Readers Are Functionals- Representation Control of the Relation Layer on Token Arenas

Abstract

Readers Are Functionals- Representation Control of the Relation Layer on Token Arenas a multiplicity table, a faithful-reader certificate, and an isotypic correction to the rank budget Driven by Dean A. Kulik October 2026 Abstract Attaching 2-cells to a token graph and computing invariant cohomology under a symmetry group yields a ledger of ranks, interactions, residues and containments. Papers I and II of this series reported such ledgers as facts about a substrate, or at best about a substrate-reader pair. We show that over ℚ with a finite reader group, the ledger is neither. Every quantity it reports at a reader H is dim VH for a module V fixed independently of H, hence the inner product dim VH = ⟨ m(V), fH ⟩ of a reader-free multiplicity vector against the reader's fixed-point profile fH(λ) = dim λH. The relation field is recorded once; a reader only evaluates it. The general layer that makes this work is classical and is used here with attribution rather than claimed: that rational representations are determined by their characters (Serre, Thm. 4 Cor. 2), that they are detected on cyclic subgroups (Artin, Serre Thm. 17), and the Burnside-ring analogue that G-sets are determined by fixed-point counts (Burnside; Bouc, Thm. 2.3.2). Assembled, these give the consequences we need. The recoverability loss satisfies LH(R|S) = dim (Q1(R|S))H, so loss is monotone along subgroup inclusion and the set of readers that see a containment is upward-closed: a containment cannot appear, vanish and reappear along a chain. Forced pairwise overlap must be counted inside each isotypic component before contraction, and the scalar rank budget understates it whenever the excess vector changes sign. Readers with fH > 0 in every component — faithful readers — form a downward-closed ideal in the subgroup lattice, and a single equality at a faithful reader certifies a module equality at every reader. A nontrivial faithful reader exists exactly when some proper H has ℚ[G/H] containing every irreducible, which fails outright for G cyclic of prime order. On a 12-vertex C4-bearing witness with Aut(X) ≅ C2 × A4 — 26 subgroups in 12 conjugacy classes — we extract the full 9 × 6 multiplicity table from twelve measured reader columns. The fit is 54-fold overdetermined and solves exactly. The table then replaces the ledger. Residue zero at every reader is one module identity. The rotation-triangle family is the isotypic complement of the commuting squares plus one copy of the central sign character, which fixes both the saturation and the additivity pattern at a stroke. The spectator-square quotient is 3+ ⊕ 3−, whose subgroup shadow is the principal filter above the normal Klein four-group. The table predicts two forced overlaps that the scalar budget reports as zero; direct row-space intersection confirms them, 14 of 14. The minimal faithful apparatus on this witness uses 152 of 300 coordinates — strictly fewer than the raw field, and strictly more than the 22 of the full-automorphism reader, which is blind in five of six sectors and for that reason reported those two overlaps as independence. We give the resulting method — inventory, deficiency, dominance shortlist, faithful-reader certification, propagation, evaluation — and show by measured counterexample that dominance is necessary and not sufficient: of four families dominating the commuting-square deficiency, two close it and two fall exactly one dimension short. All arithmetic is exact over ℚ. 1. Introduction A token arena is the state graph of K indistinguishable tokens moving on a finite simple graph X: vertices are the K-subsets of V(X), adjacent when one token slides along a base edge into an unoccupied vertex. Attaching 2-cells along chosen closed walks and computing cohomology invariant under a group G ≤ Aut(X) produces a first-cohomology residue εharm, and comparing cell families produces a ledger: which family reads how much, which families overlap, which family is indispensable, which vocabulary closes the residue. Papers I and II [1,2] built such ledgers and read them as structure of the substrate. That reading is wrong in a specific and correctable way. The same substrate read under two groups gives two different containment lattices: on the witness of this paper, spectator squares are contained in the span of spectator triangles under the full automorphism group and are not under the trivial group, by six dimensions. One might then retreat to the position that the ledger describes the pair (X, G). That retreat is also wrong, and in the more interesting direction. Over ℚ a finite group has semisimple representation theory, and taking H-invariants is exact. Every family image, every intersection, every quotient appearing in the ledger is therefore a rational G-module, fixed before any reader is chosen, and the reader enters only by selecting fixed vectors. The ledger is a function