extended over a time interval into vectors by reiterating the random selection of a number
. The pseudo-Boolean space is notated as and each individual
0 ≤ ℕ ≤ 1 X
Boolean
of possibly infinitely many pseudo-Boolean spaces may interact in D4 to D9 with each other. In this way, point values on a Calabi-Yau pseudo-Boolean manifold in a vector
space are able to represent point particles externally and strings internally, providing elegant topological and algebraic solutions to problems involving the quantum harmonic
oscillator.
The Hilbert space has varied density and separability of its constituent points: this also leads to variable volume of the Hilbert space while maintaining constant energy. It is
posited that all the points which make up the Hilbert space which are themselves the pseudo-Boolean spaces could occupy the volume in H of a single point and that volume
observed in H exists as a result of varied separability of points’ neighborhoods from each other.
1.0. The Structure of Pseudo-Boolean Spaces
1.1. Definitions of Smooth, Countable, and Separable n-Polar Pseudo-Boolean Space
Let such that
X ∃
Boolean
X
Boolean
= C
੬
ℝ ]0
1· ੬P
1
The Boolean space is defined as a single real number chosen from between 0 and 1 in any of the poles of the space. This definition makes it easy to see how the space
would appear to an observer immediately outside it, but it does not make clear
X
Boolean
the topological mechanisms by which the space chooses values. For the latter, we will use a more comprehensive definition,
X
Boolean
≡ C
{ ·1·[C ]∧[C
] ·[Σ ]}
2 n
੬P 1
੬P 1 −1
1 {ℝ
1
From this formula, some special cases occur which will be examined herein. Namely, , the pseudo-Boolean space behaves like an n2-dimensional dipolar space
∀
2 n
∈ ℤ
relative to itself, where n is poles, not dimensions. Where the number of poles in X
Boolean
is an even number, the space can be treated internally as having the number of dimensions as half of its poles because the dimension in this special case is defined as
fyb985
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
