Toward an Ontology of Updating: The Relational Lag Principle

関係lag原理による更新存在論の再定位

Abstract

This paper proposes an ontology grounded in updating rather than substance.
Existence is redefined as the irreversible topological redistribution of relational lag under a conservation constraint.

Instead of treating lag as a scalar quantity, we formalize it as a topological distribution across relational channels, governed by a conservation law $Tr ΔW = 0$.
Within this framework, structure emerges not from entities but from patterned reweighting.
Gravity is interpreted as the spatialization of update gradients; entropy as fixation density without requiring logarithmic formalism; and identity as temporary coherence within a continuously redistributed lag field.
The $R/Z$ distinction clarifies how conservation holds generatively $R$ while appearing broken in projected trace domains $Z$.
This shift allows ontology to be re-anchored in irreversible updating rather than static being.
Existence is thus not what persists as substance, but what has been topologically updated and cannot return.

1. Introduction

Traditional ontology has largely begun with substance. Whether in classical metaphysics, modern physics, or analytic philosophy, “being” is typically presupposed as something that persists and upon which change is imposed.

Even process philosophies, which sought to overcome substantialism, often treated change as a primary characteristic of entities rather than as the generative condition of existence itself.

This paper proposes a shift: existence does not precede updating. Rather, existence is the irreversible topological trace of relational updating. What we call “being” is not a static substrate nor a temporal slice, but the stabilized residue of non-synchronous relational redistribution.

To formalize this shift, I introduce the Relational Lag Principle as the minimal structural ground of an ontology of updating.

2. Genealogy of Update-Oriented Thought

Several traditions anticipated aspects of an update-oriented ontology:

Across these traditions, two structural gaps remain:

  1. The absence of a clearly defined conservation layer.

  2. The lack of a projection mechanism distinguishing generative continuity from observable trace.

The ontology of updating proposed here addresses both.

3. The Relational Lag Principle

3.1 Fundamental Assumption

Existence presupposes neither substance nor equilibrium, but relational updating.

Let:

Existence is defined as:

The irreversible topological history of non-synchronous relational updating.

Updating does not occur in a vacuum. It is mediated by lag, defined not as delay in time, but as the structural non-simultaneity between relational states.

Lag is not a scalar quantity. It is an operator of relational redistribution.

3.2 Conservation at the Generative Layer

At the generative layer $R_0$, relational updating obeys a conservation principle.

Let $W(t)$ be a redistribution operator acting on relational space.

Minimal update:

\[W(t+1) = W(t) + \Delta W(t)\]

Conservation condition:

\[\mathrm{Tr}(\Delta W) = 0\]

This does not conserve substance. It conserves total relational redistribution.

In the continuous limit:

\[\partial_t W + \nabla \cdot J = 0\]

Lag flows, but does not vanish.

3.3 Projection and Irreversibility

Irreversibility arises not at the generative layer, but at the projection layer $Z_0$.

Let $\Pi_{Z_0}$ be the projection from generative relational continuity to the minimal trace domain.
Define the $Z_0$-projected invariant measure:

\[\Lambda_{Z_0} = \int \mathrm{Tr}\left(\Pi_{Z_0}\Delta W(t)\right) d\mu, \qquad \Delta W(t):=W(t)-W(t-1).\]

At $R_0$, conservation implies $\mathrm{Tr}(\Delta W)=0$; irreversibility becomes measurable only through projection.
$\Lambda_{Z_0}$ thus measures the density of irreversible updating as trace.

Time, in this framework, is not primary. It is the ordered projection of irreversible relational change.

4. Consequences

4.1 Entropy Reinterpreted

Entropy is not logarithmic information spread.

It is:

The fixation rate of redistributed lag.

\[E_{\text{lag}} = \frac{\Delta \text{fixed lag}}{\text{total lag}}\]

Logarithmic formulations are special cases, not foundations.

4.2 Gravity Repositioned

Gravity is not a force.

\[g_{\text{lag}} \propto \nabla W\]

It is the spatialization of lag-density gradients.

Space itself is stabilized redistribution.

4.3 Self and Identity

Identity is not substance persistence.

It is a temporarily stabilized pattern of lag concentration.

Selfhood emerges as a local freezing regime within relational redistribution.

4.4 Narrative and Boundary

Narrative does not describe existence; it constrains future redistribution.

Meaning functions as a boundary condition within updating space.

5. Philosophical Repositioning

Traditional ontology:

Being → Change

Process ontology:

Process → Events

Update ontology:

Updating → Being

Existence becomes derivative.

Substance becomes projection.

Time becomes ordered irreversibility.

Lag becomes generative asymmetry.

Being becomes derivative of conserved asymmetry.

6. Conclusion

An ontology of updating requires neither substance nor teleology.

It requires:

Ontology must therefore begin not from what is, but from how updating distributes irreversibility.

