07-ilang-weight / 07-inertia-of-pair-full-grip
07inertia of pair full gripverified
Determines, for two full-grip bodies bound through the field, the pair's inertia against its parts and its cost, and whether the binding cost is weighed as it resists acceleration.
check_fixed_source.py24 KB · raw
"""Fixed held source g_s Phi_{x_s} in laboratory coordinates (07-inertia-of-pair-full-grip, N_f = 1, t_f = 1).
The source acts on the field at the fixed place x_s = 0; no P sectors. Centre contract (7.14): h_c = E_b(P) + V,
V = first-order matrix of the source between the Bloch states Psi_P of the selected level; its P = 0 symbol (row sum)
W_ex(X) = sum_P e^{iPX} <Psi_P|g_s Phi|Psi_0>, midpoint gauge, against W_0(X) = g_s rho_0(x_s - X) (7.15).
Parts (command-line argument; default: all):
exact EXACT, full laboratory space (bodies j, l, field y, branches), L = 12, 16: eigenvalues of C + g Phi_{x_s}
in the selected band against h_c with V from (ME) exact matrix elements, (mid) W_0, (held) held-field
average without kinetic part, (nobond) the two bodies without bond; row sums against W_0.
me64 EXACT first order (exact Bloch eigenvectors) and BAND (band contract (2.10) in laboratory coordinates
with the fixed source), L = 64: W_ex' against W_0', pair and bond.
me128 EXACT first order and BAND, L = 128, case R (the largest exact size run: one shift-invert solve per P).
bandS1, bandS2 BAND, L = 256: W_ex' and the lowest band eigenvalues with the fixed source.
Outputs: output/fixed_source_<part>.txt, output/fixed_source_<part>.json.
"""
import json
import sys
from pathlib import Path
import numpy as np
import scipy.sparse as sp
import scipy.sparse.linalg as spla
sys.path.insert(0, str(Path(__file__).parent))
from check_pair_full_grip import (TF, SX, shift, field, band_contract, body_kappa, body_level, pair_ops, # noqa: E402
parity_idx, params)
OUT = Path(__file__).parent / "output"
LOG = []
def log(s=""):
print(s, flush=True)
LOG.append(s)
def hsign(L, r): # signed representative of the relative place
r = np.asarray(r) % L
return np.where(r > L // 2, r - L, r)
def bloch_P(L):
P = 2 * np.pi * np.arange(L) / L
return np.where(P > np.pi + 1e-12, P - 2 * np.pi, P)
def band_setup(L, g1, g2, t1, t2):
E, Om, dd, kk, _ = band_contract(L, g1, g2, t1, t2)
ev = np.arange(0, L, 2)
low = (body_kappa(L, g1, t1) + body_kappa(L, g2, t2)) < 0
U = E[ev] + dd[0][ev] + dd[1][ev]
i0, i1 = np.arange(len(ev)), (np.arange(len(ev)) + 1) % len(ev)
def band(P):
h = np.diag(U).astype(complex)
t = kk[0][ev + 1] * np.exp(-2j * P) + kk[1][ev + 1]
h[i0, i1] += t
h[i1, i0] += t.conj()
w, v = np.linalg.eigh(h)
k = 0 if low else -1
return w[k], v[:, k]
return dict(Om=Om, ev=ev, low=low, band=band)
def sector_selected(L, g1, g2, t1, t2, bs, P, idx):
"""Exact selected level of the sector contract at P (parity class of the selected level)."""
eb, fb = bs['band'](P)
H = pair_ops(L, g1, g2, t1, t2, P)[idx][:, idx]
if len(idx) <= 3000:
w, v = np.linalg.eigh(H.toarray())
c = np.argsort(np.abs(w - eb))[:10]
w, v = w[c], v[:, c]
else:
w, v = spla.eigsh(H.tocsc(), k=8, sigma=eb, which='LM', tol=0)
best = (-1,)
for k in range(len(w)):
x = np.zeros(4 * L * L, complex)
x[idx] = v[:, k]
a = np.einsum('ry,ry->r', bs['Om'], x.reshape(2, 2, L, L)[0, 0])[bs['ev']]
ov = abs(np.vdot(fb, a)) ** 2
if ov > best[0]:
best = (ov, w[k], x.reshape(4 * L, L))
return best[1], best[2]
def mid_density(L, A): # field density at distance k from the midpoint; A: (4L, L) rows (s, r), cols y - j
pr = (np.abs(A) ** 2).reshape(4, L, L).sum(0)
dens = np.zeros(L)
for r in range(L):
h = int(hsign(L, r))
dens += np.roll(pr[r], -(h // 2)) if h % 2 == 0 else 0.5 * (np.roll(pr[r], -((h - 1) // 2)) + np.roll(pr[r], -((h + 1) // 2)))
return dens
class RowSum:
"""P = 0 symbol of the first-order source matrix in the midpoint gauge: W_ex(X)/g_s, source at x_s = 0."""
def __init__(self, L, A0, hrow):
self.L, self.A0, self.hrow, self.M = L, A0, np.asarray(hrow, float), {}
def add(self, P, A):
ph = np.exp(-0.5j * P * self.hrow)[:, None]
ov = np.vdot(ph * A, self.A0) # <u_P|u_0>
A = A * np.exp(1j * np.angle(ov))
q = np.einsum('my,my->y', A.conj(), self.A0)
self.M[float(P)] = np.sum(np.exp(1j * P * np.arange(self.L)) * q) / self.L
def W(self):
X = np.arange(self.L)
return sum(np.exp(1j * P * X) * m for P, m in self.M.items()).real
def grad_X(W, L, s): # central difference in X at X = x_s - s = -s
return (W[(-s + 1) % L] - W[(-s - 1) % L]) / 2
def grad_k(rho, L, s): # gradient in X of g_s rho(x_s - X), per unit g_s
return (rho[(s - 1) % L] - rho[(s + 1) % L]) / 2
def densities(L, g1, g2, t1, t2, bs, A0):
"""rho_mid (exact P=0), rho_held (f0^2-average of held densities), rho_nobond (bodies without bond),
single-body densities rho_1, rho_2 (exact P=0, from the body)."""
_, f0 = bs['band'](0.0)
f0 = np.abs(f0)
rb = [np.abs(u[:L]) ** 2 + np.abs(u[L:]) ** 2 for u in (body_level(L, g, t, 0.0, vec=True)[1] for g, t in ((g1, t1), (g2, t2)))]
held, nob = np.zeros(L), np.zeros(L)
for i, r in enumerate(bs['ev']):
h = int(hsign(L, r))
held += f0[i] ** 2 * np.roll(bs['Om'][r] ** 2, -(h // 2))
nob += f0[i] ** 2 * (np.roll(rb[0], -(h // 2)) + np.roll(rb[1], h // 2) - 1.0 / L)
return dict(mid=mid_density(L, A0), held=held, nobond=nob, b1=rb[0], b2=rb[1])
def single_rowsum(L, g, t): # exact first-order row sum of one body (centre = body place)
_, u0 = body_level(L, g, t, 0.0, vec=True)
rs = RowSum(L, u0.reshape(2, L), np.zeros(2))
for P in bloch_P(L):
rs.add(P, body_level(L, g, t, P, vec=True)[1].reshape(2, L))
return rs.W()
# ---------------- EXACT: full laboratory space, basis (s1, s2, j, r = l - j, y) ----------------
def lab_ops(L, g1, g2, t1, t2):
S, I2, IL = shift(L), sp.identity(2), sp.identity(L)
sg = np.array([1.0, -1.0])
s1, s2, j, r, y = np.meshgrid(*(np.arange(k) for k in (2, 2, L, L, L)), indexing='ij')
dg = sg[s1] * g1 * (y == j) + sg[s2] * g2 * (y == (j + r) % L)
mX = sp.csr_matrix(-1j * SX)
A1 = 0.5 * t1 * sp.kron(sp.kron(mX, I2), sp.kron(S.T, sp.kron(S, IL))) # j -> j+1, r -> r-1
A2 = 0.5 * t2 * sp.kron(sp.kron(I2, mX), sp.kron(IL, sp.kron(S.T, IL))) # l -> l+1: r -> r+1
H = sp.kron(sp.identity(4 * L * L), -TF * (S + S.T)) + sp.diags(dg.ravel()) + A1 + A1.conj().T + A2 + A2.conj().T
return H.tocsr(), np.flatnonzero(((s1 + s2 + r) % 2 == 0).ravel()), (y == 0).ravel().astype(float)
def centre_eigs(EP, Ps, M, g): # eigenvalues of h_c = E_b(P) + g M (plane-wave basis)
return np.sort(np.linalg.eigvalsh(np.diag(EP) + g * M))
def M_from_density(rho, Ps): # M(P,P') = (1/L) sum_X rho(x_s - X) e^{-i(P-P')X}, x_s = 0; Ps[k] = 2 pi k / L
L = len(rho)
Wq = np.fft.fft(rho[(-np.arange(L)) % L])
k = np.arange(L)
return Wq[(k[:, None] - k[None, :]) % L] / L
def part_exact():
res = []
for name, p, svals in (("EA equal bodies", (12, 0.15, 1.0, 0.3), (2, 3, 4)),
("EA equal bodies", (16, 0.15, 1.0, 0.3), (2, 3, 4, 5, 6)),
("EB unequal bodies and speeds", (12, 0.15, 0.5, 0.3, 0.15), (2, 3, 4)),
("EB unequal bodies and speeds", (16, 0.15, 0.5, 0.3, 0.15), (2, 3, 4, 5, 6))):
L, g1, g2, t1, t2 = params(*p)
bs, idx, Ps = band_setup(L, g1, g2, t1, t2), parity_idx(L), bloch_P(L)
EP, AP = [], []
for P in Ps:
e, A = sector_selected(L, g1, g2, t1, t2, bs, P, idx)
EP.append(e)
AP.append(A)
EP = np.array(EP)
H, lidx, phi = lab_ops(L, g1, g2, t1, t2)
B = np.zeros((4 * L ** 3, L), complex)
for a, (P, A) in enumerate(zip(Ps, AP)):
v = A.reshape(4, L, L)
B[:, a] = np.stack([np.roll(v, jj, axis=-1) * np.exp(1j * P * jj) for jj in range(L)], axis=1).ravel() / np.sqrt(L)
resid = float(np.max(np.linalg.norm(H @ B - B * EP[None, :], axis=0)))
out_cls = float(np.linalg.norm(np.delete(B, lidx, axis=0)))
Hc, Bc, pc = H[lidx][:, lidx].tocsc(), B[lidx], phi[lidx]
Mex = Bc.conj().T @ (pc[:, None] * Bc)
dn = densities(L, g1, g2, t1, t2, bs, AP[int(np.argmin(np.abs(Ps)))])
Mv = {"ME": Mex, "mid": M_from_density(dn['mid'], Ps), "held": M_from_density(dn['held'], Ps),
"nobond": M_from_density(dn['nobond'], Ps)}
g = 1e-7
K = L // 2 + 1
E0 = np.sort(EP)
sh, dev, nsel = {}, {}, []
for sgn in (1, -1):
w, v = spla.eigsh(Hc + sp.diags(sgn * g * pc), k=3 * L, sigma=E0.mean() + 1e-9, which='LM', tol=0)
wt = np.linalg.norm(Bc.conj().T @ v, axis=0) ** 2
sel = np.sort(w[wt > 0.5])
nsel.append(int(len(sel)))
sh[sgn] = {"exact": (sel - E0) / (sgn * g)}
for k, M in Mv.items():
sh[sgn][k] = (centre_eigs(EP, Ps, M, sgn * g) - E0) / (sgn * g)
for k in Mv:
dev[k] = [max(float(np.max(np.abs(sh[s]["exact"][:K] - sh[s][k][:K]))) for s in (1, -1)),
max(float(np.max(np.abs(sh[s]["exact"] - sh[s][k]))) for s in (1, -1))]
bond = max(float(np.max(np.abs(sh[s]["mid"][:K] - sh[s]["nobond"][:K]))) for s in (1, -1))
kin = max(float(np.max(np.abs(sh[s]["mid"][:K] - sh[s]["held"][:K]))) for s in (1, -1))
log(f"== EXACT {name}: L={L} g1={g1:.5g} g2={g2:.5g} tau1={t1:.4g} tau2={t2:.4g} (lab dimension {len(lidx)})")
log(f" Bloch states of the sector contracts in laboratory coordinates: max residual {resid:.1e}, weight outside class {out_cls:.1e}")
log(f" band levels selected with the fixed source: {nsel} of {L}; first-order shifts (eigenvalue - E_b)/g, g={g:g}")
log(f" scale over the lowest {K} levels (|P|<=pi/2): bond effect max|mid-nobond|={bond:.3e}, kinetic effect max|mid-held|={kin:.3e}")
for k in Mv:
log(f" max|exact - h_c[{k}]|: lowest {K}: {dev[k][0]:.3e} ({dev[k][0] / bond:.3f} of bond effect), all {L}: {dev[k][1]:.3e}")
rs = RowSum(L, AP[int(np.argmin(np.abs(Ps)))], np.tile(hsign(L, np.arange(L)), 4))
for P, A in zip(Ps, AP):
rs.add(P, A)
Wp, W1, W2 = rs.W(), single_rowsum(L, g1, t1), single_rowsum(L, g2, t2)
rows = []
for s in svals:
ex = grad_X(Wp, L, s)
exb = ex - grad_X(W1, L, s) - grad_X(W2, L, s)
d = {k: grad_k(dn[k], L, s) for k in ("mid", "held", "nobond", "b1", "b2")}
Wb = d["mid"] - d["b1"] - d["b2"]
rows.append(dict(s=s, Wex=ex, W0=d["mid"], Wheld=d["held"], Wex_bond=exb, Wb=Wb,
Wb_held=d["held"] - d["b1"] - d["b2"], Wex1=grad_X(W1, L, s), W1=d["b1"]))
log(f" s={s}: W_ex'/g_s={ex:.6e} W_0'/g_s={d['mid']:.6e}; bond: W_ex,b'={exb:.4e} W_b'={Wb:.4e} "
f"(diff/|W_b'|={(exb - Wb) / abs(Wb):.2e}; held-field average {(d['held'] - d['b1'] - d['b2'] - Wb) / abs(Wb):.2e}; "
f"single body W_ex,1'/W_1'-1={grad_X(W1, L, s) / d['b1'] - 1:.1e})")
res.append(dict(case=name, L=L, g1=g1, g2=g2, tau1=t1, tau2=t2, residual=resid, n_selected=nsel, K=K,
bond_effect=bond, kinetic_effect=kin, deviations=dev, gradients=rows))
return res
# ---------------- EXACT first order at larger L (exact Bloch eigenvectors, v_{-P} = Z conj v_P) ----------------
def exact_rowsums(L, g1, g2, t1, t2, bs):
idx, Ps = parity_idx(L), bloch_P(L)
Z = np.repeat([1.0, -1.0, -1.0, 1.0], L)[:, None]
E0, A0 = sector_selected(L, g1, g2, t1, t2, bs, 0.0, idx)
rs = RowSum(L, A0, np.tile(hsign(L, np.arange(L)), 4))
rs.add(0.0, A0)
for P in Ps[(Ps > 0)]:
_, A = sector_selected(L, g1, g2, t1, t2, bs, P, idx)
rs.add(P, A)
if P < np.pi - 1e-12:
rs.add(-P, Z * A.conj())
return rs.W(), A0, E0
# ---------------- BAND: band contract (2.10) in laboratory coordinates with the fixed source ----------------
def band_lab(L, g1, g2, t1, t2, gsrc):
"""Sparse band contract on (j, r/2), r = l - j even, with the source gsrc on the field at x_s = 0:
exact held blocks with the source, second-order Schrieffer-Wolff through the flipped held levels."""
ev = np.arange(0, L, 2)
nr = len(ev)
E, Om = np.zeros((L, nr)), np.zeros((L, nr, L))
for j in range(L):
e, u = np.linalg.eigh(np.array([field(L, [(j, g1), ((j + r) % L, g2), (0, gsrc)]) for r in ev]))
E[j], Om[j] = e[:, 0], u[:, :, 0] * np.sign(u[:, :, 0].sum(1))[:, None]
I = lambda j, r: (np.asarray(j) % L) * nr + (np.asarray(r) % L) // 2
diag, rows, cols, vals = E.copy(), [], [], []
ro = np.arange(1, L, 2)
for x in range(L):
for body, t in ((1, t1), (2, t2)):
if body == 1: # body 1 flipped at x, body 2 at x + r (r odd): connects (x-1, r+1) and (x+1, r-1)
Hs = [field(L, [(x, -g1), ((x + r) % L, g2), (0, gsrc)]) for r in ro]
ja, ra, jb, rb = (x - 1) % L, (ro + 1) % L, (x + 1) % L, (ro - 1) % L
else: # body 1 at x, body 2 flipped at x + r: connects (x, r-1) and (x, r+1)
Hs = [field(L, [(x, g1), ((x + r) % L, -g2), (0, gsrc)]) for r in ro]
ja, ra, jb, rb = x, (ro - 1) % L, x, (ro + 1) % L
ek, uk = np.linalg.eigh(np.array(Hs))
Oa, Ob = Om[ja, ra // 2], Om[jb, rb // 2]
Ea, Eb = E[ja, ra // 2][:, None], E[jb, rb // 2][:, None]
oa, ob = np.einsum('axk,ax->ak', uk, Oa), np.einsum('axk,ax->ak', uk, Ob)
q = t ** 2 / 4
np.add.at(diag, (np.full(len(ro), ja), ra // 2), q * np.sum(oa ** 2 / (Ea - ek), 1))
np.add.at(diag, (np.full(len(ro), jb), rb // 2), q * np.sum(ob ** 2 / (Eb - ek), 1))
k = -q * np.sum(oa * ob * 0.5 * (1 / (Ea - ek) + 1 / (Eb - ek)), 1)
a, b = I(np.full(len(ro), ja), ra), I(np.full(len(ro), jb), rb)
rows += [a, b]
cols += [b, a]
vals += [k, k]
n = L * nr
H = sp.coo_matrix((np.concatenate(vals), (np.concatenate(rows), np.concatenate(cols))), shape=(n, n)).tocsr()
return H + sp.diags(diag.ravel())
def band_lab_first_order(L, g1, g2, t1, t2):
"""The same band contract at g_s = 0 (H0) and its exact first derivative in g_s (dH), from the held spectra
without source: dE = Omega(p)^2, dOmega = R' e_p Omega(p), and dR = R e_p e_p^T R, dR/dE = -R^2 in the
resolvent form kappa = -(tau^2/8)[<b|R(E_a)|a> + <b|R(E_b)|a>], d = (tau^2/4)<a|R(E_a)|a>."""
