07-ilang-weight / 07-inertia-of-pair-full-grip
07inertia of pair full gripverified
Determines, for two full-grip bodies bound through the field, the pair's inertia against its parts and its cost, and whether the binding cost is weighed as it resists acceleration.
check_pair_full_grip.py18 KB · raw
"""Check of 07-ilang-weight/07-inertia-of-pair-full-grip (N_f = 1, t_f = 1).
Item 1: distance of the group G from all other held levels of its parity class (field excitations,
single flips, joint flip), from the exact held spectra. Item 2: curvature in P of the exact bound level
of two full-grip bodies and one field object (sector contract continued in P) against (7.3), (7.5).
Item 3: cost (7.6), (7.8). Item 4: weight of the selected level from exact eigenvalues of the sector
contract with a source at fixed distance from the pair's midpoint (first order in g_s, odd part), and
Gamma_b = W_b'/D_0 against (7.9)-(7.12); held-field averages (v1) for comparison.
Outputs: output/check_log.txt, output/check_results.json.
"""
import json
from pathlib import Path
import numpy as np
import scipy.sparse as sp
import scipy.sparse.linalg as spla
OUT = Path(__file__).parent / "output"
TF = 1.0
SX = np.array([[0.0, 1.0], [1.0, 0.0]])
GS = 1e-5 # source strength (weights are taken as odd parts in g_s)
LOG = []
def log(s=""):
print(s, flush=True)
LOG.append(s)
def shift(L): # (S g)(y) = g(y+1)
return sp.csr_matrix((np.ones(L), (np.arange(L), (np.arange(L) + 1) % L)), shape=(L, L))
def field(L, src): # one field object: C_1 + sum_x g_x Phi_x
H = -TF * (np.eye(L, k=1) + np.eye(L, k=-1))
H[0, -1] = H[-1, 0] = -TF
for x, g in src:
H[x % L, x % L] += g
return H
def eps0(L, src):
return np.linalg.eigvalsh(field(L, src))[0]
def ground(L, src): # spectrum, eigenvectors, positive ground vector
e, u = np.linalg.eigh(field(L, src))
return e, u, u[:, 0] * np.sign(u[:, 0].sum())
def ell(L, x):
x = np.mod(x, L)
return (L * L - 1) / (12 * L) - x * (L - x) / (2 * L)
def curv(f, d): # E''(0) by Richardson; E(P) = E(-P) (sigma3 x sigma3 times conjugation)
e0, e1, e2 = f(0.0), f(d), f(2 * d)
return (4 * 2 * (e1 - e0) / d**2 - 2 * (e2 - e0) / (2 * d) ** 2) / 3, e0
def mindist(a, b): # min |x - y| over x in a, y in b
b = np.sort(np.asarray(b))
i = np.clip(np.searchsorted(b, a), 1, len(b) - 1)
return float(np.min(np.minimum(np.abs(a - b[i - 1]), np.abs(a - b[i]))))
# ---------- one body (sector contract (5.1), full grip); basis (branch, field place - body place) ----------
def body_level(L, g, tau, P, vec=False):
S = shift(L)
H = sp.kron(sp.identity(2), -TF * (S + S.T)) + sp.kron(sp.diags([1.0, -1.0]), sp.diags((np.arange(L) == 0) * g))
A = 0.5 * tau * np.exp(-1j * P) * sp.kron(sp.csr_matrix(-1j * SX), S)
w, v = np.linalg.eigh((H + A + A.conj().T).toarray())
om = ground(L, [(0, g)])[2]
i = np.argmax(np.abs(om @ v[:L, :]) ** 2)
return (w[i], v[:, i]) if vec else w[i]
def body_kappa(L, g, tau): # hop amplitude of the band contract of one body (second-order Schrieffer-Wolff)
e0, _, om = ground(L, [(0, g)])
ek, uk, _ = ground(L, [(0, -g)])
return -tau**2 / 4 * np.sum((uk.T @ np.roll(om, -1)) * (uk.T @ np.roll(om, 1)) / (e0[0] - ek))
# ---------- pair: sector contract, basis ((s1,s2), r = l-j, y = field place - j) ----------
def pair_ops(L, g1, g2, t1, t2, P):
S, I2, IL = shift(L), sp.identity(2), sp.identity(L)
sg, r, y = np.array([1.0, -1.0]), np.arange(L), np.arange(L)
dg = (sg[:, None, None, None] * g1 * (y == 0)[None, None, None, :]
+ sg[None, :, None, None] * g2 * (y[None, :] == r[:, None])[None, None, :, :])
H0 = sp.kron(sp.identity(4 * L), -TF * (S + S.T)) + sp.diags(dg.ravel())
mX = sp.csr_matrix(-1j * SX)
A1 = 0.5 * t1 * np.exp(-1j * P) * sp.kron(sp.kron(mX, I2), sp.kron(S, S))
A2 = 0.5 * t2 * sp.kron(sp.kron(I2, mX), sp.kron(S.T, IL))
return (H0 + A1 + A1.conj().T + A2 + A2.conj().T).tocsr()
def parity_idx(L): # the class r + s1 + s2 even (conserved), which contains the selected level
s1, s2, r, _ = np.meshgrid(*(np.arange(k) for k in (2, 2, L, L)), indexing='ij')
return np.flatnonzero(((s1 + s2 + r) % 2 == 0).ravel())
def midpoint_weights(L, z): # source of unit strength at distance z from the pair's midpoint j + r/2
w = np.zeros((L, L))
for r in range(L):
h = r - L if r > L // 2 else r
for a in ([h // 2] if h % 2 == 0 else [(h - 1) // 2, (h + 1) // 2]):
w[r, (z + a) % L] += 1.0 if h % 2 == 0 else 0.5
return np.tile(w.ravel(), 4)
def band_contract(L, g1, g2, t1, t2):
"""Band contract (2.10) at P = 0 from the exact held blocks (class r even): E(r), d_i(r) on even r,
kappa_i(rho) on odd rho (bond between rho-1 and rho+1); held spectra of the class."""
E, Om = np.zeros(L), np.zeros((L, L))
spec = {"G": [], "exc": [], "flip": [], "joint": []}
for r in range(0, L, 2):
e, _, om = ground(L, [(0, g1), (r, g2)])
E[r], Om[r] = e[0], om
spec["G"].append(e[0])
spec["exc"] += list(e[1:])
spec["joint"] += list(np.linalg.eigvalsh(field(L, [(0, -g1), (r, -g2)])))
dd, kk = np.zeros((2, L)), np.zeros((2, L))
for rm in range(1, L, 2):
up, dn = (rm + 1) % L, rm - 1
for i, (src, t, fu, fd) in enumerate((([(0, -g1), (rm, g2)], t1, np.roll(Om[up], -1), np.roll(Om[dn], 1)),
([(0, g1), (rm, -g2)], t2, Om[up], Om[dn]))):
