08-ilang-propagation / 06-mark-inertia
06mark inertiaverified
Determines the inertia of a moving body with its own mark against its cost in oscillator cells, as a function of the ratio of body speed to field speed, and whether weight over inertia is universal.
check_mark_inertia.py10 KB · raw
"""Check script of 08-ilang-propagation/06-mark-inertia, derivation v1.
Part A (check required by the question): ring of L = 32 cells. For a square grip
and one other grip of class (G2), two clock levels, both branches and
r = tau/c0 in {0.5, 1, 1.5}: the second-order coefficients (in delta F) of
Cost_{k,sigma} and of E''_{k,sigma}(0) from the formula (6.2), (6.3), (6.5) of the
derivation are compared with an independent computation: exact diagonalisation
of the total contract restricted to states with at most two field quanta, in the
sector of the joint label P (P continued continuously, E'' from the exact second
derivative of the eigenvalue), the second-order coefficient being extracted from
several small values of the scale s of delta F.
Part B: the formula (6.3) on long rings (L = 8 ell): rho = I^mark tau^2 / C^mark
of the one-quantum (F1^2) and two-quantum (F2^2) parts against 2 - r^2, and the
held mark against its leading form in ell, as ell grows.
Units: c0 = sqrt(kappa_h) = 1, F0 = 1, so eps_k = F0 M_k = M_k.
"""
from itertools import combinations_with_replacement
from pathlib import Path
import numpy as np
OUT = Path(__file__).parent / "output"
OUT.mkdir(exist_ok=True)
LOG = []
def log(s=""):
print(s, flush=True)
LOG.append(s)
S1 = np.array([[0.0, 1.0], [1.0, 0.0]])
S3 = np.diag([1.0, -1.0])
def modes(L, c0, mh):
q = 2 * np.pi * np.arange(L) / L
Om = np.sqrt(mh**2 + 2 * c0**2 * (1 - np.cos(q)))
return q, Om, 1.0 / (2 * L * Om) # c_q^2 = 1/(2 L Omega_q)
def omega(p, tau, eps):
return np.sqrt(eps**2 + (tau * np.sin(p))**2)
def theta(p, tau, eps):
return np.arctan(tau * np.sin(p) / eps)
# ------------------------------------------------------------------ formula (6.3)
def mark_sums(P, sigma, tau, eps, q, Om, c2, chunk=256):
"""(S1, S2) with E^(2)(P) = -M_k^2 (F1^2 S1 + F2^2 S2)."""
wP, tP = omega(P, tau, eps), theta(P, tau, eps)
def S(E, Q, w):
p = P - Q
th = 0.5 * (tP + theta(p, tau, eps))
wp = omega(p, tau, eps)
return np.sum(w * (np.cos(th)**2 / (E + sigma * (wp - wP))
+ np.sin(th)**2 / (E - sigma * (wp + wP))))
s1 = S(Om, q, c2)
s2 = 0.0
for i in range(0, len(q), chunk):
sl = slice(i, i + chunk)
s2 += S(Om[sl, None] + Om[None, :], q[sl, None] + q[None, :],
2 * c2[sl, None] * c2[None, :])
return s1, s2
def curv(f, h=1e-2):
"""Second derivative at 0 of an even function, Richardson in h."""
f0 = f(0.0)
c = lambda hh: 2 * (f(hh) - f0) / hh**2
return (4 * c(h) - c(2 * h)) / 3
def formula_second_order(sigma, tau, eps, Mk, F1, F2, q, Om, c2):
"""Second-order coefficients of Cost and E''(0): mark part (6.3) plus the
second-order part of the zero-quantum level sigma*omega-bar (6.2)."""
w0 = np.sum(c2)
def E2(P):
a, b = mark_sums(P, sigma, tau, eps, q, Om, c2)
zq = sigma * (F2 * w0 * Mk)**2 * (tau * np.sin(P))**2 / (2 * omega(P, tau, eps)**3)
return -Mk**2 * (F1**2 * a + F2**2 * b) + zq
return E2(0.0), curv(E2)
# ------------------------------------------------------------------ exact diagonalisation
def field_ops(L, q, Om, c2):
states = [()] + [(i,) for i in range(L)] + list(combinations_with_replacement(range(L), 2))
idx = {s: i for i, s in enumerate(states)}
n = len(states)
A = np.zeros((n, n)) # sum_q c_q a_q on <= 2 quanta
cq = np.sqrt(c2)
for j, s in enumerate(states):
for m in set(s):
t = list(s)
t.remove(m)
A[idx[tuple(t)], j] += cq[m] * np.sqrt(s.count(m))
Ef = np.array([Om[list(s)].sum() if s else 0.0 for s in states])
Qn = np.array([q[list(s)].sum() if s else 0.0 for s in states])
q0 = A + A.T # q at the body's cell (x = 0)
nq2 = A @ A + A.T @ A.T + 2 * A.T @ A # :q0^2:, exact on <= 2 quanta
return Ef, Qn, q0, nq2, np.sum(c2)
def ed_levels(P, tau, eps, Mk, F1, F2, s, ops):
Ef, Qn, q0, nq2, w0 = ops
n = len(Ef)
sn, cs = np.sin(P - Qn), np.cos(P - Qn)
H = (np.kron(np.diag(Ef), np.eye(2)) + eps * np.kron(np.eye(n), S3)
- tau * np.kron(np.diag(sn), S1)
+ s * Mk * np.kron(F1 * q0 + F2 * (nq2 + w0 * np.eye(n)), S3))
H1 = -tau * np.kron(np.diag(cs), S1) # dH/dP
H2 = tau * np.kron(np.diag(sn), S1) # d2H/dP2
E, V = np.linalg.eigh(H)
_, u0 = np.linalg.eigh(-tau * np.sin(P) * S1 + eps * S3)
out = {}
for sigma, col in ((+1, 1), (-1, 0)):
ref = np.zeros(2 * n)
ref[:2] = u0[:, col] # field vacuum x band sigma
j = int(np.argmax(np.abs(V.T @ ref)))
psi = V[:, j]
d1 = V.T @ (H1 @ psi)
m = np.arange(len(E)) != j
Epp = psi @ H2 @ psi + 2 * np.sum(d1[m]**2 / (E[j] - E[m]))
out[sigma] = (E[j], Epp)
return out
def coeffs(svals, fpos, fneg, f0):
"""Fit even part s^2 a2 + s^4 a4 + s^6 a6 and odd part s a1 + s^3 a3 + s^5 a5."""
s = np.array(svals)
even = (np.array(fpos) + np.array(fneg)) / 2 - f0
odd = (np.array(fpos) - np.array(fneg)) / 2
a2 = np.linalg.solve(np.vstack([s**2, s**4, s**6]).T, even)[0]
a1 = np.linalg.solve(np.vstack([s, s**3, s**5]).T, odd)[0]
return a1, a2
def part_a():
L, c0, mh = 32, 1.0, 0.25
q, Om, c2 = modes(L, c0, mh)
w0 = np.sum(c2)
log(f"PART A: ring L={L}, c0={c0}, m_h={mh} (ell={c0/mh}), F0=1, "
f"G0={2*np.sum(c2/Om):.6f}, w0={w0:.6f}, Omega_max={Om.max():.4f}")
log("second-order coefficients: formula (6.2), (6.3), (6.5) vs ED restricted to <= 2 quanta")
ops = field_ops(L, q, Om, c2)
grips = {"square (F1=0.3, F2=F1^2/4=0.0225)": (0.3, 0.0225),
"other (F1=0.3, F2=-0.15)": (0.3, -0.15)}
svals = [0.01, 0.02, 0.04]
maxrel = 0.0
log(f"{'grip':36s} {'Mk':>3s} {'sig':>3s} {'r':>4s} | {'Cost2 form':>12s} {'Cost2 ED':>12s}"
f" | {'E2pp form':>12s} {'E2pp ED':>12s} | {'Cost1 ED-form':>13s} {'Epp1 ED-form':>13s}")
for gname, (F1, F2) in grips.items():
for Mk in (3.0, 4.0):
eps = Mk
for r in (0.5, 1.0, 1.5):
tau = r * c0
res = {sv: ed_levels(0.0, tau, eps, Mk, F1, F2, sv, ops)
for sv in [0.0] + svals + [-x for x in svals]}
for sigma in (+1, -1):
e0, p0 = res[0.0][sigma]
c1, c2c = coeffs(svals, [res[x][sigma][0] for x in svals],
[res[-x][sigma][0] for x in svals], e0)
d1, d2 = coeffs(svals, [res[x][sigma][1] for x in svals],
[res[-x][sigma][1] for x in svals], p0)
fc, fpp = formula_second_order(sigma, tau, eps, Mk, F1, F2, q, Om, c2)
c1f = sigma * Mk * F2 * w0
d1f = -sigma * tau**2 * F2 * w0 / Mk
rel = max(abs(c2c - fc) / abs(fc), abs(d2 - fpp) / abs(fpp))
maxrel = max(maxrel, rel)
log(f"{gname:36s} {Mk:3.0f} {sigma:+3d} {r:4.1f} | {fc:12.8f} {c2c:12.8f}"
f" | {fpp:12.8f} {d2:12.8f} | {c1-c1f:13.2e} {d1-d1f:13.2e}")
# zeroth order: bare level and curvature of (6.7) of 04-ilang-time/06-inertia-internal-energy
assert abs(e0 - sigma * eps) < 1e-10
assert abs(p0 - sigma * tau**2 / eps) < 1e-8
log(f"max relative deviation of second-order coefficients (formula vs ED): {maxrel:.2e}")
log("zeroth order: ED level = sigma*F0*M_k and E''(0) = sigma*tau^2/(F0*M_k) asserted to 1e-10 / 1e-8")
return maxrel
def part_b():
c0 = 1.0
log("")
log("PART B: formula (6.3) on rings with L = 8 ell, c0 = 1, F0 = 1;")
log("rho_1, rho_2 = I^mark tau^2/C^mark of the F1^2 and F2^2 parts; prediction (6.9): 2 - r^2")
log("held mark: g1 = G0*2c0^2/ell, g2 = J2*16c0^3/ell (prediction (6.7): -> 1);"
" Cratio = C^mark(tau)/C^mark(0) for the F1^2 part")
log("t_i = rho_i + 2 sigma eps_k R_i: rho with the term (iv) of Step 6 removed,"
" R_i = <<sin^2 Q/E^2>>/<<1/E>> of the part i")
log(f"{'ell':>5s} {'g1':>8s} {'g2':>8s} {'Mk':>3s} {'sig':>3s} {'r':>4s} {'2-r^2':>6s}"
f" {'rho_1':>9s} {'rho_2':>9s} {'t_1':>9s} {'t_2':>9s} {'Cratio':>8s}")
for ell in (16, 32, 64, 128, 256, 512):
mh, L = c0 / ell, 8 * ell
q, Om, c2 = modes(L, c0, mh)
G0 = 2 * np.sum(c2 / Om)
J2, N2 = 0.0, 0.0
for i in range(0, L, 256):
Es = Om[i:i+256, None] + Om[None, :]
ww = c2[i:i+256, None] * c2[None, :]
J2 += np.sum(ww / Es)
N2 += np.sum(ww * np.sin(q[i:i+256, None] + q[None, :])**2 / Es**2)
R1 = np.sum(c2 * np.sin(q)**2 / Om**2) / (G0 / 2)
R2 = N2 / J2
g1, g2 = G0 * 2 * c0**2 / ell, J2 * 16 * c0**3 / ell
for Mk in (3.0, 4.0):
eps = Mk
for sigma in (+1, -1):
for r in (0.5, 1.0, 1.5):
tau = r * c0
cache = {}
def ms(P):
if P not in cache:
cache[P] = mark_sums(P, sigma, tau, eps, q, Om, c2)
return cache[P]
C1, C2 = -Mk**2 * ms(0.0)[0], -Mk**2 * ms(0.0)[1]
E1 = -Mk**2 * curv(lambda P: ms(P)[0])
E2 = -Mk**2 * curv(lambda P: ms(P)[1])
I1, I2 = -(eps**2 / tau**4) * E1, -(eps**2 / tau**4) * E2
rho1, rho2 = I1 * tau**2 / C1, I2 * tau**2 / C2
cr = C1 / (-Mk**2 * G0 / 2)
t1, t2 = rho1 + 2 * sigma * eps * R1, rho2 + 2 * sigma * eps * R2
log(f"{ell:5d} {g1:8.5f} {g2:8.5f} {Mk:3.0f} {sigma:+3d} {r:4.1f} {2-r*r:6.2f}"
f" {rho1:9.5f} {rho2:9.5f} {t1:9.5f} {t2:9.5f} {cr:8.5f}")
if __name__ == "__main__":
part_a()
part_b()
(OUT / "check_mark_inertia.txt").write_text("\n".join(LOG) + "\n", encoding="utf-8")
torus_mark_inertia.py31 KB · raw
"""Computation script of 08-ilang-propagation/06-mark-inertia, derivation v4.
Changes in v4 (verification.r2.md):
- S_i''(0) by analytic differentiation of the summands in P (no finite-difference step);
Part E checks it against exact diagonalisation, Part G against Richardson finite
differences with steps h0, h0/2, h0/4 at the points of smallest isolation gap (I4).
- Crossings rho_i,- = 1 by Brent root-finding on the lattice sums, with an error estimate
(root tolerance; for the two-quantum part the binning, from a lattice with dE/2); the
windows are built from the roots and checked against every computed point (I1).
- y_ab from direct evaluation at the stated levels instead of interpolation (I1).
- Part G also compares L = 64 and L = 128 at ell = 8 at the smallest-gap points (I3).
