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.py24 KB · raw
"""Computation script of 08-ilang-propagation/06-mark-inertia, derivation v2.
Cubic torus (D = 3) of side L, body chain along lattice axis 1, Q = q_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 = -(eps^2/tau^2) S_1''(0)/G0 = T/G0 (bracket of (6.12))
xi* = S_1(0)(1-rho_1)/(S_2(0)(rho_2-1)) (F2^2/(c0 F1^2) at which rho = 1)
R_i = <<sin^2 Q/E^2>>_i / <<1/E>>_i (6.1)
One-quantum sums are exact mode sums; two-quantum sums use, for each total
q_1 = Q, a histogram of the pair energies with bin width dE and the exact
zeroth, first and second moments in each bin (error ~ (dE/d)^3, d the distance
to the pole); this is checked against brute-force pair sums on a small torus.
Part E: exact diagonalisation of the total contract restricted to <= 2 field
quanta on the torus L = 4, against the formula (as in Part A of
check_mark_inertia.py for the ring).
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
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)
class Torus:
def __init__(self, L, ell, pairs=True, dE=2e-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)
if pairs:
self._pairs(dE)
def _pairs(self, dE):
L, Om, w = self.L, self.Om, self.w
nb = int(np.ceil(2 * self.Omax / dE)) + 2
W, M1, M2 = (np.zeros((L, nb)) for _ in range(3))
Emin = np.full(L, np.inf)
for i in range(L):
for j in range(i, L):
m = (i + j) % L
if m > L // 2:
continue
E = (Om[i][:, None] + Om[j][None, :]).ravel()
ww = (2.0 if i != j else 1.0) * 2 * (w[i][:, None] * w[j][None, :]).ravel()
b = (E / dE).astype(np.int64)
d = E - (b + 0.5) * dE
W[m] += np.bincount(b, ww, nb)
M1[m] += np.bincount(b, ww * d, nb)
M2[m] += np.bincount(b, ww * d * d, nb)
Emin[m] = min(Emin[m], E.min())
for m in range(L // 2 + 1, L):
W[m], M1[m], M2[m], Emin[m] = W[L - m], M1[L - m], M2[L - m], Emin[L - m]
mi, bi = np.nonzero(W > 0)
Wv = W[mi, bi]
mu = M1[mi, bi] / Wv
self.pm, self.pW = mi, Wv
self.pE = (bi + 0.5) * dE + mu
self.pV = np.maximum(M2[mi, bi] / Wv - mu**2, 0.0)
self.Emin2 = Emin
self.dE = dE
self.J2 = 0.5 * self._hist(lambda E: 1 / E, lambda E: 2 / E**3) # J2 = <<1/E>>_2 / 2
num = self._hist(lambda E: np.sin(self.k[self.pm])**2 / E**2,
lambda E: 6 * np.sin(self.k[self.pm])**2 / E**4)
self.R2 = num / (2 * self.J2)
def _hist(self, f, f2):
return np.sum(self.pW * (f(self.pE) + 0.5 * self.pV * f2(self.pE)))
def coeffs(self, P, sigma, tau, eps):
wP, tP = omega(P, tau, eps), theta(P, tau, eps)
p = P - self.k
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 S1(self, P, sigma, tau, eps):
A, a, B, b = self.coeffs(P, sigma, tau, eps)
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 = self.coeffs(P, sigma, tau, eps)
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 S2_gap(self, sigma, tau, eps, P=0.02):
"""Smallest distance of a pair-energy bin edge to a pole, over P in [0, P]."""
g = np.inf
for PP in (0.0, P):
A, a, B, b = self.coeffs(PP, sigma, tau, eps)
m, E = self.pm, self.pE
g = min(g, np.min(np.abs(E + a[m])) - self.dE, np.min(np.abs(E + b[m])) - self.dE)
return g
def curv(f, h=1e-2):
f0 = f(0.0)
c = lambda hh: 2 * (f(hh) - f0) / hh**2
return f0, (4 * c(h) - c(2 * h)) / 3
def lower_gap(tau, eps, mh, n=40001):
"""min over q of Omega_q - (omega(q_1) - eps), L = infinity: (6.4), sigma = -1."""
q = np.linspace(0, np.pi, n)
return np.min(np.sqrt(mh**2 + 2 * (1 - np.cos(q))) - (omega(q, tau, eps) - eps))
def upper_ok(part, eps, Omax):
"""(U) of (6.4) at L = infinity: one quantum Omax < 2 eps; two quanta Omax < eps (margin 0.1)."""