ek, uk = np.linalg.eigh(field(L, src))
spec["flip"] += list(ek)
ou, od, q = uk.T @ fu, uk.T @ fd, t**2 / 4
dd[i, up] += q * np.sum(ou**2 / (E[up] - ek))
dd[i, dn] += q * np.sum(od**2 / (E[dn] - ek))
kk[i, rm] = -q * np.sum(ou * od * 0.5 * (1 / (E[up] - ek) + 1 / (E[dn] - ek)))
return E, Om, dd, kk, {k: np.array(v) for k, v in spec.items()}
def run_case(name, L, g1, g2, t1, t2, dist=(), eig_s=None, d=0.02):
n, l0 = 1.0 / L, ell(L, 0)
ev = np.arange(0, L, 2)
rs = np.where(ev > L // 2, ev - L, ev) # signed even r
E, Om, dd, kk, spec = band_contract(L, g1, g2, t1, t2)
kb1, kb2 = body_kappa(L, g1, t1), body_kappa(L, g2, t2)
low = (kb1 + kb2) < 0 # positive relative inertia: lowest edge
U = E[ev] + dd[0][ev] + dd[1][ev]
i0, i1 = np.arange(len(ev)), (np.arange(len(ev)) + 1) % len(ev)
def band(P):
h = np.diag(U).astype(complex)
t = kk[0][ev + 1] * np.exp(-2j * P) + kk[1][ev + 1] # <r|h|r+2>, bond rho = r+1
h[i0, i1] += t
h[i1, i0] += t.conj()
w, v = np.linalg.eigh(h)
k = 0 if low else -1
return w[k], v[:, k]
idx = parity_idx(L)
cache = {}
def exact(P):
if P not in cache:
eb, fb = band(P)
w, v = spla.eigsh(pair_ops(L, g1, g2, t1, t2, P)[idx][:, idx].tocsc(), k=8, sigma=eb, which='LM', tol=0)
best = (-1,)
for k in range(len(w)):
x = np.zeros(4 * L * L, complex)
x[idx] = v[:, k]
a = np.einsum('ry,ry->r', Om, x.reshape(2, 2, L, L)[0, 0])[ev]
ov = abs(np.vdot(fb, a)) ** 2
if ov > best[0]:
best = (ov, w[k], np.sum(np.abs(a) ** 2), x)
cache[P] = best
return cache[P]
Eb2, Eb0 = curv(lambda P: exact(P)[1], d)
ov0, _, gw0, v0 = exact(0.0)
ovmin = min(c[0] for c in cache.values())
Ebb2, _ = curv(lambda P: band(P)[0], d)
mu = [1.0 / curv(lambda P: body_level(L, g, t, P), d)[0] for g, t in ((g1, t1), (g2, t2))]
E1, E2 = body_level(L, g1, t1, 0.0), body_level(L, g2, t2, 0.0)
D0 = 1.0 / Eb2 - mu[0] - mu[1]
# band-contract quantities at P = 0
_, f0 = band(0.0)
f0 = (f0 * np.exp(-1j * np.angle(f0[0]))).real
rho = (ev + 1) % L
b0 = f0 * np.roll(f0, -1) # f0(rho-1) f0(rho+1), rho = r+1
k1, k2 = kk[0][ev + 1], kk[1][ev + 1]
beta0 = b0.sum()
B = lambda x: -2 * n * g1 * g2 / TF * ell(L, x)
ra = np.arange(L)
bar = lambda f: 0.25 * (f[(rho - 1) % L] + 2 * f[rho] + f[(rho + 1) % L])
Mfull = lambda a, b: np.array([eps0(L, [(0, a), (r, b)]) - eps0(L, [(0, a)]) - eps0(L, [(0, b)]) + eps0(L, []) for r in ra])
Mpp = Mfull(g1, g2)
Mo = [(Mpp - Mfull(-g1, g2)) / 2, (Mpp - Mfull(g1, -g2)) / 2] # parts of M odd in g1, in g2
sR = [(eps0(L, [(0, g)]) - eps0(L, [(0, -g)])) / 2 for g in (g1, g2)]
cb = [sR[0] * (-8 * kb1), sR[1] * (-8 * kb2)] # c_i^2 = sigma_i R_i / mu_i (band contract)
ce = [sR[0] / mu[0], sR[1] / mu[1]] # c_i^2 with the exact mu_i
Dlat = (mu[0] + mu[1]) * (1 - beta0) / beta0
Dmark2 = (1 / t1**2 + 1 / t2**2) * np.sum(b0 * bar(B(ra))) / beta0**2
dmo = bar(Mo[0]) / ce[0] + bar(Mo[1]) / ce[1]
Dmark = np.sum(b0 * dmo) / beta0**2
Dres = 1 / np.sum(b0 / (mu[0] + mu[1] + dmo)) - mu[0] - mu[1]
Dband = 1 / Ebb2 + 1 / (8 * kb1) + 1 / (8 * kb2)
Sc = abs(Dlat) + abs(Dmark)
Mavg = np.sum(f0**2 * Mpp[ev])
Moavg = [np.sum(f0**2 * Mo[0][ev]), np.sum(f0**2 * Mo[1][ev])]
K0 = 2 * (-1 / (8 * mu[0]) - 1 / (8 * mu[1])) * (beta0 - 1)
dE0 = Eb0 - E1 - E2 + (-2 * TF)
w_est = (4 * abs(kb1 + kb2) / abs(n * g1 * g2 / TF)) ** (1 / 3)
delta_m = abs(mu[0] + mu[1]) * (1 - beta0) / (abs(B(0)) * (1 / t1**2 + 1 / t2**2)) # S5 parameter
# distances of G from the other held levels of its class (item 1, S2)
G = spec["G"]
dfl, djo = mindist(G, np.concatenate([spec["flip"], spec["exc"]])), mindist(G, spec["joint"])
dall = min(dfl, djo)
djo_lead = 2 * n * abs(g1 + g2) - 1.5 * abs(B(0))
# local inertia (7.3): m_i(rho) - mubar_i against Mbar_i^odd(rho)/c_i^2 and Mbar^(2)(rho)/tau_i^2, |rho| <= 3
near = np.abs(np.where(rho > L // 2, rho - L, rho)) <= 3
loc, loc2 = [], []
for k, kb, c, Mi, t in ((k1, kb1, cb[0], Mo[0], t1), (k2, kb2, cb[1], Mo[1], t2)):
sh = (-1 / (8 * k) + 1 / (8 * kb))[near]
loc.append(float(np.max(np.abs(sh / (bar(Mi) / c)[near] - 1))))
loc2.append(float(np.max(np.abs(sh / (bar(B(ra)) / t**2)[near] - 1))))
res = dict(case=name, L=L, g1=g1, g2=g2, tau1=t1, tau2=t2, eta=max(abs(g1), abs(g2)) * l0 / TF,
w=w_est, mean_abs_r=float(np.sum(f0**2 * np.abs(rs))), overlap=ov0, min_overlap=ovmin,
group_weight=gw0, selected="lowest" if low else "highest", d_all=dall, d_flip=dfl, d_joint=djo,
d_joint_lead=djo_lead, tau_over_d=max(t1, t2) / dall, mu1=mu[0], mu2=mu[1],
mu1_band=-1 / (8 * kb1), mu2_band=-1 / (8 * kb2),
Epp_exact=Eb2, Epp_band=Ebb2, Epp_66=float(np.sum(b0 / (-1 / (8 * k1) - 1 / (8 * k2)))), beta0=beta0,
delta_m=delta_m, D0_exact=D0, D0_band=Dband, Dlat=Dlat, Dmark_odd=Dmark, Dmark_lead=Dmark2,
D0_formula=Dlat + Dmark, D0_resummed=Dres, D0_lead=Dlat + Dmark2, local_inertia_relerr_odd=loc,
local_inertia_relerr_lead=loc2, dE0_exact=dE0, M_avg=Mavg, M_odd_avg=Moavg,
M0_second_order=float(B(0)), K0=K0, dE0_formula=Mavg + K0)
log(f"== {name}: L={L} g1={g1:.5g} g2={g2:.5g} tau1={t1:.4g} tau2={t2:.4g} eta={res['eta']:.3f} w={w_est:.2f}")
log(f" held spectra of the class: d={dall:.4e} (flips and excitations {dfl:.4e}, joint flip {djo:.4e}; "
f"leading 2n|g1+g2|-1.5|B(0)|={djo_lead:.4e}); max tau/d={max(t1, t2) / dall:.3f}")
