One Inference, Four Failure Modes: Formal Models of Why Pain Location Fails

Abstract

Patient-reported pain location is diagnostically decisive for some presentations and nearly uninformative for others. A companion paper argues this is not one gradient of diagnostic utility but three distinct failures of localization. This paper gives those failures their mathematics and shows they are one object: a single Bayesian generative model failing at different nodes - the likelihood, the model class, and group- or context-dependence in that same likelihood. The count is not in dispute: the first three are failures of the inference that produces a felt location, and the fourth, added here, is a failure of reporting it. Anatomical multiplexing is a non-identifiable inverse problem: a rank-deficient referral matrix sends distinct causes to one report. Delocalized amplification is a change of generative model whose dynamics are a neural-field bifurcation, with spatial extent as the order parameter. Referred and atypical displacement is group- or context-dependence in the presentation likelihood - a probability, not a loss - whose consequence sits one layer downstream, in a decision threshold varying with group prevalence and cost. Repeated observation cannot reduce recoverable information about a fixed inferential target, an exact chain-rule identity, while practical value can fall. The fourth node is the report itself. A spatial Bayesian model factorizes mislocalization into referral blur, an anatomical offset and a precision-weighted cognitive override, and reproduces phantom-limb, mirror-box and central-post-stroke reports as regimes of one equation. It is estimable under conditions the paper states as necessary or sufficient rather than assuming them; Sec. 8 gives the design that meets them. One control is reported against the paper's own interest: the transport statistic first used to measure migration scores a stationary, spreading profile as though it had travelled.