We propose a mathematical framework for inference problems in which a hidden center state must be estimated not from direct observation but from the peripheral factors it influences. Such problems appear in many settings—financial markets, monitored systems, diagnostic contexts—yet have been treated, to date, as domain-specific methodologies rather than instances of a common structure. We term the peripheral factor stream a Digital Exhaust of the center, following usage already in use in data science and surveillance studies, and show that Digital Exhaust admits a unified probabilistic treatment independent of domain. The central definition we propose is not of the signals themselves but of the conversion mechanism that embeds them into a common inferential structure: a pair of maps (ι, λ) that indexes domain-specific factors and assigns conditional likelihoods, producing a likelihood matrix L whose posterior update rule is identical across domains. This mechanism is the paper's principal contribution. Two invariance properties follow: the inference structure is covariant under the choice of center (center covariance), and covariant under the type of neighborhood relation that defines "peripheral" (neighborhood-type covariance). These invariances establish Digital Exhaust as a structural property of inference problems rather than an observer-specific methodology. Key contributions:- Formal definition of Digital Exhaust as the image of a conversion mechanism- The conversion mechanism (ι, λ) as principal novelty- Two invariance properties (center covariance, neighborhood-type covariance)- Worked instantiation with empirical data on a factor-based system- Catalogue of extensions identifying open directions for companion papers This paper is the first in a five-paper Digital Exhaust programme, establishing the mathematical core that subsequent papers extend in temporal (collapse chains), methodological (five-step reverse methodology), and operational (MCCO, DRF) directions.
H Y Rao (Sat,) studied this question.