The Interior of a Reader - The Zero Set of an Invariant Price, Its Two Methods, and the Memory They Require

Abstract

The Interior of a Reader The Zero Set of an Invariant Price, Its Two Methods, and the Memory They Require Driven by Dean A. Kulik October 2026 Abstract Paper IV of this series showed that an exact price on a token arena, if it respects a reader's symmetry, is the difference of a potential that is constant on the reader's classes. On the 12-vertex witness of the series this leaves exactly one of 22 orbits of moves on which every such price is zero. This paper opens that orbit. We call the configurations and moves that a reader cannot price its interior, and we determine the interior of the witness completely: what it is, why it is there, what moves in it, and what a reader has to remember in order to move in it deterministically. The interior has 12 configurations and 24 moves, and the 3- and 4-cells supported on it are 8 triangles and 6 squares. Together they are a cuboctahedron: every move lies on one triangle and one square, V − E + F = 2, the 14 boundaries have rank 13, and the single relation among them is the fundamental class of a sphere, on which the reader acts by the sign character of Aut(X)/A4. The cause lies in the substrate. The six vertices of degree 7 in the witness induce an octahedron; the interior is the graph whose vertices are that octahedron's edges and whose moves pivot one token across a face; and that graph is the octahedron's medial graph. On tetrahedral, octahedral and icosahedral cores the same graph, taken with its cells of length at most L, first closes into a sphere at L = 3, 4, 5, the degree of the core, and longer cells add redundancy and no rank. The reader never exchanges the two ends of a core edge, so an interior configuration is an ordered pair (prev, cur) and every interior move runs head to tail. Each state has exactly two continuations. The two globally consistent choices, T+ and T−, are permutations of the 12 states of cycle type 34. They generate a group of order 12 that acts regularly and commutes with A4; the interior is its Cayley graph; and the relators T+3, T−3 and (T−T+−1)2 are the 4 + 4 triangles and the 6 squares. In coordinates the reader is the pyritohedral group of signed cyclic permutations of three axes, a state is a step from one axis to the next, and the two methods differ by a sign. The harmonic class that survives every square relation in Paper I is located on the interior. It takes one value t on every forward move and has period 3t; each of the 200 arena triangles carries exactly one period and each of the 1083 squares carries none. Reduced modulo its period, its potential is a clock with 24 values on which the reader acts by a character of order 3. This is the torsion companion of a lemma of Paper IV, and its kernel is the unique reader of order 8, which is also the largest reader on whose classes the two methods become a single update that still moves every class: a 3-cycle on three classes. Determinism is governed by a single standard criterion: an equivariant rule exists exactly when every stabiliser fixes one of the available continuations. On the interior, A4 admits exactly the two rules T+ and T− and no rule that reads the current vertex alone; the full reader admits none, because the element that fixes a state exchanges its two futures. On the whole arena one remembered step is enough for all 26 readers, and none is needed exactly for the readers of odd order. Every mechanism used is standard and is named as such. What the paper contributes is the exact data of one arena, the place each standard object occupies in its relation layer, the binding of four earlier measurements to one object, and a record of the claims withdrawn on the way. One generalisation is reported as failed: the number of classes a reader needs in order to price every pivot is bounded below by a chromatic number, which is attained on the tetrahedron and the octahedron and not on the icosahedron. All numbers are reproduced by one standard-library script printed in the appendix. Keywords: token graph, graph configuration space, medial graph, cuboctahedron, Cayley graph, pyritohedral symmetry, equivariant rule, second-order dynamics, relation layer MSC 2020: 05C76, 05C10, 05E18, 20B25, 55R80 License: CC BY-NC 4.0. The verification source and its transcript are Appendices B and C. 1. Introduction 1.1 The orbit that Paper IV left behind The substrate of this series is one finite object: a graph X on 12 vertices with 30 edges, its two-token arena A2(X) with 66 configurations and 300 moves, and the group G = Aut(X) ≅ C2 × A4 of order 24 [1, 3, 4]. A reader is a subgroup H ≤ G. What a reader can tell apart are its orbits on configurations, which we call its classes. Paper IV asked what a price on moves must look like if it is exact, p = dφ, and respects the reader. Its Lemma 8.1 showed that the potential φ is