Faithful on Prose, Unanchored on Reasoning: A Position-Domain Calibration of the Jacobian Lens

Abstract

The Jacobian lens (J-lens) reads which tokens a language model is positioned to verbalize, and its released artifacts are being inherited as defaults. Those artifacts are fitted with a maximum sequence length of 128 tokens, a setting that appears in the distributed configuration files and not in the main text of the canonical write-up — and 128 is also the default of the published fitting script, so it propagates to independently fitted lenses as well; readers are applying the lens thousands of tokens deep. We calibrate fidelity against position on OLMo-3-32B, with the decision table, the thresholds, and the axes of both figures committed before any measurement existed. On prose — the lens's home domain — fidelity does not decay with depth. On reasoning traces it does, but the finding does not survive contact with its own coordinate system: the same measurements read as a degradation of +0.82 dex against one baseline cell and an improvement of −0.14 dex against another. We report both coordinates and treat that dependence as a primary result rather than a nuisance. Extending the prompt axis from one prompt to three, we find the deep-prompt degradation reproduces in all three, and that in two of them the decision is carried by a deep-rank measure alone while head-of-distribution rank correlation stays flat — the failure mode we had named at design time in order to guard against it. A second campaign holds a short two-hop prompt fixed and translates it in depth behind a content-free corridor of filler, so that one read-out can be followed across the fit boundary without changing what is read. Across corridor lengths from 88 to 484 tokens, the landmark-relative onset of the read-out is unchanged in every cell that passes a frozen behavioral-invariance gate. We state this as a scope-qualified absence, not a calibration: an instrument that did not break out of its domain has not thereby been calibrated there. We give the resulting problem a name: the calibration was anchored neither outside nor inside the instrument — no external oracle to check it against, and no internal origin to measure from. The two halves differ in kind. The missing external oracle is structural: an instrument that reads disposition can only be checked against next-token logits, which differ by construction from what it reads, and that carries to any disposition-reading instrument checked the same way. The missing internal origin is a single measured instance, on this lens and this axis, and we do not claim it generalizes. We also do not claim the lens is unusable: on prose it is faithful. The claim is narrower and, we think, more useful: readings taken deep inside a reasoning trace have no verifiable anchor. That absence of an anchor is not specific to this lens — anyone who inherits a disposition-reading instrument together with its default settings inherits the same problem. Give us a ruler for the order of reasoning.