The orconion (orc for short), a map from verses to verses, was defined as the composition of the two involutions, flectope and monotope. It is a correspondence between the standard and the dual global orderings on the set of all verses. When lexors are arranged in sequence from 1 to 114 (like chps) the first verse in lexor one, 1:1, aligns with the first verse in chp one, which is also 1:1 (the two have the numbered tetron, 1:1, in common). Subsequently, under the orc, verse 1:1 is mapped to itself. It is the only fixed point. A given verse V and its image OV under the action of orc, O, defines a directed interval, DI = [V, OV]. It is the set of all consecutive verses bounded between (and including) V and OV. When V < OV the function is increasing at V and the interval runs from V to OV (left to right) and is positively ordered. When OV < V, (decreasing function at V), the interval runs in the opposite direction from OV to V (right to left) and is negatively ordered. The interval at a fixed point, OV = V, is the singleton DI = {V}. This only applies to verse 1:1.
A given verse V is commonly acted upon by a function such as the orc resulting in another verse. This may be understood as a transformation of one verse into another. Conversely, it is also conceivable to view verses as operators acting on a given function resulting in a new function, or stated differently, transforming a given function into another function. This reverse action, compatible with the function-action, establishes a form of reciprocity between verses and functions. A trivial function-action (on the verse) is to be fittingly matched with a trivial verse-action (on the function): a fixed point is matched with a fixed function, that is the function remains the same.
Of primary interest is the reverse action of a given verse V on O. This action is facilitated via an auxiliary function to be defined in terms of DI. The restriction of O to DI, is a function R, which is the same as O but is only defined on the smaller specified domain, the DI. O is a bijection, a function that is both injective (one-to-one) and surjective (onto). It is an endomap: its domain (the source) and codomain (the target) are the same, the set of all verses, W. A bijective endomap is a permutation. However, its restriction R, although, still a bijection from DI to its image R(DI) is not an endomap as it may not necessarily map back into DI: R(DI) = O(DI) ≠ DI. Note that the verses in R(DI) may not even be contiguous and therefor do not form an interval. However, a permutation P can be defined that is functionally equivalent to R. That is its action on its domain parallels that of R on DI.
The consecutive verses in DI, start and end with the bounding verses V and OV. Let N be the length of DI (its number of verses). Then, the increasingly ordered set I = {1, 2, .. , N} can be bijectively mapped into the positively ordered DI, where 1 is mapped to V, 2 to the next verse after V, continuing until N is mapped to the last verse OV in the interval. When DI is negatively ordered the mapping is reversed: 1 is mapped to OV, 2 to the next verse after OV until N is mapped to the last verse V. Similarly, the image of DI under R, R(DI) can be bijectively mapped to I while preserving order: the smallest verse in R(DI) is mapped to 1, the next smallest to 2, continuing until the last verse (the largest) is mapped to N. The compositions of the three bijections, I -> DI -> R(DI) -> I, is a bijective endomap on I, hence, a permutation. This permutation, P, on the interval I is functionally equivalent to R on DI. It is a representation of R. It acts on the ordered interval I the same way R acts on DI. The verses in DI and R(DI) are uniquely represented by their equivalent counterparts I and P(I), respectively, preserving the relative ordering of verses (increasing verses mapped to increasing numbers).
P acts on DI by permuting its N verses. P can be extended to act on the set of all verses as follows. Placing a copy of P next to itself is a permutation of twice the size, 2N, where each copy acts independently on its own interval of length N (no interference between the two P's). The enlarged domain is the new interval I = {1, 2, .. , N, N + 1, N + 2, .. , 2N}, where the first P acts on the first half {1, 2, .. , N} while the second P acts on the second half {N + 1, N + 2, .. , 2N}. The elements in the second interval are shifted by N. The result is the direct sum of permutations, P ⊕ P. Repeating the direct sum of P with itself K times, P ⊕ P ⊕ .. ⊕ P, will result in a permutation of length KN acting on the set {1, 2, .. , N, N + 1, .. , 2N, 2N + 1 .. , 3N, .. , KN}, where each P acts independently on its own distinct interval of length N that is shifted by N from its preceding interval. In order for the direct sum to act on all verses its length must be equal to the total number of all verses. Since, the number of verses, in general, is not a multiple of N, there will be a remainder sub-interval, SI. When DI is positively ordered the direct sum is performed left-to-right and SI consists of the beginning segment of DI and appears at the right end of the direct sum: P ⊕ P ⊕ .. ⊕ P ⊕ SI. When DI is negatively ordered the direct sum is performed right-to-left and SI contains the ending segment of DI and appears at the beginning of the direct sum: SI ⊕ P ⊕ P ⊕ .. ⊕ P. The two remainders although of the same length are distinct sub-intervals formed from opposite ends of the DI. When the number of verses is a multiple of N there will be no remainder and the left/right direct sums coincide.
A given verse V is mapped under O to OV. This can be thought of as O moving V from its current position to where OV is. This displacement is positive if V < OV, (O is increasing at V), negative if OV < V, (O is decreasing at V), or zero if V = OV, (a fixed point). Let d = N − 1, then the displacement is, accordingly, +d, (V moves d units to the right), −d, (V moves d units to the left), or 0, (no movement). The interval displacement, IDS, by O can be defined as the sum of all verse displacements in DI divided by N. This consolidates the movements of all verses in the DI under O into an effective displacement of the entire interval treated as a single unit. It is possible for IDS to be zero despite internal verse movements due to cancellations of opposite displacements.
The IDS is the effective left/right translation of the DI as a unit under the action of O. This translation is then incorporated into the direct sum, (RM ⊕) P ⊕ P ⊕ .. ⊕ P (⊕ RM), via a global left/right cyclic shift resulting in an auxiliary function, the inchor. This intermediate function serves as the key in the implementation of the action of a verse V on orc.