0004: Polar face directions and edge feet as the default placement

Date: 2026-10-01. Status: accepted.

Context

Every recipe needs a point per face and, for R4, a point per edge. PROGRAM.md §7 asks for polar-dual directions to be compared with projected centroids. Decision 0002 added edge feet and projected edge midpoints.

Decision

The defaults are polar face directions (n, from the plane n·x = h) and edge feet (the foot of the perpendicular from the origin). Centroid and midpoint placements remain as explicit variants (Recipe.placed) for controlled comparison.

Why

They are the only placements that commute with polarity:

• polar face directions are the dual's vertex directions;
• an edge and its dual edge share their foot direction.

So with these defaults, R1 and R2 give exactly the reciprocal function on the polar dual, and R4ve and R4fe swap. The alternatives break both relations on every fixture where they differ from the defaults (notes/2026-10-01-recipes-and-diagnostics.md). All placements are rotation-equivariant and preserve symmetry, so those criteria do not decide between them.

Reopen if

A visual or interpretive comparison (M5/M6) shows centroid or midpoint placement reading markedly better, enough to outweigh exact duality. Record that as a design preference, not a mathematical one.