HEG-10: Bounded Persistent Non-Closure (v0.2)
OS-Free Relational Dynamics:
Bounded Persistent Non-Closure as a Structural Principle
1. Introduction
Classical physical models frequently rely on implicit closure assumptions: states are fully determined, synchrony is achievable, and fixed points represent stability. However, many real systems—especially many-body and delay-driven systems—exhibit persistent non-closure without divergence. This paper proposes a minimal structural principle:
Complete synchrony does not occur in relational systems.
We formalize this using a lag function and show that stability may arise not from closure but from bounded persistent non-closure. Continuous and discrete formulations are shown to be structurally equivalent. Finally, we reinterpret many-body systems as the default physical condition and propose gravity as a geometric manifestation of persistent relational non-synchrony.
2. The Non-OS Principle
Let relational lag between two dynamically coupled entities S′ and O′ be defined as
\[\ell(t) := S′(t) - O′(t)\]We postulate the Non-OS condition:
\[\forall t,\quad \ell(t) \not\equiv 0\]That is, exact synchrony (OS) does not occur generically.
This principle does not assume divergence or instability; it simply excludes exact closure.
3. Bounded Persistent Non-Closure
We assume lag satisfies:
\[\sup_{t\ge0} |\ell(t)| < \infty\]and
\[\lim_{t\to\infty} \ell(t) \neq 0\]Lag remains bounded but non-vanishing.
We define this condition as:
\[\textbf{Bounded Persistent Non-Closure}\]This structure excludes both:
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fixed-point closure
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unbounded divergence
It characterizes systems that remain dynamically coherent without achieving synchrony.
4. Dynamical Realization: Delay Systems and Hopf Bifurcation
Consider a linear delay differential equation:
\[\dot{x}(t) = A x(t) + B x(t-\tau)\]The characteristic equation is:
\[\lambda - A - B e^{-\lambda \tau} = 0\]If
\[\lambda = i\omega\]a Hopf bifurcation occurs.
A stable limit cycle emerges:
\[\sup |x(t)| < \infty\]while
\[\lim_{t\to\infty} x(t) \neq x^*\]This provides a concrete dynamical realization of bounded persistent non-closure: stability without fixed-point closure.
5. Discrete Analogue: Residual Phase
Define discrete lag:
\[r_n := (S_n - O_n) \bmod N\]Under folding symmetry, residual phase evolves cyclically:
\[r_{n+1} = r_n + \Delta \mod N\]Residual phase remains bounded but non-zero.
Continuous lag and discrete residual phase are structurally equivalent manifestations of persistent non-closure under coarse-graining.
6. Reordering the One–Two–Many Hierarchy
Standard hierarchy:
\[\text{One-body} \rightarrow \text{Two-body} \rightarrow \text{Many-body}\]We propose the inverse structural ordering:
\[\text{Many-body (default)} \rightarrow \text{Two-body (approximation)} \rightarrow \text{One-body (limit)}\]One-body corresponds to the limiting case:
\[\ell \to 0\]Two-body solvability reflects partial lag reduction.
Many-body systems generically satisfy bounded persistent non-closure.
Thus, many-body “difficulty” is not mathematical inconvenience but structural realism.
7. Gravity as Lag-Geometry (Interpretive Extension)
In classical mechanics, gravity appears as force; in general relativity, as curvature. Within the present framework, gravity may be interpreted as the geometric imprint of persistent relational non-synchrony. If lag structures organize bounded trajectories, coherent orbital and large-scale ordering phenomena can be seen as spatial manifestations of non-vanishing relational delay. This interpretation does not modify existing field equations but repositions gravity as downstream from a more primitive relational asymmetry.
8. Discussion
The Non-OS principle redefines:
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Stability → bounded oscillation
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Time → irreversible persistence of lag
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Closure → limiting idealization
Many-body physics, delay systems, and cyclic discrete models all instantiate bounded persistent non-closure.
The framework does not replace existing physical laws but provides a structural reinterpretation applicable across dynamical regimes.
9. Conclusion
We propose that exact synchrony is not structurally realized in relational systems. Lag persists but remains bounded. Stability emerges as sustained oscillation rather than fixed closure. Continuous and discrete domains share a common remainder structure. Many-body systems constitute the physical default. Gravity may be interpreted as the geometric imprint of persistent non-synchrony.
This minimal principle offers a unifying structural lens across dynamical systems and relational ontology.
#axis-core-prelude #bounded-persistent-non-closure
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| Drafted Feb 20, 2026 · Web Feb 20, 2026 |