07-ilang-weight / 06-inertia-of-pair
06inertia of pairverified
Determines the inertia of a pair bound through the field against the inertias of its bodies and its cost, and whether weight over inertia of the pair equals that of a single body.
check_inertia.py20 KB · raw
"""Check of item 2 of 07-ilang-weight/06-inertia-of-pair (and of the numbers of items 1, 3, 5).
Two Model D bodies (one clock level each: half-gap M_i, branch sigma_i), N_f = 1 field object
on a ring of L places, couplings (a) and (c). The exact sector contract H(P) of bodies and
field (body 1 at place 0 as representative) is continued to real P; the curvature of the
exact bound level at P = 0 is compared with (6.6) evaluated on the band contract at order
tau^2 (second-order Schrieffer-Wolff on the tau = 0 levels), and with mu_1 + mu_2.
Also: the ring remainder against its bound (6.16), the inverse-expansion quantities of B2 (x_bar, A_0),
the gaps delta_1, delta of R1 from the held-sector spectra (Step 7), and the leading-order sign rule
(6.17) on a long line chain. Only the selected level (one-signed, s_beta = +1) is treated.
Branch basis (|+>, i|->): the motion term is real, -i sigma_1 -> J.
"""
import json
from pathlib import Path
import numpy as np
OUT = Path(__file__).parent / "output"
J = np.array([[0.0, 1.0], [-1.0, 0.0]])
SG = np.array([1.0, -1.0]) # branch index 0: sigma = +1, 1: sigma = -1
def bidx(sig):
return 0 if sig > 0 else 1
def ground(h):
w, v = np.linalg.eigh(h)
g = v[:, 0]
return w[0], g * np.sign(g.sum())
def field_hop(L, tf):
h = np.zeros((L, L))
for y in range(L):
h[y, (y + 1) % L] = h[(y + 1) % L, y] = -tf
return h
# ---------------------------------------------------------------- one body + field, sector
def body_sector(L, tf, b, cpl):
cc = 1.0 if cpl == "c" else 0.0
n = 2 * L
ix = lambda s, y: s * L + y
H0 = np.zeros((n, n)); B = np.zeros((n, n))
hf = field_hop(L, tf)
for s in range(2):
H0[s * L:(s + 1) * L, s * L:(s + 1) * L] += hf
for y in range(L):
a = ix(s, y)
H0[a, a] += SG[s] * b["M"] * (1 + b["gam"] * (y == 0))
f = 1 + cc * b["gam"] * 0.5 * ((y == 0) + (y == 1))
for sp in range(2): # right hop 0 -> 1, frame shift y -> y-1, phase e^{-iP}
B[ix(sp, (y - 1) % L), a] += 0.5 * b["tau"] * J[sp, s] * f
return H0, B
def curvature(H0, B, sel_vec, mode):
"""E''(0) of the eigenvector of H(0)=H0+B+B^T selected by overlap with sel_vec (G-weight)."""
H = H0 + B + B.T
w, v = np.linalg.eigh(H)
gw = np.abs(sel_vec.T @ v) ** 2 if sel_vec.ndim == 1 else (np.abs(sel_vec.T @ v) ** 2).sum(0)
cand = np.where(gw > 0.5)[0]
k = cand[np.argmin(w[cand])] if mode == "low" else cand[np.argmax(w[cand])]
H1 = -1j * (B - B.T)
H2 = -(B + B.T)
x = v[:, k]
d = w[k] - w
d[k] = np.inf
m1 = v.conj().T @ (H1 @ x)
E2 = (x @ H2 @ x) + 2 * np.sum(np.abs(m1) ** 2 / d)
return w[k], E2.real, v[:, k], gw[k]
def sw_scalar(H0, V, gvec):
"""Second-order SW energy of the single G state gvec (V complex allowed)."""
w, U = np.linalg.eigh(H0)
a = np.argmax(np.abs(U.T @ gvec))
Vt = U.T @ V @ U
d = w[a] - w
d[a] = np.inf
return w[a] + np.sum(np.abs(Vt[:, a]) ** 2 / d)
def body_data(L, tf, b, cpl):
H0, B = body_sector(L, tf, b, cpl)
s = bidx(b["sig"])
hf = field_hop(L, tf); hf0 = hf.copy(); hf0[0, 0] += b["sig"] * b["gam"] * b["M"]
e0, Om = ground(hf0)
g = np.zeros(2 * L); g[s * L:(s + 1) * L] = Om
E0, E2, _, gw = curvature(H0, B, g, "low" if b["sig"] > 0 else "high")
# band contract (order tau^2): kappa_bar from P = 0 and P = pi/2
Hs = lambda P: sw_scalar(H0, np.exp(-1j * P) * B + np.exp(1j * P) * B.T, g)
kbar = (Hs(0.0) - Hs(np.pi / 2)).real / 4
eps = lambda gg: np.linalg.eigvalsh(hf + np.diag([gg] + [0] * (L - 1)))[0]
R = b["M"] + 0.5 * (eps(b["gam"] * b["M"]) - eps(-b["gam"] * b["M"]))
c2 = b["tau"] ** 2 * (1 + (b["gam"] / L if cpl == "c" else 0.0)) ** 2
return dict(E0=E0, mu=1 / E2, mubar=-1 / (8 * kbar), kbar=kbar, R=R, c2=c2, gw=gw)
# ---------------------------------------------------------------- two bodies + field, sector
def pair_sector(L, tf, b1, b2, cpl):
cc = 1.0 if cpl == "c" else 0.0
n = 4 * L * L
ix = lambda s1, r, s2, y: ((s1 * L + r) * 2 + s2) * L + y
H0 = np.zeros((n, n)); B1 = np.zeros((n, n)); V2 = np.zeros((n, n))
for s1 in range(2):
for r in range(L):
for s2 in range(2):
for y in range(L):
a = ix(s1, r, s2, y)
H0[a, a] += SG[s1] * b1["M"] * (1 + b1["gam"] * (y == 0)) \
+ SG[s2] * b2["M"] * (1 + b2["gam"] * (y == r))
H0[ix(s1, r, s2, (y + 1) % L), a] += -tf
H0[ix(s1, r, s2, (y - 1) % L), a] += -tf
for t in range(2):
f = 1 + cc * b2["gam"] * 0.5 * ((y == r) + (y == (r + 1) % L))
V2[ix(s1, (r + 1) % L, t, y), a] += 0.5 * b2["tau"] * J[t, s2] * f
f = 1 + cc * b2["gam"] * 0.5 * ((y == (r - 1) % L) + (y == r))
V2[ix(s1, (r - 1) % L, t, y), a] += 0.5 * b2["tau"] * J[s2, t] * f
f = 1 + cc * b1["gam"] * 0.5 * ((y == 0) + (y == 1))
B1[ix(t, (r - 1) % L, s2, (y - 1) % L), a] += 0.5 * b1["tau"] * J[t, s1] * f
return H0, B1, V2, ix
def pair_block(L, tf, b1, b2, cpl, par):
"""Restrict to the block of (-1)^r s1 s2 containing the band G (branches sig_i, r = par mod 2).
Returns H0, B1, V2 on the block, the tau=0 eigenbasis U0 (block by block in (s1, r, s2)),
its eigenvalues w0, the columns gi of U0 that form G (field ground, positive), and rs."""
H0, B1, V2, ix = pair_sector(L, tf, b1, b2, cpl)
pi0 = (-1) ** par * b1["sig"] * b2["sig"]
trip = [(s1, r, s2) for s1 in range(2) for r in range(L) for s2 in range(2)
if (-1) ** r * SG[s1] * SG[s2] == pi0]
keep = np.array([ix(s1, r, s2, y) for (s1, r, s2) in trip for y in range(L)])
sub = lambda A: A[np.ix_(keep, keep)]
H0b, B1b, V2b = sub(H0), sub(B1), sub(V2)
n = len(keep)
U0 = np.zeros((n, n)); w0 = np.zeros(n)
s1g, s2g = bidx(b1["sig"]), bidx(b2["sig"])
rs, gi = [], []
for t, (s1, r, s2) in enumerate(trip):
sl = slice(t * L, (t + 1) * L)
w, v = np.linalg.eigh(H0b[sl, sl])
v[:, 0] *= np.sign(v[:, 0].sum())
U0[sl, sl] = v; w0[sl] = w
if (s1, s2) == (s1g, s2g):
rs.append(r); gi.append(t * L)
order = np.argsort(rs)
rs = [rs[i] for i in order]; gi = np.array([gi[i] for i in order])
return H0b, B1b, V2b, U0, w0, gi, rs
def band_contract(U0, w0, gi, V):
"""Second-order SW band contract on the G columns gi of the tau=0 eigenbasis U0."""
mask = np.ones(len(w0), bool); mask[gi] = False
Vgc = U0[:, gi].T @ V @ U0[:, mask]
Ea, Ec = w0[gi], w0[mask]
A = Vgc / (Ea[:, None] - Ec[None, :])
Heff = np.diag(Ea).astype(complex) + 0.5 * (A @ Vgc.conj().T + Vgc @ A.conj().T)
return Heff, Ea
def ell(r, L):
r = np.mod(r, L)
return (L * L - 1) / (12 * L) - r * (L - r) / (2 * L)
def band_data(L, tf, b1, b2, cpl, par, mode):
"""Band contract at order tau^2 at P = 0: relative level f, bonds b, kappa_i(rho), m(rho)."""
H0, B1, V2, U0, w0, gi, rs = pair_block(L, tf, b1, b2, cpl, par)
K1, _ = band_contract(U0, w0, gi, B1 + B1.T)
K2, Ea = band_contract(U0, w0, gi, V2)
K1 = (K1 - np.diag(Ea)).real; K2 = (K2 - np.diag(Ea)).real
w, v = np.linalg.eigh(np.diag(Ea) + K1 + K2)
k = 0 if mode == "low" else -1
f = v[:, k]
nr = len(rs)
nb = lambda A: np.array([A[j, (j + 1) % nr] for j in range(nr)]) # bond (r_j, r_j + 2)
k1, k2 = nb(K1), nb(K2)
b = np.array([f[j] * f[(j + 1) % nr] for j in range(nr)])
rho = np.array(rs) + 1
return dict(H0=H0, B1=B1, V2=V2, U0=U0, w0=w0, gi=gi, rs=rs, K1=K1, K2=K2, Ea=Ea, f=f, eps=w[k],
b=b, k1=k1, k2=k2, m=-1 / (8 * k1) - 1 / (8 * k2), rho=rho)
def mark_formula(L, tf, b1, b2, cpl, d1, d2, rho):
"""s_c (c1^-2 + c2^-2) Mbar2(rho), with M2(r) = -(2 n g1 g2 / t_f) ell(r) (quoted (4.3))."""