Cubic torus (D = 3) of side L, body chain along lattice axis 1, Q = q_1; in
Part F also the ring (D = 1). The lattice sums of (6.3) are evaluated for the
one-quantum (F1^2) and the two-quantum (F2^2) part separately:
S_i(P) = << cos^2(th)/(E + a) + sin^2(th)/(E + b) >>_i ,
a = sigma [omega(P-Q) - omega(P)], b = -sigma [omega(P-Q) + omega(P)],
with <<f>>_1 = sum_q c_q^2 f(Omega_q, q_1) and
<<f>>_2 = 2 sum_{q,q'} c_q^2 c_q'^2 f(Omega_q + Omega_q', q_1 + q_1'),
so that E^(2) = -M_k^2 (F1^2 S_1 + F2^2 S_2). Derived quantities:
rho_i = -(eps^2/tau^2) S_i''(0)/S_i(0) (6.6), per part
t = rho_1 S_1(0)/G0 (bracket of (6.12)-(6.13))
xi* = S_1(0)(1-rho_1)/(S_2(0)(rho_2-1)) (F2^2/(c0 F1^2) at which rho = 1)
Isolation (6.4), L = infinity: (k, sigma) is isolated iff b_k > sigma*eps_k with
b_k = min_q [Omega_q - omega_k(q_1)]; the minimum lies at q_2 = q_3 = 0, so b_k
is the same function on the ring and the torus. Thresholds: the lower branch is
isolated iff eps_k > eps_-(r), the upper branch iff eps_k < eps_+(r) < m_h/2.
Only isolated states are evaluated, except in Part E (formula check, both
branches; the upper branch there is a formal coefficient).
One-quantum sums are exact mode sums over classes. Two-quantum sums (v3): for
each q_1 the one-quantum energies are put on a grid of step dE with linear
(cloud-in-cell) weights, which keep the zeroth and first moments; the pair
measure per total q_1 is the convolution (cyclic in q_1, linear in E), done by
FFT, and the excess variance of the binning is removed to second order
(term -V f''/2). Checked against brute-force pair sums (Part A).
Units: c0 = sqrt(kappa_h) = 1, F0 = 1, so eps_k = M_k.
"""
from itertools import combinations_with_replacement
from pathlib import Path
import numpy as np
from scipy.optimize import brentq
OUT = Path(__file__).parent / "output"
OUT.mkdir(exist_ok=True)
LOG = []
def log(s=""):
print(s, flush=True)
LOG.append(s)
def omega(p, tau, eps):
return np.sqrt(eps**2 + (tau * np.sin(p))**2)
def theta(p, tau, eps):
return np.arctan(tau * np.sin(p) / eps)
# ------------------------------------------------------------------ isolation (6.4), L = infinity
_QQ = np.linspace(0.0, np.pi, 400001)
def b_min(tau, eps, mh):
"""b_k = min_q [Omega_q - omega_k(q_1)] (c0 = 1), attained at q_2 = q_3 = 0."""
return np.min(np.sqrt(mh**2 + 2 * (1 - np.cos(_QQ))) - omega(_QQ, tau, eps))
def thresholds(tau, mh):
"""(eps_-, eps_+): lower branch isolated iff eps > eps_-, upper iff eps < eps_+."""
gm = lambda e: b_min(tau, e, mh) + e # increasing in eps
gp = lambda e: b_min(tau, e, mh) - e # decreasing in eps
e0 = 1e-10
if gm(e0) > 0:
em = 0.0
else:
lo, hi = e0, 50.0
for _ in range(80):
mid = 0.5 * (lo + hi)
lo, hi = (lo, mid) if gm(mid) > 0 else (mid, hi)
em = hi
if gp(e0) <= 0:
ep = 0.0
else:
lo, hi = e0, mh
for _ in range(80):
mid = 0.5 * (lo + hi)
lo, hi = (mid, hi) if gp(mid) > 0 else (lo, mid)
ep = lo
return em, ep
def lower_gap(tau, eps, mh):
return b_min(tau, eps, mh) + eps
def upper_gap(tau, eps, mh):
return b_min(tau, eps, mh) - eps
# ------------------------------------------------------------------ lattice sums
def _terms(P, sigma, tau, eps, Q):
wP, tP = omega(P, tau, eps), theta(P, tau, eps)
p = P - Q
th = 0.5 * (tP + theta(p, tau, eps))
wp = omega(p, tau, eps)
return np.cos(th)**2, sigma * (wp - wP), np.sin(th)**2, -sigma * (wp + wP)
def _terms_d(P, sigma, tau, eps, Q):
"""As _terms, with first and second P-derivatives: (A, A', A''), (a, a', a''), (b, b', b''), B.
A = cos^2(thb), B = sin^2(thb) = 1 - A, thb = (theta(P) + theta(P - Q))/2."""
def wd(p):
s, c = np.sin(p), np.cos(p)
w = np.sqrt(eps**2 + (tau * s)**2)
w1 = tau**2 * s * c / w
w2 = tau**2 * np.cos(2 * p) / w - (tau**2 * s * c)**2 / w**3
t1 = eps * tau * c / w**2
t2 = -eps * tau * s / w**2 - 2 * eps * tau * c * w1 / w**3
return w, w1, w2, np.arctan(tau * s / eps), t1, t2
W0, W1, W2, T0, T1, T2 = wd(P)
w, w1, w2, th, t1, t2 = wd(P - Q)
tb, tb1, tb2 = 0.5 * (T0 + th), 0.5 * (T1 + t1), 0.5 * (T2 + t2)
A = (np.cos(tb)**2, -np.sin(2 * tb) * tb1, -2 * np.cos(2 * tb) * tb1**2 - np.sin(2 * tb) * tb2)
a = (sigma * (w - W0), sigma * (w1 - W1), sigma * (w2 - W2))
b = (-sigma * (w + W0), -sigma * (w1 + W1), -sigma * (w2 + W2))
return A, a, b, np.sin(tb)**2
def _frac(c, x, E, n):
"""Value and second P-derivative of c/(E + x)^n, n in {1, 3}; c, x = (f, f', f'')."""
d = E + x[0]
if n == 1:
return c[0] / d, c[2] / d - 2 * c[1] * x[1] / d**2 + 2 * c[0] * x[1]**2 / d**3 - c[0] * x[2] / d**2
return (c[0] / d**3,
c[2] / d**3 - 6 * c[1] * x[1] / d**4 + 12 * c[0] * x[1]**2 / d**5 - 3 * c[0] * x[2] / d**4)
def _sum_d(E, w, A, a, b, B0, wV=None):
"""(S, S'') of sum w [A/(E+a) + B/(E+b)] - sum wV [A/(E+a)^3 + B/(E+b)^3] (binning variance term)."""
B = (B0, -A[1], -A[2])
fa, fa2 = _frac(A, a, E, 1)
fb, fb2 = _frac(B, b, E, 1)
s0, s2 = np.sum(w * (fa + fb)), np.sum(w * (fa2 + fb2))
if wV is not None:
ga, ga2 = _frac(A, a, E, 3)
gb, gb2 = _frac(B, b, E, 3)
s0 -= np.sum(wV * (ga + gb))
s2 -= np.sum(wV * (ga2 + gb2))
return s0, s2
class Torus:
def __init__(self, L, ell, pairs=True, dE=1e-3):
self.L, self.ell, self.mh = L, ell, 1.0 / ell
k = 2 * np.pi * np.arange(L) / L
s1 = 2 * (1 - np.cos(k))
u, n = np.unique(np.round((s1[:, None] + s1[None, :]).ravel(), 12), return_counts=True)
self.k = k
self.Om = np.sqrt(self.mh**2 + s1[:, None] + u[None, :]) # (L, U) mode classes
self.w = n[None, :] / (2 * L**3 * self.Om) # multiplicity * c_q^2
self.G0 = 2 * np.sum(self.w / self.Om)
self.w0 = np.sum(self.w)
self.Omax = np.sqrt(self.mh**2 + 12)
self.R1 = np.sum(self.w * (np.sin(k)**2)[:, None] / self.Om**2) / np.sum(self.w / self.Om)
self.pairs = pairs
if pairs:
self._pairs(dE)
def _pairs(self, dE):
L = self.L
x = self.Om / dE
g = np.floor(x).astype(np.int64)
t = x - g
nb = int(g.max()) + 2
h, hv = np.zeros((L, nb)), np.zeros((L, nb))
for i in range(L):
wi, ti, gi = self.w[i], t[i], g[i]
h[i] = np.bincount(gi, wi * (1 - ti), nb) + np.bincount(gi + 1, wi * ti, nb)
vi = wi * ti * (1 - ti) * dE**2
hv[i] = np.bincount(gi, vi * (1 - ti), nb) + np.bincount(gi + 1, vi * ti, nb)
n2 = 2 * nb
Hf = np.fft.rfft2(h, s=(L, n2))
W = 2 * np.fft.irfft2(Hf * Hf, s=(L, n2))
V = 4 * np.fft.irfft2(Hf * np.fft.rfft2(hv, s=(L, n2)), s=(L, n2))
omin, omax = self.Om.min(axis=1), self.Om.max(axis=1)
E = dE * np.arange(n2)
ii = np.arange(L)
mask = np.zeros((L, n2), dtype=bool)
for m in range(L):
j = (m - ii) % L
lo, hi = np.min(omin + omin[j]), np.max(omax + omax[j])
mask[m] = (E >= lo - 3 * dE) & (E <= hi + 3 * dE)
mi, gi = np.nonzero(mask)
self.pm, self.pE, self.pW, self.pV = mi, E[gi], W[mi, gi], V[mi, gi]
self.dE = dE
sq = np.sin(self.k[self.pm])**2
self.J2 = 0.5 * np.sum(self.pW / self.pE - self.pV / self.pE**3)
self.R2 = np.sum(self.pW * sq / self.pE**2 - 3 * self.pV * sq / self.pE**4) / (2 * self.J2)
def S1(self, P, sigma, tau, eps):
A, a, B, b = _terms(P, sigma, tau, eps, self.k)
E = self.Om
return np.sum(self.w * (A[:, None] / (E + a[:, None]) + B[:, None] / (E + b[:, None])))
def S2(self, P, sigma, tau, eps):
A, a, B, b = _terms(P, sigma, tau, eps, self.k)
m, E = self.pm, self.pE
x, y = E + a[m], E + b[m]
f = A[m] / x + B[m] / y
f2 = 2 * A[m] / x**3 + 2 * B[m] / y**3
return np.sum(self.pW * f - 0.5 * self.pV * f2)
def S1d(self, sigma, tau, eps):
"""(S_1(0), S_1''(0)), analytic in P."""
A, a, b, B0 = _terms_d(0.0, sigma, tau, eps, self.k)
col = lambda t: tuple(z[:, None] for z in t)
return _sum_d(self.Om, self.w, col(A), col(a), col(b), B0[:, None])
def S2d(self, sigma, tau, eps):
"""(S_2(0), S_2''(0)), analytic in P, on the binned pair measure."""
A, a, b, B0 = _terms_d(0.0, sigma, tau, eps, self.k)
m = self.pm
sel = lambda t: tuple(z[m] for z in t)
return _sum_d(self.pE, self.pW, sel(A), sel(a), sel(b), B0[m], self.pV)
class Ring:
def __init__(self, L, ell):
self.L, self.ell, self.mh = L, ell, 1.0 / ell
self.k = 2 * np.pi * np.arange(L) / L
self.Om = np.sqrt(self.mh**2 + 2 * (1 - np.cos(self.k)))
self.c2 = 1 / (2 * L * self.Om)
self.G0 = 2 * np.sum(self.c2 / self.Om)
@staticmethod
def _S(P, sigma, tau, eps, E, Q, w):
A, a, B, b = _terms(P, sigma, tau, eps, Q)
return np.sum(w * (A / (E + a) + B / (E + b)))
def S1(self, P, sigma, tau, eps):
return self._S(P, sigma, tau, eps, self.Om, self.k, self.c2)
def S2(self, P, sigma, tau, eps, chunk=256):
s, Om, k, c2 = 0.0, self.Om, self.k, self.c2
for i in range(0, self.L, chunk):
sl = slice(i, i + chunk)
s += self._S(P, sigma, tau, eps, Om[sl, None] + Om[None, :], k[sl, None] + k[None, :],
2 * c2[sl, None] * c2[None, :])
return s
def S1d(self, sigma, tau, eps):
A, a, b, B0 = _terms_d(0.0, sigma, tau, eps, self.k)
return _sum_d(self.Om, self.c2, A, a, b, B0)
def S2d(self, sigma, tau, eps, chunk=256):
s0 = s2 = 0.0
Om, k, c2 = self.Om, self.k, self.c2
for i in range(0, self.L, chunk):
sl = slice(i, i + chunk)
A, a, b, B0 = _terms_d(0.0, sigma, tau, eps, k[sl, None] + k[None, :])
x0, x2 = _sum_d(Om[sl, None] + Om[None, :], 2 * c2[sl, None] * c2[None, :], A, a, b, B0)
s0, s2 = s0 + x0, s2 + x2
return s0, s2
def curv(f, h, f0=None):
"""Richardson-extrapolated second derivative at 0 of an even function (used only as a check)."""
f0 = f(0.0) if f0 is None else f0
c = lambda hh: 2 * (f(hh) - f0) / hh**2
return f0, (4 * c(h) - c(2 * h)) / 3
def analyse(T, sigma, tau, eps, parts=(1, 2)):
"""{i: (S_i(0), rho_i)}; S_i''(0) by analytic differentiation of the summands (v4)."""
out = {}
for i in parts:
s0, spp = T.S1d(sigma, tau, eps) if i == 1 else T.S2d(sigma, tau, eps)
out[i] = (s0, -(eps**2 / tau**2) * spp / s0)
return out
def roots(T, r, part, e, y, Th=None):
"""Roots of rho_{part,-}(eps) = 1 by Brent's method in each grid interval where y = rho - 1
changes sign. Returns (root, error estimate): residual-based for part 1 (exact mode sum);
for part 2 the binning error, |rho_2 - 1| at the root on the lattice Th with dE/2, over the slope."""
