Orientation as Holonomy of the Reader Configuration Space

Abstract

Orientation as Holonomy of the Reader Configuration Space Driven by Dean A. Kulik September 2026 Abstract A finite reader partitions a cell set into m sectors and observes a permutation T through the sector-to-sector transition table C(T). Because T is a bijection, the antisymmetric part A = C − Cᵀ has zero row sums and therefore defines a conserved directed-circulation channel. At three sectors that channel is one-dimensional and transforms by the sign representation of S₃, so its sign is a natural orientation observable. We show that this sign is not a function of T. Holding one permutation and one unordered three-sector partition fixed, the six labelings of that partition give the same magnitude and both signs. No refinement of T can supply the sign, because a function cannot take two values at one argument. The missing structure is the reader’s own space of admissible configurations. For an ambient adjacency graph X, let 𝒞(X, m) have as vertices the partitions of the cells into m connected nonempty regions, with edges given by an admissible move rule. Labelings form an Sₘ-cover, and transport around closed walks defines a monodromy ρ : π₁(𝒞(X, m)) → Sₘ, of which the orientation observable is the parity projection sgn ∘ ρ. The image of ρ depends on X and the move rule, never on T. We compute it. Under either of two natural move rules, unrestricted subsets and connected regions of a two-dimensional torus give S₃; requiring three sectors to remain contiguous arcs of an oriented cycle gives A₃; the three-dimensional cube Q₃ has full S₃ monodromy exhaustively; and the larger three-dimensional torus Z₃³ admits an explicit odd exchange loop, so its image is not contained in A₃. Restricting moves to adjacent cells does not by itself remove odd holonomy — what removes it is the admissibility condition on the partitions, not the locality of the move. Separately, a finite translation system carries a full three-sector channel while its state-transition graph is triangle-free, showing that the complete graph supporting the channel is a graph on sector labels and is independent of state adjacency. 1. Introduction A reader observes a transformation through a restricted interface. In the permutation setting the transformation is exact and finite, the reader is a partition, and the observation is a table of sector-to-sector counts. This makes the inverse question sharp: which features of the observation belong to the transformation, and which belong to the way the reader is arranged? Prior partition-reader work gives an exact accounting for coarse observations, with what the reader resolves and what remains in the fiber summing to log₂ N! for every reading [1]. Under all balanced two-block cuts the fiber is precisely the independent cycle reversals, so the unoriented cycle structure survives and orientation does not [2]. That residual motivates a finer reader. Three sectors is the first resolution at which a conserved directed circulation can exist at all, and at three sectors that circulation is a single scalar. Its sign looks like the orientation the two-sector reader could not see. This paper shows the sign is not carried by the transformation, locates what does carry it, and computes that object across several ambient geometries. Relation to known work. The behaviour we find in one versus two dimensions is not itself new. That indistinguishable objects cannot be exchanged on a line and can be exchanged in the plane, with the braid group governing the difference, is classical [5,6]. Configuration spaces of points on graphs and surfaces are well developed [5,6,7]; so is the topology of the partition lattice as a poset [8]; and the representation appearing in §3 is a standard Specht module [9]. The present work treats partitions into connected regions, under an explicit move rule, as a configuration space in its own right, computes its monodromy, and identifies a reader’s observable with that monodromy. That identification is the contribution. We make no priority claim against the literature cited above; the result stands on the constructions and computations given here. 2. Definitions 2.1 Ambient adjacency and readers Let X = (Ω, E_X) be a finite graph. Its vertices are cells. A subset is connected if it induces a connected subgraph of X. Taking X complete recovers the unconstrained case, in which every nonempty subset is connected. An m-sector reader is an ordered tuple R = (R₁, …, Rₘ) of pairwise disjoint nonempty connected subsets covering Ω. Forgetting the order gives the underlying partition. 