08-ilang-propagation / 05-anchor
05anchorverified
Determines when oscillator cells without pinning have a lowest eigenvector with held bodies and sources, what an anchor changes, and whether the mark is long-range and weighed with an anchor.
check_anchor.py34 KB · raw
"""Check for 08-ilang-propagation/05-anchor, derivation v3.
Field of oscillator cells without pinning, anchor (kappa_a/2) q_h^2, held bodies with grips
F(q) = F0 + F1 q + F2 q^2, held sources g q_xs. In a sector of the held bodies the uncompressed
contract is C_f = 1/2 p.p + 1/2 q.A q + b.q + c0, and for A > 0 its exact lowest eigenvalue is
E0 = c0 + 1/2 Tr sqrt(A) - 1/2 b.A^{-1}.b (computed here by dense or sparse linear algebra).
The script compares this with the closed forms of the derivation:
(i) existence conditions (5.4)-(5.6), (5.10), the pair criterion (5.26), and a pair whose bodies
meet the one-body condition while the pair does not;
(ii) mutual cost of two sources (5.14)-(5.16) against the separation, two distances from the anchor;
(iii) cost, weight, bond and the ratios rho_mark, rho_bond,i (5.18)-(5.22), two levels,
both branches, three distances from the anchor; bonds for common square, non-square,
8F0F2=F1^2 and linear grips and for a mixed (square, non-square) pair, with the existence of
the pair block at s = 1;
plus the asymptotic statements (5.13), (5.15), (5.23)-(5.25), the mirror placement of (5.27), the pair
control of (5.27) (N/D -> 0 while a_bond does not tend to 0) and the line formula (5.28) from the closed
forms at large L.
Outputs: output/existence.txt, output/mutual_cost.txt, output/weighing.txt, output/asymptotics.txt
"""
import numpy as np
from pathlib import Path
from scipy import linalg as sla, sparse as sp
from scipy.sparse import linalg as spla
from scipy.special import ive
from scipy.integrate import quad
from scipy.optimize import brentq
OUT = Path(__file__).parent / "output"
OUT.mkdir(parents=True, exist_ok=True)
KH, KA = 1.0, 0.7 # kappa_h, kappa_a
MK = {0: 1.0, 1: 1.5} # M_k of the two levels used (m = 1, 2 pi/(N lambda_0) = 0.5)
GRIPS = { # (F0, F1, F2)
"square": (1.0, 0.6, 0.09), # 4 F0 F2 = F1^2 (beta_hat = b_hat = 0.3)
"nonsquare": (1.0, 0.6, 0.03), # beta_hat / b_hat = 1/3
"eight": (1.0, 0.6, 0.045), # 8 F0 F2 = F1^2 (beta_hat / b_hat = 1/2)
"linear": (1.0, 0.6, 0.0),
"negF2": (1.0, 0.6, -0.05),
}
HS = 0.01 # step of the s-stencil
COMBOS_A = [((0, 1), (0, 1)), ((0, -1), (0, -1)), ((1, 1), (1, 1)), ((1, -1), (1, -1)), ((0, 1), (1, -1))]
COMBOS_B = [((0, 1), (1, -1)), ((1, -1), (0, 1))] # each body: both levels, both branches
BOND_SETS = [("square", "square", COMBOS_A), ("nonsquare", "nonsquare", COMBOS_B), ("eight", "eight", COMBOS_B),
("linear", "linear", COMBOS_B), ("square", "nonsquare", COMBOS_B)]
def crit526(a1, a2, g1, g2, g12):
"""(5.26): A > 0 for two bodies with the anchor (determinant and trace of Gamma_2^{-1} + diag(alpha))."""
return bool((1 + a1 * g1) * (1 + a2 * g2) > a1 * a2 * g12 ** 2 and g1 + g2 + (a1 + a2) * (g1 * g2 - g12 ** 2) > 0)
class Lattice:
"""Ring (D=1) or cubic torus (D=3) of side L; Laplacian and pseudo-inverse Green function."""
def __init__(self, D, L):
self.D, self.L, self.n = D, L, L ** D
idx = np.arange(self.n).reshape((L,) * D)
r = np.concatenate([idx.ravel()] * D)
c = np.concatenate([np.roll(idx, -1, axis=ax).ravel() for ax in range(D)])
adj = sp.coo_matrix((np.ones(len(r)), (r, c)), shape=(self.n, self.n))
adj = (adj + adj.T).tocsr()
self.lap = (sp.diags(np.asarray(adj.sum(axis=1)).ravel()) - adj).tocsr()
k = 2 * np.pi * np.fft.fftfreq(L)
grids = np.meshgrid(*([k] * D), indexing="ij")
dh = sum(2 - 2 * np.cos(kk) for kk in grids)
inv = np.zeros_like(dh)
inv[dh > 1e-12] = 1 / dh[dh > 1e-12]
self.g = np.real(np.fft.ifftn(inv)) / KH # g(x) = (kappa_h Delta)^+ (x)
def site(self, x):
x = np.atleast_1d(x)
return int(np.ravel_multi_index(tuple(int(c) % self.L for c in x), (self.L,) * self.D))
def gx(self, x):
x = np.atleast_1d(x)
return self.g[tuple(int(c) % self.L for c in x)]
def Ga(self, x, y, h): # (5.8)
x, y, h = (np.atleast_1d(v) for v in (x, y, h))
return 1 / KA + self.gx(x - y) - self.gx(x - h) - self.gx(y - h) + self.gx(0 * x)
def Ka_sparse(self, h):
e = np.zeros(self.n)
e[self.site(h)] = KA
return (KH * self.lap + sp.diags(e)).tocsc()
def lam_min(A):
return sla.eigh(A, eigvals_only=True, subset_by_index=[0, 0])[0]
# ---------------------------------------------------------------- infinite cubic lattice
def ginf3(x):
"""g_inf(x) of the infinite cubic lattice (kappa_h = 1): int_0^inf e^{-6t} prod_j I_{x_j}(2t) dt."""
a, b, c = (abs(int(v)) for v in x)
f = lambda t: ive(a, 2 * t) * ive(b, 2 * t) * ive(c, 2 * t)
pts = [0, 1, 10, 100, 1e3, 1e4, 1e5, 1e6, 1e7]
s = sum(quad(f, pts[i], pts[i + 1], limit=400, epsabs=1e-16, epsrel=1e-12)[0] for i in range(len(pts) - 1))
# tail t > T from ive(n, 2t) = (4 pi t)^{-1/2} (1 - (4n^2-1)/(16 t) + O(t^-2))
T, cc = pts[-1], sum((4 * n * n - 1) / 16 for n in (a, b, c))
return (s + (4 * np.pi) ** -1.5 * (2 / np.sqrt(T) - cc * (2 / 3) * T ** -1.5)) / KH
def gstencil3(terms):
"""sum_j c_j g_inf(x_j) (kappa_h = 1) for sum_j c_j = 0, from one Bessel integral of the combined integrand
(no cancellation between separately computed values); terms = [(c_j, x_j), ...]."""
cs = [c for (c, _) in terms]
xs = [tuple(abs(int(v)) for v in x) for (_, x) in terms]
f = lambda t: sum(c * ive(a, 2 * t) * ive(b, 2 * t) * ive(e, 2 * t) for c, (a, b, e) in zip(cs, xs))
pts = [0, 1, 10, 100, 1e3, 1e4, 1e5, 1e6, 1e7, 1e8]
s = sum(quad(f, pts[i], pts[i + 1], limit=800, epsabs=1e-24, epsrel=1e-11)[0] for i in range(len(pts) - 1))
T = pts[-1]
tail = -(4 * np.pi) ** -1.5 * (2 / 3) * T ** -1.5 * sum(c * sum((4 * n * n - 1) / 16 for n in x) for c, x in zip(cs, xs))
return (s + tail) / KH
# ================================================================ (i) existence
def part_i(lines):
P = lambda s: lines.append(s)
P("(i) Existence of a lowest eigenvector: lambda_min(A) > 0 against the criteria of the derivation")
P(f"kappa_h={KH}, kappa_a={KA}, levels M_k={MK}, grips (F0,F1,F2)={GRIPS}\n")
for D, Ls in ((1, (32, 64)), (3, (8, 12))):
for L in Ls:
lat = Lattice(D, L)
lapd = KH * lat.lap.toarray()
n = lat.n
P(f"--- {'ring' if D == 1 else 'cubic torus'} L={L}")
# no anchor, one body: exists iff mu F2 > 0 (5.5)
x0 = lat.site([3] * D)
P(" no anchor, one body at a place: lambda_min(A0) and prediction [mu F2 > 0]")
for gname in ("square", "negF2", "linear"):
F0, F1, F2 = GRIPS[gname]
for k in (0, 1):
for sg in (+1, -1):
mu = sg * MK[k]
A = lapd.copy(); A[x0, x0] += 2 * mu * F2
lm = lam_min(A)
P(f" grip={gname:9s} k={k} sigma={sg:+d}: lambda_min={lm:+.3e} "
f"exists={lm > 1e-10} predicted={mu * F2 > 0} b.1={mu * F1:+.2f}")
# no anchor, two bodies with mixed signs: threshold alpha2* = -alpha1/(1+alpha1 r12) (5.6)
x1, x2 = lat.site([0] * D), lat.site([4] + [0] * (D - 1))
r12 = 2 * (lat.gx([0] * D) - lat.gx([4] + [0] * (D - 1)))
a1 = 2 * MK[1] * GRIPS["square"][2]
def lm2(a2):
A = lapd.copy(); A[x1, x1] += a1; A[x2, x2] += a2
return lam_min(A)
a2num = brentq(lm2, -a1 + 1e-9, -1e-9, xtol=1e-14)
a2pred = -a1 / (1 + a1 * r12)
P(f" no anchor, two bodies (separation 4): alpha1={a1:.4f}, r12={r12:.6f}: threshold alpha2 "
f"numerical={a2num:.10f} predicted={a2pred:.10f} rel.dev={abs(a2num / a2pred - 1):.1e}")
# anchor: one body, threshold mu F2 = -1/(2 Gamma(x)) (5.10)
h = [0] * D
Ka = lapd.copy(); Ka[lat.site(h), lat.site(h)] += KA
P(" anchor at h, one body at distance d: threshold alpha* (lambda_min=0) vs -1/Gamma(x), "
"and which (k, sigma) of grip negF2 / square have a lowest eigenvector")
for d in ((2, 5, 10) if D == 1 else (1, 2, 3)):
xx = [d] + [0] * (D - 1); i = lat.site(xx)
Gam = lat.Ga(xx, xx, h)
def lm1(al):
A = Ka.copy(); A[i, i] += al
return lam_min(A)
anum = brentq(lm1, -2 / Gam, -0.2 / Gam, xtol=1e-14)
ex = []
for gname in ("negF2", "square"):
for k in (0, 1):
for sg in (+1, -1):
al = 2 * sg * MK[k] * GRIPS[gname][2]
ok_num = lm1(al) > 0
ok_pred = 1 + al * Gam > 0
ex.append(f"{gname}:k{k}{'+' if sg > 0 else '-'}={'Y' if ok_num else 'N'}"
f"{'' if ok_num == ok_pred else '(MISMATCH)'}")
P(f" d={d}: Gamma={Gam:.6f} alpha* numerical={anum:.10f} predicted={-1 / Gam:.10f} "
f"| {' '.join(ex)}")
# anchor, two bodies: A > 0 iff Gamma_2^{-1} + diag(alpha) > 0 (5.10), random samples
rng = np.random.default_rng(1)
y1, y2 = [1] + [0] * (D - 1), [3] + [1] * (D - 1)
i1, i2 = lat.site(y1), lat.site(y2)
G2 = np.array([[lat.Ga(y1, y1, h), lat.Ga(y1, y2, h)], [lat.Ga(y2, y1, h), lat.Ga(y2, y2, h)]])
g11, g22, g12 = G2[0, 0], G2[1, 1], G2[0, 1]
agree, agree26, tot = 0, 0, 0
for _ in range(40):
al = rng.uniform(-2.0, 2.0, size=2) / np.diag(G2)
A = Ka.copy(); A[i1, i1] += al[0]; A[i2, i2] += al[1]
num = lam_min(A) > 0
pred = np.linalg.eigvalsh(np.linalg.inv(G2) + np.diag(al))[0] > 0
agree += int(num == pred); agree26 += int(num == crit526(al[0], al[1], g11, g22, g12)); tot += 1
P(f" anchor, two bodies, random (alpha1, alpha2): criterion (5.10) agrees with lambda_min in {agree}/{tot}, "
f"explicit form (5.26) in {agree26}/{tot}")
# a pair whose bodies each meet the one-body condition (5.10) but which violates (5.26)
lmax = np.linalg.eigvalsh(G2)[-1]
a = 0.5 * (1 / lmax + 1 / max(g11, g22))
A = Ka.copy(); A[i1, i1] -= a; A[i2, i2] -= a
lm_single = []
for ii in (i1, i2):
A1 = Ka.copy(); A1[ii, ii] -= a
lm_single.append(lam_min(A1))
P(f" anchor, two bodies at {y1}, {y2} (Gamma = {g11:.6f}, {g22:.6f}, G12 = {g12:.6f}) with alpha1 = alpha2 = -a, "
f"a = {a:.6f} between 1/lambda_max(Gamma_2) = {1 / lmax:.6f} and 1/max Gamma = {1 / max(g11, g22):.6f}:")
P(f" one body alone: 1 - a Gamma(x_i) = {1 - a * g11:+.4f}, {1 - a * g22:+.4f}; lambda_min = "
f"{lm_single[0]:+.3e}, {lm_single[1]:+.3e} (exists)")
P(f" pair: lambda_min(A) = {lam_min(A):+.3e}; (5.26) predicts existence = {crit526(-a, -a, g11, g22, g12)}; "
f"sufficient condition a (max Gamma + G12) < 1: {a * (max(g11, g22) + g12) < 1}\n")
# ================================================================ (ii) mutual cost
def part_ii(lines):
P = lambda s: lines.append(s)
P("(ii) Mutual cost of two held sources E12 = -g1 g2 (A^{-1})_{x1 x2} from the exact E0 "
"(sparse LU of A) against (5.14) E12 = -g1 g2 G_a(x1,x2); g1 = g2 = 1")
P(f"kappa_h={KH}, kappa_a={KA}\n")
# ring
for L in (64, 256):
lat = Lattice(1, L); h = [0]
lu = spla.splu(lat.Ka_sparse(h))
P(f"--- ring L={L}: pair on one side of the anchor, x1 = h + d1, x2 = x1 + s")
for d1 in (4, 16):
e = np.zeros(lat.n); e[lat.site([d1])] = 1.0
col = lu.solve(e)
row = []
for s in (1, 2, 4, 8, 16):
ex = -col[lat.site([d1 + s])]
fo = -lat.Ga([d1], [d1 + s], h)
row.append(f"s={s}: {ex:.10f} (formula {fo:.10f})")
P(f" d1={d1}: " + "; ".join(row))
P(f" L->inf value (5.16) on the line: -(1/kappa_a + d1/kappa_h) = {-(1 / KA + d1 / KH):.10f}")
e = np.zeros(lat.n); e[lat.site([5])] = 1.0
col = lu.solve(e)
P(f" opposite sides x1=h+5, x2=h-7: exact {-col[lat.site([-7])]:.10f}, formula "
f"{-lat.Ga([5], [-7], h):.10f}, -1/kappa_a = {-1 / KA:.10f}")
# cubic torus
g0inf = ginf3([0, 0, 0])
gcache = {}
def gi(x):
key = tuple(sorted(abs(int(v)) for v in x))
if key not in gcache:
gcache[key] = ginf3(key)
return gcache[key]
for L in (16, 32):
rs = (2, 4, 6, 8) if L == 16 else (2, 4, 6, 8, 12, 16)
lat = Lattice(3, L); h = np.array([0, 0, 0])
lu = spla.splu(lat.Ka_sparse(h))
P(f"--- cubic torus L={L}: pair along x with midpoint h + D e_y, x1,2 = h + D e_y -/+ (r/2) e_x")
for Dm in (0, 6):
res = []
for r in rs:
y1 = h + np.array([-r // 2, Dm, 0]); y2 = h + np.array([r // 2, Dm, 0])
e = np.zeros(lat.n); e[lat.site(y1)] = 1.0
ex = -lu.solve(e)[lat.site(y2)]
fo = -lat.Ga(y1, y2, h)
linf = -(1 / KA + gi(y1 - y2) - gi(y1 - h) - gi(y2 - h) + g0inf)
cont = -(1 / KA + g0inf) - (1 / (4 * np.pi * KH)) * (1 / r - 2 / np.hypot(Dm, r / 2))
res.append((r, ex, fo, linf, cont))
P(f" D={Dm}: r | exact E12 | formula (5.14), this L | L=inf lattice, (5.14) with g_inf | continuum image form (5.15)")
for (r, ex, fo, linf, cont) in res:
P(f" {r:2d} | {ex:.10f} | {fo:.10f} | {linf:.10f} | {cont:.10f}")
P(" effective coefficient c(r) = [E12(r)-E12(r')] / [-(1/(4 pi kappa_h))(1/r - 1/r')], consecutive r:")
for j in range(1, len(res)):
(r0, e0, _, l0, _), (r1, e1, _, l1, _) = res[j - 1], res[j]
den = -(1 / (4 * np.pi * KH)) * (1 / r1 - 1 / r0)
pc = (1 / r1 - 2 / np.hypot(Dm, r1 / 2) - 1 / r0 + 2 / np.hypot(Dm, r0 / 2)) / (1 / r1 - 1 / r0)
P(f" r={r0}->{r1}: exact {(e1 - e0) / den:+.4f}, L=inf lattice {(l1 - l0) / den:+.4f}, "
f"continuum image form {pc:+.4f}")
# one source at the anchor
vals = []
e = np.zeros(lat.n); e[lat.site(h)] = 1.0
col = lu.solve(e)
for y in ([3, 0, 0], [5, 2, 1], [L // 2, L // 2, L // 2]):
vals.append(-col[lat.site(y)])
P(f" source 1 at the anchor, source 2 at three places: E12 = {', '.join(f'{v:.12f}' for v in vals)}; "
f"-1/kappa_a = {-1 / KA:.12f}")
P("")
# ================================================================ (iii) weighing
def stencil(f):
"""f: dict s -> value with f(0)=0. Returns (first derivative, half second derivative) at s=0."""