log(f" selected {res['selected']} level, <|r|>={res['mean_abs_r']:.2f}, overlap={ov0:.6f} (min over P {ovmin:.6f}), group weight={gw0:.6f}")
log(f" E''(0): exact={Eb2:.6e} band(2.10)={Ebb2:.6e} (6.6) sum b0/m={res['Epp_66']:.6e} rel. band-exact={(Ebb2 - Eb2) / Eb2:.2e}")
log(f" mu1={mu[0]:.6e} mu2={mu[1]:.6e} (band: {-1 / (8 * kb1):.6e}, {-1 / (8 * kb2):.6e}) beta0={beta0:.5f}")
log(f" Dlat={Dlat:.4e} Dmark(odd parts)={Dmark:.4e} Dmark(lead)={Dmark2:.4e} |Dlat/Dmark|={abs(Dlat / Dmark):.3f} delta_m(S5)={delta_m:.3f}")
log(f" D0: exact={D0:.6e} band={Dband:.6e} (7.5)={Dlat + Dmark:.6e} resummed={Dres:.6e} leading={Dlat + Dmark2:.6e}")
log(f" deviation from exact D0 / (|Dlat|+|Dmark|): band {(Dband - D0) / Sc:.2e}, (7.5) {(Dlat + Dmark - D0) / Sc:.2e}, resummed {(Dres - D0) / Sc:.2e}, leading {(Dlat + Dmark2 - D0) / Sc:.2e}")
log(f" local inertia (7.3), max rel. error near r=0: odd parts {loc[0]:.2e}, {loc[1]:.2e}; second order {loc2[0]:.2e}, {loc2[1]:.2e}")
log(f" cost: dE0 exact={dE0:.6e} <M>+K0={Mavg + K0:.6e} (<M>={Mavg:.4e}, M2(0)={B(0):.4e}, K0={K0:.4e}); (<M1odd>+<M2odd>)/<M>={(Moavg[0] + Moavg[1]) / Mavg:.4f}")
log(f" D0_exact/dE0={D0 / dE0:.6e} vs 1/tau1^2+1/tau2^2={1 / t1**2 + 1 / t2**2:.6e} (ratio {D0 / dE0 / (1 / t1**2 + 1 / t2**2):.4f}), 1/tau1^2={1 / t1**2:.6e}")
# ---------- item 4: weights ----------
# exact selected level: field density at distance z from the pair's midpoint in the exact P=0 eigenvector
# (first-order shift of the selected level by a source g_s at that place, Hellmann-Feynman)
pr = (np.abs(v0) ** 2).reshape(4, L, L).sum(0)
dens = np.zeros(L)
for r in range(L):
h = r - L if r > L // 2 else r
dens += np.roll(pr[r], -(h // 2)) if h % 2 == 0 else 0.5 * (np.roll(pr[r], -((h - 1) // 2)) + np.roll(pr[r], -((h + 1) // 2)))
dens_b = [np.abs(u[:L]) ** 2 + np.abs(u[L:]) ** 2 for u in (body_level(L, g, t, 0.0, vec=True)[1] for g, t in ((g1, t1), (g2, t2)))]
res['weight'] = []
for s in dist:
dz = lambda f: -GS * (f[(s + 1) % L] - f[(s - 1) % L]) / 2
W0p, W1p, W2p = dz(dens), dz(dens_b[0]), dz(dens_b[1])
phip = -(2 * GS / TF) * (ell(L, 1 - s) - ell(L, -1 - s)) / 2
u = s / L
ks = (L * L * (1 - 2 * u) ** 2 + 2) / (L * L - 1)
Vavg = np.sum(f0**2 * n * g1 * g2 / TF * np.mod(rs, L) * (L - np.mod(rs, L)) / L)
Cev = [-n * g**2 / TF * l0 for g in (g1, g2)]
# held-field average (v1): f0^2-average of exact held weights, no kinetic part
X, xs = 0, s
Wh = lambda src: (eps0(L, src + [(xs, GS)]) - eps0(L, src + [(xs, -GS)])) / 2
dW = lambda fun: (fun(X + 1) - fun(X - 1)) / 2
Wp = lambda Xc: np.sum([f0[i] ** 2 * Wh([(Xc - rs[i] // 2, g1), (Xc - rs[i] // 2 + rs[i], g2)]) for i in range(len(ev))])
W0h, W1h, W2h = dW(Wp), dW(lambda x: Wh([(x, g1)])), dW(lambda x: Wh([(x, g2)]))
Wbh = W0h - W1h - W2h
Wbp = W0p - W1p - W2p
Wbf = phip * (2 * ks * B(0) + Vavg - K0) # (7.10)
G1, G2, Gp, Gb = W1p / mu[0], W2p / mu[1], Eb2 * W0p, Wbp / D0
Gb_lead = 2 * ks * phip / (1 / t1**2 + 1 / t2**2) # (7.12)
Gb_f = Wbf / (Dlat + Dmark2)
Gb_r = (Wbh - K0 * phip) / (Dlat + Dmark) # same structure with exact held weights and exact odd parts of M
wr = dict(s=s, u=u, kappa_s=ks, phip=phip, W0p=W0p, W1p=W1p, W2p=W2p, Wbp=Wbp, Wbp_formula=Wbf,
W0p_heldavg=W0h, Wbp_heldavg=Wbh, kinetic=(Wbp - Wbh) / phip, kinetic_formula=-K0,
Gamma1=G1, Gamma2=G2, Gamma_pair=Gp, Gamma_bond=Gb, Gamma_bond_lead=Gb_lead,
Gamma_bond_formula=Gb_f, Gamma_bond_refined=Gb_r, eta_over_ks=res['eta'] / ks, w_over_Lks=w_est / (L * ks))
log(f" source at distance s={s} from the midpoint (u={u:.4f}, kappa_s={ks:.4f}; eta/kappa_s={wr['eta_over_ks']:.3f}, w/(L kappa_s)={wr['w_over_Lks']:.3f}):")
log(f" exact selected level: W0'/phi'={W0p / phip:.6e} Wb'/phi'={Wbp / phip:.6e} (7.10): {Wbf / phip:.6e}")
log(f" held-field average: W0'/phi'={W0h / phip:.6e} Wb'/phi'={Wbh / phip:.6e}; kinetic part (exact - held avg)/phi'={wr['kinetic']:.4e} vs -K0={-K0:.4e}")
log(f" Gamma/phi': body1={G1 / phip:.6e} (tau1^2={t1**2:.6e}) body2={G2 / phip:.6e} (tau2^2={t2**2:.6e}) pair={Gp / phip:.6e}")
log(f" Gamma_b=Wb'/D0: /phi'={Gb / phip:.6e} (7.12) lead={Gb_lead / phip:.6e} (ratio {Gb / Gb_lead:.4f}) (7.10)/(7.5)={Gb_f / phip:.6e} (ratio {Gb / Gb_f:.4f}) refined={Gb_r / phip:.6e} (ratio {Gb / Gb_r:.4f})")
log(f" Gamma_b/Gamma_1={Gb / G1:.4f} Gamma_b/Gamma_2={Gb / G2:.4f} lead 2kappa_s/(1+tau1^2/tau2^2)={2 * ks / (1 + t1**2 / t2**2):.4f}")
if s == eig_s: # the same weight from exact eigenvalues of the sector contract with the source, +-g_s
H0 = pair_ops(L, g1, g2, t1, t2, 0.0)[idx][:, idx].tocsc()
dE = {}
for z in (s - 1, s + 1):
src = midpoint_weights(L, z)[idx]
for sg in (1, -1):
w, v = spla.eigsh(H0 + sp.diags(sg * GS * src), k=6, sigma=Eb0, which='LM', tol=0)
dE[z, sg] = w[np.argmax(np.abs(v.conj().T @ v0[idx]))]
W0e = -((dE[s + 1, 1] - dE[s + 1, -1]) - (dE[s - 1, 1] - dE[s - 1, -1])) / 4
wr['W0p_eigenvalues'] = W0e
log(f" exact eigenvalues with the source (+-g_s): W0'/phi'={W0e / phip:.6e}, rel. diff. to the density form {(W0e - W0p) / W0p:.2e}; Wb'/phi' from them {(W0e - W1p - W2p) / phip:.6e}")
res['weight'].append(wr)
return res
def params(L, eta1, q, r1, r2=None):
"""g1 from eta1 = |g1| l0 / t_f, g2 = q g1, tau_i = r_i d with d = 2 n min|g_i|."""