ev = np.arange(0, L, 2)
nr = len(ev)
G = {}
for r in ev:
e, U = np.linalg.eigh(field(L, [(0, g1), (r, g2)]))
Om = U[:, 0] * np.sign(U[:, 0].sum())
G[r] = (e[0], Om, ((U[:, 1:] / (e[0] - e[1:])) @ U[:, 1:].T) * Om[None, :])
p = np.arange(L) # source place in the frame of the intermediate held level
x = (-p) % L # its origin (place of body 1)
I = lambda j, r: (np.asarray(j) % L) * nr + (np.asarray(r) % L) // 2
d0, d1 = np.zeros((L, nr)), np.zeros((L, nr))
for r in ev:
d0[:, r // 2] = G[r][0]
d1[x, r // 2] = G[r][1][p] ** 2
rows, cols, v0, v1 = [], [], [], []
for body, t in ((1, t1), (2, t2)):
q = t ** 2 / 4
for rho in range(1, L, 2):
if body == 1:
e, U = np.linalg.eigh(field(L, [(0, -g1), (rho, g2)]))
ra, rb, sa, sb, pa, pb, ja, jb = (rho + 1) % L, rho - 1, -1, 1, (p + 1) % L, (p - 1) % L, (x - 1) % L, (x + 1) % L
else:
e, U = np.linalg.eigh(field(L, [(0, g1), (rho, -g2)]))
ra, rb, sa, sb, pa, pb, ja, jb = rho - 1, (rho + 1) % L, 0, 0, p, p, x, x
(Ea, Oa, dOa), (Eb, Ob, dOb) = G[ra], G[rb]
a, b = U.T @ np.roll(Oa, sa), U.T @ np.roll(Ob, sb)
da, db = U.T @ np.roll(dOa[:, pa], sa, axis=0), U.T @ np.roll(dOb[:, pb], sb, axis=0)
dEa, dEb = Oa[pa] ** 2, Ob[pb] ** 2
ra_, rb_ = 1 / (Ea - e), 1 / (Eb - e)
def dG(ri, dE):
return (db.T @ (a * ri)) + (da.T @ (b * ri)) + (U @ (b * ri)) * (U @ (a * ri)) - dE * np.sum(a * b * ri ** 2)
k0 = -q * 0.5 * (np.sum(a * b * ra_) + np.sum(a * b * rb_))
k1 = -q * 0.5 * (dG(ra_, dEa) + dG(rb_, dEb))
for (jj, rr, o, do, ri, dE) in ((ja, ra, a, da, ra_, dEa), (jb, rb, b, db, rb_, dEb)):
np.add.at(d0, (jj, np.full(L, rr // 2)), q * np.sum(o * o * ri))
np.add.at(d1, (jj, np.full(L, rr // 2)), q * (2 * (do.T @ (o * ri)) + (U @ (o * ri)) ** 2 - dE * np.sum(o * o * ri ** 2)))
ia, ib = I(ja, ra), I(jb, rb)
rows += [ia, ib]
cols += [ib, ia]
v0 += [np.full(L, k0), np.full(L, k0)]
v1 += [k1, k1]
n = L * nr
mk = lambda vals, dg: (sp.coo_matrix((np.concatenate(vals), (np.concatenate(rows), np.concatenate(cols))), shape=(n, n)).tocsr()
+ sp.diags(dg.ravel()))
return mk(v0, d0), mk(v1, d1)
def single_band_lab(L, g, t, gsrc): # band contract of one body in laboratory coordinates with the source
E, Om = np.zeros(L), np.zeros((L, L))
for j in range(L):
e, u = np.linalg.eigh(field(L, [(j, g), (0, gsrc)]))
E[j], Om[j] = e[0], u[:, 0] * np.sign(u[:, 0].sum())
H = np.diag(E)
for x in range(L):
ek, uk = np.linalg.eigh(field(L, [(x, -g), (0, gsrc)]))
a, b = (x - 1) % L, (x + 1) % L
oa, ob, q = uk.T @ Om[a], uk.T @ Om[b], t ** 2 / 4
H[a, a] += q * np.sum(oa ** 2 / (E[a] - ek))
H[b, b] += q * np.sum(ob ** 2 / (E[b] - ek))
H[a, b] = H[b, a] = H[a, b] - q * np.sum(oa * ob * 0.5 * (1 / (E[a] - ek) + 1 / (E[b] - ek)))
return H
def band_rowsums(L, g1, g2, t1, t2, bs, gsrc=1e-5, fd=False):
"""First-order source term dH of the band contract in laboratory coordinates (odd part in gsrc), its row sum
in the midpoint gauge (pair) and per body; H0 and Bloch vectors of the band contract."""
H0, dH = band_lab_first_order(L, g1, g2, t1, t2)
if fd: # validation of the first derivative against finite differences of the band contract with the source
Hp, Hm = band_lab(L, g1, g2, t1, t2, gsrc), band_lab(L, g1, g2, t1, t2, -gsrc)
fdv = (float(abs((Hp + Hm) / 2 - H0).max() / abs(H0).max()), float(abs((Hp - Hm) / (2 * gsrc) - dH).max() / abs(dH).max()))
log(f" band contract with the source: analytic first order vs finite differences +-{gsrc:g}: rel. max diff H0 {fdv[0]:.1e}, dH {fdv[1]:.1e}")
ev, Ps = bs['ev'], bloch_P(L)
j = np.arange(L)
F = {float(P): bs['band'](P) for P in Ps}
Bl = lambda P: (np.exp(1j * P * j)[:, None] * F[float(P)][1][None, :]).ravel() / np.sqrt(L)
resid = max(float(np.linalg.norm(H0 @ Bl(P) - F[float(P)][0] * Bl(P))) for P in Ps[:4])
hr = hsign(L, ev)
f0 = F[0.0][1]
phi0 = (dH @ Bl(0.0)).reshape(L, len(ev))
FT = np.fft.fft(phi0, axis=0) # sum_j e^{-iPj} phi0(j, r), P = 2 pi k / L
M = {}
for k, P in enumerate(2 * np.pi * np.arange(L) / L):
Pr = float(bloch_P(L)[k])
f = F[Pr][1]
f = f * np.exp(1j * np.angle(np.vdot(np.exp(-0.5j * Pr * hr) * f, f0)))
M[Pr] = np.vdot(f, FT[k]) / np.sqrt(L)
W = sum(np.exp(1j * P * j) * m for P, m in M.items()).real
Ws = []
for g, t in ((g1, t1), (g2, t2)):
dHs = (single_band_lab(L, g, t, gsrc) - single_band_lab(L, g, t, -gsrc)) / (2 * gsrc)
Ws.append(dHs.sum(1)) # row sum: the Wannier states of one body are its places
return W, Ws, H0, dH, F, resid
def report(L, svals, dn, ex=None, band=None):
"""Gradients in X at distance s (per unit g_s): pair W', bond W_b' = W_pair' - W_1' - W_2', against W_0' (7.15)."""
rows = []
for s in svals:
d = {k: grad_k(dn[k], L, s) for k in ("mid", "held", "b1", "b2")}
Wb = d["mid"] - d["b1"] - d["b2"]
row = dict(s=s, W0=d["mid"], Wb=Wb, Wb_held=d["held"] - d["b1"] - d["b2"], W1=d["b1"], W2=d["b2"])
txt = f" s={s}: W_0'/g_s={d['mid']:.6e}, W_b'/g_s={Wb:.4e}; held-field average (no kinetic part): W_b,held'/W_b'-1={row['Wb_held'] / Wb - 1:.2e}"
for lab, W in (("exact", ex), ("band", band)):
if W is None:
continue
Wp, W1, W2 = (grad_X(w, L, s) for w in W)
row[lab] = dict(W=Wp, Wb=Wp - W1 - W2, W1=W1, W2=W2)
txt += (f"\n {lab} fixed source: W'/W_0'-1={Wp / d['mid'] - 1:.2e}, (W'-W_0')/|W_b'|={(Wp - d['mid']) / abs(Wb):.2e}, "
f"W_b'/W_b'(7.15)-1={(Wp - W1 - W2) / Wb - 1:.2e} (= Gamma_b ratio), W_1'/W_1'(7.15)-1={W1 / d['b1'] - 1:.1e}, W_2'/W_2'(7.15)-1={W2 / d['b2'] - 1:.1e}")
log(txt)
rows.append(row)
return rows
def part_large(name, p, svals, exact=True, band=True, eig=False):
L, g1, g2, t1, t2 = params(*p)
bs = band_setup(L, g1, g2, t1, t2)
log(f"== {name}: L={L} g1={g1:.5g} g2={g2:.5g} tau1={t1:.4g} tau2={t2:.4g}")
res = dict(case=name, L=L, g1=g1, g2=g2, tau1=t1, tau2=t2)
if exact:
Wp, A0, E0 = exact_rowsums(L, g1, g2, t1, t2, bs)
Wex = (Wp, single_rowsum(L, g1, t1), single_rowsum(L, g2, t2))
else:
E0, A0 = sector_selected(L, g1, g2, t1, t2, bs, 0.0, parity_idx(L))
Wex = None
dn = densities(L, g1, g2, t1, t2, bs, A0)
Wbd = None
if band:
Wbp, Wbs, H0, dH, F, resid = band_rowsums(L, g1, g2, t1, t2, bs, fd=(L <= 64))
Wbd = (Wbp, Wbs[0], Wbs[1])
log(f" band contract in laboratory coordinates: residual of its Bloch states {resid:.1e} (without source)")
res['gradients'] = report(L, svals, dn, Wex, Wbd)
if eig: # lowest band eigenvalues of the band contract with the fixed source against centre contracts
Ps = bloch_P(L)
j = np.arange(L)
B = np.stack([(np.exp(1j * P * j)[:, None] * F[float(P)][1][None, :]).ravel() / np.sqrt(L) for P in Ps], axis=1)
EP = np.array([F[float(P)][0] for P in Ps])
Mv = {"ME": B.conj().T @ (dH @ B), "mid": M_from_density(dn['mid'], Ps), "held": M_from_density(dn['held'], Ps)}
nob = M_from_density(dn['nobond'], Ps)
g, K = 1e-6, 21
Es = np.sort(EP)
sig = Es[0] - 1e-3 * (Es[1] - Es[0]) if bs['low'] else Es[-1] + 1e-3 * (Es[-1] - Es[-2])
w, v = spla.eigsh((H0 + g * dH).tocsc(), k=60, sigma=sig, which='LM', tol=0)
wt = np.linalg.norm(B.conj().T @ v, axis=0) ** 2
sel = np.sort(w[wt > 0.5])
sel = sel[:K] if bs['low'] else sel[-K:]
E0s = Es[:K] if bs['low'] else Es[-K:]
pick = (lambda a: a[:K]) if bs['low'] else (lambda a: a[-K:])
sh = {"band": (sel - E0s) / g}
for k, M in list(Mv.items()) + [("nobond", nob)]:
sh[k] = (pick(centre_eigs(EP, Ps, M, g)) - E0s) / g
bond = float(np.max(np.abs(sh["mid"] - sh["nobond"])))
log(f" lowest {len(sel)} band levels with the fixed source (g={g:g}), first-order shifts; bond effect max|mid-nobond|={bond:.3e}, "
f"kinetic effect max|mid-held|={float(np.max(np.abs(sh['mid'] - sh['held']))):.3e}")
res['eig'] = {}
for k in Mv:
dv = float(np.max(np.abs(sh["band"] - sh[k])))
res['eig'][k] = dv
log(f" max|band - h_c[{k}]|={dv:.3e} ({dv / bond:.3f} of bond effect)")
res['eig']['bond_effect'] = bond
return res
PARTS = {
"exact": lambda: part_exact(),
"me64": lambda: [part_large("A equal bodies", (64, 0.15, 1.0, 0.3), (4, 8, 12, 16)),
part_large("B unequal bodies and speeds", (64, 0.15, 0.5, 0.3, 0.15), (4, 8, 12, 16))],
"me128": lambda: [part_large("R equal bodies, v1 case", (128, 0.15, 1.0, 0.3), (8, 16, 32))],
"bandS1": lambda: [part_large("S1 equal bodies, regime (ii)", (256, 0.10, 1.0, 0.4), (16, 32, 64), exact=False, eig=True)],
"bandS2": lambda: [part_large("S2 equal bodies, regime (ii), smaller eta", (256, 0.05, 1.0, 0.4), (16, 32, 64), exact=False, eig=True)],
}
if __name__ == "__main__":
OUT.mkdir(exist_ok=True)
for part in (sys.argv[1:] or list(PARTS)):
LOG.clear()
log(f"# part {part}")
out = PARTS[part]()
(OUT / f"fixed_source_{part}.txt").write_text("\n".join(LOG) + "\n", encoding="utf-8")
(OUT / f"fixed_source_{part}.json").write_text(json.dumps(out, indent=1, default=float), encoding="utf-8")
check_pair_full_grip.py19 KB · raw
"""Check of 07-ilang-weight/07-inertia-of-pair-full-grip (N_f = 1, t_f = 1).
Item 1: distance of the group G from all other held levels of its parity class (field excitations,
single flips, joint flip), from the exact held spectra. Item 2: curvature in P of the exact bound level
of two full-grip bodies and one field object (sector contract continued in P) against (7.3), (7.5).
Item 3: cost (7.6), (7.8). Item 4: W_0 = g_s rho_0, rho_0 the field density at fixed distance from the
pair's midpoint in the exact P=0 eigenvector (7.15), cross-checked by the auxiliary midpoint probe (a
mathematical operator on the sector space that follows the midpoint; NOT a held source and not a contract
of the description), and Gamma_b = W_b'/D_0 against (7.9)-(7.12); held-field averages (v1) for comparison.
The fixed held source g_s Phi_{x_s} in laboratory coordinates is checked in check_fixed_source.py.
Ring flux remainder rho_L = E''(band, ring)/E''_line - 1 of (6.6), with small-w controls F (I2).
Outputs: output/check_log.txt, output/check_results.json.
"""
import json
from pathlib import Path
import numpy as np
import scipy.sparse as sp
import scipy.sparse.linalg as spla
OUT = Path(__file__).parent / "output"
TF = 1.0
SX = np.array([[0.0, 1.0], [1.0, 0.0]])
GS = 1e-5 # source strength (weights are taken as odd parts in g_s)
LOG = []
def log(s=""):
print(s, flush=True)
LOG.append(s)
def shift(L): # (S g)(y) = g(y+1)
return sp.csr_matrix((np.ones(L), (np.arange(L), (np.arange(L) + 1) % L)), shape=(L, L))
def field(L, src): # one field object: C_1 + sum_x g_x Phi_x
H = -TF * (np.eye(L, k=1) + np.eye(L, k=-1))
H[0, -1] = H[-1, 0] = -TF
for x, g in src:
H[x % L, x % L] += g
return H
def eps0(L, src):
return np.linalg.eigvalsh(field(L, src))[0]
def ground(L, src): # spectrum, eigenvectors, positive ground vector
e, u = np.linalg.eigh(field(L, src))
return e, u, u[:, 0] * np.sign(u[:, 0].sum())
def ell(L, x):
x = np.mod(x, L)
return (L * L - 1) / (12 * L) - x * (L - x) / (2 * L)
def curv(f, d): # E''(0) by Richardson; E(P) = E(-P) (sigma3 x sigma3 times conjugation)
e0, e1, e2 = f(0.0), f(d), f(2 * d)
return (4 * 2 * (e1 - e0) / d**2 - 2 * (e2 - e0) / (2 * d) ** 2) / 3, e0
def mindist(a, b): # min |x - y| over x in a, y in b
b = np.sort(np.asarray(b))
i = np.clip(np.searchsorted(b, a), 1, len(b) - 1)
return float(np.min(np.minimum(np.abs(a - b[i - 1]), np.abs(a - b[i]))))
# ---------- one body (sector contract (5.1), full grip); basis (branch, field place - body place) ----------
def body_level(L, g, tau, P, vec=False):
S = shift(L)
H = sp.kron(sp.identity(2), -TF * (S + S.T)) + sp.kron(sp.diags([1.0, -1.0]), sp.diags((np.arange(L) == 0) * g))
A = 0.5 * tau * np.exp(-1j * P) * sp.kron(sp.csr_matrix(-1j * SX), S)
w, v = np.linalg.eigh((H + A + A.conj().T).toarray())
om = ground(L, [(0, g)])[2]
i = np.argmax(np.abs(om @ v[:L, :]) ** 2)
return (w[i], v[:, i]) if vec else w[i]
def body_kappa(L, g, tau): # hop amplitude of the band contract of one body (second-order Schrieffer-Wolff)
e0, _, om = ground(L, [(0, g)])
ek, uk, _ = ground(L, [(0, -g)])
return -tau**2 / 4 * np.sum((uk.T @ np.roll(om, -1)) * (uk.T @ np.roll(om, 1)) / (e0[0] - ek))
# ---------- pair: sector contract, basis ((s1,s2), r = l-j, y = field place - j) ----------
def pair_ops(L, g1, g2, t1, t2, P):
S, I2, IL = shift(L), sp.identity(2), sp.identity(L)
sg, r, y = np.array([1.0, -1.0]), np.arange(L), np.arange(L)
dg = (sg[:, None, None, None] * g1 * (y == 0)[None, None, None, :]
+ sg[None, :, None, None] * g2 * (y[None, :] == r[:, None])[None, None, :, :])
H0 = sp.kron(sp.identity(4 * L), -TF * (S + S.T)) + sp.diags(dg.ravel())
mX = sp.csr_matrix(-1j * SX)
A1 = 0.5 * t1 * np.exp(-1j * P) * sp.kron(sp.kron(mX, I2), sp.kron(S, S))
A2 = 0.5 * t2 * sp.kron(sp.kron(I2, mX), sp.kron(S.T, IL))
return (H0 + A1 + A1.conj().T + A2 + A2.conj().T).tocsr()
def parity_idx(L): # the class r + s1 + s2 even (conserved), which contains the selected level
s1, s2, r, _ = np.meshgrid(*(np.arange(k) for k in (2, 2, L, L)), indexing='ij')
return np.flatnonzero(((s1 + s2 + r) % 2 == 0).ravel())
def midpoint_weights(L, z): # source of unit strength at distance z from the pair's midpoint j + r/2
w = np.zeros((L, L))
for r in range(L):
h = r - L if r > L // 2 else r
for a in ([h // 2] if h % 2 == 0 else [(h - 1) // 2, (h + 1) // 2]):
w[r, (z + a) % L] += 1.0 if h % 2 == 0 else 0.5
return np.tile(w.ravel(), 4)
def band_contract(L, g1, g2, t1, t2):
"""Band contract (2.10) at P = 0 from the exact held blocks (class r even): E(r), d_i(r) on even r,
kappa_i(rho) on odd rho (bond between rho-1 and rho+1); held spectra of the class."""