2.2 Two move rules Two natural notions of an elementary reader move appear in what follows, and they generate different graphs on the same vertex set. We name both and label every result with the one used. Transfer τ. Move a single cell from one sector to another. Sector sizes change. Admissible when every affected sector remains nonempty and connected. Swap σ. Exchange one cell of one sector with one cell of another. Sector sizes are preserved. Admissible under the same conditions. A swap is not a τ-edge; it is a pair of transfers whose intermediate state may itself be inadmissible, so σ is genuinely a different move rule rather than a shorthand. Where sector sizes are constrained to be equal, only σ is available. Where they are free, τ is the finer rule. §7 reports the same answer under both in the case where both apply, but this is a computed fact, not an identity. 2.3 Configuration space, cover, monodromy Fix a move rule μ ∈ {τ, σ}. The configuration space 𝒞(X, m; μ) is the graph whose vertices are the admissible underlying partitions and whose edges are the μ-moves. The ordered space 𝒞̃ has labeled tuples as vertices, with a move carrying labels along; forgetting labels gives an m!-fold covering of graphs on which Sₘ acts freely by relabeling. A closed walk in 𝒞 lifts to a walk in 𝒞̃ whose endpoint differs from its start by a relabeling, giving ρ : π₁(𝒞, p) → Sₘ, well defined up to conjugacy by the choice of base labeling. Operationally, im ρ is trivial exactly when 𝒞̃ has m! times as many connected components as 𝒞. No observer enters this definition: ρ is a property of the configuration space and its cover, present whether or not anything occupies an address in it. 2.4 Two complete graphs Two complete graphs occur in this paper and must be kept apart. The state-transition graph Γ has states as vertices and generator actions as edges. The graph Kₘ of §3 has sector labels as vertices and is complete by construction, since Cᵢⱼ is defined for every ordered pair of sectors. §10 shows the two are independent. 3. The conserved antisymmetric channel For an ordered reader and a permutation T of Ω, set Cᵢⱼ(T) = #{x ∈ Rᵢ : T(x) ∈ Rⱼ}. Bijectivity gives ∑ⱼ Cᵢⱼ = |Rᵢ| and ∑ᵢ Cᵢⱼ = |Rⱼ|, so row and column sums agree. Hence A = C − Cᵀ is antisymmetric with every row sum zero. Write 𝒜ₘ for that space. Proposition 3.1. dim 𝒜ₘ = C(m,2) − (m−1) = (m−1)(m−2)/2, and 𝒜ₘ is canonically the cycle space Z₁(Kₘ; ℝ). Proof. An antisymmetric matrix is determined by its C(m,2) entries above the diagonal. The zero-row-sum equations impose m linear conditions whose sum vanishes identically by antisymmetry, hence m−1 independent ones. Identifying Aᵢⱼ with a flow on the edge ij of Kₘ, zero row sum is zero divergence at each vertex, and the kernel of the divergence operator on a connected graph is its cycle space, of dimension |E| − |V| + 1 = C(m,2) − m + 1. ∎ Two consequences. For m = 2, K₂ is a tree and the channel vanishes identically: a balanced two-sector reader has no conserved directed circulation, which is the algebraic content of the orientation-blindness reported in [2]. For m = 3, the dimension is one and every element is A(a) = [ 0 a −a ; −a 0 a ; a −a 0 ]. Proposition 3.2. For m ≥ 3, 𝒜ₘ is an irreducible Sₘ-module, isomorphic to the Specht module S^(m−2,1,1). It contains the sign representation if and only if m = 3, where σ · A(a) = A(sgn(σ)·a). The identification is classical [9] and we use it as background rather than claim it. We verified irreducibility and the sign-isotypic dimension independently for m ≤ 6 by isotypic projection and by character inner product, obtaining agreement. The representation theory settles what the channel is. It does not settle where the sign comes from, and §4 shows the natural answer is wrong. 4. The sign is not a function of the transformation Throughout this section X is complete, so every subset is admissible. Take Ω = {0,…,5}, T = (1,2,0,4,5,3) in one-line notation, and the unordered partition {0,3} | {1,4} | {2,5}. Evaluating a = A₁₂ for each of the six labelings: labeling a |a| sgn(a) (0,1,2) 2 2 +1 (0,2,1) −2 2 −1 (1,0,2) −2 2 −1 (1,2,0) 2 2 +1 (2,0,1) 2 2 +1 (2,1,0) −2 2 −1 Proposition 4.1. sgn(a) is not a function of T. Proof. The permutation is the same in all six rows, and so is the unordered partition. Only the labeling differs, and both signs occur. ∎ This is stronger than the statement that some particular reader fails to recover orientation. It says no refinement of T can determine the sign, since a function cannot take two values at one argument. Writing ℓ for a labeling and γℓ for any path in 𝒞̃ from the base labeling to ℓ, the observation decomposes as a(T, ℓ) = |a(T)| · sgn(ρ(γℓ)). The first factor is an invariant of T; the second is fixed by the reader’s address in the cover. The reader manufactures nothing — it does not alter C(T) — but which lift of the channel is present at its address is determined by where it stands. The question becomes whether a labeling can be chosen consistently across the whole configuration space, and that is a question about ρ. 5. The exchange construction Theorem 5.1. Let B and C be sectors with |B| = |C| = k. Suppose the k swaps below are each admissible. Then they form a