all_r0_assignments_count

The module treats the O1 yardstick discussion as a finite combinatorial search. One starts from four candidate $B_{\mathrm{pow}}$ values and four candidate $r_0$ values, forms every sector-to-value assignment by permutation, then filters by the structural constraints used in the yardstick principle.

Here the object counted is the unfiltered $r_0$ side: allR0Assignments is the list of every permutation of the $r_0$ value pool that successfully converts to an R0Assignment. The companion length notion is ordinary finite-list length (the same Nat length used for traces elsewhere in the foundation). Downstream filters impose sum targets such as $V\cdot W+E_{\mathrm{passive}}=147$ and related structural constraints; this theorem only sizes the raw pool.

One-line wrapper: native_decide evaluates the concrete closed term allR0Assignments.length and checks equality with $24$. No lemmas are invoked beyond the definitional unfolding of the permutation list and the Nat length function.

In the Recognition yardstick story, particle masses sit on a $\varphi$-ladder with a sector-dependent yardstick; assigning the four $r_0$ candidates to sectors is part of fixing that ladder. This count anchors the O1 choice-set enumeration: the raw $r_0$ assignment space is $4!=24$ before structural cuts. The module claims that after those cuts the valid $r_0$ (and $B_{\mathrm{pow}}$) choice sets collapse to singletons, so the enumeration is the explicit finite witness behind uniqueness of the yardstick assignment. No downstream theorem currently depends on this lemma in the graph; it is a verification checkpoint for the combinatorial side of the mass-yardstick story rather than a step in the T0–T8 forcing chain.