dipolar, thus any pair of poles forms a dimension and even cases of the number of poles in the Pseudo-Boolean space satisfy
dimensions trivially. 2
n
poles
, where represents additional iterations of the “choice function,”
∫ X ∂C
∀
Boolean
C ∂
the pseudo-Boolean space behaves like a 1-dimensional vector relative to a space immediately external itself. This pseudo-Boolean vector extension is used as the vector
basis of the Hilbert Lie Space.
, the pseudo-Boolean space behaves like a 3-brane in a space
∭ X
∂C ∀
Boolean
immediately external itself.
1.2. Demonstration that the n=12 Case of a Pseudo-Boolean Space is a Subset of the Class, D-6 Calabi-Yau Spaces
This special case of a pseudo-Boolean space will now be demonstrated to be Calabi-Yau, since such a proof makes this theory coherent with existing M-Theory. The
proof will be accomplished by showing that the pseudo-Boolean space in question satisfies all requirements to meet the definition of a Calabi-Yau manifold.
In order for a space to be Calabi-Yau, it must be a compactified Kahler manifold whose canonical vector bundle is trivial. For a space to be a Kahler manifold, it must be
complex, Riemannian and smooth, and symplectic.
1.3. The Canonical Bundle of X Must be Trivial
The Canonical Vector will be defined,
V
ℂ
= Ω
n
Where is the cotangent bundle and n is the number of dimensions. The cotangent Ω
bundle may be described in terms of the tangent bundle, provided that
X
Boolean
⊂ T
2
Where represents the class of Hausdorff-quality separable spaces. When the above
T
2
equation is true, the tangent and cotangent bundles are mutually isomorphic. It will therefore be beneficial to prove that the pseudo-Boolean space in question is Hausdorff.
Prove:
neighborhoods of all points. Mathematically, show that for any two elements of ∃
the basis of
, .
X
Boolean
੬ ੬
m l
= ∅
A simple derivation from the given definition of the Pseudo-Boolean space and the true statement that the Pseudo-Boolean space is symplectic with a differential 2-form of ℝ
V shows that
fyb985
X
Boolean
≡ C
{ ·1·[C ]∧[C
] ·[Σ ]}
2 n
੬P 1
੬P 1 −1
1 {ℝ
1
Has a differential form which is a Lie Algebra because it is a vector space in which and
. Another demonstration that V , ] [ℝ , ]
[ ℝ + V = 0
f , ℝ]
V , ℝ ] V f, ℝ ] [ · V = f · [
+ · [ the form is Lie is that in the case of the two products,
and , the former
V , ℝ ] [
ℝ , V ] [
creates the vector basis multiplied across the set of real numbers, which is representative of the curve of
, whereas the latter represents the set of real X
Boolean
numbers multiplied by the vector basis, which represents a ring of vectors not related to the curve of
. Given that the curve of the Pseudo-Boolean is Lie, it is known that X
Boolean
the Pseudo-Boolean is Lie. may thus be represented as,
X
Boolean
[ V , ੬ ] ≡ Π [ [ , ੬ ]
X
Boolean
=
P ℝ
1 P
{ V } ∧ {
੬ ੬ ੬ } ∧ { ੬ = },
∈ ℝ
m l
= ੬
m l
m
੬
l
. ੬
੬
m l
= ∅ .
V T } ∧
⊂ {
2
T } X
Boolean
⊂ {
2
The tangent bundle, , in respect to the basis of the Pseudo-Boolean spac Ｔ
V Ｔ
=
∂
ℝ 1
∂X
Boolean
=
In the case of n=1, the cotangent bundle is equal to the canonical vector per the definition of the canonical vector. The cotangent vector is also the dual of
π .
Ｔ : E
→ X
Boolean
. Ω
Ｔ X
E T =
Boolean
: → X
Boolean
,
is trivial.
E =
X Ｔ
:
Boolean
Ω
Because of the above three logical demonstrations, we know that the Pseudo-Boolean space is complex, and the product relation with the tangent field satisfies the definition of
a smooth Riemannian manifold. In respect to a Hilbert space, , into which
is immerse, the tangent bundle will
X
Boolean
not be trivial. Instead, it will represent the curvature of the knot of the Pseudo-Boolean space’s poles, which are the vector basis of
from within the reference frame of its X
Boolean
own space.
1.4. Knottedness of a Pseudo-Boolean Space
