HEG-10: Bounded Persistent Non-Closure (v0.3)
A Structural Note on Bounded Persistent Non-Closure in Relational Dynamics
Continuous–Discrete Correspondence and Many-Body Reordering
Abstract
This note presents a minimal structural observation concerning relational dynamics: exact synchrony does not generically occur, yet divergence is not necessary. Defining lag as ℓ(t) = S′(t) − O′(t), we consider systems in which lag remains bounded but non-vanishing. This condition—termed bounded persistent non-closure—appears naturally in delay differential systems exhibiting Hopf bifurcation and sustained oscillations. A discrete analogue is formulated via residual phase under folding symmetry, establishing structural correspondence between continuous and discrete domains. Reordering the standard one–two–many hierarchy, we suggest that many-body systems may be treated as the structural baseline, with two-body and one-body descriptions corresponding to approximation and limiting reductions of lag. As a speculative remark, gravity is briefly interpreted as a geometric manifestation of persistent relational non-synchrony. The note aims to provide a minimal cross-domain structural perspective rather than a replacement for existing physical theories.
1. Introduction
Many dynamical systems exhibit long-term coherence without fixed-point closure. Classical formulations often privilege closure, synchrony, or complete determination. However, delay systems and many-body interactions frequently display persistent relational asymmetry without instability.
This note isolates a minimal structural pattern underlying such systems.
2. Non-OS Condition
Define relational lag:
\[\ell(t) := S′(t) - O′(t)\]We consider systems satisfying:
\[\forall t,\quad \ell(t) \not\equiv 0\]Exact synchrony is not generically realized.
3. Bounded Persistent Non-Closure
Assume:
\[\sup_{t\ge0} |\ell(t)| < \infty\]and
\[\lim_{t\to\infty} \ell(t) \neq 0\]Lag remains bounded but does not vanish.
This structural condition is referred to as bounded persistent non-closure.
It excludes both fixed-point closure and unbounded divergence.
4. Dynamical Realization: Delay Systems
Consider a linear delay differential equation:
\[\dot{x}(t) = A x(t) + B x(t-\tau)\]The characteristic equation is:
\[\lambda - A - B e^{-\lambda \tau} = 0\]Under suitable parameter values, purely imaginary roots
\[\lambda = i\omega\]induce Hopf bifurcation, producing a stable limit cycle.
Such systems satisfy:
\[\sup |x(t)| < \infty \quad \text{while} \quad x(t) \not\to x^*\]This provides a canonical example of bounded persistent non-closure.
5. Discrete Analogue: Residual Phase
Define discrete lag:
\[r_n := (S_n - O_n) \bmod N\]Under folding symmetry,
\[r_{n+1} = r_n + \Delta \pmod N\]Residual phase persists without closure while remaining bounded.
Continuous lag and discrete residual phase are structurally analogous remainder structures under coarse-graining.
6. Reordering the One–Two–Many Hierarchy
The conventional pedagogical hierarchy:
One-body → Two-body → Many-body
may be structurally inverted:
Many-body → Two-body → One-body
Many-body systems generically satisfy bounded persistent non-closure.
Two-body systems may represent partial lag reduction.
One-body systems correspond to the limiting case:
\[\ell \to 0\]This reordering is interpretive rather than prescriptive.
7. Speculative Remark: Gravity as Lag-Geometry
As a speculative extension, gravitational phenomena may be interpreted as geometric manifestations of persistent relational non-synchrony. In classical mechanics gravity appears as force; in general relativity, as curvature. Within the present structural perspective, stable orbital and large-scale ordering phenomena could be read as spatially legible consequences of bounded non-closure. This interpretation does not modify existing field equations and is offered only as a conceptual bridge.
8. Discussion
Bounded persistent non-closure appears in delay systems, oscillatory dynamics, and many-body interactions. The present note does not introduce new field equations but proposes a structural viewpoint that may unify continuous and discrete relational asymmetry.
9. Conclusion
Exact synchrony is not generically realized in relational systems. Lag may persist without divergence. Stability may arise as sustained bounded oscillation rather than fixed-point closure. Many-body systems can be viewed as structurally primary, with simpler systems emerging as reductions of lag. The proposed structural perspective is minimal and intended to stimulate cross-domain dialogue.
References (Minimal)
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Hale, J. K., & Lunel, S. M. V. (1993). Introduction to Functional Differential Equations.
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Strogatz, S. H. (2015). Nonlinear Dynamics and Chaos.
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Kuramoto, Y. (1984). Chemical Oscillations, Waves, and Turbulence.
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Standard many-body reference (e.g., Anderson or equivalent)
#axis-core-prelude #bounded-persistent-non-closure
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| Drafted Feb 20, 2026 · Web Feb 20, 2026 |