n = 1.0 / L
g1, g2 = b1["sig"] * b1["gam"] * b1["M"], b2["sig"] * b2["gam"] * b2["M"]
M2 = lambda r: -(2 * n * g1 * g2 / tf) * ell(r, L)
M2bar = 0.25 * (M2(rho + 1) + 2 * M2(rho) + M2(rho - 1))
sc = 1.0 if cpl == "a" else -1.0
return sc * (1 / d1["c2"] + 1 / d2["c2"]) * M2bar
def run_case(L, tf, b1, b2, cpl, par, mode):
bd = band_data(L, tf, b1, b2, cpl, par, mode)
H0, B1, V2, U0, gi = bd["H0"], bd["B1"], bd["V2"], bd["U0"], bd["gi"]
G = U0[:, gi]
# exact bound level and its curvature: H(P) = H0 + V2 + e^{-iP}B1 + e^{iP}B1^T (body 1 as representative)
Eb, Ebpp, _, gw = curvature(H0 + V2, B1, G, mode)
def lev(P): # finite-difference cross-check of the continuation in P
w, v = np.linalg.eigh(H0 + V2 + np.exp(-1j * P) * B1 + np.exp(1j * P) * B1.T)
c = np.where((np.abs(G.T @ v) ** 2).sum(0) > 0.5)[0]
return w[c].min() if mode == "low" else w[c].max()
h = 2e-3
Ebpp_fd = (lev(h) - 2 * lev(0.0) + lev(-h)) / h ** 2
# curvature of the band-contract level on the ring (same continuation)
def band_lev(P):
hP, _ = band_contract(U0, bd["w0"], gi, np.exp(-1j * P) * B1 + np.exp(1j * P) * B1.T + V2)
e = np.linalg.eigvalsh(hP)
return e[0] if mode == "low" else e[-1]
Epp_band = (band_lev(h) - 2 * band_lev(0.0) + band_lev(-h)) / h ** 2
# (6.6): sum_rho b/m plus the ring term Phi'(0)^2 d^2 eps / dPhi^2
b, m, k1, k2 = bd["b"], bd["m"], bd["k1"], bd["k2"]
Epp_66 = np.sum(b / m)
h0 = np.diag(bd["Ea"]) + bd["K1"] + bd["K2"]
nr = len(bd["rs"])
def eF(F):
hF = h0.astype(complex); hF[nr - 1, 0] *= np.exp(1j * F); hF[0, nr - 1] *= np.exp(-1j * F)
e = np.linalg.eigvalsh(hF)
return e[0] if mode == "low" else e[-1]
d2F = (eF(1e-2) - 2 * eF(0.0) + eF(-1e-2)) / 1e-4
dPhi = -2 * np.sum(k1 / (k1 + k2))
ring = dPhi ** 2 * d2F
# bound (6.16) of the ring remainder, evaluated at the bond farthest from contact
wall = np.linalg.eigvalsh(h0)
kk = 0 if mode == "low" else nr - 1
Dnu = np.min(np.abs(np.delete(wall, kk) - wall[kk]))
rho_b = (np.array(bd["rs"]) + 1) % L
js = int(np.argmin(np.abs(rho_b - L / 2)))
fa, fb, s_star = bd["f"][js], bd["f"][(js + 1) % nr], k1[js] + k2[js]
ring_bound = dPhi ** 2 * (2 * abs(s_star * fa * fb) + 4 * s_star ** 2 * (fa ** 2 + fb ** 2) / Dnu) / abs(Epp_66)
beta = b.sum()
d1, d2 = body_data(L, tf, b1, cpl), body_data(L, tf, b2, cpl)
mu, mub = d1["mu"] + d2["mu"], d1["mubar"] + d2["mubar"]
# B2: x_bar = max |m(rho) - m_0| / |m_0| and the inverse-expansion ratio |A_0| / (|m_0| beta_0)
xbar = np.max(np.abs(m - mub)) / abs(mub)
A0 = np.sum(b * (m - mub))
inv_line_expanded = mub / beta + A0 / beta ** 2 # first order in m - m_0, cf. (6.8)
Dmark = np.sum(b * mark_formula(L, tf, b1, b2, cpl, d1, d2, bd["rho"])) / beta ** 2
Dlat = mu * (1 - beta) / beta
dE = Eb - d1["E0"] - d2["E0"] + (-2 * tf) # cost difference: pair - body 1 - body 2 + field alone
n = 1.0 / L
g1, g2 = b1["sig"] * b1["gam"] * b1["M"], b2["sig"] * b2["gam"] * b2["M"]
return dict(L=L, coupling=cpl, parity=par, level=mode, Gweight=float(gw), Eb0=Eb,
inv_Epp_exact=1 / Ebpp, inv_Epp_fd=1 / Ebpp_fd, inv_Epp_band_ring=1 / Epp_band,
inv_Epp_66=1 / Epp_66, inv_Epp_66_plus_ring=1 / (Epp_66 + ring), ring_term_rel=ring / Epp_66,
ring_bound=ring_bound, f_antipode_sq=max(fa ** 2, fb ** 2), f_min_sign=float(np.min(bd["f"]) * np.sign(bd["f"].sum())),
xbar=xbar, A_rel=abs(A0) / (abs(mub) * beta),
rel_dev_expanded_vs_unexpanded=inv_line_expanded * Epp_66 - 1,
beta=beta, mu1=d1["mu"], mu2=d2["mu"], mu_sum=mu, mubar_sum=mub, mu_over_beta=mu / beta,
rel_dev_band_ring_vs_exact=Ebpp / Epp_band - 1,
rel_dev_66ring_vs_exact=Ebpp / (Epp_66 + ring) - 1,
rel_dev_mu_over_beta_vs_exact=(mu / beta) * Ebpp - 1,
D_exact=1 / Ebpp - mu, D_lat=Dlat, D_mark_formula=Dmark, D_lat_plus_mark=Dlat + Dmark,
sigR_over_c2_mu=[b1["sig"] * d1["R"] / d1["c2"] / d1["mu"], b2["sig"] * d2["R"] / d2["c2"] / d2["mu"]],
kappa_bar=[d1["kbar"], d2["kbar"]], chi=n * abs(g1 * g2) / tf / abs(d1["kbar"] + d2["kbar"]),
participation=1 / np.sum(bd["f"] ** 4), dE_cost=dE,
D_over_dE_times_c1sq=(1 / Ebpp - mu) * d1["c2"] / dE,
D_over_dE_times_c2sq=(1 / Ebpp - mu) * d2["c2"] / dE,
c1_over_c2=float(np.sqrt(d1["c2"] / d2["c2"])))
def r1_gaps(L, tf, b1, b2, par):
"""delta_1 and delta of R1 from the held-sector (tau = 0) spectra, N_f = 1, in the block of G."""
hf = field_hop(L, tf)
lev = {}
for s1 in range(2):
for s2 in range(2):
for r in range(L):
h = hf.copy(); h[0, 0] += SG[s1] * b1["gam"] * b1["M"]; h[r, r] += SG[s2] * b2["gam"] * b2["M"]
lev[(s1, s2, r)] = SG[s1] * b1["M"] + SG[s2] * b2["M"] + np.linalg.eigvalsh(h)
s1g, s2g = bidx(b1["sig"]), bidx(b2["sig"])
same = [r for r in range(L) if r % 2 == par]
other = [r for r in range(L) if r % 2 != par]
G = np.array([lev[(s1g, s2g, r)][0] for r in same])
one = np.concatenate([lev[(1 - s1g, s2g, r)] for r in other] + [lev[(s1g, 1 - s2g, r)] for r in other])
rest = np.concatenate([lev[(s1g, s2g, r)][1:] for r in same] + [lev[(1 - s1g, 1 - s2g, r)] for r in same])
d1 = np.min(np.abs(G[:, None] - one[None, :]))
d_rest = np.min(np.abs(G[:, None] - rest[None, :]))
return dict(delta1=float(d1), delta=float(min(d1, d_rest)), field_gap=float(4 * tf * np.sin(np.pi / L) ** 2),
G_width=float(G.max() - G.min()))
AIRY0 = 1.0188 # first zero of Ai'
def equality_crossing(chi=0.05, R=1.0, c=0.25, sig=-1, se=-1, Ls=range(100, 3001, 4)):
"""Leading-order sign rule (6.17) on the line: Delta_nu(L) = D_nu - Delta E_nu / c^2 from (6.8)-(6.10)
for the relative chain with uniform hopping, V(r) = s_e (F/L) r (L - r), equal speeds c, both bodies
with R_i = R and branch sig; the selected level (extreme, one-signed; highest for s_e = -1)."""
from scipy.linalg import eigh_tridiagonal
mu = sig * R / c ** 2
kb = -1 / (8 * mu)
t0 = 2 * kb
F = chi * abs(t0)
S = 2 / c ** 2
Qm1 = 4 * mu * mu * c ** 2 - 1
out = {}
for sc in (+1, -1):
prev, cross = None, []
for L in Ls:
r = 2 * np.arange(-(L // 4) + 1, L // 4 + 1) # even r (L multiple of 4)
Vf = lambda x: se * (F / L) * np.abs(x) * (L - np.abs(x))
M0 = -se * F * (L * L - 1) / (6 * L)
M2 = lambda x: M0 + Vf(x)
n = len(r)
sel = (n - 1, n - 1) if se < 0 else (0, 0)
w, v = eigh_tridiagonal(Vf(r), np.full(n - 1, t0), select="i", select_range=sel)
f = v[:, 0]
b = f[:-1] * f[1:]
rho = r[:-1] + 1
beta = b.sum()
K = 2 * t0 * (beta - 1)
Mbar = 0.25 * (M2(rho - 1) + 2 * M2(rho) + M2(rho + 1))
D = 2 * mu * (1 - beta) / beta + sc * S * np.sum(b * Mbar) / beta ** 2
dE = K + M0 + np.sum(f ** 2 * Vf(r))
Dl = D - dE / c ** 2
if prev is not None and np.sign(Dl) != np.sign(prev):
cross.append(int(L))
prev = Dl
out["coupling_a" if sc > 0 else "coupling_c"] = dict(crossings=cross)
est = 2 * 4 ** (1 / 3) * AIRY0 * Qm1 * chi ** (-1 / 3)
return dict(chi=chi, R=R, c=c, sig=sig, s_e=se, Q_minus_1=Qm1, L_star_estimate=est, **out)
def mark_check(L, tf, b1, b2, cpl, par, mode, scales=(1.0, 0.3, 0.1, 0.03)):
"""Ratio [m(rho) - mubar_1 - mubar_2] / [s_c (c1^-2+c2^-2) Mbar2(rho)] at the contact bonds, couplings scaled."""
out = []
for s in scales:
c1, c2 = dict(b1), dict(b2)
c1["gam"] *= s; c2["gam"] *= s
bd = band_data(L, tf, c1, c2, cpl, par, mode)
d1, d2 = body_data(L, tf, c1, cpl), body_data(L, tf, c2, cpl)
num = bd["m"] - d1["mubar"] - d2["mubar"]
pred = mark_formula(L, tf, c1, c2, cpl, d1, d2, bd["rho"])
j = np.argsort(np.abs(((bd["rho"] + L // 2) % L) - L // 2))[:2] # bonds nearest to contact
out.append(dict(scale=s, coupling=cpl, ratio_contact=float(np.mean(num[j] / pred[j])),
ratio_all=[float(x) for x in num / pred]))
return out
SETS = { # (body 1, body 2, parity of r, selected level, equal speeds); t_f = 1, 2 M_i > 4 t_f so that R1 holds for sigma = +1
# all sets have s_beta = +1 (selected level bound, one-signed); D has g_1 + g_2 = 0 (no first-order pair weight)
"A": (dict(M=3.0, tau=0.20, gam=0.09, sig=+1), dict(M=4.0, tau=0.20, gam=0.07, sig=+1), 0, "low", True),
"B": (dict(M=2.5, tau=0.16, gam=0.10, sig=-1), dict(M=2.0, tau=0.20, gam=-0.14, sig=-1), 1, "high", False),
"D": (dict(M=3.0, tau=0.20, gam=0.09, sig=-1), dict(M=3.0, tau=0.20, gam=-0.09, sig=-1), 0, "high", True),
}
def bodies(name, L, cpl):
"""Bodies of a set on a ring of L places. For the equal-speed sets and coupling (c), tau_2 is set so that
c_2 = tau_2 (1 + gamma_2 n) equals c_1 = tau_1 (1 + gamma_1 n) exactly (n = 1/L)."""