E, Om = np.zeros(L), np.zeros((L, L))
spec = {"G": [], "exc": [], "flip": [], "joint": []}
for r in range(0, L, 2):
e, _, om = ground(L, [(0, g1), (r, g2)])
E[r], Om[r] = e[0], om
spec["G"].append(e[0])
spec["exc"] += list(e[1:])
spec["joint"] += list(np.linalg.eigvalsh(field(L, [(0, -g1), (r, -g2)])))
dd, kk = np.zeros((2, L)), np.zeros((2, L))
for rm in range(1, L, 2):
up, dn = (rm + 1) % L, rm - 1
for i, (src, t, fu, fd) in enumerate((([(0, -g1), (rm, g2)], t1, np.roll(Om[up], -1), np.roll(Om[dn], 1)),
([(0, g1), (rm, -g2)], t2, Om[up], Om[dn]))):
ek, uk = np.linalg.eigh(field(L, src))
spec["flip"] += list(ek)
ou, od, q = uk.T @ fu, uk.T @ fd, t**2 / 4
dd[i, up] += q * np.sum(ou**2 / (E[up] - ek))
dd[i, dn] += q * np.sum(od**2 / (E[dn] - ek))
kk[i, rm] = -q * np.sum(ou * od * 0.5 * (1 / (E[up] - ek) + 1 / (E[dn] - ek)))
return E, Om, dd, kk, {k: np.array(v) for k, v in spec.items()}
def run_case(name, L, g1, g2, t1, t2, dist=(), eig_s=None, d=0.02):
n, l0 = 1.0 / L, ell(L, 0)
ev = np.arange(0, L, 2)
rs = np.where(ev > L // 2, ev - L, ev) # signed even r
E, Om, dd, kk, spec = band_contract(L, g1, g2, t1, t2)
kb1, kb2 = body_kappa(L, g1, t1), body_kappa(L, g2, t2)
low = (kb1 + kb2) < 0 # positive relative inertia: lowest edge
U = E[ev] + dd[0][ev] + dd[1][ev]
i0, i1 = np.arange(len(ev)), (np.arange(len(ev)) + 1) % len(ev)
def band(P):
h = np.diag(U).astype(complex)
t = kk[0][ev + 1] * np.exp(-2j * P) + kk[1][ev + 1] # <r|h|r+2>, bond rho = r+1
h[i0, i1] += t
h[i1, i0] += t.conj()
w, v = np.linalg.eigh(h)
k = 0 if low else -1
return w[k], v[:, k]
idx = parity_idx(L)
cache = {}
def exact(P):
if P not in cache:
eb, fb = band(P)
w, v = spla.eigsh(pair_ops(L, g1, g2, t1, t2, P)[idx][:, idx].tocsc(), k=8, sigma=eb, which='LM', tol=0)
best = (-1,)
for k in range(len(w)):
x = np.zeros(4 * L * L, complex)
x[idx] = v[:, k]
a = np.einsum('ry,ry->r', Om, x.reshape(2, 2, L, L)[0, 0])[ev]
ov = abs(np.vdot(fb, a)) ** 2
if ov > best[0]:
best = (ov, w[k], np.sum(np.abs(a) ** 2), x)
cache[P] = best
return cache[P]
Eb2, Eb0 = curv(lambda P: exact(P)[1], d)
ov0, _, gw0, v0 = exact(0.0)
ovmin = min(c[0] for c in cache.values())
Ebb2, _ = curv(lambda P: band(P)[0], d)
mu = [1.0 / curv(lambda P: body_level(L, g, t, P), d)[0] for g, t in ((g1, t1), (g2, t2))]
E1, E2 = body_level(L, g1, t1, 0.0), body_level(L, g2, t2, 0.0)
D0 = 1.0 / Eb2 - mu[0] - mu[1]
# band-contract quantities at P = 0
_, f0 = band(0.0)
f0 = (f0 * np.exp(-1j * np.angle(f0[0]))).real
rho = (ev + 1) % L
b0 = f0 * np.roll(f0, -1) # f0(rho-1) f0(rho+1), rho = r+1
k1, k2 = kk[0][ev + 1], kk[1][ev + 1]
beta0 = b0.sum()
B = lambda x: -2 * n * g1 * g2 / TF * ell(L, x)
ra = np.arange(L)
bar = lambda f: 0.25 * (f[(rho - 1) % L] + 2 * f[rho] + f[(rho + 1) % L])
Mfull = lambda a, b: np.array([eps0(L, [(0, a), (r, b)]) - eps0(L, [(0, a)]) - eps0(L, [(0, b)]) + eps0(L, []) for r in ra])
Mpp = Mfull(g1, g2)
Mo = [(Mpp - Mfull(-g1, g2)) / 2, (Mpp - Mfull(g1, -g2)) / 2] # parts of M odd in g1, in g2
sR = [(eps0(L, [(0, g)]) - eps0(L, [(0, -g)])) / 2 for g in (g1, g2)]
cb = [sR[0] * (-8 * kb1), sR[1] * (-8 * kb2)] # c_i^2 = sigma_i R_i / mu_i (band contract)
ce = [sR[0] / mu[0], sR[1] / mu[1]] # c_i^2 with the exact mu_i
Dlat = (mu[0] + mu[1]) * (1 - beta0) / beta0
Dmark2 = (1 / t1**2 + 1 / t2**2) * np.sum(b0 * bar(B(ra))) / beta0**2
dmo = bar(Mo[0]) / ce[0] + bar(Mo[1]) / ce[1]
Dmark = np.sum(b0 * dmo) / beta0**2
Dres = 1 / np.sum(b0 / (mu[0] + mu[1] + dmo)) - mu[0] - mu[1]
Dband = 1 / Ebb2 + 1 / (8 * kb1) + 1 / (8 * kb2)
Sc = abs(Dlat) + abs(Dmark)
Mavg = np.sum(f0**2 * Mpp[ev])
Moavg = [np.sum(f0**2 * Mo[0][ev]), np.sum(f0**2 * Mo[1][ev])]
K0 = 2 * (-1 / (8 * mu[0]) - 1 / (8 * mu[1])) * (beta0 - 1)
dE0 = Eb0 - E1 - E2 + (-2 * TF)
w_est = (4 * abs(kb1 + kb2) / abs(n * g1 * g2 / TF)) ** (1 / 3)
delta_m = abs(mu[0] + mu[1]) * (1 - beta0) / (abs(B(0)) * (1 / t1**2 + 1 / t2**2)) # S5 parameter
# distances of G from the other held levels of its class (item 1, S2)
G = spec["G"]
dfl, djo = mindist(G, np.concatenate([spec["flip"], spec["exc"]])), mindist(G, spec["joint"])
dall = min(dfl, djo)
djo_lead = 2 * n * abs(g1 + g2) - 1.5 * abs(B(0))
# local inertia (7.3): m_i(rho) - mubar_i against Mbar_i^odd(rho)/c_i^2 and Mbar^(2)(rho)/tau_i^2, |rho| <= 3
near = np.abs(np.where(rho > L // 2, rho - L, rho)) <= 3
loc, loc2 = [], []
for k, kb, c, Mi, t in ((k1, kb1, cb[0], Mo[0], t1), (k2, kb2, cb[1], Mo[1], t2)):
sh = (-1 / (8 * k) + 1 / (8 * kb))[near]
loc.append(float(np.max(np.abs(sh / (bar(Mi) / c)[near] - 1))))
loc2.append(float(np.max(np.abs(sh / (bar(B(ra)) / t**2)[near] - 1))))
res = dict(case=name, L=L, g1=g1, g2=g2, tau1=t1, tau2=t2, eta=max(abs(g1), abs(g2)) * l0 / TF,
w=w_est, mean_abs_r=float(np.sum(f0**2 * np.abs(rs))), overlap=ov0, min_overlap=ovmin,
group_weight=gw0, selected="lowest" if low else "highest", d_all=dall, d_flip=dfl, d_joint=djo,
d_joint_lead=djo_lead, tau_over_d=max(t1, t2) / dall, mu1=mu[0], mu2=mu[1],
mu1_band=-1 / (8 * kb1), mu2_band=-1 / (8 * kb2),
Epp_exact=Eb2, Epp_band=Ebb2, Epp_66=float(np.sum(b0 / (-1 / (8 * k1) - 1 / (8 * k2)))), beta0=beta0,
delta_m=delta_m, D0_exact=D0, D0_band=Dband, Dlat=Dlat, Dmark_odd=Dmark, Dmark_lead=Dmark2,
D0_formula=Dlat + Dmark, D0_resummed=Dres, D0_lead=Dlat + Dmark2, local_inertia_relerr_odd=loc,
local_inertia_relerr_lead=loc2, dE0_exact=dE0, M_avg=Mavg, M_odd_avg=Moavg,
M0_second_order=float(B(0)), K0=K0, dE0_formula=Mavg + K0)
log(f"== {name}: L={L} g1={g1:.5g} g2={g2:.5g} tau1={t1:.4g} tau2={t2:.4g} eta={res['eta']:.3f} w={w_est:.2f}")
log(f" held spectra of the class: d={dall:.4e} (flips and excitations {dfl:.4e}, joint flip {djo:.4e}; "
f"leading 2n|g1+g2|-1.5|B(0)|={djo_lead:.4e}); max tau/d={max(t1, t2) / dall:.3f}")
log(f" selected {res['selected']} level, <|r|>={res['mean_abs_r']:.2f}, overlap={ov0:.6f} (min over P {ovmin:.6f}), group weight={gw0:.6f}")
log(f" E''(0): exact={Eb2:.6e} band(2.10)={Ebb2:.6e} (6.6) sum b0/m={res['Epp_66']:.6e} rel. band-exact={(Ebb2 - Eb2) / Eb2:.2e}")
res['rho_L'] = Ebb2 / res['Epp_66'] - 1
res['eps_L'] = abs(res['rho_L']) / (res['Epp_66'] * abs(Dmark2)) if Dmark2 != 0 else float('nan')
log(f" ring flux remainder rho_L=E''band/E''line-1={res['rho_L']:.3e} eps_L=|rho_L|/(E''line |Dmark|)={res['eps_L']:.3e} (eta={res['eta']:.3f}, w={w_est:.2f})")
log(f" mu1={mu[0]:.6e} mu2={mu[1]:.6e} (band: {-1 / (8 * kb1):.6e}, {-1 / (8 * kb2):.6e}) beta0={beta0:.5f}")
log(f" Dlat={Dlat:.4e} Dmark(odd parts)={Dmark:.4e} Dmark(lead)={Dmark2:.4e} |Dlat/Dmark|={abs(Dlat / Dmark):.3f} delta_m(S5)={delta_m:.3f}")
log(f" D0: exact={D0:.6e} band={Dband:.6e} (7.5)={Dlat + Dmark:.6e} resummed={Dres:.6e} leading={Dlat + Dmark2:.6e}")
log(f" deviation from exact D0 / (|Dlat|+|Dmark|): band {(Dband - D0) / Sc:.2e}, (7.5) {(Dlat + Dmark - D0) / Sc:.2e}, resummed {(Dres - D0) / Sc:.2e}, leading {(Dlat + Dmark2 - D0) / Sc:.2e}")
log(f" local inertia (7.3), max rel. error near r=0: odd parts {loc[0]:.2e}, {loc[1]:.2e}; second order {loc2[0]:.2e}, {loc2[1]:.2e}")
log(f" cost: dE0 exact={dE0:.6e} <M>+K0={Mavg + K0:.6e} (<M>={Mavg:.4e}, M2(0)={B(0):.4e}, K0={K0:.4e}); (<M1odd>+<M2odd>)/<M>={(Moavg[0] + Moavg[1]) / Mavg:.4f}")
log(f" D0_exact/dE0={D0 / dE0:.6e} vs 1/tau1^2+1/tau2^2={1 / t1**2 + 1 / t2**2:.6e} (ratio {D0 / dE0 / (1 / t1**2 + 1 / t2**2):.4f}), 1/tau1^2={1 / t1**2:.6e}")
# ---------- item 4: weights ----------
# exact selected level: field density at distance z from the pair's midpoint in the exact P=0 eigenvector
# (first-order shift of the selected level by a source g_s at that place, Hellmann-Feynman)
pr = (np.abs(v0) ** 2).reshape(4, L, L).sum(0)
dens = np.zeros(L)
for r in range(L):
h = r - L if r > L // 2 else r
dens += np.roll(pr[r], -(h // 2)) if h % 2 == 0 else 0.5 * (np.roll(pr[r], -((h - 1) // 2)) + np.roll(pr[r], -((h + 1) // 2)))
dens_b = [np.abs(u[:L]) ** 2 + np.abs(u[L:]) ** 2 for u in (body_level(L, g, t, 0.0, vec=True)[1] for g, t in ((g1, t1), (g2, t2)))]
res['weight'] = []
for s in dist:
dz = lambda f: -GS * (f[(s + 1) % L] - f[(s - 1) % L]) / 2
W0p, W1p, W2p = dz(dens), dz(dens_b[0]), dz(dens_b[1])
phip = -(2 * GS / TF) * (ell(L, 1 - s) - ell(L, -1 - s)) / 2
u = s / L
ks = (L * L * (1 - 2 * u) ** 2 + 2) / (L * L - 1)
Vavg = np.sum(f0**2 * n * g1 * g2 / TF * np.mod(rs, L) * (L - np.mod(rs, L)) / L)
Cev = [-n * g**2 / TF * l0 for g in (g1, g2)]
# held-field average (v1): f0^2-average of exact held weights, no kinetic part
X, xs = 0, s
Wh = lambda src: (eps0(L, src + [(xs, GS)]) - eps0(L, src + [(xs, -GS)])) / 2
dW = lambda fun: (fun(X + 1) - fun(X - 1)) / 2
Wp = lambda Xc: np.sum([f0[i] ** 2 * Wh([(Xc - rs[i] // 2, g1), (Xc - rs[i] // 2 + rs[i], g2)]) for i in range(len(ev))])
W0h, W1h, W2h = dW(Wp), dW(lambda x: Wh([(x, g1)])), dW(lambda x: Wh([(x, g2)]))
Wbh = W0h - W1h - W2h
Wbp = W0p - W1p - W2p
Wbf = phip * (2 * ks * B(0) + Vavg - K0) # (7.10)
G1, G2, Gp, Gb = W1p / mu[0], W2p / mu[1], Eb2 * W0p, Wbp / D0
Gb_lead = 2 * ks * phip / (1 / t1**2 + 1 / t2**2) # (7.12)
Gb_f = Wbf / (Dlat + Dmark2)
Gb_r = (Wbh - K0 * phip) / (Dlat + Dmark) # same structure with exact held weights and exact odd parts of M
wr = dict(s=s, u=u, kappa_s=ks, phip=phip, W0p=W0p, W1p=W1p, W2p=W2p, Wbp=Wbp, Wbp_formula=Wbf,
W0p_heldavg=W0h, Wbp_heldavg=Wbh, kinetic=(Wbp - Wbh) / phip, kinetic_formula=-K0,
Gamma1=G1, Gamma2=G2, Gamma_pair=Gp, Gamma_bond=Gb, Gamma_bond_lead=Gb_lead,
Gamma_bond_formula=Gb_f, Gamma_bond_refined=Gb_r, eta_over_ks=res['eta'] / ks, w_over_Lks=w_est / (L * ks))
log(f" source at distance s={s} from the midpoint (u={u:.4f}, kappa_s={ks:.4f}; eta/kappa_s={wr['eta_over_ks']:.3f}, w/(L kappa_s)={wr['w_over_Lks']:.3f}):")
log(f" exact selected level: W0'/phi'={W0p / phip:.6e} Wb'/phi'={Wbp / phip:.6e} (7.10): {Wbf / phip:.6e}")
log(f" held-field average: W0'/phi'={W0h / phip:.6e} Wb'/phi'={Wbh / phip:.6e}; kinetic part (exact - held avg)/phi'={wr['kinetic']:.4e} vs -K0={-K0:.4e}")
log(f" Gamma/phi': body1={G1 / phip:.6e} (tau1^2={t1**2:.6e}) body2={G2 / phip:.6e} (tau2^2={t2**2:.6e}) pair={Gp / phip:.6e}")
log(f" Gamma_b=Wb'/D0: /phi'={Gb / phip:.6e} (7.12) lead={Gb_lead / phip:.6e} (ratio {Gb / Gb_lead:.4f}) (7.10)/(7.5)={Gb_f / phip:.6e} (ratio {Gb / Gb_f:.4f}) refined={Gb_r / phip:.6e} (ratio {Gb / Gb_r:.4f})")
log(f" Gamma_b/Gamma_1={Gb / G1:.4f} Gamma_b/Gamma_2={Gb / G2:.4f} lead 2kappa_s/(1+tau1^2/tau2^2)={2 * ks / (1 + t1**2 / t2**2):.4f}")
if s == eig_s: # auxiliary midpoint probe (not a held source): HF shift by g_s at fixed distance from the midpoint, +-g_s
H0 = pair_ops(L, g1, g2, t1, t2, 0.0)[idx][:, idx].tocsc()
dE = {}
for z in (s - 1, s + 1):
src = midpoint_weights(L, z)[idx]
for sg in (1, -1):
w, v = spla.eigsh(H0 + sp.diags(sg * GS * src), k=6, sigma=Eb0, which='LM', tol=0)
dE[z, sg] = w[np.argmax(np.abs(v.conj().T @ v0[idx]))]
W0e = -((dE[s + 1, 1] - dE[s + 1, -1]) - (dE[s - 1, 1] - dE[s - 1, -1])) / 4
wr['W0p_eigenvalues'] = W0e
log(f" auxiliary midpoint probe (sector eigenvalues, +-g_s; not a held source): W0'/phi'={W0e / phip:.6e}, rel. diff. to the density form {(W0e - W0p) / W0p:.2e}; Wb'/phi' from them {(W0e - W1p - W2p) / phip:.6e}")
res['weight'].append(wr)
return res
def params(L, eta1, q, r1, r2=None):
"""g1 from eta1 = |g1| l0 / t_f, g2 = q g1, tau_i = r_i d with d = 2 n min|g_i|."""