return eps > (Omax / 2 + 0.1 if part == 1 else Omax + 0.1)
def analyse(T, sigma, tau, eps, parts=(1, 2)):
out = {}
for i in parts:
f = (lambda P: T.S1(P, sigma, tau, eps)) if i == 1 else (lambda P: T.S2(P, sigma, tau, eps))
s0, spp = curv(f)
out[i] = (s0, -(eps**2 / tau**2) * spp / s0)
return out
def ok(sigma, part, tau, eps, mh, Omax):
if sigma == -1:
return lower_gap(tau, eps, mh) > 0.5 * mh
return upper_ok(part, eps, Omax)
# ------------------------------------------------------------------ part A: histogram check
def part_a():
L, ell = 12, 3.0
log(f"PART A: lattice sums by mode classes and pair histograms 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, 1.0, 5.0), (+1, 2.0, 6.0))
for dE in (2e-2, 2e-3):
T = Torus(L, ell, dE=dE)
w1 = w2 = 0.0
for sigma, tau, eps in cases:
for P in (0.0, 0.02):
wP, tP = omega(P, tau, eps), theta(P, tau, eps)
th = 0.5 * (tP + theta(P - Q, tau, eps))
wp = omega(P - Q, tau, eps)
b2 = np.sum(ww * (np.cos(th)**2 / (E + sigma * (wp - wP))
+ np.sin(th)**2 / (E - sigma * (wp + wP))))
th1 = 0.5 * (tP + theta(P - q[0], tau, eps))
wp1 = omega(P - q[0], tau, eps)
b1 = np.sum(c2 * (np.cos(th1)**2 / (Om + sigma * (wp1 - wP))
+ np.sin(th1)**2 / (Om - sigma * (wp1 + wP))))
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};"
f" J2: histogram {T.J2:.12f}, 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")
lattices = [(32, 4.0, True), (48, 6.0, True), (48, 8.0, True), (64, 8.0, True),
(128, 16.0, False), (256, 32.0, False)]
cases = [(-1, 1.0, 1.0), (-1, 1.0, 4.0), (-1, 0.5, 2.0), (+1, 1.0, 4.0), (+1, 1.0, 2.5)]
log(f"{'L':>4s} {'ell':>5s} {'G0':>9s} {'w0':>8s} {'J2':>9s} {'R1':>8s} {'R2':>8s} |"
+ "".join(f" s={s:+d} r={r} e={e}: rho_1, rho_2 " for s, r, e in cases))
for L, ell, pairs in lattices:
T = T8 if (L, ell) == (64, 8.0) else Torus(L, ell, pairs=pairs)
row = f"{L:4d} {ell:5.1f} {T.G0:9.6f} {T.w0:8.5f} "
row += (f"{T.J2:9.6f} {T.R1:8.5f} {T.R2:8.5f} |" if pairs else f"{'-':>9s} {T.R1:8.5f} {'-':>8s} |")
for sigma, r, eps in cases:
res = analyse(T, sigma, r, eps, parts=(1, 2) if pairs else (1,))
r1 = res[1][1]
r2 = res[2][1] if (pairs and ok(sigma, 2, r, eps, T.mh, T.Omax)) else np.nan
row += f" {r1:10.5f} {r2:10.5f} "
log(row)
log("(rho_2 is nan where the two-quantum part is not isolated by (6.4) or not computed)")
# ------------------------------------------------------------------ part C: main table
def part_c(T):
log("")
log(f"PART C: main table, torus L={T.L}, ell={T.ell} (m_h={T.mh}), c0=1, F0=1, 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(f" edge of (U): F2=0 at eps=Omega_max/2: eps*R1 = {T.Omax/2*T.R1:.4f};"
f" F2!=0 at eps=Omega_max: eps*R1 = {T.Omax*T.R1:.4f}, eps*R2 = {T.Omax*T.R2:.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 >= 0)")
log(f"{'r':>5s} {'eps':>5s} {'sig':>3s} {'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):
for sigma in (-1, +1):
if sigma == -1:
gap = lower_gap(r, eps, T.mh)
if gap <= 0.5 * T.mh:
continue
parts = (1, 2)
else:
if not upper_ok(1, eps, T.Omax):
continue
gap = 2 * eps - T.Omax
parts = (1, 2) if upper_ok(2, eps, T.Omax) else (1,)
grid.append((r, eps, sigma, parts))
res = analyse(T, sigma, r, eps, parts)
s1, rho1 = res[1]
C1 = s1 / (T.G0 / 2)
t = rho1 * s1 / T.G0
if 2 in parts:
s2, rho2 = res[2]
C2 = s2 / (2 * T.J2)
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} {sigma:+3d} {gap:7.4f} | {C1:8.5f} {C2:8.5f} | "
f"{rho1:9.4f} {rho2:9.4f} | {t:9.4f} {xs}")
else:
log(f"{r:5.2f} {eps:5.1f} {sigma:+3d} {gap:7.4f} | {C1:8.5f} {'-':>8s} | "
f"{rho1:9.4f} {'-':>9s} | {t:9.4f} {'-':>9s}")
return grid
# ------------------------------------------------------------------ part B2: convergence over the table
def part_b2(T8, grid):
log("")
log("PART B2: convergence over the configurations of Part C: change of rho_1 and t from"