b1, b2, par, mode, eq = SETS[name]
b1, b2 = dict(b1), dict(b2)
if eq and cpl == "c":
b2["tau"] = b1["tau"] * (1 + b1["gam"] / L) / (1 + b2["gam"] / L)
return b1, b2, par, mode
if __name__ == "__main__":
OUT.mkdir(exist_ok=True)
tf = 1.0
res = []
for name in SETS:
for L in (12, 16):
for cpl in ("a", "c"):
b1, b2, par, mode = bodies(name, L, cpl)
r = run_case(L, tf, b1, b2, cpl, par, mode)
r["set"] = name
res.append(r)
keys = ["set", "L", "coupling", "level", "c1_over_c2", "chi", "beta", "inv_Epp_exact", "inv_Epp_fd",
"inv_Epp_band_ring", "inv_Epp_66", "inv_Epp_66_plus_ring", "mu_sum", "mu_over_beta",
"rel_dev_band_ring_vs_exact", "rel_dev_66ring_vs_exact", "rel_dev_mu_over_beta_vs_exact",
"ring_term_rel", "ring_bound", "f_antipode_sq", "f_min_sign", "xbar", "A_rel",
"rel_dev_expanded_vs_unexpanded", "D_exact", "D_lat", "D_mark_formula",
"D_lat_plus_mark", "dE_cost", "D_over_dE_times_c1sq", "D_over_dE_times_c2sq"]
fmt = lambda k, v: f"{k}={v:.6g}" if isinstance(v, (float, np.floating)) else f"{k}={v}"
lines = [" ".join(fmt(k, r[k]) for k in keys) for r in res]
mk = []
for cpl in ("a", "c"):
b1, b2, par, mode = bodies("A", 12, cpl)
mk += mark_check(12, tf, b1, b2, cpl, par, mode)
lines.append("mark term check (set A, L=12): ratio of m(rho)-mubar_1-mubar_2 to the mark term of (6.7) at contact bonds")
lines += [f" coupling={x['coupling']} scale={x['scale']} ratio={x['ratio_contact']:.4f}" for x in mk]
gaps = []
for name in SETS:
for L in (12, 16):
b1, b2, par, mode = bodies(name, L, "a")
for t in (tf, 40.0):
g = r1_gaps(L, t, b1, b2, par)
g.update(set=name, L=L, t_f=t)
gaps.append(g)
lines.append("R1 gaps from the held-sector spectra (N_f = 1): delta_1 (one motion term), delta (all other levels)")
lines += [f" set={g['set']} L={g['L']} t_f={g['t_f']}: delta1={g['delta1']:.4f} delta={g['delta']:.4f} "
f"field_gap={g['field_gap']:.4f} width_of_G={g['G_width']:.4f}" for g in gaps]
cr = equality_crossing()
lines.append(f"leading-order sign rule (6.17) on the line: chi={cr['chi']} R={cr['R']} c={cr['c']} sigma_i={cr['sig']} "
f"s_e={cr['s_e']} Q-1={cr['Q_minus_1']:.1f}")
lines.append(f" sign changes of D_0 - dE_0/c^2 in L in [100, 3000] (selected level): coupling (a): at {cr['coupling_a']['crossings']}; "
f"coupling (c): at {cr['coupling_c']['crossings']}")
lines.append(f" estimate L*_0 = 2 4^(1/3) a_0 (Q-1) chi^(-1/3) = {cr['L_star_estimate']:.0f}")
txt = "\n".join(lines)
print(txt)
(OUT / "inertia_check.txt").write_text(txt + "\n")
conv = lambda o: o.tolist() if isinstance(o, np.ndarray) else float(o)
(OUT / "inertia_check.json").write_text(json.dumps(dict(cases=res, mark_check=mk, r1_gaps=gaps, sign_rule=cr),
indent=1, default=conv))
check_weight.py16 KB · raw
"""Check of item 4 of 07-ilang-weight/06-inertia-of-pair against exact eigenvalues with a held source.
Two Model D bodies and N_f = 1 field object on a ring of L places (set A of check_inertia.py),
plus a held source g_s Phi_{x_s}. The exact contract (all places, no sector) is diagonalized near
the bottom of the bound band, where the bound pair sits in the weight well opposite the source.
Its levels are compared with
(i) the first-order projected contract on the exact bound band: E_b(P_k) + <P',v_P'| g_s Phi_xs |P,v_P>,
with the exact P-dependent sector eigenvectors v_P (no frozen relative profile);
(ii) the frozen-profile description E_b(P) + W_nu(X) with W_nu of (6.11) (profile f_nu at P = 0);
(iii) the same with the pair band rescaled to the inertia mu_1 + mu_2.
Second part: the acceleration from rest (6.13) of the centre Y conjugate to P, computed exactly from the
double commutator in the position-space band contract (order tau^2, first-order held weights), against
-E''_line sum phi^2 W'_nu, and the acceleration of the midpoint against -(beta/2) sum phi^2 (W_1'/mu_1 + W_2'/mu_2).
Set D (g_1 + g_2 = 0): the pair's acceleration against that of body 1 alone, as the couplings in the weights are scaled.
"""
import json
from pathlib import Path
import numpy as np
import scipy.sparse as sp
import scipy.sparse.linalg as sla
from check_inertia import (OUT, J, SG, bodies, bidx, ground, field_hop, pair_block, band_data,
body_data, curvature)
def full_contract(L, tf, b1, b2, cpl, gs, xs, par):
"""Exact contract on all places, restricted to the block (-1)^(x2-x1) s1 s2 of the band G."""
cc = 1.0 if cpl == "c" else 0.0
ix = lambda x1, s1, x2, s2, y: (((x1 * 2 + s1) * L + x2) * 2 + s2) * L + y
pi0 = (-1) ** par * b1["sig"] * b2["sig"]
states = [(x1, s1, x2, s2, y) for x1 in range(L) for s1 in range(2) for x2 in range(L) for s2 in range(2)
for y in range(L) if (-1) ** ((x2 - x1) % L) * SG[s1] * SG[s2] == pi0]
pos = {ix(*s): i for i, s in enumerate(states)}
rows, cols, vals = [], [], []
def add(i, st, v):
j = pos[ix(*st)]
rows.append(j); cols.append(i); vals.append(v)
for i, (x1, s1, x2, s2, y) in enumerate(states):
add(i, (x1, s1, x2, s2, y), SG[s1] * b1["M"] * (1 + b1["gam"] * (y == x1))
+ SG[s2] * b2["M"] * (1 + b2["gam"] * (y == x2)) + gs * (y == xs))
for dy in (1, -1):
add(i, (x1, s1, x2, s2, (y + dy) % L), -tf)
t1, t2 = 1 - s1, 1 - s2 # a motion term flips the branch (J has zero diagonal)
for dx in (1, -1):
# body hop x -> x+dx (bond between x and x+dx); J for right hops, J^T for left hops
Jm = J[t1, s1] if dx == 1 else J[s1, t1]
f = 1 + cc * b1["gam"] * 0.5 * ((y == x1) + (y == (x1 + dx) % L))
add(i, ((x1 + dx) % L, t1, x2, s2, y), 0.5 * b1["tau"] * Jm * f)
Jm = J[t2, s2] if dx == 1 else J[s2, t2]
f = 1 + cc * b2["gam"] * 0.5 * ((y == x2) + (y == (x2 + dx) % L))
add(i, (x1, s1, (x2 + dx) % L, t2, y), 0.5 * b2["tau"] * Jm * f)
H = sp.csr_matrix((vals, (rows, cols)), shape=(len(states), len(states)))
assert abs(H - H.T).max() < 1e-14
return H, states, pos, ix
def sector_state(blk, mode, P):
"""Exact bound level and eigenvector of the sector contract H(P) (body 1 as representative)."""
H0, B1, V2, U0, w0, gi, rs = blk
G = U0[:, gi]
w, v = np.linalg.eigh(H0 + V2 + np.exp(-1j * P) * B1 + np.exp(1j * P) * B1.T)
c = np.where((np.abs(G.T @ v) ** 2).sum(0) > 0.5)[0]
k = c[np.argmin(w[c])] if mode == "low" else c[np.argmax(w[c])]
return w[k], v[:, k]
def held_weights(L, tf, b1, b2, gs, xs=0):
"""First-order held weights: pair W(j,l) = g_s |Omega_jl(x_s)|^2 and single bodies W_i(x)."""
