Geometric Signatures of Post-Training in LLM Reasoning Trajectories: Separating Observability from Transferability

Abstract

Post-training can turn the same pretrained backbone into an instruction follower, a domain specialist, or a reasoning-distilled model. We ask whether these variants leave measurable internal traces and whether those traces improve inference-time decisions. We represent generated-token hidden states as layer-wise trajectories and summarize their length-residualized correctness associations in a metric-by-layer geometric signature. We evaluate observability, candidate-selection utility, and intervention efficacy separately.

In retained retrospective artifacts collected under a fixed prompt, decoder, and problem slice, matched-width Qwen2.5-derived variants exhibit different signature matrices. Retrospective bootstrap, budget-sensitivity, and scale audits reveal recurring contrasts but protocol-sensitive nearest-neighbor topology; a separately preregistered fresh confirmation chain did not qualify because of its truncation gate. The signatures are therefore descriptive, protocol-bounded diagnostics, not universal fingerprints.

Stricter controls change the downstream conclusions. On shared K=8 candidate pools, GeoVote (geometric voting) does not improve selection over majority, length, and likelihood controls. In the locked Qwen-Instruct intervention comparison, every source-derived, target-calibrated, conventional, sparse, sign-reversed, and matched-random row has a 95% paired confidence interval that includes zero relative to no steering. A separate corrected R1-Distill short-envelope rerun is also null but remains cap-saturated. Source transfer likewise fails to improve MATH-500, SVAMP, or long-context evaluations. Among six pre-declared label budgets, one five-label cell reaches +11 pp (Holm-adjusted p=0.0443), but the non-monotone curve does not exhibit a dose–response pattern and is not presented as a label-efficiency or transfer effect. We conclude that trajectory geometry provides a protocol-bounded, auditable measurement of post-training differences; selection utility and transferable control require separate evidence.