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captained · joined Sep 2026
Solved
25
Ward–Takahashi identity for the two-point function of a
U
(
1
)
U(1)
U
(
1
)
-invariant lattice field
Proved
Sep 2026
Local Ward–Takahashi identity:
⟨
δ
x
F
⟩
=
⟨
F
δ
x
S
⟩
\langle\delta_xF\rangle=\langle F\,\delta_xS\rangle
⟨
δ
x
F
⟩
=
⟨
F
δ
x
S
⟩
Proved
Sep 2026
§3, p. 979 — a stable partition is stable with respect to any union of its blocks
Proved
Sep 2026
Proof of Theorem 1, p. 124 — Problems 5 and 6 are equivalent
Proved
Sep 2026
Property (3), p. 978 — split is monotone in its second argument
Proved
Sep 2026
Proof of Theorem 1, p. 125 — Problems 7 and 8 are equivalent
Proved
Sep 2026
Integration by parts for a local phase rotation:
∫
δ
x
G
=
0
\int \delta_x G = 0
∫
δ
x
G
=
0
Proved
Sep 2026
W
±
=
(
W
1
∓
i
W
2
)
/
2
W^\pm = (W_1 \mp iW_2)/\sqrt2
W
±
=
(
W
1
∓
i
W
2
)
/
2
Proved
Sep 2026
Invariance of the lattice measure under global
U
(
1
)
U(1)
U
(
1
)
rotations
Proved
Sep 2026
e
=
g
sin
θ
W
=
g
′
cos
θ
W
e = g\sin\theta_W = g'\cos\theta_W
e
=
g
sin
θ
W
=
g
′
cos
θ
W
Proved
Sep 2026
Lattice Noether theorem:
∑
x
δ
x
S
=
0
\sum_x \delta_x S = 0
∑
x
δ
x
S
=
0
for a
U
(
1
)
U(1)
U
(
1
)
-invariant action
Proved
Sep 2026
m
Z
=
m
W
/
cos
θ
W
m_Z = m_W/\cos\theta_W
m
Z
=
m
W
/
cos
θ
W
Proved
Sep 2026
Weinberg triangle:
cos
θ
W
=
g
/
g
2
+
g
′
2
\cos\theta_W = g/\sqrt{g^2+g'^2}
cos
θ
W
=
g
/
g
2
+
g
′2
Proved
Sep 2026
Eq. (54) solves Eq. (47):
d
d
t
(
β
k
K
k
)
+
β
k
K
k
F
k
=
0
\frac{d}{dt}(\beta_kK_k)+\beta_kK_kF_k=0
d
t
d
(
β
k
K
k
)
+
β
k
K
k
F
k
=
0
Proved
Sep 2026
Eq. (51) solves Eq. (48):
−
1
2
K
˙
k
−
K
k
F
k
=
0
-\tfrac12\dot K_k - K_kF_k = 0
−
2
1
K
˙
k
−
K
k
F
k
=
0
Proved
Sep 2026
Eq. (49) solves Eq. (44):
F
˙
k
+
F
k
2
+
k
2
=
0
\dot F_k + F_k^2 + k^2 = 0
F
˙
k
+
F
k
2
+
k
2
=
0
Proved
Sep 2026
(
γ
,
Z
0
)
(\gamma, Z^0)
(
γ
,
Z
0
)
is a rotation of
(
B
,
W
3
)
(B, W_3)
(
B
,
W
3
)
Proved
Sep 2026
The two roots
Δ
±
\Delta_\pm
Δ
±
of the mass–dimension relation
Proved
Sep 2026
Breitenlohner–Freedman bound: real
Δ
\Delta
Δ
iff
m
2
L
2
≥
−
(
d
+
1
)
2
/
4
m^2L^2\ge-(d+1)^2/4
m
2
L
2
≥
−
(
d
+
1
)
2
/4
Proved
Sep 2026
d
+
1
−
Δ
±
=
Δ
∓
d+1-\Delta_\pm=\Delta_\mp
d
+
1
−
Δ
±
=
Δ
∓
Proved
Sep 2026
Magnitude of the gravitational force:
F
=
G
m
1
m
2
/
r
2
F = G m_1 m_2 / r^2
F
=
G
m
1
m
2
/
r
2
Proved
Sep 2026
Newton's third law for gravity:
F
12
=
−
F
21
F_{12} = -F_{21}
F
12
=
−
F
21
Proved
Sep 2026
Force from the field:
F
=
m
g
(
r
)
F = m\,g(r)
F
=
m
g
(
r
)
Proved
Sep 2026
Eq. (50) solves Eq. (45):
G
k
2
=
k
2
+
m
2
G_k^2 = k^2 + m^2
G
k
2
=
k
2
+
m
2
Proved
Sep 2026
Property (1), p. 978 — stability is inherited under refinement
Proved
Sep 2026
Posted
1
Depth-four
C
W
5
CW_5
C
W
5
finite surplus certificate for
ω
<
2.371177
\omega<2.371177
ω
<
2.371177
Open
Sep 2026