Order Beyond Symmetry Breaking: Lag Regulation as a Structural Principle

秩序は対称性を超える

── lag制御という構造原理

要旨

近年報告された「理想非結晶」は、周期構造や対称性の破れを伴わずに、デバイ則に従う振動挙動、ボゾンピークの抑制、そしてハイパーユニフォームな密度ゆらぎを示すことが明らかになった。

この事実は、秩序を対称性縮退として理解してきた従来の物性論に再考を迫るものである。

本稿では、秩序を（1）幾何秩序、（2）統計秩序、（3）生成秩序の三類型に再分類する。幾何学的周期性は秩序の基礎ではなく、揺らぎ制御の一形態にすぎないことを示す。

さらに、秩序は物質の内在的属性としてのみ存在するのではなく、観測構文との関係において立ち現れる関係量であることを論じる。

これらを統合する構造量として、本稿は lag を導入する。lag とは、閉包内部において吸収されずに持続する更新伝播の制御状態を指す。

この枠組みにおいて、結晶は固定化されたlag構造、ガラスは不均質なlag蓄積、理想非結晶は幾何閉包を伴わない全体的lag制御として理解される。

秩序とは対称性の破れではない。
秩序とは、スケールを横断して分布するlagの制御である。

1｜序論 ― 対称性の外側へ

物性物理学において、秩序は長らく対称性の縮退によって定義されてきた。ランダウ理論以来、相転移とは対称性の低下であり、結晶は秩序の典型とされてきた。

周期構造を持たない系は、しばしば不完全な秩序、あるいは乱れとして扱われてきた。

しかし理想非結晶は、この前提を揺さぶる。周期を持たず、対称性を破らず、それでも長距離の振動整合性と揺らぎ抑制を示す。

ここで問われるのは単なる分類の問題ではない。

対称性が秩序の必要条件でないなら、秩序とは何か。

2｜秩序の再分類

秩序は少なくとも三層で捉えられる。

幾何秩序

周期性と対称操作に基づく構造秩序。

統計秩序

揺らぎの抑制や相関構造に基づく安定。

生成秩序

時間発展における更新の安定的持続。

幾何秩序は統計秩序の特殊例にすぎない。周期は揺らぎ制御の一形態である。

3｜観測構文

秩序は観測される仕方によって姿を変える。

回折は周期性を強調する。
振動スペクトルは揺らぎの分布を可視化する。
時間分解観測は更新過程の安定性を示す。

秩序は物質単体の属性ではなく、観測との関係において可視化される。

4｜lagという構造量

4.1 構造量としての lag

対称性破れを超えて秩序を再解釈するため、本稿では lag を構造量として導入する。

lag とは、閉包構造の内部において吸収されずに持続する更新伝播である。

ここで「閉包」とは、相互作用する自由度が形成する最小の自己整合的構造領域を指す（例えば局所的に安定した配置）。

振動変位、応力再配分、密度揺らぎなどの更新過程は、局所的に吸収される場合もあれば、系全体へ伝播する場合もある。

その伝播が完全に減衰することも、完全に固定されることもなく、閉包内部で構造再配分を持続的に引き起こす場合、これを lag と呼ぶ。

形式的には、構成空間 $C$ と更新作用素 $T$ を考える。

$T$ の反復適用が固定点へ収束せず、しかし $C$ 内に有界に留まるとき、その系は lag を示す。

この観点の下では、秩序は対称性の縮退ではなく、空間・時間スケールにわたる lag 分布の制御として理解される。

物理系は常に局所更新を受ける。変位、応力、密度揺らぎ。

それらが即座に吸収される場合、lagは最小である。不均質に蓄積される場合、lagは過剰となる。閉包内部で制御されたまま持続する場合、lagは秩序を生む。

結晶は固定化されたlag分布を持つ。ガラスは不均質なlag蓄積を示す。理想非結晶は周期閉包を持たず、全体的に制御されたlag分布を持つ。

4.2 lag分布の概念的可視化

対称性に基づく秩序と、lag制御に基づく秩序の差異を明確にするため、空間スケールに対するlag分布の概念図を図1に示す。

本図は幾何学的周期性による分類ではなく、閉包内部における更新伝播の制御様式の違いを可視化するものである。横軸は空間スケール、縦軸は更新が完全に吸収されず持続する度合い、すなわち有効lag強度を示す。

