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 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.
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"]
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),
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()))
AIRY = (1.0188, 2.3381) # first zero of Ai', 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; levels nu = 0, 1 counted from the extreme (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 = {0: None, 1: None}
cross = {0: [], 1: []}
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 - 2, n - 1) if se < 0 else (0, 1)
w, v = eigh_tridiagonal(Vf(r), np.full(n - 1, t0), select="i", select_range=sel)
order = [1, 0] if se < 0 else [0, 1]
for nu, k in enumerate(order):
f = v[:, k]
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[nu] is not None and np.sign(Dl) != np.sign(prev[nu][1]):
cross[nu].append(int(L))
prev[nu] = (L, Dl)
out["coupling_a" if sc > 0 else "coupling_c"] = dict(crossings_nu0=cross[0], crossings_nu1=cross[1])
est = [2 * 4 ** (1 / 3) * a * Qm1 * chi ** (-1 / 3) for a in AIRY]
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, extreme level, equal speeds); t_f = 1, 2 M_i > 4 t_f so that R1 holds for sigma = +1
"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),
"C": (dict(M=3.0, tau=0.20, gam=0.10, sig=+1), dict(M=4.0, tau=0.20, gam=-0.05, 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", "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_nu - dE_nu/c^2 in L in [100, 3000]: coupling (a): nu=0 at {cr['coupling_a']['crossings_nu0']}, "
f"nu=1 at {cr['coupling_a']['crossings_nu1']}; coupling (c): nu=0 at {cr['coupling_c']['crossings_nu0']}, nu=1 at {cr['coupling_c']['crossings_nu1']}")
lines.append(f" estimate L*_nu = 2 4^(1/3) a_nu (Q-1) chi^(-1/3): nu=0: {cr['L_star_estimate'][0]:.0f}, nu=1: {cr['L_star_estimate'][1]:.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).
"""
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"]))
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_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 name in ("A", "C"):
for cpl in ("a", "c"):
acc += accel_test(name, 24, 1.0, cpl, -0.01)
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})" 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.json21 KB · raw
{
"cases": [
{
"L": 12,
"coupling": "a",
"parity": 0,
"level": "low",
"Gweight": 0.9997316819577194,
"Eb0": 5.032130258807638,
"inv_Epp_exact": 227.88034635627238,
"inv_Epp_fd": 227.88103245814798,
"inv_Epp_band_ring": 227.68333688928885,
"inv_Epp_66": 254.73597240748586,
"inv_Epp_66_plus_ring": 227.68286354416537,
"ring_term_rel": 0.1188192578141604,
"ring_bound": 0.9808126137362798,
"f_antipode_sq": 0.009404815932082767,
"beta": 0.6931167961104955,
"mu1": 75.90797762064429,
"mu2": 101.11651860685852,
"mu_sum": 177.02449622750282,
"mubar_sum": 176.9974531366118,
"mu_over_beta": 255.40355856458265,
"rel_dev_band_ring_vs_exact": -0.0008645303122171155,
"rel_dev_66ring_vs_exact": -0.000866607477409409,
"rel_dev_mu_over_beta_vs_exact": 0.12077922755690373,
"D_exact": 50.85585012876956,
"D_lat": 78.37906233707983,
"D_mark_formula": -0.4331611470836669,
"D_lat_plus_mark": 77.94590118999616,
"sigR_over_c2_mu": [
0.9958662424874407,
0.995076547674216
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"coupling": "c",
"ratio_contact": -0.19035094178430928,
"ratio_all": [
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1.5883768060115475,
-0.704848597606271,
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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",
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},
{
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"delta": 0.12335625243097326,
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"set": "B",
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},
{
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"G_width": 0.0003278700262114853,
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},
{
