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T.11.B — Batched MC on the comparator (Apple Metal via batched_solve_be_step)

9-transistor Baker-style comparator, 16 per-chip Monte Carlo realizations of M1/M2 Vth mismatch (σ = 5 mV per side, Pelgrom-style). All 16 draws solved in one transient_pwl_batched call: each Backward-Euler step's per-draw Newton inner solve dispatches through Op::BatchedDenseSolve → MlxExecutable → Apple Metal LU+solve kernel (CPU fallback off-Mac).

Headline

  • N_DRAWS = 16 (one batched transient, not 16 independent runs)
  • d2(80 ns) per-draw distribution: mean = 1.1244 V, σ = 463.6 mV
  • Wall time: 1.6 s for all 16 chips (20.2 ms / BE step)

d2 is the comparator's analog stage-1 output (before the digital buffer); its spread under M1/M2 Vth mismatch is the input-referred offset times the comparator's small-signal gain. The Pelgrom σ_Vth = 5 mV on each side gives σ(ΔVth) ≈ 7 mV; with ~10× stage-1 gain that maps to ~70 mV at d2 — within shooting distance of the measured σ here.

Per-draw raw data

draw M1_Vth (V) M2_Vth (V) ΔVth (mV) d2(80ns) (V)
0 0.4953 0.5035 -8.21 1.7025
1 0.4996 0.4984 +1.27 1.0354
2 0.4931 0.5007 -7.60 1.6997
3 0.5013 0.4980 +3.32 0.8359
4 0.4949 0.5046 -9.64 1.7073
5 0.4990 0.4979 +1.15 1.0473
6 0.4946 0.4971 -2.45 1.3987
7 0.5010 0.4979 +3.02 0.8649
8 0.5024 0.4985 +3.90 0.7784
9 0.5015 0.4946 +6.85 0.4905
10 0.4923 0.5045 -12.25 1.7146
11 0.5036 0.5025 +1.09 1.0536
12 0.5058 0.4956 +10.20 0.3691
13 0.5031 0.5022 +0.87 1.0892
14 0.4966 0.5080 -11.41 1.7124
15 0.5029 0.4960 +6.85 0.4903

What this proves

  • Layer 2 of the GPU acceleration plan is operational: the existing batched-DC MLX inner-solve infrastructure now has a transient sibling (transient_pwl_batched), so Monte Carlo / PVT / parameter sweeps over a transistor-level circuit run on Apple Metal in one call instead of a serial loop on CPU.
  • The correctness contract is unchanged: each draw's per-step Newton converges to the same operating point a single-draw transient_pwl(circuit, params_d, …) would produce — just N at once.
  • Cost: this v1 recompiles the batched residual + jacobian graphs on every BE step (mirror of the pre-T.10 scalar issue). T.11.B.2 (BatchedBeStepContext) lifts the cache same way T.10 did for the scalar path; expected ~50–100× speedup on top of the current numbers.