The process of assigning various algebraic structures to verses and their contents begins with a suitable vector space. As a proof of concept a small subset of verses will be selected to serve as the basis. It consists of four verses, the first and last verses in chps one and nine, respectively: B = {1:1, 1:7, 9:1, 9:127}, also conveniently denoted as {E1, E2, E3, E4}, with Ei's standing for the basis verses in their natural ascending order. All remaining verses will be represented in the vector space as linear combinations of the four basis vectors.
Let V be an arbitrary verse (not in the basis). As an element in the
vector space it can be expressed as a linear combination of the four basis
vectors:
V = ∑
cj Ej, for 1 ≤ j ≤ 4, (Eq. 1),
with the scalar coefficients, cj's, from an appropriate
but yet unspecified field. They are the components of V with respect to
the basis B. They will be determined via the actions of the
verses on the orconion, O, in conjunction with the spanned intervals
formed between Ei's and V for 1 ≤ i ≤ 4. Specifically,
a pair of left and right signed weighted interval (SWI) operators are
defined: Li = ∥[Ei, V] O∥
and its transpose Ri = ∥O [Ei,
V]∥, for 1 ≤ i ≤ 4. The left/right SWI operators provide
two independent representations of a given verse V in the vector space.
The directed intervals [Ei, V] contain contiguous verses from
Ei to V, inclusively. The vertical bars, ∥ ⋅
∥, denote the signed length of the interval (its number of
verses) and the "⋅" stands for a sequence of contiguous verses.
It is positive for Ei < V, negative for
Ei > V, and ± 1 for a singleton, Ei
= V. In the exceptional case of a singleton, both positive and
negative signed lengths are valid and are considered.
The right representation is obtained by multiplying the Eq. 1 from the
right by Ri:
∑ cj
Ej Ri = V Ri,
(1 ≤ i,j ≤ 4), (Eqs. 2).
The four equations (Eqs. 2) are indexed by i, while
the components within each equation are indexed by j. Substituting
for Ri in terms of its interval form gives:
∑ cj
Ej ∥O [Ei, V]∥ =
V ∥O [Ei, V]∥, (1 ≤ i,j ≤ 4),
(Eqs. 3).
Moving Ej and V
(on the left sides of the parallel bars in Eqs. 3) inside
allows for their actions on O:
∑ cj
∥Ej O [Ei, V]∥ =
∥V O [Ei, V]∥, (1 ≤ i,j ≤ 4),
(Eqs. 4).
The terms Ej O and
V O are the verse-actions on the orc, resulting in
their respective cordisans, denoted by E'j and V':
∑ cj
∥E'j [Ei, V]∥ =
∥V' [Ei, V]∥, (1 ≤ i,j ≤ 4),
(Eqs. 5).
The cordisans, E'j and V'
in turn, act on the intervals [Ei, V] by acting on
their bounding verses, forming new intervals:
∑ cj
∥[E'j Ei, E'j V]∥
= ∥[V' Ei, V' V]∥,
(1 ≤ i,j ≤ 4), (Eqs. 6).
The original directed intervals inside the double bars in Eqs. 3 are transformed into new intervals in Eqs. 6 via the actions of the cordisans E'j and V' on verses Ei and V. Substituting the signed lengths of the new directed intervals for ∥ ⋅ ∥ in Eqs. 6 will result in a linear system (LSYS) of four equations in four unknowns, the cj's. The solutions to the LSYS equations will determine V in terms of the basis vectors, (Ej, 1 ≤ j ≤ 4). The vector representations of verses in the vector space are referred to as the versors.
All vectors in the vector space are expressed uniquely as sums of scaled basis vectors. The scalars (the components of V along the basis vectors) are the unknowns in the LSYS equations and the known quantities are the signed interval lengths. The interval lengths are strictly bounded by the total number of verses: −6346 < ∥ ⋅ ∥ < +6346. The choice of the finite field for the scalars is therefore entirely based on the only criteria that it must be sufficiently large to hold all signed lengths from −6345 to +6345. Hence, the field must contain at least 12691 (= 2 × 6345 + 1) elements. The size of a finite field, (its number of elements), is a power of a prime. The smallest prime P ≥ 12691 is 12697, leading to the choice of the prime field F12697 as the appropriate field of scalars for the vector space.
The size of the finite field F12697 above was based on the fact that the verse V was arbitrary and the field was required to be maximally large to allow representations of all verses. If, however, only a small subset of verses are to be represented in the vector space, a smaller finite field can be chosen. In general, absent other requirements, the coefficient field selected is the smallest one that can accommodate the representations of a given subset of verses. This ensures that the field is not unnecessarily large, ensuring minimality and simplicity.
The symmetric left dual representations are similarly obtained via
the application of the left SWI, Li. Multiplying the original
expression, Eq. 1 for V, this time, on the left by Li, gives:
∑ cj
Li Ej = Li V,
(1 ≤ i,j ≤ 4), (Eqs. 7),
where as before, the
equations are indexed by i while the components within each equation
are indexed by j. Similar substitution for Li
in terms of its interval form gives:
∑ cj
∥[Ei, V] O∥ Ej =
∥[Ei, V] O∥ V, (1 ≤ i,j ≤ 4),
(Eqs. 8).
Moving O into the interval [Ei, V]
in Eqs. 8 will allow for the action of the former on the latter:
∑ cj
∥[Ei O, V O]∥ Ej =
∥[Ei O, V O]∥ V, (1 ≤ i,j ≤ 4),
(Eqs. 9).
The cordisans E'i = Ei O
and V' = V O in Eqs. 9 define the dual notion of an interval operator that
consists of a pair of functions (cordisans) instead of a sequence of
contiguous verses that was the case with Ri:
∑ cj
∥[E'i, V']∥ Ej =
∥[E'i, V']∥ V, (1 ≤ i,j ≤ 4),
(Eqs. 10).
Next, Ej and V
on the right sides of the double-bars in Eqs. 10 move inside:
∑ cj
∥[E'i, V'] Ej∥ =
∥[E'i, V'] V∥, (1 ≤ i,j ≤ 4),
(Eqs. 11).
Verses Ej and V on the right side
of the cordisan interval [E'i, V'] move inside to allow for
the actions of the cordisans, forming new directed intervals:
∑ cj
∥[E'i Ej, V' Ej]∥
= ∥[E'i V, V' V]∥,
(1 ≤ i,j ≤ 4), (Eqs. 12).
The resulting LSYS equations, Eqs. 12, determine the coefficients, cj, leading to the right representation of the verse V in the vector space. The duality between the left and right (Li, and Ri) cases is apparent where the corresponding terms within the double-bars, ∥ ⋅ ∥, are transposed. In the Ri case one cordisan, E'j or V', acts on the verse-interval [Ei, V], while in the Li case a cordisan-interval [E'i, V'] acts on one verse Ej or V, displaying a reciprocal symmetry.
The left/right LSYS equations differ in one important aspect. In the
right-case, the intervals are bounded with two distinct verses (a basis
verse Ei and the verse V). They are acted by a single cordisan
which is a bijection (distinct elements are mapped to distinct elements)
resulting in an interval which is also bounded by two distinct verses. In
contrast, in the left-case, where the interval is defined by two cordisans
acting on a single verse, the possibility exists that their actions on the
verse can coincide. This is especially the case with verse one, 1:1,
(basis vector, E1), which is the fixed point of all
cordisans:
E'i E1 = V' E1
= E1, (for all verses V and 1 ≤ i ≤
4),
resulting in a directed singleton, [E1,
E1] = {E1} of signed length ± 1.
Consequently, there are two sets of left-solutions, one for each sign.