out = []
g = lambda x, TT=T: analyse(TT, -1, r, x, (part,))[part][1] - 1
for i in np.nonzero(np.sign(y[:-1]) * np.sign(y[1:]) < 0)[0]:
x = brentq(g, e[i], e[i + 1], xtol=1e-10, rtol=1e-12)
d = 1e-4 * x
slope = (g(x + d) - g(x - d)) / (2 * d)
res = abs(g(x))
if part == 2 and Th is not None:
res = max(res, abs(g(x, Th)))
out.append((x, res / abs(slope) + 1e-10))
return out
MARGIN = 0.2 # lower branch evaluated where b_k + eps_k > MARGIN * m_h
# ------------------------------------------------------------------ part A: pair-sum check
def part_a():
L, ell = 12, 3.0
log(f"PART A: lattice sums by mode classes and FFT pair convolution vs brute-force mode and ordered-pair"
f" sums, torus L={L}, ell={ell}")
g = np.array(np.meshgrid(np.arange(L), np.arange(L), np.arange(L), indexing="ij")).reshape(3, -1)
q = 2 * np.pi * g / L
Om = np.sqrt(1 / ell**2 + 2 * np.sum(1 - np.cos(q), axis=0))
c2 = 1 / (2 * L**3 * Om)
E = Om[:, None] + Om[None, :]
Q = q[0][:, None] + q[0][None, :]
ww = 2 * c2[:, None] * c2[None, :]
cases = ((-1, 1.0, 1.0), (-1, 1.5, 3.0), (-1, 0.5, 0.2), (+1, 0.5, 0.05))
for dE in (1e-2, 1e-3):
T = Torus(L, ell, dE=dE)
w1 = w2 = 0.0
for sigma, tau, eps in cases:
for P in (0.0, 0.02):
A, a, B, b = _terms(P, sigma, tau, eps, Q)
b2 = np.sum(ww * (A / (E + a) + B / (E + b)))
A1, a1, B1, b1_ = _terms(P, sigma, tau, eps, q[0])
b1 = np.sum(c2 * (A1 / (Om + a1) + B1 / (Om + b1_)))
w1 = max(w1, abs(T.S1(P, sigma, tau, eps) / b1 - 1))
w2 = max(w2, abs(T.S2(P, sigma, tau, eps) / b2 - 1))
log(f" dE={dE}: max relative deviation S_1 {w1:.2e}, S_2 {w2:.2e}"
f" (cases (sigma,r,eps) = {cases}, P = 0, 0.02)")
bR1 = np.sum(c2 * np.sin(q[0])**2 / Om**2) / np.sum(c2 / Om)
log(f" R_1: classes {T.R1:.12f}, brute force {bR1:.12f}; J2: FFT {T.J2:.12f},"
f" brute force {np.sum(c2[:, None]*c2[None, :]/E):.12f}")
# ------------------------------------------------------------------ part B: convergence in L and ell
def part_b(T8):
log("")
log("PART B: convergence in ell and L (c0 = 1); rho_1, rho_2 of the F1^2 and F2^2 parts, lower branch")
lattices = [(32, 4.0), (48, 6.0), (64, 8.0), (128, 16.0), (256, 32.0)]
cases = [(1.0, 1.0), (1.0, 4.0), (0.5, 2.0), (2.0, 2.0), (3.0, 4.0)]
log(f"{'L':>4s} {'ell':>5s} {'G0':>9s} {'w0':>8s} {'J2':>9s} {'R1':>8s} {'R2':>8s} |"
+ "".join(f" r={r} e={e}: rho_1, rho_2 " for r, e in cases))
Ts = {}
for L, ell in lattices:
T = T8 if (L, ell) == (64, 8.0) else Torus(L, ell)
Ts[(L, ell)] = T
row = f"{L:4d} {ell:5.1f} {T.G0:9.6f} {T.w0:8.5f} {T.J2:9.6f} {T.R1:8.5f} {T.R2:8.5f} |"
for r, eps in cases:
res = analyse(T, -1, r, eps)
row += f" {res[1][1]:9.5f} {res[2][1]:9.5f} "
log(row)
return Ts
# ------------------------------------------------------------------ part C: main table
def part_c(T):
log("")
log(f"PART C: main table, lower branch (isolated), torus L={T.L}, ell={T.ell} (m_h={T.mh}), c0=1, F0=1,"
f" eps_k=M_k")
log(f" G0={T.G0:.6f}, w0={T.w0:.6f}, J2={T.J2:.6f}, R1={T.R1:.6f}, R2={T.R2:.6f}, "
f"Omega_max={T.Omax:.4f}")
log(" C_i = S_i(0)/S_i(0)|_{tau=0} (moving mark over held mark), rho_i of the parts, "
"t = T/G0, xi* = F2^2/(c0 F1^2) at which rho = 1 (- if none finite >= 0)")
log(f"{'r':>5s} {'eps':>5s} {'gap':>7s} | {'C_1':>8s} {'C_2':>8s} | "
f"{'rho_1':>9s} {'rho_2':>9s} | {'t':>9s} {'xi*':>9s}")
grid = []
for r in (0.25, 0.5, 1.0, 1.5, 2.0):
for eps in (0.5, 1.0, 2.0, 3.0, 4.0, 6.0, 8.0, 16.0, 32.0):
gap = lower_gap(r, eps, T.mh)
if gap <= MARGIN * T.mh:
continue
grid.append((r, eps))
res = analyse(T, -1, r, eps)
(s1, rho1), (s2, rho2) = res[1], res[2]
xi = s1 * (1 - rho1) / (s2 * (rho2 - 1))
xs = f"{xi:9.4f}" if xi >= 0 else f"{'-':>9s}"
log(f"{r:5.2f} {eps:5.1f} {gap:7.4f} | {s1/(T.G0/2):8.5f} {s2/(2*T.J2):8.5f} | "
f"{rho1:9.4f} {rho2:9.4f} | {rho1*s1/T.G0:9.4f} {xs}")
return grid
# ------------------------------------------------------------------ part B2: convergence over the table
def part_b2(Ts, grid):
log("")
log("PART B2: convergence over the configurations of Part C: change of rho_1, rho_2 and t from"
" (L,ell)=(64,8) to (128,16) and to (256,32); absolute for eps <= 4, relative for eps > 4")
T8 = Ts[(64, 8.0)]
for key in ((128, 16.0), (256, 32.0)):
To = Ts[key]
ab = {"rho_1": 0.0, "rho_2": 0.0, "t": 0.0}
rl = {"rho_1": 0.0, "rho_2": 0.0, "t": 0.0}
for r, eps in grid:
a8, ao = analyse(T8, -1, r, eps), analyse(To, -1, r, eps)
vals = {"rho_1": (a8[1][1], ao[1][1]), "rho_2": (a8[2][1], ao[2][1]),
"t": (a8[1][1] * a8[1][0] / T8.G0, ao[1][1] * ao[1][0] / To.G0)}
for k_, (x8, xo) in vals.items():
if eps <= 4.0:
ab[k_] = max(ab[k_], abs(x8 - xo))
else:
rl[k_] = max(rl[k_], abs(x8 - xo) / abs(x8))
log(f" to {key}: eps <= 4 max |change|: " + ", ".join(f"{k_} {v:.4f}" for k_, v in ab.items())
+ "; eps > 4 max relative change: " + ", ".join(f"{k_} {v:.4f}" for k_, v in rl.items()))
# ------------------------------------------------------------------ part D: fine grid, items 2 and 3
def grid_tests(T, rs, eg, tag, Th, members=True):
log(f" thresholds (L = infinity): eps_-(r) [lower isolated above], eps_+(r) [upper isolated below],"
f" m_h/2 = {T.mh/2:.4f}")
tabs, smin = [], np.inf
for r in rs:
em, ep = thresholds(r, T.mh)
lo = {e: analyse(T, -1, r, e) for e in eg if lower_gap(r, e, T.mh) > MARGIN * T.mh}
e = np.array(sorted(lo))
s1 = np.array([lo[x][1][0] for x in e]); r1 = np.array([lo[x][1][1] for x in e])
s2 = np.array([lo[x][2][0] for x in e]); r2 = np.array([lo[x][2][1] for x in e])
smin = min(smin, np.min(s1 / (T.G0 / 2)), np.min(s2 / (2 * T.J2)))
t = r1 * s1 / T.G0
phi = e * t
d2 = 2 * np.diff(np.diff(phi) / np.diff(e)) / (e[2:] - e[:-2])
R1, R2 = roots(T, r, 1, e, r1 - 1), roots(T, r, 2, e, r2 - 1, Th)
log(f"r={r:4.2f}: eps_-={em:.4f}, eps_+={ep:.4f}; lower computed on eps in [{e.min():.3f},{e.max():.1f}]"
f" ({len(e)} pts); rho_1,- {r1[0]:.4f}..{r1[-1]:.3f} incr {bool(np.all(np.diff(r1) > 0))};"
f" rho_2,- {r2[0]:.4f}..{r2[-1]:.3f} incr {bool(np.all(np.diff(r2) > 0))}")
log(" roots (Brent, +- error estimate): rho_1,- = 1 at eps "
+ (", ".join(f"{x:.4f} (+-{dx:.0e})" for x, dx in R1) or "none")
+ "; rho_2,- = 1 at eps " + (", ".join(f"{x:.4f} (+-{dx:.0e})" for x, dx in R2) or "none"))
win = (r1 - 1) * (r2 - 1) < 0
pts = [(x, bool(w_)) for x, w_ in zip(e, win)]
if np.any(win):
ia, ib = np.nonzero(win)[0][[0, -1]]
fine = np.linspace(e[max(ia - 1, 0)], e[min(ib + 1, len(e) - 1)], 81)
fx = []
for x in fine:
if lower_gap(r, x, T.mh) <= MARGIN * T.mh:
continue
a_ = analyse(T, -1, r, x)
pts.append((x, bool((a_[1][1] - 1) * (a_[2][1] - 1) < 0)))
if (a_[1][1] - 1) * (a_[2][1] - 1) < 0:
fx.append((x, a_[1][0] * (1 - a_[1][1]) / (a_[2][0] * (a_[2][1] - 1))))
fe, fv = np.array([u for u, _ in fx]), np.array([v for _, v in fx])
log(f" finite xi* >= 0 [(rho_1-1)(rho_2-1) < 0] on eps in [{fe.min():.3f},{fe.max():.3f}]"
f" (81-point refinement, {len(fe)} pts): xi* {fv[0]:.4f} at {fe[0]:.3f} .. {fv[-1]:.2f} at"
f" {fe[-1]:.3f}; increasing {bool(np.all(np.diff(fv) > 0))},"
f" decreasing {bool(np.all(np.diff(fv) < 0))}")
else:
log(" (rho_1-1)(rho_2-1) >= 0 at every computed point")
bnd = ([(e[0] - 1e-12, "edge of computed range")]
+ sorted([(x, "rho_1=1") for x, _ in R1] + [(x, "rho_2=1") for x, _ in R2])
+ [(e[-1] + 1e-12, "edge of computed range")])
wins, ok = [], True
for (a0, na), (b0, nb_) in zip(bnd[:-1], bnd[1:]):
ins = [w_ for x, w_ in pts if a0 < x < b0]
ok = ok and (all(ins) or not any(ins))
if ins and all(ins):
wins.append(f"[{a0:.4f} ({na}), {b0:.4f} ({nb_})]")
log(" windows of finite xi* >= 0 between consecutive roots: " + ("; ".join(wins) or "none")
+ f"; all {len(pts)} computed points (grid and refinement) consistent with them: {ok}")
neg = e[1:-1][d2 <= 0]
log(f" phi increasing {bool(np.all(np.diff(phi) > 0))}; phi'' min {d2.min():.3f} max {d2.max():.3f}"
+ ("" if neg.size == 0 else f"; phi'' <= 0 at eps in [{neg.min():.3f},{neg.max():.3f}]")
+ f"; t_- {t[0]:.4f}..{t[-1]:.3f}; at eps={e[-1]:.1f}: rho_1,- - 2 eps R1 = {r1[-1]-2*e[-1]*T.R1:.4f},"
f" rho_1,-/(2 eps R1) = {r1[-1]/(2*e[-1]*T.R1):.4f}")
tabs.append((r, e, t, lo))
log(f" min over all computed points of S_i(0)/S_i(0)|_(tau=0): {smin:.4f}")
log(" grip y = F0 F2/F1^2 that equalises the bracket of (6.12) for two lower levels (eps_a, eps_b),"
" direct evaluation at the stated levels:")
tval = lambda r, x: (lambda a_: a_[1][1] * a_[1][0] / T.G0)(analyse(T, -1, r, x, (1,)))
for r, e, t, _ in tabs:
out = f" r={r:4.2f}:"
for ea, eb in ((0.25, 0.5), (0.5, 1.0), (1.0, 2.0), (2.0, 4.0), (4.0, 8.0), (1.0, 8.0)):
if min(lower_gap(r, ea, T.mh), lower_gap(r, eb, T.mh)) > MARGIN * T.mh:
ta, tb = tval(r, ea), tval(r, eb)
out += f" ({ea:g},{eb:g}) {(ea*ta - eb*tb)/(2*(ea - eb)):.4f}"
log(out)
if members:
log(f" members with stated margins (exact evaluation at the stated eps, r = 1), {tag}:")
a4 = analyse(T, -1, 1.0, 4.0)
log(f" item 2: level eps=4: rho_1,- = {a4[1][1]:.4f}, rho_2,- = {a4[2][1]:.4f}"
f" -> rho_- >= {min(a4[1][1], a4[2][1]):.4f} > 1 for every grip")
xs = []
for x in (1.2, 2.0):
a_ = analyse(T, -1, 1.0, x)
xs.append(a_[1][0] * (1 - a_[1][1]) / (a_[2][0] * (a_[2][1] - 1)))
log(f" item 2: levels eps=1.2, 2.0: xi* = {xs[0]:.4f}, {xs[1]:.4f}")
tt = {}
for x in (1.0, 2.0, 3.0):
a_ = analyse(T, -1, 1.0, x, (1,))
tt[x] = a_[1][1] * a_[1][0] / T.G0
y12 = (2 * tt[2.0] - 1 * tt[1.0]) / 2
y23 = (3 * tt[3.0] - 2 * tt[2.0]) / 2
log(f" item 3: levels eps=1,2,3: t_- = {tt[1.0]:.4f}, {tt[2.0]:.4f}, {tt[3.0]:.4f};"
f" y_12 = {y12:.4f}, y_23 = {y23:.4f}")
def part_d(T):
log("")
log(f"PART D: fine grid, 71 values of eps in [0.1, 64] (geometric), torus L={T.L}, ell={T.ell};"
f" lower branch where b_k + eps_k > {MARGIN} m_h")
log(" item 2: rho_{i,-} increasing on the grid? crossings of rho_i,- = 1; points with finite xi* >= 0;"
" item 3: phi = eps*t_- increasing and convex at the grid points?")