r2 = r1 if r2 is None else r2
g1 = eta1 * TF / ell(L, 0)
d = 2 * min(abs(g1), abs(q * g1)) / L
return L, g1, q * g1, r1 * d, r2 * d
CASES = [("A equal bodies", (32, 0.15, 1.0, 0.3), (), None),
("A equal bodies", (64, 0.15, 1.0, 0.3), (8, 16), 8),
("A' equal bodies, smaller eta", (64, 0.05, 1.0, 0.3), (8, 16), None),
("A'' equal bodies, smaller eta and tau/d", (64, 0.05, 1.0, 0.15), (), None),
("B unequal bodies and speeds", (32, 0.15, 0.5, 0.3, 0.15), (), None),
("B unequal bodies and speeds", (64, 0.15, 0.5, 0.3, 0.15), (8, 16), None),
("R equal bodies, v1 case", (128, 0.15, 1.0, 0.3), (8, 16, 32), None),
("S1 equal bodies, regime (ii)", (256, 0.10, 1.0, 0.4), (16, 32, 64), 32),
("S2 equal bodies, regime (ii), smaller eta", (256, 0.05, 1.0, 0.4), (16, 32, 64), None),
("C mixed signs, bound at top", (48, 0.075, -2.0, 0.15, 0.3), (), None),
("D control: both negative, not bound", (48, -0.15, 1.0, 0.3), (), None),
("F control: small w, L=6", (6, 0.15, 1.0, 0.1), (), None),
("F control: small w, L=8", (8, 0.15, 1.0, 0.1), (), None)]
if __name__ == "__main__":
OUT.mkdir(exist_ok=True)
results = [run_case(name, *params(*p), dist=dist, eig_s=es) for name, p, dist, es in CASES]
(OUT / "check_log.txt").write_text("\n".join(LOG) + "\n", encoding="utf-8")
(OUT / "check_results.json").write_text(json.dumps(results, indent=1, default=float), encoding="utf-8")
output/check_log.txt28 KB · raw
== A equal bodies: L=32 g1=0.056305 g2=0.056305 tau1=0.001056 tau2=0.001056 eta=0.150 w=1.83
held spectra of the class: d=2.7731e-03 (flips and excitations 2.7731e-03, joint flip 6.4749e-03; leading 2n|g1+g2|-1.5|B(0)|=6.2463e-03); max tau/d=0.381
selected lowest level, <|r|>=1.34, overlap=0.984031 (min over P 0.983940), group weight=0.984065
E''(0): exact=3.147829e-04 band(2.10)=3.291362e-04 (6.6) sum b0/m=3.291357e-04 rel. band-exact=4.56e-02
ring flux remainder rho_L=E''band/E''line-1=1.430e-06 eps_L=|rho_L|/(E''line |Dmark|)=5.583e-06 (eta=0.150, w=1.83)
mu1=1.639644e+03 mu2=1.639644e+03 (band: 1.639640e+03, 1.639640e+03) beta0=0.87303
Dlat=4.7693e+02 Dmark(odd parts)=-6.9527e+02 Dmark(lead)=-7.7827e+02 |Dlat/Dmark|=0.686 delta_m(S5)=0.440
D0: exact=-1.024964e+02 band=-2.410248e+02 (7.5)=-2.183369e+02 resummed=-2.265128e+02 leading=-3.013431e+02
deviation from exact D0 / (|Dlat|+|Dmark|): band -1.18e-01, (7.5) -9.88e-02, resummed -1.06e-01, leading -1.70e-01
local inertia (7.3), max rel. error near r=0: odd parts 2.15e-02, 2.15e-02; second order 9.31e-02, 9.31e-02
cost: dE0 exact=-2.510425e-04 <M>+K0=-2.591129e-04 (<M>=-2.9783e-04, M2(0)=-5.2786e-04, K0=3.8719e-05); (<M1odd>+<M2odd>)/<M>=2.3765
D0_exact/dE0=4.082829e+05 vs 1/tau1^2+1/tau2^2=1.794460e+06 (ratio 0.2275), 1/tau1^2=8.972299e+05
== A equal bodies: L=64 g1=0.028132 g2=0.028132 tau1=0.0002637 tau2=0.0002637 eta=0.150 w=2.31
held spectra of the class: d=6.8091e-04 (flips and excitations 6.8091e-04, joint flip 1.6178e-03; leading 2n|g1+g2|-1.5|B(0)|=1.5604e-03); max tau/d=0.387
selected lowest level, <|r|>=1.85, overlap=0.989809 (min over P 0.989709), group weight=0.989820
E''(0): exact=8.707640e-05 band(2.10)=8.958103e-05 (6.6) sum b0/m=8.958104e-05 rel. band-exact=2.88e-02
ring flux remainder rho_L=E''band/E''line-1=-9.376e-08 eps_L=|rho_L|/(E''line |Dmark|)=3.122e-07 (eta=0.150, w=2.31)
mu1=6.564038e+03 mu2=6.564038e+03 (band: 6.564034e+03, 6.564034e+03) beta0=0.92320
Dlat=1.0921e+03 Dmark(odd parts)=-2.9762e+03 Dmark(lead)=-3.3527e+03 |Dlat/Dmark|=0.367 delta_m(S5)=0.266
D0: exact=-1.643909e+03 band=-1.964989e+03 (7.5)=-1.884097e+03 resummed=-1.900351e+03 leading=-2.260628e+03
deviation from exact D0 / (|Dlat|+|Dmark|): band -7.89e-02, (7.5) -5.90e-02, resummed -6.30e-02, leading -1.52e-01
local inertia (7.3), max rel. error near r=0: odd parts 2.20e-02, 2.20e-02; second order 9.87e-02, 9.87e-02
cost: dE0 exact=-7.278875e-05 <M>+K0=-7.423568e-05 (<M>=-8.0086e-05, M2(0)=-1.3187e-04, K0=5.8499e-06); (<M1odd>+<M2odd>)/<M>=2.3900
D0_exact/dE0=2.258465e+07 vs 1/tau1^2+1/tau2^2=2.875347e+07 (ratio 0.7855), 1/tau1^2=1.437674e+07
source at distance s=8 from the midpoint (u=0.1250, kappa_s=0.5631; eta/kappa_s=0.266, w/(L kappa_s)=0.064):
exact selected level: W0'/phi'=6.396094e-04 Wb'/phi'=-1.049401e-04 (7.10): -1.326958e-04
held-field average: W0'/phi'=6.469698e-04 Wb'/phi'=-9.757629e-05; kinetic part (exact - held avg)/phi'=-7.3638e-06 vs -K0=-5.8499e-06
Gamma/phi': body1=5.671429e-08 (tau1^2=6.955682e-08) body2=5.671429e-08 (tau2^2=6.955682e-08) pair=5.569489e-08
Gamma_b=Wb'/D0: /phi'=6.383572e-08 (7.12) lead=3.916924e-08 (ratio 1.6297) (7.10)/(7.5)=5.869865e-08 (ratio 1.0875) refined=5.489432e-08 (ratio 1.1629)
Gamma_b/Gamma_1=1.1256 Gamma_b/Gamma_2=1.1256 lead 2kappa_s/(1+tau1^2/tau2^2)=0.5631
auxiliary midpoint probe (sector eigenvalues, +-g_s; not a held source): W0'/phi'=6.396094e-04, rel. diff. to the density form 5.93e-09; Wb'/phi' from them -1.049401e-04
source at distance s=16 from the midpoint (u=0.2500, kappa_s=0.2505; eta/kappa_s=0.599, w/(L kappa_s)=0.144):
exact selected level: W0'/phi'=7.492984e-04 Wb'/phi'=-6.124736e-05 (7.10): -5.025811e-05
held-field average: W0'/phi'=7.576416e-04 Wb'/phi'=-5.289702e-05; kinetic part (exact - held avg)/phi'=-8.3503e-06 vs -K0=-5.8499e-06
Gamma/phi': body1=6.174139e-08 (tau1^2=6.955682e-08) body2=6.174139e-08 (tau2^2=6.955682e-08) pair=6.524620e-08
Gamma_b=Wb'/D0: /phi'=3.725716e-08 (7.12) lead=1.742742e-08 (ratio 2.1378) (7.10)/(7.5)=2.223192e-08 (ratio 1.6758) refined=3.118042e-08 (ratio 1.1949)
Gamma_b/Gamma_1=0.6034 Gamma_b/Gamma_2=0.6034 lead 2kappa_s/(1+tau1^2/tau2^2)=0.2505
== A' equal bodies, smaller eta: L=64 g1=0.0093773 g2=0.0093773 tau1=8.791e-05 tau2=8.791e-05 eta=0.050 w=3.37
held spectra of the class: d=2.6730e-04 (flips and excitations 2.6730e-04, joint flip 5.6646e-04; leading 2n|g1+g2|-1.5|B(0)|=5.6410e-04); max tau/d=0.329
selected lowest level, <|r|>=2.42, overlap=0.995019 (min over P 0.994939), group weight=0.995019
E''(0): exact=2.660595e-05 band(2.10)=2.699290e-05 (6.6) sum b0/m=2.699290e-05 rel. band-exact=1.45e-02
ring flux remainder rho_L=E''band/E''line-1=-1.587e-07 eps_L=|rho_L|/(E''line |Dmark|)=1.909e-06 (eta=0.050, w=3.37)
mu1=1.903544e+04 mu2=1.903544e+04 (band: 1.903544e+04, 1.903544e+04) beta0=0.95175
Dlat=1.9301e+03 Dmark(odd parts)=-2.9375e+03 Dmark(lead)=-3.0812e+03 |Dlat/Dmark|=0.657 delta_m(S5)=0.484
D0: exact=-4.853071e+02 band=-1.024096e+03 (7.5)=-1.007359e+03 resummed=-1.016930e+03 leading=-1.151092e+03
deviation from exact D0 / (|Dlat|+|Dmark|): band -1.11e-01, (7.5) -1.07e-01, resummed -1.09e-01, leading -1.37e-01
local inertia (7.3), max rel. error near r=0: odd parts 2.63e-03, 2.63e-03; second order 4.81e-02, 4.81e-02
cost: dE0 exact=-9.041333e-06 <M>+K0=-9.135153e-06 (<M>=-1.0403e-05, M2(0)=-1.4652e-05, K0=1.2674e-06); (<M1odd>+<M2odd>)/<M>=2.1118
D0_exact/dE0=5.367650e+07 vs 1/tau1^2+1/tau2^2=2.587813e+08 (ratio 0.2074), 1/tau1^2=1.293906e+08
source at distance s=8 from the midpoint (u=0.1250, kappa_s=0.5631; eta/kappa_s=0.089, w/(L kappa_s)=0.093):
exact selected level: W0'/phi'=2.630864e-04 Wb'/phi'=-1.397144e-05 (7.10): -1.466959e-05
held-field average: W0'/phi'=2.644494e-04 Wb'/phi'=-1.260842e-05; kinetic part (exact - held avg)/phi'=-1.3630e-06 vs -K0=-1.2674e-06
Gamma/phi': body1=7.277422e-09 (tau1^2=7.728535e-09) body2=7.277422e-09 (tau2^2=7.728535e-09) pair=6.999665e-09
Gamma_b=Wb'/D0: /phi'=2.878888e-08 (7.12) lead=4.352137e-09 (ratio 6.6149) (7.10)/(7.5)=1.274406e-08 (ratio 2.2590) refined=1.377446e-08 (ratio 2.0900)
Gamma_b/Gamma_1=3.9559 Gamma_b/Gamma_2=3.9559 lead 2kappa_s/(1+tau1^2/tau2^2)=0.5631
source at distance s=16 from the midpoint (u=0.2500, kappa_s=0.2505; eta/kappa_s=0.200, w/(L kappa_s)=0.210):
exact selected level: W0'/phi'=2.791184e-04 Wb'/phi'=-6.442146e-06 (7.10): -5.509840e-06
held-field average: W0'/phi'=2.805571e-04 Wb'/phi'=-5.003244e-06; kinetic part (exact - held avg)/phi'=-1.4389e-06 vs -K0=-1.2674e-06
Gamma/phi': body1=7.500760e-09 (tau1^2=7.728535e-09) body2=7.500760e-09 (tau2^2=7.728535e-09) pair=7.426210e-09
Gamma_b=Wb'/D0: /phi'=1.327437e-08 (7.12) lead=1.936380e-09 (ratio 6.8553) (7.10)/(7.5)=4.786620e-09 (ratio 2.7732) refined=6.224842e-09 (ratio 2.1325)
Gamma_b/Gamma_1=1.7697 Gamma_b/Gamma_2=1.7697 lead 2kappa_s/(1+tau1^2/tau2^2)=0.2505
== A'' equal bodies, smaller eta and tau/d: L=64 g1=0.0093773 g2=0.0093773 tau1=4.396e-05 tau2=4.396e-05 eta=0.050 w=2.12
held spectra of the class: d=2.6730e-04 (flips and excitations 2.6730e-04, joint flip 5.6646e-04; leading 2n|g1+g2|-1.5|B(0)|=5.6410e-04); max tau/d=0.164
selected lowest level, <|r|>=1.28, overlap=0.996424 (min over P 0.996404), group weight=0.996424
E''(0): exact=6.119499e-06 band(2.10)=6.180253e-06 (6.6) sum b0/m=6.180249e-06 rel. band-exact=9.93e-03
ring flux remainder rho_L=E''band/E''line-1=6.787e-07 eps_L=|rho_L|/(E''line |Dmark|)=7.328e-06 (eta=0.050, w=2.12)
mu1=7.614176e+04 mu2=7.614176e+04 (band: 7.614176e+04, 7.614176e+04) beta0=0.86471
Dlat=2.3826e+04 Dmark(odd parts)=-1.4252e+04 Dmark(lead)=-1.4986e+04 |Dlat/Dmark|=1.672 delta_m(S5)=1.358
D0: exact=1.112856e+04 band=9.522158e+03 (7.5)=9.573626e+03 resummed=9.558786e+03 leading=8.839926e+03
deviation from exact D0 / (|Dlat|+|Dmark|): band -4.22e-02, (7.5) -4.08e-02, resummed -4.12e-02, leading -6.01e-02
local inertia (7.3), max rel. error near r=0: odd parts 2.63e-03, 2.63e-03; second order 4.81e-02, 4.81e-02
cost: dE0 exact=-1.063404e-05 <M>+K0=-1.071125e-05 (<M>=-1.1600e-05, M2(0)=-1.4652e-05, K0=8.8841e-07); (<M1odd>+<M2odd>)/<M>=2.1189
D0_exact/dE0=-1.046504e+09 vs 1/tau1^2+1/tau2^2=1.035125e+09 (ratio -1.0110), 1/tau1^2=5.175625e+08
== B unequal bodies and speeds: L=32 g1=0.056305 g2=0.028152 tau1=0.0005279 tau2=0.0002639 eta=0.150 w=1.33
held spectra of the class: d=1.3636e-03 (flips and excitations 1.3636e-03, joint flip 4.9956e-03; leading 2n|g1+g2|-1.5|B(0)|=4.8827e-03); max tau/d=0.387
selected lowest level, <|r|>=0.57, overlap=0.989682 (min over P 0.989658), group weight=0.989686
E''(0): exact=4.097370e-05 band(2.10)=4.211294e-05 (6.6) sum b0/m=4.211294e-05 rel. band-exact=2.78e-02
ring flux remainder rho_L=E''band/E''line-1=1.351e-07 eps_L=|rho_L|/(E''line |Dmark|)=5.757e-07 (eta=0.150, w=1.33)
mu1=6.558566e+03 mu2=1.274622e+04 (band: 6.558562e+03, 1.274622e+04) beta0=0.67184
Dlat=9.4292e+03 Dmark(odd parts)=-4.9274e+03 Dmark(lead)=-5.5740e+03 |Dlat/Dmark|=1.914 delta_m(S5)=1.338
D0: exact=5.101118e+03 band=4.440890e+03 (7.5)=4.501822e+03 resummed=4.488702e+03 leading=3.855201e+03
deviation from exact D0 / (|Dlat|+|Dmark|): band -4.60e-02, (7.5) -4.17e-02, resummed -4.27e-02, leading -8.68e-02
local inertia (7.3), max rel. error near r=0: odd parts 2.30e-02, 6.15e-03; second order 2.01e-02, 1.31e-01
cost: dE0 exact=-1.627373e-04 <M>+K0=-1.660816e-04 (<M>=-1.8503e-04, M2(0)=-2.6393e-04, K0=1.8945e-05); (<M1odd>+<M2odd>)/<M>=2.2971
D0_exact/dE0=-3.134572e+07 vs 1/tau1^2+1/tau2^2=1.794460e+07 (ratio -1.7468), 1/tau1^2=3.588920e+06
== B unequal bodies and speeds: L=64 g1=0.028132 g2=0.014066 tau1=0.0001319 tau2=6.593e-05 eta=0.150 w=1.67
held spectra of the class: d=3.3549e-04 (flips and excitations 3.3549e-04, joint flip 1.2481e-03; leading 2n|g1+g2|-1.5|B(0)|=1.2198e-03); max tau/d=0.393
selected lowest level, <|r|>=1.00, overlap=0.993976 (min over P 0.993949), group weight=0.993978
E''(0): exact=1.276961e-05 band(2.10)=1.298382e-05 (6.6) sum b0/m=1.298382e-05 rel. band-exact=1.68e-02
ring flux remainder rho_L=E''band/E''line-1=-2.440e-08 eps_L=|rho_L|/(E''line |Dmark|)=9.245e-08 (eta=0.150, w=1.67)
mu1=2.625614e+04 mu2=5.102339e+04 (band: 2.625613e+04, 5.102339e+04) beta0=0.81280
Dlat=1.7798e+04 Dmark(odd parts)=-1.7860e+04 Dmark(lead)=-2.0330e+04 |Dlat/Dmark|=0.997 delta_m(S5)=0.763
D0: exact=1.031368e+03 band=-2.605738e+02 (7.5)=-6.138594e+01 resummed=-8.967688e+01 leading=-2.532073e+03
deviation from exact D0 / (|Dlat|+|Dmark|): band -3.62e-02, (7.5) -3.06e-02, resummed -3.14e-02, leading -9.99e-02
local inertia (7.3), max rel. error near r=0: odd parts 2.50e-02, 5.36e-03; second order 2.02e-02, 1.39e-01
cost: dE0 exact=-4.346807e-05 <M>+K0=-4.397624e-05 (<M>=-4.6676e-05, M2(0)=-6.5934e-05, K0=2.6996e-06); (<M1odd>+<M2odd>)/<M>=2.2972
D0_exact/dE0=-2.372703e+07 vs 1/tau1^2+1/tau2^2=2.875347e+08 (ratio -0.0825), 1/tau1^2=5.750694e+07
source at distance s=8 from the midpoint (u=0.1250, kappa_s=0.5631; eta/kappa_s=0.266, w/(L kappa_s)=0.046):
exact selected level: W0'/phi'=5.157547e-04 Wb'/phi'=-5.861085e-05 (7.10): -7.099485e-05
held-field average: W0'/phi'=5.189716e-04 Wb'/phi'=-5.539348e-05; kinetic part (exact - held avg)/phi'=-3.2174e-06 vs -K0=-2.6996e-06
Gamma/phi': body1=1.417853e-08 (tau1^2=1.738920e-08) body2=3.960774e-09 (tau2^2=4.347301e-09) pair=6.585988e-09
Gamma_b=Wb'/D0: /phi'=-5.682826e-08 (7.12) lead=3.916924e-09 (ratio -14.5084) (7.10)/(7.5)=2.803824e-08 (ratio -2.0268) refined=9.463588e-07 (ratio -0.0600)
Gamma_b/Gamma_1=-4.0081 Gamma_b/Gamma_2=-14.3478 lead 2kappa_s/(1+tau1^2/tau2^2)=0.2253
source at distance s=16 from the midpoint (u=0.2500, kappa_s=0.2505; eta/kappa_s=0.599, w/(L kappa_s)=0.104):
exact selected level: W0'/phi'=5.830558e-04 Wb'/phi'=-3.352277e-05 (7.10): -2.977600e-05