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"delta": 0.22977519002333135,
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"G_width": 0.013619015725279127,
"set": "C",
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},
{
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"delta": 2.7149836697954868,
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"G_width": 0.0003740889035697137,
"set": "C",
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{
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{
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}
],
"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,
1269.3955046884137
],
"coupling_a": {
"crossings_nu0": [
564
],
"crossings_nu1": [
1216
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},
"coupling_c": {
"crossings_nu0": [],
"crossings_nu1": []
}
}
}output/inertia_check.txt9 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 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 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 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 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 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 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 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 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=C L=12 coupling=a level=high c1_over_c2=1 chi=1.7286 beta=-0.576737 inv_Epp_exact=-289.006 inv_Epp_fd=-289.006 inv_Epp_band_ring=-288.271 inv_Epp_66=-306.072 inv_Epp_66_plus_ring=-288.27 mu_sum=176.192 mu_over_beta=-305.498 rel_dev_band_ring_vs_exact=-0.00254237 rel_dev_66ring_vs_exact=-0.0025439 rel_dev_mu_over_beta_vs_exact=0.057065 ring_term_rel=0.061755 ring_bound=0.429752 f_antipode_sq=0.00435964 D_exact=-465.197 D_lat=-481.689 D_mark_formula=-0.437579 D_lat_plus_mark=-482.127 dE_cost=0.0167361 D_over_dE_times_c1sq=-1111.84 D_over_dE_times_c2sq=-1111.84
set=C L=12 coupling=c level=high c1_over_c2=1 chi=1.70608 beta=-0.581149 inv_Epp_exact=-281.917 inv_Epp_fd=-281.918 inv_Epp_band_ring=-281.164 inv_Epp_66=-298.893 inv_Epp_66_plus_ring=-281.164 mu_sum=173.836 mu_over_beta=-299.125 rel_dev_band_ring_vs_exact=-0.00267193 rel_dev_66ring_vs_exact=-0.00267358 rel_dev_mu_over_beta_vs_exact=0.0610369 ring_term_rel=0.0630575 ring_bound=0.439796 f_antipode_sq=0.00449956 D_exact=-455.753 D_lat=-472.961 D_mark_formula=0.426336 D_lat_plus_mark=-472.534 dE_cost=0.0168845 D_over_dE_times_c1sq=-1097.77 D_over_dE_times_c2sq=-1097.77
set=C L=16 coupling=a level=high c1_over_c2=1 chi=1.29558 beta=-0.651475 inv_Epp_exact=-269.923 inv_Epp_fd=-269.924 inv_Epp_band_ring=-269.287 inv_Epp_66=-271.2 inv_Epp_66_plus_ring=-269.287 mu_sum=176.22 mu_over_beta=-270.494 rel_dev_band_ring_vs_exact=-0.00235643 rel_dev_66ring_vs_exact=-0.00235768 rel_dev_mu_over_beta_vs_exact=0.00211469 ring_term_rel=0.0071041 ring_bound=0.0579808 f_antipode_sq=0.00031463 D_exact=-446.144 D_lat=-446.715 D_mark_formula=-0.462319 D_lat_plus_mark=-447.177 dE_cost=0.0169812 D_over_dE_times_c1sq=-1050.91 D_over_dE_times_c2sq=-1050.91
set=C L=16 coupling=c level=high c1_over_c2=1 chi=1.28453 beta=-0.654037 inv_Epp_exact=-265.673 inv_Epp_fd=-265.674 inv_Epp_band_ring=-265.02 inv_Epp_66=-266.924 inv_Epp_66_plus_ring=-265.019 mu_sum=174.662 mu_over_beta=-267.053 rel_dev_band_ring_vs_exact=-0.00246045 rel_dev_66ring_vs_exact=-0.00246123 rel_dev_mu_over_beta_vs_exact=0.00519182 ring_term_rel=0.00718622 ring_bound=0.0586878 f_antipode_sq=0.000321502 D_exact=-440.336 D_lat=-441.715 D_mark_formula=0.454412 D_lat_plus_mark=-441.26 dE_cost=0.0170946 D_over_dE_times_c1sq=-1043.27 D_over_dE_times_c2sq=-1043.27
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=C L=12 t_f=1.0: delta1=2.0232 delta=0.2298 field_gap=0.2679 width_of_G=0.0136
set=C L=12 t_f=40.0: delta1=2.7150 delta=2.7150 field_gap=10.7180 width_of_G=0.0004
set=C L=16 t_f=1.0: delta1=2.0116 delta=0.1216 field_gap=0.1522 width_of_G=0.0132
set=C L=16 t_f=40.0: delta1=0.0230 delta=0.0230 field_gap=6.0896 width_of_G=0.0004
