Where to Spend the Bits: Causal Importance, Curvature, and the Rugged Allocation Problem for Correcting Ternary LLMs

Ternary-weight LLMs (w ∈ {-1,0,+1}, ≈1.58 bits) are nearly lossless on average, but a small residual error can be bought back with a tiny full-precision (FP16) budget—e.g. a rank-8 low-rank correction on a few decoder layers. Which layers should get the budget? The classical answer is curvature: spend bits where the Hessian is large (HAWQ-style). A competing hypothesis is causal importance: spend bits where the layer matters most to the loss, measured by attribution/ablation. We test both on the deployed BitNet-b1.58-2B-4T, where—because there is no usable FP twin—the only quality-adding mechanism is a trained low-rank correction. Findings, each labelled Fact/null/refuted with caveats inline. (1) The Hessian-trace criterion is worse than random at 1.58 bits: ranking layers by Hessian trace (the trace term of HAWQ-V2, not its full trace×perturbation form—which we now test directly in a controlled re-run and find only marginally better and still worse than random) is the worst strategy at every budget B≥3 (B=2 is a tie by identical allocation)—worse than random—because it concentrates the budget on high-curvature but redundant early layers. (2) Causal importance cℓ Pareto-dominates the Hessian-trace ranking on seed-means (6/7 budgets; sign-consistent across all 3 seeds for B≥5), but its edge over a random spread is modest (decisive only at mid budget): how many layers you correct dominates which. (3) A combinatorial greedy oracle beats every cheap ranking and random at every budget, so "which" does matter—but no cheap per-layer scalar (curvature, causal-importance, or init-gradient) tracks it; the gold-standard exact task importance (itself an O(L) ablation) is the best per-layer signal (mean gap 0.90 vs. random 1.97) while its scalable approximation (EAP-IG) collapses to ≈random. Two faithful greedy searches converge to disjoint layer sets of similar quality (the warm search stays within 1.1–1.3 PPL of the cold oracle at B≤3, below every cheap scalar): the objective is rugged and the optimum non-unique. (4) Honesty bounds: the perplexity recovery is largely domain-adaptation to the correction's training corpus, not a transferable quantization fix; and the downstream-accuracy half is a floor—even the oracle ceiling does not lift MMLU or GSM8K (the lone moving number is a measured output-format artefact). (5) Two actionable positives survive: at matched memory, correcting one sub-block (attention or MLP) across more layers beats both across fewer ("spread > concentrate"), and a learned gate on the correction gives consistent, near-free gains. The robust, falsifiable object is the ordering (causal > the Hessian-trace criterion, Hessian-trace < random); the magnitudes are pilot-scale and corpus-conditioned.