e-03 -5.0000e-02
s=+1 j= 3 -1.1026e-03 -1.1012e-03 -4.8669e-02 -5.1156e-02 -1.3055e-03 -5.0000e-02
s=+1 j= 4 -3.4119e-04 -3.3992e-04 -4.7944e-02 -5.0357e-02 -6.0565e-04 -5.0000e-02
s=+1 j= 5 1.0698e-04 1.0808e-04 -4.7517e-02 -4.9887e-02 -2.1404e-04 -5.0000e-02
s=+1 j= 6 4.0166e-04 4.0261e-04 -4.7236e-02 -4.9577e-02 2.7876e-05 -5.0000e-02
s=+1 j= 7 6.0608e-04 6.0691e-04 -4.7042e-02 -4.9363e-02 1.8294e-04 -5.0000e-02
s=+1 j= 8 7.4717e-04 7.4789e-04 -4.6907e-02 -4.9215e-02 2.8014e-04 -5.0000e-02
s=+1 j= 9 8.3330e-04 8.3396e-04 -4.6825e-02 -4.9124e-02 3.3392e-04 -5.0000e-02
s=+1 j=10 8.6280e-04 8.6343e-04 -4.6797e-02 -4.9093e-02 3.5117e-04 -5.0000e-02
s=-1 j= 1 7.7183e-03 7.7206e-03 6.0753e-02 5.7335e-02 7.8022e-03 5.0000e-02
s=-1 j= 2 2.9489e-03 2.9509e-03 5.5733e-02 5.2803e-02 3.1034e-03 5.0000e-02
s=-1 j= 3 1.2153e-03 1.2171e-03 5.3908e-02 5.1156e-02 1.4429e-03 5.0000e-02
s=-1 j= 4 3.7416e-04 3.7571e-04 5.3023e-02 5.0357e-02 6.6940e-04 5.0000e-02
s=-1 j= 5 -1.2080e-04 -1.1946e-04 5.2502e-02 4.9887e-02 2.3657e-04 5.0000e-02
s=-1 j= 6 -4.4615e-04 -4.4499e-04 5.2159e-02 4.9577e-02 -3.0810e-05 5.0000e-02
s=-1 j= 7 -6.7180e-04 -6.7079e-04 5.1922e-02 4.9363e-02 -2.0219e-04 5.0000e-02
s=-1 j= 8 -8.2750e-04 -8.2662e-04 5.1758e-02 4.9215e-02 -3.0963e-04 5.0000e-02
s=-1 j= 9 -9.2255e-04 -9.2174e-04 5.1658e-02 4.9124e-02 -3.6907e-04 5.0000e-02
s=-1 j=10 -9.5509e-04 -9.5432e-04 5.1623e-02 4.9093e-02 -3.8813e-04 5.0000e-02
(R) 1D L=32: full contract vs reduction (3.1), max |diff|: (a) 8.88e-15, full grip 3.55e-15
(R) 2D L=8: full contract vs reduction (3.1), max |diff|: (a) 1.24e-14, full grip 1.29e-14
(R) 3D L=6: full contract vs reduction (3.1), max |diff|: (a) 3.55e-14, full grip 1.64e-14
(S) asymptotics of ell: {"2D_L64_ell0_minus_lnL_over_2pi": 0.04875850142855509, "2D_L128_ell0_minus_lnL_over_2pi": 0.048763848880206684, "2D_L256_ell0_minus_lnL_over_2pi": 0.04876518701309496, "2D_L512_ell0_minus_lnL_over_2pi": 0.048765521625585895, "2D_L512_ell(j)_minus_(3.10)_form": {"2": 0.00428, "4": 0.00099, "8": 0.00021, "16": 5e-05, "32": 1e-05, "64": 3e-05}, "3D_L32_ell0": 0.2456765511865084, "3D_L32_ell0_plus_xi3_over_L": 0.2527323299766616, "3D_L64_ell0": 0.24920328389380492, "3D_L64_ell0_plus_xi3_over_L": 0.2527311732888815, "3D_L96_ell0": 0.25037913173616777, "3D_L96_ell0_plus_xi3_over_L": 0.2527310579995522, "3D_L96_rel_dev_of_ell(j)_from_(3.11)_form": {"2": 0.07793, "4": 0.01983, "8": 0.00413, "16": 0.0014, "24": 0.00353}}
output/check_results.json27 KB · raw
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