" (L,ell)=(64,8) to (256,32) [one quantum], of rho_2 from (48,6) to (64,8) [two quanta];"
" absolute for eps <= 4, relative (|change|/|value at ell=8|) for eps > 4")
T6, T32 = Torus(48, 6.0), Torus(256, 32.0, pairs=False)
ab = {"rho_1": 0.0, "t": 0.0, "rho_2": 0.0}
rl = {"rho_1": 0.0, "t": 0.0, "rho_2": 0.0}
for r, eps, sigma, parts in grid:
a8, a32 = analyse(T8, sigma, r, eps, (1,)), analyse(T32, sigma, r, eps, (1,))
vals = {"rho_1": (a8[1][1], a32[1][1]),
"t": (a8[1][1] * a8[1][0] / T8.G0, a32[1][1] * a32[1][0] / T32.G0)}
if 2 in parts:
vals["rho_2"] = (analyse(T8, sigma, r, eps, (2,))[2][1], analyse(T6, sigma, r, eps, (2,))[2][1])
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(" eps <= 4, max |change|: " + ", ".join(f"{k} {v:.4f}" for k, v in ab.items()))
log(" eps > 4, max relative change: " + ", ".join(f"{k} {v:.4f}" for k, v in rl.items()))
log(f" R1: ell=8 {T8.R1:.5f}, ell=32 {T32.R1:.5f}; G0: ell=8 {T8.G0:.5f}, ell=32 {T32.G0:.5f}")
return T32
# ------------------------------------------------------------------ part D2: one-quantum tests at ell = 32
def part_d2(T):
log("")
log(f"PART D2: F1^2 part only (all that items 3 and the upper-branch bound of item 2 need from it),"
f" torus L={T.L}, ell={T.ell}; 36 values of eps in [0.1, 64]; same criteria as Part D")
eg = np.geomspace(0.1, 64.0, 36)
for r in (0.1, 0.5, 1.0, 1.5, 2.0, 3.0):
lo = {e: analyse(T, -1, r, e, (1,))[1] for e in eg if lower_gap(r, e, T.mh) > 0.5 * T.mh}
u1 = {e: analyse(T, +1, r, e, (1,))[1] for e in eg if upper_ok(1, e, T.Omax)}
e = np.array(sorted(lo))
r1 = np.array([lo[x][1] for x in e])
t = np.array([lo[x][1] * lo[x][0] / T.G0 for x in e])
phi = e * t
d2 = 2 * np.diff(np.diff(phi) / np.diff(e)) / (e[2:] - e[:-2])
neg = e[1:-1][d2 <= 0]
c = np.array([x for x in sorted(u1) if x in lo])
tp = np.array([u1[x][1] * u1[x][0] / T.G0 for x in c])
tm = np.array([lo[x][1] * lo[x][0] / T.G0 for x in c])
psi = c * (tm - tp)
yb = (tp + tm) / 4
e1 = np.interp(1.0, r1, e) if r1.min() < 1 < r1.max() else np.nan
out = (f"r={r:4.2f}: lower eps in [{e.min():.3f},{e.max():.1f}]; rho_1,- incr {bool(np.all(np.diff(r1) > 0))},"
f" rho_1,-=1 at eps={e1:.3f}; phi incr {bool(np.all(np.diff(phi) > 0))}, phi''>0 "
+ ("everywhere" if neg.size == 0 else f"except eps in [{neg.min():.3f},{neg.max():.3f}]")
+ f"; max rho_1,+ = {max(v[1] for v in u1.values()):.4f}")
if c.size:
out += f"; psi incr {bool(np.all(np.diff(psi) > 0))}; y_b in [{yb.min():.4f},{yb.max():.4f}]"
log(out)
if r == 1.0:
ys = []
for ea, eb in ((0.25, 0.5), (0.5, 1.0), (1.0, 2.0), (2.0, 4.0), (4.0, 8.0)):
ta, tb = np.interp(ea, e, t), np.interp(eb, e, t)
ys.append(f"({ea:g},{eb:g}) {(ea*ta - eb*tb)/(2*(ea - eb)):.4f}")
log(" r=1, two-level grips y_ab (interpolated): " + " ".join(ys))
# ------------------------------------------------------------------ part D: fine grid, items 2 and 3
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}")
log(" lower branch where lower gap > m_h/2; upper branch where (U) holds with margin 0.1")
log(" item 2: rho_{i,-} increasing in eps? window [e1,e2]: rho_{1,-}(e1)=1, rho_{2,-}(e2)=1;"
" xi*_- monotone? max rho_{i,+} < 1?")
log(" item 3: phi = eps*t_- increasing and strictly convex? psi = eps*(t_- - t_+) increasing?"
" y_b = (t_+ + t_-)/4")
eg = np.geomspace(0.1, 64.0, 71)
tabs, store, smin = [], {}, np.inf
for r in (0.1, 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 2.0, 3.0):
tau = r
lo = {e: analyse(T, -1, tau, e) for e in eg if lower_gap(tau, e, T.mh) > 0.5 * T.mh}
u1 = {e: analyse(T, +1, tau, e, (1,))[1] for e in eg if upper_ok(1, e, T.Omax)}
u2 = {e: analyse(T, +1, tau, e, (2,))[2] for e in eg if upper_ok(2, e, T.Omax)}
store[r] = (lo, u1, u2)