grid_tests(T, (0.1, 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 2.0, 3.0), np.geomspace(0.1, 64.0, 71),
f"L={T.L}, ell={T.ell}", Torus(T.L, T.ell, dE=T.dE / 2))
def part_d2(T):
log("")
log(f"PART D2: as Part D, both parts, torus L={T.L}, ell={T.ell}; 36 values of eps in [0.1, 64]")
grid_tests(T, (0.1, 0.5, 1.0, 1.5, 2.0, 3.0), np.geomspace(0.1, 64.0, 36), f"L={T.L}, ell={T.ell}",
Torus(T.L, T.ell, dE=T.dE / 2))
# ------------------------------------------------------------------ part E: ED on torus L = 4
def part_e():
log("")
L, ell = 4, 2.0
mh = 1 / ell
g = np.array(np.meshgrid(np.arange(L), np.arange(L), np.arange(L), indexing="ij")).reshape(3, -1)
q = 2 * np.pi * g / L
Om = np.sqrt(mh**2 + 2 * np.sum(1 - np.cos(q), axis=0))
c2 = 1 / (2 * L**3 * Om)
n = len(Om)
states = [()] + [(i,) for i in range(n)] + list(combinations_with_replacement(range(n), 2))
idx = {s: i for i, s in enumerate(states)}
ns = len(states)
A = np.zeros((ns, ns))
cq = np.sqrt(c2)
for j, s in enumerate(states):
for m in set(s):
t = list(s)
t.remove(m)
A[idx[tuple(t)], j] += cq[m] * np.sqrt(s.count(m))
Ef = np.array([Om[list(s)].sum() if s else 0.0 for s in states])
Qn = np.array([q[0][list(s)].sum() if s else 0.0 for s in states])
q0 = A + A.T
nq2 = A @ A + A.T @ A.T + 2 * A.T @ A
w0 = np.sum(c2)
S1m, S3m = np.array([[0.0, 1.0], [1.0, 0.0]]), np.diag([1.0, -1.0])
Mk, F1, F2, tau = 5.0, 0.3, 0.1, 1.0
eps = Mk
log(f"PART E: ED of the contract restricted to <= 2 field quanta, cubic torus L={L}, ell={ell}, "
f"dimension {2*ns}; M_k={Mk}, F1={F1}, F2={F2}, r={tau}, Omega_max={Om.max():.4f}"
f" (sigma=+1: formal coefficients, the upper branch is not isolated here)")
base = (np.kron(np.diag(Ef), np.eye(2)) + eps * np.kron(np.eye(ns), S3m)
- tau * np.kron(np.diag(np.sin(-Qn)), S1m))
V = Mk * np.kron(F1 * q0 + F2 * (nq2 + w0 * np.eye(ns)), S3m)
H1 = -tau * np.kron(np.diag(np.cos(-Qn)), S1m)
H2 = tau * np.kron(np.diag(np.sin(-Qn)), S1m)
svals = [0.01, 0.02, 0.04]
res = {}
for s in [0.0] + svals + [-x for x in svals]:
E, Vv = np.linalg.eigh(base + s * V)
out = {}
for sigma in (+1, -1):
ref = np.zeros(2 * ns)
ref[0 if sigma == +1 else 1] = 1.0
j = int(np.argmax(np.abs(Vv.T @ ref)))
psi = Vv[:, j]
d1 = Vv.T @ (H1 @ psi)
msk = np.arange(len(E)) != j
out[sigma] = (E[j], psi @ H2 @ psi + 2 * np.sum(d1[msk]**2 / (E[j] - E[msk])))
res[s] = out
sarr = np.array(svals)
T = Torus(L, ell)
worst = 0.0
for sigma in (+1, -1):
for comp in (0, 1):
f0 = res[0.0][sigma][comp]
even = np.array([(res[x][sigma][comp] + res[-x][sigma][comp]) / 2 - f0 for x in svals])
a2 = np.linalg.solve(np.vstack([sarr**2, sarr**4, sarr**6]).T, even)[0]
if comp == 0:
form = -Mk**2 * (F1**2 * T.S1(0.0, sigma, tau, eps) + F2**2 * T.S2(0.0, sigma, tau, eps))
else:
_, p1 = T.S1d(sigma, tau, eps)
_, p2 = T.S2d(sigma, tau, eps)
form = -Mk**2 * (F1**2 * p1 + F2**2 * p2) + sigma * (F2 * w0 * Mk)**2 * tau**2 / eps**3
worst = max(worst, abs(a2 / form - 1))
label = "Cost" if comp == 0 else "E''(0)"
log(f" sigma={sigma:+d} {label:6s} second order: ED {a2:.10f} formula {form:.10f}")
log(f" max relative deviation: {worst:.2e}")
# ------------------------------------------------------------------ part F: light levels
def part_f():
log("")
ell = 8.0
mh = 1 / ell
log(f"PART F: light levels eps_k < eps_+(r) (upper branch isolated), ell={ell} (m_h={mh}),"
f" eps = f*eps_+, both branches, both parts; ring and cubic torus")
lat = [("ring", Ring(2048, ell)), ("ring", Ring(4096, ell)), ("torus", Torus(128, ell)),
("torus", Torus(256, ell))]
log(f"{'graph':>6s} {'L':>5s} {'r':>5s} {'eps_+':>8s} {'f':>4s} {'eps':>8s} | {'rho_1,+':>9s} {'rho_2,+':>9s}"
f" {'rho_1,-':>9s} {'rho_2,-':>9s}")
ext = {}
for r in (0.25, 0.5, 0.75, 1.0):
em, ep = thresholds(r, mh)
for f in (0.3, 0.5, 0.7, 0.9):
eps = f * ep
for name, T in lat:
if name == "ring" and T.L == 4096:
parts = (1,)
else:
parts = (1, 2)
up, dn = analyse(T, +1, r, eps, parts), analyse(T, -1, r, eps, parts)
v = [up[1][1], up[2][1] if 2 in parts else np.nan, dn[1][1], dn[2][1] if 2 in parts else np.nan]
log(f"{name:>6s} {T.L:5d} {r:5.2f} {ep:8.5f} {f:4.1f} {eps:8.5f} | "
+ " ".join(f"{x:9.4f}" for x in v))
key = (name, T.L)
lo_, hi_ = ext.get(key, (np.inf, -np.inf))
vv = [x for x in v if np.isfinite(x)]
ext[key] = (min(lo_, min(vv)), max(hi_, max(vv)))
log(" range of all rho_{i,sigma} above, per lattice: "
+ "; ".join(f"{k[0]} L={k[1]}: [{a:.4f}, {b:.4f}]" for k, (a, b) in ext.items()))
return lat
# ------------------------------------------------------------------ part G: curvature and finite-size checks
def part_g(T8, T32, lat):
log("")
log("PART G: accuracy of S_i''(0) (I4) and finite size (I3) at the points of smallest isolation gap."
" Analytic rho_i against Richardson finite differences with h0 = min(1e-2, 0.05 eps/tau) (step of v3),"
" h0/2, h0/4: max |rho_i(h) - rho_i(analytic)| over the cases")
ring, torus = lat[0][1], lat[2][1]
cases = []
for r in (0.25, 0.5, 0.75, 1.0):
ep = thresholds(r, 1 / 8.0)[1]
for name, T in (("ring", ring), ("torus", torus)):
for sg in (+1, -1):
cases.append((f"light {name} L={T.L}", T, sg, r, 0.9 * ep))
eg8, eg32 = np.geomspace(0.1, 64.0, 71), np.geomspace(0.1, 64.0, 36)
edge = {}
for T, eg, rs in ((T8, eg8, (1.25, 1.5, 2.0, 3.0)), (T32, eg32, (1.5, 2.0, 3.0))):
for r in rs:
x = min(z for z in eg if lower_gap(r, z, T.mh) > MARGIN * T.mh)
edge[(T.L, r)] = x
cases.append((f"lower edge torus L={T.L} ell={T.ell}", T, -1, r, x))
dev = {}
for name, T, sg, r, eps in cases:
an = analyse(T, sg, r, eps)
gap = (upper_gap(r, eps, T.mh) if sg == 1 else lower_gap(r, eps, T.mh)) / T.mh
h0 = min(1e-2, 0.05 * eps / r)
for i in (1, 2):
f = (lambda P: T.S1(P, sg, r, eps)) if i == 1 else (lambda P: T.S2(P, sg, r, eps))
f0 = an[i][0]
for j, h in enumerate((h0, h0 / 2, h0 / 4)):
_, spp = curv(f, h, f0)
d = abs(-(eps**2 / r**2) * spp / f0 - an[i][1])
key = (name, i, j)
dev[key] = max(dev.get(key, 0.0), d)
log(f" {name:28s} sigma={sg:+d} r={r:4.2f} eps={eps:.5f} gap/m_h={gap:.3f}:"
f" rho_1={an[1][1]:.6f} rho_2={an[2][1]:.6f}")
for name in dict.fromkeys(c[0] for c in cases):
log(f" {name:28s}: " + "; ".join(
f"rho_{i}: " + ", ".join(f"{dev[(name, i, j)]:.1e}" for j in range(3)) for i in (1, 2))
+ " (h0, h0/2, h0/4)")
T128 = torus if torus.L == 128 else Torus(128, 8.0)
dmax = 0.0
for r in (1.25, 1.5, 2.0, 3.0):
x = edge[(64, r)]
a64, a128 = analyse(T8, -1, r, x), analyse(T128, -1, r, x)
dmax = max(dmax, abs(a64[1][1] - a128[1][1]), abs(a64[2][1] - a128[2][1]))
log(f" finite size, lower edge r={r:4.2f} eps={x:.4f}: rho_1 {a64[1][1]:.6f} (L=64) {a128[1][1]:.6f}"
f" (L=128); rho_2 {a64[2][1]:.6f} (L=64) {a128[2][1]:.6f} (L=128)")
log(f" max |rho_i(L=64) - rho_i(L=128)| at these points: {dmax:.2e}")
if __name__ == "__main__":
part_a()
T8 = Torus(64, 8.0)
Ts = part_b(T8)
grid = part_c(T8)
part_b2(Ts, grid)
part_d(T8)
part_d2(Ts[(256, 32.0)])
part_e()
lat = part_f()
part_g(T8, Ts[(256, 32.0)], lat)
(OUT / "torus_mark_inertia.txt").write_text("\n".join(LOG) + "\n", encoding="utf-8")
output/check_mark_inertia.txt11 KB · raw
PART A: ring L=32, c0=1.0, m_h=0.25 (ell=4.0), F0=1, G0=1.985916, w0=0.550116, Omega_max=2.0156
second-order coefficients: formula (6.2), (6.3), (6.5) vs ED restricted to <= 2 quanta
grip Mk sig r | Cost2 form Cost2 ED | E2pp form E2pp ED | Cost1 ED-form Epp1 ED-form
square (F1=0.3, F2=F1^2/4=0.0225) 3 +1 0.5 | -0.79709146 -0.79709146 | 0.00272483 0.00272483 | -5.88e-11 -7.56e-13
square (F1=0.3, F2=F1^2/4=0.0225) 3 -1 0.5 | -0.81355379 -0.81355379 | 0.06469109 0.06469109 | 6.38e-11 -1.30e-11
square (F1=0.3, F2=F1^2/4=0.0225) 3 +1 1.0 | -0.77188596 -0.77188596 | -0.01790101 -0.01790101 | -5.00e-11 -1.63e-11
square (F1=0.3, F2=F1^2/4=0.0225) 3 -1 1.0 | -0.83695453 -0.83695453 | 0.24473099 0.24473099 | 6.85e-11 -5.10e-11
square (F1=0.3, F2=F1^2/4=0.0225) 3 +1 1.5 | -0.73621933 -0.73621933 | -0.12253651 -0.12253651 | -3.93e-11 -5.93e-11
square (F1=0.3, F2=F1^2/4=0.0225) 3 -1 1.5 | -0.88257932 -0.88257932 | 0.47513821 0.47513820 | 7.74e-11 -1.10e-10
square (F1=0.3, F2=F1^2/4=0.0225) 4 +1 0.5 | -1.42156754 -1.42156754 | -0.00839540 -0.00839540 | -4.41e-10 -1.88e-11
square (F1=0.3, F2=F1^2/4=0.0225) 4 -1 0.5 | -1.44333641 -1.44333642 | 0.07578325 0.07578325 | 4.69e-10 -6.91e-11
square (F1=0.3, F2=F1^2/4=0.0225) 4 +1 1.0 | -1.38830326 -1.38830326 | -0.06039710 -0.06039710 | -3.94e-10 -1.21e-10
square (F1=0.3, F2=F1^2/4=0.0225) 4 -1 1.0 | -1.47480499 -1.47480499 | 0.28739467 0.28739468 | 4.99e-10 -2.70e-10
square (F1=0.3, F2=F1^2/4=0.0225) 4 +1 1.5 | -1.33921483 -1.33921484 | -0.21748355 -0.21748356 | -3.33e-10 -3.65e-10
square (F1=0.3, F2=F1^2/4=0.0225) 4 -1 1.5 | -1.53381757 -1.53381758 | 0.57028552 0.57028551 | 5.55e-10 -5.68e-10
