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Non-Markov Corridor: Memory Windows, Process-Tensor Class, and the Minimal Nonlocal-in-Time Slot

Kenneth Searcy

Published Feb 19, 2026• 0 citations
Publisher

Abstract

Markov compression can fail even when the present reduced state matches. This paper forces the minimal bookkeeping slot required when state-only (Markov) update laws are not corridor-admissible. We formalize operational Markov failure: corridor-equivalent present reduced states can diverge under a declared future interface (single-time instrument or multi-time strategy), so any state-only update would encode a forbidden tie-break. A conditional finite-depth result follows: whenever bounded past depth suffices at the effective tier, there exists a minimal memory window k>=1 enabling refinement-stable prediction without hidden history tags. In the general adaptive setting, the corridor pressures a multi-time process-tensor / quantum-comb class object: a CP-under-extension, multilinear, coarse-graining-stable, causal/normalized functional on instrument slots. Finally, we show containment: in compressible regimes the multi-time slot reduces (up to corridor equivalence) to CPTP state-only dynamics and, under homogeneity plus generator stabilization, to the Lindblad-class semigroup slot.

DOI
10.5281/zenodo.18701817

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Markov compression can fail even when the present reduced state matches. This paper forces the minimal bookkeeping slot required when state-only (Markov) update laws are not corridor-admissible. We formalize operational Markov failure: corridor-equivalent present reduced states can diverge under a declared future interface (single-time instrument or multi-time strategy), so any state-only update would encode a forbidden tie-break. A conditional finite-depth result follows: whenever bounded past depth suffices at the effective tier, there exists a minimal memory window k>=1 enabling refinement-stable prediction without hidden history tags. In the general adaptive setting, the corridor pressures a multi-time process-tensor / quantum-comb class object: a CP-under-extension, multilinear, coarse-graining-stable, causal/normalized functional on instrument slots. Finally, we show containment: in compressible regimes the multi-time slot reduces (up to corridor equivalence) to CPTP state-only dynamics and, under homogeneity plus generator stabilization, to the Lindblad-class semigroup slot.
Effective open dynamics emerges from compression, not mystery. This paper continues a forcing sequence that derives quantum-formal structure from refinement-stable bookkeeping constraints. Building on prior results (operator corridor, measurement corridor, composite-system corridor, open-system corridor, and open-system update corridor), it shows how a subsystem can exhibit a stable open evolution while the larger ledger evolves by closed norm-preserving updates. The core pressures are: (1) closed-to-open induction (reduction of a closed composite to a subsystem), (2) open-tier no-full-recompute (effective updates cannot depend on unbounded environment history), and (3) time-homogeneous coarse-graining at the effective tier. These pressures drive the effective family of subsystem updates into a Markov/semigroup structure (up to corridor equivalence). Under a stabilized first-order content assumption at the effective tier, the evolution occupies a time-local master-equation slot. Finally, enforcing the already-derived CPTP constraints on the effective semigroup identifies the Lindblad-class (GKSL) generator slot as the conventional representation of the corridor-forced constraint class. The paper is representation-agnostic: it forces the constraint slot and clarifies what is derived versus deferred (conditions for Markov validity, non-Markov regimes, locality and symmetry constraints, and relativistic compatibility).
A forced-structure derivation of corridor-consistent measurement updating from the Born slot. This preprint extends the Forced Sequence / Corridor program beyond closed evolution. Given the Born extraction rule and the operator corridor (local, linear, norm-preserving closed updates), it derives a refinement-stable conditioning update for registered outcomes: the state updates by normalized outcome filtering under an interface-cut. The paper defines a collapse-class slot by four constraints: (1) Born-consistent outcome weights, (2) coarse-graining invariance (no partition-sensitive tie-break dependence), (3) exhaustiveness as weight preservation at the prediction tier, and (4) single-summary bookkeeping closure at this stage of the series (with mixed-state extensions deferred to composite/open-system corridors).
Locality is a bookkeeping constraint: dynamics must avoid global ledger access. This paper continues the forced “corridor” sequence and derives an interaction–locality corridor for composite open-system dynamics. Building on tensor-class composition, entanglement, CPTP update laws, and the Lindblad-class generator slot, it shows: (i) in the non-interacting case, composite generators are pressured to be additive on subsystems (no hidden coupling); (ii) interaction enters only through a minimal coupling-residual slot beyond local parts; (iii) “locality” is forced as bounded ledger access, yielding k-local / bounded-degree generator structure on an interaction graph (a constraint graph, not a spacetime embedding); (iv) marginal consistency and corridor no-signaling constrain admissible local operations and interaction terms; and (v) coarse-graining stability requires coupling descriptions to be invariant under regrouping of outcomes and subsystem refinements. The result is a refinement-stable structural slot for interaction and locality without importing metric or relativistic axioms; specific physical Hamiltonians and spacetime locality are deferred.
This paper extends the Forced Sequence / Corridor program to composite systems. After deriving the Born slot, local linear norm-preserving closed evolution (Schrodinger-class slot), and interface-cut conditioning (collapse-class slot), the next forced step is composition: how two subsystem ledgers combine without introducing forbidden tie-break structure. The paper shows that subsystem superposition compatibility, composite superposition closure over product generators, and Born-consistent factorization on product states force a tensor-class product slot for the composite carrier. This yields an entanglement slot: composite summaries need not factor into a single product, and non-factorability is the corridor definition of entanglement. The paper also derives the corridor form of local conditioning in composites: a local outcome registration updates the joint ledger by normalized filtering, changing conditional predictions for the other subsystem as bookkeeping rather than superluminal dynamics. Open-system reductions, density-matrix style bookkeeping, and interaction-generator selection are deferred to the next forced corridors.
Entanglement forces mixed states: density-class reduction is not optional. This preprint derives the Open-System Corridor as the next forced step after tensor-class composition and entanglement. Composite systems immediately create partial-access cases: a joint ledger for AB may exist while only subsystem A is accessible at an interface, with B functioning as an environment or untracked ledger. Corridor admissibility therefore requires a reduction rule that preserves all A-outcome weights under any declared A-interface without retaining B. The paper shows that entanglement makes pure amplitude summaries on A insufficient for this role, forcing an enlarged reduced-state class. The minimal forced reduced object is a density-class state: a positive, coarse-graining additive, weight-normalized quadratic-functional object rho_A that generates Born-consistent outcome weights for any A-interface. The paper then forces a trace-slot reduction (partial trace analogue) from composite density-class objects to subsystem density-class states, establishing existence and minimality of a reduction map that preserves all local A-outcome weights (with uniqueness deferred until sequential reduction compatibilities are forced). Standard formulas using operators and trace pairing are presented only as representations of the corridor-forced role, not as axioms. The next forced papers are Open-System Update Corridor (Kraus-class update slot) and Effective Dynamics Corridor (interaction-induced open evolution and Lindblad-class pressure under added continuity/Markov assumptions).
Open systems force CPTP updates: Kraus-class laws from corridor stability. This paper extends the Forced Sequence by deriving the open-system update slot required once density-class subsystem states and trace-slot reduction exist. Coarse-graining invariance forces an outcome-indexed instrument structure and a non-selective branch-sum channel. Normalization preservation requires weight preservation for exhaustive interfaces. Because accessible subsystems can be entangled with inaccessible references, extension consistency forces complete positivity rather than mere positivity. Together these constraints yield the CPTP slot for non-selective open-system updates, with Kraus-form expressions treated as conventional representations of the forced properties. A sequential reduction compatibility principle is introduced as the lever that upgrades trace-slot reduction from existence/minimality to corridor-uniqueness at the active tier.