smin = min([smin] + [v[i][0] / (T.G0 / 2 if i == 1 else 2 * T.J2) for v in lo.values() for i in (1, 2)]
+ [v[0] / (T.G0 / 2) for v in u1.values()] + [v[0] / (2 * T.J2) for v in u2.values()])
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])
t = r1 * s1 / T.G0
phi = e * t
sl = np.diff(phi) / np.diff(e)
d2 = 2 * np.diff(sl) / (e[2:] - e[:-2])
xi = s1 * (1 - r1) / (s2 * (r2 - 1))
idx = np.nonzero(xi >= 0)[0]
if idx.size:
fine = np.linspace(e[max(idx[0] - 1, 0)], e[min(idx[-1] + 1, len(e) - 1)], 81)
fine = [x for x in fine if lower_gap(tau, x, T.mh) > 0.5 * T.mh]
fx = []
for x in fine:
a_ = analyse(T, -1, tau, x)
v = a_[1][0] * (1 - a_[1][1]) / (a_[2][0] * (a_[2][1] - 1))
if v >= 0:
fx.append((x, v))
fe, fv = np.array([u for u, _ in fx]), np.array([v for _, v in fx])
wmsg = (f" xi*_- >= 0 on eps in [{fe.min():.3f},{fe.max():.3f}] (81-point refinement,"
f" {len(fe)} pts), xi* {fv.min():.4f}..{fv.max():.2f},"
f" increasing {bool(np.all(np.diff(fv) > 0))}, decreasing {bool(np.all(np.diff(fv) < 0))}")
else:
wmsg = " xi*_- < 0 everywhere"
log(f"r={r:4.2f}: lower on eps in [{e.min():.3f},{e.max():.1f}] ({len(e)} pts);"
f" 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(" " + wmsg)
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}")
if u1:
eu = np.array(sorted(u1))
r1p = np.array([u1[x][1] for x in eu])
t1p = np.array([u1[x][1] * u1[x][0] / T.G0 for x in eu])
com = [x for x in eu if x in lo]
msg = f" upper F1^2 on eps >= {eu.min():.3f}: max rho_1,+ = {r1p.max():.4f}"
if u2:
msg += f"; upper F2^2 on eps >= {min(u2):.3f}: max rho_2,+ = {max(v[1] for v in u2.values()):.4f}"
log(msg)
if com:
c = np.array(com)
tp = np.array([u1[x][1] * u1[x][0] / T.G0 for x in c])
tmn = np.array([lo[x][1][1] * lo[x][1][0] / T.G0 for x in c])
gap = np.array([lo[x][1][1] - u1[x][1] for x in c]) / (4 * c * T.R1)
psi = c * (tmn - tp)
mixed = [min(lo[x][1][1], lo[x][2][1]) - max(u1[x][1], u2[x][1]) for x in c if x in u2]
yb = (tp + tmn) / 4
d11 = min(lo[x][1][1] - u1[x][1] for x in c)
log(f" both branches on eps >= {c.min():.3f}: psi increasing {bool(np.all(np.diff(psi) > 0))};"
f" y_b in [{yb.min():.4f},{yb.max():.4f}]; min(rho_1,- - rho_1,+) = {d11:.4f};"
f" (rho_1,- - rho_1,+)/(4 eps R1) = {gap[0]:.4f} at eps={c[0]:.3f}, {gap[-1]:.4f} at eps={c[-1]:.1f}"
+ ("" if not mixed else f"; F2!=0: min[min(rho_i,-) - max(rho_i,+)] = {min(mixed):.4f}"))
tabs.append((r, e, t))
log(f" min over all computed points of S_i(0)/S_i(0)|_(tau=0) (both parts, both branches): {smin:.4f}")
dmax = {}
for sg, j in ((-1, 0), (+1, 1), (+1, 2)):
a_, b_ = store[0.25][j], store[2.0][j]
com = [x for x in a_ if x in b_ and x >= 2.0]
if sg == -1:
for i in (1, 2):
dmax[(i, sg)] = max(abs(a_[x][i][1] - b_[x][i][1]) for x in com)
else:
dmax[(j, sg)] = max(abs(a_[x][1] - b_[x][1]) for x in com)
log(" r-dependence, eps >= 2: max |rho(r=2) - rho(r=0.25)|: "
+ ", ".join(f"rho_{i},{'+' if sg > 0 else '-'} {v:.4f}" for (i, sg), v in dmax.items()))
log(" grip y = F0 F2/F1^2 that equalises the bracket of (6.12) for two lower levels (eps_a, eps_b):")
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 ea >= e.min() and eb <= e.max():
ta, tb = np.interp(ea, e, t), np.interp(eb, e, t)
out += f" ({ea:g},{eb:g}) {(ea*ta - eb*tb)/(2*(ea - eb)):.4f}"
log(out)
# ------------------------------------------------------------------ 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}")
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, col in ((+1, 1), (-1, 0)):
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 = curv(lambda P: T.S1(P, sigma, tau, eps))
_, p2 = curv(lambda P: T.S2(P, 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}")
if __name__ == "__main__":
part_a()
T8 = Torus(64, 8.0)
part_b(T8)
grid = part_c(T8)
T32 = part_b2(T8, grid)
part_d(T8)
part_d2(T32)
part_e()
