license: apache-2.0
language:

• en
  tags:
• interpretability
• activation-steering
• weight-space
• mathematical-reasoning
• lora
• qwen
  pretty_name: Correctness as a Causal Weight-Space Direction (MATH)
  size_categories:
• 1M<n<10M

Correctness as a causal weight-space direction — MATH

Artifacts for the experiment: is "being correct" a single, causal direction in a language model's
weights? We train 12,976 rank-1 LoRA adapters on correct vs. wrong solutions to MATH problems with
Qwen/Qwen3-4B-Base, extract a direction from them, and show that steering the model's weights
along it causally controls accuracy on held-out MATH-500 — ablation collapses accuracy 62%→5%,
amplification lifts it to 77%, and a matched-KL null control confirms the effect is
correctness-specific.

Headline results (held-out MATH-500, never seen when building the direction)

• Necessary & sufficient. Forward−backward swing +63 pts (p≈9e-126). Full dose-response peaks at
  +8 then over-steering collapses the model (see math_steering.png).
• Correctness-specific. A same-construction null direction (split axis, orthogonal, no
  correctness label) steered at matched KL reproduces neither the gain nor the collapse.
• Construction-robust. PC1, the mean, and plain difference-of-means are the same vector
  (cos 0.99+); all steer identically.
• Cross-benchmark. The direction is cos +0.52 with the independently-built direction from a
  different math benchmark (MathNet) — shared A0 gate puts them in the same 1,105,920-dim space.

Method (1 paragraph)

Rank-1 LoRA on 252 modules with a frozen shared random gate A0 (seed 20260709, checksum
145.648114), so every adapter is one point in the same weight-space and only the trained B varies.
Per problem, a 2×2 design (class {correct,wrong} × split {1,2}, 5 traces/cell) → 4 adapters;
d_p = mean(correct) − mean(wrong), averaged/PCA'd over 3,244 held-out-from-test train problems →
the global direction. Loss masks the answer span (so the direction can't just encode the answer
token). Steering builds an adapter B = coef·d, runs it in vLLM, and grades with math_verify. All
steering comparisons are KL-matched.

Repository structure

• directions/d_bar_math_pc1.npy — the direction to steer with (also _lora, _pc2, _diffmean,
  _null). Each is a float32 vector of length 1,105,920 in the lora_B space.
• pca/ — top-20 principal components of the per-problem correctness contrast, in the same
  weight space: pca_basis.npy (20 × 1,105,920), per-problem loadings + keys, eigenvalues,
  split-half reliability, KL-per-coefficient calibration, and the module layout that splits a vector
  into lora_B blocks. PC1–PC4 clear a permutation null; PC7+ do not. Mind the sign convention in
  pca/README.md — raw pca_basis[0] is the negative of the correctness direction.
• loras_math/ — the 12,976 trained adapters (B_shard_*.npy + meta_shard_*.json), 8 shards.
• gen_all/ — the graded rollout corpus: 12,340 MATH train+test problems × 100 rollouts.
• steer_eval/{pc1,pc1_high,diffmean,null}/ — steering sweeps (per-coef pass@1/commit on MATH-500).
• steer_adapters/ — reshaped steering adapters at each coefficient.
• data/ — problem manifests (problems_math_all.jsonl w/ split+level+in_math500), the 2×2 trace
  set, counts_math.json (per-problem correct/wrong base rates), usable sets.
• code/ — full pipeline (prep → grade → build 2×2 → train adapters → build direction → steer →
  analyze → figures). code/PIPELINE.md documents it.
• FINDINGS.md — the complete writeup; math_steering.png — the dose-response + null-control figure;
  steer_examples.md — qualitative steered traces.

Reproduce / use the direction

import numpy as np
d = np.load("directions/d_bar_math_pc1.npy")   # (1105920,), reshape to per-module [d_out] blocks
# steer: LoRA with frozen gate A0 (seed 20260709) and lora_B = coef * d ; see code/steer_build_math.py

Attribution & licenses

• Problems from MATH (Hendrycks et al., 2021), MIT-licensed; MATH-500 held-out subset via
  HuggingFaceH4/MATH-500.
• Base model Qwen/Qwen3-4B-Base, Apache-2.0. Adapters and directions are derivatives of it.
• Grading via math_verify. Traces are model-generated; no human/PII content.

This is scientific-artifact release: the value is a locatable, causal geometry of correctness, not a
training recipe (as a method it is a weak open-loop cousin of GRPO).