hf = field_hop(L, tf)
Om, Wpair = {}, np.zeros((L, L))
for j in range(L):
for l in range(L):
h = hf.copy(); h[j, j] += b1["sig"] * b1["gam"] * b1["M"]; h[l, l] += b2["sig"] * b2["gam"] * b2["M"]
_, o = ground(h)
Om[(j, l)] = o
Wpair[j, l] = gs * o[xs] ** 2
W1, W2 = np.zeros(L), np.zeros(L)
for x in range(L):
h = hf.copy(); h[x, x] += b1["sig"] * b1["gam"] * b1["M"]; W1[x] = gs * ground(h)[1][xs] ** 2
h = hf.copy(); h[x, x] += b2["sig"] * b2["gam"] * b2["M"]; W2[x] = gs * ground(h)[1][xs] ** 2
return Om, Wpair, W1, W2
def run(L, tf, cpl, gs, nlev=4):
b1, b2, par, mode = bodies("A", L, cpl)
xs = 0
s1g, s2g = bidx(b1["sig"]), bidx(b2["sig"])
Om, Wpair, W1, W2 = held_weights(L, tf, b1, b2, gs, xs)
# relative level f_nu (band contract, P = 0) and the pair weight W_nu(X), (6.11)
bd = band_data(L, tf, b1, b2, cpl, par, mode)
rsg = [((r + L // 2) % L) - L // 2 for r in bd["rs"]] # signed separations
Wnu = np.array([sum(bd["f"][k] ** 2 * Wpair[(X - r // 2) % L, (X + r // 2) % L] for k, r in enumerate(rsg))
for X in range(L)])
# exact bound band: levels E_b(P_k) and P-dependent sector eigenvectors v_P
blk = pair_block(L, tf, b1, b2, cpl, par)
Ps = 2 * np.pi * np.arange(L) / L
sts = [sector_state(blk, mode, P) for P in Ps]
Eb = np.array([s[0] for s in sts]); Vb = np.array([s[1] for s in sts]).T
# (i) projected contract: <P',v'|Phi_xs|P,v> = (1/L) sum_x e^{i(P-P')x} <v'|Phi_{xs-x}|v>_frame
yof = np.tile(np.arange(L), Vb.shape[0] // L)
Wp = np.zeros((L, L), complex)
for x in range(L):
sel = yof == (xs - x) % L
rho = Vb[sel].conj().T @ Vb[sel]
Wp += np.exp(1j * (Ps[None, :] - Ps[:, None]) * x) * rho
Wp *= gs / L
assert np.abs(Wp - Wp.conj().T).max() < 1e-12
Eproj = np.linalg.eigvalsh(np.diag(Eb) + Wp)[:nlev]
# (ii), (iii) frozen-profile descriptions
F = np.exp(1j * np.outer(np.arange(L), Ps)) / np.sqrt(L) # <X|P_k>
_, Epp, _, _ = curvature(bd["H0"] + bd["V2"], bd["B1"], bd["U0"][:, bd["gi"]], mode)
d1, d2 = body_data(L, tf, b1, cpl), body_data(L, tf, b2, cpl)
mu = d1["mu"] + d2["mu"]
s_free = 1 / (mu * Epp) # rescales the band curvature to 1/(mu_1 + mu_2)
model = lambda Eband: np.linalg.eigvalsh(F @ np.diag(Eband) @ F.conj().T + np.diag(Wnu))[:nlev]
Em = model(Eb)
Ealt = model(Eb[0] + s_free * (Eb - Eb[0]))
# exact contract with the source, levels near the bottom of the bound band, filtered by G-weight
H, states, pos, ix = full_contract(L, tf, b1, b2, cpl, gs, xs, par)
target = Em[0] - 0.3 * (Em[1] - Em[0])
w, v = sla.eigsh(H, k=3 * nlev, sigma=target, which="LM")
Gw = np.zeros(len(w))
for (j, l), o in Om.items():
if (l - j) % 2 != par:
continue
idx = [pos[ix(j, s1g, l, s2g, y)] for y in range(L)]
Gw += np.abs(o @ v[idx, :]) ** 2
ex = np.sort(w[Gw > 0.5])[:nlev]
X = L // 4
dWnu = (Wnu[X + 1] - Wnu[X - 1]) / 2
dW1, dW2 = (W1[X + 1] - W1[X - 1]) / 2, (W2[X + 1] - W2[X - 1]) / 2
return dict(L=L, coupling=cpl, g_s=gs, exact_levels=ex.tolist(), proj_levels=Eproj.tolist(),
model_levels=Em.tolist(), alt_levels=Ealt.tolist(), exact_spacings=np.diff(ex).tolist(),
proj_spacings=np.diff(Eproj).tolist(), model_spacings=np.diff(Em).tolist(),
alt_spacings=np.diff(Ealt).tolist(), beta=float(bd["b"].sum()), s_free=float(s_free),
max_abs_dev_proj=float(np.max(np.abs(ex - Eproj))), max_abs_dev_model=float(np.max(np.abs(ex - Em))),
G_weights=np.sort(Gw)[::-1][:nlev].tolist(),
dWnu_over_dW1_plus_dW2=float(dWnu / (dW1 + dW2)), Epp_times_mu_sum=float(Epp * mu),
Pi_pair_over_weighted_single=float(Epp * dWnu * mu / (dW1 + dW2)))
def accel_test(name, L, tf, cpl, gs, widths=(1.0, 1.5, 2.0), wscales=(1.0, 0.05)):
"""Acceleration from rest (6.13) in the position-space band contract with first-order held weights.
wscale < 1 scales the couplings g_i inside the held weights only (band contract and f_nu unchanged), to
isolate the first order in g_i of the weights, on which the last step of (6.13) rests."""
b1, b2, par, mode = bodies(name, L, cpl)
bd = band_data(L, tf, b1, b2, cpl, par, mode)
rs = list(bd["rs"]); nr = len(rs); idx = {r: k for k, r in enumerate(rs)}
K1, K2, Ea, f = bd["K1"], bd["K2"], bd["Ea"], bd["f"]
I = lambda j, r: (j % L) * nr + idx[r % L]
N = L * nr
H = np.zeros((N, N))
for j in range(L):
for k, r in enumerate(rs):
a = I(j, r)
H[a, a] += Ea[k] + K1[k, k] + K2[k, k]
H[I(j + 2, r - 2), a] += K1[idx[(r - 2) % L], k] # body 1 by +2
H[I(j - 2, r + 2), a] += K1[idx[(r + 2) % L], k] # body 1 by -2
H[I(j, r + 2), a] += K2[idx[(r + 2) % L], k] # body 2 by +2
H[I(j, r - 2), a] += K2[idx[(r - 2) % L], k] # body 2 by -2
assert np.abs(H - H.T).max() < 1e-15
H -= np.mean(np.diag(H)) * np.eye(N) # a constant does not enter the commutators; reduces round-off
data = {}
for ws in wscales:
b1w, b2w = dict(b1, gam=b1["gam"] * ws), dict(b2, gam=b2["gam"] * ws)
data[ws] = held_weights(L, tf, b1w, b2w, gs)[1:]
# theta'(r): theta'(r+2) - theta'(r) = (k1 - k2)/(k1 + k2) at rho = r + 1, theta' odd (even class)
k1, k2 = bd["k1"], bd["k2"]
step = {(rs[j] + 1) % L: (k1[j] - k2[j]) / (k1[j] + k2[j]) for j in range(nr)}
sgn = lambda r: ((r + L // 2) % L) - L // 2
th = {0: 0.0}
for r in range(2, L // 2 + 1, 2):
th[r] = th[r - 2] + step[(r - 1) % L]
th[-r] = th[-r + 2] - step[(-r + 1) % L]
x0 = L // 4
Xu = np.zeros(N); Y = np.zeros(N); Fr = np.zeros(N); Rs = np.zeros(N)
for j in range(L):
for k, r in enumerate(rs):
rsn = sgn(r)
Xc = j + rsn / 2
Xc = x0 + ((Xc - x0 + L / 2) % L) - L / 2
a = I(j, r)
Xu[a], Y[a], Fr[a], Rs[a] = Xc, Xc + th[rsn], f[k], rsn
rsg = [sgn(r) for r in rs]
Xg = np.arange(x0 - L // 2 + 2, x0 + L // 2 - 1)
beta, Eline = bd["b"].sum(), np.sum(bd["b"] / bd["m"])
d1, d2 = body_data(L, tf, b1, cpl), body_data(L, tf, b2, cpl)
out = []
for ws, (Wpair, W1, W2) in data.items():
Wb = np.array([Wpair[j, (j + r) % L] for j in range(L) for r in rs])
Wnu = lambda X: sum(f[k] ** 2 * Wpair[int(X - rr // 2) % L, int(X + rr // 2) % L] for k, rr in enumerate(rsg))
dWnu = np.array([(Wnu(X + 1) - Wnu(X - 1)) / 2 for X in Xg])
dW1 = np.array([(W1[(X + 1) % L] - W1[(X - 1) % L]) / 2 for X in Xg])
dW2 = np.array([(W2[(X + 1) % L] - W2[(X - 1) % L]) / 2 for X in Xg])
for wd in widths:
psi = np.exp(-(Y - x0) ** 2 / (4 * wd ** 2)) * Fr
psi /= np.linalg.norm(psi)
acc = lambda Ht, Z: -psi @ (Ht @ Ht @ (Z * psi) - 2 * Ht @ (Z * (Ht @ psi)) + Z * (Ht @ Ht @ psi))
Ht = H + np.diag(Wb)
# a is linear in W exactly ([W,[W,Z]] = 0 for diagonal W, Z): a = a(g_s=0) + first order in g_s
aY, a0, amid, am0 = acc(Ht, Y - x0), acc(H, Y - x0), acc(Ht, Xu - x0), acc(H, Xu - x0)
# the g_s-independent part with the relative profile cut away from the relative cut r = L/2
psi_c = np.exp(-(Y - x0) ** 2 / (4 * wd ** 2)) * Fr * (np.abs(Rs) < L // 3)
psi_c /= np.linalg.norm(psi_c)
a0cut = -psi_c @ (H @ H @ ((Y - x0) * psi_c) - 2 * H @ ((Y - x0) * (H @ psi_c)) + (Y - x0) * (H @ H @ psi_c))
p2 = psi ** 2
predY = -Eline * np.sum(p2 * np.interp(Y, Xg, dWnu))
predmid = -0.5 * beta * np.sum(p2 * (np.interp(Xu, Xg, dW1) / d1["mu"] + np.interp(Xu, Xg, dW2) / d2["mu"]))
pred1 = -np.sum(p2 * np.interp(Y, Xg, dW1)) / d1["mu"] # body 1 alone at the centre, leading form of (7.7)
out.append(dict(set=name, L=L, coupling=cpl, weight_scale=ws, width=wd, beta=float(beta), a_Y=float(aY),
a_Y_gs0=float(a0), a_Y_gs0_profile_cut=float(a0cut), pred_Y=float(predY), ratio_Y=float((aY - a0) / predY), a_mid=float(amid),
a_mid_gs0=float(am0), pred_mid=float(predmid), ratio_mid=float((amid - am0) / predmid),
aY_over_amid=float((aY - a0) / (amid - am0)), pred_single1=float(pred1),
aY_over_single1=float((aY - a0) / pred1),
pred_ratio=float(predY / predmid)))
return out
def weight_ratio_scaling(L, tf, cpl, gs, scales=(1.0, 0.3, 0.1, 0.03)):
"""dW_nu/(dW_1+dW_2) at X = L/4 with the relative profile f_nu of set A kept fixed and the
couplings gamma_i scaled: tends to 1 as the couplings go to zero (first order in g_i)."""