r2 = r1 if r2 is None else r2
g1 = eta1 * TF / ell(L, 0)
d = 2 * min(abs(g1), abs(q * g1)) / L
return L, g1, q * g1, r1 * d, r2 * d
CASES = [("A equal bodies", (32, 0.15, 1.0, 0.3), (), None),
("A equal bodies", (64, 0.15, 1.0, 0.3), (8, 16), 8),
("A' equal bodies, smaller eta", (64, 0.05, 1.0, 0.3), (8, 16), None),
("A'' equal bodies, smaller eta and tau/d", (64, 0.05, 1.0, 0.15), (), None),
("B unequal bodies and speeds", (32, 0.15, 0.5, 0.3, 0.15), (), None),
("B unequal bodies and speeds", (64, 0.15, 0.5, 0.3, 0.15), (8, 16), None),
("R equal bodies, v1 case", (128, 0.15, 1.0, 0.3), (8, 16, 32), None),
("S1 equal bodies, regime (ii)", (256, 0.10, 1.0, 0.4), (16, 32, 64), 32),
("S2 equal bodies, regime (ii), smaller eta", (256, 0.05, 1.0, 0.4), (16, 32, 64), None),
("C mixed signs, bound at top", (48, 0.075, -2.0, 0.15, 0.3), (), None),
("D control: both negative, not bound", (48, -0.15, 1.0, 0.3), (), None)]
if __name__ == "__main__":
OUT.mkdir(exist_ok=True)
results = [run_case(name, *params(*p), dist=dist, eig_s=es) for name, p, dist, es in CASES]
(OUT / "check_log.txt").write_text("\n".join(LOG) + "\n", encoding="utf-8")
(OUT / "check_results.json").write_text(json.dumps(results, indent=1, default=float), encoding="utf-8")
output/check_log.txt24 KB · raw
== A equal bodies: L=32 g1=0.056305 g2=0.056305 tau1=0.001056 tau2=0.001056 eta=0.150 w=1.83
held spectra of the class: d=2.7731e-03 (flips and excitations 2.7731e-03, joint flip 6.4749e-03; leading 2n|g1+g2|-1.5|B(0)|=6.2463e-03); max tau/d=0.381
selected lowest level, <|r|>=1.34, overlap=0.984031 (min over P 0.983940), group weight=0.984065
E''(0): exact=3.147829e-04 band(2.10)=3.291362e-04 (6.6) sum b0/m=3.291357e-04 rel. band-exact=4.56e-02
mu1=1.639644e+03 mu2=1.639644e+03 (band: 1.639640e+03, 1.639640e+03) beta0=0.87303
Dlat=4.7693e+02 Dmark(odd parts)=-6.9527e+02 Dmark(lead)=-7.7827e+02 |Dlat/Dmark|=0.686 delta_m(S5)=0.440
D0: exact=-1.024964e+02 band=-2.410248e+02 (7.5)=-2.183369e+02 resummed=-2.265128e+02 leading=-3.013431e+02
deviation from exact D0 / (|Dlat|+|Dmark|): band -1.18e-01, (7.5) -9.88e-02, resummed -1.06e-01, leading -1.70e-01
local inertia (7.3), max rel. error near r=0: odd parts 2.15e-02, 2.15e-02; second order 9.31e-02, 9.31e-02
cost: dE0 exact=-2.510425e-04 <M>+K0=-2.591129e-04 (<M>=-2.9783e-04, M2(0)=-5.2786e-04, K0=3.8719e-05); (<M1odd>+<M2odd>)/<M>=2.3765
D0_exact/dE0=4.082829e+05 vs 1/tau1^2+1/tau2^2=1.794460e+06 (ratio 0.2275), 1/tau1^2=8.972299e+05
== A equal bodies: L=64 g1=0.028132 g2=0.028132 tau1=0.0002637 tau2=0.0002637 eta=0.150 w=2.31
held spectra of the class: d=6.8091e-04 (flips and excitations 6.8091e-04, joint flip 1.6178e-03; leading 2n|g1+g2|-1.5|B(0)|=1.5604e-03); max tau/d=0.387
selected lowest level, <|r|>=1.85, overlap=0.989809 (min over P 0.989709), group weight=0.989820
E''(0): exact=8.707640e-05 band(2.10)=8.958103e-05 (6.6) sum b0/m=8.958104e-05 rel. band-exact=2.88e-02
mu1=6.564038e+03 mu2=6.564038e+03 (band: 6.564034e+03, 6.564034e+03) beta0=0.92320
Dlat=1.0921e+03 Dmark(odd parts)=-2.9762e+03 Dmark(lead)=-3.3527e+03 |Dlat/Dmark|=0.367 delta_m(S5)=0.266
D0: exact=-1.643909e+03 band=-1.964989e+03 (7.5)=-1.884097e+03 resummed=-1.900351e+03 leading=-2.260628e+03
deviation from exact D0 / (|Dlat|+|Dmark|): band -7.89e-02, (7.5) -5.90e-02, resummed -6.30e-02, leading -1.52e-01
local inertia (7.3), max rel. error near r=0: odd parts 2.20e-02, 2.20e-02; second order 9.87e-02, 9.87e-02
cost: dE0 exact=-7.278875e-05 <M>+K0=-7.423568e-05 (<M>=-8.0086e-05, M2(0)=-1.3187e-04, K0=5.8499e-06); (<M1odd>+<M2odd>)/<M>=2.3900
D0_exact/dE0=2.258465e+07 vs 1/tau1^2+1/tau2^2=2.875347e+07 (ratio 0.7855), 1/tau1^2=1.437674e+07
source at distance s=8 from the midpoint (u=0.1250, kappa_s=0.5631; eta/kappa_s=0.266, w/(L kappa_s)=0.064):
exact selected level: W0'/phi'=6.396094e-04 Wb'/phi'=-1.049401e-04 (7.10): -1.326958e-04
held-field average: W0'/phi'=6.469698e-04 Wb'/phi'=-9.757629e-05; kinetic part (exact - held avg)/phi'=-7.3638e-06 vs -K0=-5.8499e-06
Gamma/phi': body1=5.671429e-08 (tau1^2=6.955682e-08) body2=5.671429e-08 (tau2^2=6.955682e-08) pair=5.569489e-08
Gamma_b=Wb'/D0: /phi'=6.383572e-08 (7.12) lead=3.916924e-08 (ratio 1.6297) (7.10)/(7.5)=5.869865e-08 (ratio 1.0875) refined=5.489432e-08 (ratio 1.1629)
Gamma_b/Gamma_1=1.1256 Gamma_b/Gamma_2=1.1256 lead 2kappa_s/(1+tau1^2/tau2^2)=0.5631
exact eigenvalues with the source (+-g_s): W0'/phi'=6.396094e-04, rel. diff. to the density form 5.93e-09; Wb'/phi' from them -1.049401e-04
source at distance s=16 from the midpoint (u=0.2500, kappa_s=0.2505; eta/kappa_s=0.599, w/(L kappa_s)=0.144):
exact selected level: W0'/phi'=7.492984e-04 Wb'/phi'=-6.124736e-05 (7.10): -5.025811e-05
held-field average: W0'/phi'=7.576416e-04 Wb'/phi'=-5.289702e-05; kinetic part (exact - held avg)/phi'=-8.3503e-06 vs -K0=-5.8499e-06
Gamma/phi': body1=6.174139e-08 (tau1^2=6.955682e-08) body2=6.174139e-08 (tau2^2=6.955682e-08) pair=6.524620e-08
Gamma_b=Wb'/D0: /phi'=3.725716e-08 (7.12) lead=1.742742e-08 (ratio 2.1378) (7.10)/(7.5)=2.223192e-08 (ratio 1.6758) refined=3.118042e-08 (ratio 1.1949)
Gamma_b/Gamma_1=0.6034 Gamma_b/Gamma_2=0.6034 lead 2kappa_s/(1+tau1^2/tau2^2)=0.2505
== A' equal bodies, smaller eta: L=64 g1=0.0093773 g2=0.0093773 tau1=8.791e-05 tau2=8.791e-05 eta=0.050 w=3.37
held spectra of the class: d=2.6730e-04 (flips and excitations 2.6730e-04, joint flip 5.6646e-04; leading 2n|g1+g2|-1.5|B(0)|=5.6410e-04); max tau/d=0.329
selected lowest level, <|r|>=2.42, overlap=0.995019 (min over P 0.994939), group weight=0.995019
E''(0): exact=2.660595e-05 band(2.10)=2.699290e-05 (6.6) sum b0/m=2.699290e-05 rel. band-exact=1.45e-02
mu1=1.903544e+04 mu2=1.903544e+04 (band: 1.903544e+04, 1.903544e+04) beta0=0.95175
Dlat=1.9301e+03 Dmark(odd parts)=-2.9375e+03 Dmark(lead)=-3.0812e+03 |Dlat/Dmark|=0.657 delta_m(S5)=0.484
D0: exact=-4.853071e+02 band=-1.024096e+03 (7.5)=-1.007359e+03 resummed=-1.016930e+03 leading=-1.151092e+03
deviation from exact D0 / (|Dlat|+|Dmark|): band -1.11e-01, (7.5) -1.07e-01, resummed -1.09e-01, leading -1.37e-01
local inertia (7.3), max rel. error near r=0: odd parts 2.63e-03, 2.63e-03; second order 4.81e-02, 4.81e-02
