perf(elastic): ~1.5-1.8x faster DP alignment via integer sub-interval walk

[sipemu]

Summary

The elastic warp search (dp_grid_solve / dp_edge_weight in alignment/mod.rs) dominates all elastic FDA cost — pairwise alignment, distance matrices, and Karcher means. Profiling put ~99% of the time in the DP edge-weight inner loop. This PR makes that loop ~1.5–1.8× faster with no API change and no accuracy regression.

Main win — dp_edge_weight integer walk

The inner loop computed four i/n float divisions per sub-interval step to place breakpoint fractions, plus float max/min. Since the coprime neighborhood bounds n1, n2 ≤ 7, we instead walk the merged breakpoints in integer units of 1/(n1·n2) (integer max/min, no per-step division) and hoist the single /(n1·n2) divisor out of the loop.

Control flow and tie-handling are exact; only the dt representation changes (integer-exact rational vs float fraction), so all tolerance-based tests — including the R-validation suites — pass unchanged.

Measured (release, 20 cores)

Benchmark	Baseline	Now	Speedup
align_pair m=50	2.28 ms	1.46 ms	1.56×
align_pair m=100	10.03 ms	6.41 ms	1.56×
align_pair m=200	42.1 ms	26.7 ms	1.58×
distance_matrix n=30	139 ms	76.2 ms	1.82×
distance_matrix n=50	374 ms	205 ms	1.82×
karcher_mean n=10	109 ms	53.4 ms	2.04×
karcher_mean n=20	112 ms	69.9 ms	1.60×

Benefits every DP caller (constrained, partial_match, nd, closed) since they share dp_edge_weight / dp_grid_solve.

Supporting cleanups (correctness-preserving, perf-neutral on their own)

• dp_grid_solve reuses a thread-local scratch (cost grid + parent pointers) instead of allocating two m² buffers per alignment; compatible with the parallel outer loops (one scratch per rayon worker).
• karcher_mean caches each curve's SRSF once (invariant across iterations) rather than recomputing srsf_single every iteration in the coarse and fine loops.

Tests

238 alignment + 116 elastic + 245 R-validation tests pass unchanged; clippy clean.

🤖 Generated with Claude Code

[sipemu]

Added two follow-up micro-opts (commit 6797876):

• Scalar-on-function regression with mixed scalar/functional covariates #4 — rslope table for uniform grids: dp_edge_weight split into a shared dp_edge_weight_core taking precomputed rslope/span; on a uniform grid the slope is exactly dr/dc, so the 35 √(dr/dc) values are precomputed once per alignment instead of a sqrt+division on every one of the m²·35 edges. Non-uniform grids unchanged.
• Function-on-scalar regression and functional ANOVA #5 — hoist simpsons_weights: both distance matrices now compute the integration weights once instead of on every O(n²) pair.

Incremental effect (on top of the integer-walk commit):

Benchmark	integer-walk	+#4+#5
align_pair m=100	6.41 ms	6.14 ms (−4%)
align_pair m=200	26.7 ms	25.8 ms (−3%)
distance_matrix n=10	20.0 ms	17.7 ms (−12%)
karcher n=20	69.9 ms	61.5 ms (−12%)
karcher n=10	53.4 ms	49.4 ms (−8%)

#4 landed below estimate (~3–4% on the pure DP pair cost, not the hoped 10–20% — sqrt is cheap once the per-step divisions were already gone). Cumulative vs the original baseline is now ~1.6× on pair alignment and up to ~2.2× on karcher n=10. All tests pass; clippy clean.

[sipemu]

Added the opt-in Sakoe–Chiba band (#1) — the algorithmic lever (commit 8572b2d):

• Core: dp_grid_solve_banded restricts the fill to |tr − tc| ≤ r; band_radius(band_frac, m) maps a domain-fraction to an index radius. Off by default — existing functions forward band = None and are byte-for-byte unchanged.
• New opt-in API: elastic_align_pair_banded, elastic_distance_banded, elastic_self/cross_distance_matrix_banded, karcher_mean_banded (band applied on both coarse and fine grids).

Speedup at band_frac = 0.1 (same-run ratios):

Benchmark	unbanded	band 10%	Speedup
align_pair m=200	51.7 ms	11.25 ms	4.6×
self_distance_matrix n=50	757 ms	187 ms	4.0×

Tests: wide band reproduces the unbanded result bit-for-bit; narrow band gives a valid monotone warp with distance ≥ the unbounded optimum (constrained-optimum invariant); karcher_mean_banded warps stay non-decreasing. Full suite green; clippy (all-targets) + rustfmt clean.

This grew past the original inner-loop scope into the full perf set — happy to split the band into its own PR if you'd prefer a tighter diff.