tickActivations_le_two

In the Recognition Science cosmogenesis model, the open-system engine posts at most one recognition event per tick (the T7 eight-tick cadence). A cycle is $T$ ticks with resolution map sending each tick to an optional edge on an arbitrary region-index type $\iota$. Each resolved edge activates its two endpoints (double-entry posting).

Region-activations on a tick are the unit-cost case of tick work: $0$ if the tick resolves nothing, $2$ if it resolves an edge. The sibling bound states that when every endpoint costs at most $P$ to expand, a tick costs at most $2P$ recognition work. Specializing to unit cost yields the pure activation count.

The module's goal is that per-cycle recognition cost stays bounded by the cadence while the world grows linearly, so engine cost localizes to a sub-extensive interface and never expands the locked interior.

One-line wrapper around the general per-tick work bound. Instantiate that bound at ceiling $P = 1$ with the constant unit cost on every region. The hypothesis that each region costs at most $1$ is reflexivity. Simplification rewrites unit-cost tick work as the activation count and $2 \cdot 1$ as $2$.

Activation-count half of the Phase-11 recognition-work localization. The module proves per-cycle activations $\le 2T$ and work $\le 2PT$, mentioning only the tick count $T$ and per-region ceiling $P$, never the population type $\iota$. That independence is the formal core of "the recognition-active fraction falls toward zero": numerator capped by cadence, denominator (volume) grows.

It sits beside schedule-independence of the scale-adaptive engine and conjugate-birth charge conservation ($\sigma = 0$ through every birth). Together those say the driven engine is literal, conserves $\sigma$ through growth, and pays a per-cycle recognition cost bounded by the eight-tick octave no matter how large reality grows. Sibling cycle-level bounds build directly on this per-tick ceiling.