0005: Phase 1 selection: R2 by default, R4 as the alternative

Date: 2026-10-01. Status: accepted by the owner.

This is the Phase 1 decision record that PROGRAM.md §2 and §13 ask for. It selects a default recipe and one alternative, records the trade-offs that decided them, the failure cases, and what remains open. It builds on decisions 0002 (R4 as a candidate), 0003 (chordal normalization) and 0004 (polar/foot placement).

Decision

1. Default recipe: R2, the incidence recipe. Each vertex direction gets a zero of order equal to its valence, and each polar face direction a pole of order equal to its number of sides. Orders are reduced by their gcd.
2. Alternative: R4, the edge-incidence family. Use R4ve (vertices against edges) or R4fe (faces against edges); they are swapped by polarity. Present R4 as a genuinely different construction, not a variant of R2.
3. Placement: polar face directions and edge feet (decision 0004). Centroid and midpoint placement is a labelled option for strongly oblique solids, where a polar point can leave its own cell (diagnostics.placement_report).
4. Display for comparing recipes: order-tuned sharpness, k = 2 × max |order| (DisplaySettings.tuned_for), labelled as tuned. Fixed-sharpness panels remain the controlled view and are kept alongside.
5. Not selected: R1 (uniform) stays as a baseline in comparisons. The flag recipe is recorded but not pursued. R3 (small orbit weights) was not needed: no weakness of R2 or R4 that it would address turned up.

Why: the criteria of PROGRAM.md §5, reported separately

criterion	R1	R2	R4	evidence
geometric correspondence	each zero a vertex, each pole a face	same, weighted by incidence	each pole an edge; vertices or faces as zeros	E001
rotation consistency	exact	exact	exact	M4 tests, rotation_report
symmetry preservation	full group, mirrors included	full group	full group	M4, E001 (55/55)
duality (polar dual)	reciprocal function	reciprocal function	R4ve ↔ R4fe swap	decision 0004, E001
degree economy	equal to R2 on Platonic solids, 2–3.25× R2 elsewhere, but lower on the hexagonal pyramid (7 vs 8)	never above R1 except on the hexagonal pyramid: equal on the 5 Platonic solids, lower on the 5 other solids	varies: 12–96, sometimes below R2, sometimes 3×	E001
stability: deformation	continuous	continuous	continuous	E003
stability: combinatorial transitions	jumps (globally)	jumps at every transition	continuous under its matching transition (vertex truncation for R4ve, the polar one for R4fe), jumps otherwise	E002
numerical reliability	fine to degree 312	fine	fine; near zero–pole pairs at short edges	M3 tests, E001
interpretability, display	the R1/R2 relief gap is mostly display; phase winds 3× as often	readable phase	different field (correlation with R2 0.55–0.92)	E004

R2 wins on degree economy, exact reciprocal duality and the absence of near pairs. After dividing by the degree, R1 and R2 have nearly the same shape, so R1 offers nothing R2 lacks except on rare low-degree inputs such as the hexagonal pyramid (7 against 8).

R4 is kept because it is the only recipe with any transition continuity. It also reproduces three of the six baseline pieces, and its field differs from R2's in a way no display removes.

Failure cases and costs

• R2 jumps at every combinatorial transition (E002). A tiny face or a short edge changes the function by a fixed, nonzero divisor, for example +3 at a cut corner and −1 on each adjacent face.
• No member of the incidence family is continuous under an edge bevel (E002). The R4ve and R4fe jumps there are not proportional.
• R4 makes tight zero–pole pairs at short edges and small faces (E001: chordal distances 0.04–0.09 on irregular-9). They are what makes it continuous, but in a print they are a spike beside a pit.
• Polar placement can put a pole outside the face it stands for on oblique solids: at shear σ > 1 on the sheared box (E003), and on 2 of irregular-9's faces. Centroid placement avoids that, but breaks duality.
• Degree is not continuous. It counts nearly cancelling structure that is invisible in the field (E002), so degree economy must always be reported with that caveat.
• Fixed-display comparisons overstate differences between recipes of different local order (E004).

Open questions

• Full-function symmetry. Magnitude symmetry is settled. Whether f itself is invariant (not just |f|) under the symmetry group, and how phase behaves under rotation of the chart, is unexplored. The chart phase is a convention (decision 0003).
• Continuity under edge bevels. Is there any natural recipe outside the incidence lattice that is continuous there, and under all three transitions?
• Edge placement when the foot leaves its edge (irregular-9, the sheared box). Is a midpoint fallback rule better, at the cost of duality?
• Non-geometric-mean sea levels. The baseline's pole flowers normalized along the equator. If a recipe ever needs that, decision 0003 says to add a named convention.
• Novelty. No literature review has been done yet. Nothing here claims that R2 or R4 is new (PROGRAM.md §12).
• Physical evidence. Photos of the four printed baseline pieces are deferred. No piece has been printed from R2 or R4 on a non-Platonic solid.

Reopen if

• Phase 2 printing shows R4's near pairs to be unprintable at practical sizes.
• A review finds prior work that settles the choice.
• A recipe outside the incidence lattice turns out to be continuous across transitions.

Clarifications (2026-10-03, after review)

The decision itself stands. Four statements above are made precise here. The narrative (docs/research/narrative.md) carries the same corrections.

1. Unreduced multiplicities define a recipe. Identities between recipes, duality comparisons and transition comparisons refer to unreduced divisors (RecipeResult.unreduced). gcd reduction is a per-solid economy step, so the "degree" entries in the table are reduced degrees. Reduction can break continuity: reduced R4fe jumps at the raised-face transition, where the gcd changes from 1 to 2 (tests/test_families_and_transitions.py).
2. "Full group" means symmetry of the relief. What was tested is that the divisor is preserved, which is the same as |f| being invariant. The function itself transforms by a phase character: f(gx) = e^{iθ_g} f(x) for rotations and e^{iθ_g} conj f(x) for reflections. θ_g is nontrivial in several cases, for example cube roots of unity for R2 on the tetrahedron, which includes the existing tetrahedral dual (diagnostics.character_report).
3. Corpus observations are not universal.
4. Proved generally: R2 jumps under every vertex truncation, and R4ve is continuous under it.
5. Shown for the tested cube cases only: the edge-bevel and raised-face results.
6. Observations of the tested corpus, not proofs: "no near zero–pole pairs" for R2, and "the only recipe with transition continuity" for R4.
7. The order-tuned display equalizes only the highest-order features. With k = 2 μ_max, a feature of order μ has tip exponent μ / (2 μ_max). It is 1/2 only where μ = μ_max.

Erratum (2026-10-03)

The criteria table gives R4's correlation with R2 as 0.55–0.92. Across the E001 corpus it is 0.48–0.92: the dodecahedron is at 0.48. The site's claims test (tests/test_site_claims.py) caught the error. The conclusion is unchanged, and if anything stronger: R4's field differs from R2's.