cost: dE0 exact=-9.041333e-06 <M>+K0=-9.135153e-06 (<M>=-1.0403e-05, M2(0)=-1.4652e-05, K0=1.2674e-06); (<M1odd>+<M2odd>)/<M>=2.1118
D0_exact/dE0=5.367650e+07 vs 1/tau1^2+1/tau2^2=2.587813e+08 (ratio 0.2074), 1/tau1^2=1.293906e+08
source at distance s=8 from the midpoint (u=0.1250, kappa_s=0.5631; eta/kappa_s=0.089, w/(L kappa_s)=0.093):
exact selected level: W0'/phi'=2.630864e-04 Wb'/phi'=-1.397144e-05 (7.10): -1.466959e-05
held-field average: W0'/phi'=2.644494e-04 Wb'/phi'=-1.260842e-05; kinetic part (exact - held avg)/phi'=-1.3630e-06 vs -K0=-1.2674e-06
Gamma/phi': body1=7.277422e-09 (tau1^2=7.728535e-09) body2=7.277422e-09 (tau2^2=7.728535e-09) pair=6.999665e-09
Gamma_b=Wb'/D0: /phi'=2.878888e-08 (7.12) lead=4.352137e-09 (ratio 6.6149) (7.10)/(7.5)=1.274406e-08 (ratio 2.2590) refined=1.377446e-08 (ratio 2.0900)
Gamma_b/Gamma_1=3.9559 Gamma_b/Gamma_2=3.9559 lead 2kappa_s/(1+tau1^2/tau2^2)=0.5631
source at distance s=16 from the midpoint (u=0.2500, kappa_s=0.2505; eta/kappa_s=0.200, w/(L kappa_s)=0.210):
exact selected level: W0'/phi'=2.791184e-04 Wb'/phi'=-6.442146e-06 (7.10): -5.509840e-06
held-field average: W0'/phi'=2.805571e-04 Wb'/phi'=-5.003244e-06; kinetic part (exact - held avg)/phi'=-1.4389e-06 vs -K0=-1.2674e-06
Gamma/phi': body1=7.500760e-09 (tau1^2=7.728535e-09) body2=7.500760e-09 (tau2^2=7.728535e-09) pair=7.426210e-09
Gamma_b=Wb'/D0: /phi'=1.327437e-08 (7.12) lead=1.936380e-09 (ratio 6.8553) (7.10)/(7.5)=4.786620e-09 (ratio 2.7732) refined=6.224842e-09 (ratio 2.1325)
Gamma_b/Gamma_1=1.7697 Gamma_b/Gamma_2=1.7697 lead 2kappa_s/(1+tau1^2/tau2^2)=0.2505
== A'' equal bodies, smaller eta and tau/d: L=64 g1=0.0093773 g2=0.0093773 tau1=4.396e-05 tau2=4.396e-05 eta=0.050 w=2.12
held spectra of the class: d=2.6730e-04 (flips and excitations 2.6730e-04, joint flip 5.6646e-04; leading 2n|g1+g2|-1.5|B(0)|=5.6410e-04); max tau/d=0.164
selected lowest level, <|r|>=1.28, overlap=0.996424 (min over P 0.996404), group weight=0.996424
E''(0): exact=6.119499e-06 band(2.10)=6.180253e-06 (6.6) sum b0/m=6.180249e-06 rel. band-exact=9.93e-03
mu1=7.614176e+04 mu2=7.614176e+04 (band: 7.614176e+04, 7.614176e+04) beta0=0.86471
Dlat=2.3826e+04 Dmark(odd parts)=-1.4252e+04 Dmark(lead)=-1.4986e+04 |Dlat/Dmark|=1.672 delta_m(S5)=1.358
D0: exact=1.112856e+04 band=9.522158e+03 (7.5)=9.573626e+03 resummed=9.558786e+03 leading=8.839926e+03
deviation from exact D0 / (|Dlat|+|Dmark|): band -4.22e-02, (7.5) -4.08e-02, resummed -4.12e-02, leading -6.01e-02
local inertia (7.3), max rel. error near r=0: odd parts 2.63e-03, 2.63e-03; second order 4.81e-02, 4.81e-02
cost: dE0 exact=-1.063404e-05 <M>+K0=-1.071125e-05 (<M>=-1.1600e-05, M2(0)=-1.4652e-05, K0=8.8841e-07); (<M1odd>+<M2odd>)/<M>=2.1189
D0_exact/dE0=-1.046504e+09 vs 1/tau1^2+1/tau2^2=1.035125e+09 (ratio -1.0110), 1/tau1^2=5.175625e+08
== B unequal bodies and speeds: L=32 g1=0.056305 g2=0.028152 tau1=0.0005279 tau2=0.0002639 eta=0.150 w=1.33
held spectra of the class: d=1.3636e-03 (flips and excitations 1.3636e-03, joint flip 4.9956e-03; leading 2n|g1+g2|-1.5|B(0)|=4.8827e-03); max tau/d=0.387
selected lowest level, <|r|>=0.57, overlap=0.989682 (min over P 0.989658), group weight=0.989686
E''(0): exact=4.097370e-05 band(2.10)=4.211294e-05 (6.6) sum b0/m=4.211294e-05 rel. band-exact=2.78e-02
mu1=6.558566e+03 mu2=1.274622e+04 (band: 6.558562e+03, 1.274622e+04) beta0=0.67184
Dlat=9.4292e+03 Dmark(odd parts)=-4.9274e+03 Dmark(lead)=-5.5740e+03 |Dlat/Dmark|=1.914 delta_m(S5)=1.338
D0: exact=5.101118e+03 band=4.440890e+03 (7.5)=4.501822e+03 resummed=4.488702e+03 leading=3.855201e+03
deviation from exact D0 / (|Dlat|+|Dmark|): band -4.60e-02, (7.5) -4.17e-02, resummed -4.27e-02, leading -8.68e-02
local inertia (7.3), max rel. error near r=0: odd parts 2.30e-02, 6.15e-03; second order 2.01e-02, 1.31e-01
cost: dE0 exact=-1.627373e-04 <M>+K0=-1.660816e-04 (<M>=-1.8503e-04, M2(0)=-2.6393e-04, K0=1.8945e-05); (<M1odd>+<M2odd>)/<M>=2.2971
D0_exact/dE0=-3.134572e+07 vs 1/tau1^2+1/tau2^2=1.794460e+07 (ratio -1.7468), 1/tau1^2=3.588920e+06
== B unequal bodies and speeds: L=64 g1=0.028132 g2=0.014066 tau1=0.0001319 tau2=6.593e-05 eta=0.150 w=1.67
held spectra of the class: d=3.3549e-04 (flips and excitations 3.3549e-04, joint flip 1.2481e-03; leading 2n|g1+g2|-1.5|B(0)|=1.2198e-03); max tau/d=0.393
selected lowest level, <|r|>=1.00, overlap=0.993976 (min over P 0.993949), group weight=0.993978
E''(0): exact=1.276961e-05 band(2.10)=1.298382e-05 (6.6) sum b0/m=1.298382e-05 rel. band-exact=1.68e-02
mu1=2.625614e+04 mu2=5.102339e+04 (band: 2.625613e+04, 5.102339e+04) beta0=0.81280
Dlat=1.7798e+04 Dmark(odd parts)=-1.7860e+04 Dmark(lead)=-2.0330e+04 |Dlat/Dmark|=0.997 delta_m(S5)=0.763
D0: exact=1.031368e+03 band=-2.605738e+02 (7.5)=-6.138594e+01 resummed=-8.967688e+01 leading=-2.532073e+03
deviation from exact D0 / (|Dlat|+|Dmark|): band -3.62e-02, (7.5) -3.06e-02, resummed -3.14e-02, leading -9.99e-02
local inertia (7.3), max rel. error near r=0: odd parts 2.50e-02, 5.36e-03; second order 2.02e-02, 1.39e-01
cost: dE0 exact=-4.346807e-05 <M>+K0=-4.397624e-05 (<M>=-4.6676e-05, M2(0)=-6.5934e-05, K0=2.6996e-06); (<M1odd>+<M2odd>)/<M>=2.2972
D0_exact/dE0=-2.372703e+07 vs 1/tau1^2+1/tau2^2=2.875347e+08 (ratio -0.0825), 1/tau1^2=5.750694e+07
source at distance s=8 from the midpoint (u=0.1250, kappa_s=0.5631; eta/kappa_s=0.266, w/(L kappa_s)=0.046):
exact selected level: W0'/phi'=5.157547e-04 Wb'/phi'=-5.861085e-05 (7.10): -7.099485e-05
held-field average: W0'/phi'=5.189716e-04 Wb'/phi'=-5.539348e-05; kinetic part (exact - held avg)/phi'=-3.2174e-06 vs -K0=-2.6996e-06
Gamma/phi': body1=1.417853e-08 (tau1^2=1.738920e-08) body2=3.960774e-09 (tau2^2=4.347301e-09) pair=6.585988e-09
Gamma_b=Wb'/D0: /phi'=-5.682826e-08 (7.12) lead=3.916924e-09 (ratio -14.5084) (7.10)/(7.5)=2.803824e-08 (ratio -2.0268) refined=9.463588e-07 (ratio -0.0600)
Gamma_b/Gamma_1=-4.0081 Gamma_b/Gamma_2=-14.3478 lead 2kappa_s/(1+tau1^2/tau2^2)=0.2253
source at distance s=16 from the midpoint (u=0.2500, kappa_s=0.2505; eta/kappa_s=0.599, w/(L kappa_s)=0.104):
exact selected level: W0'/phi'=5.830558e-04 Wb'/phi'=-3.352277e-05 (7.10): -2.977600e-05
