Open question · OPEN-DS-STABLE-BASINS

Can constrained profiled values be attained?

Openpublication: open

Meaning

Wasted-cell tracking and constrained-value attainment remain unresolved beyond the established obstructions.

Scientific context

Two further questions remain about the margin-carrying regime for the profiled criterion: whether stable empirical sequences can track the degenerate configurations the theory allows, and whether the constrained optimal values are attained at all.

Statement

Resolve the two academic remainders left by DS17 on the scalar conditionally centered class: whether ordinary exchange-stable empirical sequences can track (M5)-free coincident-centroid wasted-cell configurations, and whether the constrained values v*(kappa) and v*+(kappa) are attained and one-sided continuous under their distinct DS16 conventions. The Gaussian sign-split construction proves only nonemptiness of the closed population feasible class for kappa<=1/pi. Deployment and non-centered Delta-selection are closed out of an earlier line of work by DS19 and are not part of this “Foundations” question.

Ruled out by

The (M5)-free population classification supplies neither empirical stable sequences nor attainment. Keep v*(kappa), defined by a closed full-eigenvalue floor, distinct from the strict nuisance-floor value v*+(kappa); nonemptiness proves neither attainment nor one-sided continuity.

Relationships

Related settled results

The full chain beneath it

Where it comes from

Raised by

Local graph

Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

The question in full

Inhabitation and selection of margin-retaining stable states

After DS17: (M5)-free tracking of coincident-projected-centroid wasted-cell configurations by empirical exchange-stable sequences; attainment and one-sided continuity of $v^*(\kappa)$ and $v^{*+}(\kappa)$ under their DS16 conventions (the Gaussian sign-split family proves nonemptiness for $\kappa\le1/\pi$ , not attainment). No wasted-cell state becomes deployable by retaining a nuisance floor. Target claim: OPEN-DS-STABLE-BASINS.

From OPEN_PROBLEMS.md