other (F1=0.3, F2=-0.15) 3 +1 0.5 | -0.88255089 -0.88255088 | 0.00244496 0.00244496 | 1.97e-09 1.22e-10
other (F1=0.3, F2=-0.15) 3 -1 0.5 | -0.90108819 -0.90108819 | 0.07033359 0.07033359 | -2.17e-09 6.41e-10
other (F1=0.3, F2=-0.15) 3 +1 1.0 | -0.85386846 -0.85386846 | -0.01963079 -0.01963079 | 1.67e-09 8.37e-10
other (F1=0.3, F2=-0.15) 3 -1 1.0 | -0.92705583 -0.92705583 | 0.26807672 0.26807672 | -2.47e-09 2.53e-09
other (F1=0.3, F2=-0.15) 3 +1 1.5 | -0.81331009 -0.81331009 | -0.12802719 -0.12802719 | 1.32e-09 2.48e-09
other (F1=0.3, F2=-0.15) 3 -1 1.5 | -0.97758359 -0.97758359 | 0.53083127 0.53083125 | -3.16e-09 4.96e-09
other (F1=0.3, F2=-0.15) 4 +1 0.5 | -1.57419026 -1.57419024 | -0.01025575 -0.01025575 | 1.49e-08 1.09e-09
other (F1=0.3, F2=-0.15) 4 -1 0.5 | -1.59864379 -1.59864377 | 0.08292655 0.08292654 | -1.60e-08 3.27e-09
other (F1=0.3, F2=-0.15) 4 +1 1.0 | -1.53652382 -1.53652380 | -0.06819105 -0.06819104 | 1.32e-08 5.66e-09
other (F1=0.3, F2=-0.15) 4 -1 1.0 | -1.63363731 -1.63363729 | 0.31662572 0.31662569 | -1.77e-08 1.28e-08
other (F1=0.3, F2=-0.15) 4 +1 1.5 | -1.48093983 -1.48093982 | -0.23595428 -0.23595426 | 1.11e-08 1.53e-08
other (F1=0.3, F2=-0.15) 4 -1 1.5 | -1.69916986 -1.69916983 | 0.63864993 0.63864987 | -2.13e-08 2.58e-08
max relative deviation of second-order coefficients (formula vs ED): 2.26e-07
zeroth order: ED level = sigma*F0*M_k and E''(0) = sigma*tau^2/(F0*M_k) asserted to 1e-10 / 1e-8
PART B: formula (6.3) on rings with L = 8 ell, c0 = 1, F0 = 1;
rho_1, rho_2 = I^mark tau^2/C^mark of the F1^2 and F2^2 parts; prediction (6.9): 2 - r^2
held mark: g1 = G0*2c0^2/ell, g2 = J2*16c0^3/ell (prediction (6.7): -> 1); Cratio = C^mark(tau)/C^mark(0) for the F1^2 part
t_i = rho_i + 2 sigma eps_k R_i: rho with the term (iv) of Step 6 removed, R_i = <<sin^2 Q/E^2>>/<<1/E>> of the part i
ell g1 g2 Mk sig r 2-r^2 rho_1 rho_2 t_1 t_2 Cratio
16 1.00018 0.96912 3 +1 0.5 1.75 1.02947 0.27353 1.71048 1.55478 0.99488
16 1.00018 0.96912 3 +1 1.0 1.00 0.42042 -0.09180 1.10143 1.18945 0.98053
16 1.00018 0.96912 3 +1 1.5 -0.25 -0.49901 -0.61591 0.18200 0.66535 0.95935
16 1.00018 0.96912 3 -1 0.5 1.75 2.34232 2.68110 1.66131 1.39984 1.00443
16 1.00018 0.96912 3 -1 1.0 1.00 1.77840 2.40227 1.09739 1.12101 1.01848
16 1.00018 0.96912 3 -1 1.5 -0.25 0.72788 1.84891 0.04687 0.56766 1.04497
16 1.00018 0.96912 4 +1 0.5 1.75 0.79609 -0.17873 1.70411 1.52961 0.99624
16 1.00018 0.96912 4 +1 1.0 1.00 0.19283 -0.53165 1.10085 1.17669 0.98551
16 1.00018 0.96912 4 +1 1.5 -0.25 -0.73980 -1.05627 0.16821 0.65207 0.96919
16 1.00018 0.96912 4 -1 0.5 1.75 2.57525 3.12404 1.66724 1.41570 1.00337
16 1.00018 0.96912 4 -1 1.0 1.00 2.00585 2.83561 1.09784 1.12727 1.01394
16 1.00018 0.96912 4 -1 1.5 -0.25 0.97571 2.28919 0.06769 0.58085 1.03324
32 1.00055 0.98473 3 +1 0.5 1.75 1.30991 0.66603 1.73271 1.63703 0.99686
32 1.00055 0.98473 3 +1 1.0 1.00 0.63602 0.17247 1.05882 1.14347 0.98799
32 1.00055 0.98473 3 +1 1.5 -0.25 -0.41564 -0.56204 0.00716 0.40895 0.97466
32 1.00055 0.98473 3 -1 0.5 1.75 2.12805 2.50760 1.70526 1.53660 1.00278
32 1.00055 0.98473 3 -1 1.0 1.00 1.48043 2.07982 1.05764 1.10882 1.01154
32 1.00055 0.98473 3 -1 1.5 -0.25 0.31874 1.26875 -0.10406 0.29775 1.02776
32 1.00055 0.98473 4 +1 0.5 1.75 1.16546 0.32687 1.72918 1.62153 0.99769
32 1.00055 0.98473 4 +1 1.0 1.00 0.49491 -0.15757 1.05863 1.13709 0.99105
32 1.00055 0.98473 4 +1 1.5 -0.25 -0.56826 -0.89831 -0.00454 0.39635 0.98081
32 1.00055 0.98473 4 -1 0.5 1.75 2.27233 2.84200 1.70860 1.54734 1.00211
32 1.00055 0.98473 4 -1 1.0 1.00 1.62149 2.40670 1.05776 1.11204 1.00869
32 1.00055 0.98473 4 -1 1.5 -0.25 0.47630 1.60897 -0.08742 0.31431 1.02055
64 1.00064 0.99254 3 +1 0.5 1.75 1.49021 0.99884 1.74289 1.68674 0.99814
64 1.00064 0.99254 3 +1 1.0 1.00 0.78115 0.41067 1.03383 1.09857 0.99285
64 1.00064 0.99254 3 +1 1.5 -0.25 -0.35207 -0.49218 -0.09939 0.19572 0.98478
64 1.00064 0.99254 3 -1 0.5 1.75 1.98048 2.31156 1.72780 1.62366 1.00168
64 1.00064 0.99254 3 -1 1.0 1.00 1.28617 1.76929 1.03349 1.08139 1.00692
64 1.00064 0.99254 3 -1 1.5 -0.25 0.07381 0.77769 -0.17887 0.08979 1.01650
64 1.00064 0.99254 4 +1 0.5 1.75 1.40406 0.76016 1.74097 1.67736 0.99863
64 1.00064 0.99254 4 +1 1.0 1.00 0.69686 0.17823 1.03377 1.09543 0.99467
64 1.00064 0.99254 4 +1 1.5 -0.25 -0.44489 -0.73290 -0.10798 0.18430 0.98849
64 1.00064 0.99254 4 -1 0.5 1.75 2.06655 2.54781 1.72965 1.63061 1.00127
64 1.00064 0.99254 4 -1 1.0 1.00 1.37043 2.00020 1.03352 1.08300 1.00521
64 1.00064 0.99254 4 -1 1.5 -0.25 0.16961 1.02296 -0.16730 0.10576 1.01223
128 1.00066 0.99645 3 +1 0.5 1.75 1.60052 1.25165 1.74753 1.71538 0.99893
128 1.00066 0.99645 3 +1 1.0 1.00 0.87314 0.59979 1.02014 1.06352 0.99585
128 1.00066 0.99645 3 +1 1.5 -0.25 -0.30818 -0.42551 -0.16118 0.03822 0.99111
128 1.00066 0.99645 3 -1 0.5 1.75 1.88631 2.14051 1.73931 1.67678 1.00098
128 1.00066 0.99645 3 -1 1.0 1.00 1.16705 1.51888 1.02005 1.05514 1.00403
128 1.00066 0.99645 3 -1 1.5 -0.25 -0.06643 0.41326 -0.21343 -0.05048 1.00955
128 1.00066 0.99645 4 +1 0.5 1.75 1.55048 1.09149 1.74648 1.70980 0.99921
128 1.00066 0.99645 4 +1 1.0 1.00 0.82412 0.44367 1.02012 1.06198 0.99690
128 1.00066 0.99645 4 +1 1.5 -0.25 -0.36296 -0.58958 -0.16696 0.02873 0.99328
128 1.00066 0.99645 4 -1 0.5 1.75 1.93632 2.29943 1.74032 1.68112 1.00074
128 1.00066 0.99645 4 -1 1.0 1.00 1.21606 1.67424 1.02005 1.05592 1.00303
128 1.00066 0.99645 4 -1 1.5 -0.25 -0.00998 0.58111 -0.20599 -0.03720 1.00709
256 1.00067 0.99840 3 +1 0.5 1.75 1.66579 1.43035 1.74963 1.73137 0.99939
256 1.00067 0.99840 3 +1 1.0 1.00 0.92909 0.73812 1.01293 1.03914 0.99764
256 1.00067 0.99840 3 +1 1.5 -0.25 -0.27981 -0.37076 -0.19597 -0.06974 0.99491
256 1.00067 0.99840 3 -1 0.5 1.75 1.82902 2.00926 1.74519 1.70824 1.00056
256 1.00067 0.99840 3 -1 1.0 1.00 1.09673 1.33612 1.01290 1.03510 1.00230
256 1.00067 0.99840 3 -1 1.5 -0.25 -0.14469 0.16335 -0.22853 -0.13767 1.00542
256 1.00067 0.99840 4 +1 0.5 1.75 1.63728 1.32673 1.74906 1.72810 0.99955
256 1.00067 0.99840 4 +1 1.0 1.00 0.90114 0.63703 1.01292 1.03839 0.99824
256 1.00067 0.99840 4 +1 1.5 -0.25 -0.31142 -0.47842 -0.19964 -0.07706 0.99616
256 1.00067 0.99840 4 -1 0.5 1.75 1.85751 2.11225 1.74573 1.71089 1.00043
256 1.00067 0.99840 4 -1 1.0 1.00 1.12468 1.43684 1.01290 1.03548 1.00173
256 1.00067 0.99840 4 -1 1.5 -0.25 -0.11219 0.27373 -0.22397 -0.12763 1.00403
512 1.00067 0.99938 3 +1 0.5 1.75 1.70349 1.55041 1.75057 1.74011 0.99966
512 1.00067 0.99938 3 +1 1.0 1.00 0.96211 0.83367 1.00920 1.02337 0.99868
512 1.00067 0.99938 3 +1 1.5 -0.25 -0.26229 -0.32977 -0.21521 -0.14007 0.99713
512 1.00067 0.99938 3 -1 0.5 1.75 1.79528 1.91619 1.74819 1.72649 1.00032
512 1.00067 0.99938 3 -1 1.0 1.00 1.05627 1.21113 1.00919 1.02143 1.00130
512 1.00067 0.99938 3 -1 1.5 -0.25 -0.18766 0.00087 -0.23475 -0.18883 1.00303
512 1.00067 0.99938 4 +1 0.5 1.75 1.68749 1.48528 1.75027 1.73821 0.99975
512 1.00067 0.99938 4 +1 1.0 1.00 0.94642 0.77007 1.00920 1.02300 0.99901
512 1.00067 0.99938 4 +1 1.5 -0.25 -0.28022 -0.39831 -0.21744 -0.14537 0.99784
512 1.00067 0.99938 4 -1 0.5 1.75 1.81127 1.98101 1.74849 1.72807 1.00024
512 1.00067 0.99938 4 -1 1.0 1.00 1.07197 1.27454 1.00919 1.02160 1.00097
512 1.00067 0.99938 4 -1 1.5 -0.25 -0.16927 0.07122 -0.23205 -0.18171 1.00226
output/torus_mark_inertia.txt30 KB · raw
PART A: lattice sums by mode classes and FFT pair convolution vs brute-force mode and ordered-pair sums, torus L=12, ell=3.0
dE=0.01: max relative deviation S_1 4.20e-14, S_2 7.33e-10 (cases (sigma,r,eps) = ((-1, 1.0, 1.0), (-1, 1.5, 3.0), (-1, 0.5, 0.2), (1, 0.5, 0.05)), P = 0, 0.02)
dE=0.001: max relative deviation S_1 4.20e-14, S_2 2.46e-13 (cases (sigma,r,eps) = ((-1, 1.0, 1.0), (-1, 1.5, 3.0), (-1, 0.5, 0.2), (1, 0.5, 0.05)), P = 0, 0.02)
R_1: classes 0.229832921964, brute force 0.229832921964; J2: FFT 0.011638555558, brute force 0.011638555558
PART B: convergence in ell and L (c0 = 1); rho_1, rho_2 of the F1^2 and F2^2 parts, lower branch
L ell G0 w0 J2 R1 R2 | r=1.0 e=1.0: rho_1, rho_2 r=1.0 e=4.0: rho_1, rho_2 r=0.5 e=2.0: rho_1, rho_2 r=2.0 e=2.0: rho_1, rho_2 r=3.0 e=4.0: rho_1, rho_2
32 4.0 0.232239 0.22480 0.011949 0.23251 0.12474 | 1.03105 0.45186 2.60889 1.38144 1.57378 0.78110 1.70071 0.85177 2.42262 1.56652
48 6.0 0.239180 0.22625 0.012219 0.23251 0.12634 | 1.04864 0.45934 2.62184 1.39875 1.59684 0.79160 1.64517 0.86582 2.11221 1.59060
64 8.0 0.242616 0.22682 0.012329 0.23200 0.12701 | 1.05570 0.46260 2.62347 1.40610 1.60686 0.79611 1.60931 0.87171 1.94764 1.59989
128 16.0 0.247714 0.22742 0.012450 0.23035 0.12778 | 1.06325 0.46651 2.61780 1.41470 1.61911 0.80147 1.54831 0.87840 1.71425 1.60921
256 32.0 0.250233 0.22760 0.012487 0.22901 0.12802 | 1.06527 0.46781 2.61022 1.41747 1.62351 0.80323 1.51653 0.88045 1.61492 1.61155