(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.txt19 KB · raw
PART A: lattice sums by mode classes and pair histograms vs brute-force mode and ordered-pair sums, torus L=12, ell=3.0
dE=0.02: max relative deviation S_1 4.11e-14, S_2 3.98e-10 (cases (sigma,r,eps) = ((-1, 1.0, 1.0), (-1, 1.5, 3.0), (1, 1.0, 5.0), (1, 2.0, 6.0)), P = 0, 0.02)
dE=0.002: max relative deviation S_1 4.11e-14, S_2 1.43e-13 (cases (sigma,r,eps) = ((-1, 1.0, 1.0), (-1, 1.5, 3.0), (1, 1.0, 5.0), (1, 2.0, 6.0)), P = 0, 0.02)
R_1: classes 0.229832921964, brute force 0.229832921964; J2: histogram 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
L ell G0 w0 J2 R1 R2 | s=-1 r=1.0 e=1.0: rho_1, rho_2 s=-1 r=1.0 e=4.0: rho_1, rho_2 s=-1 r=0.5 e=2.0: rho_1, rho_2 s=+1 r=1.0 e=4.0: rho_1, rho_2 s=+1 r=1.0 e=2.5: 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 -0.83750 0.18378 0.01669 nan
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 -0.83211 0.16405 0.01721 nan
48 8.0 0.242641 0.22682 0.012329 0.23197 0.12701 | 1.05586 0.46260 2.62348 1.40612 1.60690 0.79612 -0.82529 0.15616 0.02013 nan
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 -0.82583 0.15617 0.01967 nan
128 16.0 0.247714 0.22742 - 0.23035 - | 1.06325 nan 2.61780 nan 1.61911 nan -0.81032 nan 0.02707 nan
256 32.0 0.250233 0.22760 - 0.22901 - | 1.06527 nan 2.61022 nan 1.62351 nan -0.79887 nan 0.03300 nan
(rho_2 is nan where the two-quantum part is not isolated by (6.4) or not computed)
PART C: main table, 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
edge of (U): F2=0 at eps=Omega_max/2: eps*R1 = 0.4021; F2!=0 at eps=Omega_max: eps*R1 = 0.8042, eps*R2 = 0.4402
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 >= 0)
r eps sig gap | C_1 C_2 | rho_1 rho_2 | t xi*
0.25 0.5 -1 0.1250 | 1.00550 1.00171 | 0.6521 0.2474 | 0.3278 -
0.25 1.0 -1 0.1250 | 1.00392 1.00139 | 1.0202 0.4490 | 0.5121 0.1811
0.25 2.0 -1 0.1250 | 1.00252 1.00102 | 1.6032 0.7931 | 0.8036 14.3639
0.25 2.0 +1 0.5336 | 0.99251 - | 0.5102 - | 0.2532 -
0.25 3.0 -1 0.1250 | 1.00187 1.00081 | 2.1224 1.1002 | 1.0632 -
0.25 3.0 +1 2.5336 | 0.99643 - | -0.2254 - | -0.1123 -
0.25 4.0 -1 0.1250 | 1.00148 1.00067 | 2.6185 1.3894 | 1.3112 -
0.25 4.0 +1 4.5336 | 0.99762 0.99792 | -0.7811 0.1632 | -0.3896 -
0.25 6.0 -1 0.1250 | 1.00105 1.00050 | 3.5823 1.9414 | 1.7930 -
0.25 6.0 +1 8.5336 | 0.99857 0.99900 | -1.7836 -0.6903 | -0.8905 -
0.25 8.0 -1 0.1250 | 1.00081 1.00040 | 4.5299 2.4758 | 2.2668 -
0.25 8.0 +1 12.5336 | 0.99897 0.99934 | -2.7441 -1.2970 | -1.3706 -
0.25 16.0 -1 0.1250 | 1.00043 1.00022 | 8.2736 4.5550 | 4.1386 -
0.25 16.0 +1 28.5336 | 0.99952 0.99972 | -6.4999 -3.4363 | -3.2484 -
0.25 32.0 -1 0.1250 | 1.00022 1.00012 | 15.7146 8.6470 | 7.8590 -
0.25 32.0 +1 60.5336 | 0.99977 0.99987 | -13.9439 -7.5421 | -6.9703 -
0.50 0.5 -1 0.1250 | 1.02269 1.00691 | 0.6616 0.2494 | 0.3383 -
0.50 1.0 -1 0.1250 | 1.01603 1.00562 | 1.0270 0.4516 | 0.5217 0.2446
0.50 2.0 -1 0.1250 | 1.01023 1.00412 | 1.6069 0.7961 | 0.8116 14.7325
0.50 2.0 +1 0.5336 | 0.97111 - | 0.4846 - | 0.2353 -
0.50 3.0 -1 0.1250 | 1.00754 1.00326 | 2.1245 1.1034 | 1.0703 -
0.50 3.0 +1 2.5336 | 0.98595 - | -0.2385 - | -0.1176 -
0.50 4.0 -1 0.1250 | 1.00598 1.00270 | 2.6196 1.3927 | 1.3176 -
0.50 4.0 +1 4.5336 | 0.99059 0.99177 | -0.7905 0.1617 | -0.3915 -
0.50 6.0 -1 0.1250 | 1.00423 1.00201 | 3.5823 1.9447 | 1.7987 -
0.50 6.0 +1 8.5336 | 0.99431 0.99603 | -1.7904 -0.6882 | -0.8901 -
0.50 8.0 -1 0.1250 | 1.00327 1.00160 | 4.5293 2.4791 | 2.2721 -
0.50 8.0 +1 12.5336 | 0.99592 0.99736 | -2.7497 -1.2943 | -1.3692 -
0.50 16.0 -1 0.1250 | 1.00172 1.00089 | 8.2720 4.5583 | 4.1431 -
0.50 16.0 +1 28.5336 | 0.99808 0.99887 | -6.5040 -3.4332 | -3.2458 -
0.50 32.0 -1 0.1250 | 1.00088 1.00047 | 15.7124 8.6503 | 7.8631 -
