Transformation without loss

Abstract

Transformation without loss Residual cohomology of token-configuration arenas under change of substrate, symmetry, and admissibility Driven by Dean A. Kulik September 2026 Lead-in Put some indistinguishable tokens on the vertices of a finite graph, one per vertex, and let a token step to an empty neighbour. That is the whole system. The graph is the substrate; the space of configurations and legal steps is the arena. Now change something about it — add an edge, subdivide one, glue two vertices together, allow a site to hold two tokens instead of one, or simply look at the system with a coarser symmetry. Each of those changes alters which transformations exist. The question this paper asks is narrow and answerable: When the rules of movement change, what happens to the structure that was there before? The naive answers are that structure is created or destroyed. Both turn out to be wrong in a precise and checkable sense. Every elementary change either extends the transformation structure, re-expresses it, or hides it from a particular observer. Structure is never annihilated. The one operation that does produce a genuine kernel — restricting which configurations are admissible — produces it as a relative cohomology class: something present in the larger admissible space and absent from the smaller one, with an exact dimension formula. The headline object is a single integer attached to each such restriction. For a distinguished site allowed to hold tokens instead of , that integer is the first Betti number of a small quotient graph, and it can be computed without ever building the arena. A concrete instance, which the reader can verify in Section 9: on the 7-cycle with four tokens, the number of independent transformation channels visible to a symmetric observer goes as the distinguished site’s capacity is relaxed from one token to four. The increments are the capacity defects. The final zero is not a saturation effect — it is a statement about the local symmetry at that site, and on other substrates it is nonzero. 0. Scope This is a mathematics paper. It is part of a larger programme (NEXUS) whose ambition is explanatory rather than descriptive: not another representation of physical regularity, but an account of what transformation requires. That ambition does not appear in the theorems below and is not used to support any of them. The rule followed throughout is prove downward, interpret upward, never prove upward by interpretation. Section 11 is the only interpretive section and proves nothing. One point of orientation, because it affects how the results should be read. A complete explanation of transformation is not the same thing as a terminal state of the system being explained. The objects here are finite and the theorems are closed; the programme they belong to expects its own descriptions to keep transforming. Nothing in Sections 1–10 depends on that expectation. 1. Setup is a finite connected simple graph, a token count, . Arena. has as vertices the -subsets and as edges the moves with , , . Lemma 1.1 (edge parametrisation). Arena edges are in -equivariant bijection with pairs , , , . The two orientations are the two choices of which endpoint of carries the token at the source. Proof. and is the moved token’s base edge; the correspondence is manifestly equivariant. Everything downstream follows from this. It is the reason the base graph, not the arena, controls most of the answers. Residual. With the real cochain groups of the arena 1-skeleton ( antisymmetric under edge reversal), Averaging over the finite group gives over ; the quotient presentation is the one used below. Three distinct objects appear in the literature on this and must not be conflated: , its invariant part , and of the complex with commuting squares filled. Every statement here names which one it means. 2. What is Theorem 2.1. , the number of -orbits of arena edges whose stabiliser contains no orientation-reversing element. Proof. Orbit–stabiliser on oriented edges. If some reverses then invariance forces , hence ; otherwise one scalar determines on the whole orbit. Theorem 2.2. For connected , , the number of configuration orbits. Hence where is the quotient graph with the reversible edge orbits deleted. The three descriptions — orbit counting, cochain dimensions, quotient-graph topology — are the same integer. Measured across 30 arena/predicate combinations with no discrepancy. Theorem 2.3 (closed form). Writing , for edges of fixed pointwise by and for edges whose endpoints swaps, Cycle structure of on plus two edge counts per group element. No arena is built. Verified against direct orbit counting on 106 (substrate, , block-profile) cases across five base graphs including a non-vertex-transitive one, : no discrepancies. Two immediate consequences. for all under the full symmetric group — is transitive on and the transposition reverses while fixing pointwise, so the single edge orbit is reversible. And for the trivial group there is no quotient at all, so exactly — not a limit. Measured, not proved: . Across three arenas and both standard 2-cell predicates (disjoint-support commuting squares; all arena 4-cycles), the invariant harmonic space was zero in every case, so the entire invariant non-exact sector is coexact. Cell counts moved by a factor of 5.7 without moving by one dimension: the residual is a property of the 1-skeleton and the group, and the 2-cells only decide how it splits. No case with is known. 