h = HS
d1 = (8 * (f[h] - f[-h]) - (f[2 * h] - f[-2 * h])) / (12 * h)
d2 = (16 * (f[h] + f[-h]) - (f[2 * h] + f[-2 * h])) / (24 * h * h)
return d1, d2
def part_iii(lines):
P = lambda s: lines.append(s)
P("(iii) Cost, weight, bond and weighing ratios at order s^2 from the exact E0 (dense; s-stencil "
f"step {HS}) against (5.18)-(5.22); g_s = +-1 (the g_s-linear part is exact)")
P(f"kappa_h={KH}, kappa_a={KA}, levels M_k={MK}\n")
S = (-2 * HS, -HS, HS, 2 * HS)
combos1 = [(k, sg) for k in (0, 1) for sg in (+1, -1)]
for D, Ls, ds, Rp in ((1, (32, 64), (2, 5, 10), 3), (3, (8, 12), (1, 2, 3), 1)):
for L in Ls:
lat = Lattice(D, L); n = lat.n
h = np.zeros(D, dtype=int)
Ka = KH * lat.lap.toarray(); Ka[lat.site(h), lat.site(h)] += KA
lam, U = np.linalg.eigh(Ka); om = np.sqrt(lam)
Mab = 1 / (om[:, None] * om[None, :] * (om[:, None] + om[None, :]))
w = lambda i: 0.5 * np.sum(U[i] ** 2 / om)
Sig = lambda i, j: (U[i] * U[j]) @ Mab @ (U[i] * U[j])
Tc, Ec = {}, {}
def E(bodies, s, xs):
"""Exact E0 = c0 + 1/2 Tr sqrt(A) - 1/2 b.A^-1.b with source +1, -1, 0 at site xs."""
key = (tuple(bodies), s, xs)
if key not in Ec:
A = Ka.copy(); b = np.zeros(n); c0 = 0.0
for (i, mu, g) in bodies:
F0, F1, F2 = GRIPS[g]
A[i, i] += 2 * s * mu * F2; b[i] += s * mu * F1; c0 += s * mu * F0
tkey = tuple(sorted((i, 2 * s * mu * GRIPS[g][2]) for (i, mu, g) in bodies if GRIPS[g][2] != 0))
if tkey not in Tc:
lam_ = sla.eigvalsh(A)
assert lam_[0] > 0
Tc[tkey] = 0.5 * np.sqrt(lam_).sum()
cf = sla.cho_factor(A)
es = np.zeros(n); es[xs] = 1.0
Ec[key] = [c0 + Tc[tkey] - 0.5 * bb @ sla.cho_solve(cf, bb) for bb in (b + es, b - es, b)]
return Ec[key]
def single(i, mu, g, xs):
"""C1, Cmark, W1, Wmark of one body at site i (source at xs, g_s = 1)."""
C = {s: E([(i, mu, g)], s, xs)[2] - E([], 0.0, xs)[2] for s in S}
W = {s: 0.5 * ((E([(i, mu, g)], s, xs)[0] - E([], 0.0, xs)[0])
- (E([(i, mu, g)], s, xs)[1] - E([], 0.0, xs)[1])) for s in S}
return stencil(C) + stencil(W)
P(f"=== {'ring' if D == 1 else 'cubic torus'} L={L}, anchor h=0; body placements x=h+d e, x'=h+(d+1) e, "
f"source at h+(d+1+{Rp}) e")
maxdev, maxdev20, maxdev19, maxdev21 = 0.0, 0.0, 0.0, 0.0
for d in ds:
ex = np.array([1] + [0] * (D - 1))
X, Xp, Xs = h + d * ex, h + (d + 1) * ex, h + (d + 1 + Rp) * ex
i, ip, isrc = lat.site(X), lat.site(Xp), lat.site(Xs)
Gm = {i: lat.Ga(X, X, h), ip: lat.Ga(Xp, Xp, h)}
ph = {i: -lat.Ga(X, Xs, h), ip: -lat.Ga(Xp, Xs, h)}
Gb = 0.5 * (Gm[i] + Gm[ip]); pb = 0.5 * (ph[i] + ph[ip])
amark = pb * (Gm[i] - Gm[ip]) / (Gb * (ph[i] - ph[ip]))
P(f" d={d}: Gamma(x)={Gm[i]:.6f}, w(x)={w(i):.6f}, Sigma(x)={Sig(i, i):.6f}, phi_a(x)={ph[i]:.6f}; "
f"a_mark={amark:+.6f}")
for g in ("square", "nonsquare", "eight", "linear"):
F0, F1, F2 = GRIPS[g]
for (k, sg) in combos1:
mu = sg * MK[k]
r = {j: single(j, mu, g, isrc) for j in (i, ip)}
# closed forms (5.18)
for j in (i, ip):
pred = (mu * (F0 + F2 * w(j)), -0.5 * mu ** 2 * (F1 ** 2 * Gm[j] + F2 ** 2 * Sig(j, j)),
mu * F1 * ph[j], -2 * mu ** 2 * F1 * F2 * Gm[j] * ph[j])
for a, bb in zip(r[j], pred):
maxdev = max(maxdev, abs(a - bb) / max(abs(bb), 1e-12) if abs(bb) > 1e-12 else abs(a))
exist = "Y" if all(1 + 2 * mu * F2 * Gm[j] > 0 for j in (i, ip)) else "N"
if F2 == 0:
rho_num = (r[i][3] - r[ip][3]) / (r[i][2] - r[ip][2]) * \
(0.5 * (r[i][0] + r[ip][0])) / (0.5 * (r[i][1] + r[ip][1]))
maxdev20 = max(maxdev20, abs(rho_num))
P(f" grip={g:9s} k={k} sigma={sg:+d}: rho_mark numerical={rho_num:+.3e} predicted=0 "
f"[block at s=1 by (5.10): {exist}]")
continue
rho_num = (r[i][3] - r[ip][3]) / (r[i][2] - r[ip][2]) * \
(0.5 * (r[i][0] + r[ip][0])) / (0.5 * (r[i][1] + r[ip][1]))
wb = 0.5 * (w(i) + w(ip)); Sb = 0.5 * (Sig(i, i) + Sig(ip, ip))
rho_pred = 4 * F2 * (F0 + F2 * wb) / (F1 ** 2 * Gb + F2 ** 2 * Sb) * \
(Gm[i] * ph[i] - Gm[ip] * ph[ip]) / (ph[i] - ph[ip])
maxdev20 = max(maxdev20, abs(rho_num / rho_pred - 1))
lead = 4 * F0 * F2 / F1 ** 2 * (1 + amark)
P(f" grip={g:9s} k={k} sigma={sg:+d}: rho_mark numerical={rho_num:.8f} "
f"predicted (5.20)={rho_pred:.8f} leading (4F0F2/F1^2)(1+a_mark)={lead:.6f} "
f"[block at s=1 by (5.10): {exist}]")
# bond: pair x1 = X, x2 = X + 2 e2 (ring: X + 2 e), translated by e
e2 = np.array([0, 1, 0]) if D == 3 else ex
Y1, Y2 = X, X + 2 * e2
Y1p, Y2p = Y1 + ex, Y2 + ex
Ym, Ymp = X + e2, X + e2 + ex
Ys = Y2p + 2 * ex if D == 1 else Ymp + Rp * ex
js = [lat.site(v) for v in (Y1, Y2, Y1p, Y2p, Ym, Ymp)]
jsrc = lat.site(Ys)
phi = lambda v: -lat.Ga(v, Ys, h)
G12, G12p = lat.Ga(Y1, Y2, h), lat.Ga(Y1p, Y2p, h)
Gam1, Gam2, Gam1p, Gam2p = (lat.Ga(v, v, h) for v in (Y1, Y2, Y1p, Y2p))
dphi1, dphi2 = phi(Y1) - phi(Y1p), phi(Y2) - phi(Y2p)
dphim = phi(Ym) - phi(Ymp)
mr = (dphi1 + dphi2) / (2 * dphim)
Gbar, dG = 0.5 * (G12 + G12p), G12 - G12p
pb1, pb2 = 0.5 * (phi(Y1) + phi(Y1p)), 0.5 * (phi(Y2) + phi(Y2p))
abond = (pb1 + pb2) * dG / (2 * Gbar * dphim)
S12b = 0.5 * (Sig(js[0], js[1]) + Sig(js[2], js[3]))
wmb = 0.5 * (w(js[4]) + w(js[5]))
P(f" bond, pair separation 2, source at {[int(c) for c in Ys]}: m_r={mr:.6f}, a_bond={abond:+.6f}")
for (g1, g2, combos) in BOND_SETS:
(F01, F11, F21), (F02, F12, F22) = GRIPS[g1], GRIPS[g2]
be1, be2 = 2 * F21 / F11, 2 * F22 / F12
Psi = lambda y1, y2: be1 * phi(y1) + be2 * phi(y2)
dPsi, Psib = Psi(Y1, Y2) - Psi(Y1p, Y2p), 0.5 * (Psi(Y1, Y2) + Psi(Y1p, Y2p))
P(f" grips (body 1, body 2) = ({g1}, {g2}):")
for (c1, c2) in combos:
mu1, mu2 = c1[1] * MK[c1[0]], c2[1] * MK[c2[0]]
al1, al2 = 2 * mu1 * F21, 2 * mu2 * F22
# existence of the pair block at s = 1, both placements: lambda_min(A) against (5.26)
ex_num, ex_pred = [], []
for (a_, b_, ga, gb, gab) in ((js[0], js[1], Gam1, Gam2, G12), (js[2], js[3], Gam1p, Gam2p, G12p)):
A = Ka.copy(); A[a_, a_] += al1; A[b_, b_] += al2
ex_num.append("Y" if lam_min(A) > 0 else "N")
ex_pred.append("Y" if crit526(al1, al2, ga, gb, gab) else "N")
def bond(a_, b_):
B = {}; Wb = {}
for s in S:
e12 = E([(a_, mu1, g1), (b_, mu2, g2)], s, jsrc)
e1 = E([(a_, mu1, g1)], s, jsrc); e2_ = E([(b_, mu2, g2)], s, jsrc)
e0 = E([], 0.0, jsrc)
bb = [e12[t] - e1[t] - e2_[t] + e0[t] for t in range(3)]
B[s] = bb[2]; Wb[s] = 0.5 * (bb[0] - bb[1])
return stencil(B)[1], stencil(Wb)[1]
B2, W2 = bond(js[0], js[1]); B2p, W2p = bond(js[2], js[3])
# (5.19) for different grips
B19 = lambda y1, y2, j1, j2: -mu1 * mu2 * (F11 * F12 * lat.Ga(y1, y2, h) + F21 * F22 * Sig(j1, j2))
W19 = lambda y1, y2: -2 * mu1 * mu2 * lat.Ga(y1, y2, h) * (F11 * F22 * phi(y2) + F21 * F12 * phi(y1))
for num, pred in ((B2, B19(Y1, Y2, js[0], js[1])), (B2p, B19(Y1p, Y2p, js[2], js[3])),
(W2, W19(Y1, Y2)), (W2p, W19(Y1p, Y2p))):
maxdev19 = max(maxdev19, abs(num - pred) / abs(pred) if abs(pred) > 1e-12 else abs(num))
out = []
for ii, mui, gi in ((1, mu1, g1), (2, mu2, g2)):
F0i, F1i, F2i = GRIPS[gi]
bhi, bei = F1i / (2 * F0i), 2 * F2i / F1i
rm = single(js[4], mui, gi, jsrc); rmp = single(js[5], mui, gi, jsrc)
rho_num = ((W2 - W2p) / (0.5 * (B2 + B2p))) / ((rm[2] - rmp[2]) / (0.5 * (rm[0] + rmp[0])))
rho_pred = (Gbar * dPsi + Psib * dG) / (Gbar + 0.25 * be1 * be2 * S12b) * \
(1 + bei * bhi * wmb) / (2 * bhi * dphim) # (5.21)
lead = (be1 * dphi1 + be2 * dphi2) / (2 * bhi * dphim) + \
(be1 * pb1 + be2 * pb2) * dG / (2 * bhi * Gbar * dphim) # (5.22)
maxdev21 = max(maxdev21, abs(rho_num / rho_pred - 1) if abs(rho_pred) > 1e-12 else abs(rho_num))
out.append(f"rho_bond,{ii} num={rho_num:+.8f} (5.21)={rho_pred:+.8f} lead(5.22)={lead:+.6f}")
P(f" (k,sigma)=({c1[0]},{c1[1]:+d}),({c2[0]},{c2[1]:+d}): pair block at s=1 (both placements) "
f"lambda_min>0: {'/'.join(ex_num)}, (5.26): {'/'.join(ex_pred)}; B={B2:+.6e}; " + "; ".join(out))
P(f" max relative deviation over this lattice: C1, Cmark, W1, Wmark from (5.18): {maxdev:.2e}; "
f"rho_mark from (5.20): {maxdev20:.2e}; B, W_bond from (5.19): {maxdev19:.2e}; "
f"rho_bond,i from (5.21): {maxdev21:.2e} (absolute deviation where the prediction is 0)\n")
# ================================================================ asymptotics from closed forms
def part_asym(lines):
P = lambda s: lines.append(s)
P("Asymptotic statements from the closed forms (no eigenvalue computation)")
g0 = ginf3([0, 0, 0])
P(f"gamma_3 = kappa_h g_inf(0) = {g0 * KH:.10f}")
for x in ([4, 0, 0], [8, 0, 0], [16, 0, 0], [6, 6, 6]):
r = np.linalg.norm(x)
P(f" g_inf({x}) * 4 pi kappa_h |x| = {ginf3(x) * 4 * np.pi * KH * r:.6f}")
P("\n(5.15) orders of the limits on the infinite cubic lattice (L -> inf first), anchor at 0, g1 = g2 = 1;"
" c(r) as in mutual_cost.txt")
E12 = lambda y1, y2: -(1 / KA + g0 + ginf3(np.subtract(y1, y2)) - ginf3(y1) - ginf3(y2))
for lab, Dm, rr in (("(a) midpoint (0,40,0), r << D", 40, (4, 8, 16)), ("(c) midpoint at the anchor", 0, (8, 16, 32, 64))):
vals = [E12([-r // 2, Dm, 0], [r // 2, Dm, 0]) for r in rr]
cs = [(vals[j] - vals[j - 1]) / (-(1 / (4 * np.pi * KH)) * (1 / rr[j] - 1 / rr[j - 1])) for j in range(1, len(rr))]
P(f" {lab}: " + ", ".join(f"c({rr[j - 1]}->{rr[j]})={cs[j - 1]:+.4f}" for j in range(1, len(rr))))
lim = -(1 / KA + g0 - ginf3([3, 0, 0]))
P(" (b) x1=(3,0,0) fixed, x2=(3+r,0,0): 4 pi kappa_h r^2 [E12 - E12(r=inf)] -> -(x1-h).r_hat = -3: " +
", ".join(f"r={r}: {4 * np.pi * KH * r * r * (E12([3, 0, 0], [3 + r, 0, 0]) - lim):+.4f}" for r in (8, 16, 32, 64)))
cphi = 1 / KA + g0
Ginf = 1 / KA + 2 * g0
P(f"\n(5.23) a_mark on the infinite cubic lattice (L -> inf first), collinear: anchor 0, body d, d+1, "
f"source d+1+R'; prediction a_mark ~ (2 c_phi/Gamma_inf) (R/d)^2 with c_phi=1/kappa_a+gamma_3/kappa_h="
f"{cphi:.6f}, Gamma_inf={Ginf:.6f}, R=R'+1/2, d->d+1/2")
for Rp in (3, 6):
for d in (8, 16, 32, 64):
Gm = lambda x: 1 / KA + 2 * (g0 - ginf3([x, 0, 0]))
ph = lambda x: -(1 / KA + g0 + ginf3([x - (d + 1 + Rp), 0, 0]) - ginf3([x, 0, 0]) - ginf3([d + 1 + Rp, 0, 0]))
a = 0.5 * (ph(d) + ph(d + 1)) * (Gm(d) - Gm(d + 1)) / (0.5 * (Gm(d) + Gm(d + 1)) * (ph(d) - ph(d + 1)))
pred = 2 * cphi / Ginf * ((Rp + 0.5) / (d + 0.5)) ** 2
P(f" R'={Rp} d={d:3d}: a_mark={a:+.6e} prediction={pred:+.6e} ratio={a / pred:.4f}")
P("\n(5.24) a_bond on the infinite cubic lattice: pair (d,0,0),(d,2,0) -> translated by e_x, source (d+1+R',1,0)")
for Rp in (3, 6):
for d in (8, 16, 32, 64):
xs = np.array([d + 1 + Rp, 1, 0])
phi = lambda v: -(1 / KA + g0 + ginf3(np.array(v) - xs) - ginf3(v) - ginf3(xs))
G12 = lambda v1, v2: 1 / KA + g0 + ginf3(np.array(v1) - np.array(v2)) - ginf3(v1) - ginf3(v2)
Y1, Y2, Y1p, Y2p = [d, 0, 0], [d, 2, 0], [d + 1, 0, 0], [d + 1, 2, 0]
Gb = 0.5 * (G12(Y1, Y2) + G12(Y1p, Y2p))
dphim = phi([d, 1, 0]) - phi([d + 1, 1, 0])
a = 0.5 * (phi(Y1) + phi(Y1p) + phi(Y2) + phi(Y2p)) * (G12(Y1, Y2) - G12(Y1p, Y2p)) / (2 * Gb * dphim)
R = Rp + 0.5; dd = np.hypot(d + 0.5, 1.0)
pred = 2 * cphi / (cphi + ginf3([0, 2, 0])) * (R / dd) ** 2 * ((d + 0.5) / dd)
P(f" R'={Rp} d={d:3d}: a_bond={a:+.6e} prediction={pred:+.6e} ratio={a / pred:.4f}")
P("\n(5.27) mirror placement on the infinite cubic lattice (control): anchor 0, body at (d,0,0) and (d+2,0,0), "
"source at (d+1,R,0); Delta g_s = 0 by reflection, so a_mark = -2 phi_bar/(g_s Gamma_bar) -> "
"2 [c_phi + g_inf((1,R,0))]/Gamma_inf as d -> inf")
for R in (3, 6):
lim = 2 * (cphi + ginf3([1, R, 0])) / Ginf
for d in (8, 16, 32, 64):
xs = np.array([d + 1, R, 0])
Gm = lambda v: 1 / KA + 2 * (g0 - ginf3(v))
ph = lambda v: -(1 / KA + g0 + ginf3(np.array(v) - xs) - ginf3(v) - ginf3(xs))
x, xp = [d, 0, 0], [d + 2, 0, 0]
dgs = ginf3(np.array(x) - xs) - ginf3(np.array(xp) - xs)
dph = ph(x) - ph(xp)
a = 0.5 * (ph(x) + ph(xp)) * (Gm(x) - Gm(xp)) / (0.5 * (Gm(x) + Gm(xp)) * dph)
P(f" R={R} d={d:3d}: Delta g_s={dgs:+.1e}, Delta phi_a={dph:+.3e}, a_mark={a:+.6f}, limit={lim:.6f}")
glim = ginf3([1, 0, 0])
P("\n(5.27) pair control on the infinite cubic lattice: anchor 0, source e_x, pair x_1,2=(d,+-t,0) translated by e_x, "
"d=n^3, t=n^4; N=Dg_h(x1)+Dg_h(x2), D=Dg_s(xbar), H=Dg_h(xbar) at the midpoint xbar=(d,0,0) (differences by "
"combined Bessel integrals); a_bond = (phibar1+phibar2) DeltaG12 / (2 Gbar12 Delta phibar) with DeltaG12=-N, "
f"Delta phibar=-g_s(D-H), g_s=1; predicted: N/D->0, H/D->1, N/(D-H)->1, m_r->0, a_bond -> -(c_phi-g_inf(e_x))/c_phi "
f"= {-(cphi - glim) / cphi:.6f}")
for nn in (2, 3, 4, 5, 6):
d, t = nn ** 3, nn ** 4
N = 2 * gstencil3([(1, [d, t, 0]), (-1, [d + 1, t, 0])])
Dd = gstencil3([(1, [d - 1, 0, 0]), (-1, [d, 0, 0])])
H = gstencil3([(1, [d, 0, 0]), (-1, [d + 1, 0, 0])])
DmH = gstencil3([(1, [d - 1, 0, 0]), (-2, [d, 0, 0]), (1, [d + 1, 0, 0])])
dphj = gstencil3([(1, [d - 1, t, 0]), (-2, [d, t, 0]), (1, [d + 1, t, 0])]) # = -Delta phi_j / g_s
phi = lambda v: -(cphi + ginf3(np.subtract(v, [1, 0, 0])) - ginf3(v) - glim)
G12 = lambda v1, v2: cphi + ginf3(np.subtract(v1, v2)) - ginf3(v1) - ginf3(v2)
X1, X2, X1p, X2p = [d, t, 0], [d, -t, 0], [d + 1, t, 0], [d + 1, -t, 0]
pbsum = 0.5 * (phi(X1) + phi(X1p) + phi(X2) + phi(X2p))
Gb = 0.5 * (G12(X1, X2) + G12(X1p, X2p))
ab = pbsum * (-N) / (2 * Gb * (-DmH))
mr = (2 * dphj) / (2 * DmH)
P(f" n={nn} d={d:4d} t={t:5d}: N/D={N / Dd:.4e} H/D={H / Dd:.6f} N/(D-H)={N / DmH:.6f} "
f"(D-H)/(D-H directly)={(Dd - H) / DmH:.6f} m_r={mr:+.4e} a_bond={ab:+.6f}")
P("\n(5.25) ring: a_mark for a body between anchor and source (d=10, d+1, source at d+1+3) -> 1 as L -> inf")
for L in (32, 64, 256, 1024, 4096):
Gm = lambda x: 1 / KA + x * (L - x) / (KH * L)
xs = 14
ph = lambda x: -(1 / KA + (x * (L - x) + xs * (L - xs) - abs(xs - x) * (L - abs(xs - x))) / (2 * KH * L))
a = 0.5 * (ph(10) + ph(11)) * (Gm(10) - Gm(11)) / (0.5 * (Gm(10) + Gm(11)) * (ph(10) - ph(11)))
P(f" L={L:5d}: a_mark={a:.6f}")
L, xs, Kr = 4096, 20, KH / KA
P(f"\n(5.28) line, every comparison: a_mark from the ring closed forms (5.8), (5.11) at L={L} against (5.28); "
f"anchor 0, source at {xs}")
rr = lambda a_, b_: (abs(a_ - b_) % L) * (L - abs(a_ - b_) % L) / (KH * L)
Gr = lambda x: 1 / KA + rr(x, 0)
phr = lambda x: -(1 / KA + 0.5 * (rr(x, 0) + rr(xs, 0) - rr(x, xs)))
def line528(x, xp):
dm = lambda v: (abs(v), min(abs(v), xs) if v > 0 else 0)
(d1, m1), (d2, m2) = dm(x), dm(xp)
return (Kr + 0.5 * (m1 + m2)) * (d1 - d2) / ((Kr + 0.5 * (d1 + d2)) * (m1 - m2))
