Layered Observation Systems: The Five-Step Lattice and Its Digital Exhaust at Each Node

The Five-Step Methodology of [tonyrao 2026c] treats each entry of the state-observation matrix as an atomic judgement: each (state, observation) pair is classified as classical or non-classical via the interference parameter κ. The Market Observation Equivalence Hypothesis [tonyrao 2026d] grounds this judgement in an isomorphism between Hidden Markov Model Bayesian update and quantum POVM measurement collapse, but only under a diagonal restriction ρ = diag(π) corresponding to fully decohered systems. The diagonal restriction is equivalent, this paper observes, to assuming that each lattice entry has no internal sub-structure—each (state, observation) pair is treated as atomic. This paper proposes that each entry of the Five-Step lattice should not be treated as atomic. Each entry (s_i, o_j) is itself a centred observation system in the sense of the Digital Exhaust framework [tonyrao 2026a]: it has a centre, peripheral factors, and a likelihood matrix Li,j that records how the centre is to be inferred from the peripherals. The result is a layered observation system: the outer layer is the Five-Step lattice, and each entry Mi,j = Li,j is itself a Digital Exhaust likelihood matrix. Three claims follow. First, the layered structure is well-defined, with variable-Ni,j per cell paralleling the variable-N_t structure of [tonyrao 2026b]. Second, the inner layer discloses structure that the outer-only summary suppresses: κᵢⱼ = f(Li,j, πi,j). Third, MOEH is recovered as a degenerate corner case in which each Li,j collapses to a scalar; lifting the atomic-entry assumption admits microscopic observation capability via an explicitly constructed induced density matrix on S × O, whose off-diagonal block (and its Frobenius norm α) the diagonal restriction excludes by construction. Part of the NSD / MOEH research programme. Third paper in the Digital Exhaust series.