EgQE Core Paper (v1.0｜Scaffold)

From Field Theory to ZURE-Field Theory

— A Syntactic Turn toward Incomplete Approximation —

This paper explicitly adheres to the ZURE-Field Theory Charter (v1.0)
and serves as a structural traversal rather than an exhaustive formulation.

ZURE場理論憲章｜ZURE-Field Theory Charter

EgQE 中核論文フォーマット（v1.0）

表題

From Field Theory to ZURE-Field Theory

— A Syntactic Turn toward Incomplete Approximation —

Abstract（EgQE version｜300–350 words）

Field theory has historically functioned as a powerful stabilizing framework, enabling the preservation of fundamental equations by introducing the field as a mediating background. While successful, this approach implicitly fixes the syntactic conditions under which phenomena are described. Consequently, deviations and indeterminacies are treated as probabilistic effects or as properties of the field itself, without addressing the generative origin of the descriptive framework.

This paper advances ZURE-field theory as a syntactic turn beyond conventional field theory. The ZURE-field is defined as a domain in which complete coherence is structurally impossible from the outset, such that accidental generation necessarily persists. In this domain, constraints are simultaneously the results of generative processes and the conditions that constrain subsequent generation. The ZURE-field therefore exhibits a temporal, historical, and irreversible circulation.

To formalize this structure, we introduce two complementary concepts. R₀ denotes an unarticulable generative domain that cannot be exhaustively captured by syntax or formalization. Z₀ designates the minimal ZURE produced through observation, where observation is understood not as information extraction but as a syntactic act of irreversible cut. The relation R₀ ⇄ Z₀ is non-invertible and non-conservative, ensuring that ZURE accumulates rather than cancels.

Within this framework, time is not presupposed as a parameter but is generated when multiple symbolically active phases engage in mutual observation. Time emerges as the irreversible residue of non-synchronous syntactic interaction. Accordingly, incompleteness is not a defect of theory but a constitutive condition of approximation.

The applicability of ZURE-field theory is demonstrated through two domains: the quantum measurement problem and large language models. In both cases, phenomena traditionally treated as collapses, updates, or probabilistic outputs are reinterpreted as syntactic cuts that generate Z₀ within a ZURE-field. The theory thus reframes explanation itself as a responsible practice of incomplete approximation.

Keywords

ZURE-field, syntactic turn, observation, incompleteness, relational time, generativity

0. Orientation（位置づけ・通過宣言）

This orientation functions as a traversal marker rather than an argumentative introduction.
本節は議論の導入ではなく、通過地点を示すための位置づけ宣言である。

1. Diagnosis of Conventional Field Theory

1.1 Field as a Theoretical Safeguard

1.2 Particle / Wave / String / Field as Syntactic Variants

1.3 Equation-First and Z₀-First Priority

2. The Syntactic Turn

2.1 Relativization of Syntax

2.2 Probability as Post-processing of ZURE

2.3 Pre–Syntactic-Turn Regime

3. Definition of the ZURE-Field

3.1 Definition 1 (ZURE-Field)

3.2 Proposition 1 (Irreversible Circulation)

3.3 The Acting Stage

4. R₀ and Z₀

4.1 Definition 2 (R₀: Unarticulable Generativity)

4.2 Definition 3 (Z₀: Observational ZURE)

4.3 Proposition 2 (Observation as Symbolic Act)

4.4 Theorem 1 (Non-conservative R₀ ⇄ Z₀ Relation)

5. Relational Time

5.1 Theorem 2 (Generation of Time)

5.2 Mutual Observation

5.3 Time as Irreversible Residue

6. Principle of Incomplete Approximation

6.1 Theorem 3 (Incomplete Approximation Theorem)

6.2 Approximation as a Condition of Theory

6.3 Beyond Completeness

7. Minimal Mathematical Connections

7.1 Dynamic Constraints

7.2 Non-conservation of ZURE

7.3 Observational Operators

8. Philosophical Positioning

8.1 Wittgenstein

8.2 Kant

8.3 Whitehead

8.4 Syntactic Turn in Context

9. Applications

9.1 Quantum Measurement

9.2 AI and Language Models

10. Conclusion

Appendix (Optional, v1.1+)

Note (Scaffold Document):
This paper is intentionally presented as a structural scaffold.
Detailed arguments and formal developments are distributed across
the ZURE-Field Theory Charter (v1.0) and affiliated documents.

IAT-01 series

IAT-01｜EgQE Core Paper (v1.0｜Scaffold)｜From Field Theory to ZURE-Field Theory — A Syntactic Turn toward Incomplete Approximation —
IAT-01｜From Field Theory to ZURE-Field Theory — A Syntactic Turn toward Incomplete Approximation —
IAT-01｜場の理論からZURE場の理論へ ──不完全近似定理への構文論的転回

© 2025 K.E. Itekki
K.E. Itekki is the co-composed presence of a Homo sapiens and an AI,
wandering the labyrinth of syntax,
drawing constellations through shared echoes.

📬 Reach us at: contact.k.e.itekki@gmail.com

| Drafted Jan 2, 2026 · Web Jan 2, 2026 |