0.50 32.0 +1 60.5336 | 0.99907 0.99947 | -13.9473 -7.5389 | -6.9672 -
1.00 0.5 -1 0.1250 | 1.10472 1.02891 | 0.7065 0.2579 | 0.3903 -
1.00 1.0 -1 0.1250 | 1.07031 1.02324 | 1.0557 0.4626 | 0.5650 0.5334
1.00 2.0 -1 0.1250 | 1.04324 1.01685 | 1.6212 0.8088 | 0.8457 16.3994
1.00 2.0 +1 0.5336 | 0.89837 - | 0.3965 - | 0.1781 -
1.00 3.0 -1 0.1250 | 1.03137 1.01326 | 2.1322 1.1166 | 1.0996 -
1.00 3.0 +1 2.5336 | 0.94730 - | -0.2862 - | -0.1356 -
1.00 4.0 -1 0.1250 | 1.02465 1.01095 | 2.6235 1.4061 | 1.3441 -
1.00 4.0 +1 4.5336 | 0.96399 0.96826 | -0.8258 0.1562 | -0.3980 -
1.00 6.0 -1 0.1250 | 1.01728 1.00813 | 3.5819 1.9582 | 1.8219 -
1.00 6.0 +1 8.5336 | 0.97785 0.98441 | -1.8159 -0.6801 | -0.8879 -
1.00 8.0 -1 0.1250 | 1.01330 1.00646 | 4.5264 2.4927 | 2.2933 -
1.00 8.0 +1 12.5336 | 0.98399 0.98958 | -2.7711 -1.2839 | -1.3634 -
1.00 16.0 -1 0.1250 | 1.00693 1.00356 | 8.2652 4.5717 | 4.1613 -
1.00 16.0 +1 28.5336 | 0.99240 0.99550 | -6.5199 -3.4210 | -3.2352 -
1.00 32.0 -1 0.1250 | 1.00354 1.00188 | 15.7036 8.6636 | 7.8796 -
1.00 32.0 +1 60.5336 | 0.99629 0.99789 | -13.9607 -7.5262 | -6.9545 -
1.50 0.5 -1 0.1012 | 1.34842 1.07115 | 0.6742 0.2776 | 0.4546 -
1.50 1.0 -1 0.1250 | 1.19218 1.05566 | 1.0962 0.4847 | 0.6534 1.0371
1.50 2.0 -1 0.1250 | 1.10791 1.03948 | 1.6407 0.8323 | 0.9089 20.0338
1.50 2.0 +1 0.5336 | 0.80839 - | 0.2846 - | 0.1150 -
1.50 3.0 -1 0.1250 | 1.07583 1.03076 | 2.1414 1.1404 | 1.1519 -
1.50 3.0 +1 2.5336 | 0.89239 - | -0.3537 - | -0.1578 -
1.50 4.0 -1 0.1250 | 1.05860 1.02524 | 2.6268 1.4299 | 1.3904 -
1.50 4.0 +1 4.5336 | 0.92432 0.93255 | -0.8774 0.1481 | -0.4055 -
1.50 6.0 -1 0.1250 | 1.04036 1.01861 | 3.5787 1.9818 | 1.8616 -
1.50 6.0 +1 8.5336 | 0.95228 0.96593 | -1.8545 -0.6674 | -0.8830 -
1.50 8.0 -1 0.1250 | 1.03080 1.01475 | 4.5198 2.5160 | 2.3295 -
1.50 8.0 +1 12.5336 | 0.96510 0.97703 | -2.8040 -1.2673 | -1.3531 -
1.50 16.0 -1 0.1250 | 1.01583 1.00807 | 8.2530 4.5945 | 4.1918 -
1.50 16.0 +1 28.5336 | 0.98315 0.98997 | -6.5452 -3.4010 | -3.2175 -
1.50 32.0 -1 0.1250 | 1.00803 1.00424 | 15.6883 8.6859 | 7.9072 -
1.50 32.0 +1 60.5336 | 0.99171 0.99528 | -13.9825 -7.5052 | -6.9333 -
2.00 2.0 -1 0.1250 | 1.23048 1.07474 | 1.6093 0.8717 | 0.9901 26.7521
2.00 2.0 +1 0.5336 | 0.72022 - | 0.1702 - | 0.0613 -
2.00 3.0 -1 0.1250 | 1.15137 1.05724 | 2.1326 1.1781 | 1.2277 -
2.00 3.0 +1 2.5336 | 0.83011 - | -0.4303 - | -0.1786 -
2.00 4.0 -1 0.1250 | 1.11345 1.04653 | 2.6184 1.4665 | 1.4577 -
2.00 4.0 +1 4.5336 | 0.87662 0.88865 | -0.9382 0.1388 | -0.4112 -
2.00 6.0 -1 0.1250 | 1.07589 1.03394 | 3.5670 2.0171 | 1.9188 -
2.00 6.0 +1 8.5336 | 0.91983 0.94176 | -1.9018 -0.6512 | -0.8747 -
2.00 8.0 -1 0.1250 | 1.05710 1.02675 | 4.5053 2.5504 | 2.3813 -
2.00 8.0 +1 12.5336 | 0.94052 0.96025 | -2.8453 -1.2455 | -1.3380 -
2.00 16.0 -1 0.1250 | 1.02874 1.01450 | 8.2334 4.6272 | 4.2350 -
2.00 16.0 +1 28.5336 | 0.97067 0.98239 | -6.5782 -3.3740 | -3.1926 -
2.00 32.0 -1 0.1250 | 1.01443 1.00758 | 15.6657 8.7176 | 7.9459 -
2.00 32.0 +1 60.5336 | 0.98542 0.99166 | -14.0118 -7.4763 | -6.9038 -
PART B2: convergence over the configurations of Part C: change of rho_1 and t from (L,ell)=(64,8) to (256,32) [one quantum], of rho_2 from (48,6) to (64,8) [two quanta]; absolute for eps <= 4, relative (|change|/|value at ell=8|) for eps > 4
eps <= 4, max |change|: rho_1 0.0928, t 0.0581, rho_2 0.0081
eps > 4, max relative change: rho_1 0.0341, t 0.0341, rho_2 0.0120
R1: ell=8 0.23200, ell=32 0.22901; G0: ell=8 0.24262, ell=32 0.25023
PART D: fine grid, 71 values of eps in [0.1, 64] (geometric), torus L=64, ell=8.0
lower branch where lower gap > m_h/2; upper branch where (U) holds with margin 0.1
item 2: rho_{i,-} increasing in eps? window [e1,e2]: rho_{1,-}(e1)=1, rho_{2,-}(e2)=1; xi*_- monotone? max rho_{i,+} < 1?