then constant on classes, and its Theorem 8.2 drew the consequence: exactly one of the 22 orbits of moves has both ends in a single class, so every such price vanishes on it. The observation was used there once, in an argument about the unweighted Laplacian whose scope is corrected in §13.3, and then set aside. Table 11 of the same paper shows the orbit refusing to go away: each of the five candidate potentials tried there zeroes at least those 24 moves, and the best of them zeroes exactly those. This paper is about that orbit. The question is deliberately small: what are the 24 moves on which an invariant price is silent, and what is on them? The answer ties together four measurements that the earlier papers reported separately: the octahedron inside the witness [1, Appendix A]; the harmonic class that survives every square relation [1, §3.3], with the unexplained circulation 22587 printed beside it [1, §5.1 and Appendix A]; the one-dimensional sign line shared by the commuting squares and the rotation triangles [3, §8.1]; and the forced zero of the gradient price [4, §8]. The working picture is that potential flows out of a domain and implementation flows in. An invariant potential registers a move only when the move leaves a class of the reader. Whatever happens inside a class is invisible to it and has to be supplied from somewhere else. The sections below find three suppliers, in this order: the substrate, which fixes the shape of the interior (§§4–5); the reader's own symmetry, which fixes what a state is and which continuations it has (§§6–7); and memory, which decides whether a next step can be chosen at all (§9). The sentence is an organising picture for one witness. Each clause of it is a finite computation stated below, and nothing more is claimed for it. 1.2 Summary of results statement section status 1 An exact H-invariant price vanishes on the moves internal to an H-class, and only there when its potential separates classes. Interiors and blind cells grow with the reader. §3 proved; checked on 100 reader pairs 2 Under G the interior is 12 configurations, 24 moves, 8 triangles and 6 squares forming a cuboctahedron. The single relation among the 14 boundaries carries the sign character 1−. §4 computed 3 The interior is the pivot graph of the octahedral core, which is the core's medial graph. §4 lemma proved; identification computed 4 On tetrahedral, octahedral and icosahedral cores the pivot graph, with its cells of length ≤ L, first forms a sphere at L = 3, 4, 5. At the trivial reader the whole arena closes at 3, 4, 4. §5 computed; saturation proved 5 The reader never reverses a core edge. An interior configuration is a state (prev, cur), all 24 moves are head-to-tail, and each state has two continuations. §6 computed 6 T+ and T− have cycle type 34 and generate a regular group of order 12 commuting with A4. Its Cayley graph is the interior and its relators are the 14 cells. §7 computed 7 The surviving harmonic class is t on every forward move and has period 3t: one period on each of 200 triangles, zero on each of 1083 squares. §8 computed 8 Its potential modulo the period is a clock with 24 values, a character of order 3, and kernel the reader of order 8. §8 lemma proved; values computed 9 An equivariant rule exists iff every stabiliser fixes a continuation. A4 admits exactly T+ and T−; G admits none. §9 proved; census of 26 readers computed 10 The reader of order 8 is the largest on whose classes the two methods induce one update that moves every class. §9.3 computed 11 On the whole arena one remembered step suffices for every reader, and none is needed exactly for readers of odd order. §9.4 computed 12 Pricing every pivot takes at least 3 classes, the chromatic number of the pivot graph. A reader attains 3 on the tetrahedron and octahedron and needs 9 on the icosahedron. §10 computed; the general claim is refuted Table 1. The results, with the status of each. Every computed entry is reproduced by the script of Appendix B. 1.3 What is new and what is not Nothing in Table 1 is a new theorem of graph theory or of group theory. Medial graphs, Cayley complexes, the pyritohedral group, the abelianisation of A4, fixed points of stabilisers and second-order reversible rules are all standard, and §12 names each one together with the status of the source. What the paper contributes is narrower: the exact data of one arena; the identification of where each standard object sits in the relation layer of that arena; the binding of four published measurements to a single object; one generalisation that was tested and failed; and the record of what was claimed on the way and then withdrawn (§13.2). Three status words are used throughout. Proved means that a short argument is given here; all of them are elementary. Computed means an exact finite computation over a stated range, reproduced by the script in Appendix B, whose transcript is Appendix C. Every val