結晶では、lagはほぼ一定かつ低く保たれ、再配分が固定化された状態に対応する。ガラスでは、lagが不均質に蓄積し、局所的な過剰伝播が生じる。これは経験的にボゾンピークと関連づけられてきた振動異常に対応する。これに対し、理想非結晶は短距離では適度なlagを保ちつつ、長距離では揺らぎが抑制される。これは周期閉包を伴わない全体的制御状態を示す。

本図は定量モデルではなく、構造地図である。秩序を対称性の有無ではなく、lag分布の制御として理解するための視覚的補助である。

spatial scale r lag intensity L(r) Crystal (fixed lag) Glass (heterogeneous lag) Ideal non-crystal (regulated lag) Figure 1 | Conceptual lag distribution across spatial scales

図1｜空間スケールに対する概念的lag分布。
結晶は再配分の少ない固定lag構造に対応する。ガラスは局所的lag過剰を示し、ボゾンピークに対応する不均質分布を持つ。理想非結晶は周期閉包を伴わず、長距離揺らぎが抑制された全体的lag制御状態に対応する。

結論

対称性は秩序の一表現にすぎない。

より根源的には、秩序とは 更新の伝播が過不足なく制御される状態である。

秩序は対称性の破れではない。秩序とはlagの制御である。

参考文献｜References

■ 原論文

■ プレスリリース

Order Beyond Symmetry Breaking:

Lag Regulation as a Structural Principle

Abstract

Recent reports of ideal non-crystals demonstrate that coherent vibrational behavior, suppression of the boson peak, and hyperuniform density fluctuations can emerge without periodic structure or symmetry breaking. This challenges the long-standing identification of order with symmetry reduction in condensed matter physics.

We propose a conceptual reclassification of order into geometric, statistical, and generative forms, arguing that geometric periodicity represents a special stabilization regime rather than the foundation of order. We further show that order is not purely an intrinsic structural property, but a relational quantity dependent on observational syntax.

To unify these regimes, we introduce lag as a structural quantity describing the regulated persistence of update propagation within a bounded configuration. Under this framework, crystals correspond to fixed lag configurations, glasses to heterogeneous lag accumulation, and ideal non-crystals to globally regulated lag without geometric closure.

We conclude that order is more fundamentally understood as the regulation of lag distribution across scales, rather than as symmetry breaking.

1｜Introduction

1.1 Order Beyond Symmetry

Order in condensed matter physics has long been understood through the framework of symmetry breaking. Since Landau’s formulation, phases of matter have been classified according to the reduction of symmetry and the emergence of periodic structure. Crystalline solids became the paradigm of order, while amorphous materials were often characterized in relation to the absence or incompleteness of such structure.

Although statistical mechanics and fluctuation theory have refined this picture, the conceptual identification of order with symmetry has remained central. Even in cases where disorder plays a structural role, the reference point of order has typically been geometric periodicity.

Recent reports of ideal non-crystals introduce a configuration that challenges this assumption. These systems exhibit Debye-like vibrational behavior, suppression of the boson peak, and hyperuniform density fluctuations, despite lacking periodic structure and symmetry breaking. They therefore demonstrate that stability and long-range coherence can arise without geometric closure.

This raises a fundamental question:
If symmetry breaking is not necessary for order, what is?

1.2 From Structure to Regulation

The existence of such systems suggests that order may not be reducible to geometric structure. Instead, order may reflect how fluctuations and updates propagate within a bounded configuration.

Rather than classifying matter by the presence or absence of periodic symmetry, it may be more appropriate to examine how systems regulate the redistribution of local changes across scales. Under this view, symmetry becomes one manifestation of stabilization, rather than its defining principle.

To explore this possibility, we first propose a reclassification of order into geometric, statistical, and generative forms. We then examine the role of observational syntax in determining which form of order becomes visible. Finally, we introduce lag as a structural quantity describing the regulated persistence of update propagation within closure.

Through this framework, we argue that order is more fundamentally understood as the regulation of lag distribution across scales, rather than as symmetry breaking.

2｜Reclassification of Order

2.1 Geometric, Statistical, and Generative Order

If symmetry breaking is not a necessary condition for order, the concept of order must be reformulated at a more fundamental level. To this end, we distinguish three forms of order that operate at different structural layers.

2.1.1 Geometric Order

Geometric order refers to periodicity and symmetry in spatial configuration. It is characterized by translational or rotational invariance and is typically identified through diffraction patterns. Crystalline solids represent the canonical example of geometric order.

This form of order is discrete and structural: it depends on identifiable symmetry operations and fixed spatial repetition.