for (x, xp, lab) in ((12, 11, "between anchor and source"), (21, 19, "straddling the source"),
(25, 15, "straddling the source"), (5, -7, "across the anchor"), (8, -3, "across the anchor")):
a = 0.5 * (phr(x) + phr(xp)) * (Gr(x) - Gr(xp)) / (0.5 * (Gr(x) + Gr(xp)) * (phr(x) - phr(xp)))
P(f" x={x:3d}, x'={xp:3d} ({lab}): a_mark ring={a:+.6f} (5.28)={line528(x, xp):+.6f}")
P("\n(5.13) anchored line (L = inf): w(d) and Sigma(d) from the resolvent (5.9); predictions "
"w(10d)-w(d) -> ln10/(2 pi sqrt(kappa_h)), Sigma(d)/d -> 2 ln2/(pi kappa_h^{3/2})")
def Rdd(nu, d):
t = nu * nu / (2 * KH)
sq = np.sqrt(t * t + 2 * t)
g0n = 1 / (KH * 2 * sq) # 1/(kappa_h (1/z - z)), 1/z = 1 + t + sq
one_m = -np.expm1(-2 * d * np.log1p(t + sq)) # 1 - z^{2d}
return (g0n + KA * g0n * g0n * one_m) / (1 + KA * g0n)
def wS(d):
f1 = lambda u: np.exp(u) * Rdd(np.exp(u), d)
f2 = lambda u: np.exp(u) * Rdd(np.exp(u), d) ** 2
lo, hi = -40.0, 25.0
w_ = quad(f1, lo, hi, limit=800, epsabs=1e-13)[0] / np.pi
S_ = 2 * quad(f2, lo, hi, limit=800, epsabs=1e-13)[0] / np.pi
return w_, S_
prev = None
for d in (10, 100, 1000, 10000):
w_, S_ = wS(d)
inc = "" if prev is None else f" w(d)-w(d/10)={w_ - prev:.6f}"
P(f" d={d:6d}: w={w_:.6f} Sigma/d={S_ / d:.6f} Gamma={1 / KA + d / KH:.3f}{inc}")
prev = w_
P(f" predictions: ln10/(2 pi)={np.log(10) / (2 * np.pi):.6f}, 2 ln2/pi={2 * np.log(2) / np.pi:.6f}")
if __name__ == "__main__":
for name, fn in (("existence.txt", part_i), ("mutual_cost.txt", part_ii),
("weighing.txt", part_iii), ("asymptotics.txt", part_asym)):
lines = []
fn(lines)
(OUT / name).write_text("\n".join(lines) + "\n", encoding="utf-8")
print("wrote", OUT / name, flush=True)
output/asymptotics.txt6 KB · raw
Asymptotic statements from the closed forms (no eigenvalue computation)
gamma_3 = kappa_h g_inf(0) = 0.2527310099
g_inf([4, 0, 0]) * 4 pi kappa_h |x| = 1.019830
g_inf([8, 0, 0]) * 4 pi kappa_h |x| = 1.004108
g_inf([16, 0, 0]) * 4 pi kappa_h |x| = 1.000988
g_inf([6, 6, 6]) * 4 pi kappa_h |x| = 0.998451
(5.15) orders of the limits on the infinite cubic lattice (L -> inf first), anchor at 0, g1 = g2 = 1; c(r) as in mutual_cost.txt
(a) midpoint (0,40,0), r << D: c(4->8)=+1.0341, c(8->16)=+0.9956
(c) midpoint at the anchor: c(8->16)=-3.1350, c(16->32)=-3.0272, c(32->64)=-3.0065
(b) x1=(3,0,0) fixed, x2=(3+r,0,0): 4 pi kappa_h r^2 [E12 - E12(r=inf)] -> -(x1-h).r_hat = -3: r=8: -2.2024, r=16: -2.5327, r=32: -2.7447, r=64: -2.8662
(5.23) a_mark on the infinite cubic lattice (L -> inf first), collinear: anchor 0, body d, d+1, source d+1+R'; prediction a_mark ~ (2 c_phi/Gamma_inf) (R/d)^2 with c_phi=1/kappa_a+gamma_3/kappa_h=1.681302, Gamma_inf=1.934033, R=R'+1/2, d->d+1/2
R'=3 d= 8: a_mark=+2.354491e-01 prediction=+2.947882e-01 ratio=0.7987
R'=3 d= 16: a_mark=+6.851490e-02 prediction=+7.823121e-02 ratio=0.8758
R'=3 d= 32: a_mark=+1.813394e-02 prediction=+2.016421e-02 ratio=0.8993
R'=3 d= 64: a_mark=+4.636675e-03 prediction=+5.119511e-03 ratio=0.9057
R'=6 d= 8: a_mark=+6.421039e-01 prediction=+1.016718e+00 ratio=0.6315
R'=6 d= 16: a_mark=+2.306867e-01 prediction=+2.698179e-01 ratio=0.8550
R'=6 d= 32: a_mark=+6.571165e-02 prediction=+6.954595e-02 ratio=0.9449
R'=6 d= 64: a_mark=+1.715213e-02 prediction=+1.765709e-02 ratio=0.9714
(5.24) a_bond on the infinite cubic lattice: pair (d,0,0),(d,2,0) -> translated by e_x, source (d+1+R',1,0)
R'=3 d= 8: a_bond=+2.539632e-01 prediction=+3.239171e-01 ratio=0.7840
R'=3 d= 16: a_bond=+7.600590e-02 prediction=+8.727102e-02 ratio=0.8709
R'=3 d= 32: a_bond=+2.027305e-02 prediction=+2.258620e-02 ratio=0.8976
R'=3 d= 64: a_bond=+5.193783e-03 prediction=+5.740509e-03 ratio=0.9048
R'=6 d= 8: a_bond=+6.966915e-01 prediction=+1.117183e+00 ratio=0.6236
R'=6 d= 16: a_bond=+2.562273e-01 prediction=+3.009960e-01 ratio=0.8513
R'=6 d= 32: a_bond=+7.351920e-02 prediction=+7.789935e-02 ratio=0.9438
R'=6 d= 64: a_bond=+1.922696e-02 prediction=+1.979890e-02 ratio=0.9711
(5.27) mirror placement on the infinite cubic lattice (control): anchor 0, body at (d,0,0) and (d+2,0,0), source at (d+1,R,0); Delta g_s = 0 by reflection, so a_mark = -2 phi_bar/(g_s Gamma_bar) -> 2 [c_phi + g_inf((1,R,0))]/Gamma_inf as d -> inf
R=3 d= 8: Delta g_s=+0.0e+00, Delta phi_a=+2.010e-03, a_mark=+1.763447, limit=1.765042
R=3 d= 16: Delta g_s=+0.0e+00, Delta phi_a=+5.541e-04, a_mark=+1.763986, limit=1.765042
R=3 d= 32: Delta g_s=+0.0e+00, Delta phi_a=+1.464e-04, a_mark=+1.764467, limit=1.765042
R=3 d= 64: Delta g_s=+0.0e+00, Delta phi_a=+3.769e-05, a_mark=+1.764746, limit=1.765042
R=6 d= 8: Delta g_s=+0.0e+00, Delta phi_a=+2.010e-03, a_mark=+1.751637, limit=1.752262
R=6 d= 16: Delta g_s=+0.0e+00, Delta phi_a=+5.541e-04, a_mark=+1.751348, limit=1.752262
R=6 d= 32: Delta g_s=+0.0e+00, Delta phi_a=+1.464e-04, a_mark=+1.751685, limit=1.752262
R=6 d= 64: Delta g_s=+0.0e+00, Delta phi_a=+3.769e-05, a_mark=+1.751954, limit=1.752262
(5.27) pair control on the infinite cubic lattice: anchor 0, source e_x, pair x_1,2=(d,+-t,0) translated by e_x, d=n^3, t=n^4; N=Dg_h(x1)+Dg_h(x2), D=Dg_s(xbar), H=Dg_h(xbar) at the midpoint xbar=(d,0,0) (differences by combined Bessel integrals); a_bond = (phibar1+phibar2) DeltaG12 / (2 Gbar12 Delta phibar) with DeltaG12=-N, Delta phibar=-g_s(D-H), g_s=1; predicted: N/D->0, H/D->1, N/(D-H)->1, m_r->0, a_bond -> -(c_phi-g_inf(e_x))/c_phi = -0.948811
n=2 d= 8 t= 16: N/D=1.5818e-01 H/D=0.774972 N/(D-H)=0.702914 (D-H)/(D-H directly)=1.000000 m_r=-1.6854e-02 a_bond=-0.669486
n=3 d= 27 t= 81: N/D=6.1634e-02 H/D=0.928500 N/(D-H)=0.862012 (D-H)/(D-H directly)=1.000000 m_r=-1.1028e-02 a_bond=-0.818555
n=4 d= 64 t= 256: N/D=2.8264e-02 H/D=0.969225 N/(D-H)=0.918405 (D-H)/(D-H directly)=1.000000 m_r=-5.8710e-03 a_bond=-0.871625
n=5 d= 125 t= 625: N/D=1.5017e-02 H/D=0.984126 N/(D-H)=0.946054 (D-H)/(D-H directly)=1.000000 m_r=-3.3358e-03 a_bond=-0.897726
n=6 d= 216 t= 1296: N/D=8.8640e-03 H/D=0.990783 N/(D-H)=0.961726 (D-H)/(D-H directly)=1.000000 m_r=-2.0414e-03 a_bond=-0.912545
(5.25) ring: a_mark for a body between anchor and source (d=10, d+1, source at d+1+3) -> 1 as L -> inf
L= 32: a_mark=0.528868
L= 64: a_mark=0.811924
L= 256: a_mark=0.959032
L= 1024: a_mark=0.990075
L= 4096: a_mark=0.997538
(5.28) line, every comparison: a_mark from the ring closed forms (5.8), (5.11) at L=4096 against (5.28); anchor 0, source at 20
x= 12, x'= 11 (between anchor and source): a_mark ring=+0.997420 (5.28)=+1.000000
x= 21, x'= 19 (straddling the source): a_mark ring=+1.953142 (5.28)=+1.953333
x= 25, x'= 15 (straddling the source): a_mark ring=+1.766104 (5.28)=+1.766667
x= 5, x'= -7 (across the anchor): a_mark ring=-0.213945 (5.28)=-0.211538
x= 8, x'= -3 (across the anchor): a_mark ring=+0.491203 (5.28)=+0.489691
(5.13) anchored line (L = inf): w(d) and Sigma(d) from the resolvent (5.9); predictions w(10d)-w(d) -> ln10/(2 pi sqrt(kappa_h)), Sigma(d)/d -> 2 ln2/(pi kappa_h^{3/2})
d= 10: w=0.809467 Sigma/d=0.489480 Gamma=11.429
d= 100: w=1.157998 Sigma/d=0.446296 Gamma=101.429 w(d)-w(d/10)=0.348531
d= 1000: w=1.522451 Sigma/d=0.441776 Gamma=1001.429 w(d)-w(d/10)=0.364453
d= 10000: w=1.888714 Sigma/d=0.441322 Gamma=10001.429 w(d)-w(d/10)=0.366263
predictions: ln10/(2 pi)=0.366468, 2 ln2/pi=0.441271
output/existence.txt11 KB · raw
(i) Existence of a lowest eigenvector: lambda_min(A) > 0 against the criteria of the derivation
kappa_h=1.0, kappa_a=0.7, levels M_k={0: 1.0, 1: 1.5}, grips (F0,F1,F2)={'square': (1.0, 0.6, 0.09), 'nonsquare': (1.0, 0.6, 0.03), 'eight': (1.0, 0.6, 0.045), 'linear': (1.0, 0.6, 0.0), 'negF2': (1.0, 0.6, -0.05)}
--- ring L=32
no anchor, one body at a place: lambda_min(A0) and prediction [mu F2 > 0]
grip=square k=0 sigma=+1: lambda_min=+3.717e-03 exists=True predicted=True b.1=+0.60
grip=square k=0 sigma=-1: lambda_min=-9.615e-03 exists=False predicted=False b.1=-0.60
grip=square k=1 sigma=+1: lambda_min=+4.725e-03 exists=True predicted=True b.1=+0.90
grip=square k=1 sigma=-1: lambda_min=-1.904e-02 exists=False predicted=False b.1=-0.90
grip=negF2 k=0 sigma=+1: lambda_min=-4.161e-03 exists=False predicted=False b.1=+0.60
grip=negF2 k=0 sigma=-1: lambda_min=+2.445e-03 exists=True predicted=True b.1=-0.60
grip=negF2 k=1 sigma=+1: lambda_min=-7.289e-03 exists=False predicted=False b.1=+0.90
grip=negF2 k=1 sigma=-1: lambda_min=+3.291e-03 exists=True predicted=True b.1=-0.90
grip=linear k=0 sigma=+1: lambda_min=+7.846e-17 exists=False predicted=False b.1=+0.60
grip=linear k=0 sigma=-1: lambda_min=+7.846e-17 exists=False predicted=False b.1=-0.60
grip=linear k=1 sigma=+1: lambda_min=+7.846e-17 exists=False predicted=False b.1=+0.90
grip=linear k=1 sigma=-1: lambda_min=+7.846e-17 exists=False predicted=False b.1=-0.90
no anchor, two bodies (separation 4): alpha1=0.2700, r12=3.500000: threshold alpha2 numerical=-0.1388174807 predicted=-0.1388174807 rel.dev=1.8e-14
anchor at h, one body at distance d: threshold alpha* (lambda_min=0) vs -1/Gamma(x), and which (k, sigma) of grip negF2 / square have a lowest eigenvector
d=2: Gamma=3.303571 alpha* numerical=-0.3027027027 predicted=-0.3027027027 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=Y negF2:k1-=Y square:k0+=Y square:k0-=Y square:k1+=Y square:k1-=Y
d=5: Gamma=5.647321 alpha* numerical=-0.1770750988 predicted=-0.1770750988 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=Y negF2:k1-=Y square:k0+=Y square:k0-=N square:k1+=Y square:k1-=N
d=10: Gamma=8.303571 alpha* numerical=-0.1204301075 predicted=-0.1204301075 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=N negF2:k1-=Y square:k0+=Y square:k0-=N square:k1+=Y square:k1-=N
anchor, two bodies, random (alpha1, alpha2): criterion (5.10) agrees with lambda_min in 40/40, explicit form (5.26) in 40/40
anchor, two bodies at [1], [3] (Gamma = 2.397321, 4.147321, G12 = 2.334821) with alpha1 = alpha2 = -a, a = 0.207279 between 1/lambda_max(Gamma_2) = 0.173439 and 1/max Gamma = 0.241119:
one body alone: 1 - a Gamma(x_i) = +0.5031, +0.1403; lambda_min = +5.295e-03, +1.888e-03 (exists)
pair: lambda_min(A) = -3.507e-03; (5.26) predicts existence = False; sufficient condition a (max Gamma + G12) < 1: False
--- ring L=64
no anchor, one body at a place: lambda_min(A0) and prediction [mu F2 > 0]
grip=square k=0 sigma=+1: lambda_min=+1.363e-03 exists=True predicted=True b.1=+0.60
grip=square k=0 sigma=-1: lambda_min=-8.183e-03 exists=False predicted=False b.1=-0.60
grip=square k=1 sigma=+1: lambda_min=+1.606e-03 exists=True predicted=True b.1=+0.90
grip=square k=1 sigma=-1: lambda_min=-1.816e-02 exists=False predicted=False b.1=-0.90
grip=negF2 k=0 sigma=+1: lambda_min=-2.849e-03 exists=False predicted=False b.1=+0.60
grip=negF2 k=0 sigma=-1: lambda_min=+9.931e-04 exists=True predicted=True b.1=-0.60
grip=negF2 k=1 sigma=+1: lambda_min=-5.792e-03 exists=False predicted=False b.1=+0.90
grip=negF2 k=1 sigma=-1: lambda_min=+1.248e-03 exists=True predicted=True b.1=-0.90
grip=linear k=0 sigma=+1: lambda_min=+2.264e-16 exists=False predicted=False b.1=+0.60
grip=linear k=0 sigma=-1: lambda_min=+2.264e-16 exists=False predicted=False b.1=-0.60
grip=linear k=1 sigma=+1: lambda_min=+2.264e-16 exists=False predicted=False b.1=+0.90
grip=linear k=1 sigma=-1: lambda_min=+2.264e-16 exists=False predicted=False b.1=-0.90
no anchor, two bodies (separation 4): alpha1=0.2700, r12=3.750000: threshold alpha2 numerical=-0.1341614907 predicted=-0.1341614907 rel.dev=7.3e-14
anchor at h, one body at distance d: threshold alpha* (lambda_min=0) vs -1/Gamma(x), and which (k, sigma) of grip negF2 / square have a lowest eigenvector
d=2: Gamma=3.366071 alpha* numerical=-0.2970822281 predicted=-0.2970822281 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=Y negF2:k1-=Y square:k0+=Y square:k0-=Y square:k1+=Y square:k1-=Y
d=5: Gamma=6.037946 alpha* numerical=-0.1656192237 predicted=-0.1656192237 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=Y negF2:k1-=Y square:k0+=Y square:k0-=N square:k1+=Y square:k1-=N
d=10: Gamma=9.866071 alpha* numerical=-0.1013574661 predicted=-0.1013574661 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=N negF2:k1-=Y square:k0+=Y square:k0-=N square:k1+=Y square:k1-=N
anchor, two bodies, random (alpha1, alpha2): criterion (5.10) agrees with lambda_min in 40/40, explicit form (5.26) in 40/40
anchor, two bodies at [1], [3] (Gamma = 2.412946, 4.287946, G12 = 2.381696) with alpha1 = alpha2 = -a, a = 0.201208 between 1/lambda_max(Gamma_2) = 0.169204 and 1/max Gamma = 0.233212:
one body alone: 1 - a Gamma(x_i) = +0.5145, +0.1372; lambda_min = +1.739e-03, +7.853e-04 (exists)
pair: lambda_min(A) = -2.105e-03; (5.26) predicts existence = False; sufficient condition a (max Gamma + G12) < 1: False
--- cubic torus L=8
no anchor, one body at a place: lambda_min(A0) and prediction [mu F2 > 0]
grip=square k=0 sigma=+1: lambda_min=+3.379e-04 exists=True predicted=True b.1=+0.60
grip=square k=0 sigma=-1: lambda_min=-3.664e-04 exists=False predicted=False b.1=-0.60
grip=square k=1 sigma=+1: lambda_min=+4.972e-04 exists=True predicted=True b.1=+0.90
grip=square k=1 sigma=-1: lambda_min=-5.614e-04 exists=False predicted=False b.1=-0.90
grip=negF2 k=0 sigma=+1: lambda_min=-1.998e-04 exists=False predicted=False b.1=+0.60
grip=negF2 k=0 sigma=-1: lambda_min=+1.910e-04 exists=True predicted=True b.1=-0.60
grip=negF2 k=1 sigma=+1: lambda_min=-3.032e-04 exists=False predicted=False b.1=+0.90
grip=negF2 k=1 sigma=-1: lambda_min=+2.834e-04 exists=True predicted=True b.1=-0.90
grip=linear k=0 sigma=+1: lambda_min=-8.867e-16 exists=False predicted=False b.1=+0.60
grip=linear k=0 sigma=-1: lambda_min=-8.867e-16 exists=False predicted=False b.1=-0.60
grip=linear k=1 sigma=+1: lambda_min=-8.867e-16 exists=False predicted=False b.1=+0.90
grip=linear k=1 sigma=-1: lambda_min=-8.867e-16 exists=False predicted=False b.1=-0.90
no anchor, two bodies (separation 4): alpha1=0.2700, r12=0.449684: threshold alpha2 numerical=-0.2407673308 predicted=-0.2407673308 rel.dev=2.1e-13