held-field average: W0'/phi'=5.865617e-04 Wb'/phi'=-3.001577e-05; kinetic part (exact - held avg)/phi'=-3.5070e-06 vs -K0=-2.6996e-06
Gamma/phi': body1=1.543525e-08 (tau1^2=1.738920e-08) body2=4.141402e-09 (tau2^2=4.347301e-09) pair=7.445398e-09
Gamma_b=Wb'/D0: /phi'=-3.250321e-08 (7.12) lead=1.742742e-09 (ratio -18.6506) (7.10)/(7.5)=1.175954e-08 (ratio -2.7640) refined=5.329462e-07 (ratio -0.0610)
Gamma_b/Gamma_1=-2.1058 Gamma_b/Gamma_2=-7.8484 lead 2kappa_s/(1+tau1^2/tau2^2)=0.1002
== R equal bodies, v1 case: L=128 g1=0.014063 g2=0.014063 tau1=6.592e-05 tau2=6.592e-05 eta=0.150 w=2.91
held spectra of the class: d=1.6868e-04 (flips and excitations 1.6868e-04, joint flip 4.0439e-04; leading 2n|g1+g2|-1.5|B(0)|=3.9004e-04); max tau/d=0.391
selected lowest level, <|r|>=2.45, overlap=0.993446 (min over P 0.993339), group weight=0.993450
E''(0): exact=2.315069e-05 band(2.10)=2.357385e-05 (6.6) sum b0/m=2.357385e-05 rel. band-exact=1.83e-02
mu1=2.626162e+04 mu2=2.626162e+04 (band: 2.626161e+04, 2.626161e+04) beta0=0.95282
Dlat=2.6009e+03 Dmark(odd parts)=-1.2403e+04 Dmark(lead)=-1.4037e+04 |Dlat/Dmark|=0.210 delta_m(S5)=0.163
D0: exact=-9.327978e+03 band=-1.010333e+04 (7.5)=-9.801789e+03 resummed=-9.831106e+03 leading=-1.143575e+04
deviation from exact D0 / (|Dlat|+|Dmark|): band -5.17e-02, (7.5) -3.16e-02, resummed -3.35e-02, leading -1.40e-01
local inertia (7.3), max rel. error near r=0: odd parts 2.20e-02, 2.20e-02; second order 1.02e-01, 1.02e-01
cost: dE0 exact=-1.995791e-05 <M>+K0=-2.020866e-05 (<M>=-2.1107e-05, M2(0)=-3.2961e-05, K0=8.9831e-07); (<M1odd>+<M2odd>)/<M>=2.4030
D0_exact/dE0=4.673824e+08 vs 1/tau1^2+1/tau2^2=4.602241e+08 (ratio 1.0156), 1/tau1^2=2.301120e+08
source at distance s=8 from the midpoint (u=0.0625, kappa_s=0.7658; eta/kappa_s=0.196, w/(L kappa_s)=0.030):
exact selected level: W0'/phi'=1.433803e-04 Wb'/phi'=-3.246042e-05 (7.10): -4.772353e-05
held-field average: W0'/phi'=1.444181e-04 Wb'/phi'=-3.142246e-05; kinetic part (exact - held avg)/phi'=-1.0380e-06 vs -K0=-8.9831e-07
Gamma/phi': body1=3.347865e-09 (tau1^2=4.345709e-09) body2=3.347865e-09 (tau2^2=4.345709e-09) pair=3.319352e-09
Gamma_b=Wb'/D0: /phi'=3.479899e-09 (7.12) lead=3.327917e-09 (ratio 1.0457) (7.10)/(7.5)=4.173188e-09 (ratio 0.8339) refined=3.297436e-09 (ratio 1.0553)
Gamma_b/Gamma_1=1.0394 Gamma_b/Gamma_2=1.0394 lead 2kappa_s/(1+tau1^2/tau2^2)=0.7658
source at distance s=16 from the midpoint (u=0.1250, kappa_s=0.5627; eta/kappa_s=0.267, w/(L kappa_s)=0.040):
exact selected level: W0'/phi'=1.597441e-04 Wb'/phi'=-2.638446e-05 (7.10): -3.433231e-05
held-field average: W0'/phi'=1.608867e-04 Wb'/phi'=-2.524156e-05; kinetic part (exact - held avg)/phi'=-1.1429e-06 vs -K0=-8.9831e-07
Gamma/phi': body1=3.543738e-09 (tau1^2=4.345709e-09) body2=3.543738e-09 (tau2^2=4.345709e-09) pair=3.698186e-09
Gamma_b=Wb'/D0: /phi'=2.828529e-09 (7.12) lead=2.445141e-09 (ratio 1.1568) (7.10)/(7.5)=3.002192e-09 (ratio 0.9422) refined=2.666847e-09 (ratio 1.0606)
Gamma_b/Gamma_1=0.7982 Gamma_b/Gamma_2=0.7982 lead 2kappa_s/(1+tau1^2/tau2^2)=0.5627
source at distance s=32 from the midpoint (u=0.2500, kappa_s=0.2501; eta/kappa_s=0.600, w/(L kappa_s)=0.091):
exact selected level: W0'/phi'=1.868889e-04 Wb'/phi'=-1.573377e-05 (7.10): -1.373043e-05
held-field average: W0'/phi'=1.881882e-04 Wb'/phi'=-1.443405e-05; kinetic part (exact - held avg)/phi'=-1.2997e-06 vs -K0=-8.9831e-07
Gamma/phi': body1=3.857771e-09 (tau1^2=4.345709e-09) body2=3.857771e-09 (tau2^2=4.345709e-09) pair=4.326606e-09
Gamma_b=Wb'/D0: /phi'=1.686729e-09 (7.12) lead=1.087024e-09 (ratio 1.5517) (7.10)/(7.5)=1.200659e-09 (ratio 1.4048) refined=1.564241e-09 (ratio 1.0783)
Gamma_b/Gamma_1=0.4372 Gamma_b/Gamma_2=0.4372 lead 2kappa_s/(1+tau1^2/tau2^2)=0.2501
== S1 equal bodies, regime (ii): L=256 g1=0.0046876 g2=0.0046876 tau1=1.465e-05 tau2=1.465e-05 eta=0.100 w=5.12
held spectra of the class: d=3.0428e-05 (flips and excitations 3.0428e-05, joint flip 6.8870e-05; leading 2n|g1+g2|-1.5|B(0)|=6.7750e-05); max tau/d=0.481
selected lowest level, <|r|>=4.13, overlap=0.996119 (min over P 0.995947), group weight=0.996120
E''(0): exact=6.694777e-06 band(2.10)=6.769117e-06 (6.6) sum b0/m=6.769122e-06 rel. band-exact=1.11e-02
mu1=8.675530e+04 mu2=8.675530e+04 (band: 8.675528e+04, 8.675528e+04) beta0=0.98232
Dlat=3.1235e+03 Dmark(odd parts)=-2.8583e+04 Dmark(lead)=-3.1396e+04 |Dlat/Dmark|=0.109 delta_m(S5)=0.090
D0: exact=-2.414042e+04 band=-2.578081e+04 (7.5)=-2.545921e+04 resummed=-2.549118e+04 leading=-2.827274e+04
deviation from exact D0 / (|Dlat|+|Dmark|): band -5.17e-02, (7.5) -4.16e-02, resummed -4.26e-02, leading -1.30e-01
local inertia (7.3), max rel. error near r=0: odd parts 1.03e-02, 1.03e-02; second order 8.43e-02, 8.43e-02
cost: dE0 exact=-2.533427e-06 <M>+K0=-2.552421e-06 (<M>=-2.6543e-06, M2(0)=-3.6622e-06, K0=1.0192e-07); (<M1odd>+<M2odd>)/<M>=2.2556
D0_exact/dE0=9.528760e+09 vs 1/tau1^2+1/tau2^2=9.320391e+09 (ratio 1.0224), 1/tau1^2=4.660196e+09
source at distance s=16 from the midpoint (u=0.0625, kappa_s=0.7657; eta/kappa_s=0.131, w/(L kappa_s)=0.026):
exact selected level: W0'/phi'=2.737495e-05 Wb'/phi'=-4.139222e-06 (7.10): -5.364688e-06
held-field average: W0'/phi'=2.748649e-05 Wb'/phi'=-4.027682e-06; kinetic part (exact - held avg)/phi'=-1.1154e-07 vs -K0=-1.0192e-07
Gamma/phi': body1=1.816268e-10 (tau1^2=2.145833e-10) body2=1.816268e-10 (tau2^2=2.145833e-10) pair=1.832692e-10
Gamma_b=Wb'/D0: /phi'=1.714644e-10 (7.12) lead=1.642994e-10 (ratio 1.0436) (7.10)/(7.5)=1.897478e-10 (ratio 0.9036) refined=1.622045e-10 (ratio 1.0571)
Gamma_b/Gamma_1=0.9440 Gamma_b/Gamma_2=0.9440 lead 2kappa_s/(1+tau1^2/tau2^2)=0.7657
source at distance s=32 from the midpoint (u=0.1250, kappa_s=0.5625; eta/kappa_s=0.178, w/(L kappa_s)=0.036):
exact selected level: W0'/phi'=2.950067e-05 Wb'/phi'=-3.259660e-06 (7.10): -3.876911e-06
held-field average: W0'/phi'=2.962028e-05 Wb'/phi'=-3.140019e-06; kinetic part (exact - held avg)/phi'=-1.1964e-07 vs -K0=-1.0192e-07
Gamma/phi': body1=1.888088e-10 (tau1^2=2.145833e-10) body2=1.888088e-10 (tau2^2=2.145833e-10) pair=1.975004e-10
Gamma_b=Wb'/D0: /phi'=1.350291e-10 (7.12) lead=1.207115e-10 (ratio 1.1186) (7.10)/(7.5)=1.371254e-10 (ratio 0.9847) refined=1.273384e-10 (ratio 1.0604)
