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.
