S′–O′ Lag and the Three Regimes of Observation

— A Syntactic Reclassification of Light, Gravity, and Attraction

Abstract

This paper proposes a syntactic reclassification of several foundational physical phenomena based on the notion of lag between subject (S′) and object (O′).
Lag is defined as an irreducible delay between generation and registration that does not constitute a physical force, spacetime geometry, or dynamical variable.

We show that three observational regimes naturally emerge from different modes of lag processing:
(i) pass-through lag, corresponding to the observed invariance of the speed of light;
(ii) sedimented lag, corresponding to gravitational acceleration; and
(iii) circulating lag, corresponding to stable attraction under uniform motion.

This framework preserves all empirical predictions of existing theories while reframing light-speed invariance, gravity, and attraction as effects of observational structure rather than fundamental physical primitives.
No modification of spacetime geometry or introduction of additional forces is required.

This reclassification clarifies the conceptual status of several invariant observational phenomena without altering their operational content.

1. Introduction

Foundational physical theories often rely on a small number of organizing assumptions that stabilize description before any specific dynamics are introduced.
Such assumptions are frequently presented as principles or laws, not because they describe physical causes, but because they fix the representational framework within which physical quantities can be compared.
Over time, however, the conceptual distinction between physical mechanisms and descriptive structure tends to erode, allowing organizing assumptions to acquire unintended ontological weight.

Several central phenomena in modern physics—most notably the observed invariance of the speed of light, gravitational acceleration, and stable attraction under uniform motion—are commonly treated as distinct in origin and explanatory status.
Despite their different theoretical treatments, these phenomena share a common feature: each reflects how observational differences are registered, stabilized, or rendered invariant across frames.
This paper argues that these phenomena can be reclassified at the level of observational structure, without modifying empirical predictions or introducing new physical entities.

We introduce the notion of lag as an irreducible delay between generation and registration, understood not as a dynamical quantity but as a syntactic condition of observation.
Lag does not act, propagate, or curve; it constrains how events become comparable as traces.
When treated explicitly, lag allows several apparently disparate observational phenomena to be described within a single framework.

By analyzing how lag is processed—whether it passes through observation, accumulates, or circulates—we identify three distinct observational regimes corresponding to light-speed invariance, gravitational acceleration, and uniform attraction.
The goal of this paper is not to challenge established theories, but to clarify the descriptive role played by assumptions that have historically been elevated to principles.
In doing so, we aim to restore conceptual proportionality between empirical content and the syntactic structures that support it.

2. Lag as a Syntactic Quantity

We define lag as an irreducible delay between generation and registration—that is, between the occurrence of an event and its stabilization as an observable trace.
Lag is not introduced as a physical interaction, temporal interval, or dynamical variable.
Rather, it specifies a structural condition under which observations become comparable.

Lag does not act on objects, propagate through space, or induce change.
It does not curve spacetime, transmit forces, or store energy.
Its role is purely syntactic: it constrains how events are recorded, aligned, and related within an observational framework.

Importantly, lag is not optional.
Any act of observation presupposes a distinction between what is generated and what is registered.
Lag formalizes this distinction without assigning it causal efficacy.

Lag must therefore be distinguished from quantities traditionally treated as physical:

Lag exists prior to such operational refinements.
It marks the minimal asymmetry between event and trace that makes observation possible at all.

We further assume that lag admits a lower bound, denoted Z₀.
This bound does not represent a smallest time, length, or energy scale.
Instead, Z₀ specifies the threshold below which lag becomes unobservable and cannot be isolated, compared, or synchronized.

For lag ≥ Z₀, delay may appear explicitly in observation.
For lag < Z₀, delay persists but loses representability.
Below this threshold, lag does not vanish; it becomes structurally implicit.

This distinction is essential.
Many phenomena traditionally treated as invariant or geometric emerge precisely where lag falls below the Z₀ threshold.
Rather than appearing as delay, lag is stabilized through alternative descriptive structures.

In the sections that follow, we analyze how different modes of lag processing—pass-through, sedimentation, and circulation—give rise to distinct observational regimes.

3. The S′–O′ Framework

To articulate how lag manifests in observation, we introduce a minimal relational framework consisting of a subject (S′), an object (O′), and the lag between them.

Here, S′ denotes the observing subject, understood not as a psychological agent but as the locus at which observations are registered and compared.
O′ denotes the observed object or event, not as an ontological entity but as that which becomes available to observation as a trace.

The primes indicate that neither S′ nor O′ is taken as an absolute or pre-given entity.
Both are defined only through their relational position within an observational act.

Lag is situated between S′ and O′ as a structural asymmetry.
It does not belong exclusively to either side, nor does it mediate interaction in a causal sense.
Instead, lag specifies how the relation between S′ and O′ is processed and stabilized.

The S′–O′ relation is therefore not symmetric.
Even when observational outcomes appear invariant or stable, the underlying relation retains a directional structure determined by lag.

Crucially, the S′–O′ framework does not presuppose spacetime geometry, force laws, or dynamical evolution.
It precedes such structures and constrains how they may later be introduced.

Different observational phenomena arise depending on how lag is resolved within the S′–O′ relation.
When lag passes through observation, accumulates, or circulates, distinct observational regimes emerge.

This framework allows phenomena traditionally treated as unrelated to be analyzed within a common descriptive structure.
In the following section, we classify these regimes and show how they correspond to light-speed invariance, gravitational acceleration, and uniform attraction.

