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

BENCHMARK.md

@@ -264,6 +264,51 @@ Representative Base2_64 results:
 | `12288/8192` | 44.41 | 7.94 | BZ |
 | `20480/2048` (`10n/n`) | 19.15 | 19.19 | Newton |
 
+### Near-balanced division — Newton balanced band (2026-05-31)
+
+Profiling the dispatch path on near-balanced (ratio ≈ 2) division at large
+divisor sizes surfaced two compounding problems, both fixed in
+`perf/newton-balanced-division`:
+
+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)`, which reaches ~`B` once the
+   chunk exceeds `2n` limbs (the `+1` limb routinely appears from the Knuth
+   normalize shift on `a ≈ 2n`). The estimate then underestimated `Q` by more
+   than 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 length boundaries
+   for non-power-of-2 divisor sizes, where the transform length doubles. The
+   constant factor compounds across the recursion depth into a **5–60× slowdown**
+   versus Newton, worst right above a power of two (`n = 2^k + 1`). Newton pads
+   once to the working size and stays flat. Fix: a Newton *balanced band* —
+   `b ≥ NEWTON_BALANCED_B (98304)` and `a ≥ 2b` — takes the near-balanced large
+   case that BZ previously owned. (Exact-power-of-2 divisor sizes, BZ's best
+   case where it ties Newton, regress ~4 %; they are rare in practice.)
+
+Dispatch wall-clock, BigMath Base2_64, M1 Max, before (`main` `d38f3c8`, → BZ)
+vs after (Newton balanced band):
+
+| shape (limbs) | divisor | before ms | after ms | speedup |
+|---|---|---:|---:|---:|
+| `200000/100000` | non-pow2 | 496.6 | 249.9 | **1.99×** |
+| `500000/250000` | non-pow2 | 5 177.7 | 588.3 | **8.80×** |
+| `1000000/500000` | non-pow2 | 10 554.7 | 1 485.7 | **7.10×** |
+| `2000000/1000000` | non-pow2 | 21 006.9 | 3 747.6 | **5.61×** |
+| `524288/262144` | pow2 (BZ best) | 656.4 | 684.0 | 0.96× |
+| `524290/262145` | 2¹⁸+1 (BZ worst) | 92 060.5 | 941.9 | **97.7×** |
+| `3000000/1000000` | ratio 3 (control) | 4 687.7 | 4 635.7 | 1.01× |
+
+![Near-balanced division dispatch — Newton balanced band vs Burnikel-Ziegler](docs/images/division_balanced_speedup.png)
+
+Harness: `tests/performance/division_balanced_bench.cpp`. Plot regenerated by
+`docs/images/make_division_balanced_plot.py`. Cross-checked against BZ
+limb-for-limb across `na ∈ {2n-1, 2n, 2n+1, 2n+2, 3n}` × `nb` near the boundary
+(120 cases) and the canonical `div_correctness` suite — all match.
+
 ---
 
 ## Decimal parse (string → BigInteger)

CMakeLists.txt

@@ -90,6 +90,9 @@ if(BIGMATH_BUILD_TESTS)
     add_executable(mul_xl_bench tests/performance/mul_xl_bench.cpp)
     target_link_libraries(mul_xl_bench PRIVATE bigmath::bigmath)
 
+    add_executable(division_balanced_bench tests/performance/division_balanced_bench.cpp)
+    target_link_libraries(division_balanced_bench PRIVATE bigmath::bigmath)
+
     add_executable(mfa_correctness tests/performance/mfa_correctness.cpp)
     target_link_libraries(mfa_correctness PRIVATE bigmath::bigmath)