held-field average: W0'/phi'=5.865617e-04 Wb'/phi'=-3.001577e-05; kinetic part (exact - held avg)/phi'=-3.5070e-06 vs -K0=-2.6996e-06
Gamma/phi': body1=1.543525e-08 (tau1^2=1.738920e-08) body2=4.141402e-09 (tau2^2=4.347301e-09) pair=7.445398e-09
Gamma_b=Wb'/D0: /phi'=-3.250321e-08 (7.12) lead=1.742742e-09 (ratio -18.6506) (7.10)/(7.5)=1.175954e-08 (ratio -2.7640) refined=5.329462e-07 (ratio -0.0610)
Gamma_b/Gamma_1=-2.1058 Gamma_b/Gamma_2=-7.8484 lead 2kappa_s/(1+tau1^2/tau2^2)=0.1002
== R equal bodies, v1 case: L=128 g1=0.014063 g2=0.014063 tau1=6.592e-05 tau2=6.592e-05 eta=0.150 w=2.91
held spectra of the class: d=1.6868e-04 (flips and excitations 1.6868e-04, joint flip 4.0439e-04; leading 2n|g1+g2|-1.5|B(0)|=3.9004e-04); max tau/d=0.391
selected lowest level, <|r|>=2.45, overlap=0.993446 (min over P 0.993339), group weight=0.993450
E''(0): exact=2.315069e-05 band(2.10)=2.357385e-05 (6.6) sum b0/m=2.357385e-05 rel. band-exact=1.83e-02
ring flux remainder rho_L=E''band/E''line-1=2.142e-07 eps_L=|rho_L|/(E''line |Dmark|)=6.473e-07 (eta=0.150, w=2.91)
mu1=2.626162e+04 mu2=2.626162e+04 (band: 2.626161e+04, 2.626161e+04) beta0=0.95282
Dlat=2.6009e+03 Dmark(odd parts)=-1.2403e+04 Dmark(lead)=-1.4037e+04 |Dlat/Dmark|=0.210 delta_m(S5)=0.163
D0: exact=-9.327978e+03 band=-1.010333e+04 (7.5)=-9.801789e+03 resummed=-9.831106e+03 leading=-1.143575e+04
deviation from exact D0 / (|Dlat|+|Dmark|): band -5.17e-02, (7.5) -3.16e-02, resummed -3.35e-02, leading -1.40e-01
local inertia (7.3), max rel. error near r=0: odd parts 2.20e-02, 2.20e-02; second order 1.02e-01, 1.02e-01
cost: dE0 exact=-1.995791e-05 <M>+K0=-2.020866e-05 (<M>=-2.1107e-05, M2(0)=-3.2961e-05, K0=8.9831e-07); (<M1odd>+<M2odd>)/<M>=2.4030
D0_exact/dE0=4.673824e+08 vs 1/tau1^2+1/tau2^2=4.602241e+08 (ratio 1.0156), 1/tau1^2=2.301120e+08
source at distance s=8 from the midpoint (u=0.0625, kappa_s=0.7658; eta/kappa_s=0.196, w/(L kappa_s)=0.030):
exact selected level: W0'/phi'=1.433803e-04 Wb'/phi'=-3.246042e-05 (7.10): -4.772353e-05
held-field average: W0'/phi'=1.444181e-04 Wb'/phi'=-3.142246e-05; kinetic part (exact - held avg)/phi'=-1.0380e-06 vs -K0=-8.9831e-07
Gamma/phi': body1=3.347865e-09 (tau1^2=4.345709e-09) body2=3.347865e-09 (tau2^2=4.345709e-09) pair=3.319352e-09
Gamma_b=Wb'/D0: /phi'=3.479899e-09 (7.12) lead=3.327917e-09 (ratio 1.0457) (7.10)/(7.5)=4.173188e-09 (ratio 0.8339) refined=3.297436e-09 (ratio 1.0553)
Gamma_b/Gamma_1=1.0394 Gamma_b/Gamma_2=1.0394 lead 2kappa_s/(1+tau1^2/tau2^2)=0.7658
source at distance s=16 from the midpoint (u=0.1250, kappa_s=0.5627; eta/kappa_s=0.267, w/(L kappa_s)=0.040):
exact selected level: W0'/phi'=1.597441e-04 Wb'/phi'=-2.638446e-05 (7.10): -3.433231e-05
held-field average: W0'/phi'=1.608867e-04 Wb'/phi'=-2.524156e-05; kinetic part (exact - held avg)/phi'=-1.1429e-06 vs -K0=-8.9831e-07
Gamma/phi': body1=3.543738e-09 (tau1^2=4.345709e-09) body2=3.543738e-09 (tau2^2=4.345709e-09) pair=3.698186e-09
Gamma_b=Wb'/D0: /phi'=2.828529e-09 (7.12) lead=2.445141e-09 (ratio 1.1568) (7.10)/(7.5)=3.002192e-09 (ratio 0.9422) refined=2.666847e-09 (ratio 1.0606)
Gamma_b/Gamma_1=0.7982 Gamma_b/Gamma_2=0.7982 lead 2kappa_s/(1+tau1^2/tau2^2)=0.5627
source at distance s=32 from the midpoint (u=0.2500, kappa_s=0.2501; eta/kappa_s=0.600, w/(L kappa_s)=0.091):
exact selected level: W0'/phi'=1.868889e-04 Wb'/phi'=-1.573377e-05 (7.10): -1.373043e-05
held-field average: W0'/phi'=1.881882e-04 Wb'/phi'=-1.443405e-05; kinetic part (exact - held avg)/phi'=-1.2997e-06 vs -K0=-8.9831e-07
Gamma/phi': body1=3.857771e-09 (tau1^2=4.345709e-09) body2=3.857771e-09 (tau2^2=4.345709e-09) pair=4.326606e-09
Gamma_b=Wb'/D0: /phi'=1.686729e-09 (7.12) lead=1.087024e-09 (ratio 1.5517) (7.10)/(7.5)=1.200659e-09 (ratio 1.4048) refined=1.564241e-09 (ratio 1.0783)
Gamma_b/Gamma_1=0.4372 Gamma_b/Gamma_2=0.4372 lead 2kappa_s/(1+tau1^2/tau2^2)=0.2501
== S1 equal bodies, regime (ii): L=256 g1=0.0046876 g2=0.0046876 tau1=1.465e-05 tau2=1.465e-05 eta=0.100 w=5.12
held spectra of the class: d=3.0428e-05 (flips and excitations 3.0428e-05, joint flip 6.8870e-05; leading 2n|g1+g2|-1.5|B(0)|=6.7750e-05); max tau/d=0.481
selected lowest level, <|r|>=4.13, overlap=0.996119 (min over P 0.995947), group weight=0.996120
E''(0): exact=6.694777e-06 band(2.10)=6.769117e-06 (6.6) sum b0/m=6.769122e-06 rel. band-exact=1.11e-02
ring flux remainder rho_L=E''band/E''line-1=-8.189e-07 eps_L=|rho_L|/(E''line |Dmark|)=3.853e-06 (eta=0.100, w=5.12)
mu1=8.675530e+04 mu2=8.675530e+04 (band: 8.675528e+04, 8.675528e+04) beta0=0.98232
Dlat=3.1235e+03 Dmark(odd parts)=-2.8583e+04 Dmark(lead)=-3.1396e+04 |Dlat/Dmark|=0.109 delta_m(S5)=0.090
D0: exact=-2.414042e+04 band=-2.578081e+04 (7.5)=-2.545921e+04 resummed=-2.549118e+04 leading=-2.827274e+04
deviation from exact D0 / (|Dlat|+|Dmark|): band -5.17e-02, (7.5) -4.16e-02, resummed -4.26e-02, leading -1.30e-01
local inertia (7.3), max rel. error near r=0: odd parts 1.03e-02, 1.03e-02; second order 8.43e-02, 8.43e-02
cost: dE0 exact=-2.533427e-06 <M>+K0=-2.552421e-06 (<M>=-2.6543e-06, M2(0)=-3.6622e-06, K0=1.0192e-07); (<M1odd>+<M2odd>)/<M>=2.2556
D0_exact/dE0=9.528760e+09 vs 1/tau1^2+1/tau2^2=9.320391e+09 (ratio 1.0224), 1/tau1^2=4.660196e+09
source at distance s=16 from the midpoint (u=0.0625, kappa_s=0.7657; eta/kappa_s=0.131, w/(L kappa_s)=0.026):
exact selected level: W0'/phi'=2.737495e-05 Wb'/phi'=-4.139222e-06 (7.10): -5.364688e-06
held-field average: W0'/phi'=2.748649e-05 Wb'/phi'=-4.027682e-06; kinetic part (exact - held avg)/phi'=-1.1154e-07 vs -K0=-1.0192e-07
Gamma/phi': body1=1.816268e-10 (tau1^2=2.145833e-10) body2=1.816268e-10 (tau2^2=2.145833e-10) pair=1.832692e-10
Gamma_b=Wb'/D0: /phi'=1.714644e-10 (7.12) lead=1.642994e-10 (ratio 1.0436) (7.10)/(7.5)=1.897478e-10 (ratio 0.9036) refined=1.622045e-10 (ratio 1.0571)
Gamma_b/Gamma_1=0.9440 Gamma_b/Gamma_2=0.9440 lead 2kappa_s/(1+tau1^2/tau2^2)=0.7657
source at distance s=32 from the midpoint (u=0.1250, kappa_s=0.5625; eta/kappa_s=0.178, w/(L kappa_s)=0.036):
exact selected level: W0'/phi'=2.950067e-05 Wb'/phi'=-3.259660e-06 (7.10): -3.876911e-06
held-field average: W0'/phi'=2.962028e-05 Wb'/phi'=-3.140019e-06; kinetic part (exact - held avg)/phi'=-1.1964e-07 vs -K0=-1.0192e-07
Gamma/phi': body1=1.888088e-10 (tau1^2=2.145833e-10) body2=1.888088e-10 (tau2^2=2.145833e-10) pair=1.975004e-10
Gamma_b=Wb'/D0: /phi'=1.350291e-10 (7.12) lead=1.207115e-10 (ratio 1.1186) (7.10)/(7.5)=1.371254e-10 (ratio 0.9847) refined=1.273384e-10 (ratio 1.0604)
Gamma_b/Gamma_1=0.7152 Gamma_b/Gamma_2=0.7152 lead 2kappa_s/(1+tau1^2/tau2^2)=0.5625
auxiliary midpoint probe (sector eigenvalues, +-g_s; not a held source): W0'/phi'=2.950067e-05, rel. diff. to the density form -4.67e-08; Wb'/phi' from them -3.259661e-06
source at distance s=64 from the midpoint (u=0.2500, kappa_s=0.2500; eta/kappa_s=0.400, w/(L kappa_s)=0.080):
exact selected level: W0'/phi'=3.295124e-05 Wb'/phi'=-1.781834e-06 (7.10): -1.588023e-06
held-field average: W0'/phi'=3.308310e-05 Wb'/phi'=-1.649848e-06; kinetic part (exact - held avg)/phi'=-1.3199e-07 vs -K0=-1.0192e-07
Gamma/phi': body1=2.001784e-10 (tau1^2=2.145833e-10) body2=2.001784e-10 (tau2^2=2.145833e-10) pair=2.206012e-10
Gamma_b=Wb'/D0: /phi'=7.381122e-11 (7.12) lead=5.365318e-11 (ratio 1.3757) (7.10)/(7.5)=5.616799e-11 (ratio 1.3141) refined=6.880666e-11 (ratio 1.0727)
Gamma_b/Gamma_1=0.3687 Gamma_b/Gamma_2=0.3687 lead 2kappa_s/(1+tau1^2/tau2^2)=0.2500
== S2 equal bodies, regime (ii), smaller eta: L=256 g1=0.0023438 g2=0.0023438 tau1=7.324e-06 tau2=7.324e-06 eta=0.050 w=6.48
held spectra of the class: d=1.6639e-05 (flips and excitations 1.6639e-05, joint flip 3.5396e-05; leading 2n|g1+g2|-1.5|B(0)|=3.5248e-05); max tau/d=0.440
selected lowest level, <|r|>=4.83, overlap=0.997536 (min over P 0.997388), group weight=0.997536
E''(0): exact=3.122416e-06 band(2.10)=3.144965e-06 (6.6) sum b0/m=3.144960e-06 rel. band-exact=7.22e-03
ring flux remainder rho_L=E''band/E''line-1=1.589e-06 eps_L=|rho_L|/(E''line |Dmark|)=1.644e-05 (eta=0.050, w=6.48)
mu1=1.713587e+05 mu2=1.713587e+05 (band: 1.713587e+05, 1.713587e+05) beta0=0.98691
Dlat=4.5440e+03 Dmark(odd parts)=-2.9193e+04 Dmark(lead)=-3.0730e+04 |Dlat/Dmark|=0.156 delta_m(S5)=0.131
D0: exact=-2.245256e+04 band=-2.474878e+04 (7.5)=-2.464917e+04 resummed=-2.467203e+04 leading=-2.618559e+04
deviation from exact D0 / (|Dlat|+|Dmark|): band -6.81e-02, (7.5) -6.51e-02, resummed -6.58e-02, leading -1.11e-01
local inertia (7.3), max rel. error near r=0: odd parts 2.70e-03, 2.70e-03; second order 5.02e-02, 5.02e-02
cost: dE0 exact=-6.878422e-07 <M>+K0=-6.912139e-07 (<M>=-7.2940e-07, M2(0)=-9.1554e-07, K0=3.8181e-08); (<M1odd>+<M2odd>)/<M>=2.1189
D0_exact/dE0=3.264201e+10 vs 1/tau1^2+1/tau2^2=3.728156e+10 (ratio 0.8756), 1/tau1^2=1.864078e+10
source at distance s=16 from the midpoint (u=0.0625, kappa_s=0.7657; eta/kappa_s=0.065, w/(L kappa_s)=0.033):
exact selected level: W0'/phi'=1.578923e-05 Wb'/phi'=-1.183756e-06 (7.10): -1.339821e-06
held-field average: W0'/phi'=1.582897e-05 Wb'/phi'=-1.144013e-06; kinetic part (exact - held avg)/phi'=-3.9743e-08 vs -K0=-3.8181e-08
Gamma/phi': body1=4.952472e-11 (tau1^2=5.364582e-11) body2=4.952472e-11 (tau2^2=5.364582e-11) pair=4.930054e-11
Gamma_b=Wb'/D0: /phi'=5.272256e-11 (7.12) lead=4.107484e-11 (ratio 1.2836) (7.10)/(7.5)=5.116636e-11 (ratio 1.0304) refined=4.796082e-11 (ratio 1.0993)
Gamma_b/Gamma_1=1.0646 Gamma_b/Gamma_2=1.0646 lead 2kappa_s/(1+tau1^2/tau2^2)=0.7657
source at distance s=32 from the midpoint (u=0.1250, kappa_s=0.5625; eta/kappa_s=0.089, w/(L kappa_s)=0.045):
exact selected level: W0'/phi'=1.641720e-05 Wb'/phi'=-8.959661e-07 (7.10): -9.678770e-07
held-field average: W0'/phi'=1.645848e-05 Wb'/phi'=-8.547355e-07; kinetic part (exact - held avg)/phi'=-4.1231e-08 vs -K0=-3.8181e-08
Gamma/phi': body1=5.051732e-11 (tau1^2=5.364582e-11) body2=5.051732e-11 (tau2^2=5.364582e-11) pair=5.126132e-11
Gamma_b=Wb'/D0: /phi'=3.990486e-11 (7.12) lead=3.017787e-11 (ratio 1.3223) (7.10)/(7.5)=3.696220e-11 (ratio 1.0796) refined=3.622502e-11 (ratio 1.1016)
Gamma_b/Gamma_1=0.7899 Gamma_b/Gamma_2=0.7899 lead 2kappa_s/(1+tau1^2/tau2^2)=0.5625
source at distance s=64 from the midpoint (u=0.2500, kappa_s=0.2500; eta/kappa_s=0.200, w/(L kappa_s)=0.101):
exact selected level: W0'/phi'=1.741125e-05 Wb'/phi'=-4.331032e-07 (7.10): -3.956550e-07
held-field average: W0'/phi'=1.745487e-05 Wb'/phi'=-3.894352e-07; kinetic part (exact - held avg)/phi'=-4.3668e-08 vs -K0=-3.8181e-08
Gamma/phi': body1=5.206725e-11 (tau1^2=5.364582e-11) body2=5.206725e-11 (tau2^2=5.364582e-11) pair=5.436516e-11
Gamma_b=Wb'/D0: /phi'=1.928971e-11 (7.12) lead=1.341330e-11 (ratio 1.4381) (7.10)/(7.5)=1.510965e-11 (ratio 1.2766) refined=1.734811e-11 (ratio 1.1119)
Gamma_b/Gamma_1=0.3705 Gamma_b/Gamma_2=0.3705 lead 2kappa_s/(1+tau1^2/tau2^2)=0.2500
== C mixed signs, bound at top: L=48 g1=0.018758 g2=-0.037516 tau1=0.0001172 tau2=0.0002345 eta=0.150 w=1.04
held spectra of the class: d=6.2169e-04 (flips and excitations 6.5773e-04, joint flip 6.2169e-04; leading 2n|g1+g2|-1.5|B(0)|=6.0573e-04); max tau/d=0.377
selected highest level, <|r|>=0.14, overlap=0.987247 (min over P 0.987241), group weight=0.987251
E''(0): exact=1.418394e-05 band(2.10)=1.463866e-05 (6.6) sum b0/m=1.463867e-05 rel. band-exact=3.21e-02
ring flux remainder rho_L=E''band/E''line-1=-3.613e-07 eps_L=|rho_L|/(E''line |Dmark|)=9.579e-07 (eta=0.150, w=1.04)
mu1=2.869504e+04 mu2=-1.471817e+04 (band: 2.869504e+04, -1.471817e+04) beta0=0.36195
Dlat=2.4639e+04 Dmark(odd parts)=2.9466e+04 Dmark(lead)=2.5768e+04 |Dlat/Dmark|=0.836 delta_m(S5)=0.836
D0: exact=5.652541e+04 band=5.433537e+04 (7.5)=5.410423e+04 resummed=5.408365e+04 leading=5.040706e+04
deviation from exact D0 / (|Dlat|+|Dmark|): band -4.05e-02, (7.5) -4.48e-02, resummed -4.51e-02, leading -1.13e-01
local inertia (7.3), max rel. error near r=0: odd parts 6.24e-03, 2.04e-02; second order 1.99e-01, 2.71e-02
cost: dE0 exact=1.185405e-04 <M>+K0=1.214758e-04 (<M>=1.2675e-04, M2(0)=1.1724e-04, K0=-5.2789e-06); (<M1odd>+<M2odd>)/<M>=1.9451
D0_exact/dE0=4.768447e+08 vs 1/tau1^2+1/tau2^2=9.094323e+07 (ratio 5.2433), 1/tau1^2=7.275458e+07
== D control: both negative, not bound: L=48 g1=-0.037516 g2=-0.037516 tau1=0.000469 tau2=0.000469 eta=0.150 w=2.10
held spectra of the class: d=1.3363e-03 (flips and excitations 1.3363e-03, joint flip 2.8765e-03; leading 2n|g1+g2|-1.5|B(0)|=2.7746e-03); max tau/d=0.351
selected highest level, <|r|>=21.42, overlap=0.994877 (min over P 0.994788), group weight=0.994878
E''(0): exact=-1.511180e-04 band(2.10)=-1.533368e-04 (6.6) sum b0/m=-1.533368e-04 rel. band-exact=1.47e-02
ring flux remainder rho_L=E''band/E''line-1=-2.277e-08 eps_L=|rho_L|/(E''line |Dmark|)=-1.415e-07 (eta=0.150, w=2.10)
mu1=-3.679545e+03 mu2=-3.679545e+03 (band: -3.679543e+03, -3.679543e+03) beta0=0.95602
Dlat=-3.3855e+02 Dmark(odd parts)=1.1696e+03 Dmark(lead)=1.0493e+03 |Dlat/Dmark|=0.289 delta_m(S5)=0.152
D0: exact=7.417469e+02 band=8.374918e+02 (7.5)=8.310697e+02 resummed=8.322954e+02 leading=7.106973e+02
deviation from exact D0 / (|Dlat|+|Dmark|): band 6.35e-02, (7.5) 5.92e-02, resummed 6.00e-02, leading -2.06e-02
local inertia (7.3), max rel. error near r=0: odd parts 2.45e-02, 2.45e-02; second order 2.47e-01, 2.47e-01