anchor at h, one body at distance d: threshold alpha* (lambda_min=0) vs -1/Gamma(x), and which (k, sigma) of grip negF2 / square have a lowest eigenvector
d=1: Gamma=1.761254 alpha* numerical=-0.5677773671 predicted=-0.5677773671 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=Y negF2:k1-=Y square:k0+=Y square:k0-=Y square:k1+=Y square:k1-=Y
d=2: Gamma=1.845431 alpha* numerical=-0.5418788955 predicted=-0.5418788955 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=Y negF2:k1-=Y square:k0+=Y square:k0-=Y square:k1+=Y square:k1-=Y
d=3: Gamma=1.871738 alpha* numerical=-0.5342627406 predicted=-0.5342627406 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=Y negF2:k1-=Y square:k0+=Y square:k0-=Y square:k1+=Y square:k1-=Y
anchor, two bodies, random (alpha1, alpha2): criterion (5.10) agrees with lambda_min in 40/40, explicit form (5.26) in 40/40
anchor, two bodies at [1, 0, 0], [3, 1, 1] (Gamma = 1.761254, 1.878598, G12 = 1.600983) with alpha1 = alpha2 = -a, a = 0.412270 between 1/lambda_max(Gamma_2) = 0.292228 and 1/max Gamma = 0.532312:
one body alone: 1 - a Gamma(x_i) = +0.2739, +0.2255; lambda_min = +3.563e-04, +2.936e-04 (exists)
pair: lambda_min(A) = -5.359e-04; (5.26) predicts existence = False; sufficient condition a (max Gamma + G12) < 1: False
--- cubic torus L=12
no anchor, one body at a place: lambda_min(A0) and prediction [mu F2 > 0]
grip=square k=0 sigma=+1: lambda_min=+9.996e-05 exists=True predicted=True b.1=+0.60
grip=square k=0 sigma=-1: lambda_min=-1.087e-04 exists=False predicted=False b.1=-0.60
grip=square k=1 sigma=+1: lambda_min=+1.470e-04 exists=True predicted=True b.1=+0.90
grip=square k=1 sigma=-1: lambda_min=-1.668e-04 exists=False predicted=False b.1=-0.90
grip=negF2 k=0 sigma=+1: lambda_min=-5.926e-05 exists=False predicted=False b.1=+0.60
grip=negF2 k=0 sigma=-1: lambda_min=+5.655e-05 exists=True predicted=True b.1=-0.60
grip=negF2 k=1 sigma=+1: lambda_min=-8.996e-05 exists=False predicted=False b.1=+0.90
grip=negF2 k=1 sigma=-1: lambda_min=+8.386e-05 exists=True predicted=True b.1=-0.90
grip=linear k=0 sigma=+1: lambda_min=-1.988e-16 exists=False predicted=False b.1=+0.60
grip=linear k=0 sigma=-1: lambda_min=-1.988e-16 exists=False predicted=False b.1=-0.60
grip=linear k=1 sigma=+1: lambda_min=-1.988e-16 exists=False predicted=False b.1=+0.90
grip=linear k=1 sigma=-1: lambda_min=-1.988e-16 exists=False predicted=False b.1=-0.90
no anchor, two bodies (separation 4): alpha1=0.2700, r12=0.461270: threshold alpha2 numerical=-0.2400975469 predicted=-0.2400975468 rel.dev=1.3e-11
anchor at h, one body at distance d: threshold alpha* (lambda_min=0) vs -1/Gamma(x), and which (k, sigma) of grip negF2 / square have a lowest eigenvector
d=1: Gamma=1.761712 alpha* numerical=-0.5676297142 predicted=-0.5676297142 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=Y negF2:k1-=Y square:k0+=Y square:k0-=Y square:k1+=Y square:k1-=Y
d=2: Gamma=1.847457 alpha* numerical=-0.5412844656 predicted=-0.5412844656 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=Y negF2:k1-=Y square:k0+=Y square:k0-=Y square:k1+=Y square:k1-=Y
d=3: Gamma=1.877051 alpha* numerical=-0.5327505289 predicted=-0.5327505289 | negF2:k0+=Y negF2:k0-=Y negF2:k1+=Y negF2:k1-=Y square:k0+=Y square:k0-=Y square:k1+=Y square:k1-=Y
anchor, two bodies, random (alpha1, alpha2): criterion (5.10) agrees with lambda_min in 40/40, explicit form (5.26) in 40/40
anchor, two bodies at [1, 0, 0], [3, 1, 1] (Gamma = 1.761712, 1.883815, G12 = 1.602561) with alpha1 = alpha2 = -a, a = 0.411341 between 1/lambda_max(Gamma_2) = 0.291844 and 1/max Gamma = 0.530838:
one body alone: 1 - a Gamma(x_i) = +0.2753, +0.2251; lambda_min = +1.059e-04, +8.670e-05 (exists)
pair: lambda_min(A) = -1.586e-04; (5.26) predicts existence = False; sufficient condition a (max Gamma + G12) < 1: False
output/mutual_cost.txt6 KB · raw
(ii) Mutual cost of two held sources E12 = -g1 g2 (A^{-1})_{x1 x2} from the exact E0 (sparse LU of A) against (5.14) E12 = -g1 g2 G_a(x1,x2); g1 = g2 = 1
kappa_h=1.0, kappa_a=0.7
--- ring L=64: pair on one side of the anchor, x1 = h + d1, x2 = x1 + s
d1=4: s=1: -5.1160714286 (formula -5.1160714286); s=2: -5.0535714286 (formula -5.0535714286); s=4: -4.9285714286 (formula -4.9285714286); s=8: -4.6785714286 (formula -4.6785714286); s=16: -4.1785714286 (formula -4.1785714286)
L->inf value (5.16) on the line: -(1/kappa_a + d1/kappa_h) = -5.4285714286
d1=16: s=1: -13.1785714286 (formula -13.1785714286); s=2: -12.9285714286 (formula -12.9285714286); s=4: -12.4285714286 (formula -12.4285714286); s=8: -11.4285714286 (formula -11.4285714286); s=16: -9.4285714286 (formula -9.4285714286)
L->inf value (5.16) on the line: -(1/kappa_a + d1/kappa_h) = -17.4285714286
opposite sides x1=h+5, x2=h-7: exact -1.9754464286, formula -1.9754464286, -1/kappa_a = -1.4285714286
--- ring L=256: pair on one side of the anchor, x1 = h + d1, x2 = x1 + s
d1=4: s=1: -5.3504464286 (formula -5.3504464286); s=2: -5.3348214286 (formula -5.3348214286); s=4: -5.3035714286 (formula -5.3035714286); s=8: -5.2410714286 (formula -5.2410714286); s=16: -5.1160714286 (formula -5.1160714286)
L->inf value (5.16) on the line: -(1/kappa_a + d1/kappa_h) = -5.4285714286
d1=16: s=1: -16.3660714286 (formula -16.3660714286); s=2: -16.3035714286 (formula -16.3035714286); s=4: -16.1785714286 (formula -16.1785714286); s=8: -15.9285714286 (formula -15.9285714286); s=16: -15.4285714286 (formula -15.4285714286)
L->inf value (5.16) on the line: -(1/kappa_a + d1/kappa_h) = -17.4285714286
opposite sides x1=h+5, x2=h-7: exact -1.5652901786, formula -1.5652901786, -1/kappa_a = -1.4285714286
--- cubic torus L=16: pair along x with midpoint h + D e_y, x1,2 = h + D e_y -/+ (r/2) e_x
D=0: r | exact E12 | formula (5.14), this L | L=inf lattice, (5.14) with g_inf | continuum image form (5.15)
2 | -1.5521474018 | -1.5521474018 | -1.5520630666 | -1.5619362311
4 | -1.6161920237 | -1.6161920237 | -1.6158126766 | -1.6216193348
6 | -1.6405966215 | -1.6405966215 | -1.6395768761 | -1.6415137027
8 | -1.6529995167 | -1.6529995167 | -1.6507127513 | -1.6514608866
effective coefficient c(r) = [E12(r)-E12(r')] / [-(1/(4 pi kappa_h))(1/r - 1/r')], consecutive r:
r=2->4: exact -3.2192, L=inf lattice -3.2044, continuum image form -3.0000
r=4->6: exact -3.6801, L=inf lattice -3.5836, continuum image form -3.0000
r=6->8: exact -3.7406, L=inf lattice -3.3585, continuum image form -3.0000
D=6: r | exact E12 | formula (5.14), this L | L=inf lattice, (5.14) with g_inf | continuum image form (5.15)
2 | -1.6944344012 | -1.6944344012 | -1.6978633698 | -1.6949262627
4 | -1.6733874597 | -1.6733874597 | -1.6763388724 | -1.6760322003
6 | -1.6688972054 | -1.6688972054 | -1.6709151156 | -1.6708399322
8 | -1.6688691370 | -1.6688691370 | -1.6692251878 | -1.6691788028
effective coefficient c(r) = [E12(r)-E12(r')] / [-(1/(4 pi kappa_h))(1/r - 1/r')], consecutive r:
r=2->4: exact +1.0579, L=inf lattice +1.0819, continuum image form +0.9497
r=4->6: exact +0.6771, L=inf lattice +0.8179, continuum image form +0.7830
r=6->8: exact +0.0085, L=inf lattice +0.5097, continuum image form +0.5010
source 1 at the anchor, source 2 at three places: E12 = -1.428571428571, -1.428571428571, -1.428571428571; -1/kappa_a = -1.428571428571
--- cubic torus L=32: pair along x with midpoint h + D e_y, x1,2 = h + D e_y -/+ (r/2) e_x
D=0: r | exact E12 | formula (5.14), this L | L=inf lattice, (5.14) with g_inf | continuum image form (5.15)
2 | -1.5520733285 | -1.5520733285 | -1.5520630666 | -1.5619362311
4 | -1.6158549799 | -1.6158549799 | -1.6158126766 | -1.6216193348
6 | -1.6396768051 | -1.6396768051 | -1.6395768761 | -1.6415137027
8 | -1.6509023983 | -1.6509023983 | -1.6507127513 | -1.6514608866
12 | -1.6617240446 | -1.6617240446 | -1.6612162537 | -1.6614080705
16 | -1.6674389658 | -1.6674389659 | -1.6663048488 | -1.6663816625
effective coefficient c(r) = [E12(r)-E12(r')] / [-(1/(4 pi kappa_h))(1/r - 1/r')], consecutive r:
r=2->4: exact -3.2060, L=inf lattice -3.2044, continuum image form -3.0000
r=4->6: exact -3.5922, L=inf lattice -3.5836, continuum image form -3.0000
r=6->8: exact -3.3856, L=inf lattice -3.3585, continuum image form -3.0000
r=8->12: exact -3.2637, L=inf lattice -3.1678, continuum image form -3.0000
r=12->16: exact -3.4472, L=inf lattice -3.0694, continuum image form -3.0000
D=6: r | exact E12 | formula (5.14), this L | L=inf lattice, (5.14) with g_inf | continuum image form (5.15)
2 | -1.6974901645 | -1.6974901645 | -1.6978633698 | -1.6949262627
4 | -1.6760026402 | -1.6760026402 | -1.6763388724 | -1.6760322003
6 | -1.6706447335 | -1.6706447335 | -1.6709151156 | -1.6708399322
8 | -1.6690560582 | -1.6690560582 | -1.6692251878 | -1.6691788028
12 | -1.6693862736 | -1.6693862736 | -1.6692040382 | -1.6691773045
16 | -1.6712274436 | -1.6712274436 | -1.6703711253 | -1.6703605361
effective coefficient c(r) = [E12(r)-E12(r')] / [-(1/(4 pi kappa_h))(1/r - 1/r')], consecutive r:
r=2->4: exact +1.0801, L=inf lattice +1.0819, continuum image form +0.9497
r=4->6: exact +0.8080, L=inf lattice +0.8179, continuum image form +0.7830
r=6->8: exact +0.4791, L=inf lattice +0.5097, continuum image form +0.5010
r=8->12: exact -0.0996, L=inf lattice +0.0064, continuum image form +0.0005
r=12->16: exact -1.1106, L=inf lattice -0.7040, continuum image form -0.7137
source 1 at the anchor, source 2 at three places: E12 = -1.428571428560, -1.428571428560, -1.428571428560; -1/kappa_a = -1.428571428571
output/weighing.txt72 KB · raw
(iii) Cost, weight, bond and weighing ratios at order s^2 from the exact E0 (dense; s-stencil step 0.01) against (5.18)-(5.22); g_s = +-1 (the g_s-linear part is exact)
kappa_h=1.0, kappa_a=0.7, levels M_k={0: 1.0, 1: 1.5}
=== ring L=32, anchor h=0; body placements x=h+d e, x'=h+(d+1) e, source at h+(d+1+3) e
d=2: Gamma(x)=3.303571, w(x)=0.611203, Sigma(x)=1.350120, phi_a(x)=-3.053571; a_mark=+0.964419
grip=square k=0 sigma=+1: rho_mark numerical=2.05681151 predicted (5.20)=2.05681147 leading (4F0F2/F1^2)(1+a_mark)=1.964419 [block at s=1 by (5.10): Y]
grip=square k=0 sigma=-1: rho_mark numerical=2.05681151 predicted (5.20)=2.05681147 leading (4F0F2/F1^2)(1+a_mark)=1.964419 [block at s=1 by (5.10): Y]
grip=square k=1 sigma=+1: rho_mark numerical=2.05681164 predicted (5.20)=2.05681147 leading (4F0F2/F1^2)(1+a_mark)=1.964419 [block at s=1 by (5.10): Y]
grip=square k=1 sigma=-1: rho_mark numerical=2.05681164 predicted (5.20)=2.05681147 leading (4F0F2/F1^2)(1+a_mark)=1.964419 [block at s=1 by (5.10): N]
grip=nonsquare k=0 sigma=+1: rho_mark numerical=0.66654228 predicted (5.20)=0.66654228 leading (4F0F2/F1^2)(1+a_mark)=0.654806 [block at s=1 by (5.10): Y]
grip=nonsquare k=0 sigma=-1: rho_mark numerical=0.66654228 predicted (5.20)=0.66654228 leading (4F0F2/F1^2)(1+a_mark)=0.654806 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=+1: rho_mark numerical=0.66654228 predicted (5.20)=0.66654228 leading (4F0F2/F1^2)(1+a_mark)=0.654806 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=-1: rho_mark numerical=0.66654228 predicted (5.20)=0.66654228 leading (4F0F2/F1^2)(1+a_mark)=0.654806 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=+1: rho_mark numerical=1.00780344 predicted (5.20)=1.00780344 leading (4F0F2/F1^2)(1+a_mark)=0.982210 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=-1: rho_mark numerical=1.00780344 predicted (5.20)=1.00780344 leading (4F0F2/F1^2)(1+a_mark)=0.982210 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=+1: rho_mark numerical=1.00780344 predicted (5.20)=1.00780344 leading (4F0F2/F1^2)(1+a_mark)=0.982210 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=-1: rho_mark numerical=1.00780344 predicted (5.20)=1.00780344 leading (4F0F2/F1^2)(1+a_mark)=0.982210 [block at s=1 by (5.10): Y]
grip=linear k=0 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=0 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
bond, pair separation 2, source at [7]: m_r=1.000000, a_bond=+1.166354
grips (body 1, body 2) = (square, square):
(k,sigma)=(0,+1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-1.150152e+00; rho_bond,1 num=+2.28470542 (5.21)=+2.28470692 lead(5.22)=+2.166354; rho_bond,2 num=+2.28470542 (5.21)=+2.28470692 lead(5.22)=+2.166354
(k,sigma)=(0,-1),(0,-1): pair block at s=1 (both placements) lambda_min>0: N/N, (5.26): N/N; B=-1.150152e+00; rho_bond,1 num=+2.28470542 (5.21)=+2.28470692 lead(5.22)=+2.166354; rho_bond,2 num=+2.28470542 (5.21)=+2.28470692 lead(5.22)=+2.166354
(k,sigma)=(1,+1),(1,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-2.587840e+00; rho_bond,1 num=+2.28469931 (5.21)=+2.28470692 lead(5.22)=+2.166354; rho_bond,2 num=+2.28469931 (5.21)=+2.28470692 lead(5.22)=+2.166354
(k,sigma)=(1,-1),(1,-1): pair block at s=1 (both placements) lambda_min>0: N/N, (5.26): N/N; B=-2.587840e+00; rho_bond,1 num=+2.28469931 (5.21)=+2.28470692 lead(5.22)=+2.166354; rho_bond,2 num=+2.28469931 (5.21)=+2.28470692 lead(5.22)=+2.166354
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: N/N, (5.26): N/N; B=+1.725229e+00; rho_bond,1 num=+2.28470649 (5.21)=+2.28470692 lead(5.22)=+2.166354; rho_bond,2 num=+2.28470711 (5.21)=+2.28470692 lead(5.22)=+2.166354
grips (body 1, body 2) = (nonsquare, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+1.717406e+00; rho_bond,1 num=+0.73622386 (5.21)=+0.73622386 lead(5.22)=+0.722118; rho_bond,2 num=+0.73622386 (5.21)=+0.73622386 lead(5.22)=+0.722118
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+1.717406e+00; rho_bond,1 num=+0.73622386 (5.21)=+0.73622386 lead(5.22)=+0.722118; rho_bond,2 num=+0.73622386 (5.21)=+0.73622386 lead(5.22)=+0.722118
grips (body 1, body 2) = (eight, eight):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+1.718629e+00; rho_bond,1 num=+1.11439011 (5.21)=+1.11439013 lead(5.22)=+1.083177; rho_bond,2 num=+1.11439013 (5.21)=+1.11439013 lead(5.22)=+1.083177
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+1.718629e+00; rho_bond,1 num=+1.11439014 (5.21)=+1.11439013 lead(5.22)=+1.083177; rho_bond,2 num=+1.11439013 (5.21)=+1.11439013 lead(5.22)=+1.083177
