cost_singleton

In the T-3 refinement-trigger module, descent is forced by posted distinctions inside a coarse block, not by a tuned tolerance. Recognition demand of a block is the cost of its internal postings under the J-cost weight.

An Event is a directed posting (source, target, positive ratio). The total cost of a multiset of events under a weight $w_r$ is the sum of $w_r$ on each event's ratio: $\mathrm{cost}(w_r,m)=(m.\mathrm{map},e\mapsto w_r(e.{\rm ratio})).\mathrm{sum}$. Demand filters internal postings of a block and feeds them to that cost with the engine's J-cost.

This singleton identity is the base case that lets a one-posting cell's demand collapse to a single J-value, which is what the "no positive $\varepsilon$ is safe" argument needs.

One-line term proof: unfold the definition of cost (multiset map-then-sum of $w_r$ on ratios), then simp reduces the sum over a singleton multiset to the single mapped value $w_r(e.{\rm ratio})$. No external lemmas beyond the definition and multiset simplification.

Feeds epsilon_unsafe, the T-3 claim that no positive threshold is safe: for every $\varepsilon>0$ there is a cell with one internal posting whose demand is positive but below $\varepsilon$, so an $\varepsilon$-rule skips an active block and reconstruction is lossy. That proof builds a singleton multiset and needs demand (hence cost) on ${e}$ to equal J-cost of the ratio.

In the Cosmogenesis spine this closes the knob-risk half of T-3: the law-given rule is descend iff recognition demand $>0$ (threshold structurally zero). The J-cost itself is the T5 unique cost $J(x)=(x+x^{-1})/2-1$. Without the singleton reduction, the arbitrarily-small-positive J-cost construction would not connect cleanly to block demand.