b1, b2, par, mode = bodies("A", L, cpl)
bd = band_data(L, tf, b1, b2, cpl, par, mode)
rsg = [((r + L // 2) % L) - L // 2 for r in bd["rs"]]
hf, X, out = field_hop(L, tf), L // 4, []
for sc in scales:
g1 = sc * b1["sig"] * b1["gam"] * b1["M"]; g2 = sc * b2["sig"] * b2["gam"] * b2["M"]
def w(pots):
h = hf.copy()
for x, g in pots:
h[x % L, x % L] += g
return gs * ground(h)[1][0] ** 2
Wnu = lambda Y: sum(bd["f"][k] ** 2 * w([(Y - r // 2, g1), (Y + r // 2, g2)]) for k, r in enumerate(rsg))
dWnu = Wnu(X + 1) - Wnu(X - 1)
dWs = w([(X + 1, g1)]) - w([(X - 1, g1)]) + w([(X + 1, g2)]) - w([(X - 1, g2)])
out.append(dict(scale=sc, ratio=float(dWnu / dWs)))
return out
if __name__ == "__main__":
OUT.mkdir(exist_ok=True)
res = [run(L, 1.0, cpl, -0.01) for L in (16, 20) for cpl in ("a", "c")]
res.append(run(16, 1.0, "a", -0.005))
lines = []
for r in res:
lines.append(f"L={r['L']} coupling={r['coupling']} g_s={r['g_s']} beta={r['beta']:.5f} "
f"E''(mu1+mu2)={r['Epp_times_mu_sum']:.5f} s_free={r['s_free']:.5f}")
lines.append(" exact spacings " + " ".join(f"{x:.6e}" for x in r["exact_spacings"]))
lines.append(" projected spacings " + " ".join(f"{x:.6e}" for x in r["proj_spacings"]))
lines.append(" frozen spacings " + " ".join(f"{x:.6e}" for x in r["model_spacings"]))
lines.append(" alt spacings " + " ".join(f"{x:.6e}" for x in r["alt_spacings"]))
lines.append(f" max |exact - projected| level = {r['max_abs_dev_proj']:.3e} max |exact - frozen| level = "
f"{r['max_abs_dev_model']:.3e} G-weights {np.round(r['G_weights'], 4).tolist()}")
lines.append(f" dW_nu/(dW_1+dW_2) = {r['dWnu_over_dW1_plus_dW2']:.5f} "
f"Pi_pair / inertia-weighted Pi_i = {r['Pi_pair_over_weighted_single']:.5f}")
sc = weight_ratio_scaling(16, 1.0, "a", -0.01)
lines.append("dW_nu/(dW_1+dW_2) at X=L/4, L=16, f_nu of set A fixed, couplings scaled: "
+ " ".join(f"scale={x['scale']}: {x['ratio']:.5f}" for x in sc))
acc = []
for cpl in ("a", "c"):
acc += accel_test("A", 24, 1.0, cpl, -0.01)
acc += accel_test("D", 24, 1.0, cpl, -0.01, wscales=(1.0, 0.3, 0.1))
lines.append("acceleration from rest (6.13), position-space band contract, L=24, g_s=-0.01, envelope centre L/4; "
"ratios use the first-order part a - a(g_s=0):")
lines += [f" set={x['set']} coupling={x['coupling']} weight_scale={x['weight_scale']} w={x['width']} beta={x['beta']:.4f}: a_Y={x['a_Y']:.6e} "
f"(g_s=0: {x['a_Y_gs0']:.1e}, profile cut at |r|<L/3: {x['a_Y_gs0_profile_cut']:.1e}) pred_Y={x['pred_Y']:.6e} ratio={x['ratio_Y']:.5f} | a_mid={x['a_mid']:.6e} (g_s=0: {x['a_mid_gs0']:.1e}) "
f"pred_mid={x['pred_mid']:.6e} ratio={x['ratio_mid']:.5f} | a_Y/a_mid={x['aY_over_amid']:.4f} (pred {x['pred_ratio']:.4f}) "
f"| a_Y/(body 1 alone)={x['aY_over_single1']:.5f}" for x in acc]
txt = "\n".join(lines)
print(txt)
(OUT / "weight_check.txt").write_text(txt + "\n")
(OUT / "weight_check.json").write_text(json.dumps(dict(cases=res, weight_ratio_scaling=sc, acceleration=acc),
indent=1))
output/inertia_check.json23 KB · raw
{
"cases": [
{
"L": 12,
"coupling": "a",
"parity": 0,
"level": "low",
"Gweight": 0.9997316819577194,
"Eb0": 5.032130258807638,
"inv_Epp_exact": 227.88034635627238,
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0.0016241755190924412,
0.00166311357680482
],
"chi": 1.3860204767771962,
"participation": 1.451900849255391,
"dE_cost": 0.008258284715316755,
"D_over_dE_times_c1sq": -563.0096989594944,
"D_over_dE_times_c2sq": -563.0096989594944,
"c1_over_c2": 1.0,
"set": "D"
},
{
"L": 16,
"coupling": "c",
"parity": 0,
"level": "high",
"Gweight": 0.9994810896452992,
"Eb0": -8.004580816490956,
"inv_Epp_exact": -263.8443775070864,
"inv_Epp_fd": -263.84467747806366,
"inv_Epp_band_ring": -263.101051666476,
"inv_Epp_66": -263.6569275780088,
"inv_Epp_66_plus_ring": -263.10075921907395,
"ring_term_rel": 0.00211389872300494,
"ring_bound": 0.015516696920546282,
"f_antipode_sq": 0.00012315703846460462,
"f_min_sign": 0.003652812551268295,
"xbar": 0.005352202819800602,
"A_rel": 0.004985473902126727,
"rel_dev_expanded_vs_unexpanded": 1.9106617163977546e-06,
"beta": 0.5702370787831708,
"mu1": -75.52084829502913,
"mu2": -74.01155729696538,
"mu_sum": -149.53240559199452,
"mubar_sum": -149.60140952230824,
"mu_over_beta": -262.22848558196495,
"rel_dev_band_ring_vs_exact": -0.002817288917177807,
"rel_dev_66ring_vs_exact": -0.0028183973258723016,
"rel_dev_mu_over_beta_vs_exact": -0.006124412960355818,
"D_exact": -114.31197191509187,
"D_lat": -112.69607998997046,
"D_mark_formula": -0.6514420267405332,
"D_lat_plus_mark": -113.347522016711,
"sigR_over_c2_mu": [
0.9881119453239511,
0.9958387934320369
],
"kappa_bar": [
0.001654224455900355,
0.0016883382036072536
],
"chi": 1.3631008492960248,
"participation": 1.458347746751434,
"dE_cost": 0.008232485525650368,
"D_over_dE_times_c1sq": -561.6850373617414,
"D_over_dE_times_c2sq": -561.6850373617413,
"c1_over_c2": 1.0,
"set": "D"
}
],
"mark_check": [
{
"scale": 1.0,
"coupling": "a",
"ratio_contact": 1.400643091214988,
"ratio_all": [
1.400643091215232,
-1.0654491181891712,
2.2191205421886404,
2.2191205421886404,
-1.0654491181879528,
1.4006430912147445
]
},
{
"scale": 0.3,
"coupling": "a",
"ratio_contact": 1.3096894943591075,
"ratio_all": [
1.3096894943595587,
0.4419096256653923,
1.479130018902239,
1.4791300189123895,
0.4419096256924606,
1.3096894943586563
]
},
{
"scale": 0.1,
"coupling": "a",
"ratio_contact": 1.1320744440622226,
"ratio_all": [
1.1320744440662829,
0.8407940132672468,
1.171640696741504,
1.1716406967415038,
0.8407940133078493,
1.1320744440581623
]
},
{
"scale": 0.03,
"coupling": "a",
"ratio_contact": 1.0430236745237598,
"ratio_all": [
1.043023674704215,
0.9559654934885242,
1.0527707309231507,
1.0527707310359349,
0.9559654894282803,
1.0430236743433043
]
},
{
"scale": 1.0,
"coupling": "c",
"ratio_contact": -0.19035094178430928,
"ratio_all": [
-0.19035094178455655,
1.5883768060115475,
-0.704848597606271,
-0.704848597606271,
1.5883768060107233,
-0.19035094178406198
]
},
{
"scale": 0.3,
"coupling": "c",
"ratio_contact": 0.4156587020246604,
"ratio_all": [
0.4156587020246603,
0.9826982223941745,
0.3572495600450063,
0.35724956003479813,
0.9826982223669528,
0.41565870202466043
]
},
{
"scale": 0.1,
"coupling": "c",
"ratio_contact": 0.7736068425396266,
"ratio_all": [
0.7736068425477715,
0.9527029521821109,
0.7719644806400368,
0.7719644806298557,
0.952702952141387,
0.7736068425314818
]
},
{
"scale": 0.03,
"coupling": "c",
"ratio_contact": 0.9284491467165781,
"ratio_all": [
0.928449146490545,
0.9804793794027348,
0.9301118790472036,
0.9301118790472036,
0.9804793843754607,
0.928449146942611
]
}
],
"r1_gaps": [
{
"delta1": 2.0093948956940566,
"delta": 0.22767970925219227,
"field_gap": 0.26794919243112264,
"G_width": 0.011467767370039184,
"set": "A",
"L": 12,
"t_f": 1.0
},
{
"delta1": 2.6255743391509867,
"delta": 2.6255743391509867,
"field_gap": 10.717967697244905,
"G_width": 0.0004662311858822932,
"set": "A",
"L": 12,
"t_f": 40.0
},
{
"delta1": 1.9996153917738986,
"delta": 0.12329213131151917,
"field_gap": 0.15224093497742647,
"G_width": 0.00985989972467749,
"set": "A",
"L": 16,
"t_f": 1.0
},
{
"delta1": 0.02417838132411987,
"delta": 0.02417838132411987,
"field_gap": 6.089637399097059,
"G_width": 0.000464061919117853,
"set": "A",
"L": 16,
"t_f": 40.0
},
{
"delta1": 3.923793969574124,
"delta": 0.228842419649963,
"field_gap": 0.26794919243112264,
"G_width": 0.011235284738974727,
"set": "B",
"L": 12,
"t_f": 1.0
},
{
"delta1": 3.952881536478415,
"delta": 3.952881536478415,
"field_gap": 10.717967697244905,
"G_width": 0.0002915115450008443,
"set": "B",
"L": 12,
"t_f": 40.0
},
{
"delta1": 3.9294842268699854,
"delta": 0.12335625243097326,
"field_gap": 0.15224093497742647,