grips (body 1, body 2) = (linear, linear):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+1.716429e+00; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+1.716429e+00; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
grips (body 1, body 2) = (square, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+1.719362e+00; rho_bond,1 num=+1.45153426 (5.21)=+1.45153418 lead(5.22)=+1.371274; rho_bond,2 num=+1.39633795 (5.21)=+1.39633796 lead(5.22)=+1.371274
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+1.719362e+00; rho_bond,1 num=+1.45153454 (5.21)=+1.45153418 lead(5.22)=+1.371274; rho_bond,2 num=+1.39633783 (5.21)=+1.39633796 lead(5.22)=+1.371274
d=5: Gamma(x)=5.647321, w(x)=0.715303, Sigma(x)=2.643709, phi_a(x)=-5.022321; a_mark=+0.822319
grip=square k=0 sigma=+1: rho_mark numerical=1.92089161 predicted (5.20)=1.92089145 leading (4F0F2/F1^2)(1+a_mark)=1.822319 [block at s=1 by (5.10): Y]
grip=square k=0 sigma=-1: rho_mark numerical=1.92089161 predicted (5.20)=1.92089145 leading (4F0F2/F1^2)(1+a_mark)=1.822319 [block at s=1 by (5.10): N]
grip=square k=1 sigma=+1: rho_mark numerical=1.92089226 predicted (5.20)=1.92089145 leading (4F0F2/F1^2)(1+a_mark)=1.822319 [block at s=1 by (5.10): Y]
grip=square k=1 sigma=-1: rho_mark numerical=1.92089226 predicted (5.20)=1.92089145 leading (4F0F2/F1^2)(1+a_mark)=1.822319 [block at s=1 by (5.10): N]
grip=nonsquare k=0 sigma=+1: rho_mark numerical=0.61993590 predicted (5.20)=0.61993590 leading (4F0F2/F1^2)(1+a_mark)=0.607440 [block at s=1 by (5.10): Y]
grip=nonsquare k=0 sigma=-1: rho_mark numerical=0.61993590 predicted (5.20)=0.61993590 leading (4F0F2/F1^2)(1+a_mark)=0.607440 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=+1: rho_mark numerical=0.61993590 predicted (5.20)=0.61993590 leading (4F0F2/F1^2)(1+a_mark)=0.607440 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=-1: rho_mark numerical=0.61993590 predicted (5.20)=0.61993590 leading (4F0F2/F1^2)(1+a_mark)=0.607440 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=+1: rho_mark numerical=0.93842570 predicted (5.20)=0.93842570 leading (4F0F2/F1^2)(1+a_mark)=0.911159 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=-1: rho_mark numerical=0.93842570 predicted (5.20)=0.93842570 leading (4F0F2/F1^2)(1+a_mark)=0.911159 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=+1: rho_mark numerical=0.93842570 predicted (5.20)=0.93842570 leading (4F0F2/F1^2)(1+a_mark)=0.911159 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=-1: rho_mark numerical=0.93842570 predicted (5.20)=0.93842570 leading (4F0F2/F1^2)(1+a_mark)=0.911159 [block at s=1 by (5.10): Y]
grip=linear k=0 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=0 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
bond, pair separation 2, source at [10]: m_r=1.000000, a_bond=+0.904371
grips (body 1, body 2) = (square, square):
(k,sigma)=(0,+1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-1.933962e+00; rho_bond,1 num=+2.01765675 (5.21)=+2.01766300 lead(5.22)=+1.904371; rho_bond,2 num=+2.01765675 (5.21)=+2.01766300 lead(5.22)=+1.904371
(k,sigma)=(0,-1),(0,-1): pair block at s=1 (both placements) lambda_min>0: N/N, (5.26): N/N; B=-1.933962e+00; rho_bond,1 num=+2.01765675 (5.21)=+2.01766300 lead(5.22)=+1.904371; rho_bond,2 num=+2.01765675 (5.21)=+2.01766300 lead(5.22)=+1.904371
(k,sigma)=(1,+1),(1,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-4.351387e+00; rho_bond,1 num=+2.01763119 (5.21)=+2.01766300 lead(5.22)=+1.904371; rho_bond,2 num=+2.01763119 (5.21)=+2.01766300 lead(5.22)=+1.904371
(k,sigma)=(1,-1),(1,-1): pair block at s=1 (both placements) lambda_min>0: N/N, (5.26): N/N; B=-4.351387e+00; rho_bond,1 num=+2.01763119 (5.21)=+2.01766300 lead(5.22)=+1.904371; rho_bond,2 num=+2.01763119 (5.21)=+2.01766300 lead(5.22)=+1.904371
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: N/N, (5.26): N/N; B=+2.900947e+00; rho_bond,1 num=+2.01766198 (5.21)=+2.01766300 lead(5.22)=+1.904371; rho_bond,2 num=+2.01766404 (5.21)=+2.01766300 lead(5.22)=+1.904371
grips (body 1, body 2) = (nonsquare, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+2.883042e+00; rho_bond,1 num=+0.64847446 (5.21)=+0.64847446 lead(5.22)=+0.634790; rho_bond,2 num=+0.64847446 (5.21)=+0.64847446 lead(5.22)=+0.634790
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+2.883042e+00; rho_bond,1 num=+0.64847446 (5.21)=+0.64847446 lead(5.22)=+0.634790; rho_bond,2 num=+0.64847446 (5.21)=+0.64847446 lead(5.22)=+0.634790
grips (body 1, body 2) = (eight, eight):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+2.885840e+00; rho_bond,1 num=+0.98237335 (5.21)=+0.98237338 lead(5.22)=+0.952185; rho_bond,2 num=+0.98237339 (5.21)=+0.98237338 lead(5.22)=+0.952185
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+2.885840e+00; rho_bond,1 num=+0.98237341 (5.21)=+0.98237338 lead(5.22)=+0.952185; rho_bond,2 num=+0.98237337 (5.21)=+0.98237338 lead(5.22)=+0.952185
grips (body 1, body 2) = (linear, linear):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+2.880804e+00; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+2.880804e+00; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
grips (body 1, body 2) = (square, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+2.887518e+00; rho_bond,1 num=+1.31414245 (5.21)=+1.31414220 lead(5.22)=+1.234437; rho_bond,2 num=+1.25903352 (5.21)=+1.25903359 lead(5.22)=+1.234437
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: N/N, (5.26): N/N; B=+2.887518e+00; rho_bond,1 num=+1.31414311 (5.21)=+1.31414220 lead(5.22)=+1.234437; rho_bond,2 num=+1.25903286 (5.21)=+1.25903359 lead(5.22)=+1.234437
d=10: Gamma(x)=8.303571, w(x)=0.792249, Sigma(x)=4.263750, phi_a(x)=-7.053571; a_mark=+0.528868
grip=square k=0 sigma=+1: rho_mark numerical=1.61964455 predicted (5.20)=1.61964428 leading (4F0F2/F1^2)(1+a_mark)=1.528868 [block at s=1 by (5.10): Y]
grip=square k=0 sigma=-1: rho_mark numerical=1.61964455 predicted (5.20)=1.61964428 leading (4F0F2/F1^2)(1+a_mark)=1.528868 [block at s=1 by (5.10): N]
grip=square k=1 sigma=+1: rho_mark numerical=1.61964568 predicted (5.20)=1.61964428 leading (4F0F2/F1^2)(1+a_mark)=1.528868 [block at s=1 by (5.10): Y]
grip=square k=1 sigma=-1: rho_mark numerical=1.61964568 predicted (5.20)=1.61964428 leading (4F0F2/F1^2)(1+a_mark)=1.528868 [block at s=1 by (5.10): N]
grip=nonsquare k=0 sigma=+1: rho_mark numerical=0.52112633 predicted (5.20)=0.52112633 leading (4F0F2/F1^2)(1+a_mark)=0.509623 [block at s=1 by (5.10): Y]
grip=nonsquare k=0 sigma=-1: rho_mark numerical=0.52112633 predicted (5.20)=0.52112633 leading (4F0F2/F1^2)(1+a_mark)=0.509623 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=+1: rho_mark numerical=0.52112632 predicted (5.20)=0.52112633 leading (4F0F2/F1^2)(1+a_mark)=0.509623 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=-1: rho_mark numerical=0.52112632 predicted (5.20)=0.52112633 leading (4F0F2/F1^2)(1+a_mark)=0.509623 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=+1: rho_mark numerical=0.78953802 predicted (5.20)=0.78953802 leading (4F0F2/F1^2)(1+a_mark)=0.764434 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=-1: rho_mark numerical=0.78953802 predicted (5.20)=0.78953802 leading (4F0F2/F1^2)(1+a_mark)=0.764434 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=+1: rho_mark numerical=0.78953801 predicted (5.20)=0.78953802 leading (4F0F2/F1^2)(1+a_mark)=0.764434 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=-1: rho_mark numerical=0.78953801 predicted (5.20)=0.78953802 leading (4F0F2/F1^2)(1+a_mark)=0.764434 [block at s=1 by (5.10): N]
grip=linear k=0 sigma=+1: rho_mark numerical=-7.188e-13 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=0 sigma=-1: rho_mark numerical=-7.188e-13 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
bond, pair separation 2, source at [15]: m_r=1.000000, a_bond=+0.510369
grips (body 1, body 2) = (square, square):
(k,sigma)=(0,+1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-2.787577e+00; rho_bond,1 num=+1.60596890 (5.21)=+1.60598131 lead(5.22)=+1.510369; rho_bond,2 num=+1.60596890 (5.21)=+1.60598131 lead(5.22)=+1.510369
(k,sigma)=(0,-1),(0,-1): pair block at s=1 (both placements) lambda_min>0: N/N, (5.26): N/N; B=-2.787577e+00; rho_bond,1 num=+1.60596890 (5.21)=+1.60598131 lead(5.22)=+1.510369; rho_bond,2 num=+1.60596890 (5.21)=+1.60598131 lead(5.22)=+1.510369
(k,sigma)=(1,+1),(1,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-6.271894e+00; rho_bond,1 num=+1.60591794 (5.21)=+1.60598131 lead(5.22)=+1.510369; rho_bond,2 num=+1.60591794 (5.21)=+1.60598131 lead(5.22)=+1.510369
(k,sigma)=(1,-1),(1,-1): pair block at s=1 (both placements) lambda_min>0: N/N, (5.26): N/N; B=-6.271894e+00; rho_bond,1 num=+1.60591794 (5.21)=+1.60598131 lead(5.22)=+1.510369; rho_bond,2 num=+1.60591794 (5.21)=+1.60598131 lead(5.22)=+1.510369
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: N/N, (5.26): N/N; B=+4.181385e+00; rho_bond,1 num=+1.60598013 (5.21)=+1.60598131 lead(5.22)=+1.510369; rho_bond,2 num=+1.60598322 (5.21)=+1.60598131 lead(5.22)=+1.510369
grips (body 1, body 2) = (nonsquare, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+4.150313e+00; rho_bond,1 num=+0.51511119 (5.21)=+0.51511119 lead(5.22)=+0.503456; rho_bond,2 num=+0.51511119 (5.21)=+0.51511119 lead(5.22)=+0.503456
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+4.150313e+00; rho_bond,1 num=+0.51511118 (5.21)=+0.51511119 lead(5.22)=+0.503456; rho_bond,2 num=+0.51511119 (5.21)=+0.51511119 lead(5.22)=+0.503456
grips (body 1, body 2) = (eight, eight):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+4.155169e+00; rho_bond,1 num=+0.78084263 (5.21)=+0.78084267 lead(5.22)=+0.755185; rho_bond,2 num=+0.78084267 (5.21)=+0.78084267 lead(5.22)=+0.755185
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+4.155169e+00; rho_bond,1 num=+0.78084266 (5.21)=+0.78084267 lead(5.22)=+0.755185; rho_bond,2 num=+0.78084262 (5.21)=+0.78084267 lead(5.22)=+0.755185
grips (body 1, body 2) = (linear, linear):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+4.146429e+00; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+4.146429e+00; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
grips (body 1, body 2) = (square, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+4.158082e+00; rho_bond,1 num=+1.06388910 (5.21)=+1.06388882 lead(5.22)=+0.994923; rho_bond,2 num=+1.01603576 (5.21)=+1.01603598 lead(5.22)=+0.994923
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: N/N, (5.26): N/N; B=+4.158079e+00; rho_bond,1 num=+1.06388927 (5.21)=+1.06388882 lead(5.22)=+0.994923; rho_bond,2 num=+1.01603397 (5.21)=+1.01603598 lead(5.22)=+0.994923
max relative deviation over this lattice: C1, Cmark, W1, Wmark from (5.18): 1.19e-06; rho_mark from (5.20): 8.67e-07; B, W_bond from (5.19): 3.53e-05; rho_bond,i from (5.21): 3.95e-05 (absolute deviation where the prediction is 0)
=== ring L=64, anchor h=0; body placements x=h+d e, x'=h+(d+1) e, source at h+(d+1+3) e
d=2: Gamma(x)=3.366071, w(x)=0.609472, Sigma(x)=1.332455, phi_a(x)=-3.241071; a_mark=+0.981939
grip=square k=0 sigma=+1: rho_mark numerical=2.07559516 predicted (5.20)=2.07559512 leading (4F0F2/F1^2)(1+a_mark)=1.981939 [block at s=1 by (5.10): Y]
grip=square k=0 sigma=-1: rho_mark numerical=2.07559516 predicted (5.20)=2.07559512 leading (4F0F2/F1^2)(1+a_mark)=1.981939 [block at s=1 by (5.10): Y]
grip=square k=1 sigma=+1: rho_mark numerical=2.07559532 predicted (5.20)=2.07559512 leading (4F0F2/F1^2)(1+a_mark)=1.981939 [block at s=1 by (5.10): Y]
grip=square k=1 sigma=-1: rho_mark numerical=2.07559532 predicted (5.20)=2.07559512 leading (4F0F2/F1^2)(1+a_mark)=1.981939 [block at s=1 by (5.10): N]
grip=nonsquare k=0 sigma=+1: rho_mark numerical=0.67248224 predicted (5.20)=0.67248224 leading (4F0F2/F1^2)(1+a_mark)=0.660646 [block at s=1 by (5.10): Y]
grip=nonsquare k=0 sigma=-1: rho_mark numerical=0.67248224 predicted (5.20)=0.67248224 leading (4F0F2/F1^2)(1+a_mark)=0.660646 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=+1: rho_mark numerical=0.67248224 predicted (5.20)=0.67248224 leading (4F0F2/F1^2)(1+a_mark)=0.660646 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=-1: rho_mark numerical=0.67248224 predicted (5.20)=0.67248224 leading (4F0F2/F1^2)(1+a_mark)=0.660646 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=+1: rho_mark numerical=1.01681087 predicted (5.20)=1.01681087 leading (4F0F2/F1^2)(1+a_mark)=0.990970 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=-1: rho_mark numerical=1.01681087 predicted (5.20)=1.01681087 leading (4F0F2/F1^2)(1+a_mark)=0.990970 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=+1: rho_mark numerical=1.01681088 predicted (5.20)=1.01681087 leading (4F0F2/F1^2)(1+a_mark)=0.990970 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=-1: rho_mark numerical=1.01681088 predicted (5.20)=1.01681087 leading (4F0F2/F1^2)(1+a_mark)=0.990970 [block at s=1 by (5.10): Y]
grip=linear k=0 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=0 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
bond, pair separation 2, source at [7]: m_r=1.000000, a_bond=+1.212563
grips (body 1, body 2) = (square, square):
(k,sigma)=(0,+1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-1.195053e+00; rho_bond,1 num=+2.33396220 (5.21)=+2.33396410 lead(5.22)=+2.212563; rho_bond,2 num=+2.33396220 (5.21)=+2.33396410 lead(5.22)=+2.212563
(k,sigma)=(0,-1),(0,-1): pair block at s=1 (both placements) lambda_min>0: N/N, (5.26): N/N; B=-1.195053e+00; rho_bond,1 num=+2.33396220 (5.21)=+2.33396410 lead(5.22)=+2.212563; rho_bond,2 num=+2.33396220 (5.21)=+2.33396410 lead(5.22)=+2.212563
(k,sigma)=(1,+1),(1,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-2.688865e+00; rho_bond,1 num=+2.33395446 (5.21)=+2.33396410 lead(5.22)=+2.212563; rho_bond,2 num=+2.33395446 (5.21)=+2.33396410 lead(5.22)=+2.212563
(k,sigma)=(1,-1),(1,-1): pair block at s=1 (both placements) lambda_min>0: N/N, (5.26): N/N; B=-2.688865e+00; rho_bond,1 num=+2.33395446 (5.21)=+2.33396410 lead(5.22)=+2.212563; rho_bond,2 num=+2.33395446 (5.21)=+2.33396410 lead(5.22)=+2.212563