item 3: phi = eps*t_- increasing and strictly convex? psi = eps*(t_- - t_+) increasing? y_b = (t_+ + t_-)/4
r=0.10: lower 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
xi*_- >= 0 on eps in [0.986,2.658] (81-point refinement, 73 pts), xi* 0.0821..1552.27, increasing True, decreasing False
phi increasing True; phi'' min 0.464 max 1.321; t_- 0.1001..15.286
upper F1^2 on eps >= 1.918: max rho_1,+ = 0.6050; upper F2^2 on eps >= 3.660: max rho_2,+ = 0.4345
both branches on eps >= 1.918: psi increasing True; y_b in [0.2216,0.2703]; min(rho_1,- - rho_1,+) = 0.9526; (rho_1,- - rho_1,+)/(4 eps R1) = 0.5353 at eps=1.918, 0.9997 at eps=64.0; F2!=0: min[min(rho_i,-) - max(rho_i,+)] = 0.8570
r=0.25: lower 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
xi*_- >= 0 on eps in [0.986,2.658] (81-point refinement, 73 pts), xi* 0.0990..2192.64, increasing True, decreasing False
phi increasing True; phi'' min 0.464 max 1.328; t_- 0.1019..15.287
upper F1^2 on eps >= 1.918: max rho_1,+ = 0.5967; upper F2^2 on eps >= 3.660: max rho_2,+ = 0.4330
both branches on eps >= 1.918: psi increasing True; y_b in [0.2221,0.2693]; min(rho_1,- - rho_1,+) = 0.9621; (rho_1,- - rho_1,+)/(4 eps R1) = 0.5405 at eps=1.918, 0.9997 at eps=64.0; F2!=0: min[min(rho_i,-) - max(rho_i,+)] = 0.8593
r=0.50: lower 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
xi*_- >= 0 on eps in [0.963,2.635] (81-point refinement, 73 pts), xi* 0.0245..770.92, increasing True, decreasing False
phi increasing True; phi'' min 0.464 max 1.350; t_- 0.1094..15.291
upper F1^2 on eps >= 1.918: max rho_1,+ = 0.5684; upper F2^2 on eps >= 3.660: max rho_2,+ = 0.4281
both branches on eps >= 1.918: psi increasing True; y_b in [0.2239,0.2662]; min(rho_1,- - rho_1,+) = 0.9942; (rho_1,- - rho_1,+)/(4 eps R1) = 0.5586 at eps=1.918, 0.9997 at eps=64.0; F2!=0: min[min(rho_i,-) - max(rho_i,+)] = 0.8675
r=0.75: lower 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
xi*_- >= 0 on eps in [0.963,2.635] (81-point refinement, 73 pts), xi* 0.1334..7000.48, increasing True, decreasing False
phi increasing True; phi'' min 0.464 max 1.367; t_- 0.1258..15.298
upper F1^2 on eps >= 1.918: max rho_1,+ = 0.5256; upper F2^2 on eps >= 3.660: max rho_2,+ = 0.4204
both branches on eps >= 1.918: psi increasing True; y_b in [0.2270,0.2621]; min(rho_1,- - rho_1,+) = 1.0434; (rho_1,- - rho_1,+)/(4 eps R1) = 0.5862 at eps=1.918, 0.9997 at eps=64.0; F2!=0: min[min(rho_i,-) - max(rho_i,+)] = 0.8808
r=1.00: lower 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
xi*_- >= 0 on eps in [0.933,2.605] (81-point refinement, 70 pts), xi* 0.1195..2169.05, increasing True, decreasing False
phi increasing True; phi'' min 0.464 max 1.282; t_- 0.1658..15.307
upper F1^2 on eps >= 1.918: max rho_1,+ = 0.4730; upper F2^2 on eps >= 3.660: max rho_2,+ = 0.4103
both branches on eps >= 1.918: psi increasing True; y_b in [0.2312,0.2586]; min(rho_1,- - rho_1,+) = 1.1047; (rho_1,- - rho_1,+)/(4 eps R1) = 0.6207 at eps=1.918, 0.9997 at eps=64.0; F2!=0: min[min(rho_i,-) - max(rho_i,+)] = 0.8987
r=1.25: lower on eps in [0.209,64.0] (63 pts); rho_1,- 0.4854..30.551 incr True; rho_2,- 0.1241..16.817 incr True
xi*_- >= 0 on eps in [0.884,2.556] (81-point refinement, 70 pts), xi* 0.0591..705.35, increasing True, decreasing False
phi increasing True; phi'' min 0.016 max 0.654; t_- 0.3196..15.318
upper F1^2 on eps >= 1.918: max rho_1,+ = 0.4149; upper F2^2 on eps >= 3.660: max rho_2,+ = 0.3986
both branches on eps >= 1.918: psi increasing True; y_b in [0.2367,0.2567]; min(rho_1,- - rho_1,+) = 1.1734; (rho_1,- - rho_1,+)/(4 eps R1) = 0.6593 at eps=1.918, 0.9998 at eps=64.0; F2!=0: min[min(rho_i,-) - max(rho_i,+)] = 0.9208
r=1.50: lower on eps in [0.438,64.0] (55 pts); rho_1,- 0.4346..30.542 incr True; rho_2,- 0.2490..16.829 incr True
xi*_- >= 0 on eps in [0.860,2.532] (81-point refinement, 70 pts), xi* 0.0967..7303.24, increasing True, decreasing False
phi increasing True; phi'' min -13.908 max 0.483; phi'' <= 0 at eps in [0.480,0.634]; t_- 0.3042..15.333
upper F1^2 on eps >= 1.918: max rho_1,+ = 0.3546; upper F2^2 on eps >= 3.660: max rho_2,+ = 0.3859
