In order to develop more extensive mathematical structures among chps and verses, the mirror symmetry will be appropriately adapted and applied. Although, an involution, having too few distinct states (only two), is insufficient, the set of states can readily be enlarged via the composition of symmetries. The inclusion of an additional mirror allows for the reflections back and forth across the two mirror to combine leading to a large set of states.
The (discrete) reflection symmetry will be applied to the set of verses. The first verse, 1:1, is exceptionally unique among all verses. It consists of four words which come in multiples of 19, throughout. They are also the gematric values of four specific divine attributes. As such it will be referred to as the tetron, thereafter. It is the only numbered tetron. Since, this verse is absent in chp 9, the pair of chps 1 and 9 (which incidentally form the digits of 19) together have only one (numbered) tetron, an indirect hint at the "one"ness of the author. The compensating one in 27:30 is unique in that although it is a "numbered tetron" it is not the entire verse. It forms the second half of the verse. The remaining 112 ones are unnumbered or can conveniently be considered 0-numbered.
The verses, therefore, are either numbered or unnumbered. This, in turn, will allow to implement the mirrors acting on verses. Accordingly, the set of verses is first partitioned into disjoint blocks. Let T = {2:0 - 8:0, 10:0 - 114:0} be the set of tetrons. And similarly, V = {1:1 - 114:6} be the set of remaining verses. The two sets are disjoint: T ∩ V = ∅. Combining the tetrons and the basic verses gives the adjoin set, W = T ∪ V, with their corresponding cardinalities, ∣W∣ = ∣T∣ + ∣V∣ = 112 + 6234 = 6346. Note, the sum of the digits on both sides is 19.
A pair of mirrors, A and B, act on two overlapping set of verses: A acts on the set of all verses, W, while B only acts on the set of numbered verses, V. The process begins with the action of A on the first tetron, 2:0, followed by B, moving ahead alternating between the two until the next tetron is reached (via A), indicating the end of the sequence. Continuing with A again, (won't bring in any new elements), it retraces the steps back, (being an involution), to the previously obtained elements, leading to a cycle (tecot, hereinafter). Appropriately, a biflection (bif) is defined as the combined action of A followed by B (composition of reflections). The tecot thus obtained is of even number of elements since each bif action corresponds to a pair of elements. The procedure then continues with the next smallest available tetron until all the tetrons are accounted for. A total of 56, (112/2), tecots are thus obtained.
Each tecot, (bounded by a pair of tetrons), in turn, splits into a mutually symmetric pair of subsets as follows: starting at the left-end of the tecot with the smaller tetron, a sequence of elements is selected via repeated applications of bif, (one element for each composition of A followed by B), until the rightmost element is reached. Symmetrically, the exact same procedure is repeated with the larger tetron at the right-end of the tecot until the left-most element is reached. The two half-subsets so obtained are, henceforth, designated as the left and right lexors, resp. Thus, lexors and, consequently, their tetrons and verses acquire handedness attribute. The handedness attribute can then be extended to chapters, as well: each chapter acquires the handedness of its tetron.
The left-right symmetry between the pair of lexors and their corresponding verses can be formalized via an involution (qupode, thereafter) between them: a given verse in the left lexor is mapped to its symmetric counterpart in the right lexor, (same distance from their respective tetrons). The qupode so defined reverses the handedness of lexors and their verses. Since chapters, by definition, acquire the handedness of their tetrons, the qupode pairs can be extended to chapters as well. A pair of chapters form a qupode pair whenever their tetrons do. Note that the qupode involution is a map between a pair of lexors and all their verses. On the other hand it is only a map between a pair of chapter. It does not extend to their verses. This is because chapters, unlike lexors, do not come in symmetric pairs and are of different lengths, so a map between their verses can not be established. Also, a given lexor and its verses are of the same handedness, while a chapter can have verses of opposite handedness.
Lexors are structurally similar to chapters in that they start with a tetron followed by (numbered) verses. They are the biflective analogue of chapters. However, this analogy is not complete thus far as currently there are two less lexors than there are chapters. There are no lexors 1 and 9, as there are only 112 tetrons. Nevertheless, the two missing lexors can be constructed ad hoc for now until the general biflectivity mechanism is developed. The two lexors (1 and 9) are put together as follows: in the derivations of the tecots above not all verses participated in the biflection process. A small subset of the outermost verses are acted identically under the actions of the two mirrors rendering bif ineffective to form a tecot. This remaining subset, however, can be split symmetrically into a pair of left and right lexors (1 and 9, resp.) based on their handedness, and sequenced compatibly with the general ordering of verses in the other lexors. This completes the correspondence between lexors and chapters. Furthermore, lexors attain the same titles as their corresponding chapters; lexor N (containing tetron N:0) corresponds to chp N, 1 ≤ N ≤ 114, and hence, aquires the same title as chp N.
Lexors, by construction, come in symmetric pairs (and hence, of equal lengths). This implies there are fewer distinct lexor-lengths compared to distinct chapter-lengths, as chapters vary greatly in lenght and there is no comparable pair-symmetry among chapters.