of X alone, recorded as a table of multiplicities, and G is a point of evaluation. Nothing in the field is created or destroyed by the reader; what changes is which isotypic components have a fixed readout. This has operational teeth, not only interpretive ones. Three errors follow from ignoring it, and all three were present in the earlier ledgers. A reader-invisible module gets called absent. Slack in a scalar rank budget gets called genuine independence. Multiplicity coverage gets called an actual span. Sections 4, 5 and 10 separate these into three distinct algebraic conditions, and Section 9 exhibits a case where the second error silently reports two families as exactly independent when the underlying modules are forced to share a dimension. Sections 2–5 are general: the apparatus, the evaluation theorem, the isotypic capacity bound, and the faithful-reader lemma. Sections 6–10 are a single worked witness, chosen in Paper II as a counterexample to a different conjecture and reused here because its automorphism group is large enough to have a nontrivial subgroup lattice and small enough to enumerate exactly. Section 11 states the method the general theory licenses; Section 12 records scope, including two corrections to the framing of Papers I and II. 2. The apparatus 2.1 Arena, cells, invariant complex Let X be a finite simple graph on n vertices and fix 1 ≤ K ≤ n. The arena AK(X) has vertex set the K-subsets of V(X), with S ~ T when |S △ T| = 2 and the symmetric difference is an edge of X. Fix G ≤ Aut(X); it acts on the arena. Let C1G be the space of G-invariant rational 1-cochains on the arena: one coordinate per G-orbit of oriented arena edges, with an orbit containing an edge and its reverse forced to zero. Let d0 be the coboundary from G-invariant 0-cochains, i.e. one row per vertex orbit, and set ε = dim C1G − rank d0 . A relation family 𝒦 is a set of closed arena walks (here: 3-cycles and 4-cycles). Each cell c ∈ 𝒦 evaluates a 1-cochain by summing it with sign along c; this gives a row d1(c) ∈ (C1G)*, and because d1d0 = 0 every such row annihilates im d0. Writing AG = ( C1G / im d0 )* , dim AG = ε , every family determines a subspace WF ≤ AG, the span of its cell rows, with rank d1(𝒦) = dim WF and εharm(𝒦) = dim ℋ1G(𝒦) = ε − rank d1(𝒦) = dim AG/W𝒦 . So the residue is a quotient, and ε is a hard ceiling on any family's rank. We call AG the ambient module and the WF the family submodules. Both are G-stable. 2.2 The five families A 4-cycle of the arena is one of three kinds, classified by the multiset of base edges its four moves use. If exactly two distinct base edges appear, each twice, the cycle is a commuting square: two tokens move along edges of disjoint closed support, in either order. Otherwise one token circulates a base 4-cycle while the others sit still; call it a spectator square if all four arena vertices share a common occupied vertex and a rotation square otherwise. Triangles split the same way into spectator and rotation 3-cycles. We write 𝔉 = { 𝒦disj, spec4, rot4, spec3, rot3 } , and 𝒦4 = 𝒦disj ∪ spec4 ∪ rot4 for all squares, 𝒦3,4 for the union of all five. Paper II showed 𝒦4 = 𝒦disj exactly when X has no C4 subgraph [2]; the present paper works on a substrate where they differ, so all five families are populated. 2.3 The ledger quantities For families R, S and reader H ≤ G, write rH(R) = dim WHR and DH = dim AH. The ledger reports IH(R,S) = rH(R) + rH(S) − rH(R ∪ S) , LH(R|S) = rH(R ∪ S) − rH(S) , the interaction and the loss. By the dimension formula for subspace sums, IH is the dimension of the intersection of the two row spaces, so LH(R|S) = rH(R) − IH(R,S) = dim QH(R|S) , QH(R|S) = WHR / ( WHR ∩ WHS ) . This is worth stating because it closes off a line of enquiry before it is opened. One naturally asks, of a redundant vocabulary, which other families determine a given one; for two families that question returns IH read from the other side, and is not a second measurement. Only leave-one-out against three or more families carries information the pairwise table does not, because inclusion–exclusion fails there. Finally, R ⪯H S means WHR ≤ WHS, equivalently LH(R|S) = 0. The subscript is not decoration: Section 8 exhibits a pair with spec4 ⪯ spec3 under one reader and not under another on the same substrate. 3. Reader evaluation This section and the two that follow are standard representation theory, assembled in the form the application needs. Nothing in Sections 3–5 is claimed as new; the contribution of this paper begins at Section 6. We state the results with proofs because the exact hypotheses matter downstream and because two of them were, in earlier drafts of this work, used in incorrect forms. Let Irrℚ(G) be the rational irreducible representations of G, and for a finite-dimensional rational G-module V let m(V) ∈ ℤ≥0Irr be its multiplicity vector. For H ≤ G define the fixed-point profile fH(λ) = dim λH = (1/|H|) Σh∈H χλ(h) . Theorem 1. (reader evaluation) For every finite-