cost: dE0 exact=1.268030e-04 <M>+K0=1.278234e-04 (<M>=1.3380e-04, M2(0)=-2.3448e-04, K0=-5.9765e-06); (<M1odd>+<M2odd>)/<M>=1.8116
D0_exact/dE0=5.849600e+06 vs 1/tau1^2+1/tau2^2=9.094323e+06 (ratio 0.6432), 1/tau1^2=4.547161e+06
== F control: small w, L=6: L=6 g1=0.30857 g2=0.30857 tau1=0.01029 tau2=0.01029 eta=0.150 w=0.50
held spectra of the class: d=9.1350e-02 (flips and excitations 9.1350e-02, joint flip 1.9069e-01; leading 2n|g1+g2|-1.5|B(0)|=1.8257e-01); max tau/d=0.113
selected lowest level, <|r|>=0.01, overlap=0.988633 (min over P 0.988632), group weight=0.988645
E''(0): exact=8.500661e-05 band(2.10)=8.506993e-05 (6.6) sum b0/m=7.563210e-05 rel. band-exact=7.45e-04
ring flux remainder rho_L=E''band/E''line-1=1.248e-01 eps_L=|rho_L|/(E''line |Dmark|)=1.878e+00 (eta=0.150, w=0.50)
mu1=5.029825e+02 mu2=5.029825e+02 (band: 5.029796e+02, 5.029796e+02) beta0=0.07191
Dlat=1.2982e+04 Dmark(odd parts)=-8.0556e+02 Dmark(lead)=-8.7862e+02 |Dlat/Dmark|=16.116 delta_m(S5)=3.201
D0: exact=1.075783e+04 band=1.074908e+04 (7.5)=1.217673e+04 resummed=1.216967e+04 leading=1.210367e+04
deviation from exact D0 / (|Dlat|+|Dmark|): band -6.35e-04, (7.5) 1.03e-01, resummed 1.02e-01, leading 9.76e-02
local inertia (7.3), max rel. error near r=0: odd parts 5.54e-02, 5.54e-02; second order 1.34e-01, 1.34e-01
cost: dE0 exact=-9.988143e-03 <M>+K0=-1.011942e-02 (<M>=-1.1042e-02, M2(0)=-1.5429e-02, K0=9.2258e-04); (<M1odd>+<M2odd>)/<M>=2.4010
D0_exact/dE0=-1.077060e+06 vs 1/tau1^2+1/tau2^2=1.890432e+04 (ratio -56.9743), 1/tau1^2=9.452160e+03
== F control: small w, L=8: L=8 g1=0.22857 g2=0.22857 tau1=0.005714 tau2=0.005714 eta=0.150 w=0.55
held spectra of the class: d=4.9178e-02 (flips and excitations 4.9178e-02, joint flip 1.0479e-01; leading 2n|g1+g2|-1.5|B(0)|=1.0143e-01); max tau/d=0.116
selected lowest level, <|r|>=0.01, overlap=0.988164 (min over P 0.988162), group weight=0.988175
E''(0): exact=5.512182e-05 band(2.10)=5.596583e-05 (6.6) sum b0/m=5.554283e-05 rel. band-exact=1.53e-02
ring flux remainder rho_L=E''band/E''line-1=7.616e-03 eps_L=|rho_L|/(E''line |Dmark|)=6.250e-02 (eta=0.150, w=0.55)
mu1=9.069252e+02 mu2=9.069252e+02 (band: 9.069219e+02, 9.069219e+02) beta0=0.09074
Dlat=1.8175e+04 Dmark(odd parts)=-1.9989e+03 Dmark(lead)=-2.1937e+03 |Dlat/Dmark|=9.093 delta_m(S5)=3.141
D0: exact=1.632778e+04 band=1.605420e+04 (7.5)=1.617616e+04 resummed=1.617446e+04 leading=1.598128e+04
deviation from exact D0 / (|Dlat|+|Dmark|): band -1.36e-02, (7.5) -7.52e-03, resummed -7.60e-03, leading -1.72e-02
local inertia (7.3), max rel. error near r=0: odd parts 1.04e-02, 1.04e-02; second order 9.60e-02, 9.60e-02
cost: dE0 exact=-5.488281e-03 <M>+K0=-5.578652e-03 (<M>=-6.0799e-03, M2(0)=-8.5714e-03, K0=5.0129e-04); (<M1odd>+<M2odd>)/<M>=2.4142
D0_exact/dE0=-2.975027e+06 vs 1/tau1^2+1/tau2^2=6.125000e+04 (ratio -48.5719), 1/tau1^2=3.062500e+04
output/check_results.json35 KB · raw
[
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{
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]
},
{
"case": "A'' equal bodies, smaller eta and tau/d",
"L": 64,
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"weight": []
},
{
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"Wbp_heldavg": 1.9471760990714756e-12,
"kinetic": -4.366798290051431e-08,
"kinetic_formula": -3.818117832923612e-08,
"Gamma1": -2.60336232512965e-16,
"Gamma2": -2.60336232512965e-16,
"Gamma_pair": -2.7182578027896295e-16,
"Gamma_bond": -9.644853173010335e-17,
"Gamma_bond_lead": -6.706648083763806e-17,
"Gamma_bond_formula": -7.554823024524214e-17,
"Gamma_bond_refined": -8.674052532453165e-17,
"eta_over_ks": 0.19997253753203956,
"w_over_Lks": 0.10121845987868938
}
]
},
{
"case": "C mixed signs, bound at top",
"L": 48,
"g1": 0.018758141554494137,
"g2": -0.037516283108988274,
"tau1": 0.00011723838471558836,
"tau2": 0.00023447676943117671,
"eta": 0.15,
"w": 1.041158777266362,
"mean_abs_r": 0.13810469705569056,
"overlap": 0.9872470164252335,
"min_overlap": 0.987240797521818,
"group_weight": 0.9872508634337063,
"selected": "highest",
"d_all": 0.0006216899326016989,
"d_flip": 0.0006577313873510349,
"d_joint": 0.0006216899326016989,
"d_joint_lead": 0.0006057316543638732,
"tau_over_d": 0.37716031277829953,
"mu1": 28695.039374734657,
"mu2": -14718.174725381527,
"mu1_band": 28695.03965007095,
"mu2_band": -14718.170084167125,
"Epp_exact": 1.41839403185647e-05,
"Epp_band": 1.4638664354767647e-05,
"Epp_66": 1.463866964395878e-05,
"beta0": 0.36194934332835926,
"delta_m": 0.8364206236342893,
"D0_exact": 56525.40605543449,
"D0_band": 54335.37366647218,
"Dlat": 24638.66237668533,
"Dmark_odd": 29465.566724283017,
"Dmark_lead": 25768.396683671108,
"D0_formula": 54104.22910096835,
"D0_resummed": 54083.65251672134,
"D0_lead": 50407.059060356434,
"local_inertia_relerr_odd": [
0.0062409256729234475,
0.020386497433878548
],
"local_inertia_relerr_lead": [
0.1989633907136068,
0.02712052085998584
],
"dE0_exact": 0.0001185404929811007,
"M_avg": 0.00012675474438470904,
"M_odd_avg": [
0.00013876363463858333,
0.00010778416296791952
],
"M0_second_order": 0.00011723838471558834,
"K0": -5.278908675824563e-06,
"dE0_formula": 0.00012147583570888447,
"rho_L": -3.613163805926689e-07,
"eps_L": 9.578525144483581e-07,
"weight": []
},
{
"case": "D control: both negative, not bound",
"L": 48,
"g1": -0.037516283108988274,
"g2": -0.037516283108988274,
"tau1": 0.00046895353886235343,
"tau2": 0.00046895353886235343,
"eta": 0.15,
"w": 2.1005640771124123,
"mean_abs_r": 21.417931625743467,
"overlap": 0.994877036956525,
"min_overlap": 0.9947877042006043,
"group_weight": 0.9948781454123341,
"selected": "highest",
"d_all": 0.0013362737410593795,
"d_flip": 0.0013362737410593795,
"d_joint": 0.002876515657451506,
"d_joint_lead": 0.0027746417716022578,
"tau_over_d": 0.3509412214375877,
"mu1": -3679.5454725828235,
"mu2": -3679.5454725828235,
"mu1_band": -3679.542521041781,
"mu2_band": -3679.542521041781,
"Epp_exact": -0.00015111803123672254,
"Epp_band": -0.00015333676254651132,
"Epp_66": -0.0001533367660377337,
"beta0": 0.9560185993341996,
"delta_m": 0.1517829689913267,
"D0_exact": 741.7469272732651,
"D0_band": 837.491763389733,
"Dlat": -338.55317001238535,
"Dmark_odd": 1169.6228608353165,
"Dmark_lead": 1049.2504842404715,
"D0_formula": 831.0696908229311,
"D0_resummed": 832.2954273517444,
"D0_lead": 710.697314228086,
"local_inertia_relerr_odd": [
0.024542144961622547,
0.024542144961106294
],
"local_inertia_relerr_lead": [
0.24719812611666314,
0.24719812611659608
],
"dE0_exact": 0.000126803026265776,
"M_avg": 0.00013379986126172325,
"M_odd_avg": [
0.00012119358067063575,
0.00012119358067065425
],
"M0_second_order": -0.0002344767694311767,
"K0": -5.9764719574083725e-06,
"dE0_formula": 0.00012782338930431489,
"rho_L": -2.2768331775502304e-08,
"eps_L": -1.4151606153500945e-07,
"weight": []
},
{
"case": "F control: small w, L=6",
"L": 6,
"g1": 0.30857142857142855,
"g2": 0.30857142857142855,
"tau1": 0.010285714285714287,
"tau2": 0.010285714285714287,
"eta": 0.15,
"w": 0.5003760013350822,
"mean_abs_r": 0.005005857871816854,
"overlap": 0.9886334697183945,
"min_overlap": 0.9886319734349777,
"group_weight": 0.9886447935735377,
"selected": "lowest",
"d_all": 0.09134992510580386,
"d_flip": 0.09134992510580386,
"d_joint": 0.19068902145378464,
"d_joint_lead": 0.18257142857142855,
"tau_over_d": 0.11259685515670763,
"mu1": 502.98251603803766,
"mu2": 502.98251603803766,
"mu1_band": 502.97964669442575,
"mu2_band": 502.97964669442575,
"Epp_exact": 8.500660943674039e-05,
"Epp_band": 8.506992656605779e-05,
"Epp_66": 7.563210105765028e-05,
"beta0": 0.07191495267297024,
"delta_m": 3.2009866436696965,
"D0_exact": 10757.826120471575,
"D0_band": 10749.076127071849,
"Dlat": 12982.294636961775,
"Dmark_odd": -805.563074176013,
"Dmark_lead": -878.6248622095832,
"D0_formula": 12176.731562785762,
"D0_resummed": 12169.674153105903,
"D0_lead": 12103.669774752192,
"local_inertia_relerr_odd": [
0.055449168177904906,
0.05544916817799128
],
"local_inertia_relerr_lead": [
0.13416723511632223,
0.13416723511632567
],
"dE0_exact": -0.009988142727931137,
"M_avg": -0.011041999706228668,
"M_odd_avg": [
-0.01325568340744365,
-0.01325568340744365
],
"M0_second_order": -0.015428571428571427,
"K0": 0.0009225818171946598,
"dE0_formula": -0.010119417889034008,
"rho_L": 0.12478597548431924,
"eps_L": 1.8778293595137685,
"weight": []
},
{
"case": "F control: small w, L=8",
"L": 8,
"g1": 0.22857142857142856,
"g2": 0.22857142857142856,
"tau1": 0.005714285714285714,
"tau2": 0.005714285714285714,
"eta": 0.15,
"w": 0.5527033173028901,
"mean_abs_r": 0.008251601031074002,
"overlap": 0.9881642098810868,
"min_overlap": 0.9881624990846023,
"group_weight": 0.9881751263245574,
"selected": "lowest",
"d_all": 0.049177846510910506,
"d_flip": 0.049177846510910506,
"d_joint": 0.10478526197676352,
"d_joint_lead": 0.10142857142857142,
"tau_over_d": 0.11619633879287401,
"mu1": 906.9251555308517,
"mu2": 906.9251555308517,
"mu1_band": 906.9218976749206,
"mu2_band": 906.9218976749206,
"Epp_exact": 5.512182321698589e-05,
"Epp_band": 5.596582678772203e-05,
"Epp_66": 5.554282542554698e-05,
"beta0": 0.09074296813241921,
"delta_m": 3.141440285882005,
"D0_exact": 16327.784737247952,
"D0_band": 16054.20278587896,
"Dlat": 18175.02980155255,
"Dmark_odd": -1998.8651072143596,
"Dmark_lead": -2193.7476301489314,
"D0_formula": 16176.164694338191,
"D0_resummed": 16174.46336166016,
"D0_lead": 15981.28217140362,
"local_inertia_relerr_odd": [
0.01039618280925203,
0.010396182809403465
],
"local_inertia_relerr_lead": [
0.09604581118332234,
0.09604581118319166
],
"dE0_exact": -0.0054882811232948026,
"M_avg": -0.006079937656836436,
"M_odd_avg": [
-0.007339035014989627,
-0.007339035014989627
],
"M0_second_order": -0.00857142857142857,
"K0": 0.0005012855946946163,
"dE0_formula": -0.005578652062141819,
"rho_L": 0.007615769614422341,
"eps_L": 0.06250275559936878,
"weight": []
}
]output/fixed_source_bandS1.json1 KB · raw
[
{
"case": "S1 equal bodies, regime (ii)",
"L": 256,
"g1": 0.004687571526665141,
"g2": 0.004687571526665141,
"tau1": 1.4648661020828566e-05,
"tau2": 1.4648661020828566e-05,
"gradients": [
{
"s": 16,
"W0": -2.3953082142124434e-05,
"Wb": 3.621819089213215e-06,
"Wb_held": 3.5242286614599674e-06,
"W1": -1.3787450615668825e-05,
"W2": -1.3787450615668825e-05,
"band": {
"W": -2.3946101047880516e-05,
"Wb": 3.628598413240279e-06,
"W1": -1.3787349730560398e-05,
"W2": -1.3787349730560398e-05
}
},
{
"s": 32,
"W0": -2.212550246445807e-05,
"Wb": 2.444744941014574e-06,
"Wb_held": 2.355045521955021e-06,
"W1": -1.2285123702736322e-05,
"W2": -1.2285123702736322e-05,
"band": {
"W": -2.2121608738817763e-05,
"Wb": 2.448499464737528e-06,
"W1": -1.2285054101777645e-05,
"W2": -1.2285054101777645e-05
}
},
{
"s": 64,
"W0": -1.647562035527947e-05,
"Wb": 8.909168509709312e-07,
"Wb_held": 8.249915647261241e-07,
"W1": -8.6832686031252e-06,
"W2": -8.6832686031252e-06,
"band": {
"W": -1.6473938923684817e-05,
"Wb": 8.924835108812396e-07,
"W1": -8.683211217283028e-06,
"W2": -8.683211217283028e-06
}
}
],
"eig": {
"ME": 1.7763568394002505e-09,
"mid": 1.0258460747536446e-07,
"held": 1.3811174426336947e-06,
"bond_effect": 2.6793900431698603e-05
}
}
]output/fixed_source_bandS1.txt1 KB · raw
# part bandS1
== S1 equal bodies, regime (ii): L=256 g1=0.0046876 g2=0.0046876 tau1=1.465e-05 tau2=1.465e-05
band contract in laboratory coordinates: residual of its Bloch states 1.3e-15 (without source)
s=16: W_0'/g_s=-2.395308e-05, W_b'/g_s=3.6218e-06; held-field average (no kinetic part): W_b,held'/W_b'-1=-2.69e-02
band fixed source: W'/W_0'-1=-2.91e-04, (W'-W_0')/|W_b'|=1.93e-03, W_b'/W_b'(7.15)-1=1.87e-03 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-7.3e-06, W_2'/W_2'(7.15)-1=-7.3e-06
s=32: W_0'/g_s=-2.212550e-05, W_b'/g_s=2.4447e-06; held-field average (no kinetic part): W_b,held'/W_b'-1=-3.67e-02
band fixed source: W'/W_0'-1=-1.76e-04, (W'-W_0')/|W_b'|=1.59e-03, W_b'/W_b'(7.15)-1=1.54e-03 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-5.7e-06, W_2'/W_2'(7.15)-1=-5.7e-06
s=64: W_0'/g_s=-1.647562e-05, W_b'/g_s=8.9092e-07; held-field average (no kinetic part): W_b,held'/W_b'-1=-7.40e-02
band fixed source: W'/W_0'-1=-1.02e-04, (W'-W_0')/|W_b'|=1.89e-03, W_b'/W_b'(7.15)-1=1.76e-03 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-6.6e-06, W_2'/W_2'(7.15)-1=-6.6e-06
lowest 21 band levels with the fixed source (g=1e-06), first-order shifts; bond effect max|mid-nobond|=2.679e-05, kinetic effect max|mid-held|=1.279e-06
max|band - h_c[ME]|=1.776e-09 (0.000 of bond effect)
max|band - h_c[mid]|=1.026e-07 (0.004 of bond effect)
max|band - h_c[held]|=1.381e-06 (0.052 of bond effect)
output/fixed_source_bandS2.json1 KB · raw
[
{
"case": "S2 equal bodies, regime (ii), smaller eta",
"L": 256,
"g1": 0.0023437857633325705,
"g2": 0.0023437857633325705,
"tau1": 7.324330510414283e-06,
"tau2": 7.324330510414283e-06,
"gradients": [
{
"s": 16,
"W0": -1.3815572963403857e-05,
"Wb": 1.0357867669941439e-06,
"Wb_held": 1.0009966347997775e-06,
"W1": -7.425679865199e-06,
"W2": -7.425679865199e-06,
"band": {
"W": -1.3812994775526909e-05,
"Wb": 1.038325927145596e-06,
"W1": -7.425660351336252e-06,
"W2": -7.425660351336252e-06
}
},
{
"s": 32,
"W0": -1.2312896355648993e-05,
"Wb": 6.719745690463887e-07,
"Wb_held": 6.410160077038526e-07,
