# Recursive Field Persistence and Memory-Bearing Systems Retained-source reading edition · 18 October 2026 Source author: A. Keddeh. Source version: 19 July 2026. Publisher: Keddeh Systems. ## Reading context Series Paper 18 supplies the recurrence and discrete-time local-stability context used in Section 5.5 of the central study. Setting a memory coefficient to zero does not remove independently retained forcing or process noise. Local deterministic attraction is distinct from a stochastic stationarity theorem. The complete retained plain-text capture follows. Paragraph spacing and heading levels are editorial reading aids. No source wording, mathematical expression or line order has been changed; the original capture is also retained byte for byte. This is the recovered text layer, not a claim to reproduce unavailable original document layout, omitted tables or media. [Source document](https://docs.google.com/document/d/1kcRBlMBkSIufe67NCh5fl6l76Ui54daj/edit). ## Retained source KEX-WP-18 Recursive Field Persistence and Memory-Bearing Systems State-space stability, retained residue, wake topology, and cross-scale analogy ### Abstract This paper extracts the monograph’s recursive persistence branch into a state-space research program. It defines source state, motion, pressure or coupling, boundaries, retained residue, memory, and recursive equilibrium. Linearised stability is analysed through the transition Jacobian; persistence requires bounded or attracting modes under the declared environment. The same abstract recurrence can describe simulation memory, distributed systems, or physical hypotheses, but cross-domain analogy is not physical equivalence. ### 1. Problem and scope The monograph repeatedly uses S(t+Δt)=T(S,M,P,B,R), with residue feeding future equilibrium. This becomes mathematically usable only after each term has type, unit, dimension, and update law. A fixed point S* satisfies S*=T(S*,...). Local stability depends on eigenvalues of the Jacobian J=∂T/∂S at S*. Discrete-time attraction requires spectral radius below one. A wake or shell is a spatial structure requiring geometry, boundary conditions, medium, and conservation laws. The recurrence alone does not establish a physical wake in vacuum or cosmology. ### 2. Formal model S_{t+1}=T(S_t,M_t,P_t,B_t,R_t;θ) (1) R_{t+1}=λR_t+κ·residual_t+η_t (2) local persistence/stability: ρ(J)<1 (3) All symbols in these equations are local to this paper unless explicitly imported from Series Paper 01. A compatibility numeral such as zero may appear in ranks, matrices, encodings, distances, statistics, or projected outputs without becoming an untyped KEX active state. ### 3. Results and claim register ### 4. Architecture or state mapping ### 5. Application and cross-domain translation In virtual worlds residue may be a delta log; in BRAINK it may be authorised memory; in distributed systems it may be replica state; in a physical hypothesis it must be a measured field or latent variable with units. The architecture can run parameter sweeps and sensitivity analysis without promoting proxy variables to physical discovery. Proxy-to-observable mapping is a separate gate. Cross-domain reuse is accepted only as a structure-preserving mapping. Similar words or diagrams are insufficient: the mapping must preserve identity, allowed transitions, observer/environment terms where relevant, and the authoritative proof boundary. ### 6. Experimental programme ### 7. Falsifiers and failure conditions • If the recurrence is unstable where persistence is claimed, the parameterisation fails. • If a memoryless model predicts equally well, residue is unsupported. • If proxies lack measured mapping, physical interpretation remains unresolved. ### 8. Conclusion Recursive persistence is a useful common mathematical architecture. The discipline of this paper is to retain the recurrence while requiring each domain to earn its own variables, measurements, and physical interpretation. References [1] A. Keddeh, KEX_BRAINK_MONOGRAPH_DRAFT_XXIII(2), 2026. Internal research corpus. [2] Planck Collaboration, 'Planck 2018 results. VI. Cosmological parameters,' Astronomy & Astrophysics 641, A6 (2020). https://doi.org/10.1051/0004-6361/201833910 [3] KEX/BRAINK White Paper Series v1.0, KEX-WP-18, calculation receipt 31ba82c00739db6ef07e724a863dc317b885622d4eb7a1360bb7702a28b79513. CENTRAL THESIS Persistent structure can be modeled as recursive state evolution, but physical correspondence requires domain equations and measured stability parameters. Series field Value Author and origin authority A. Keddeh System lineage KEX / KEDDEH THEOREM / BRAINK / BRAINK⁶ Series KEX/BRAINK Modular White Paper Series v1.0 Publication date 19 July 2026 Prerequisite Series Paper 01 Next research route Series Paper 21 STATUS DISCIPLINE Mathematical validity, engineering performance, and physical correspondence are separate evidence lanes. This paper promotes only the claims that pass its declared gates. PAPER BOUNDARY Persistent structure can be modeled as recursive state evolution, but physical correspondence requires domain equations and measured stability parameters. ID Result Status RF-01 Recursive memory can be represented as a state-space system. FORMALISED RF-02 Stability conditions are parameter- and domain-dependent. MATHEMATICAL RF-03 Shared recurrence does not prove shared physics. BOUNDARY Component/state Function/value Evidence or interpretation Source state S current configuration Motion/coupling M,P domain dynamics Boundary B geometry and constraints Residue R retained state/memory Transition T typed update operator Falsifier observables rejects parameterisation Experiment Dataset/workload Acceptance criterion Stability Parameter sweep Predicted vs observed convergence Identifiability Synthetic known systems Recover θ and R Comparator Memoryless λ=0 model Out-of-sample improvement Physical branch Domain data Units, uncertainty, independent replication NON-AGGREGATE GATE One failed mandatory falsifier blocks promotion even if other tests score highly. Results remain at the highest evidence state for which every required gate passes.