"G_width": 0.012519759295922306,
"set": "B",
"L": 16,
"t_f": 1.0
},
{
"delta1": 3.964513119814299,
"delta": 3.964513119814299,
"field_gap": 6.089637399097059,
"G_width": 0.0003278700262114853,
"set": "B",
"L": 16,
"t_f": 40.0
},
{
"delta1": 5.931465266486541,
"delta": 0.2250676621861638,
"field_gap": 0.26794919243112264,
"G_width": 0.018142710513801674,
"set": "D",
"L": 12,
"t_f": 1.0
},
{
"delta1": 5.954532728947015,
"delta": 5.954532728947015,
"field_gap": 10.717967697244905,
"G_width": 0.0004556237025212795,
"set": "D",
"L": 12,
"t_f": 40.0
},
{
"delta1": 5.9380388057013,
"delta": 0.11868015548718969,
"field_gap": 0.15224093497742647,
"G_width": 0.01814271051380345,
"set": "D",
"L": 16,
"t_f": 1.0
},
{
"delta1": 5.965746421707351,
"delta": 5.965746421707351,
"field_gap": 6.089637399097059,
"G_width": 0.0004556237024928578,
"set": "D",
"L": 16,
"t_f": 40.0
}
],
"sign_rule": {
"chi": 0.05,
"R": 1.0,
"c": 0.25,
"sig": -1,
"s_e": -1,
"Q_minus_1": 63.0,
"L_star_estimate": 553.1243916755296,
"coupling_a": {
"crossings": [
564
]
},
"coupling_c": {
"crossings": []
}
}
}output/inertia_check.txt10 KB · raw
set=A L=12 coupling=a level=low c1_over_c2=1 chi=2.18493 beta=0.693117 inv_Epp_exact=227.88 inv_Epp_fd=227.881 inv_Epp_band_ring=227.683 inv_Epp_66=254.736 inv_Epp_66_plus_ring=227.683 mu_sum=177.024 mu_over_beta=255.404 rel_dev_band_ring_vs_exact=-0.00086453 rel_dev_66ring_vs_exact=-0.000866607 rel_dev_mu_over_beta_vs_exact=0.120779 ring_term_rel=0.118819 ring_bound=0.980813 f_antipode_sq=0.00940482 f_min_sign=0.0399792 xbar=0.00351052 A_rel=0.00246055 rel_dev_expanded_vs_unexpanded=9.07857e-07 D_exact=50.8559 D_lat=78.3791 D_mark_formula=-0.433161 D_lat_plus_mark=77.9459 dE_cost=-0.00353514 D_over_dE_times_c1sq=-575.432 D_over_dE_times_c2sq=-575.432
set=A L=12 coupling=c level=low c1_over_c2=1 chi=2.15978 beta=0.695604 inv_Epp_exact=224.102 inv_Epp_fd=224.102 inv_Epp_band_ring=223.92 inv_Epp_66=251.399 inv_Epp_66_plus_ring=223.92 mu_sum=174.96 mu_over_beta=251.522 rel_dev_band_ring_vs_exact=-0.000810322 rel_dev_66ring_vs_exact=-0.000812665 rel_dev_mu_over_beta_vs_exact=0.122358 ring_term_rel=0.122721 ring_bound=1.01868 f_antipode_sq=0.00964338 f_min_sign=0.0408688 xbar=0.0011114 A_rel=0.00038387 rel_dev_expanded_vs_unexpanded=3.18981e-08 D_exact=49.1417 D_lat=76.5623 D_mark_formula=0.424144 D_lat_plus_mark=76.9865 dE_cost=-0.0035177 D_over_dE_times_c1sq=-567.207 D_over_dE_times_c2sq=-567.207
set=A L=16 coupling=a level=low c1_over_c2=1 chi=1.63512 beta=0.79213 inv_Epp_exact=215.788 inv_Epp_fd=215.788 inv_Epp_band_ring=215.589 inv_Epp_66=222.405 inv_Epp_66_plus_ring=215.589 mu_sum=176.675 mu_over_beta=223.038 rel_dev_band_ring_vs_exact=-0.000922874 rel_dev_66ring_vs_exact=-0.000924078 rel_dev_mu_over_beta_vs_exact=0.0335974 ring_term_rel=0.0316186 ring_bound=0.340164 f_antipode_sq=0.00151956 f_min_sign=0.0171615 xbar=0.00446272 A_rel=0.00252832 rel_dev_expanded_vs_unexpanded=4.19049e-07 D_exact=39.113 D_lat=46.363 D_mark_formula=-0.440081 D_lat_plus_mark=45.9229 dE_cost=-0.00312846 D_over_dE_times_c1sq=-500.093 D_over_dE_times_c2sq=-500.093
set=A L=16 coupling=c level=low c1_over_c2=1 chi=1.62269 beta=0.793088 inv_Epp_exact=214.057 inv_Epp_fd=214.057 inv_Epp_band_ring=213.867 inv_Epp_66=220.83 inv_Epp_66_plus_ring=213.866 mu_sum=175.303 mu_over_beta=221.039 rel_dev_band_ring_vs_exact=-0.000889061 rel_dev_66ring_vs_exact=-0.000890678 rel_dev_mu_over_beta_vs_exact=0.0326181 ring_term_rel=0.0325613 ring_bound=0.352012 f_antipode_sq=0.00155458 f_min_sign=0.0174664 xbar=0.00225855 A_rel=0.000686161 rel_dev_expanded_vs_unexpanded=3.29449e-08 D_exact=38.7535 D_lat=45.7357 D_mark_formula=0.434106 D_lat_plus_mark=46.1698 dE_cost=-0.00312016 D_over_dE_times_c1sq=-502.419 D_over_dE_times_c2sq=-502.419
set=B L=12 coupling=a level=high c1_over_c2=0.8 chi=1.55239 beta=0.771791 inv_Epp_exact=-182.829 inv_Epp_fd=-182.829 inv_Epp_band_ring=-182.62 inv_Epp_66=-193.47 inv_Epp_66_plus_ring=-182.62 mu_sum=-149.732 mu_over_beta=-194.007 rel_dev_band_ring_vs_exact=-0.00113989 rel_dev_66ring_vs_exact=-0.00114076 rel_dev_mu_over_beta_vs_exact=0.0611391 ring_term_rel=0.0594129 ring_bound=0.339697 f_antipode_sq=0.00428455 f_min_sign=0.0654565 xbar=0.00413987 A_rel=0.00285439 rel_dev_expanded_vs_unexpanded=2.71453e-06 D_exact=-33.0961 D_lat=-44.2741 D_mark_formula=0.481902 D_lat_plus_mark=-43.7922 dE_cost=0.00364334 D_over_dE_times_c1sq=-232.55 D_over_dE_times_c2sq=-363.36
set=B L=12 coupling=c level=high c1_over_c2=0.816189 chi=1.55919 beta=0.770603 inv_Epp_exact=-182.059 inv_Epp_fd=-182.06 inv_Epp_band_ring=-181.861 inv_Epp_66=-193.079 inv_Epp_66_plus_ring=-181.861 mu_sum=-148.439 mu_over_beta=-192.627 rel_dev_band_ring_vs_exact=-0.0010876 rel_dev_66ring_vs_exact=-0.00108869 rel_dev_mu_over_beta_vs_exact=0.0580434 ring_term_rel=0.0616834 ring_bound=0.354983 f_antipode_sq=0.00422347 f_min_sign=0.0649882 xbar=0.00361246 A_rel=0.00217061 rel_dev_expanded_vs_unexpanded=2.99912e-06 D_exact=-33.6207 D_lat=-44.188 D_mark_formula=-0.483418 D_lat_plus_mark=-44.6714 dE_cost=0.00365744 D_over_dE_times_c1sq=-239.264 D_over_dE_times_c2sq=-359.166
set=B L=16 coupling=a level=high c1_over_c2=0.8 chi=1.16692 beta=0.776915 inv_Epp_exact=-191.462 inv_Epp_fd=-191.462 inv_Epp_band_ring=-191.165 inv_Epp_66=-192.381 inv_Epp_66_plus_ring=-191.165 mu_sum=-149.957 mu_over_beta=-193.016 rel_dev_band_ring_vs_exact=-0.00155024 rel_dev_66ring_vs_exact=-0.00155098 rel_dev_mu_over_beta_vs_exact=0.00811379 ring_term_rel=0.0063618 ring_bound=0.036297 f_antipode_sq=0.000245782 f_min_sign=0.0156774 xbar=0.00508554 A_rel=0.0036114 rel_dev_expanded_vs_unexpanded=3.07637e-06 D_exact=-41.5054 D_lat=-43.0589 D_mark_formula=0.578926 D_lat_plus_mark=-42.48 dE_cost=0.00509914 D_over_dE_times_c1sq=-208.376 D_over_dE_times_c2sq=-325.588
set=B L=16 coupling=c level=high c1_over_c2=0.812106 chi=1.16811 beta=0.776063 inv_Epp_exact=-191.061 inv_Epp_fd=-191.061 inv_Epp_band_ring=-190.773 inv_Epp_66=-192.03 inv_Epp_66_plus_ring=-190.773 mu_sum=-148.578 mu_over_beta=-191.452 rel_dev_band_ring_vs_exact=-0.00150643 rel_dev_66ring_vs_exact=-0.00150725 rel_dev_mu_over_beta_vs_exact=0.00204295 ring_term_rel=0.00658724 ring_bound=0.0378638 f_antipode_sq=0.000243974 f_min_sign=0.0156197 xbar=0.0034698 A_rel=0.00258131 rel_dev_expanded_vs_unexpanded=1.33362e-06 D_exact=-42.4827 D_lat=-42.873 D_mark_formula=-0.579869 D_lat_plus_mark=-43.4529 dE_cost=0.00510877 D_over_dE_times_c1sq=-215.55 D_over_dE_times_c2sq=-326.83
set=D L=12 coupling=a level=high c1_over_c2=1 chi=1.84462 beta=0.50723 inv_Epp_exact=-291.922 inv_Epp_fd=-291.923 inv_Epp_band_ring=-291.327 inv_Epp_66=-298.987 inv_Epp_66_plus_ring=-291.326 mu_sum=-151.825 mu_over_beta=-299.322 rel_dev_band_ring_vs_exact=-0.00204124 rel_dev_66ring_vs_exact=-0.00204284 rel_dev_mu_over_beta_vs_exact=0.0253465 ring_term_rel=0.0262962 ring_bound=0.169125 f_antipode_sq=0.0024589 f_min_sign=0.0163468 xbar=0.00168391 A_rel=0.00122634 rel_dev_expanded_vs_unexpanded=1.0803e-07 D_exact=-140.098 D_lat=-147.497 D_mark_formula=0.621176 D_lat_plus_mark=-146.876 dE_cost=0.00763991 D_over_dE_times_c1sq=-733.504 D_over_dE_times_c2sq=-733.504
set=D L=12 coupling=c level=high c1_over_c2=1 chi=1.80915 beta=0.511624 inv_Epp_exact=-284.569 inv_Epp_fd=-284.57 inv_Epp_band_ring=-283.986 inv_Epp_66=-291.964 inv_Epp_66_plus_ring=-283.986 mu_sum=-148.891 mu_over_beta=-291.017 rel_dev_band_ring_vs_exact=-0.00204783 rel_dev_66ring_vs_exact=-0.00204926 rel_dev_mu_over_beta_vs_exact=0.0226591 ring_term_rel=0.0280927 ring_bound=0.181834 f_antipode_sq=0.00258689 f_min_sign=0.0170553 xbar=0.00331197 A_rel=0.00307313 rel_dev_expanded_vs_unexpanded=9.74202e-07 D_exact=-135.678 D_lat=-142.126 D_mark_formula=-0.605468 D_lat_plus_mark=-142.731 dE_cost=0.00759982 D_over_dE_times_c1sq=-724.862 D_over_dE_times_c2sq=-724.862