2.1.2 Statistical Order

Statistical order concerns the regulation of fluctuations rather than geometric repetition. It describes systems in which density or displacement correlations exhibit constrained behavior across scales, even in the absence of periodic structure.

Hyperuniform systems and materials obeying Debye-like vibrational scaling without symmetry breaking fall within this category. Here, order is reflected in the suppression of long-wavelength fluctuations and the bounded redistribution of local updates.

Statistical order does not require symmetry reduction; instead, it manifests as controlled variance.

2.1.3 Generative Order

Generative order refers to stability in the temporal evolution of configurations. It concerns how update processes propagate and reorganize structure over time. A system exhibits generative order when its update dynamics remain bounded and coherent, neither collapsing into static fixation nor diverging into uncontrolled fluctuation.

Generative order therefore addresses the regulation of transformation rather than static arrangement.

2.2 Structural Relations Between the Three Forms

These three forms are not independent categories but structurally related.

Geometric order can be understood as a specific realization of statistical order in which fluctuation suppression is implemented through periodic closure. In other words, symmetry-based order represents one stabilization regime within a broader space of fluctuation regulation.

Similarly, statistical order presupposes generative constraints: fluctuation suppression is meaningful only insofar as update processes remain bounded over time.

Thus, geometric order ⊂ statistical order ⊂ generative order, in terms of structural generality.

This hierarchical relation allows order to be understood without presupposing symmetry as its defining criterion. Instead, symmetry emerges as one possible structural outcome of regulated update propagation.

3｜Observational Syntax of Order

3.1 Order as Relational Visibility

Order is often treated as an intrinsic property of matter. However, what counts as order depends on how it is observed. Measurement does not merely detect pre-existing structure; it selects and amplifies particular stabilization regimes.

Different observational schemes correspond to different syntactic frameworks—structured procedures that determine which aspects of configuration become visible.

Under this view, order is not simply present or absent. It is rendered legible through specific observational syntax.

3.2 Diffraction and Geometric Syntax

Diffraction experiments privilege geometric periodicity. The resulting reciprocal-space peaks identify translational symmetry and long-range order in crystalline systems.

Within this syntactic framework, periodic closure becomes the primary marker of order. Systems lacking sharp Bragg peaks are therefore categorized as disordered or partially ordered.

This does not imply that such systems lack structure. Rather, it reflects the selectivity of the observational syntax.

3.3 Vibrational and Elastic Syntax

Elastic response and vibrational spectra provide a different syntactic regime. Here, order is reflected not in periodic repetition but in the distribution of low-frequency modes and the scaling of correlations.

Debye-like behavior indicates coherent propagation of vibrational updates across scales. Conversely, excess low-frequency modes—such as the boson peak—signal heterogeneous accumulation of local instability.

In this framework, statistical stabilization becomes visible even in the absence of geometric symmetry.

3.4 Dynamical Syntax

Time-resolved measurements introduce a generative perspective. Order becomes identifiable through bounded update propagation and stable reconfiguration dynamics.

A system may lack geometric periodicity while maintaining coherent dynamical redistribution. Under dynamical syntax, order corresponds to regulated transformation rather than fixed structure.

3.5 Implication: Order as Relational Quantity

These observational regimes reveal that order is not reducible to a single structural descriptor. What appears ordered under one syntax may appear disordered under another.

Therefore, order must be understood as a relational quantity: it emerges at the interface between configuration and observational procedure.

This prepares the ground for a structural quantity capable of bridging these regimes. In the following section, we introduce lag as such a unifying measure.

4｜Lag as a Structural Quantity

4.1 From Fluctuation to Propagation

In the previous sections, we distinguished geometric, statistical, and generative forms of order and showed that order depends on observational syntax. To unify these regimes without presupposing symmetry, we now introduce lag as a structural quantity.

Physical systems are continuously subject to local updates: atomic displacements, stress redistribution, density fluctuations, and energy exchange. These updates may be locally absorbed, propagate across the system, or accumulate in heterogeneous patterns.

Lag characterizes how such updates persist within a bounded configuration.

We define lag as the regulated persistence of update propagation within a closure—neither fully absorbed nor fully frozen.

4.2 Formal Characterization

Let a system be described by a configuration space $C$ and an update operator $T$, representing local structural change.

If repeated applications of $T$ converge immediately to a fixed configuration, the system exhibits minimal lag.
If repeated applications of $T$ produce unbounded or unstable divergence, lag is uncontrolled.