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: N/N, (5.26): N/N; B=+1.792579e+00; rho_bond,1 num=+2.33396348 (5.21)=+2.33396410 lead(5.22)=+2.212563; rho_bond,2 num=+2.33396427 (5.21)=+2.33396410 lead(5.22)=+2.212563
grips (body 1, body 2) = (nonsquare, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+1.784890e+00; rho_bond,1 num=+0.75193242 (5.21)=+0.75193242 lead(5.22)=+0.737521; rho_bond,2 num=+0.75193242 (5.21)=+0.75193242 lead(5.22)=+0.737521
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+1.784890e+00; rho_bond,1 num=+0.75193242 (5.21)=+0.75193242 lead(5.22)=+0.737521; rho_bond,2 num=+0.75193242 (5.21)=+0.75193242 lead(5.22)=+0.737521
grips (body 1, body 2) = (eight, eight):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+1.786091e+00; rho_bond,1 num=+1.13820041 (5.21)=+1.13820043 lead(5.22)=+1.106282; rho_bond,2 num=+1.13820043 (5.21)=+1.13820043 lead(5.22)=+1.106282
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+1.786091e+00; rho_bond,1 num=+1.13820044 (5.21)=+1.13820043 lead(5.22)=+1.106282; rho_bond,2 num=+1.13820043 (5.21)=+1.13820043 lead(5.22)=+1.106282
grips (body 1, body 2) = (linear, linear):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+1.783929e+00; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+1.783929e+00; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
grips (body 1, body 2) = (square, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+1.786812e+00; rho_bond,1 num=+1.47758261 (5.21)=+1.47758250 lead(5.22)=+1.395852; rho_bond,2 num=+1.42147068 (5.21)=+1.42147069 lead(5.22)=+1.395852
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+1.786812e+00; rho_bond,1 num=+1.47758298 (5.21)=+1.47758250 lead(5.22)=+1.395852; rho_bond,2 num=+1.42147055 (5.21)=+1.42147069 lead(5.22)=+1.395852
d=5: Gamma(x)=6.037946, w(x)=0.716014, Sigma(x)=2.674050, phi_a(x)=-5.725446; a_mark=+0.919297
grip=square k=0 sigma=+1: rho_mark numerical=2.02458539 predicted (5.20)=2.02458512 leading (4F0F2/F1^2)(1+a_mark)=1.919297 [block at s=1 by (5.10): Y]
grip=square k=0 sigma=-1: rho_mark numerical=2.02458539 predicted (5.20)=2.02458512 leading (4F0F2/F1^2)(1+a_mark)=1.919297 [block at s=1 by (5.10): N]
grip=square k=1 sigma=+1: rho_mark numerical=2.02458649 predicted (5.20)=2.02458512 leading (4F0F2/F1^2)(1+a_mark)=1.919297 [block at s=1 by (5.10): Y]
grip=square k=1 sigma=-1: rho_mark numerical=2.02458649 predicted (5.20)=2.02458512 leading (4F0F2/F1^2)(1+a_mark)=1.919297 [block at s=1 by (5.10): N]
grip=nonsquare k=0 sigma=+1: rho_mark numerical=0.65300027 predicted (5.20)=0.65300027 leading (4F0F2/F1^2)(1+a_mark)=0.639766 [block at s=1 by (5.10): Y]
grip=nonsquare k=0 sigma=-1: rho_mark numerical=0.65300027 predicted (5.20)=0.65300027 leading (4F0F2/F1^2)(1+a_mark)=0.639766 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=+1: rho_mark numerical=0.65300027 predicted (5.20)=0.65300027 leading (4F0F2/F1^2)(1+a_mark)=0.639766 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=-1: rho_mark numerical=0.65300027 predicted (5.20)=0.65300027 leading (4F0F2/F1^2)(1+a_mark)=0.639766 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=+1: rho_mark numerical=0.98858086 predicted (5.20)=0.98858086 leading (4F0F2/F1^2)(1+a_mark)=0.959648 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=-1: rho_mark numerical=0.98858086 predicted (5.20)=0.98858086 leading (4F0F2/F1^2)(1+a_mark)=0.959648 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=+1: rho_mark numerical=0.98858088 predicted (5.20)=0.98858086 leading (4F0F2/F1^2)(1+a_mark)=0.959648 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=-1: rho_mark numerical=0.98858088 predicted (5.20)=0.98858086 leading (4F0F2/F1^2)(1+a_mark)=0.959648 [block at s=1 by (5.10): Y]
grip=linear k=0 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=0 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
bond, pair separation 2, source at [10]: m_r=1.000000, a_bond=+1.039611
grips (body 1, body 2) = (square, square):
(k,sigma)=(0,+1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-2.131345e+00; rho_bond,1 num=+2.16246098 (5.21)=+2.16247190 lead(5.22)=+2.039611; rho_bond,2 num=+2.16246098 (5.21)=+2.16247190 lead(5.22)=+2.039611
(k,sigma)=(0,-1),(0,-1): pair block at s=1 (both placements) lambda_min>0: N/N, (5.26): N/N; B=-2.131345e+00; rho_bond,1 num=+2.16246098 (5.21)=+2.16247190 lead(5.22)=+2.039611; rho_bond,2 num=+2.16246098 (5.21)=+2.16247190 lead(5.22)=+2.039611
(k,sigma)=(1,+1),(1,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-4.795482e+00; rho_bond,1 num=+2.16241626 (5.21)=+2.16247190 lead(5.22)=+2.039611; rho_bond,2 num=+2.16241626 (5.21)=+2.16247190 lead(5.22)=+2.039611
(k,sigma)=(1,-1),(1,-1): pair block at s=1 (both placements) lambda_min>0: N/N, (5.26): N/N; B=-4.795482e+00; rho_bond,1 num=+2.16241626 (5.21)=+2.16247190 lead(5.22)=+2.039611; rho_bond,2 num=+2.16241626 (5.21)=+2.16247190 lead(5.22)=+2.039611
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: N/N, (5.26): N/N; B=+3.197023e+00; rho_bond,1 num=+2.16246979 (5.21)=+2.16247190 lead(5.22)=+2.039611; rho_bond,2 num=+2.16247357 (5.21)=+2.16247190 lead(5.22)=+2.039611
grips (body 1, body 2) = (nonsquare, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+3.178439e+00; rho_bond,1 num=+0.69462897 (5.21)=+0.69462898 lead(5.22)=+0.679870; rho_bond,2 num=+0.69462898 (5.21)=+0.69462898 lead(5.22)=+0.679870
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+3.178439e+00; rho_bond,1 num=+0.69462899 (5.21)=+0.69462898 lead(5.22)=+0.679870; rho_bond,2 num=+0.69462898 (5.21)=+0.69462898 lead(5.22)=+0.679870
grips (body 1, body 2) = (eight, eight):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+3.181343e+00; rho_bond,1 num=+1.05240510 (5.21)=+1.05240517 lead(5.22)=+1.019805; rho_bond,2 num=+1.05240518 (5.21)=+1.05240517 lead(5.22)=+1.019805
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+3.181343e+00; rho_bond,1 num=+1.05240523 (5.21)=+1.05240517 lead(5.22)=+1.019805; rho_bond,2 num=+1.05240515 (5.21)=+1.05240517 lead(5.22)=+1.019805
grips (body 1, body 2) = (linear, linear):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+3.176116e+00; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+3.176116e+00; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
grips (body 1, body 2) = (square, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+3.183086e+00; rho_bond,1 num=+1.40311379 (5.21)=+1.40311331 lead(5.22)=+1.317444; rho_bond,2 num=+1.34401696 (5.21)=+1.34401706 lead(5.22)=+1.317444
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: N/N, (5.26): N/N; B=+3.183085e+00; rho_bond,1 num=+1.40311524 (5.21)=+1.40311331 lead(5.22)=+1.317444; rho_bond,2 num=+1.34401598 (5.21)=+1.34401706 lead(5.22)=+1.317444
d=10: Gamma(x)=9.866071, w(x)=0.805930, Sigma(x)=4.741418, phi_a(x)=-9.241071; a_mark=+0.811924
grip=square k=0 sigma=+1: rho_mark numerical=1.92343554 predicted (5.20)=1.92343421 leading (4F0F2/F1^2)(1+a_mark)=1.811924 [block at s=1 by (5.10): Y]
grip=square k=0 sigma=-1: rho_mark numerical=1.92343554 predicted (5.20)=1.92343421 leading (4F0F2/F1^2)(1+a_mark)=1.811924 [block at s=1 by (5.10): N]
grip=square k=1 sigma=+1: rho_mark numerical=1.92344096 predicted (5.20)=1.92343421 leading (4F0F2/F1^2)(1+a_mark)=1.811924 [block at s=1 by (5.10): Y]
grip=square k=1 sigma=-1: rho_mark numerical=1.92344096 predicted (5.20)=1.92343421 leading (4F0F2/F1^2)(1+a_mark)=1.811924 [block at s=1 by (5.10): N]
grip=nonsquare k=0 sigma=+1: rho_mark numerical=0.61794256 predicted (5.20)=0.61794256 leading (4F0F2/F1^2)(1+a_mark)=0.603975 [block at s=1 by (5.10): Y]
grip=nonsquare k=0 sigma=-1: rho_mark numerical=0.61794256 predicted (5.20)=0.61794256 leading (4F0F2/F1^2)(1+a_mark)=0.603975 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=+1: rho_mark numerical=0.61794255 predicted (5.20)=0.61794256 leading (4F0F2/F1^2)(1+a_mark)=0.603975 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=-1: rho_mark numerical=0.61794255 predicted (5.20)=0.61794256 leading (4F0F2/F1^2)(1+a_mark)=0.603975 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=+1: rho_mark numerical=0.93652337 predicted (5.20)=0.93652336 leading (4F0F2/F1^2)(1+a_mark)=0.905962 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=-1: rho_mark numerical=0.93652337 predicted (5.20)=0.93652336 leading (4F0F2/F1^2)(1+a_mark)=0.905962 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=+1: rho_mark numerical=0.93652343 predicted (5.20)=0.93652336 leading (4F0F2/F1^2)(1+a_mark)=0.905962 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=-1: rho_mark numerical=0.93652343 predicted (5.20)=0.93652336 leading (4F0F2/F1^2)(1+a_mark)=0.905962 [block at s=1 by (5.10): N]
grip=linear k=0 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=0 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
bond, pair separation 2, source at [15]: m_r=1.000000, a_bond=+0.867189
grips (body 1, body 2) = (square, square):
(k,sigma)=(0,+1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-3.467098e+00; rho_bond,1 num=+1.98924985 (5.21)=+1.98930136 lead(5.22)=+1.867189; rho_bond,2 num=+1.98924985 (5.21)=+1.98930136 lead(5.22)=+1.867189
(k,sigma)=(0,-1),(0,-1): pair block at s=1 (both placements) lambda_min>0: N/N, (5.26): N/N; B=-3.467098e+00; rho_bond,1 num=+1.98924985 (5.21)=+1.98930136 lead(5.22)=+1.867189; rho_bond,2 num=+1.98924985 (5.21)=+1.98930136 lead(5.22)=+1.867189
(k,sigma)=(1,+1),(1,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-7.800528e+00; rho_bond,1 num=+1.98903680 (5.21)=+1.98930136 lead(5.22)=+1.867189; rho_bond,2 num=+1.98903680 (5.21)=+1.98930136 lead(5.22)=+1.867189
(k,sigma)=(1,-1),(1,-1): pair block at s=1 (both placements) lambda_min>0: N/N, (5.26): N/N; B=-7.800528e+00; rho_bond,1 num=+1.98903680 (5.21)=+1.98930136 lead(5.22)=+1.867189; rho_bond,2 num=+1.98903680 (5.21)=+1.98930136 lead(5.22)=+1.867189
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: N/N, (5.26): N/N; B=+5.200703e+00; rho_bond,1 num=+1.98929451 (5.21)=+1.98930136 lead(5.22)=+1.867189; rho_bond,2 num=+1.98930983 (5.21)=+1.98930136 lead(5.22)=+1.867189
grips (body 1, body 2) = (nonsquare, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+5.163572e+00; rho_bond,1 num=+0.63719761 (5.21)=+0.63719764 lead(5.22)=+0.622396; rho_bond,2 num=+0.63719763 (5.21)=+0.63719764 lead(5.22)=+0.622396
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+5.163572e+00; rho_bond,1 num=+0.63719764 (5.21)=+0.63719764 lead(5.22)=+0.622396; rho_bond,2 num=+0.63719762 (5.21)=+0.63719764 lead(5.22)=+0.622396
grips (body 1, body 2) = (eight, eight):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+5.169375e+00; rho_bond,1 num=+0.96622197 (5.21)=+0.96622221 lead(5.22)=+0.933594; rho_bond,2 num=+0.96622229 (5.21)=+0.96622221 lead(5.22)=+0.933594
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+5.169375e+00; rho_bond,1 num=+0.96622240 (5.21)=+0.96622221 lead(5.22)=+0.933594; rho_bond,2 num=+0.96622208 (5.21)=+0.96622221 lead(5.22)=+0.933594
grips (body 1, body 2) = (linear, linear):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+5.158929e+00; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+5.158929e+00; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
grips (body 1, body 2) = (square, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+5.172858e+00; rho_bond,1 num=+1.31027380 (5.21)=+1.31027201 lead(5.22)=+1.223166; rho_bond,2 num=+1.24997527 (5.21)=+1.24997587 lead(5.22)=+1.223166
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: N/N, (5.26): N/N; B=+5.172851e+00; rho_bond,1 num=+1.31027819 (5.21)=+1.31027201 lead(5.22)=+1.223166; rho_bond,2 num=+1.24996979 (5.21)=+1.24997587 lead(5.22)=+1.223166
max relative deviation over this lattice: C1, Cmark, W1, Wmark from (5.18): 2.63e-06; rho_mark from (5.20): 3.51e-06; B, W_bond from (5.19): 9.09e-05; rho_bond,i from (5.21): 1.33e-04 (absolute deviation where the prediction is 0)
=== cubic torus L=8, anchor h=0; body placements x=h+d e, x'=h+(d+1) e, source at h+(d+1+1) e
d=1: Gamma(x)=1.761254, w(x)=0.250142, Sigma(x)=0.110614, phi_a(x)=-1.608066; a_mark=+0.915054
grip=square k=0 sigma=+1: rho_mark numerical=1.95555323 predicted (5.20)=1.95555323 leading (4F0F2/F1^2)(1+a_mark)=1.915054 [block at s=1 by (5.10): Y]
grip=square k=0 sigma=-1: rho_mark numerical=1.95555323 predicted (5.20)=1.95555323 leading (4F0F2/F1^2)(1+a_mark)=1.915054 [block at s=1 by (5.10): Y]
grip=square k=1 sigma=+1: rho_mark numerical=1.95555323 predicted (5.20)=1.95555323 leading (4F0F2/F1^2)(1+a_mark)=1.915054 [block at s=1 by (5.10): Y]
grip=square k=1 sigma=-1: rho_mark numerical=1.95555323 predicted (5.20)=1.95555323 leading (4F0F2/F1^2)(1+a_mark)=1.915054 [block at s=1 by (5.10): Y]
grip=nonsquare k=0 sigma=+1: rho_mark numerical=0.64305813 predicted (5.20)=0.64305813 leading (4F0F2/F1^2)(1+a_mark)=0.638351 [block at s=1 by (5.10): Y]
grip=nonsquare k=0 sigma=-1: rho_mark numerical=0.64305813 predicted (5.20)=0.64305813 leading (4F0F2/F1^2)(1+a_mark)=0.638351 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=+1: rho_mark numerical=0.64305811 predicted (5.20)=0.64305813 leading (4F0F2/F1^2)(1+a_mark)=0.638351 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=-1: rho_mark numerical=0.64305811 predicted (5.20)=0.64305813 leading (4F0F2/F1^2)(1+a_mark)=0.638351 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=+1: rho_mark numerical=0.96800210 predicted (5.20)=0.96800210 leading (4F0F2/F1^2)(1+a_mark)=0.957527 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=-1: rho_mark numerical=0.96800210 predicted (5.20)=0.96800210 leading (4F0F2/F1^2)(1+a_mark)=0.957527 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=+1: rho_mark numerical=0.96800208 predicted (5.20)=0.96800210 leading (4F0F2/F1^2)(1+a_mark)=0.957527 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=-1: rho_mark numerical=0.96800208 predicted (5.20)=0.96800210 leading (4F0F2/F1^2)(1+a_mark)=0.957527 [block at s=1 by (5.10): Y]