Gamma_b/Gamma_1=0.7152 Gamma_b/Gamma_2=0.7152 lead 2kappa_s/(1+tau1^2/tau2^2)=0.5625
exact eigenvalues with the source (+-g_s): W0'/phi'=2.950067e-05, rel. diff. to the density form -4.67e-08; Wb'/phi' from them -3.259661e-06
source at distance s=64 from the midpoint (u=0.2500, kappa_s=0.2500; eta/kappa_s=0.400, w/(L kappa_s)=0.080):
exact selected level: W0'/phi'=3.295124e-05 Wb'/phi'=-1.781834e-06 (7.10): -1.588023e-06
held-field average: W0'/phi'=3.308310e-05 Wb'/phi'=-1.649848e-06; kinetic part (exact - held avg)/phi'=-1.3199e-07 vs -K0=-1.0192e-07
Gamma/phi': body1=2.001784e-10 (tau1^2=2.145833e-10) body2=2.001784e-10 (tau2^2=2.145833e-10) pair=2.206012e-10
Gamma_b=Wb'/D0: /phi'=7.381122e-11 (7.12) lead=5.365318e-11 (ratio 1.3757) (7.10)/(7.5)=5.616799e-11 (ratio 1.3141) refined=6.880666e-11 (ratio 1.0727)
Gamma_b/Gamma_1=0.3687 Gamma_b/Gamma_2=0.3687 lead 2kappa_s/(1+tau1^2/tau2^2)=0.2500
== S2 equal bodies, regime (ii), smaller eta: L=256 g1=0.0023438 g2=0.0023438 tau1=7.324e-06 tau2=7.324e-06 eta=0.050 w=6.48
held spectra of the class: d=1.6639e-05 (flips and excitations 1.6639e-05, joint flip 3.5396e-05; leading 2n|g1+g2|-1.5|B(0)|=3.5248e-05); max tau/d=0.440
selected lowest level, <|r|>=4.83, overlap=0.997536 (min over P 0.997388), group weight=0.997536
E''(0): exact=3.122416e-06 band(2.10)=3.144965e-06 (6.6) sum b0/m=3.144960e-06 rel. band-exact=7.22e-03
mu1=1.713587e+05 mu2=1.713587e+05 (band: 1.713587e+05, 1.713587e+05) beta0=0.98691
Dlat=4.5440e+03 Dmark(odd parts)=-2.9193e+04 Dmark(lead)=-3.0730e+04 |Dlat/Dmark|=0.156 delta_m(S5)=0.131
D0: exact=-2.245256e+04 band=-2.474878e+04 (7.5)=-2.464917e+04 resummed=-2.467203e+04 leading=-2.618559e+04
deviation from exact D0 / (|Dlat|+|Dmark|): band -6.81e-02, (7.5) -6.51e-02, resummed -6.58e-02, leading -1.11e-01
local inertia (7.3), max rel. error near r=0: odd parts 2.70e-03, 2.70e-03; second order 5.02e-02, 5.02e-02
cost: dE0 exact=-6.878422e-07 <M>+K0=-6.912139e-07 (<M>=-7.2940e-07, M2(0)=-9.1554e-07, K0=3.8181e-08); (<M1odd>+<M2odd>)/<M>=2.1189
D0_exact/dE0=3.264201e+10 vs 1/tau1^2+1/tau2^2=3.728156e+10 (ratio 0.8756), 1/tau1^2=1.864078e+10
source at distance s=16 from the midpoint (u=0.0625, kappa_s=0.7657; eta/kappa_s=0.065, w/(L kappa_s)=0.033):
exact selected level: W0'/phi'=1.578923e-05 Wb'/phi'=-1.183756e-06 (7.10): -1.339821e-06
held-field average: W0'/phi'=1.582897e-05 Wb'/phi'=-1.144013e-06; kinetic part (exact - held avg)/phi'=-3.9743e-08 vs -K0=-3.8181e-08
Gamma/phi': body1=4.952472e-11 (tau1^2=5.364582e-11) body2=4.952472e-11 (tau2^2=5.364582e-11) pair=4.930054e-11
Gamma_b=Wb'/D0: /phi'=5.272256e-11 (7.12) lead=4.107484e-11 (ratio 1.2836) (7.10)/(7.5)=5.116636e-11 (ratio 1.0304) refined=4.796082e-11 (ratio 1.0993)
Gamma_b/Gamma_1=1.0646 Gamma_b/Gamma_2=1.0646 lead 2kappa_s/(1+tau1^2/tau2^2)=0.7657
source at distance s=32 from the midpoint (u=0.1250, kappa_s=0.5625; eta/kappa_s=0.089, w/(L kappa_s)=0.045):
exact selected level: W0'/phi'=1.641720e-05 Wb'/phi'=-8.959661e-07 (7.10): -9.678770e-07
held-field average: W0'/phi'=1.645848e-05 Wb'/phi'=-8.547355e-07; kinetic part (exact - held avg)/phi'=-4.1231e-08 vs -K0=-3.8181e-08
Gamma/phi': body1=5.051732e-11 (tau1^2=5.364582e-11) body2=5.051732e-11 (tau2^2=5.364582e-11) pair=5.126132e-11
Gamma_b=Wb'/D0: /phi'=3.990486e-11 (7.12) lead=3.017787e-11 (ratio 1.3223) (7.10)/(7.5)=3.696220e-11 (ratio 1.0796) refined=3.622502e-11 (ratio 1.1016)
Gamma_b/Gamma_1=0.7899 Gamma_b/Gamma_2=0.7899 lead 2kappa_s/(1+tau1^2/tau2^2)=0.5625
source at distance s=64 from the midpoint (u=0.2500, kappa_s=0.2500; eta/kappa_s=0.200, w/(L kappa_s)=0.101):
exact selected level: W0'/phi'=1.741125e-05 Wb'/phi'=-4.331032e-07 (7.10): -3.956550e-07
held-field average: W0'/phi'=1.745487e-05 Wb'/phi'=-3.894352e-07; kinetic part (exact - held avg)/phi'=-4.3668e-08 vs -K0=-3.8181e-08
Gamma/phi': body1=5.206725e-11 (tau1^2=5.364582e-11) body2=5.206725e-11 (tau2^2=5.364582e-11) pair=5.436516e-11
Gamma_b=Wb'/D0: /phi'=1.928971e-11 (7.12) lead=1.341330e-11 (ratio 1.4381) (7.10)/(7.5)=1.510965e-11 (ratio 1.2766) refined=1.734811e-11 (ratio 1.1119)
Gamma_b/Gamma_1=0.3705 Gamma_b/Gamma_2=0.3705 lead 2kappa_s/(1+tau1^2/tau2^2)=0.2500
== C mixed signs, bound at top: L=48 g1=0.018758 g2=-0.037516 tau1=0.0001172 tau2=0.0002345 eta=0.150 w=1.04
held spectra of the class: d=6.2169e-04 (flips and excitations 6.5773e-04, joint flip 6.2169e-04; leading 2n|g1+g2|-1.5|B(0)|=6.0573e-04); max tau/d=0.377
selected highest level, <|r|>=0.14, overlap=0.987247 (min over P 0.987241), group weight=0.987251
E''(0): exact=1.418394e-05 band(2.10)=1.463866e-05 (6.6) sum b0/m=1.463867e-05 rel. band-exact=3.21e-02
mu1=2.869504e+04 mu2=-1.471817e+04 (band: 2.869504e+04, -1.471817e+04) beta0=0.36195
Dlat=2.4639e+04 Dmark(odd parts)=2.9466e+04 Dmark(lead)=2.5768e+04 |Dlat/Dmark|=0.836 delta_m(S5)=0.836
D0: exact=5.652541e+04 band=5.433537e+04 (7.5)=5.410423e+04 resummed=5.408365e+04 leading=5.040706e+04
deviation from exact D0 / (|Dlat|+|Dmark|): band -4.05e-02, (7.5) -4.48e-02, resummed -4.51e-02, leading -1.13e-01
local inertia (7.3), max rel. error near r=0: odd parts 6.24e-03, 2.04e-02; second order 1.99e-01, 2.71e-02
cost: dE0 exact=1.185405e-04 <M>+K0=1.214758e-04 (<M>=1.2675e-04, M2(0)=1.1724e-04, K0=-5.2789e-06); (<M1odd>+<M2odd>)/<M>=1.9451
D0_exact/dE0=4.768447e+08 vs 1/tau1^2+1/tau2^2=9.094323e+07 (ratio 5.2433), 1/tau1^2=7.275458e+07
== D control: both negative, not bound: L=48 g1=-0.037516 g2=-0.037516 tau1=0.000469 tau2=0.000469 eta=0.150 w=2.10
held spectra of the class: d=1.3363e-03 (flips and excitations 1.3363e-03, joint flip 2.8765e-03; leading 2n|g1+g2|-1.5|B(0)|=2.7746e-03); max tau/d=0.351
selected highest level, <|r|>=21.42, overlap=0.994877 (min over P 0.994788), group weight=0.994878
E''(0): exact=-1.511180e-04 band(2.10)=-1.533368e-04 (6.6) sum b0/m=-1.533368e-04 rel. band-exact=1.47e-02
mu1=-3.679545e+03 mu2=-3.679545e+03 (band: -3.679543e+03, -3.679543e+03) beta0=0.95602
Dlat=-3.3855e+02 Dmark(odd parts)=1.1696e+03 Dmark(lead)=1.0493e+03 |Dlat/Dmark|=0.289 delta_m(S5)=0.152