"W1": -6.492435462347691e-06,
"W2": -6.492435462347691e-06,
"band": {
"W": -1.2311526896927657e-05,
"Wb": 6.732928553745791e-07,
"W1": -6.492409876151118e-06,
"W2": -6.492409876151118e-06
}
},
{
"s": 64,
"W0": -8.7056228768893e-06,
"Wb": 2.1655160135827106e-07,
"Wb_held": 1.9474499092769076e-07,
"W1": -4.461087239123786e-06,
"W2": -4.461087239123786e-06,
"band": {
"W": -8.705014882909486e-06,
"Wb": 2.1714368148104818e-07,
"W1": -4.461079282195267e-06,
"W2": -4.461079282195267e-06
}
}
],
"eig": {
"ME": 1.5543122344752192e-09,
"mid": 6.616929226765933e-08,
"held": 5.566658245470535e-07,
"bond_effect": 7.989386929807551e-06
}
}
]output/fixed_source_bandS2.txt1 KB · raw
# part bandS2
== S2 equal bodies, regime (ii), smaller eta: L=256 g1=0.0023438 g2=0.0023438 tau1=7.324e-06 tau2=7.324e-06
band contract in laboratory coordinates: residual of its Bloch states 7.3e-16 (without source)
s=16: W_0'/g_s=-1.381557e-05, W_b'/g_s=1.0358e-06; held-field average (no kinetic part): W_b,held'/W_b'-1=-3.36e-02
band fixed source: W'/W_0'-1=-1.87e-04, (W'-W_0')/|W_b'|=2.49e-03, W_b'/W_b'(7.15)-1=2.45e-03 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-2.6e-06, W_2'/W_2'(7.15)-1=-2.6e-06
s=32: W_0'/g_s=-1.231290e-05, W_b'/g_s=6.7197e-07; held-field average (no kinetic part): W_b,held'/W_b'-1=-4.61e-02
band fixed source: W'/W_0'-1=-1.11e-04, (W'-W_0')/|W_b'|=2.04e-03, W_b'/W_b'(7.15)-1=1.96e-03 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-3.9e-06, W_2'/W_2'(7.15)-1=-3.9e-06
s=64: W_0'/g_s=-8.705623e-06, W_b'/g_s=2.1655e-07; held-field average (no kinetic part): W_b,held'/W_b'-1=-1.01e-01
band fixed source: W'/W_0'-1=-6.98e-05, (W'-W_0')/|W_b'|=2.81e-03, W_b'/W_b'(7.15)-1=2.73e-03 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-1.8e-06, W_2'/W_2'(7.15)-1=-1.8e-06
lowest 21 band levels with the fixed source (g=1e-06), first-order shifts; bond effect max|mid-nobond|=7.989e-06, kinetic effect max|mid-held|=4.927e-07
max|band - h_c[ME]|=1.554e-09 (0.000 of bond effect)
max|band - h_c[mid]|=6.617e-08 (0.008 of bond effect)
max|band - h_c[held]|=5.567e-07 (0.070 of bond effect)
output/fixed_source_exact.json8 KB · raw
[
{
"case": "EA equal bodies",
"L": 12,
"g1": 0.15104895104895105,
"g2": 0.15104895104895105,
"tau1": 0.0075524475524475524,
"tau2": 0.0075524475524475524,
"residual": 2.9643681602056813e-15,
"n_selected": [
12,
12
],
"K": 7,
"bond_effect": 0.0009024869740414942,
"kinetic_effect": 7.263967205517474e-05,
"deviations": {
"ME": [
1.554312234475219e-08,
1.554312234475219e-08
],
"mid": [
0.0009763745367763477,
0.0009763745367763477
],
"held": [
0.0009103917619768254,
0.001029938356822413
],
"nobond": [
0.0010571610253862218,
0.00132595268098612
]
},
"gradients": [
{
"s": 2,
"Wex": -0.01176501592683838,
"W0": -0.012397110019173513,
"Wheld": -0.01287403480989752,
"Wex_bond": 0.0026318482701482085,
"Wb": 0.002216353253660569,
"Wb_held": 0.0017394284629365618,
"Wex1": -0.0071984320984932945,
"W1": -0.007306731636417041
},
{
"s": 3,
"Wex": -0.010440408497595859,
"W0": -0.0104758520807412,
"Wheld": -0.01087668037658028,
"Wex_bond": 0.0010797987472643336,
"Wb": 0.0010803320302118086,
"Wb_held": 0.0006795037343727273,
"Wex1": -0.005760103622430096,
"W1": -0.005778092055476504
},
{
"s": 4,
"Wex": -0.007450604141238888,
"W0": -0.007503837704122163,
"Wheld": -0.007783976402854886,
"Wex_bond": 0.0005254725441979843,
"Wb": 0.0004917671499865967,
"Wb_held": 0.00021162845125387297,
"Wex1": -0.003988038342718436,
"W1": -0.00399780242705438
}
]
},
{
"case": "EA equal bodies",
"L": 16,
"g1": 0.11294117647058823,
"g2": 0.11294117647058823,
"tau1": 0.0042352941176470585,
"tau2": 0.0042352941176470585,
"residual": 5.716660380179216e-15,
"n_selected": [
16,
16
],
"K": 9,
"bond_effect": 0.0005270894831710393,
"kinetic_effect": 4.935163389063746e-05,
"deviations": {
"ME": [
1.7763568394002505e-08,
2.220446049250313e-08
],
"mid": [
0.0004564504330062391,
0.0007073808205859677
],
"held": [
0.00041654679705516173,
0.0007567235726924082
],
"nobond": [
0.0007277467517496916,
0.0010862999388905337
]
},
"gradients": [
{
"s": 2,
"Wex": -0.0070071849718532885,
"W0": -0.007387699790295819,
"Wheld": -0.007618015816994568,
"Wex_bond": 0.0018189307718413407,
"Wb": 0.001555275437123027,
"Wb_held": 0.0013249594104242784,
"Wex1": -0.004413057871847315,
"W1": -0.004471487613709423
},
{
"s": 3,
"Wex": -0.00689054728319189,
"W0": -0.006910858351792941,
"Wheld": -0.007129294692266294,
"Wex_bond": 0.0009047035963062611,
"Wb": 0.0009026939013020291,
"Wb_held": 0.0006842575608286758,
"Wex1": -0.0038976254397490757,
"W1": -0.003906776126547485
},
{
"s": 4,
"Wex": -0.005921834427690526,
"W0": -0.005951966897423534,
"Wheld": -0.006136620779759839,
"Wex_bond": 0.0005603037672305029,
"Wb": 0.0005411805974624359,
"Wb_held": 0.0003565267151261313,
"Wex1": -0.0032410690974605146,
"W1": -0.003246573747442985
},
{
"s": 5,
"Wex": -0.0047207874593494445,
"W0": -0.004715467234239927,
"Wheld": -0.004859147041736969,
"Wex_bond": 0.0002890311026776185,
"Wb": 0.00029856754843122363,
"Wb_held": 0.00015488774093418145,
"Wex1": -0.0025049092810135315,
"W1": -0.0025070173913355753
},
{
"s": 6,
"Wex": -0.003265277386309974,
"W0": -0.003266452552892034,
"Wheld": -0.0033646655561178976,
"Wex_bond": 0.00014468540929048895,
"Wb": 0.00014591459296234982,
"Wb_held": 4.770158973648603e-05,
"Wex1": -0.0017049813978002315,
"W1": -0.0017061835729271918
}
]
},
{
"case": "EB unequal bodies and speeds",
"L": 12,
"g1": 0.15104895104895105,
"g2": 0.07552447552447553,
"tau1": 0.0037762237762237762,
"tau2": 0.0018881118881118881,
"residual": 2.8066189335470666e-15,
"n_selected": [
12,
12
],
"K": 7,
"bond_effect": 0.0006637801419628886,
"kinetic_effect": 6.693312570860144e-05,
"deviations": {
"ME": [
1.7763568394002505e-08,
1.7763568394002505e-08
],
"mid": [
0.00019033219444963834,
0.00019033219444963834
],
"held": [
0.0001727618048619206,
0.000195012894721458
],
"nobond": [
0.0005028244487448319,
0.0005028244487448319
]
},
"gradients": [
{
"s": 2,
"Wex": -0.009896412620824274,
"W0": -0.010071845362658961,
"Wheld": -0.010273223047263233,
"Wex_bond": 0.0012789146806656063,
"Wb": 0.001148394930511061,
"Wb_held": 0.0009470172459067894,
"Wex1": -0.007276683583876675,
"W1": -0.00730582482398047
},
{
"s": 3,
"Wex": -0.008150813702160038,
"W0": -0.008158319460803407,
"Wheld": -0.00831708236006138,
"Wex_bond": 0.0006408396948097209,
"Wb": 0.0006374975174898642,
"Wb_held": 0.0004787346182318908,
"Wex1": -0.0057740557798723335,
"W1": -0.005777069563723501
},
{
"s": 4,
"Wex": -0.005729501948446153,
"W0": -0.005736101186648267,
"Wheld": -0.005845256330324601,
"Wex_bond": 0.00031771905402936534,
"Wb": 0.0003134019693157575,
"Wb_held": 0.0002042468256394231,
"Wex1": -0.003995336047367945,
"W1": -0.0039969495905000235
}
]
},
{
"case": "EB unequal bodies and speeds",
"L": 16,
"g1": 0.11294117647058823,
"g2": 0.05647058823529411,
"tau1": 0.0021176470588235292,
"tau2": 0.0010588235294117646,
"residual": 3.793158767957919e-15,
"n_selected": [
16,
16
],
"K": 9,
"bond_effect": 0.00037928105101059373,
"kinetic_effect": 3.395728143118504e-05,
"deviations": {
"ME": [
2.886579864025407e-08,
2.886579864025407e-08
],
"mid": [
0.0001091970958100319,
0.00013530510045711708
],
"held": [
0.00012722267683784594,
0.00015592860336255399
],
"nobond": [
0.00030943247963932663,
0.00034122704661854186
]
},
"gradients": [
{
"s": 2,
"Wex": -0.005957060035404738,
"W0": -0.0060999612406282475,
"Wheld": -0.006207292152193365,
"Wex_bond": 0.0009204120528948957,
"Wb": 0.0008018869852101027,
"Wb_held": 0.0006945560736449849,
"Wex1": -0.004455529022210799,
"W1": -0.004471249415422311
},
{
"s": 3,
"Wex": -0.005448963232384991,
"W0": -0.005451659731967366,
"Wheld": -0.005545612748129813,
"Wex_bond": 0.0005317243940681816,
"Wb": 0.0005310131233448104,
"Wb_held": 0.0004370601071823632,
"Wex1": -0.0039050179049951103,
"W1": -0.003906462735008071
},
{
"s": 4,
"Wex": -0.004603394593434538,
"W0": -0.004607278278034418,
"Wheld": -0.00468510121085243,
"Wex_bond": 0.00033619309712421344,
"Wb": 0.00033356951746319266,
"Wb_held": 0.00025574658464518063,
"Wex1": -0.003245332498447523,
"W1": -0.0032462442730518647
},
{
"s": 5,
"Wex": -0.003599881598526672,
"W0": -0.0036030993768586955,
"Wheld": -0.0036629524743044956,
"Wex_bond": 0.00019699960726270171,
"Wb": 0.00019440492278575972,
"Wb_held": 0.00013455182533995969,
"Wex1": -0.0025062643859193393,
"W1": -0.0025067226421388797
},
{
"s": 6,
"Wex": -0.0024716996616103507,
"W0": -0.0024738982399042653,
"Wheld": -0.0025144838741332776,
"Wex_bond": 0.00010393523743952582,
"Wb": 0.00010210392375919336,
"Wb_held": 6.151828953018107e-05,
"Wex1": -0.0017057010449389731,
"W1": -0.0017059637682780501
}
]
}
]output/fixed_source_exact.txt6 KB · raw
# part exact
== EXACT EA equal bodies: L=12 g1=0.15105 g2=0.15105 tau1=0.007552 tau2=0.007552 (lab dimension 3456)
Bloch states of the sector contracts in laboratory coordinates: max residual 3.0e-15, weight outside class 0.0e+00
band levels selected with the fixed source: [12, 12] of 12; first-order shifts (eigenvalue - E_b)/g, g=1e-07
scale over the lowest 7 levels (|P|<=pi/2): bond effect max|mid-nobond|=9.025e-04, kinetic effect max|mid-held|=7.264e-05
max|exact - h_c[ME]|: lowest 7: 1.554e-08 (0.000 of bond effect), all 12: 1.554e-08
max|exact - h_c[mid]|: lowest 7: 9.764e-04 (1.082 of bond effect), all 12: 9.764e-04
max|exact - h_c[held]|: lowest 7: 9.104e-04 (1.009 of bond effect), all 12: 1.030e-03
max|exact - h_c[nobond]|: lowest 7: 1.057e-03 (1.171 of bond effect), all 12: 1.326e-03
s=2: W_ex'/g_s=-1.176502e-02 W_0'/g_s=-1.239711e-02; bond: W_ex,b'=2.6318e-03 W_b'=2.2164e-03 (diff/|W_b'|=1.87e-01; held-field average -2.15e-01; single body W_ex,1'/W_1'-1=-1.5e-02)
s=3: W_ex'/g_s=-1.044041e-02 W_0'/g_s=-1.047585e-02; bond: W_ex,b'=1.0798e-03 W_b'=1.0803e-03 (diff/|W_b'|=-4.94e-04; held-field average -3.71e-01; single body W_ex,1'/W_1'-1=-3.1e-03)
s=4: W_ex'/g_s=-7.450604e-03 W_0'/g_s=-7.503838e-03; bond: W_ex,b'=5.2547e-04 W_b'=4.9177e-04 (diff/|W_b'|=6.85e-02; held-field average -5.70e-01; single body W_ex,1'/W_1'-1=-2.4e-03)
== EXACT EA equal bodies: L=16 g1=0.11294 g2=0.11294 tau1=0.004235 tau2=0.004235 (lab dimension 8192)
Bloch states of the sector contracts in laboratory coordinates: max residual 5.7e-15, weight outside class 0.0e+00
band levels selected with the fixed source: [16, 16] of 16; first-order shifts (eigenvalue - E_b)/g, g=1e-07
scale over the lowest 9 levels (|P|<=pi/2): bond effect max|mid-nobond|=5.271e-04, kinetic effect max|mid-held|=4.935e-05
max|exact - h_c[ME]|: lowest 9: 1.776e-08 (0.000 of bond effect), all 16: 2.220e-08
max|exact - h_c[mid]|: lowest 9: 4.565e-04 (0.866 of bond effect), all 16: 7.074e-04
max|exact - h_c[held]|: lowest 9: 4.165e-04 (0.790 of bond effect), all 16: 7.567e-04
max|exact - h_c[nobond]|: lowest 9: 7.277e-04 (1.381 of bond effect), all 16: 1.086e-03
s=2: W_ex'/g_s=-7.007185e-03 W_0'/g_s=-7.387700e-03; bond: W_ex,b'=1.8189e-03 W_b'=1.5553e-03 (diff/|W_b'|=1.70e-01; held-field average -1.48e-01; single body W_ex,1'/W_1'-1=-1.3e-02)
s=3: W_ex'/g_s=-6.890547e-03 W_0'/g_s=-6.910858e-03; bond: W_ex,b'=9.0470e-04 W_b'=9.0269e-04 (diff/|W_b'|=2.23e-03; held-field average -2.42e-01; single body W_ex,1'/W_1'-1=-2.3e-03)
s=4: W_ex'/g_s=-5.921834e-03 W_0'/g_s=-5.951967e-03; bond: W_ex,b'=5.6030e-04 W_b'=5.4118e-04 (diff/|W_b'|=3.53e-02; held-field average -3.41e-01; single body W_ex,1'/W_1'-1=-1.7e-03)
s=5: W_ex'/g_s=-4.720787e-03 W_0'/g_s=-4.715467e-03; bond: W_ex,b'=2.8903e-04 W_b'=2.9857e-04 (diff/|W_b'|=-3.19e-02; held-field average -4.81e-01; single body W_ex,1'/W_1'-1=-8.4e-04)
s=6: W_ex'/g_s=-3.265277e-03 W_0'/g_s=-3.266453e-03; bond: W_ex,b'=1.4469e-04 W_b'=1.4591e-04 (diff/|W_b'|=-8.42e-03; held-field average -6.73e-01; single body W_ex,1'/W_1'-1=-7.0e-04)
== EXACT EB unequal bodies and speeds: L=12 g1=0.15105 g2=0.075524 tau1=0.003776 tau2=0.001888 (lab dimension 3456)
Bloch states of the sector contracts in laboratory coordinates: max residual 2.8e-15, weight outside class 0.0e+00
band levels selected with the fixed source: [12, 12] of 12; first-order shifts (eigenvalue - E_b)/g, g=1e-07
scale over the lowest 7 levels (|P|<=pi/2): bond effect max|mid-nobond|=6.638e-04, kinetic effect max|mid-held|=6.693e-05
max|exact - h_c[ME]|: lowest 7: 1.776e-08 (0.000 of bond effect), all 12: 1.776e-08
max|exact - h_c[mid]|: lowest 7: 1.903e-04 (0.287 of bond effect), all 12: 1.903e-04
max|exact - h_c[held]|: lowest 7: 1.728e-04 (0.260 of bond effect), all 12: 1.950e-04
max|exact - h_c[nobond]|: lowest 7: 5.028e-04 (0.758 of bond effect), all 12: 5.028e-04
s=2: W_ex'/g_s=-9.896413e-03 W_0'/g_s=-1.007185e-02; bond: W_ex,b'=1.2789e-03 W_b'=1.1484e-03 (diff/|W_b'|=1.14e-01; held-field average -1.75e-01; single body W_ex,1'/W_1'-1=-4.0e-03)
s=3: W_ex'/g_s=-8.150814e-03 W_0'/g_s=-8.158319e-03; bond: W_ex,b'=6.4084e-04 W_b'=6.3750e-04 (diff/|W_b'|=5.24e-03; held-field average -2.49e-01; single body W_ex,1'/W_1'-1=-5.2e-04)
s=4: W_ex'/g_s=-5.729502e-03 W_0'/g_s=-5.736101e-03; bond: W_ex,b'=3.1772e-04 W_b'=3.1340e-04 (diff/|W_b'|=1.38e-02; held-field average -3.48e-01; single body W_ex,1'/W_1'-1=-4.0e-04)
== EXACT EB unequal bodies and speeds: L=16 g1=0.11294 g2=0.056471 tau1=0.002118 tau2=0.001059 (lab dimension 8192)