PART C: main table, lower branch (isolated), torus L=64, ell=8.0 (m_h=0.125), c0=1, F0=1, eps_k=M_k
G0=0.242616, w0=0.226815, J2=0.012329, R1=0.231996, R2=0.127005, Omega_max=3.4664
C_i = S_i(0)/S_i(0)|_{tau=0} (moving mark over held mark), rho_i of the parts, t = T/G0, xi* = F2^2/(c0 F1^2) at which rho = 1 (- if none finite >= 0)
r eps gap | C_1 C_2 | rho_1 rho_2 | t xi*
0.25 0.5 0.1250 | 1.00550 1.00171 | 0.6521 0.2474 | 0.3278 -
0.25 1.0 0.1250 | 1.00392 1.00139 | 1.0202 0.4490 | 0.5121 0.1811
0.25 2.0 0.1250 | 1.00252 1.00102 | 1.6032 0.7931 | 0.8036 14.3639
0.25 3.0 0.1250 | 1.00187 1.00081 | 2.1224 1.1002 | 1.0632 -
0.25 4.0 0.1250 | 1.00148 1.00067 | 2.6185 1.3894 | 1.3112 -
0.25 6.0 0.1250 | 1.00105 1.00050 | 3.5823 1.9414 | 1.7930 -
0.25 8.0 0.1250 | 1.00081 1.00040 | 4.5299 2.4758 | 2.2668 -
0.25 16.0 0.1250 | 1.00043 1.00022 | 8.2736 4.5550 | 4.1386 -
0.25 32.0 0.1250 | 1.00022 1.00012 | 15.7146 8.6470 | 7.8590 -
0.50 0.5 0.1250 | 1.02269 1.00691 | 0.6616 0.2494 | 0.3383 -
0.50 1.0 0.1250 | 1.01603 1.00562 | 1.0270 0.4516 | 0.5217 0.2446
0.50 2.0 0.1250 | 1.01023 1.00412 | 1.6069 0.7961 | 0.8116 14.7325
0.50 3.0 0.1250 | 1.00754 1.00326 | 2.1245 1.1034 | 1.0703 -
0.50 4.0 0.1250 | 1.00598 1.00270 | 2.6196 1.3927 | 1.3176 -
0.50 6.0 0.1250 | 1.00423 1.00201 | 3.5823 1.9447 | 1.7987 -
0.50 8.0 0.1250 | 1.00327 1.00160 | 4.5293 2.4791 | 2.2721 -
0.50 16.0 0.1250 | 1.00172 1.00089 | 8.2720 4.5583 | 4.1431 -
0.50 32.0 0.1250 | 1.00088 1.00047 | 15.7124 8.6503 | 7.8631 -
1.00 0.5 0.1250 | 1.10472 1.02891 | 0.7065 0.2579 | 0.3903 -
1.00 1.0 0.1250 | 1.07031 1.02324 | 1.0557 0.4626 | 0.5650 0.5334
1.00 2.0 0.1250 | 1.04324 1.01685 | 1.6212 0.8088 | 0.8457 16.3994
1.00 3.0 0.1250 | 1.03137 1.01326 | 2.1322 1.1166 | 1.0996 -
1.00 4.0 0.1250 | 1.02465 1.01095 | 2.6235 1.4061 | 1.3441 -
1.00 6.0 0.1250 | 1.01728 1.00813 | 3.5819 1.9582 | 1.8219 -
1.00 8.0 0.1250 | 1.01330 1.00646 | 4.5264 2.4927 | 2.2933 -
1.00 16.0 0.1250 | 1.00693 1.00356 | 8.2652 4.5717 | 4.1613 -
1.00 32.0 0.1250 | 1.00354 1.00188 | 15.7036 8.6636 | 7.8796 -
1.50 0.5 0.1012 | 1.34842 1.07115 | 0.6742 0.2776 | 0.4546 -
1.50 1.0 0.1250 | 1.19218 1.05566 | 1.0962 0.4847 | 0.6534 1.0371
1.50 2.0 0.1250 | 1.10791 1.03948 | 1.6407 0.8323 | 0.9089 20.0338
1.50 3.0 0.1250 | 1.07583 1.03076 | 2.1414 1.1404 | 1.1519 -
1.50 4.0 0.1250 | 1.05860 1.02524 | 2.6268 1.4299 | 1.3904 -
1.50 6.0 0.1250 | 1.04036 1.01861 | 3.5787 1.9818 | 1.8616 -
1.50 8.0 0.1250 | 1.03080 1.01475 | 4.5198 2.5160 | 2.3295 -
1.50 16.0 0.1250 | 1.01583 1.00807 | 8.2530 4.5945 | 4.1918 -
1.50 32.0 0.1250 | 1.00803 1.00424 | 15.6883 8.6859 | 7.9072 -
2.00 2.0 0.1250 | 1.23048 1.07474 | 1.6093 0.8717 | 0.9901 26.7521
2.00 3.0 0.1250 | 1.15137 1.05724 | 2.1326 1.1781 | 1.2277 -
2.00 4.0 0.1250 | 1.11345 1.04653 | 2.6184 1.4665 | 1.4577 -
2.00 6.0 0.1250 | 1.07589 1.03394 | 3.5670 2.0171 | 1.9188 -
2.00 8.0 0.1250 | 1.05710 1.02675 | 4.5053 2.5504 | 2.3813 -
2.00 16.0 0.1250 | 1.02874 1.01450 | 8.2334 4.6272 | 4.2350 -
2.00 32.0 0.1250 | 1.01443 1.00758 | 15.6657 8.7176 | 7.9459 -
PART B2: convergence over the configurations of Part C: change of rho_1, rho_2 and t from (L,ell)=(64,8) to (128,16) and to (256,32); absolute for eps <= 4, relative for eps > 4
to (128, 16.0): eps <= 4 max |change|: rho_1 0.0634, rho_2 0.0093, t 0.0412; eps > 4 max relative change: rho_1 0.0163, rho_2 0.0060, t 0.0167
to (256, 32.0): eps <= 4 max |change|: rho_1 0.0928, rho_2 0.0122, t 0.0581; eps > 4 max relative change: rho_1 0.0269, rho_2 0.0079, t 0.0277
PART D: fine grid, 71 values of eps in [0.1, 64] (geometric), torus L=64, ell=8.0; lower branch where b_k + eps_k > 0.2 m_h
item 2: rho_{i,-} increasing on the grid? crossings of rho_i,- = 1; points with finite xi* >= 0; item 3: phi = eps*t_- increasing and convex at the grid points?
thresholds (L = infinity): eps_-(r) [lower isolated above], eps_+(r) [upper isolated below], m_h/2 = 0.0625
r=0.10: eps_-=0.0000, eps_+=0.0625; lower computed on eps in [0.100,64.0] (71 pts); rho_1,- 0.1999..30.572 incr True; rho_2,- 0.0553..16.790 incr True
roots (Brent, +- error estimate): rho_1,- = 1 at eps 0.9721 (+-1e-10); rho_2,- = 1 at eps 2.6683 (+-1e-10)
finite xi* >= 0 [(rho_1-1)(rho_2-1) < 0] on eps in [0.986,2.658] (81-point refinement, 73 pts): xi* 0.0821 at 0.986 .. 1552.27 at 2.658; increasing True, decreasing False
windows of finite xi* >= 0 between consecutive roots: [0.9721 (rho_1=1), 2.6683 (rho_2=1)]; all 152 computed points (grid and refinement) consistent with them: True
phi increasing True; phi'' min 0.464 max 1.321; t_- 0.1001..15.286; at eps=64.0: rho_1,- - 2 eps R1 = 0.8764, rho_1,-/(2 eps R1) = 1.0295
r=0.25: eps_-=0.0000, eps_+=0.0625; lower computed on eps in [0.100,64.0] (71 pts); rho_1,- 0.2022..30.571 incr True; rho_2,- 0.0555..16.791 incr True
roots (Brent, +- error estimate): rho_1,- = 1 at eps 0.9692 (+-1e-10); rho_2,- = 1 at eps 2.6654 (+-1e-10)
finite xi* >= 0 [(rho_1-1)(rho_2-1) < 0] on eps in [0.986,2.658] (81-point refinement, 73 pts): xi* 0.0990 at 0.986 .. 2192.64 at 2.658; increasing True, decreasing False
windows of finite xi* >= 0 between consecutive roots: [0.9692 (rho_1=1), 2.6654 (rho_2=1)]; all 152 computed points (grid and refinement) consistent with them: True
phi increasing True; phi'' min 0.464 max 1.328; t_- 0.1019..15.287; at eps=64.0: rho_1,- - 2 eps R1 = 0.8757, rho_1,-/(2 eps R1) = 1.0295
r=0.50: eps_-=0.0000, eps_+=0.0625; lower computed on eps in [0.100,64.0] (71 pts); rho_1,- 0.2112..30.569 incr True; rho_2,- 0.0561..16.794 incr True
roots (Brent, +- error estimate): rho_1,- = 1 at eps 0.9587 (+-1e-10); rho_2,- = 1 at eps 2.6550 (+-1e-10)
finite xi* >= 0 [(rho_1-1)(rho_2-1) < 0] on eps in [0.963,2.635] (81-point refinement, 73 pts): xi* 0.0245 at 0.963 .. 770.92 at 2.635; increasing True, decreasing False
windows of finite xi* >= 0 between consecutive roots: [0.9587 (rho_1=1), 2.6550 (rho_2=1)]; all 152 computed points (grid and refinement) consistent with them: True
phi increasing True; phi'' min 0.464 max 1.350; t_- 0.1094..15.291; at eps=64.0: rho_1,- - 2 eps R1 = 0.8732, rho_1,-/(2 eps R1) = 1.0294
r=0.75: eps_-=0.0000, eps_+=0.0620; lower computed on eps in [0.100,64.0] (71 pts); rho_1,- 0.2310..30.565 incr True; rho_2,- 0.0573..16.799 incr True
roots (Brent, +- error estimate): rho_1,- = 1 at eps 0.9402 (+-1e-10); rho_2,- = 1 at eps 2.6373 (+-1e-10)
finite xi* >= 0 [(rho_1-1)(rho_2-1) < 0] on eps in [0.963,2.635] (81-point refinement, 73 pts): xi* 0.1334 at 0.963 .. 7000.51 at 2.635; increasing True, decreasing False
windows of finite xi* >= 0 between consecutive roots: [0.9402 (rho_1=1), 2.6373 (rho_2=1)]; all 152 computed points (grid and refinement) consistent with them: True
phi increasing True; phi'' min 0.464 max 1.367; t_- 0.1258..15.298; at eps=64.0: rho_1,- - 2 eps R1 = 0.8691, rho_1,-/(2 eps R1) = 1.0293
r=1.00: eps_-=0.0000, eps_+=0.0260; lower computed on eps in [0.100,64.0] (71 pts); rho_1,- 0.2783..30.559 incr True; rho_2,- 0.0593..16.807 incr True
roots (Brent, +- error estimate): rho_1,- = 1 at eps 0.9123 (+-1e-10); rho_2,- = 1 at eps 2.6119 (+-1e-10)
finite xi* >= 0 [(rho_1-1)(rho_2-1) < 0] on eps in [0.933,2.605] (81-point refinement, 70 pts): xi* 0.1195 at 0.933 .. 2169.06 at 2.605; increasing True, decreasing False
windows of finite xi* >= 0 between consecutive roots: [0.9123 (rho_1=1), 2.6119 (rho_2=1)]; all 152 computed points (grid and refinement) consistent with them: True
phi increasing True; phi'' min 0.464 max 1.283; t_- 0.1658..15.307; at eps=64.0: rho_1,- - 2 eps R1 = 0.8633, rho_1,-/(2 eps R1) = 1.0291
r=1.25: eps_-=0.1170, eps_+=0.0000; lower computed on eps in [0.159,64.0] (66 pts); rho_1,- 0.3458..30.551 incr True; rho_2,- 0.0966..16.817 incr True
roots (Brent, +- error estimate): rho_1,- = 1 at eps 0.8740 (+-1e-10); rho_2,- = 1 at eps 2.5779 (+-1e-10)
finite xi* >= 0 [(rho_1-1)(rho_2-1) < 0] on eps in [0.884,2.556] (81-point refinement, 70 pts): xi* 0.0591 at 0.884 .. 705.35 at 2.556; increasing True, decreasing False
windows of finite xi* >= 0 between consecutive roots: [0.8740 (rho_1=1), 2.5779 (rho_2=1)]; all 147 computed points (grid and refinement) consistent with them: True
phi increasing True; phi'' min -10.935 max 0.655; phi'' <= 0 at eps in [0.174,0.209]; t_- 0.2388..15.318; at eps=64.0: rho_1,- - 2 eps R1 = 0.8558, rho_1,-/(2 eps R1) = 1.0288
r=1.50: eps_-=0.3460, eps_+=0.0000; lower computed on eps in [0.399,64.0] (56 pts); rho_1,- -0.1158..30.542 incr True; rho_2,- 0.2307..16.829 incr True
roots (Brent, +- error estimate): rho_1,- = 1 at eps 0.8447 (+-1e-10); rho_2,- = 1 at eps 2.5342 (+-1e-10)