D0: exact=7.417469e+02 band=8.374918e+02 (7.5)=8.310697e+02 resummed=8.322954e+02 leading=7.106973e+02
deviation from exact D0 / (|Dlat|+|Dmark|): band 6.35e-02, (7.5) 5.92e-02, resummed 6.00e-02, leading -2.06e-02
local inertia (7.3), max rel. error near r=0: odd parts 2.45e-02, 2.45e-02; second order 2.47e-01, 2.47e-01
cost: dE0 exact=1.268030e-04 <M>+K0=1.278234e-04 (<M>=1.3380e-04, M2(0)=-2.3448e-04, K0=-5.9765e-06); (<M1odd>+<M2odd>)/<M>=1.8116
D0_exact/dE0=5.849600e+06 vs 1/tau1^2+1/tau2^2=9.094323e+06 (ratio 0.6432), 1/tau1^2=4.547161e+06
output/check_results.json31 KB · raw
[
{
"case": "A equal bodies",
"L": 32,
"g1": 0.0563049853372434,
"g2": 0.0563049853372434,
"tau1": 0.0010557184750733138,
"tau2": 0.0010557184750733138,
"eta": 0.15,
"w": 1.8327471329658809,
"mean_abs_r": 1.3449393969979602,
"overlap": 0.9840311641666921,
"min_overlap": 0.9839401351422165,
"group_weight": 0.9840650866868894,
"selected": "lowest",
"d_all": 0.0027731216888811083,
"d_flip": 0.0027731216888811083,
"d_joint": 0.006474870145792178,
"d_joint_lead": 0.00624633431085044,
"tau_over_d": 0.3806967719109623,
"mu1": 1639.6442960847244,
"mu2": 1639.6442960847244,
"mu1_band": 1639.6404827348354,
"mu2_band": 1639.6404827348354,
"Epp_exact": 0.00031478294008439417,
"Epp_band": 0.00032913617433368475,
"Epp_66": 0.00032913570363746355,
"beta0": 0.8730288754581559,
"delta_m": 0.43957473808855857,
"D0_exact": -102.4963564406089,
"D0_band": -241.02483326409856,
"Dlat": 476.9314875484351,
"Dmark_odd": -695.2683465833387,
"Dmark_lead": -778.2745416727439,
"D0_formula": -218.33685903490363,
"D0_resummed": -226.5127572911092,
"D0_lead": -301.3430541243088,
"local_inertia_relerr_odd": [
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],
"local_inertia_relerr_lead": [
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],
"dE0_exact": -0.00025104249557394986,
"M_avg": -0.0002978319839226902,
"M_odd_avg": [
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-0.0003538940645471576
],
"M0_second_order": -0.0005278592375366569,
"K0": 3.871910659069045e-05,
"dE0_formula": -0.00025911287733199973,
"weight": []
},
{
"case": "A equal bodies",
"L": 64,
"g1": 0.028131868131868132,
"g2": 0.028131868131868132,
"tau1": 0.0002637362637362637,
"tau2": 0.0002637362637362637,
"eta": 0.15,
"w": 2.309603035702898,
"mean_abs_r": 1.8522804401625805,
"overlap": 0.9898088992269419,
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"group_weight": 0.9898197554585997,
"selected": "lowest",
"d_all": 0.0006809090838195697,
"d_flip": 0.0006809090838195697,
"d_joint": 0.00161780310028381,
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"tau_over_d": 0.38732963034775686,
"mu1": 6564.0381995259195,
"mu2": 6564.0381995259195,
"mu1_band": 6564.033712456998,
"mu2_band": 6564.033712456998,
"Epp_exact": 8.707640025695795e-05,
"Epp_band": 8.958102685567108e-05,
"Epp_66": 8.958103525479998e-05,
"beta0": 0.9232020058717421,
"delta_m": 0.2659015211102517,
"D0_exact": -1643.9085068439417,
"D0_band": -1964.989314510607,
"Dlat": 1092.0794450156043,
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"Dmark_lead": -3352.7076936436397,
"D0_formula": -1884.0965942360215,
"D0_resummed": -1900.3513595762251,
"D0_lead": -2260.628248628035,
"local_inertia_relerr_odd": [
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0.02197773341299425
],
"local_inertia_relerr_lead": [
0.0987031491693745,
0.09870314917125822
],
"dE0_exact": -7.278875329741474e-05,
"M_avg": -8.0085588320732e-05,
"M_odd_avg": [
-9.570033724676371e-05,
-9.570033724649829e-05
],
"M0_second_order": -0.00013186813186813188,
"K0": 5.849904570467353e-06,
"dE0_formula": -7.423568375026466e-05,
"weight": [
{
"s": 8,
"u": 0.125,
"kappa_s": 0.5631257631257631,
"phip": -7.500000000000001e-06,
"W0p": -4.797070823218553e-09,
"W1p": -2.7920607106964907e-09,
"W2p": -2.7920607106964907e-09,
"Wbp": 7.870505981744284e-10,
"Wbp_formula": 9.952186747161408e-10,
"W0p_heldavg": -4.8522735330443126e-09,
"Wbp_heldavg": 7.318222083178924e-10,
"kinetic": -7.363785314204793e-06,
"kinetic_formula": -5.849904570467353e-06,
"Gamma1": -4.2535716975232475e-13,
"Gamma2": -4.2535716975232475e-13,
"Gamma_pair": -4.177116590635535e-13,
"Gamma_bond": -4.787678845250626e-13,
"Gamma_bond_lead": -2.9376926659863505e-13,
"Gamma_bond_formula": -4.4023986487832947e-13,
"Gamma_bond_refined": -4.1170739067703334e-13,
"eta_over_ks": 0.26637033824804857,
"w_over_Lks": 0.06408434810821882,
"W0p_eigenvalues": -4.797070851658702e-09
},
{
"s": 16,
"u": 0.25,
"kappa_s": 0.25054945054945055,
"phip": -5e-06,
"W0p": -3.74649178294495e-09,
"W1p": -2.0263642911491804e-09,
"W2p": -2.0263642911491804e-09,
"Wbp": 3.062367993534109e-10,
"Wbp_formula": 2.5129057242075107e-10,
"W0p_heldavg": -3.788207872029568e-09,
"Wbp_heldavg": 2.6448508697055774e-10,
"kinetic": -8.350342476570628e-06,
"kinetic_formula": -5.849904570467353e-06,
"Gamma1": -3.0870696201852274e-13,
"Gamma2": -3.0870696201852274e-13,
"Gamma_pair": -3.262310180511185e-13,
"Gamma_bond": -1.8628579271807512e-13,
"Gamma_bond_lead": -8.713711116802531e-14,
"Gamma_bond_formula": -1.1115961793950781e-13,
"Gamma_bond_refined": -1.5590209691027033e-13,
"eta_over_ks": 0.5986842105263158,
"w_over_Lks": 0.14403363229780958
}
]
},
{
"case": "A' equal bodies, smaller eta",
"L": 64,
"g1": 0.009377289377289377,
"g2": 0.009377289377289377,
"tau1": 8.791208791208791e-05,
"tau2": 8.791208791208791e-05,
"eta": 0.05,
"w": 3.368895005628803,
"mean_abs_r": 2.418395197942303,
"overlap": 0.995018659396197,
"min_overlap": 0.9949393308958033,
"group_weight": 0.9950194788404344,
"selected": "lowest",
"d_all": 0.0002673014846679145,
"d_flip": 0.0002673014846679145,
"d_joint": 0.0005664572838339765,
"d_joint_lead": 0.0005641025641025641,
"tau_over_d": 0.32888739103453407,
"mu1": 19035.44187708198,
"mu2": 19035.44187708198,
"mu1_band": 19035.439535714922,
"mu2_band": 19035.439535714922,
"Epp_exact": 2.6605950683341224e-05,
"Epp_band": 2.699289671411383e-05,
"Epp_66": 2.6992900999228454e-05,
"beta0": 0.9517487811729664,
"delta_m": 0.48447469264898246,
"D0_exact": -485.3070573639561,
"D0_band": -1024.0955938667248,
"Dlat": 1930.0960288037227,
"Dmark_odd": -2937.454723013449,
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