Fix near-balanced division: Newton balanced band + single-block pathology

[mmurshed]

Summary

Profiling the division dispatch path on near-balanced (ratio ≈ 2) operands at large divisor sizes surfaced two compounding problems. Both fixed here.

1. Newton single-block pathology

The single-block path (na ≤ 2n+1) ran a 2n+1-limb chunk through the truncated (chunk·R) >> 2n quotient estimate. The truncation error scales with chunk/B^(2n), reaching ~B once the chunk exceeds 2n limbs — and the +1 limb routinely appears from the Knuth normalize shift on a ≈ 2n. The estimate then undershot Q past the 8-step fixup cap and bailed to quadratic FastDivision. At nb=50000 this was a 23× spike (5200 ms vs ~360 ms at neighbouring sizes).

Fix: route na > 2n through the blockwise path so every chunk stays ≤ 2n.

2. Burnikel-Ziegler non-power-of-2 blowup

BZ's recursive 2n/n halving lands its intermediate NTT multiplies just over power-of-2 transform-length boundaries for non-power-of-2 divisor sizes, doubling the FFT length. The constant factor compounds across recursion depth into a 5–60× slowdown vs Newton, worst at n = 2^k + 1. Newton pads once and stays flat.

Fix: a Newton balanced band — b ≥ NEWTON_BALANCED_B (98304) and a ≥ 2b — takes the near-balanced large case BZ previously owned. Exact-power-of-2 divisor sizes (BZ's best case, where it ties Newton) regress ~4 %; rare in practice. The existing ratio-≥3 Newton band is untouched.

Measured (Base2_64, M1 Max, dispatch wall-clock; before = main d38f3c8 → BZ)

shape (limbs)	divisor	before	after	speedup
500000/250000	non-pow2	5177.7 ms	588.3 ms	8.80×
1000000/500000	non-pow2	10554.7 ms	1485.7 ms	7.10×
2000000/1000000	non-pow2	21006.9 ms	3747.6 ms	5.61×
524290/262145	2¹⁸+1 (BZ worst)	92060.5 ms	941.9 ms	97.7×
524288/262144	pow2 (BZ best)	656.4 ms	684.0 ms	0.96×
3000000/1000000	ratio 3 (control)	4687.7 ms	4635.7 ms	1.01×

Validation

• div_correctness (canonical): pass, 0 mismatches
• unit_tests: 246 passed, 0 failed
• Newton vs BZ cross-check limb-for-limb across na ∈ {2n-1,2n,2n+1,2n+2,3n} × nb near the boundary (120 cases): all match
• Adds tests/performance/division_balanced_bench.cpp + docs/images/make_division_balanced_plot.py

🤖 Generated with Claude Code