3. What a reader can see A reader coarse-grains the substrate into blocks and records block-to-block flow. Two natural responses exist, and both are blind to the residual, for different reasons. The scalar response , , is exact, so its integral round any loop vanishes — in the arena and in any quotient of it. The matrix response is subtler. Contracting the antisymmetric block-flow table with any fixed antisymmetric gives for a move : it depends only on the moved base edge and its orientation. Hence every matrix-reader cochain is a pullback along the equivariant map from arena directed edges to base directed edges. Theorem 3.1 (reader bound). . Proof. Equivariance gives , so only invariant base forms contribute. And with , so exact base forms pull back to coboundaries. What remains is the base graph’s own residual at one token, and . Measured: the bound is attained and is independent of . Nineteen cases; on every symmetric substrate the reach is — of channels on at , of at , of on and Petersen — and on deliberately constructed trivial-automorphism bases with the reach is exactly and across against arena residuals of and . 4. The four motions Let be an elementary change. The residual is only defined relative to a group, and the group changes too, so comparisons pass through . There is no map ; there is a span. motion what happens to a transformation induced map on residuals measured extension — add a base edge survives verbatim; new ones appear onto, 6 pairs re-factorisation — subdivide an edge onto, 5 pairs, to hiding — enlarge the group unchanged , 5 pairs identification — contract an edge configurations merge, many-to-one injective; kernel only via admissibility Section 5 No motion removes a channel. The one case that looks like removal — takes the residual from 5 to 0 — added six transformations and destroyed none: the five channels persist as -invariant cochains, and is a statement about the enlarged observer. The same correction applies to a number that motivated much of this work. for looks like loops appearing as the cycle lengthens. Pulling the channel back along the subdivision chain map gives rank of , landing inside at the common group; likewise of and of . The cochains are present throughout. The shorter cycle’s dihedral group cannot see them. 5. Admissibility: the capacity filtration Fix a distinguished site and let . Let be the arena in which may hold up to tokens and every other site at most one. Then , with restriction and newly admitted layer Theorem 5.1. is surjective; . Proof. Extend an invariant 1-cochain by on the new edges and average; the extension is invariant and restricts correctly. For a graph , so every 1-cochain is a cocycle and the extension descends. Definition. , the capacity defect, and . Theorem 5.2. . Proof. If then with invariant; extend and average to , which restricts to because preserves the layers. Then vanishes on , so every kernel class has a representative supported on the new layer. The converse is immediate. 5.1 The cone theorem The relative complex is the reduced complex of the quotient space: is collapsed to a single point , not deleted. Define the graph : • vertices: the -orbits of , plus the cone point ; • edges: the -orbits of arena moves with at least one endpoint in (cross moves land on ; internal moves join two layer orbits), with the reversible orbits deleted. Theorem 5.3 (cone). . Proof. By Theorem 2.1 applied relatively, . Relative 0-cochains are functions on the non-cone vertices, so . The relative coboundary is the ordinary graph coboundary with the value at pinned to zero, so its kernel is the constants on each component with the -component forced to zero: , hence . Since , . Since and : The cone point is not bookkeeping. Without it the saturated case would return . 5.2 Connectivity is automatic Lemma 5.4 (reversible orbits are loops). If reverses the arena edge then , so and the orbit is a self-loop in the quotient. Deleting self-loops cannot change components, so , the count before deletion. Lemma 5.5 (vacancy propagation). Let with a vacancy . If there is no immediate move to , but along a substrate path from to the first occupied-to-vacant edge has , so the move p