D2a — σ₃ and step: The Observable and the Breaking Step

D2a builds the dynamics of the discrete regime on the socle of D1′ v1 §7. There the mechanism is extension — pairs are added to the relation, any set of them —, and the non-freezing filter MP(III) retains the living trajectories: no symmetry returns (clause (i)), and infinitely many breaking steps occur. D2a defines the observable σ₃^obs, the number of indistinguishability classes that a ball meets, shows how it behaves under the steps that the filter admits, and defines the step of the dynamics as a breaking extension. No base point, horizon, weight or order among pairs enters any definition.
Derived results. Clause (i) protects every count: along a step that satisfies it, the classes refine and no count of any observer decreases (Lemma 3.2); a count decreases only when the step creates a symmetry (Corollary 3.4). A state with trivial automorphism group has no breaking extension; a homogeneous state always has one; a loop at a point x of a non-trivial orbit is a breaking step whenever no point of the orbit of x under the automorphisms of the loopless part carries a loop — in particular on every loopless state with a non-trivial automorphism group (Proposition 4.3). Along a trajectory that satisfies (i), no state recurs, the automorphism group never grows and strictly decreases at every breaking step, and no count decreases (Proposition 6.1).
Negative results. Counts and breakings are independent. On the chain ℕ, a trajectory along which the count of the end point rises at every step freezes, its automorphism group being trivial throughout; on two copies of ℕ, a loop at one end point is a breaking step that changes no count of any observer (Proposition 5.1). The filter therefore cannot be stated as a criterion on counts. σ₃^obs is not local: a breaking step at any distance from an observer changes its count while its ball is unchanged (Proposition 2.5). A non-trivial automorphism group does not guarantee a breaking extension: on three points, a relation with a symmetry of order 2 has no extension that satisfies (i) and refines its classes (Proposition 4.3(d)). Neither richness nor life implies connectivity (Proposition 1.6). OPEN: the existence of a breaking extension on infinite locally finite states with a non-trivial automorphism group, beyond the cases of Proposition 4.3.