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ADR-0009: Method (B) TOI Correction Deferred from Concept Proof

Status

Accepted

Context

The original concept proof plan called for three methods: (A) smoothing, (B) TOI correction, (C) stratum-aware tracking, compared head-to-head on the disk-disk union area problem.

During the mathematical derivation of Method (B) (see docs/explanation/toi_derivation.md), a key finding emerged: the gradient jump at external tangent is second-order. At \(d = r_1 + r_2\), the area gradient is continuous -- the intersecting formula reduces exactly to the disjoint formula. This means Method (B)'s TOI correction term vanishes to first order at external tangent.

At internal tangent (\(d = |r_1 - r_2|\)), the gradient jump is genuine first-order, but Method (C) also handles this case via its stratum-aware formulas.

Separately, the boundary proximity benchmark revealed that Method (A)'s primary failure mode is smoothing bias (systematic error from temperature parameters), not gradient discontinuity. Method (C) eliminates smoothing bias entirely by using exact formulas, making it the direct comparison partner for Method (A).

Decision

Defer Method (B) implementation from the concept proof. Proceed with Method (A) vs Method (C) comparison only.

Rationale:

  • Method (B) provides no benefit at external tangent (correction is first-order zero)
  • At internal tangent, Method (C) is equally effective and has a cleaner implementation (no root-finding for TOI)
  • Reducing from 3 methods to 2 shrinks the benchmark matrix from 9 cells to 6 cells
  • The concept proof hypothesis ("stratum-aware beats smoothing") is testable with 2 methods

What is preserved

  • docs/explanation/toi_derivation.md: Complete mathematical derivation, including the second-order finding
  • src/brepax/boolean/toi.py: Skeleton module, ready for future implementation
  • Contact dynamics correspondence table in the derivation doc

Re-evaluation conditions

Method (B) should be reconsidered if:

  • 3D primitive Boolean operations produce higher-order boundary transitions where TOI correction becomes non-trivial at first order
  • Optimization trajectories require explicit boundary-crossing awareness (e.g., path-dependent stratum tracking)
  • A use case emerges where Method (C) is insufficient but Method (B)'s root-finding approach adds value

Empirical Validation

3D sphere-sphere and cylinder-sphere gradient benchmarks confirmed the deferral decision with an additional finding:

Method B does not address the primary Stratum 1 gradient error. Forced-label experiments (bypassing stratum detection, directly computing STE gradient with correct stratum=intersecting) showed that STE accuracy at near-tangent configurations degrades to 25-542% error regardless of detection quality. The root cause is the sigmoid kernel width exceeding the intersection feature size, not stratum misclassification at the boundary.

Method B corrects the gradient jump at stratum transitions. The observed error is a stratum-internal integration problem that Method B cannot address.

Interior STE accuracy is < 1% (sphere-sphere) and < 10% (cylinder-sphere) at resolution 128, and optimization converges despite near-tangent error.

See tests/benchmarks/test_gradient_accuracy_3d.py for data.

Consequences

  • Concept proof benchmark compares Method (A) vs Method (C) only
  • Gate criteria apply to this 2-method comparison
  • boolean/toi.py remains as skeleton; NotImplementedError in union_area() dispatch
  • The TOI derivation document serves as architectural knowledge for future extensions