grip=linear k=0 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=0 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
bond, pair separation 2, source at [3, 1, 0]: m_r=0.710282, a_bond=+0.826912
grips (body 1, body 2) = (square, square):
(k,sigma)=(0,+1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-5.769611e-01; rho_bond,1 num=+1.57103889 (5.21)=+1.57103895 lead(5.22)=+1.537194; rho_bond,2 num=+1.57103889 (5.21)=+1.57103895 lead(5.22)=+1.537194
(k,sigma)=(0,-1),(0,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-5.769611e-01; rho_bond,1 num=+1.57103889 (5.21)=+1.57103895 lead(5.22)=+1.537194; rho_bond,2 num=+1.57103889 (5.21)=+1.57103895 lead(5.22)=+1.537194
(k,sigma)=(1,+1),(1,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-1.298162e+00; rho_bond,1 num=+1.57103876 (5.21)=+1.57103895 lead(5.22)=+1.537194; rho_bond,2 num=+1.57103876 (5.21)=+1.57103895 lead(5.22)=+1.537194
(k,sigma)=(1,-1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-1.298162e+00; rho_bond,1 num=+1.57103876 (5.21)=+1.57103895 lead(5.22)=+1.537194; rho_bond,2 num=+1.57103876 (5.21)=+1.57103895 lead(5.22)=+1.537194
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.654416e-01; rho_bond,1 num=+1.57103897 (5.21)=+1.57103895 lead(5.22)=+1.537194; rho_bond,2 num=+1.57103897 (5.21)=+1.57103895 lead(5.22)=+1.537194
grips (body 1, body 2) = (nonsquare, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.649560e-01; rho_bond,1 num=+0.51623367 (5.21)=+0.51623367 lead(5.22)=+0.512398; rho_bond,2 num=+0.51623367 (5.21)=+0.51623367 lead(5.22)=+0.512398
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.649560e-01; rho_bond,1 num=+0.51623366 (5.21)=+0.51623367 lead(5.22)=+0.512398; rho_bond,2 num=+0.51623366 (5.21)=+0.51623367 lead(5.22)=+0.512398
grips (body 1, body 2) = (eight, eight):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.650319e-01; rho_bond,1 num=+0.77718550 (5.21)=+0.77718547 lead(5.22)=+0.768597; rho_bond,2 num=+0.77718550 (5.21)=+0.77718547 lead(5.22)=+0.768597
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.650319e-01; rho_bond,1 num=+0.77718547 (5.21)=+0.77718547 lead(5.22)=+0.768597; rho_bond,2 num=+0.77718547 (5.21)=+0.77718547 lead(5.22)=+0.768597
grips (body 1, body 2) = (linear, linear):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.648953e-01; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.648953e-01; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
grips (body 1, body 2) = (square, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.650774e-01; rho_bond,1 num=+1.14386952 (5.21)=+1.14386951 lead(5.22)=+1.118749; rho_bond,2 num=+1.12696288 (5.21)=+1.12696287 lead(5.22)=+1.118749
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.650774e-01; rho_bond,1 num=+1.14386953 (5.21)=+1.14386951 lead(5.22)=+1.118749; rho_bond,2 num=+1.12696288 (5.21)=+1.12696287 lead(5.22)=+1.118749
d=2: Gamma(x)=1.845431, w(x)=0.252020, Sigma(x)=0.116884, phi_a(x)=-1.653413; a_mark=+0.430726
grip=square k=0 sigma=+1: rho_mark numerical=1.46112323 predicted (5.20)=1.46112323 leading (4F0F2/F1^2)(1+a_mark)=1.430726 [block at s=1 by (5.10): Y]
grip=square k=0 sigma=-1: rho_mark numerical=1.46112323 predicted (5.20)=1.46112323 leading (4F0F2/F1^2)(1+a_mark)=1.430726 [block at s=1 by (5.10): Y]
grip=square k=1 sigma=+1: rho_mark numerical=1.46112322 predicted (5.20)=1.46112323 leading (4F0F2/F1^2)(1+a_mark)=1.430726 [block at s=1 by (5.10): Y]
grip=square k=1 sigma=-1: rho_mark numerical=1.46112322 predicted (5.20)=1.46112323 leading (4F0F2/F1^2)(1+a_mark)=1.430726 [block at s=1 by (5.10): Y]
grip=nonsquare k=0 sigma=+1: rho_mark numerical=0.48044164 predicted (5.20)=0.48044166 leading (4F0F2/F1^2)(1+a_mark)=0.476909 [block at s=1 by (5.10): Y]
grip=nonsquare k=0 sigma=-1: rho_mark numerical=0.48044164 predicted (5.20)=0.48044166 leading (4F0F2/F1^2)(1+a_mark)=0.476909 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=+1: rho_mark numerical=0.48044166 predicted (5.20)=0.48044166 leading (4F0F2/F1^2)(1+a_mark)=0.476909 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=-1: rho_mark numerical=0.48044166 predicted (5.20)=0.48044166 leading (4F0F2/F1^2)(1+a_mark)=0.476909 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=+1: rho_mark numerical=0.72322553 predicted (5.20)=0.72322555 leading (4F0F2/F1^2)(1+a_mark)=0.715363 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=-1: rho_mark numerical=0.72322553 predicted (5.20)=0.72322555 leading (4F0F2/F1^2)(1+a_mark)=0.715363 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=+1: rho_mark numerical=0.72322554 predicted (5.20)=0.72322555 leading (4F0F2/F1^2)(1+a_mark)=0.715363 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=-1: rho_mark numerical=0.72322554 predicted (5.20)=0.72322555 leading (4F0F2/F1^2)(1+a_mark)=0.715363 [block at s=1 by (5.10): Y]
grip=linear k=0 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=0 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
bond, pair separation 2, source at [4, 1, 0]: m_r=0.535514, a_bond=+0.349449
grips (body 1, body 2) = (square, square):
(k,sigma)=(0,+1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-5.946991e-01; rho_bond,1 num=+0.90447490 (5.21)=+0.90447492 lead(5.22)=+0.884964; rho_bond,2 num=+0.90447490 (5.21)=+0.90447492 lead(5.22)=+0.884964
(k,sigma)=(0,-1),(0,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-5.946991e-01; rho_bond,1 num=+0.90447490 (5.21)=+0.90447492 lead(5.22)=+0.884964; rho_bond,2 num=+0.90447490 (5.21)=+0.90447492 lead(5.22)=+0.884964
(k,sigma)=(1,+1),(1,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-1.338073e+00; rho_bond,1 num=+0.90447484 (5.21)=+0.90447492 lead(5.22)=+0.884964; rho_bond,2 num=+0.90447484 (5.21)=+0.90447492 lead(5.22)=+0.884964
(k,sigma)=(1,-1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-1.338073e+00; rho_bond,1 num=+0.90447484 (5.21)=+0.90447492 lead(5.22)=+0.884964; rho_bond,2 num=+0.90447484 (5.21)=+0.90447492 lead(5.22)=+0.884964
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.920486e-01; rho_bond,1 num=+0.90447494 (5.21)=+0.90447492 lead(5.22)=+0.884964; rho_bond,2 num=+0.90447495 (5.21)=+0.90447492 lead(5.22)=+0.884964
grips (body 1, body 2) = (nonsquare, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.915329e-01; rho_bond,1 num=+0.29719994 (5.21)=+0.29719992 lead(5.22)=+0.294988; rho_bond,2 num=+0.29719994 (5.21)=+0.29719992 lead(5.22)=+0.294988
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.915329e-01; rho_bond,1 num=+0.29719989 (5.21)=+0.29719992 lead(5.22)=+0.294988; rho_bond,2 num=+0.29719989 (5.21)=+0.29719992 lead(5.22)=+0.294988
grips (body 1, body 2) = (eight, eight):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.916135e-01; rho_bond,1 num=+0.44743438 (5.21)=+0.44743436 lead(5.22)=+0.442482; rho_bond,2 num=+0.44743438 (5.21)=+0.44743436 lead(5.22)=+0.442482
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.916135e-01; rho_bond,1 num=+0.44743437 (5.21)=+0.44743436 lead(5.22)=+0.442482; rho_bond,2 num=+0.44743437 (5.21)=+0.44743436 lead(5.22)=+0.442482
grips (body 1, body 2) = (linear, linear):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.914684e-01; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.914684e-01; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
grips (body 1, body 2) = (square, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.916618e-01; rho_bond,1 num=+0.63247077 (5.21)=+0.63247074 lead(5.22)=+0.618557; rho_bond,2 num=+0.62310525 (5.21)=+0.62310522 lead(5.22)=+0.618557
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.916618e-01; rho_bond,1 num=+0.63247074 (5.21)=+0.63247074 lead(5.22)=+0.618557; rho_bond,2 num=+0.62310521 (5.21)=+0.62310522 lead(5.22)=+0.618557
d=3: Gamma(x)=1.871738, w(x)=0.252479, Sigma(x)=0.118775, phi_a(x)=-1.663309; a_mark=+0.129224
grip=square k=0 sigma=+1: rho_mark numerical=1.15324262 predicted (5.20)=1.15324253 leading (4F0F2/F1^2)(1+a_mark)=1.129224 [block at s=1 by (5.10): Y]
grip=square k=0 sigma=-1: rho_mark numerical=1.15324262 predicted (5.20)=1.15324253 leading (4F0F2/F1^2)(1+a_mark)=1.129224 [block at s=1 by (5.10): Y]
grip=square k=1 sigma=+1: rho_mark numerical=1.15324251 predicted (5.20)=1.15324253 leading (4F0F2/F1^2)(1+a_mark)=1.129224 [block at s=1 by (5.10): Y]
grip=square k=1 sigma=-1: rho_mark numerical=1.15324251 predicted (5.20)=1.15324253 leading (4F0F2/F1^2)(1+a_mark)=1.129224 [block at s=1 by (5.10): Y]
grip=nonsquare k=0 sigma=+1: rho_mark numerical=0.37919960 predicted (5.20)=0.37919966 leading (4F0F2/F1^2)(1+a_mark)=0.376408 [block at s=1 by (5.10): Y]
grip=nonsquare k=0 sigma=-1: rho_mark numerical=0.37919960 predicted (5.20)=0.37919966 leading (4F0F2/F1^2)(1+a_mark)=0.376408 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=+1: rho_mark numerical=0.37919966 predicted (5.20)=0.37919966 leading (4F0F2/F1^2)(1+a_mark)=0.376408 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=-1: rho_mark numerical=0.37919966 predicted (5.20)=0.37919966 leading (4F0F2/F1^2)(1+a_mark)=0.376408 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=+1: rho_mark numerical=0.57082472 predicted (5.20)=0.57082469 leading (4F0F2/F1^2)(1+a_mark)=0.564612 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=-1: rho_mark numerical=0.57082472 predicted (5.20)=0.57082469 leading (4F0F2/F1^2)(1+a_mark)=0.564612 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=+1: rho_mark numerical=0.57082468 predicted (5.20)=0.57082469 leading (4F0F2/F1^2)(1+a_mark)=0.564612 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=-1: rho_mark numerical=0.57082468 predicted (5.20)=0.57082469 leading (4F0F2/F1^2)(1+a_mark)=0.564612 [block at s=1 by (5.10): Y]
grip=linear k=0 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=0 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
bond, pair separation 2, source at [5, 1, 0]: m_r=0.463091, a_bond=+0.105633
grips (body 1, body 2) = (square, square):
(k,sigma)=(0,+1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-6.010252e-01; rho_bond,1 num=+0.58126935 (5.21)=+0.58126936 lead(5.22)=+0.568724; rho_bond,2 num=+0.58126935 (5.21)=+0.58126936 lead(5.22)=+0.568724
(k,sigma)=(0,-1),(0,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-6.010252e-01; rho_bond,1 num=+0.58126935 (5.21)=+0.58126936 lead(5.22)=+0.568724; rho_bond,2 num=+0.58126935 (5.21)=+0.58126936 lead(5.22)=+0.568724
(k,sigma)=(1,+1),(1,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-1.352307e+00; rho_bond,1 num=+0.58126936 (5.21)=+0.58126936 lead(5.22)=+0.568724; rho_bond,2 num=+0.58126936 (5.21)=+0.58126936 lead(5.22)=+0.568724
(k,sigma)=(1,-1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-1.352307e+00; rho_bond,1 num=+0.58126936 (5.21)=+0.58126936 lead(5.22)=+0.568724; rho_bond,2 num=+0.58126936 (5.21)=+0.58126936 lead(5.22)=+0.568724
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+9.015378e-01; rho_bond,1 num=+0.58126939 (5.21)=+0.58126936 lead(5.22)=+0.568724; rho_bond,2 num=+0.58126939 (5.21)=+0.58126936 lead(5.22)=+0.568724
grips (body 1, body 2) = (nonsquare, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+9.010112e-01; rho_bond,1 num=+0.19099703 (5.21)=+0.19099705 lead(5.22)=+0.189575; rho_bond,2 num=+0.19099703 (5.21)=+0.19099705 lead(5.22)=+0.189575
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+9.010112e-01; rho_bond,1 num=+0.19099703 (5.21)=+0.19099705 lead(5.22)=+0.189575; rho_bond,2 num=+0.19099703 (5.21)=+0.19099705 lead(5.22)=+0.189575
grips (body 1, body 2) = (eight, eight):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+9.010935e-01; rho_bond,1 num=+0.28754659 (5.21)=+0.28754658 lead(5.22)=+0.284362; rho_bond,2 num=+0.28754659 (5.21)=+0.28754658 lead(5.22)=+0.284362
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+9.010935e-01; rho_bond,1 num=+0.28754658 (5.21)=+0.28754658 lead(5.22)=+0.284362; rho_bond,2 num=+0.28754658 (5.21)=+0.28754658 lead(5.22)=+0.284362
grips (body 1, body 2) = (linear, linear):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+9.009454e-01; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+9.009454e-01; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
grips (body 1, body 2) = (square, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+9.011429e-01; rho_bond,1 num=+0.39459878 (5.21)=+0.39459879 lead(5.22)=+0.385913; rho_bond,2 num=+0.38875154 (5.21)=+0.38875155 lead(5.22)=+0.385913
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+9.011429e-01; rho_bond,1 num=+0.39459872 (5.21)=+0.39459879 lead(5.22)=+0.385913; rho_bond,2 num=+0.38875148 (5.21)=+0.38875155 lead(5.22)=+0.385913
max relative deviation over this lattice: C1, Cmark, W1, Wmark from (5.18): 1.50e-08; rho_mark from (5.20): 1.37e-07; B, W_bond from (5.19): 7.32e-08; rho_bond,i from (5.21): 1.71e-07 (absolute deviation where the prediction is 0)
=== cubic torus L=12, anchor h=0; body placements x=h+d e, x'=h+(d+1) e, source at h+(d+1+1) e
d=1: Gamma(x)=1.761712, w(x)=0.239778, Sigma(x)=0.083083, phi_a(x)=-1.609939; a_mark=+0.915896
grip=square k=0 sigma=+1: rho_mark numerical=1.95528543 predicted (5.20)=1.95528538 leading (4F0F2/F1^2)(1+a_mark)=1.915896 [block at s=1 by (5.10): Y]
grip=square k=0 sigma=-1: rho_mark numerical=1.95528543 predicted (5.20)=1.95528538 leading (4F0F2/F1^2)(1+a_mark)=1.915896 [block at s=1 by (5.10): Y]
grip=square k=1 sigma=+1: rho_mark numerical=1.95528541 predicted (5.20)=1.95528538 leading (4F0F2/F1^2)(1+a_mark)=1.915896 [block at s=1 by (5.10): Y]
grip=square k=1 sigma=-1: rho_mark numerical=1.95528541 predicted (5.20)=1.95528538 leading (4F0F2/F1^2)(1+a_mark)=1.915896 [block at s=1 by (5.10): Y]
grip=nonsquare k=0 sigma=+1: rho_mark numerical=0.64316301 predicted (5.20)=0.64316296 leading (4F0F2/F1^2)(1+a_mark)=0.638632 [block at s=1 by (5.10): Y]
