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Band Switching and Expectation Symmetry

Abstract

The actual switch of two band pairs exchanges good and regular and yields equal expectations under an invariant finite law.

Let X be a finite flag carrier. The permutations s and t are involutions without fixed points, representing corner and band matchings. For four distinct points a,b,t(a),t(b), let u swap t(a) and t(b), and let t’=utu. This replaces a–t(a), b–t(b) by a–t(b), b–t(a) and leaves all other pairs unchanged. Good(s,t,a,b) means that a and b are in the same integer-power orbit of s*t. Bad(s,t,a,b) means that s(a) and b are in the same such orbit. Regular(s,t,a,b) means that neither orbit relation holds. The two-matching connectivity theorem identifies Regular with distinct combinatorial boundary components.

Theorem 1.1 (Actual Band Surgery Exchanges the Categories).

Proof. Machine-checked in Lean as D5/S3/Combinatorics/PerfectMatchings/FiniteBandSwitch.band_switch (✓ std3). ∎

Citation. Babak Ghanbari, Robert Šámal (2026). Facial diagrams and cycle double cover. DOI: 10.48550/arXiv.2605.01410. URL: https://arxiv.org/abs/2605.01410v1.

Commentary.

Delete the two selected band pairs by fixing their four endpoints. First-return paths in the remaining finite matching system construct a fixed-point-free endpoint pairing. Its value at one endpoint and involutivity force one of the three pairings of four points. The proved reattachment orbit formula then gives good-to-regular, regular-to-good, and bad-to-bad. The cut and both reattachments are explicitly identified with t and utu; their orbit effect is not a premise.

For the expectation statement, Omega is a nonempty finite set of equally weighted states. The corner matching s is fixed and t(omega) is a fixed-point-free involutive band matching for every state. The four marked points are distinct in every state. A state bijection tau satisfies t(tau(omega)) = switch(t(omega),a,b) for each omega, where switch is the actual matching operation above. MeanGood and MeanPhysicalRegular are the rational sums of the corresponding zero-one indicators divided by card(Omega). PhysicalRegular is the negation of corner/band connectivity.

Theorem 1.2 (Equal Good and Regular Expectations Under the Stated Law).

Proof. Machine-checked in Lean as D5/S3/Combinatorics/PerfectMatchings/FiniteBandSwitch.twist_invariant_uniform_expectation (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Babak Ghanbari, Robert Šámal (2026). Facial diagrams and cycle double cover. DOI: 10.48550/arXiv.2605.01410. URL: https://arxiv.org/abs/2605.01410v1.

Commentary.

Band surgery proves that a state’s regular indicator equals the good indicator of its image under tau. Reindexing the finite sum gives equality of the two means. The statement makes the state-space and actual matching-transport assumptions explicit. It does not identify an unspecified distribution on unlabelled embeddings, prove surface realization, or imply exact thirds for any category.

Application to Independent Signed Rotations

Let G be a finite simple cubic graph, with vertex set V, edge set E, and dart set D consisting of ordered adjacent vertex pairs. At each vertex choose a labelling of its three neighbours. The forward and inverse 3-cycles are its two cyclic orders; among the six shuffled neighbour lists, each cyclic order has three representatives. Thus independent uniform cyclic orders and independent fair edge signs have the uniform state space Omega = Bool^V times Bool^E, of size 2^(card(V)+card(E)). This is the full labelled signed-rotation law, with every rotation and sign retained.

For a sampled rotation rho, write R(d) for the dart obtained by rotating the neighbour at the initial vertex of d. On X = D times Bool define P(d,1) = (R(d),0), P(d,0) = (R^(-1)(d),1), and L(d,b) = (reverse(d), b xor lambda(edge(d))). Here lambda is the sign bit in the neighbour-list sampler: zero preserves side across a band and one reverses it. P and L are involutions. P has no fixed points because it reverses side; L has none because the adjacent vertices of a simple graph are distinct. The P/L components are the physical facial boundaries, and each has the two disjoint oriented P*L orbits given by the matching-component decomposition.

Fix an edge and choose either of its darts d. Put a=(d,0) and b=(d,1). The four flags a,b,L(a),L(b) are distinct, independently of its sign. GoodAt(P,L,a,b) says the two edge sides occur in one oriented boundary orbit, so their directed traversals agree. BadAt uses the opposite oriented orbit, represented by P(a), so the traversals oppose. PhysicalRegularAt says the sides lie in different P/L components. The disjoint two-orbit decomposition makes these three categories mutually exclusive and exhaustive, including singleton product orbits.

Toggle only the chosen edge sign. On the same flag carrier and with the same corner matching this changes L to uLu, where u swaps L(a) and L(b). Every other band pair is unchanged. The actual cut and its two reattachments therefore satisfy band_switch, which exchanges good and regular and preserves bad. In particular the third endpoint pairing is bad under both signs; components missing the four flags remain present. For each fixed rotation, apply twist_invariant_uniform_expectation to the whole sign-vector space and its one-coordinate toggle. Summing this equality over all rotation vectors gives equal good and regular probabilities for every edge under the full uniform law.

Let G_count, B_count and R_count count edges in the three physical categories. For each state their sum is card(E). Interchanging the finite state and edge sums in the per-edge equality gives E[G_count]=E[R_count]. Averaging the partition identity then gives E[G_count]=E[R_count]=(card(E)-E[B_count])/2. These finite averages are exactly the expectations under the stated uniform law. There is no connectedness, bridgelessness, orientability or edge-transitivity premise. The empty graph has vacuous edge assertions and all three counts zero. The conclusion imposes no exact-thirds identity.

References