extends_trans

In the primitive recognition calculus, a finite trace is the empty history or a history grown by one distinction act at a time (K2.4). Distinction acts are the atomic recognition steps; traces are their ordered finite compositions.

Syntactic composition is append: glue a second trace onto the end of a first by recursing on the second argument. Extension (K2.5) is the induced prefix relation: $U$ extends $T$ when some suffix $V$ satisfies $\mathrm{append}(T,V)=U$. Reflexivity is the empty-suffix case; the present result is the matching transitivity law (R4).

The only algebraic fact needed upstream is associativity of append, so chained suffixes compose unambiguously.

Unpack both extension hypotheses to suffix witnesses $A$ and $B$ with $\mathrm{append}(T,A)=U$ and $\mathrm{append}(U,B)=V$. Exhibit the concatenated suffix $\mathrm{append}(A,B)$ as the witness that $V$ extends $T$. Rewrite with append associativity, then substitute the two equalities, yielding $\mathrm{append}(T,\mathrm{append}(A,B))=V$.

Transitivity is the structural glue for any reasoning that quantifies over future extensions of a trace. Downstream, TraceLogic defines implication as: $P$ implies $Q$ at $T$ when every extension $U$ of $T$ that satisfies $P$ also satisfies $Q$. Stability of that implication under further extension is proved by composing the two extension edges via this lemma ("implication itself stable under extension").

In the broader Recognition stack this sits at the foundation layer: traces are the raw histories on which later cost, forcing, and logic are hung. Without transitive extension, persistence and implication would not be well-defined relations on growing recognition histories. It is pure order-theoretic scaffolding, not a physics forcing step (T5–T8), but every stable predicate built on traces depends on it.