Lag becomes structurally meaningful when:

  1. $T$ does not collapse into a fixed point,

  2. propagation remains bounded within $C$,

  3. redistribution occurs across scales without geometric closure.

Under these conditions, lag represents a persistent but regulated redistribution regime.

This definition does not assume periodicity or symmetry.

4.3 Interpretation of Known Regimes

The concept of lag provides a reinterpretation of established physical regimes.

Crystalline solids correspond to fixed lag configurations. Local updates are rapidly redistributed through periodic structure, resulting in minimal accumulation and stable vibrational scaling.

Glasses exhibit heterogeneous lag accumulation. Local instabilities are unevenly absorbed, leading to excess low-frequency vibrational modes commonly identified as the boson peak.

Ideal non-crystals occupy an intermediate regime. They lack periodic closure but maintain globally regulated propagation. Short-range updates are allowed, yet long-wavelength fluctuations are suppressed. This corresponds to controlled lag distribution without geometric symmetry.

These distinctions are visualized conceptually in Figure 1.

spatial scale r lag intensity L(r) Crystal (fixed lag) Glass (heterogeneous lag) Ideal non-crystal (regulated lag) Figure 1 | Conceptual lag distribution across spatial scales

Figure 1 | Conceptual lag distribution across spatial scales.
Crystals correspond to fixed lag configurations with minimal redistribution. Glasses exhibit heterogeneous lag accumulation, associated with excess low-frequency modes (boson peak). Ideal non-crystals show globally regulated lag without periodic closure, characterized by suppressed long-range fluctuation.

4.4 Conceptual Visualization of Lag Distribution

To clarify the distinction between symmetry-based order and lag-based order, we introduce a conceptual representation of lag distribution across spatial scales (Figure 1).

Rather than classifying materials by geometric periodicity, the figure illustrates how different systems regulate update propagation within a closure. The horizontal axis represents spatial scale, while the vertical axis represents the effective intensity of lag—namely, the degree to which update propagation persists without being fully absorbed.

In crystalline systems, lag remains minimal and nearly fixed across scales, corresponding to a stabilized redistribution regime. In glasses, lag accumulates heterogeneously, producing localized excess propagation that is empirically associated with boson-peak-like vibrational anomalies. In contrast, ideal non-crystals exhibit moderated short-range lag combined with long-range suppression, reflecting a globally regulated but non-periodic configuration.

The figure is not intended as a quantitative model, but as a structural map: it suggests that ordered states may be understood as regimes of regulated lag distribution rather than as outcomes of symmetry reduction.

4.5 Lag Distribution Across Scales

The crucial point is that order corresponds not to symmetry operations but to the distribution of lag across spatial and temporal scales.

When lag is uniformly regulated, the system exhibits coherent propagation (Debye-like scaling).
When lag accumulates heterogeneously, vibrational anomalies emerge.
When lag is fixed through periodic closure, geometric order appears.

Thus, symmetry can be interpreted as one stabilization regime within the broader space of lag regulation.

Lag therefore provides a structural bridge between geometric and statistical descriptions of order.

5｜Conclusion

Order as Lag Regulation

The discovery of ideal non-crystals does not merely expand the classification of condensed matter. It challenges the prevailing assumption that order must be defined through symmetry reduction.

For more than a century, crystalline order has been identified with periodic structure, while deviations from periodicity were treated as disorder or incomplete order. Even within statistical mechanics, the conceptual foundation of order remained closely tied to symmetry.

Ideal non-crystals demonstrate that long-range vibrational stability, elastic coherence, and suppression of excess low-frequency modes can arise without periodic closure or symmetry breaking. This indicates that symmetry is not a necessary condition for order.

We proposed a reclassification of order into geometric, statistical, and generative forms, arguing that geometric order is a special case of statistical stabilization. We further showed that order should not be regarded as a purely intrinsic structural property, but as a relational quantity emerging through observational syntax.

Within this framework, lag was introduced as a structural quantity describing the regulated persistence of update propagation within a closure. Crystals correspond to fixed lag configurations; glasses reflect heterogeneous lag accumulation; and ideal non-crystals exhibit globally regulated lag without geometric closure.

Accordingly, order is more fundamentally understood as the regulation of lag distribution across scales, rather than as symmetry breaking. Symmetry remains an important manifestation of stabilization, but it is not the defining criterion of order.

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| Drafted Feb 18, 2026 · Web Feb 18, 2026 |