[SandObserver]

Summary

• Replace the closest-to-centroid dot selection in paintRegions with stratified sampling: one dot per occupied cell of a grid over the country's dot footprint, at a resolution that grows until it can supply the number of dots needed.
• Draw a share of each country's dots from its capital's neighbourhood, with the share scaling down as the dot count rises, so one user reads as a point on the capital and a busy country as nationwide coverage.
• Add a CAP table of capital coordinates for the countries in the embedded dot set, at a tenth of a degree; countries with no entry fall back to their dot-cloud centroid as the anchor.
• Keep the existing dot-count formula, the active/activeInfo keying, and the poll and redraw lifecycle unchanged.

Testing
All green: backend and frontend suites, coverage gate, lint, format, both typechecks, path map and cache-busting checks. Verified the new selection against the embedded dot data for all 123 countries across connection counts from 1 to 500: exact dot counts, no duplicates, every dot inside its own country, and a full-world repaint in under a millisecond.

Checklist

• Stays within the project constraints: one container, no runtime dependencies, no build step (see CONTRIBUTING.md).
• Tests added or updated for new logic, and the full suite, lint, and typechecks pass.
• Docs updated if behavior or configuration changed.
• CHANGELOG.md updated under Unreleased if the change is user-visible.