finite xi* >= 0 [(rho_1-1)(rho_2-1) < 0] on eps in [0.860,2.532] (81-point refinement, 70 pts): xi* 0.0967 at 0.860 .. 7303.39 at 2.532; increasing True, decreasing False
windows of finite xi* >= 0 between consecutive roots: [0.8447 (rho_1=1), 2.5342 (rho_2=1)]; all 137 computed points (grid and refinement) consistent with them: True
phi increasing True; phi'' min -65.091 max 0.483; phi'' <= 0 at eps in [0.438,0.634]; t_- -0.0839..15.333; at eps=64.0: rho_1,- - 2 eps R1 = 0.8465, rho_1,-/(2 eps R1) = 1.0285
r=2.00: eps_-=0.9846, eps_+=0.0000; lower computed on eps in [1.102,64.0] (45 pts); rho_1,- -0.8525..30.518 incr True; rho_2,- 0.5649..16.860 incr True
roots (Brent, +- error estimate): rho_1,- = 1 at eps 1.3385 (+-1e-10); rho_2,- = 1 at eps 2.4089 (+-1e-10)
finite xi* >= 0 [(rho_1-1)(rho_2-1) < 0] on eps in [1.341,2.394] (81-point refinement, 71 pts): xi* 0.0853 at 1.341 .. 1011.36 at 2.394; increasing True, decreasing False
windows of finite xi* >= 0 between consecutive roots: [1.3385 (rho_1=1), 2.4089 (rho_2=1)]; all 126 computed points (grid and refinement) consistent with them: True
phi increasing True; phi'' min -67.350 max 0.464; phi'' <= 0 at eps in [1.209,1.749]; t_- -0.6397..15.369; at eps=64.0: rho_1,- - 2 eps R1 = 0.8223, rho_1,-/(2 eps R1) = 1.0277
r=3.00: eps_-=2.8771, eps_+=0.0000; lower computed on eps in [3.043,64.0] (34 pts); rho_1,- -10.0497..30.445 incr True; rho_2,- 1.3281..16.950 incr True
roots (Brent, +- error estimate): rho_1,- = 1 at eps 3.5649 (+-1e-10); rho_2,- = 1 at eps none
finite xi* >= 0 [(rho_1-1)(rho_2-1) < 0] on eps in [3.043,3.559] (81-point refinement, 68 pts): xi* 234.2429 at 3.043 .. 0.26 at 3.559; increasing False, decreasing True
windows of finite xi* >= 0 between consecutive roots: [3.0428 (edge of computed range), 3.5649 (rho_1=1)]; all 115 computed points (grid and refinement) consistent with them: True
phi increasing True; phi'' min -223.950 max 0.464; phi'' <= 0 at eps in [3.337,4.827]; t_- -8.1671..15.474; at eps=64.0: rho_1,- - 2 eps R1 = 0.7492, rho_1,-/(2 eps R1) = 1.0252
min over all computed points of S_i(0)/S_i(0)|_(tau=0): 1.0000
grip y = F0 F2/F1^2 that equalises the bracket of (6.12) for two lower levels (eps_a, eps_b), direct evaluation at the stated levels:
r=0.10: (0.25,0.5) 0.2242 (0.5,1) 0.3470 (1,2) 0.5467 (2,4) 0.9087 (4,8) 1.6106 (1,8) 1.2581
r=0.25: (0.25,0.5) 0.2257 (0.5,1) 0.3482 (1,2) 0.5476 (2,4) 0.9094 (4,8) 1.6112 (1,8) 1.2587
r=0.50: (0.25,0.5) 0.2312 (0.5,1) 0.3526 (1,2) 0.5508 (2,4) 0.9118 (4,8) 1.6132 (1,8) 1.2610
r=0.75: (0.25,0.5) 0.2407 (0.5,1) 0.3599 (1,2) 0.5561 (2,4) 0.9158 (4,8) 1.6166 (1,8) 1.2649
r=1.00: (0.25,0.5) 0.2544 (0.5,1) 0.3698 (1,2) 0.5632 (2,4) 0.9212 (4,8) 1.6213 (1,8) 1.2701
r=1.25: (0.25,0.5) 0.2715 (0.5,1) 0.3814 (1,2) 0.5718 (2,4) 0.9280 (4,8) 1.6272 (1,8) 1.2766
r=1.50: (0.5,1) 0.4261 (1,2) 0.5821 (2,4) 0.9359 (4,8) 1.6343 (1,8) 1.2845
r=2.00: (2,4) 0.9627 (4,8) 1.6524
r=3.00: (4,8) 1.8593
members with stated margins (exact evaluation at the stated eps, r = 1), L=64, ell=8.0:
item 2: level eps=4: rho_1,- = 2.6235, rho_2,- = 1.4061 -> rho_- >= 1.4061 > 1 for every grip
item 2: levels eps=1.2, 2.0: xi* = 1.9596, 16.3994
item 3: levels eps=1,2,3: t_- = 0.5650, 0.8457, 1.0996; y_12 = 0.5632, y_23 = 0.8037
PART D2: as Part D, both parts, torus L=256, ell=32.0; 36 values of eps in [0.1, 64]
thresholds (L = infinity): eps_-(r) [lower isolated above], eps_+(r) [upper isolated below], m_h/2 = 0.0156
r=0.10: eps_-=0.0000, eps_+=0.0156; lower computed on eps in [0.100,64.0] (36 pts); rho_1,- 0.2289..30.219 incr True; rho_2,- 0.0563..16.922 incr True
roots (Brent, +- error estimate): rho_1,- = 1 at eps 0.9274 (+-1e-10); rho_2,- = 1 at eps 2.6406 (+-1e-10)
finite xi* >= 0 [(rho_1-1)(rho_2-1) < 0] on eps in [0.940,2.635] (81-point refinement, 74 pts): xi* 0.0718 at 0.940 .. 2833.10 at 2.635; increasing True, decreasing False
windows of finite xi* >= 0 between consecutive roots: [0.9274 (rho_1=1), 2.6406 (rho_2=1)]; all 117 computed points (grid and refinement) consistent with them: True
phi increasing True; phi'' min 0.458 max 1.273; t_- 0.1146..15.110; at eps=64.0: rho_1,- - 2 eps R1 = 0.9056, rho_1,-/(2 eps R1) = 1.0309
r=0.50: eps_-=0.0000, eps_+=0.0156; lower computed on eps in [0.100,64.0] (36 pts); rho_1,- 0.2403..30.211 incr True; rho_2,- 0.0572..16.927 incr True
roots (Brent, +- error estimate): rho_1,- = 1 at eps 0.9201 (+-1e-10); rho_2,- = 1 at eps 2.6272 (+-1e-10)
finite xi* >= 0 [(rho_1-1)(rho_2-1) < 0] on eps in [0.940,2.612] (81-point refinement, 73 pts): xi* 0.1150 at 0.940 .. 1007.75 at 2.612; increasing True, decreasing False
windows of finite xi* >= 0 between consecutive roots: [0.9201 (rho_1=1), 2.6272 (rho_2=1)]; all 117 computed points (grid and refinement) consistent with them: True
phi increasing True; phi'' min 0.458 max 1.281; t_- 0.1246..15.112; at eps=64.0: rho_1,- - 2 eps R1 = 0.8973, rho_1,-/(2 eps R1) = 1.0306
r=1.00: eps_-=0.0000, eps_+=0.0034; lower computed on eps in [0.100,64.0] (36 pts); rho_1,- 0.3070..30.184 incr True; rho_2,- 0.0609..16.940 incr True
roots (Brent, +- error estimate): rho_1,- = 1 at eps 0.8957 (+-1e-10); rho_2,- = 1 at eps 2.5838 (+-1e-10)
finite xi* >= 0 [(rho_1-1)(rho_2-1) < 0] on eps in [0.913,2.573] (81-point refinement, 67 pts): xi* 0.1014 at 0.913 .. 1438.91 at 2.573; increasing True, decreasing False
windows of finite xi* >= 0 between consecutive roots: [0.8957 (rho_1=1), 2.5838 (rho_2=1)]; all 117 computed points (grid and refinement) consistent with them: True
phi increasing True; phi'' min 0.458 max 1.174; t_- 0.1841..15.119; at eps=64.0: rho_1,- - 2 eps R1 = 0.8713, rho_1,-/(2 eps R1) = 1.0297
r=1.50: eps_-=0.3578, eps_+=0.0000; lower computed on eps in [0.438,64.0] (28 pts); rho_1,- 0.2193..30.141 incr True; rho_2,- 0.2535..16.962 incr True
roots (Brent, +- error estimate): rho_1,- = 1 at eps 0.8909 (+-1e-10); rho_2,- = 1 at eps 2.5053 (+-1e-10)
finite xi* >= 0 [(rho_1-1)(rho_2-1) < 0] on eps in [0.913,2.498] (81-point refinement, 64 pts): xi* 0.1483 at 0.913 .. 1949.26 at 2.498; increasing True, decreasing False
windows of finite xi* >= 0 between consecutive roots: [0.8909 (rho_1=1), 2.5053 (rho_2=1)]; all 109 computed points (grid and refinement) consistent with them: True
phi increasing True; phi'' min -9.444 max 0.470; phi'' <= 0 at eps in [0.527,0.634]; t_- 0.1546..15.131; at eps=64.0: rho_1,- - 2 eps R1 = 0.8275, rho_1,-/(2 eps R1) = 1.0282
r=2.00: eps_-=0.9990, eps_+=0.0000; lower computed on eps in [1.102,64.0] (23 pts); rho_1,- -1.6875..30.079 incr True; rho_2,- 0.5678..16.993 incr True
roots (Brent, +- error estimate): rho_1,- = 1 at eps 1.4187 (+-1e-10); rho_2,- = 1 at eps 2.3785 (+-1e-10)
finite xi* >= 0 [(rho_1-1)(rho_2-1) < 0] on eps in [1.434,2.376] (81-point refinement, 53 pts): xi* 0.5056 at 1.434 .. 5387.87 at 2.376; increasing True, decreasing False
windows of finite xi* >= 0 between consecutive roots: [1.4187 (rho_1=1), 2.3785 (rho_2=1)]; all 104 computed points (grid and refinement) consistent with them: True
phi increasing True; phi'' min -30.764 max 0.458; phi'' <= 0 at eps in [1.326,1.595]; t_- -1.2715..15.147; at eps=64.0: rho_1,- - 2 eps R1 = 0.7653, rho_1,-/(2 eps R1) = 1.0261
r=3.00: eps_-=2.9015, eps_+=0.0000; lower computed on eps in [3.337,64.0] (17 pts); rho_1,- -1.0119..29.896 incr True; rho_2,- 1.4204..17.084 incr True
roots (Brent, +- error estimate): rho_1,- = 1 at eps 3.6997 (+-1e-10); rho_2,- = 1 at eps none
finite xi* >= 0 [(rho_1-1)(rho_2-1) < 0] on eps in [3.337,3.692] (81-point refinement, 43 pts): xi* 31.4242 at 3.337 .. 0.25 at 3.692; increasing False, decreasing True
windows of finite xi* >= 0 between consecutive roots: [3.3370 (edge of computed range), 3.6997 (rho_1=1)]; all 98 computed points (grid and refinement) consistent with them: True
phi increasing True; phi'' min -8.825 max 0.458; phi'' <= 0 at eps in [4.014,4.827]; t_- -0.7566..15.192; at eps=64.0: rho_1,- - 2 eps R1 = 0.5824, rho_1,-/(2 eps R1) = 1.0199
min over all computed points of S_i(0)/S_i(0)|_(tau=0): 1.0000
grip y = F0 F2/F1^2 that equalises the bracket of (6.12) for two lower levels (eps_a, eps_b), direct evaluation at the stated levels:
r=0.10: (0.25,0.5) 0.2324 (0.5,1) 0.3533 (1,2) 0.5501 (2,4) 0.9072 (4,8) 1.6000 (1,8) 1.2521
r=0.50: (0.25,0.5) 0.2382 (0.5,1) 0.3576 (1,2) 0.5528 (2,4) 0.9090 (4,8) 1.6013 (1,8) 1.2537
r=1.00: (0.25,0.5) 0.2569 (0.5,1) 0.3703 (1,2) 0.5608 (2,4) 0.9141 (4,8) 1.6050 (1,8) 1.2584
r=1.50: (0.5,1) 0.4381 (1,2) 0.5733 (2,4) 0.9217 (4,8) 1.6109 (1,8) 1.2657
r=2.00: (2,4) 0.9418 (4,8) 1.6194
r=3.00: (4,8) 1.8315
members with stated margins (exact evaluation at the stated eps, r = 1), L=256, ell=32.0:
item 2: level eps=4: rho_1,- = 2.6102, rho_2,- = 1.4175 -> rho_- >= 1.4175 > 1 for every grip
item 2: levels eps=1.2, 2.0: xi* = 2.1050, 17.3835