set=D L=16 coupling=a level=high c1_over_c2=1 chi=1.38602 beta=0.5673 inv_Epp_exact=-268.304 inv_Epp_fd=-268.304 inv_Epp_band_ring=-267.556 inv_Epp_66=-268.077 inv_Epp_66_plus_ring=-267.556 mu_sum=-152.066 mu_over_beta=-268.053 rel_dev_band_ring_vs_exact=-0.00278599 rel_dev_66ring_vs_exact=-0.00278629 rel_dev_mu_over_beta_vs_exact=-0.000934643 ring_term_rel=0.00194464 ring_bound=0.0141968 f_antipode_sq=0.00011551 f_min_sign=0.00348661 xbar=0.00165576 A_rel=0.000280005 rel_dev_expanded_vs_unexpanded=1.32067e-08 D_exact=-116.237 D_lat=-115.987 D_mark_formula=0.662951 D_lat_plus_mark=-115.324 dE_cost=0.00825828 D_over_dE_times_c1sq=-563.01 D_over_dE_times_c2sq=-563.01
set=D L=16 coupling=c level=high c1_over_c2=1 chi=1.3631 beta=0.570237 inv_Epp_exact=-263.844 inv_Epp_fd=-263.845 inv_Epp_band_ring=-263.101 inv_Epp_66=-263.657 inv_Epp_66_plus_ring=-263.101 mu_sum=-149.532 mu_over_beta=-262.228 rel_dev_band_ring_vs_exact=-0.00281729 rel_dev_66ring_vs_exact=-0.0028184 rel_dev_mu_over_beta_vs_exact=-0.00612441 ring_term_rel=0.0021139 ring_bound=0.0155167 f_antipode_sq=0.000123157 f_min_sign=0.00365281 xbar=0.0053522 A_rel=0.00498547 rel_dev_expanded_vs_unexpanded=1.91066e-06 D_exact=-114.312 D_lat=-112.696 D_mark_formula=-0.651442 D_lat_plus_mark=-113.348 dE_cost=0.00823249 D_over_dE_times_c1sq=-561.685 D_over_dE_times_c2sq=-561.685
mark term check (set A, L=12): ratio of m(rho)-mubar_1-mubar_2 to the mark term of (6.7) at contact bonds
coupling=a scale=1.0 ratio=1.4006
coupling=a scale=0.3 ratio=1.3097
coupling=a scale=0.1 ratio=1.1321
coupling=a scale=0.03 ratio=1.0430
coupling=c scale=1.0 ratio=-0.1904
coupling=c scale=0.3 ratio=0.4157
coupling=c scale=0.1 ratio=0.7736
coupling=c scale=0.03 ratio=0.9284
R1 gaps from the held-sector spectra (N_f = 1): delta_1 (one motion term), delta (all other levels)
set=A L=12 t_f=1.0: delta1=2.0094 delta=0.2277 field_gap=0.2679 width_of_G=0.0115
set=A L=12 t_f=40.0: delta1=2.6256 delta=2.6256 field_gap=10.7180 width_of_G=0.0005
set=A L=16 t_f=1.0: delta1=1.9996 delta=0.1233 field_gap=0.1522 width_of_G=0.0099
set=A L=16 t_f=40.0: delta1=0.0242 delta=0.0242 field_gap=6.0896 width_of_G=0.0005
set=B L=12 t_f=1.0: delta1=3.9238 delta=0.2288 field_gap=0.2679 width_of_G=0.0112
set=B L=12 t_f=40.0: delta1=3.9529 delta=3.9529 field_gap=10.7180 width_of_G=0.0003
set=B L=16 t_f=1.0: delta1=3.9295 delta=0.1234 field_gap=0.1522 width_of_G=0.0125
set=B L=16 t_f=40.0: delta1=3.9645 delta=3.9645 field_gap=6.0896 width_of_G=0.0003
set=D L=12 t_f=1.0: delta1=5.9315 delta=0.2251 field_gap=0.2679 width_of_G=0.0181
set=D L=12 t_f=40.0: delta1=5.9545 delta=5.9545 field_gap=10.7180 width_of_G=0.0005
set=D L=16 t_f=1.0: delta1=5.9380 delta=0.1187 field_gap=0.1522 width_of_G=0.0181
set=D L=16 t_f=40.0: delta1=5.9657 delta=5.9657 field_gap=6.0896 width_of_G=0.0005
leading-order sign rule (6.17) on the line: chi=0.05 R=1.0 c=0.25 sigma_i=-1 s_e=-1 Q-1=63.0
sign changes of D_0 - dE_0/c^2 in L in [100, 3000] (selected level): coupling (a): at [564]; coupling (c): at []
estimate L*_0 = 2 4^(1/3) a_0 (Q-1) chi^(-1/3) = 553
output/weight_check.json26 KB · raw
{
"cases": [
{
"L": 16,
"coupling": "a",
"g_s": -0.01,
"exact_levels": [
5.020932168593384,
5.020932866817886,
5.021348684865307,
5.021348858838961
],
"proj_levels": [
5.020942073578656,
5.020942775733207,
5.021358225261265,
5.02135841668269
],
"model_levels": [
5.020942215972731,
5.020942284482317,
5.021357976664915,
5.021358196156728
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]
}output/weight_check.txt12 KB · raw
L=16 coupling=a g_s=-0.01 beta=0.79213 E''(mu1+mu2)=0.81874 s_free=1.22138
exact spacings 6.982245e-07 4.158180e-04 1.739737e-07
projected spacings 7.021546e-07 4.154495e-04 1.914214e-07
frozen spacings 6.850959e-08 4.156922e-04 2.194918e-07
alt spacings 1.110095e-07 4.813398e-04 2.713286e-07
max |exact - projected| level = 9.909e-06 max |exact - frozen| level = 1.005e-05 G-weights [0.9998, 0.9998, 0.9997, 0.9997]
dW_nu/(dW_1+dW_2) = 0.86564 Pi_pair / inertia-weighted Pi_i = 0.70874
L=16 coupling=c g_s=-0.01 beta=0.79309 E''(mu1+mu2)=0.81896 s_free=1.22107
exact spacings 7.098712e-07 4.180159e-04 1.762302e-07
projected spacings 7.137852e-07 4.176475e-04 1.936438e-07
frozen spacings 6.911595e-08 4.178890e-04 2.197695e-07
alt spacings 1.116386e-07 4.839770e-04 2.712574e-07
max |exact - projected| level = 9.905e-06 max |exact - frozen| level = 1.008e-05 G-weights [0.9998, 0.9998, 0.9997, 0.9997]
dW_nu/(dW_1+dW_2) = 0.86585 Pi_pair / inertia-weighted Pi_i = 0.70909
L=20 coupling=a g_s=-0.01 beta=0.85027 E''(mu1+mu2)=0.86001 s_free=1.16277
exact spacings 1.873251e-07 3.194330e-04 1.331089e-07
projected spacings 1.895163e-07 3.189730e-04 1.378310e-07
frozen spacings 5.961929e-09 3.189027e-04 2.893572e-08
alt spacings 9.357663e-09 3.513435e-04 3.554769e-08
max |exact - projected| level = 1.021e-05 max |exact - frozen| level = 1.032e-05 G-weights [0.9998, 0.9998, 0.9998, 0.9998]
dW_nu/(dW_1+dW_2) = 0.84245 Pi_pair / inertia-weighted Pi_i = 0.72452
L=20 coupling=c g_s=-0.01 beta=0.85071 E''(mu1+mu2)=0.85933 s_free=1.16370
exact spacings 1.896182e-07 3.203664e-04 1.340624e-07
projected spacings 1.918173e-07 3.199055e-04 1.387595e-07
frozen spacings 5.984500e-09 3.198343e-04 2.890636e-08
alt spacings 9.403794e-09 3.526181e-04 3.551818e-08
max |exact - projected| level = 1.021e-05 max |exact - frozen| level = 1.033e-05 G-weights [0.9998, 0.9998, 0.9998, 0.9998]
dW_nu/(dW_1+dW_2) = 0.84259 Pi_pair / inertia-weighted Pi_i = 0.72406
L=16 coupling=a g_s=-0.005 beta=0.79213 E''(mu1+mu2)=0.81874 s_free=1.22138
exact spacings 1.133454e-06 3.606300e-04 9.206063e-08
projected spacings 1.131071e-06 3.606023e-04 9.903086e-08
frozen spacings 1.375967e-07 3.610983e-04 1.954679e-07
alt spacings 1.794658e-07 4.320447e-04 2.164773e-07
max |exact - projected| level = 2.347e-06 max |exact - frozen| level = 2.814e-06 G-weights [0.9998, 0.9998, 0.9998, 0.9998]
dW_nu/(dW_1+dW_2) = 0.86564 Pi_pair / inertia-weighted Pi_i = 0.70874
dW_nu/(dW_1+dW_2) at X=L/4, L=16, f_nu of set A fixed, couplings scaled: scale=1.0: 0.86564 scale=0.3: 0.96158 scale=0.1: 0.98843 scale=0.03: 0.99655
acceleration from rest (6.13), position-space band contract, L=24, g_s=-0.01, envelope centre L/4; ratios use the first-order part a - a(g_s=0):
set=A coupling=a weight_scale=1.0 w=1.0 beta=0.8864: a_Y=3.025839e-07 (g_s=0: -5.0e-10, profile cut at |r|<L/3: 5.1e-20) pred_Y=3.360821e-07 ratio=0.90181 | a_mid=3.116739e-07 (g_s=0: 7.3e-10) pred_mid=4.208025e-07 ratio=0.73893 | a_Y/a_mid=0.9747 (pred 0.7987) | a_Y/(body 1 alone)=0.56511
set=A coupling=a weight_scale=1.0 w=1.5 beta=0.8864: a_Y=3.143013e-07 (g_s=0: -1.1e-09, profile cut at |r|<L/3: -2.7e-16) pred_Y=3.262163e-07 ratio=0.96688 | a_mid=3.228798e-07 (g_s=0: 2.5e-10) pred_mid=4.132573e-07 ratio=0.78070 | a_Y/a_mid=0.9776 (pred 0.7894) | a_Y/(body 1 alone)=0.59876
set=A coupling=a weight_scale=1.0 w=2.0 beta=0.8864: a_Y=3.068671e-07 (g_s=0: -1.7e-09, profile cut at |r|<L/3: -2.0e-11) pred_Y=3.116323e-07 ratio=0.99018 | a_mid=3.148809e-07 (g_s=0: -3.5e-10) pred_mid=4.015528e-07 ratio=0.78504 | a_Y/a_mid=0.9789 (pred 0.7761) | a_Y/(body 1 alone)=0.60286
set=A coupling=a weight_scale=0.05 w=1.0 beta=0.8864: a_Y=2.437713e-08 (g_s=0: -5.0e-10, profile cut at |r|<L/3: 5.1e-20) pred_Y=2.820650e-08 ratio=0.88188 | a_mid=2.618880e-08 (g_s=0: 7.3e-10) pred_mid=2.888528e-08 ratio=0.88139 | a_Y/a_mid=0.9770 (pred 0.9765) | a_Y/(body 1 alone)=0.67848
