Spacetime Digital Exhaust: A Common Generalisation of Multi-Factor Inference and Sequential Collapse

We present a framework that subsumes two previously distinct inferential structures: the multi-factor Digital Exhaust framework (Rao, 2026, DOI: 10. 5281/zenodo. 19748680), in which a hidden center state is inferred at a single time slice from N peripheral factors, and the Collapse Chain (Rao, 2026, DOI: 10. 5281/zenodo. 19456756), in which a single observation drives sequential evolution of the state distribution across T time steps. We show that both are corner cases of a single spacetime conversion mechanism that indexes factors along both the factor axis (cardinality Nₜ, possibly varying with t) and the time axis (length T). The mechanism produces a likelihood tensor L of shape Nₘax × K × T, whose update rule πₜ ∝ A πₓ-₁ ⊙ ∏₈=₁^Nₜ L₈, ·, ₓ reduces to multi-factor inference when T=1 and A=I, and to the collapse-chain update when Nₜ=1 for all t. The framework is not a hybrid of the two parents; rather, the two parents are degenerate slices of a single object that varies along both axes. The principal contribution is the spacetime conversion mechanism, which extends the single-slice mechanism into the temporal direction without recourse to operator-algebraic machinery. Three covariance properties follow---time-translation, factor-permutation, and joint spacetime---establishing that the framework is structural rather than parameterisation-specific. Key contributions: - Formal definition of the spacetime conversion mechanism (ι, λ) as the principal novelty- Spacetime likelihood tensor L and a single-equation update rule- Two-way degeneration theorem: parents recovered as corner cases at T=1 and Nₜ=1 respectively- Three covariance properties (time-translation, factor-permutation, joint spacetime) - Worked numerical example with K=2, T=4, varying Nₜ demonstrating the joint regime native to neither parent The paper is foundational: it establishes the mathematical structure on which subsequent work in domains, methodologies, and operational settings can build. The framework operates under three deliberate restrictions (time-homogeneous transition, no control, no branching), each identified as a separate research direction. Keywords: digital exhaust, collapse chain, sequential inference, factor-based inference, spacetime tensor, hidden Markov models, posterior update, multi-factor extension