grip=nonsquare k=0 sigma=-1: rho_mark numerical=0.64316301 predicted (5.20)=0.64316296 leading (4F0F2/F1^2)(1+a_mark)=0.638632 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=+1: rho_mark numerical=0.64316298 predicted (5.20)=0.64316296 leading (4F0F2/F1^2)(1+a_mark)=0.638632 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=-1: rho_mark numerical=0.64316298 predicted (5.20)=0.64316296 leading (4F0F2/F1^2)(1+a_mark)=0.638632 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=+1: rho_mark numerical=0.96805669 predicted (5.20)=0.96805671 leading (4F0F2/F1^2)(1+a_mark)=0.957948 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=-1: rho_mark numerical=0.96805669 predicted (5.20)=0.96805671 leading (4F0F2/F1^2)(1+a_mark)=0.957948 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=+1: rho_mark numerical=0.96805668 predicted (5.20)=0.96805671 leading (4F0F2/F1^2)(1+a_mark)=0.957948 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=-1: rho_mark numerical=0.96805668 predicted (5.20)=0.96805671 leading (4F0F2/F1^2)(1+a_mark)=0.957948 [block at s=1 by (5.10): Y]
grip=linear k=0 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=0 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
bond, pair separation 2, source at [3, 1, 0]: m_r=0.714916, a_bond=+0.828812
grips (body 1, body 2) = (square, square):
(k,sigma)=(0,+1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-5.769264e-01; rho_bond,1 num=+1.57666083 (5.21)=+1.57666085 lead(5.22)=+1.543728; rho_bond,2 num=+1.57666083 (5.21)=+1.57666085 lead(5.22)=+1.543728
(k,sigma)=(0,-1),(0,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-5.769264e-01; rho_bond,1 num=+1.57666083 (5.21)=+1.57666085 lead(5.22)=+1.543728; rho_bond,2 num=+1.57666083 (5.21)=+1.57666085 lead(5.22)=+1.543728
(k,sigma)=(1,+1),(1,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-1.298084e+00; rho_bond,1 num=+1.57666064 (5.21)=+1.57666085 lead(5.22)=+1.543728; rho_bond,2 num=+1.57666064 (5.21)=+1.57666085 lead(5.22)=+1.543728
(k,sigma)=(1,-1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-1.298084e+00; rho_bond,1 num=+1.57666064 (5.21)=+1.57666085 lead(5.22)=+1.543728; rho_bond,2 num=+1.57666064 (5.21)=+1.57666085 lead(5.22)=+1.543728
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.653896e-01; rho_bond,1 num=+1.57666079 (5.21)=+1.57666085 lead(5.22)=+1.543728; rho_bond,2 num=+1.57666080 (5.21)=+1.57666085 lead(5.22)=+1.543728
grips (body 1, body 2) = (nonsquare, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.651221e-01; rho_bond,1 num=+0.51827678 (5.21)=+0.51827670 lead(5.22)=+0.514576; rho_bond,2 num=+0.51827678 (5.21)=+0.51827670 lead(5.22)=+0.514576
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.651220e-01; rho_bond,1 num=+0.51827663 (5.21)=+0.51827670 lead(5.22)=+0.514576; rho_bond,2 num=+0.51827663 (5.21)=+0.51827670 lead(5.22)=+0.514576
grips (body 1, body 2) = (eight, eight):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.651638e-01; rho_bond,1 num=+0.78016746 (5.21)=+0.78016750 lead(5.22)=+0.771864; rho_bond,2 num=+0.78016746 (5.21)=+0.78016750 lead(5.22)=+0.771864
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.651639e-01; rho_bond,1 num=+0.78016754 (5.21)=+0.78016750 lead(5.22)=+0.771864; rho_bond,2 num=+0.78016754 (5.21)=+0.78016750 lead(5.22)=+0.771864
grips (body 1, body 2) = (linear, linear):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.650886e-01; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.650886e-01; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
grips (body 1, body 2) = (square, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.651889e-01; rho_bond,1 num=+1.14568944 (5.21)=+1.14568942 lead(5.22)=+1.121494; rho_bond,2 num=+1.12947121 (5.21)=+1.12947119 lead(5.22)=+1.121494
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.651889e-01; rho_bond,1 num=+1.14568945 (5.21)=+1.14568942 lead(5.22)=+1.121494; rho_bond,2 num=+1.12947123 (5.21)=+1.12947119 lead(5.22)=+1.121494
d=2: Gamma(x)=1.847457, w(x)=0.241121, Sigma(x)=0.086833, phi_a(x)=-1.659206; a_mark=+0.465154
grip=square k=0 sigma=+1: rho_mark numerical=1.49539019 predicted (5.20)=1.49539026 leading (4F0F2/F1^2)(1+a_mark)=1.465154 [block at s=1 by (5.10): Y]
grip=square k=0 sigma=-1: rho_mark numerical=1.49539019 predicted (5.20)=1.49539026 leading (4F0F2/F1^2)(1+a_mark)=1.465154 [block at s=1 by (5.10): Y]
grip=square k=1 sigma=+1: rho_mark numerical=1.49539022 predicted (5.20)=1.49539026 leading (4F0F2/F1^2)(1+a_mark)=1.465154 [block at s=1 by (5.10): Y]
grip=square k=1 sigma=-1: rho_mark numerical=1.49539022 predicted (5.20)=1.49539026 leading (4F0F2/F1^2)(1+a_mark)=1.465154 [block at s=1 by (5.10): Y]
grip=nonsquare k=0 sigma=+1: rho_mark numerical=0.49186198 predicted (5.20)=0.49186203 leading (4F0F2/F1^2)(1+a_mark)=0.488385 [block at s=1 by (5.10): Y]
grip=nonsquare k=0 sigma=-1: rho_mark numerical=0.49186198 predicted (5.20)=0.49186203 leading (4F0F2/F1^2)(1+a_mark)=0.488385 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=+1: rho_mark numerical=0.49186203 predicted (5.20)=0.49186203 leading (4F0F2/F1^2)(1+a_mark)=0.488385 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=-1: rho_mark numerical=0.49186203 predicted (5.20)=0.49186203 leading (4F0F2/F1^2)(1+a_mark)=0.488385 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=+1: rho_mark numerical=0.74033528 predicted (5.20)=0.74033546 leading (4F0F2/F1^2)(1+a_mark)=0.732577 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=-1: rho_mark numerical=0.74033528 predicted (5.20)=0.74033546 leading (4F0F2/F1^2)(1+a_mark)=0.732577 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=+1: rho_mark numerical=0.74033547 predicted (5.20)=0.74033546 leading (4F0F2/F1^2)(1+a_mark)=0.732577 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=-1: rho_mark numerical=0.74033547 predicted (5.20)=0.74033546 leading (4F0F2/F1^2)(1+a_mark)=0.732577 [block at s=1 by (5.10): Y]
grip=linear k=0 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=0 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
bond, pair separation 2, source at [4, 1, 0]: m_r=0.551950, a_bond=+0.387599
grips (body 1, body 2) = (square, square):
(k,sigma)=(0,+1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-5.951142e-01; rho_bond,1 num=+0.95961375 (5.21)=+0.95961365 lead(5.22)=+0.939549; rho_bond,2 num=+0.95961375 (5.21)=+0.95961365 lead(5.22)=+0.939549
(k,sigma)=(0,-1),(0,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-5.951142e-01; rho_bond,1 num=+0.95961375 (5.21)=+0.95961365 lead(5.22)=+0.939549; rho_bond,2 num=+0.95961375 (5.21)=+0.95961365 lead(5.22)=+0.939549
(k,sigma)=(1,+1),(1,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-1.339007e+00; rho_bond,1 num=+0.95961348 (5.21)=+0.95961365 lead(5.22)=+0.939549; rho_bond,2 num=+0.95961348 (5.21)=+0.95961365 lead(5.22)=+0.939549
(k,sigma)=(1,-1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-1.339007e+00; rho_bond,1 num=+0.95961348 (5.21)=+0.95961365 lead(5.22)=+0.939549; rho_bond,2 num=+0.95961348 (5.21)=+0.95961365 lead(5.22)=+0.939549
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.926713e-01; rho_bond,1 num=+0.95961365 (5.21)=+0.95961365 lead(5.22)=+0.939549; rho_bond,2 num=+0.95961365 (5.21)=+0.95961365 lead(5.22)=+0.939549
grips (body 1, body 2) = (nonsquare, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.923868e-01; rho_bond,1 num=+0.31543824 (5.21)=+0.31543832 lead(5.22)=+0.313183; rho_bond,2 num=+0.31543824 (5.21)=+0.31543832 lead(5.22)=+0.313183
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.923868e-01; rho_bond,1 num=+0.31543838 (5.21)=+0.31543832 lead(5.22)=+0.313183; rho_bond,2 num=+0.31543838 (5.21)=+0.31543832 lead(5.22)=+0.313183
grips (body 1, body 2) = (eight, eight):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.924312e-01; rho_bond,1 num=+0.47483444 (5.21)=+0.47483449 lead(5.22)=+0.469775; rho_bond,2 num=+0.47483444 (5.21)=+0.47483449 lead(5.22)=+0.469775
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.924312e-01; rho_bond,1 num=+0.47483454 (5.21)=+0.47483449 lead(5.22)=+0.469775; rho_bond,2 num=+0.47483454 (5.21)=+0.47483449 lead(5.22)=+0.469775
grips (body 1, body 2) = (linear, linear):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.923512e-01; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.923512e-01; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
grips (body 1, body 2) = (square, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.924579e-01; rho_bond,1 num=+0.66951541 (5.21)=+0.66951546 lead(5.22)=+0.655359; rho_bond,2 num=+0.66002531 (5.21)=+0.66002536 lead(5.22)=+0.655359
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+8.924579e-01; rho_bond,1 num=+0.66951558 (5.21)=+0.66951546 lead(5.22)=+0.655359; rho_bond,2 num=+0.66002547 (5.21)=+0.66002536 lead(5.22)=+0.655359
d=3: Gamma(x)=1.877051, w(x)=0.241437, Sigma(x)=0.088031, phi_a(x)=-1.676908; a_mark=+0.234536
grip=square k=0 sigma=+1: rho_mark numerical=1.26003887 predicted (5.20)=1.26003894 leading (4F0F2/F1^2)(1+a_mark)=1.234536 [block at s=1 by (5.10): Y]
grip=square k=0 sigma=-1: rho_mark numerical=1.26003887 predicted (5.20)=1.26003894 leading (4F0F2/F1^2)(1+a_mark)=1.234536 [block at s=1 by (5.10): Y]
grip=square k=1 sigma=+1: rho_mark numerical=1.26003895 predicted (5.20)=1.26003894 leading (4F0F2/F1^2)(1+a_mark)=1.234536 [block at s=1 by (5.10): Y]
grip=square k=1 sigma=-1: rho_mark numerical=1.26003895 predicted (5.20)=1.26003894 leading (4F0F2/F1^2)(1+a_mark)=1.234536 [block at s=1 by (5.10): Y]
grip=nonsquare k=0 sigma=+1: rho_mark numerical=0.41444475 predicted (5.20)=0.41444465 leading (4F0F2/F1^2)(1+a_mark)=0.411512 [block at s=1 by (5.10): Y]
grip=nonsquare k=0 sigma=-1: rho_mark numerical=0.41444475 predicted (5.20)=0.41444465 leading (4F0F2/F1^2)(1+a_mark)=0.411512 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=+1: rho_mark numerical=0.41444467 predicted (5.20)=0.41444465 leading (4F0F2/F1^2)(1+a_mark)=0.411512 [block at s=1 by (5.10): Y]
grip=nonsquare k=1 sigma=-1: rho_mark numerical=0.41444467 predicted (5.20)=0.41444465 leading (4F0F2/F1^2)(1+a_mark)=0.411512 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=+1: rho_mark numerical=0.62381146 predicted (5.20)=0.62381137 leading (4F0F2/F1^2)(1+a_mark)=0.617268 [block at s=1 by (5.10): Y]
grip=eight k=0 sigma=-1: rho_mark numerical=0.62381146 predicted (5.20)=0.62381137 leading (4F0F2/F1^2)(1+a_mark)=0.617268 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=+1: rho_mark numerical=0.62381145 predicted (5.20)=0.62381137 leading (4F0F2/F1^2)(1+a_mark)=0.617268 [block at s=1 by (5.10): Y]
grip=eight k=1 sigma=-1: rho_mark numerical=0.62381145 predicted (5.20)=0.62381137 leading (4F0F2/F1^2)(1+a_mark)=0.617268 [block at s=1 by (5.10): Y]
grip=linear k=0 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=0 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=+1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
grip=linear k=1 sigma=-1: rho_mark numerical=-0.000e+00 predicted=0 [block at s=1 by (5.10): Y]
bond, pair separation 2, source at [5, 1, 0]: m_r=0.497556, a_bond=+0.206349
grips (body 1, body 2) = (square, square):
(k,sigma)=(0,+1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-6.024264e-01; rho_bond,1 num=+0.71894417 (5.21)=+0.71894422 lead(5.22)=+0.703905; rho_bond,2 num=+0.71894417 (5.21)=+0.71894422 lead(5.22)=+0.703905
(k,sigma)=(0,-1),(0,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-6.024264e-01; rho_bond,1 num=+0.71894417 (5.21)=+0.71894422 lead(5.22)=+0.703905; rho_bond,2 num=+0.71894417 (5.21)=+0.71894422 lead(5.22)=+0.703905
(k,sigma)=(1,+1),(1,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-1.355459e+00; rho_bond,1 num=+0.71894419 (5.21)=+0.71894422 lead(5.22)=+0.703905; rho_bond,2 num=+0.71894419 (5.21)=+0.71894422 lead(5.22)=+0.703905
(k,sigma)=(1,-1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=-1.355459e+00; rho_bond,1 num=+0.71894419 (5.21)=+0.71894422 lead(5.22)=+0.703905; rho_bond,2 num=+0.71894419 (5.21)=+0.71894422 lead(5.22)=+0.703905
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+9.036396e-01; rho_bond,1 num=+0.71894432 (5.21)=+0.71894422 lead(5.22)=+0.703905; rho_bond,2 num=+0.71894432 (5.21)=+0.71894422 lead(5.22)=+0.703905
grips (body 1, body 2) = (nonsquare, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+9.033481e-01; rho_bond,1 num=+0.23632562 (5.21)=+0.23632557 lead(5.22)=+0.234635; rho_bond,2 num=+0.23632562 (5.21)=+0.23632557 lead(5.22)=+0.234635
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+9.033481e-01; rho_bond,1 num=+0.23632566 (5.21)=+0.23632557 lead(5.22)=+0.234635; rho_bond,2 num=+0.23632566 (5.21)=+0.23632557 lead(5.22)=+0.234635
grips (body 1, body 2) = (eight, eight):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+9.033937e-01; rho_bond,1 num=+0.35574559 (5.21)=+0.35574539 lead(5.22)=+0.351953; rho_bond,2 num=+0.35574559 (5.21)=+0.35574539 lead(5.22)=+0.351953
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+9.033937e-01; rho_bond,1 num=+0.35574536 (5.21)=+0.35574539 lead(5.22)=+0.351953; rho_bond,2 num=+0.35574536 (5.21)=+0.35574539 lead(5.22)=+0.351953
grips (body 1, body 2) = (linear, linear):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+9.033117e-01; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+9.033117e-01; rho_bond,1 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000; rho_bond,2 num=+0.00000000 (5.21)=+0.00000000 lead(5.22)=+0.000000
grips (body 1, body 2) = (square, nonsquare):
(k,sigma)=(0,+1),(1,-1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+9.034210e-01; rho_bond,1 num=+0.48968334 (5.21)=+0.48968329 lead(5.22)=+0.479324; rho_bond,2 num=+0.48273812 (5.21)=+0.48273808 lead(5.22)=+0.479324
(k,sigma)=(1,-1),(0,+1): pair block at s=1 (both placements) lambda_min>0: Y/Y, (5.26): Y/Y; B=+9.034210e-01; rho_bond,1 num=+0.48968320 (5.21)=+0.48968329 lead(5.22)=+0.479324; rho_bond,2 num=+0.48273799 (5.21)=+0.48273808 lead(5.22)=+0.479324
max relative deviation over this lattice: C1, Cmark, W1, Wmark from (5.18): 1.93e-08; rho_mark from (5.20): 2.47e-07; B, W_bond from (5.19): 7.74e-08; rho_bond,i from (5.21): 5.76e-07 (absolute deviation where the prediction is 0)
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