set=A coupling=a weight_scale=0.05 w=1.5 beta=0.8864: a_Y=2.543175e-08 (g_s=0: -1.1e-09, profile cut at |r|<L/3: -2.7e-16) pred_Y=2.804659e-08 ratio=0.94639 | a_mid=2.733612e-08 (g_s=0: 2.5e-10) pred_mid=2.883917e-08 ratio=0.93917 | a_Y/a_mid=0.9800 (pred 0.9725) | a_Y/(body 1 alone)=0.72517
set=A coupling=a weight_scale=0.05 w=2.0 beta=0.8864: a_Y=2.500308e-08 (g_s=0: -1.7e-09, profile cut at |r|<L/3: -2.0e-11) pred_Y=2.752737e-08 ratio=0.97023 | a_mid=2.686336e-08 (g_s=0: -3.5e-10) pred_mid=2.860463e-08 ratio=0.95144 | a_Y/a_mid=0.9813 (pred 0.9623) | a_Y/(body 1 alone)=0.73594
set=D coupling=a weight_scale=1.0 w=1.0 beta=0.6433: a_Y=6.639336e-08 (g_s=0: -1.4e-11, profile cut at |r|<L/3: -5.3e-21) pred_Y=3.289029e-08 ratio=2.01906 | a_mid=6.431882e-08 (g_s=0: -1.4e-11) pred_mid=8.202624e-08 ratio=0.78430 | a_Y/a_mid=1.0322 (pred 0.4010) | a_Y/(body 1 alone)=0.08356
set=D coupling=a weight_scale=1.0 w=1.5 beta=0.6433: a_Y=7.255820e-08 (g_s=0: -1.2e-11, profile cut at |r|<L/3: -2.1e-16) pred_Y=3.333758e-08 ratio=2.17683 | a_mid=7.029147e-08 (g_s=0: -1.3e-11) pred_mid=9.845764e-08 ratio=0.71405 | a_Y/a_mid=1.0322 (pred 0.3386) | a_Y/(body 1 alone)=0.08678
set=D coupling=a weight_scale=1.0 w=2.0 beta=0.6433: a_Y=7.425992e-08 (g_s=0: -2.8e-11, profile cut at |r|<L/3: -1.8e-11) pred_Y=3.318638e-08 ratio=2.23851 | a_mid=7.191015e-08 (g_s=0: -2.9e-11) pred_mid=1.187907e-07 ratio=0.60559 | a_Y/a_mid=1.0327 (pred 0.2794) | a_Y/(body 1 alone)=0.08398
set=D coupling=a weight_scale=0.3 w=1.0 beta=0.6433: a_Y=5.835273e-09 (g_s=0: -1.4e-11, profile cut at |r|<L/3: -5.3e-21) pred_Y=2.925295e-09 ratio=1.99954 | a_mid=5.206001e-09 (g_s=0: -1.4e-11) pred_mid=1.031765e-08 ratio=0.50597 | a_Y/a_mid=1.1205 (pred 0.2835) | a_Y/(body 1 alone)=0.02476
set=D coupling=a weight_scale=0.3 w=1.5 beta=0.6433: a_Y=6.343591e-09 (g_s=0: -1.2e-11, profile cut at |r|<L/3: -2.1e-16) pred_Y=2.919537e-09 ratio=2.17696 | a_mid=5.668725e-09 (g_s=0: -1.3e-11) pred_mid=1.154503e-08 ratio=0.49210 | a_Y/a_mid=1.1187 (pred 0.2529) | a_Y/(body 1 alone)=0.02664
set=D coupling=a weight_scale=0.3 w=2.0 beta=0.6433: a_Y=6.397806e-09 (g_s=0: -2.8e-11, profile cut at |r|<L/3: -1.8e-11) pred_Y=2.861939e-09 ratio=2.24535 | a_mid=5.712721e-09 (g_s=0: -2.9e-11) pred_mid=1.299476e-08 ratio=0.44183 | a_Y/a_mid=1.1192 (pred 0.2202) | a_Y/(body 1 alone)=0.02678
set=D coupling=a weight_scale=0.1 w=1.0 beta=0.6433: a_Y=5.985289e-10 (g_s=0: -1.4e-11, profile cut at |r|<L/3: -5.3e-21) pred_Y=3.235397e-10 ratio=1.89310 | a_mid=3.891199e-10 (g_s=0: -1.4e-11) pred_mid=7.619453e-10 ratio=0.52961 | a_Y/a_mid=1.5178 (pred 0.4246) | a_Y/(body 1 alone)=0.00817
set=D coupling=a weight_scale=0.1 w=1.5 beta=0.6433: a_Y=6.741436e-10 (g_s=0: -1.2e-11, profile cut at |r|<L/3: -2.1e-16) pred_Y=3.225347e-10 ratio=2.12775 | a_mid=4.498680e-10 (g_s=0: -1.3e-11) pred_mid=8.949571e-10 ratio=0.51678 | a_Y/a_mid=1.4838 (pred 0.3604) | a_Y/(body 1 alone)=0.00913
set=D coupling=a weight_scale=0.1 w=2.0 beta=0.6433: a_Y=6.732493e-10 (g_s=0: -2.8e-11, profile cut at |r|<L/3: -1.8e-11) pred_Y=3.158159e-10 ratio=2.22122 | a_mid=4.459405e-10 (g_s=0: -2.9e-11) pred_mid=1.053808e-09 ratio=0.45045 | a_Y/a_mid=1.4778 (pred 0.2997) | a_Y/(body 1 alone)=0.00936
set=A coupling=c weight_scale=1.0 w=1.0 beta=0.8866: a_Y=3.035332e-07 (g_s=0: -5.4e-10, profile cut at |r|<L/3: -4.8e-20) pred_Y=3.371727e-07 ratio=0.90182 | a_mid=3.126140e-07 (g_s=0: 7.2e-10) pred_mid=4.226280e-07 ratio=0.73798 | a_Y/a_mid=0.9749 (pred 0.7978) | a_Y/(body 1 alone)=0.56484
set=A coupling=c weight_scale=1.0 w=1.5 beta=0.8866: a_Y=3.152710e-07 (g_s=0: -1.2e-09, profile cut at |r|<L/3: -2.7e-16) pred_Y=3.272718e-07 ratio=0.96688 | a_mid=3.238463e-07 (g_s=0: 2.3e-10) pred_mid=4.150500e-07 ratio=0.77969 | a_Y/a_mid=0.9778 (pred 0.7885) | a_Y/(body 1 alone)=0.59846
set=A coupling=c weight_scale=1.0 w=2.0 beta=0.8866: a_Y=3.077972e-07 (g_s=0: -1.8e-09, profile cut at |r|<L/3: -2.0e-11) pred_Y=3.126361e-07 ratio=0.99018 | a_mid=3.158101e-07 (g_s=0: -3.8e-10) pred_mid=4.032949e-07 ratio=0.78402 | a_Y/a_mid=0.9790 (pred 0.7752) | a_Y/(body 1 alone)=0.60255
set=A coupling=c weight_scale=0.05 w=1.0 beta=0.8866: a_Y=2.441800e-08 (g_s=0: -5.4e-10, profile cut at |r|<L/3: -4.8e-20) pred_Y=2.829501e-08 ratio=0.88189 | a_mid=2.625762e-08 (g_s=0: 7.2e-10) pred_mid=2.901059e-08 ratio=0.88018 | a_Y/a_mid=0.9772 (pred 0.9753) | a_Y/(body 1 alone)=0.67809
set=A coupling=c weight_scale=0.05 w=1.5 beta=0.8866: a_Y=2.546310e-08 (g_s=0: -1.2e-09, profile cut at |r|<L/3: -2.7e-16) pred_Y=2.813432e-08 ratio=0.94640 | a_mid=2.739908e-08 (g_s=0: 2.3e-10) pred_mid=2.896429e-08 ratio=0.93788 | a_Y/a_mid=0.9802 (pred 0.9713) | a_Y/(body 1 alone)=0.72473
set=A coupling=c weight_scale=0.05 w=2.0 beta=0.8866: a_Y=2.502292e-08 (g_s=0: -1.8e-09, profile cut at |r|<L/3: -2.0e-11) pred_Y=2.761296e-08 ratio=0.97023 | a_mid=2.691481e-08 (g_s=0: -3.8e-10) pred_mid=2.872876e-08 ratio=0.95012 | a_Y/a_mid=0.9815 (pred 0.9612) | a_Y/(body 1 alone)=0.73548
set=D coupling=c weight_scale=1.0 w=1.0 beta=0.6449: a_Y=6.742810e-08 (g_s=0: -1.6e-11, profile cut at |r|<L/3: 1.4e-20) pred_Y=3.342997e-08 ratio=2.01746 | a_mid=6.610455e-08 (g_s=0: -1.6e-11) pred_mid=8.478602e-08 ratio=0.77985 | a_Y/a_mid=1.0200 (pred 0.3943) | a_Y/(body 1 alone)=0.08338
set=D coupling=c weight_scale=1.0 w=1.5 beta=0.6449: a_Y=7.356673e-08 (g_s=0: -1.4e-11, profile cut at |r|<L/3: -2.2e-16) pred_Y=3.388404e-08 ratio=2.17153 | a_mid=7.212066e-08 (g_s=0: -1.4e-11) pred_mid=1.015293e-07 ratio=0.71048 | a_Y/a_mid=1.0200 (pred 0.3337) | a_Y/(body 1 alone)=0.08646
set=D coupling=c weight_scale=1.0 w=2.0 beta=0.6449: a_Y=7.523200e-08 (g_s=0: -3.0e-11, profile cut at |r|<L/3: -1.9e-11) pred_Y=3.372807e-08 ratio=2.23143 | a_mid=7.373322e-08 (g_s=0: -3.0e-11) pred_mid=1.222423e-07 ratio=0.60342 | a_Y/a_mid=1.0203 (pred 0.2759) | a_Y/(body 1 alone)=0.08360
set=D coupling=c weight_scale=0.3 w=1.0 beta=0.6449: a_Y=5.967265e-09 (g_s=0: -1.6e-11, profile cut at |r|<L/3: 1.4e-20) pred_Y=2.973757e-09 ratio=2.01187 | a_mid=5.565624e-09 (g_s=0: -1.6e-11) pred_mid=1.094378e-08 ratio=0.51002 | a_Y/a_mid=1.0719 (pred 0.2717) | a_Y/(body 1 alone)=0.02488
set=D coupling=c weight_scale=0.3 w=1.5 beta=0.6449: a_Y=6.450600e-09 (g_s=0: -1.4e-11, profile cut at |r|<L/3: -2.2e-16) pred_Y=2.967823e-09 ratio=2.17807 | a_mid=6.019886e-09 (g_s=0: -1.4e-11) pred_mid=1.219284e-08 ratio=0.49487 | a_Y/a_mid=1.0713 (pred 0.2434) | a_Y/(body 1 alone)=0.02662
set=D coupling=c weight_scale=0.3 w=2.0 beta=0.6449: a_Y=6.490784e-09 (g_s=0: -3.0e-11, profile cut at |r|<L/3: -1.9e-11) pred_Y=2.909075e-09 ratio=2.24145 | a_mid=6.053620e-09 (g_s=0: -3.0e-11) pred_mid=1.366554e-08 ratio=0.44519 | a_Y/a_mid=1.0718 (pred 0.2129) | a_Y/(body 1 alone)=0.02670
set=D coupling=c weight_scale=0.1 w=1.0 beta=0.6449: a_Y=6.243318e-10 (g_s=0: -1.6e-11, profile cut at |r|<L/3: 1.4e-20) pred_Y=3.289022e-10 ratio=1.94547 | a_mid=4.906209e-10 (g_s=0: -1.6e-11) pred_mid=9.258975e-10 ratio=0.54708 | a_Y/a_mid=1.2632 (pred 0.3552) | a_Y/(body 1 alone)=0.00839
set=D coupling=c weight_scale=0.1 w=1.5 beta=0.6449: a_Y=6.906040e-10 (g_s=0: -1.4e-11, profile cut at |r|<L/3: -2.2e-16) pred_Y=3.278715e-10 ratio=2.14758 | a_mid=5.474054e-10 (g_s=0: -1.4e-11) pred_mid=1.061170e-09 ratio=0.52900 | a_Y/a_mid=1.2543 (pred 0.3090) | a_Y/(body 1 alone)=0.00921
set=D coupling=c weight_scale=0.1 w=2.0 beta=0.6449: a_Y=6.850440e-10 (g_s=0: -3.0e-11, profile cut at |r|<L/3: -1.9e-11) pred_Y=3.210199e-10 ratio=2.22669 | a_mid=5.399314e-10 (g_s=0: -3.0e-11) pred_mid=1.221811e-09 ratio=0.46663 | a_Y/a_mid=1.2538 (pred 0.2627) | a_Y/(body 1 alone)=0.00937