item 3: levels eps=1,2,3: t_- = 0.5699, 0.8458, 1.0959; y_12 = 0.5608, y_23 = 0.7980
PART E: ED of the contract restricted to <= 2 field quanta, cubic torus L=4, ell=2.0, dimension 4290; M_k=5.0, F1=0.3, F2=0.1, r=1.0, Omega_max=3.5000 (sigma=+1: formal coefficients, the upper branch is not isolated here)
sigma=+1 Cost second order: ED -0.2793475807 formula -0.2793475807
sigma=+1 E''(0) second order: ED -0.0038368899 formula -0.0038368899
sigma=-1 Cost second order: ED -0.2892384565 formula -0.2892384576
sigma=-1 E''(0) second order: ED 0.0307210119 formula 0.0307210119
max relative deviation: 3.84e-09
PART F: light levels eps_k < eps_+(r) (upper branch isolated), ell=8.0 (m_h=0.125), eps = f*eps_+, both branches, both parts; ring and cubic torus
graph L r eps_+ f eps | rho_1,+ rho_2,+ rho_1,- rho_2,-
ring 2048 0.25 0.06250 0.3 0.01875 | -0.5163 -0.1403 0.3143 0.1159
ring 4096 0.25 0.06250 0.3 0.01875 | -0.5163 nan 0.3143 nan
torus 128 0.25 0.06250 0.3 0.01875 | -0.0536 -0.0109 0.0463 0.0107
torus 256 0.25 0.06250 0.3 0.01875 | -0.0536 -0.0109 0.0463 0.0107
ring 2048 0.25 0.06250 0.5 0.03125 | -1.1159 -0.2526 0.4656 0.1831
ring 4096 0.25 0.06250 0.5 0.03125 | -1.1159 nan 0.4656 nan
torus 128 0.25 0.06250 0.5 0.03125 | -0.0959 -0.0182 0.0742 0.0178
torus 256 0.25 0.06250 0.5 0.03125 | -0.0959 -0.0182 0.0742 0.0178
ring 2048 0.25 0.06250 0.7 0.04375 | -2.2855 -0.3854 0.5875 0.2437
ring 4096 0.25 0.06250 0.7 0.04375 | -2.2855 nan 0.5875 nan
torus 128 0.25 0.06250 0.7 0.04375 | -0.1471 -0.0257 0.1004 0.0248
torus 256 0.25 0.06250 0.7 0.04375 | -0.1471 -0.0257 0.1004 0.0248
ring 2048 0.25 0.06250 0.9 0.05625 | -6.2948 -0.5466 0.6883 0.2990
ring 4096 0.25 0.06250 0.9 0.05625 | -6.2948 nan 0.6883 nan
torus 128 0.25 0.06250 0.9 0.05625 | -0.2198 -0.0332 0.1250 0.0317
torus 256 0.25 0.06250 0.9 0.05625 | -0.2197 -0.0332 0.1250 0.0317
ring 2048 0.50 0.06250 0.3 0.01875 | -0.5776 -0.1541 0.3346 0.1247
ring 4096 0.50 0.06250 0.3 0.01875 | -0.5776 nan 0.3346 nan
torus 128 0.50 0.06250 0.3 0.01875 | -0.0590 -0.0110 0.0495 0.0109
torus 256 0.50 0.06250 0.3 0.01875 | -0.0590 -0.0110 0.0495 0.0109
ring 2048 0.50 0.06250 0.5 0.03125 | -1.2864 -0.2800 0.4907 0.1958
ring 4096 0.50 0.06250 0.5 0.03125 | -1.2864 nan 0.4907 nan
torus 128 0.50 0.06250 0.5 0.03125 | -0.1070 -0.0185 0.0790 0.0180
torus 256 0.50 0.06250 0.5 0.03125 | -0.1070 -0.0185 0.0790 0.0180
ring 2048 0.50 0.06250 0.7 0.04375 | -2.7657 -0.4321 0.6139 0.2595
ring 4096 0.50 0.06250 0.7 0.04375 | -2.7657 nan 0.6139 nan
torus 128 0.50 0.06250 0.7 0.04375 | -0.1683 -0.0261 0.1064 0.0251
torus 256 0.50 0.06250 0.7 0.04375 | -0.1682 -0.0261 0.1064 0.0251
ring 2048 0.50 0.06250 0.9 0.05625 | -8.5004 -0.6212 0.7140 0.3169
ring 4096 0.50 0.06250 0.9 0.05625 | -8.5004 nan 0.7140 nan
torus 128 0.50 0.06250 0.9 0.05625 | -0.2651 -0.0337 0.1320 0.0321
torus 256 0.50 0.06250 0.9 0.05625 | -0.2650 -0.0337 0.1320 0.0321
ring 2048 0.75 0.06202 0.3 0.01861 | -0.7723 -0.1906 0.3866 0.1457
ring 4096 0.75 0.06202 0.3 0.01861 | -0.7723 nan 0.3866 nan
torus 128 0.75 0.06202 0.3 0.01861 | -0.0741 -0.0113 0.0574 0.0110
torus 256 0.75 0.06202 0.3 0.01861 | -0.0741 -0.0113 0.0574 0.0110
ring 2048 0.75 0.06202 0.5 0.03101 | -1.9041 -0.3557 0.5517 0.2257
ring 4096 0.75 0.06202 0.5 0.03101 | -1.9041 nan 0.5517 nan
torus 128 0.75 0.06202 0.5 0.03101 | -0.1419 -0.0189 0.0902 0.0183
torus 256 0.75 0.06202 0.5 0.03101 | -0.1419 -0.0189 0.0902 0.0183
ring 2048 0.75 0.06202 0.7 0.04341 | -4.9499 -0.5674 0.6754 0.2955
ring 4096 0.75 0.06202 0.7 0.04341 | -4.9499 nan 0.6754 nan
torus 128 0.75 0.06202 0.7 0.04341 | -0.2466 -0.0266 0.1200 0.0255
torus 256 0.75 0.06202 0.7 0.04341 | -0.2466 -0.0266 0.1200 0.0255
ring 2048 0.75 0.06202 0.9 0.05582 | -27.2162 -0.8510 0.7715 0.3571
ring 4096 0.75 0.06202 0.9 0.05582 | -27.2162 nan 0.7715 nan
torus 128 0.75 0.06202 0.9 0.05582 | -0.5329 -0.0345 0.1473 0.0326
torus 256 0.75 0.06202 0.9 0.05582 | -0.5301 -0.0345 0.1473 0.0326
ring 2048 1.00 0.02600 0.3 0.00780 | -0.8806 -0.1338 0.3795 0.1065
ring 4096 1.00 0.02600 0.3 0.00780 | -0.8806 nan 0.3795 nan
torus 128 1.00 0.02600 0.3 0.00780 | -0.0676 -0.0049 0.0449 0.0049
torus 256 1.00 0.02600 0.3 0.00780 | -0.0676 -0.0049 0.0449 0.0049
ring 2048 1.00 0.02600 0.5 0.01300 | -2.4095 -0.2441 0.5271 0.1664
ring 4096 1.00 0.02600 0.5 0.01300 | -2.4095 nan 0.5271 nan
torus 128 1.00 0.02600 0.5 0.01300 | -0.1447 -0.0082 0.0686 0.0081
torus 256 1.00 0.02600 0.5 0.01300 | -0.1446 -0.0082 0.0686 0.0081
ring 2048 1.00 0.02600 0.7 0.01820 | -7.5371 -0.3778 0.6313 0.2193
ring 4096 1.00 0.02600 0.7 0.01820 | -7.5371 nan 0.6313 nan
torus 128 1.00 0.02600 0.7 0.01820 | -0.3093 -0.0116 0.0891 0.0113
torus 256 1.00 0.02600 0.7 0.01820 | -0.3091 -0.0116 0.0891 0.0113
ring 2048 1.00 0.02600 0.9 0.02340 | -65.2061 -0.5435 0.7084 0.2664
ring 4096 1.00 0.02600 0.9 0.02340 | -65.2061 nan 0.7084 nan
torus 128 1.00 0.02600 0.9 0.02340 | -1.2769 -0.0149 0.1074 0.0145
torus 256 1.00 0.02600 0.9 0.02340 | -1.2097 -0.0149 0.1074 0.0145
range of all rho_{i,sigma} above, per lattice: ring L=2048: [-65.2061, 0.7715]; ring L=4096: [-65.2061, 0.7715]; torus L=128: [-1.2769, 0.1473]; torus L=256: [-1.2097, 0.1473]
PART G: accuracy of S_i''(0) (I4) and finite size (I3) at the points of smallest isolation gap. Analytic rho_i against Richardson finite differences with h0 = min(1e-2, 0.05 eps/tau) (step of v3), h0/2, h0/4: max |rho_i(h) - rho_i(analytic)| over the cases
light ring L=2048 sigma=+1 r=0.25 eps=0.05625 gap/m_h=0.100: rho_1=-6.294773 rho_2=-0.546641
light ring L=2048 sigma=-1 r=0.25 eps=0.05625 gap/m_h=1.000: rho_1=0.688284 rho_2=0.298995
light torus L=128 sigma=+1 r=0.25 eps=0.05625 gap/m_h=0.100: rho_1=-0.219772 rho_2=-0.033221
light torus L=128 sigma=-1 r=0.25 eps=0.05625 gap/m_h=1.000: rho_1=0.124991 rho_2=0.031706
light ring L=2048 sigma=+1 r=0.50 eps=0.05625 gap/m_h=0.100: rho_1=-8.500448 rho_2=-0.621246
light ring L=2048 sigma=-1 r=0.50 eps=0.05625 gap/m_h=1.000: rho_1=0.714030 rho_2=0.316921
light torus L=128 sigma=+1 r=0.50 eps=0.05625 gap/m_h=0.100: rho_1=-0.265073 rho_2=-0.033725
light torus L=128 sigma=-1 r=0.50 eps=0.05625 gap/m_h=1.000: rho_1=0.131961 rho_2=0.032095
light ring L=2048 sigma=+1 r=0.75 eps=0.05582 gap/m_h=0.085: rho_1=-27.216244 rho_2=-0.850951
light ring L=2048 sigma=-1 r=0.75 eps=0.05582 gap/m_h=0.978: rho_1=0.771491 rho_2=0.357115
light torus L=128 sigma=+1 r=0.75 eps=0.05582 gap/m_h=0.085: rho_1=-0.532888 rho_2=-0.034455
light torus L=128 sigma=-1 r=0.75 eps=0.05582 gap/m_h=0.978: rho_1=0.147277 rho_2=0.032581
light ring L=2048 sigma=+1 r=1.00 eps=0.02340 gap/m_h=0.022: rho_1=-65.206114 rho_2=-0.543476
light ring L=2048 sigma=-1 r=1.00 eps=0.02340 gap/m_h=0.397: rho_1=0.708414 rho_2=0.266437
light torus L=128 sigma=+1 r=1.00 eps=0.02340 gap/m_h=0.022: rho_1=-1.276852 rho_2=-0.014933
light torus L=128 sigma=-1 r=1.00 eps=0.02340 gap/m_h=0.397: rho_1=0.107429 rho_2=0.014463
lower edge torus L=64 ell=8.0 sigma=-1 r=1.25 eps=0.15865 gap/m_h=0.279: rho_1=0.345796 rho_2=0.096638
lower edge torus L=64 ell=8.0 sigma=-1 r=1.50 eps=0.39932 gap/m_h=0.297: rho_1=-0.115816 rho_2=0.230717
lower edge torus L=64 ell=8.0 sigma=-1 r=2.00 eps=1.10229 gap/m_h=0.455: rho_1=-0.852547 rho_2=0.564905
lower edge torus L=64 ell=8.0 sigma=-1 r=3.00 eps=3.04275 gap/m_h=0.352: rho_1=-10.049746 rho_2=1.328066
lower edge torus L=256 ell=32.0 sigma=-1 r=1.50 eps=0.43794 gap/m_h=1.000: rho_1=0.219344 rho_2=0.253451
lower edge torus L=256 ell=32.0 sigma=-1 r=2.00 eps=1.10229 gap/m_h=1.000: rho_1=-1.687452 rho_2=0.567791
lower edge torus L=256 ell=32.0 sigma=-1 r=3.00 eps=3.33699 gap/m_h=1.000: rho_1=-1.011914 rho_2=1.420400
light ring L=2048 : rho_1: 1.5e+01, 3.6e-01, 1.9e-02; rho_2: 5.5e-06, 3.4e-07, 2.2e-08 (h0, h0/2, h0/4)
light torus L=128 : rho_1: 1.1e-01, 2.7e-03, 1.5e-04; rho_2: 3.1e-07, 1.9e-08, 1.2e-09 (h0, h0/2, h0/4)
lower edge torus L=64 ell=8.0: rho_1: 8.1e-03, 4.7e-04, 2.9e-05; rho_2: 9.6e-07, 6.0e-08, 3.8e-09 (h0, h0/2, h0/4)
lower edge torus L=256 ell=32.0: rho_1: 1.2e-03, 7.2e-05, 4.5e-06; rho_2: 6.4e-07, 4.0e-08, 2.5e-09 (h0, h0/2, h0/4)
finite size, lower edge r=1.25 eps=0.1587: rho_1 0.345796 (L=64) 0.345841 (L=128); rho_2 0.096638 (L=64) 0.096637 (L=128)
finite size, lower edge r=1.50 eps=0.3993: rho_1 -0.115816 (L=64) -0.115803 (L=128); rho_2 0.230717 (L=64) 0.230717 (L=128)
finite size, lower edge r=2.00 eps=1.1023: rho_1 -0.852547 (L=64) -0.852570 (L=128); rho_2 0.564905 (L=64) 0.564905 (L=128)
finite size, lower edge r=3.00 eps=3.0428: rho_1 -10.049746 (L=64) -10.049904 (L=128); rho_2 1.328066 (L=64) 1.328071 (L=128)
max |rho_i(L=64) - rho_i(L=128)| at these points: 1.59e-04
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