4. Three Regimes of Lag Processing

Within the S′–O′ framework, observational phenomena arise according to how lag is processed.
We identify three distinct regimes: pass-through, sedimentation, and circulation.
These regimes do not correspond to different physical laws or forces, but to different syntactic resolutions of lag within observation.

4.1 S′ ≫!≫ O′ — Lag Pass-Through

In the first regime, lag is present but passes through observation without becoming representable.
The delay falls below the threshold of observability (Z₀) and therefore cannot be isolated, compared, or accumulated.

As a result, all observers register events after an effectively identical lag.
No relative difference in delay can be detected.

This regime corresponds to the observed invariance of the speed of light.
The invariance does not arise because light enforces a universal constraint, but because lag becomes observationally invisible once it passes below Z₀.

In this sense, light-speed invariance reflects the pass-through of lag rather than a physical principle governing propagation.

4.2 S′ ≪ O′ — Lag Sedimentation

In the second regime, lag does not pass through observation.
Instead, it accumulates and becomes representable as a difference in observational outcomes.

Here, delay exceeds the Z₀ threshold and appears explicitly.
The unresolved lag manifests as a persistent discrepancy between expected and registered positions of events.

This regime corresponds to gravitational acceleration as observed.
Acceleration is not introduced as a force but emerges as the observational expression of sedimented lag.

What appears as a continuous pull or curvature is, in this framework, the trace of lag that cannot be resolved through pass-through processing.

4.3 S′ ≒ O′ — Lag Circulation

In the third regime, lag neither disappears nor accumulates.
Instead, it circulates within the S′–O′ relation, forming a closed loop.

The delay is preserved but balanced.
No net accumulation occurs, and observational outcomes remain stable over time.

This regime corresponds to uniform motion under attraction.
The observed stability does not indicate the absence of lag, but its continuous circulation and cancellation.

Attraction, in this sense, is not a cause but an effect of lag stabilization.

Summary of Regimes

The three regimes may be summarized as follows:

All three arise from the same syntactic structure and differ only in how lag is resolved.

Closing Remark

What has traditionally been described as distinct physical phenomena can thus be reinterpreted as manifestations of a single observational structure.
The apparent diversity of effects reflects not multiple forces or geometries, but multiple resolutions of lag.

5. Implications and Scope

The framework presented in this paper has several implications for how foundational physical phenomena are conceptually organized.

First, it reduces the number of primitive explanatory categories.
Light-speed invariance, gravitational acceleration, and stable attraction need not be treated as independent principles, forces, or geometric effects.
They can be understood as different resolutions of a single syntactic structure governing observation.

Second, the framework clarifies the epistemic status of several long-standing assumptions.
By reclassifying these phenomena at the level of observational structure, the present approach avoids attributing causal efficacy to descriptive constraints.
This distinction helps prevent the unintended reification of principles into physical agents.

Third, the scope of the framework is deliberately limited.
It does not propose new dynamics, predict novel empirical effects, or replace existing theories.
Its contribution lies in reorganizing the conceptual roles played by assumptions already implicit in observational practice.

Finally, the framework offers a unified language for comparing phenomena that have traditionally been treated separately.
This unification occurs not through mathematical reduction, but through syntactic alignment.

6. Conclusion

This paper has proposed a syntactic reclassification of three foundational observational phenomena—light-speed invariance, gravitational acceleration, and stable attraction—using the notion of lag between subject (S′) and object (O′).

By defining lag as an irreducible delay between generation and registration, and by distinguishing three modes of lag processing—pass-through, sedimentation, and circulation—we have shown that these phenomena can be understood as different observational regimes arising from a single structural condition.

The framework preserves all empirical content of existing theories while relocating explanatory emphasis from physical primitives to observational structure.
What changes is not what is observed, but how the conditions of observation are described.

In this sense, the proposed classification does not compete with established theories.
It precedes them.

7. Limitations and Relation to Existing Theories

The present framework is intentionally non-dynamical.
It does not specify microscopic mechanisms, field equations, or spacetime metrics.
As such, it should not be read as an alternative physical theory, but as a conceptual scaffold.

Because lag is defined syntactically rather than physically, the framework does not directly address questions of causation, energy transfer, or interaction strength.
These questions remain within the domain of existing physical theories.

With respect to established frameworks such as special and general relativity, the present approach is complementary rather than adversarial.
It neither modifies their mathematical structure nor disputes their empirical success.
Instead, it clarifies the descriptive role of assumptions that have historically been elevated to principles or geometric entities.

The framework may therefore be read as a meta-level reorganization of foundational concepts, rather than as a proposal for theoretical replacement.

Appendix: Clarifying Remarks

Q1. Does lag introduce a new physical quantity?

A.
No. Lag is not introduced as a physical quantity, force, or dynamical variable.
It specifies a structural condition of observation and has no causal or energetic role.

Q2. Does this framework modify or contradict relativity?

A.
No. All empirical predictions of special and general relativity remain unchanged.
The present framework addresses the descriptive status of certain assumptions, not their operational validity.

Q3. Is spacetime curvature denied or rejected?

A.
No. Spacetime curvature is not denied as a mathematical structure.
The proposal concerns its interpretive role as a stabilizing description of unresolved lag, not its formal correctness.

SAW-11｜S′–O′ lag による三態分類｜観測構文の三態｜ミニマル定義
SAW-11｜Observation as a Relation-Update Process: Three Regimes of Lag Processing (S′–O′ Framework)｜v0.9

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| Drafted Jan 22, 2026 · Web Jan 22, 2026 |