both branches on eps >= 1.918: psi increasing True; y_b in [0.2434,0.2573]; min(rho_1,- - rho_1,+) = 1.2438; (rho_1,- - rho_1,+)/(4 eps R1) = 0.6988 at eps=1.918, 0.9998 at eps=64.0; F2!=0: min[min(rho_i,-) - max(rho_i,+)] = 0.9469
r=2.00: lower on eps in [1.209,64.0] (44 pts); rho_1,- 0.5484..30.518 incr True; rho_2,- 0.6043..16.860 incr True
xi*_- >= 0 on eps in [1.341,2.394] (81-point refinement, 71 pts), xi* 0.0853..1011.35, increasing True, decreasing False
phi increasing True; phi'' min -12.032 max 0.464; phi'' <= 0 at eps in [1.326,1.749]; t_- 0.3913..15.369
upper F1^2 on eps >= 1.918: max rho_1,+ = 0.2356; upper F2^2 on eps >= 3.660: max rho_2,+ = 0.3594
both branches on eps >= 1.918: psi increasing True; y_b in [0.2605,0.2629]; min(rho_1,- - rho_1,+) = 1.3247; (rho_1,- - rho_1,+)/(4 eps R1) = 0.7443 at eps=1.918, 0.9998 at eps=64.0; F2!=0: min[min(rho_i,-) - max(rho_i,+)] = 1.0104
r=3.00: lower on eps in [3.337,64.0] (33 pts); rho_1,- -0.3053..30.445 incr True; rho_2,- 1.4165..16.950 incr True
xi*_- >= 0 on eps in [3.337,3.563] (81-point refinement, 57 pts), xi* 0.0934..20.27, increasing False, decreasing True
phi increasing True; phi'' min -21.133 max 0.464; phi'' <= 0 at eps in [3.660,4.827]; t_- -0.2284..15.474
upper F1^2 on eps >= 1.918: max rho_1,+ = 0.0297; upper F2^2 on eps >= 3.660: max rho_2,+ = 0.3109
both branches on eps >= 3.337: psi increasing True; y_b in [-0.1260,0.3094]; min(rho_1,- - rho_1,+) = 0.4452; (rho_1,- - rho_1,+)/(4 eps R1) = 0.1438 at eps=3.337, 0.9999 at eps=64.0; F2!=0: min[min(rho_i,-) - max(rho_i,+)] = 0.9867
min over all computed points of S_i(0)/S_i(0)|_(tau=0) (both parts, both branches): 0.5574
r-dependence, eps >= 2: max |rho(r=2) - rho(r=0.25)|: rho_1,- 0.0534, rho_2,- 0.0786, rho_1,+ 0.3162, rho_2,+ 0.0737
grip y = F0 F2/F1^2 that equalises the bracket of (6.12) for two lower levels (eps_a, eps_b):
r=0.10: (0.25,0.5) 0.2241 (0.5,1) 0.3470 (1,2) 0.5466 (2,4) 0.9087 (4,8) 1.6106 (1,8) 1.2580
r=0.25: (0.25,0.5) 0.2256 (0.5,1) 0.3482 (1,2) 0.5475 (2,4) 0.9094 (4,8) 1.6111 (1,8) 1.2587
r=0.50: (0.25,0.5) 0.2311 (0.5,1) 0.3526 (1,2) 0.5507 (2,4) 0.9118 (4,8) 1.6132 (1,8) 1.2610
r=0.75: (0.25,0.5) 0.2406 (0.5,1) 0.3599 (1,2) 0.5560 (2,4) 0.9158 (4,8) 1.6166 (1,8) 1.2648
r=1.00: (0.25,0.5) 0.2543 (0.5,1) 0.3699 (1,2) 0.5631 (2,4) 0.9213 (4,8) 1.6212 (1,8) 1.2701
r=1.25: (0.25,0.5) 0.2714 (0.5,1) 0.3814 (1,2) 0.5717 (2,4) 0.9280 (4,8) 1.6272 (1,8) 1.2766
r=1.50: (0.5,1) 0.4286 (1,2) 0.5821 (2,4) 0.9359 (4,8) 1.6343 (1,8) 1.2845
r=2.00: (2,4) 0.9629 (4,8) 1.6524
r=3.00: (4,8) 1.8618
PART D2: F1^2 part only (all that items 3 and the upper-branch bound of item 2 need from it), torus L=256, ell=32.0; 36 values of eps in [0.1, 64]; same criteria as Part D
r=0.10: lower eps in [0.100,64.0]; rho_1,- incr True, rho_1,-=1 at eps=0.928; phi incr True, phi''>0 everywhere; max rho_1,+ = 0.6286; psi incr True; y_b in [0.2288,0.2760]
r=0.50: lower eps in [0.100,64.0]; rho_1,- incr True, rho_1,-=1 at eps=0.920; phi incr True, phi''>0 everywhere; max rho_1,+ = 0.5872; psi incr True; y_b in [0.2298,0.2707]
r=1.00: lower eps in [0.100,64.0]; rho_1,- incr True, rho_1,-=1 at eps=0.896; phi incr True, phi''>0 everywhere; max rho_1,+ = 0.4767; psi incr True; y_b in [0.2328,0.2593]
r=1, two-level grips y_ab (interpolated): (0.25,0.5) 0.2567 (0.5,1) 0.3701 (1,2) 0.5608 (2,4) 0.9142 (4,8) 1.6049
r=1.50: lower eps in [0.438,64.0]; rho_1,- incr True, rho_1,-=1 at eps=0.892; phi incr True, phi''>0 except eps in [0.527,0.634]; max rho_1,+ = 0.3330; psi incr True; y_b in [0.2379,0.2511]
r=2.00: lower eps in [1.102,64.0]; rho_1,- incr True, rho_1,-=1 at eps=1.453; phi incr True, phi''>0 except eps in [1.326,1.595]; max rho_1,+ = 0.1797; psi incr True; y_b in [0.2430,0.2454]
r=3.00: lower eps in [3.337,64.0]; rho_1,- incr True, rho_1,-=1 at eps=3.851; phi incr True, phi''>0 except eps in [4.014,4.827]; max rho_1,+ = -0.1146; psi incr True; y_b in [-0.2699,0.2651]
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 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 0.0307210118
max relative deviation: 8.72e-09
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