The dual action of a verse on a function reciprocates the action of the latter on the former. A displacement of a given verse into a new verse by the function is to be matched by a similar and appropriate displacement of the function into a new function under the reciprocal action of the verse. In particular, a trivial function action is to be paired with a trivial verse action. In other words, if the verse remains fixed by the function, then, by symmetry, the function, likewise, is to remain fixed by the verse. A function of particular interest is the orconion, O, (orc for short) which transforms under the action of a given verse V into a new function, Ov, referred to as the cordisan associated with V. Let O(V) be the orc-action on V and, similarly, V(O), be the verse-action on O. Since, O fixes verse one, O(1:1) = 1:1, it follows that verse one fixes O, 1:1(O) = O. Hence, the orconion is also the cordisan of verse one: O1:1 = O.
Since, the orc, as stated above, is also the cordisan of verse one, it follows that the cordisans associated with other verses should have general characteristics similar to those of orc. This suggests that the derivation of a cordisan should follow a procedure similar to the derivation of orc. Recall that the orc was the correspondence between the two global orderings on verses, the standard order set by chps and the dual order set by lexors. The correspondence between the symmetrically paired verses of the left and right lexors formed an involution, the qupode. Thus, the procedure calls for the derivation of a new qupode from the original, resulting in new symmetric pairs of lexors, and hence a new global dual order on verses. The correlation between the standard and the new dual order defines a new orconion, which is designated as the cordisan.
The transformation of the original qupode under the action of a verse is achieved via the auxiliary function, the inchor, that was derived previously. However, this transformation is subject to the constraint that the new qupode must also be an involution. Furthermore, the overall structures of lexors derived in the (co)bif scheme are to be preserved. This is made precise as follows. Let L = (L1, L2, .... , L114), be the length vector of lexors, where Li is the length of lexor i, (its number of verses), for 1 ≤ i ≤ 114. The admissible action of the inchor on the qupode permutes the components of L, in addition to a permutation of verses within and across the lexors. This block permutation (preserving the overall structures of lexors) moves a contiguous group of elements (a "block") as a single unit within a sequence, while keeping the internal order of the elements within the block unchanged.
The block transposition, moreover is required to preserve another key aspect of the qupode: the two principal lexors 1 and 9 must remain a qupode-pair. This is because the two lexors are exceptional in that they are headed by the numbered verses 1:1 and 9:1 (each other's qupode) and as such can not be paired with other lexors that are bounded by (0-numbered) tetrons. Stated differently, 1:1 and 9:1, as numbered verses, can not be paired with 0-numbered tetrons. They can only form a qupode-pair together. This is accomplished first via the permutation of tecots, where the pair of lexors in one tecot is interchanged in parallel with a lexor-pair in another tecot. At this stage the (1, 9) pair permutes with other pairs. In the second phase lexors 1 and 9 may permute with each other while the remaining lexors permute among themselves. The inchor action on the qupode is thus carried out at three stages: first on tecots, then on lexors, and finally on verses.
Let Q and N be the qupode and the inchor of some verse V, respectively. Q is to transform into a new involution under the action of N. As an involution Q acts on verses via disjoint transpositions, (cycles of length two), where the two verses in a given pair are interchanged. There are no trivial cycles (cycles of length one) or fixed points. This cycle type, therefore, must be preserved. That is the transformed qupode also acts via disjoint transpositions with no trivial cycles. This is achieved via conjugation, Q → N−1QN, where Q is sandwiched between N and its inverse N−1. Under conjugation a cycle transforms to another cycle of the same length. The qupode-pair (U, V), (a transposition, where U = Q(V) and V = Q(U)), transforms to (N(U), N(V)) under conjugation by N. Each component of the pair is replaced by its image under N. The derived lexors so obtained will induce a new global dual order on verses, leading to a new orc which is defined as the cordisan of the verse V.
Under the conjugation action of the inchor on the qupode, the trivial actions, appropriately, result in corresponding trivial actions. To see this, let OV = V, where orc acts trivially on the verse V. Then, the displacement interval at V (a fixed point) is a singleton, [V, OV] = [V, V] = {V}. The only function/permutation possible on a singleton is the trivial one, the identity map. Consequently, the direct-sum of the identity map is the identity map, again. Since, the interval displacement is zero (no need for global cyclic shift) the direct-sum (the identity map) is the inchor of V. The conjugation by the identity map, in turn, leaves the qupode intact, Q → Q, resulting in the same orc: VO = O. Hence, the conjugation action conforms to the reciprocity between fixed points and their fixed functions.