Bloch states of the sector contracts in laboratory coordinates: max residual 3.8e-15, weight outside class 0.0e+00
band levels selected with the fixed source: [16, 16] of 16; first-order shifts (eigenvalue - E_b)/g, g=1e-07
scale over the lowest 9 levels (|P|<=pi/2): bond effect max|mid-nobond|=3.793e-04, kinetic effect max|mid-held|=3.396e-05
max|exact - h_c[ME]|: lowest 9: 2.887e-08 (0.000 of bond effect), all 16: 2.887e-08
max|exact - h_c[mid]|: lowest 9: 1.092e-04 (0.288 of bond effect), all 16: 1.353e-04
max|exact - h_c[held]|: lowest 9: 1.272e-04 (0.335 of bond effect), all 16: 1.559e-04
max|exact - h_c[nobond]|: lowest 9: 3.094e-04 (0.816 of bond effect), all 16: 3.412e-04
s=2: W_ex'/g_s=-5.957060e-03 W_0'/g_s=-6.099961e-03; bond: W_ex,b'=9.2041e-04 W_b'=8.0189e-04 (diff/|W_b'|=1.48e-01; held-field average -1.34e-01; single body W_ex,1'/W_1'-1=-3.5e-03)
s=3: W_ex'/g_s=-5.448963e-03 W_0'/g_s=-5.451660e-03; bond: W_ex,b'=5.3172e-04 W_b'=5.3101e-04 (diff/|W_b'|=1.34e-03; held-field average -1.77e-01; single body W_ex,1'/W_1'-1=-3.7e-04)
s=4: W_ex'/g_s=-4.603395e-03 W_0'/g_s=-4.607278e-03; bond: W_ex,b'=3.3619e-04 W_b'=3.3357e-04 (diff/|W_b'|=7.87e-03; held-field average -2.33e-01; single body W_ex,1'/W_1'-1=-2.8e-04)
s=5: W_ex'/g_s=-3.599882e-03 W_0'/g_s=-3.603099e-03; bond: W_ex,b'=1.9700e-04 W_b'=1.9440e-04 (diff/|W_b'|=1.33e-02; held-field average -3.08e-01; single body W_ex,1'/W_1'-1=-1.8e-04)
s=6: W_ex'/g_s=-2.471700e-03 W_0'/g_s=-2.473898e-03; bond: W_ex,b'=1.0394e-04 W_b'=1.0210e-04 (diff/|W_b'|=1.79e-02; held-field average -3.97e-01; single body W_ex,1'/W_1'-1=-1.5e-04)
output/fixed_source_me128.json2 KB · raw
[
{
"case": "R equal bodies, v1 case",
"L": 128,
"g1": 0.014063358359274858,
"g2": 0.014063358359274858,
"tau1": 6.592199230910089e-05,
"tau2": 6.592199230910089e-05,
"gradients": [
{
"s": 8,
"W0": -0.00012545773244848363,
"Wb": 2.8402868025854608e-05,
"Wb_held": 2.7494726919669293e-05,
"W1": -7.693030023716912e-05,
"W2": -7.693030023716912e-05,
"exact": {
"W": -0.00012514853165823637,
"Wb": 2.8707864722935395e-05,
"W1": -7.692819819058588e-05,
"W2": -7.692819819058588e-05
},
"band": {
"W": -0.0001252194676341625,
"Wb": 2.8637239754181545e-05,
"W1": -7.692835369417202e-05,
"W2": -7.692835369417202e-05
}
},
{
"s": 16,
"W0": -0.00011980806921786614,
"Wb": 1.9788343703851154e-05,
"Wb_held": 1.8931388177748868e-05,
"W1": -6.979820646085865e-05,
"W2": -6.979820646085865e-05,
"exact": {
"W": -0.00011971147328134228,
"Wb": 1.9882082987084364e-05,
"W1": -6.979677813421332e-05,
"W2": -6.979677813421332e-05
},
"band": {
"W": -0.0001197127502425635,
"Wb": 1.9880803791518485e-05,
"W1": -6.9796777017041e-05,
"W2": -6.9796777017041e-05
}
},
{
"s": 32,
"W0": -9.344442587778425e-05,
"Wb": 7.866885802737186e-06,
"Wb_held": 7.217196711391434e-06,
"W1": -5.0655655840260716e-05,
"W2": -5.0655655840260716e-05,
"exact": {
"W": -9.342642974670672e-05,
"Wb": 7.883552923678393e-06,
"W1": -5.065499133519256e-05,
"W2": -5.065499133519256e-05
},
"band": {
"W": -9.341715092464446e-05,
"Wb": 7.892816531435143e-06,
"W1": -5.06549837280398e-05,
"W2": -5.06549837280398e-05
}
}
]
}
]output/fixed_source_me128.txt2 KB · raw
# part me128
== R equal bodies, v1 case: L=128 g1=0.014063 g2=0.014063 tau1=6.592e-05 tau2=6.592e-05
band contract in laboratory coordinates: residual of its Bloch states 1.1e-15 (without source)
s=8: W_0'/g_s=-1.254577e-04, W_b'/g_s=2.8403e-05; held-field average (no kinetic part): W_b,held'/W_b'-1=-3.20e-02
exact fixed source: W'/W_0'-1=-2.46e-03, (W'-W_0')/|W_b'|=1.09e-02, W_b'/W_b'(7.15)-1=1.07e-02 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-2.7e-05, W_2'/W_2'(7.15)-1=-2.7e-05
band fixed source: W'/W_0'-1=-1.90e-03, (W'-W_0')/|W_b'|=8.39e-03, W_b'/W_b'(7.15)-1=8.25e-03 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-2.5e-05, W_2'/W_2'(7.15)-1=-2.5e-05
s=16: W_0'/g_s=-1.198081e-04, W_b'/g_s=1.9788e-05; held-field average (no kinetic part): W_b,held'/W_b'-1=-4.33e-02
exact fixed source: W'/W_0'-1=-8.06e-04, (W'-W_0')/|W_b'|=4.88e-03, W_b'/W_b'(7.15)-1=4.74e-03 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-2.0e-05, W_2'/W_2'(7.15)-1=-2.0e-05
band fixed source: W'/W_0'-1=-7.96e-04, (W'-W_0')/|W_b'|=4.82e-03, W_b'/W_b'(7.15)-1=4.67e-03 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-2.0e-05, W_2'/W_2'(7.15)-1=-2.0e-05
s=32: W_0'/g_s=-9.344443e-05, W_b'/g_s=7.8669e-06; held-field average (no kinetic part): W_b,held'/W_b'-1=-8.26e-02
exact fixed source: W'/W_0'-1=-1.93e-04, (W'-W_0')/|W_b'|=2.29e-03, W_b'/W_b'(7.15)-1=2.12e-03 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-1.3e-05, W_2'/W_2'(7.15)-1=-1.3e-05
band fixed source: W'/W_0'-1=-2.92e-04, (W'-W_0')/|W_b'|=3.47e-03, W_b'/W_b'(7.15)-1=3.30e-03 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-1.3e-05, W_2'/W_2'(7.15)-1=-1.3e-05
output/fixed_source_me64.json5 KB · raw
[
{
"case": "A equal bodies",
"L": 64,
"g1": 0.028131868131868132,
"g2": 0.028131868131868132,
"tau1": 0.0002637362637362637,
"tau2": 0.0002637362637362637,
"gradients": [
{
"s": 4,
"W0": -0.0004993572051967205,
"Wb": 0.00011610559539964194,
"Wb_held": 0.00011027419996338401,
"W1": -0.0003077314002981812,
"W2": -0.0003077314002981812,
"exact": {
"W": -0.0004925522190041155,
"Wb": 0.0001225182604415731,
"W1": -0.0003075352397228443,
"W2": -0.0003075352397228443
},
"band": {
"W": -0.0004934991841182602,
"Wb": 0.00012190097947607781,
"W1": -0.000307700081797169,
"W2": -0.000307700081797169
}
},
{
"s": 8,
"W0": -0.00047970708232185355,
"Wb": 7.87050598174445e-05,
"Wb_held": 7.318476723386976e-05,
"W1": -0.000279206071069649,
"W2": -0.000279206071069649,
"exact": {
"W": -0.0004786650405929835,
"Wb": 7.970014095542988e-05,
"W1": -0.0002791825907742067,
"W2": -0.0002791825907742067
},
"band": {
"W": -0.0004786603183616518,
"Wb": 7.970608700739665e-05,
"W1": -0.00027918320268452423,
"W2": -0.00027918320268452423
}
},
{
"s": 12,
"W0": -0.00043668435613435086,
"Wb": 5.1091641479702327e-05,
"Wb_held": 4.61499882787467e-05,
"W1": -0.0002438879988070266,
"W2": -0.0002438879988070266,
"exact": {
"W": -0.00043619953017712265,
"Wb": 5.1544424931465604e-05,
"W1": -0.00024387197755429413,
"W2": -0.00024387197755429413
},
"band": {
"W": -0.00043611687858278753,
"Wb": 5.1627092079385185e-05,
"W1": -0.00024387198533108636,
"W2": -0.00024387198533108636
}
},
{
"s": 16,
"W0": -0.0003746491782944932,
"Wb": 3.062367993534286e-05,
"Wb_held": 2.645205308511242e-05,
"W1": -0.00020263642911491804,
"W2": -0.00020263642911491804,
"exact": {
"W": -0.00037440241597521025,
"Wb": 3.084911674900749e-05,
"W1": -0.00020262576636210887,
"W2": -0.00020262576636210887
},
"band": {
"W": -0.0003743245259925695,
"Wb": 3.092704912708959e-05,
"W1": -0.00020262578755982953,
"W2": -0.00020262578755982953
}
}
]
},
{
"case": "B unequal bodies and speeds",
"L": 64,
"g1": 0.028131868131868132,
"g2": 0.014065934065934066,
"tau1": 0.00013186813186813186,
"tau2": 6.593406593406593e-05,
"gradients": [
{
"s": 4,
"W0": -0.0004151787844054195,
"Wb": 6.425436647457088e-05,
"Wb_held": 6.161283970673032e-05,
"W1": -0.0003077309628114547,
"W2": -0.0001717021880685357,
"exact": {
"W": -0.0004143460125404924,
"Wb": 6.505757539227667e-05,
"W1": -0.00030771084766646577,
"W2": -0.00017169274026630332
},
"band": {
"W": -0.0004143730175079933,
"Wb": 6.505041353787981e-05,
"W1": -0.0003077231260763865,
"W2": -0.00017170030496948662
}
},
{
"s": 8,
"W0": -0.0003868160341753326,
"Wb": 4.3958139756287463e-05,
"Wb_held": 4.1545475770871955e-05,
"W1": -0.0002792050962841658,
"W2": -0.00015156907764745427,
"exact": {
"W": -0.000386716099425111,
"Wb": 4.405083839175653e-05,
"W1": -0.0002791993710445457,
"W2": -0.0001515675667723218
},
"band": {
"W": -0.0003867058150384198,
"Wb": 4.406115448125653e-05,
"W1": -0.0002791993868088774,
"W2": -0.0001515675827107989
}
},
{
"s": 12,
"W0": -0.0003451094973421767,
"Wb": 2.823407457810262e-05,
"Wb_held": 2.6122463490657263e-05,
"W1": -0.00024388673900923186,
"W2": -0.00012945683291104744,
"exact": {
"W": -0.0003450398819406404,
"Wb": 2.8298522637770253e-05,
"W1": -0.00024388273617060186,
"W2": -0.0001294556684078088
},
"band": {
"W": -0.00034503599339885584,
"Wb": 2.830242446295525e-05,
"W1": -0.00024388273524373824,
"W2": -0.00012945568261807285
}
},
{
"s": 16,
"W0": -0.00029152791733656584,
"Wb": 1.676138484625264e-05,
"Wb_held": 1.5008425381638282e-05,
"W1": -0.00020263511461186476,
"W2": -0.00010565418757095373,
"exact": {
"W": -0.0002914727095105641,
"Wb": 1.6813060077638248e-05,
"W1": -0.00020263245048560315,
"W2": -0.0001056533191025992
},
"band": {
"W": -0.00029147137181677676,
"Wb": 1.6814400143096153e-05,
"W1": -0.00020263247171026753,
"W2": -0.00010565330024960538
}
}
]
}
]output/fixed_source_me64.txt4 KB · raw
# part me64
== A equal bodies: L=64 g1=0.028132 g2=0.028132 tau1=0.0002637 tau2=0.0002637
band contract with the source: analytic first order vs finite differences +-1e-05: rel. max diff H0 4.7e-12, dH 5.2e-09
band contract in laboratory coordinates: residual of its Bloch states 2.3e-15 (without source)
s=4: W_0'/g_s=-4.993572e-04, W_b'/g_s=1.1611e-04; held-field average (no kinetic part): W_b,held'/W_b'-1=-5.02e-02
exact fixed source: W'/W_0'-1=-1.36e-02, (W'-W_0')/|W_b'|=5.86e-02, W_b'/W_b'(7.15)-1=5.52e-02 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-6.4e-04, W_2'/W_2'(7.15)-1=-6.4e-04
band fixed source: W'/W_0'-1=-1.17e-02, (W'-W_0')/|W_b'|=5.05e-02, W_b'/W_b'(7.15)-1=4.99e-02 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-1.0e-04, W_2'/W_2'(7.15)-1=-1.0e-04
s=8: W_0'/g_s=-4.797071e-04, W_b'/g_s=7.8705e-05; held-field average (no kinetic part): W_b,held'/W_b'-1=-7.01e-02
exact fixed source: W'/W_0'-1=-2.17e-03, (W'-W_0')/|W_b'|=1.32e-02, W_b'/W_b'(7.15)-1=1.26e-02 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-8.4e-05, W_2'/W_2'(7.15)-1=-8.4e-05
band fixed source: W'/W_0'-1=-2.18e-03, (W'-W_0')/|W_b'|=1.33e-02, W_b'/W_b'(7.15)-1=1.27e-02 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-8.2e-05, W_2'/W_2'(7.15)-1=-8.2e-05
s=12: W_0'/g_s=-4.366844e-04, W_b'/g_s=5.1092e-05; held-field average (no kinetic part): W_b,held'/W_b'-1=-9.67e-02
exact fixed source: W'/W_0'-1=-1.11e-03, (W'-W_0')/|W_b'|=9.49e-03, W_b'/W_b'(7.15)-1=8.86e-03 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-6.6e-05, W_2'/W_2'(7.15)-1=-6.6e-05
band fixed source: W'/W_0'-1=-1.30e-03, (W'-W_0')/|W_b'|=1.11e-02, W_b'/W_b'(7.15)-1=1.05e-02 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-6.6e-05, W_2'/W_2'(7.15)-1=-6.6e-05
s=16: W_0'/g_s=-3.746492e-04, W_b'/g_s=3.0624e-05; held-field average (no kinetic part): W_b,held'/W_b'-1=-1.36e-01
exact fixed source: W'/W_0'-1=-6.59e-04, (W'-W_0')/|W_b'|=8.06e-03, W_b'/W_b'(7.15)-1=7.36e-03 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-5.3e-05, W_2'/W_2'(7.15)-1=-5.3e-05
band fixed source: W'/W_0'-1=-8.67e-04, (W'-W_0')/|W_b'|=1.06e-02, W_b'/W_b'(7.15)-1=9.91e-03 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-5.3e-05, W_2'/W_2'(7.15)-1=-5.3e-05
== B unequal bodies and speeds: L=64 g1=0.028132 g2=0.014066 tau1=0.0001319 tau2=6.593e-05
band contract with the source: analytic first order vs finite differences +-1e-05: rel. max diff H0 4.6e-12, dH 5.9e-09
band contract in laboratory coordinates: residual of its Bloch states 1.2e-15 (without source)
s=4: W_0'/g_s=-4.151788e-04, W_b'/g_s=6.4254e-05; held-field average (no kinetic part): W_b,held'/W_b'-1=-4.11e-02
exact fixed source: W'/W_0'-1=-2.01e-03, (W'-W_0')/|W_b'|=1.30e-02, W_b'/W_b'(7.15)-1=1.25e-02 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-6.5e-05, W_2'/W_2'(7.15)-1=-5.5e-05
band fixed source: W'/W_0'-1=-1.94e-03, (W'-W_0')/|W_b'|=1.25e-02, W_b'/W_b'(7.15)-1=1.24e-02 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-2.5e-05, W_2'/W_2'(7.15)-1=-1.1e-05
s=8: W_0'/g_s=-3.868160e-04, W_b'/g_s=4.3958e-05; held-field average (no kinetic part): W_b,held'/W_b'-1=-5.49e-02
exact fixed source: W'/W_0'-1=-2.58e-04, (W'-W_0')/|W_b'|=2.27e-03, W_b'/W_b'(7.15)-1=2.11e-03 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-2.1e-05, W_2'/W_2'(7.15)-1=-1.0e-05
band fixed source: W'/W_0'-1=-2.85e-04, (W'-W_0')/|W_b'|=2.51e-03, W_b'/W_b'(7.15)-1=2.34e-03 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-2.0e-05, W_2'/W_2'(7.15)-1=-9.9e-06
s=12: W_0'/g_s=-3.451095e-04, W_b'/g_s=2.8234e-05; held-field average (no kinetic part): W_b,held'/W_b'-1=-7.48e-02
exact fixed source: W'/W_0'-1=-2.02e-04, (W'-W_0')/|W_b'|=2.47e-03, W_b'/W_b'(7.15)-1=2.28e-03 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-1.6e-05, W_2'/W_2'(7.15)-1=-9.0e-06
band fixed source: W'/W_0'-1=-2.13e-04, (W'-W_0')/|W_b'|=2.60e-03, W_b'/W_b'(7.15)-1=2.42e-03 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-1.6e-05, W_2'/W_2'(7.15)-1=-8.9e-06
s=16: W_0'/g_s=-2.915279e-04, W_b'/g_s=1.6761e-05; held-field average (no kinetic part): W_b,held'/W_b'-1=-1.05e-01
exact fixed source: W'/W_0'-1=-1.89e-04, (W'-W_0')/|W_b'|=3.29e-03, W_b'/W_b'(7.15)-1=3.08e-03 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-1.3e-05, W_2'/W_2'(7.15)-1=-8.2e-06
band fixed source: W'/W_0'-1=-1.94e-04, (W'-W_0')/|W_b'|=3.37e-03, W_b'/W_b'(7.15)-1=3.16e-03 (= Gamma_b ratio), W_1'/W_1'(7.15)-1=-1.3e-05, W_2'/W_2'(7.15)-1=-8.4e-06