Existence is not what persists.

Existence is what has been irreversibly updated.

The Relational Lag Principle offers a minimal structural ground for this repositioning.

HEG-8｜関係lag原理による更新存在論の再定位

Appendix A

Toward a Genealogy of Updating Ontology

Although the present work formulates an ontology grounded in irreversible relational updating, it does not emerge in isolation. Various philosophical traditions have approached aspects of becoming, process, or temporalization. However, none has fully articulated updating itself as the primary ontological operation.

This brief genealogy situates updating ontology in relation to several influential streams:

1. Heraclitus and Flux

Heraclitus foregrounded change (panta rhei), but flux remained a descriptive motif rather than a structural principle. Change was observed; it was not formalized as conserved redistribution.

2. Aristotle and Actualization

Aristotle’s distinction between potentiality and actuality introduced dynamic unfolding, yet substance remained primary. Becoming was still tethered to an underlying stable ousia.

3. Process Philosophy (Whitehead)

Alfred North Whitehead’s process metaphysics moved closer to event-ontology. Actual occasions replaced substance. Yet the conservation logic of relational redistribution was not made explicit, and irreversibility was not structurally grounded.

4. Bergson and Duration

Henri Bergson emphasized durée as lived temporality. However, duration functioned as qualitative continuity rather than as a measurable redistribution principle.

5. Four-Dimensionalism

Contemporary metaphysics often models persistence as temporal parts distributed along a time axis. But time is presupposed as a container. Updating ontology instead asks: how does temporal asymmetry arise?

Distinctive Move of Updating Ontology

Updating ontology introduces three decisive shifts:

  1. Updating as primitive
    Being is not substance, nor event, nor time — but irreversible redistribution.

  2. Relational Lag Principle
    Non-synchronous relational updating generates temporal structure.

  3. Conservation without substance
    What is conserved is not matter, nor information, but the total relational redistribution.

Thus, updating ontology does not merely join process traditions.
It repositions irreversibility and conservation at the ontological foundation.

Time becomes derivative.
Substance becomes stabilized redistribution.
Identity becomes temporary coherence under lag constraints.

This marks a structural continuation — and transformation — of process-oriented thought.

Appendix B

Heidegger’s Entwurf and The Relational Lag Principle in Ontology of Updating

Martin Heidegger’s late work, especially the notion of Entwurf (“project”) as developed in Being and Time and surrounding fragments, emphasizes existence as temporal unfolding and projection rather than as a static presence. In Entwurf, Dasein does not have a ready-made “nature” — rather, it projects itself into its possibilities. This is not a becoming of substance, but a relational temporal “thrown-projection”.

This orientation is crucial:
Heidegger does not describe change in the ordinary sense.
He describes existence as the unfolding of possibilities — as projection.

In the context of updating ontology, this resonates strongly with:

But there is a further shift:

  1. Entwurf emphasizes temporal projection as existential structure
    — Heidegger implicitly privileges temporality as the horizon of Being

  2. Updating ontology makes irreversible redistribution primary
    — Time is a result of irreversible redistribution, not the prior horizon

  3. R/Z distinction can be understood as:

    • R₀: Entwurf’s temporalizing field

    • Z₀: trace domain that makes projection visible

  4. lag is not Heideggerian temporality per se
    — it is the structural non-simultaneity that generates such temporality

  5. Therefore updating ontology extends Heidegger by:

    • preserving existential insight (projection, thrownness)

    • relocating time from primary to derivative

    • grounding projection’s trace $Z₀$ in a conservation framework

On Heidegger’s terms, Dasein’s Being is disclosed in its temporal unfolding —
but updating ontology shows that this unfolding is not anchored in pre-given time;
instead it is generated out of the non-synchronous redistribution of relational states.

Thus:

Heidegger’s Entwurf anticipates the relational-temporal direction,
but updating ontology formalizes it, relocates time as derivative,
and grounds projection in a generative-conserved network of updates.

This resolves a long-standing tension in Heidegger studies:

It paves the way for a philosophically rigorous continuation of Heidegger’s insight into Dasein as project — not by reinserting traditional temporality, but by situating projection in preserved relational updating.

Heidegger’s existential analytics predispose us to understand being as temporal unfolding.
Updating ontology retains this existential insight, relocates time as emergent, and provides a structural grounding — the Relational Lag Principle — for projection as trace.
Thus, Heidegger’s Entwurf and updating ontology are not in conflict, but in continuity:
both reject fixed substance, and updating ontology completes Heidegger by placing irreversibility and conservation at the foundations of existence.

AIP-01｜存在論から構文論へ：lagへの作法と関係更新の Osborne── ハイデガー投企論の遅延更新

HEG-8｜更新存在論序説 ── 位相履歴としての存在と非可逆宇宙

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| Drafted Feb 13, 2026 · Web Feb 13, 2026 |