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_nu - dE_nu/c^2 in L in [100, 3000]: coupling (a): nu=0 at [564], nu=1 at [1216]; coupling (c): nu=0 at [], nu=1 at []
estimate L*_nu = 2 4^(1/3) a_nu (Q-1) chi^(-1/3): nu=0: 553, nu=1: 1269
output/weight_check.json21 KB · raw
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"a_Y_gs0_profile_cut": -1.6455604386311734e-11,
"pred_Y": -4.252574691183626e-09,
"ratio_Y": 0.9859659861942981,
"a_mid": -7.217582275229117e-09,
"a_mid_gs0": -1.0324272361924302e-11,
"pred_mid": -7.272849585959628e-09,
"ratio_mid": 0.9909813090017616,
"aY_over_amid": 0.5817599422123864,
"pred_ratio": 0.5847191861898672
}
]
}output/weight_check.txt9 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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
set=C coupling=a weight_scale=1.0 w=1.0 beta=-0.7461: a_Y=-2.745196e-08 (g_s=0: 4.5e-12, profile cut at |r|<L/3: -1.4e-19) pred_Y=-4.217647e-08 ratio=0.65099 | a_mid=-7.739272e-08 (g_s=0: 3.4e-11) pred_mid=-3.998116e-08 ratio=1.93658 | a_Y/a_mid=0.3546 (pred 1.0549)
set=C coupling=a weight_scale=1.0 w=1.5 beta=-0.7461: a_Y=-2.191562e-08 (g_s=0: -1.1e-11, profile cut at |r|<L/3: -2.0e-16) pred_Y=-4.180738e-08 ratio=0.52395 | a_mid=-7.543235e-08 (g_s=0: 2.2e-11) pred_mid=-3.026213e-08 ratio=2.49337 | a_Y/a_mid=0.2903 (pred 1.3815)
set=C coupling=a weight_scale=1.0 w=2.0 beta=-0.7461: a_Y=-1.953707e-08 (g_s=0: -4.3e-11, profile cut at |r|<L/3: -1.6e-11) pred_Y=-4.134966e-08 ratio=0.47145 | a_mid=-7.396119e-08 (g_s=0: -1.0e-11) pred_mid=-1.798354e-08 ratio=4.11216 | a_Y/a_mid=0.2636 (pred 2.2993)
set=C coupling=a weight_scale=0.05 w=1.0 beta=-0.7461: a_Y=-4.348062e-09 (g_s=0: 4.5e-12, profile cut at |r|<L/3: -1.4e-19) pred_Y=-4.264482e-09 ratio=1.02066 | a_mid=-7.109098e-09 (g_s=0: 3.4e-11) pred_mid=-7.358091e-09 ratio=0.97079 | a_Y/a_mid=0.6093 (pred 0.5796)
set=C coupling=a weight_scale=0.05 w=1.5 beta=-0.7461: a_Y=-4.272892e-09 (g_s=0: -1.1e-11, profile cut at |r|<L/3: -2.0e-16) pred_Y=-4.258600e-09 ratio=1.00086 | a_mid=-7.210603e-09 (g_s=0: 2.2e-11) pred_mid=-7.325492e-09 ratio=0.98735 | a_Y/a_mid=0.5893 (pred 0.5813)
set=C coupling=a weight_scale=0.05 w=2.0 beta=-0.7461: a_Y=-4.198441e-09 (g_s=0: -4.3e-11, profile cut at |r|<L/3: -1.6e-11) pred_Y=-4.214671e-09 ratio=0.98605 | a_mid=-7.168725e-09 (g_s=0: -1.0e-11) pred_mid=-7.240728e-09 ratio=0.98868 | a_Y/a_mid=0.5805 (pred 0.5821)
set=C coupling=c weight_scale=1.0 w=1.0 beta=-0.7470: a_Y=-2.762754e-08 (g_s=0: 4.4e-12, profile cut at |r|<L/3: -7.5e-20) pred_Y=-4.242556e-08 ratio=0.65130 | a_mid=-7.776088e-08 (g_s=0: 3.4e-11) pred_mid=-4.006964e-08 ratio=1.94149 | a_Y/a_mid=0.3552 (pred 1.0588)
set=C coupling=c weight_scale=1.0 w=1.5 beta=-0.7470: a_Y=-2.207095e-08 (g_s=0: -1.1e-11, profile cut at |r|<L/3: -2.1e-16) pred_Y=-4.205433e-08 ratio=0.52456 | a_mid=-7.579603e-08 (g_s=0: 2.2e-11) pred_mid=-3.029658e-08 ratio=2.50253 | a_Y/a_mid=0.2910 (pred 1.3881)
set=C coupling=c weight_scale=1.0 w=2.0 beta=-0.7470: a_Y=-1.968500e-08 (g_s=0: -4.3e-11, profile cut at |r|<L/3: -1.6e-11) pred_Y=-4.159598e-08 ratio=0.47221 | a_mid=-7.432153e-08 (g_s=0: -1.0e-11) pred_mid=-1.795041e-08 ratio=4.13981 | a_Y/a_mid=0.2643 (pred 2.3173)
set=C coupling=c weight_scale=0.05 w=1.0 beta=-0.7470: a_Y=-4.384610e-09 (g_s=0: 4.4e-12, profile cut at |r|<L/3: -7.5e-20) pred_Y=-4.302900e-09 ratio=1.02000 | a_mid=-7.156214e-09 (g_s=0: 3.4e-11) pred_mid=-7.390794e-09 ratio=0.97286 | a_Y/a_mid=0.6104 (pred 0.5822)
set=C coupling=c weight_scale=0.05 w=1.5 beta=-0.7470: a_Y=-4.310288e-09 (g_s=0: -1.1e-11, profile cut at |r|<L/3: -2.1e-16) pred_Y=-4.296953e-09 ratio=1.00060 | a_mid=-7.259336e-09 (g_s=0: 2.2e-11) pred_mid=-7.358023e-09 ratio=0.98959 | a_Y/a_mid=0.5905 (pred 0.5840)
set=C coupling=c weight_scale=0.05 w=2.0 beta=-0.7470: a_Y=-4.235848e-09 (g_s=0: -4.3e-11, profile cut at |r|<L/3: -1.6e-11) pred_Y=-4.252575e-09 ratio=0.98597 | a_mid=-7.217582e-09 (g_s=0: -1.0e-11) pred_mid=-7.272850e-09 ratio=0.99098 | a_Y/a_mid=0.5818 (pred 0.5847)