Dimension as a Hierarchy of Rules: A Conceptual Model of Causality, Spacetime, and Black Holes

Authors: David Yan, Zonghan Wen

Editor's note

This article presents a speculative conceptual framework rather than an established physical theory. Terms such as “rule-strings,” “time-strings,” and “large/small causal strings” are constructs introduced by the authors and should not be interpreted as objects or structures established in current physics. The JYP recognizes the originality of the young authors’ conceptual proposal and considers it an interesting hypothesis to think about further, particularly by exploring possible mathematical formulations, consequences, and potential falsifiability.

Abstract

This article questions the traditional view that "three dimensions plus time equals four dimensions." We propose a conceptual model in which dimension is not the number of spatial axes, but the hierarchy of rule-strings. Rule-strings produce large causal strings; large causal strings produce time-strings and space-strings; the intersection of time-strings and space-strings produces the "present," and within the "present," small causal strings arise. From this, we derive several core propositions: causality precedes temporal sequence; black holes contain large causal strings but lack small causal strings; white holes likewise lack small causal strings. This article does not replace relativity, but treats it as a local rule within lower-dimensional strings. Finally, we discuss the relationship between this model and existing theories, and respond to possible objections.

Keywords: dimension; rule-string; causality; spacetime; black hole; white hole

Introduction

In the traditional concept of dimension, one dimension is a line, two dimensions are a plane, three dimensions are a body, and four dimensions are the "time-body" formed by adding time to the body. But this conception raises a problem: if three dimensions to four dimensions merely adds time, where should "causality" be placed? If causality is also added, does that imply five dimensions? Is the fourth dimension thereby cancelled?

This article argues that the problem is not "adding one more dimension," but reunderstanding "dimension" itself. We propose dimension is not the number of spatial axes, but the hierarchy of rule-strings, a hierarchy of modification authority. We attempt to use the structure of "rule-string - large causal string - time-string/space-string - small causal string" to reinterpret dimension, causality, spacetime, black holes and white holes.

Conceptual Framework

1. Rule-Strings

The "string" in this article is not the physical string of string theory, but a rule unit or an object of theoretical physics. A rule-string is the rule itself.

Rule-strings are denoted Ln, and there are infinitely many. They stand in both a containment relation and a priority relation: outer contains inner (Ln ... L1), and inner prioritizes over outer (L1 > ... > Ln). The further outward, the more fundamental the rule, and the harder it is to break. The outermost Ln is then the most fundamental and hardest to break; the innermost L1 has the highest priority and applies first in concrete situations.

Two distinctions are needed: applicability priority (inner first, L1 acts first) and fundamental constraint strength (outer hardest to break, Ln most fundamental).

2. Large Causal Strings and Small Causal Strings

Rule-strings directly produce large causal strings. A large causal string means a certain rule necessarily produces one or more causal relations. Large causal strings do not depend on time-strings and space-strings, and therefore can interfere with spacetime.

Large causal strings contain small causal strings. Large causal strings produce time-strings and space-strings; the intersection of time-strings and space-strings produces the "present," and within the "present," small causal strings arise. Small causal strings are causal relations produced by the choices of a subject within the "present" woven by time-strings and space-strings. They depend on time and space, and are what we ordinarily call causality.

Therefore: Rule-string → Large causal string → Time-string and Space-string → Present → Small causal string

Large causal strings have direction, from cause to effect. Small causal strings have no independent direction; their sequence is unfolded by the intersection of time-strings and space-strings.

3. Time-Strings and Space-Strings

Time-strings and space-strings are parallel, intersecting, and co-operative. The intersection point of time-strings and space-strings is the "present," which is a spacetime event. A single intersection point is a single spacetime.

4. Dimension as a Hierarchy of Modification

Our definition of dimension is: An n-dimensional being is a modifier that can modify (n-1)-dimensional and lower beings, but cannot modify n- and higher-dimensional beings, and can be modified by (n+1)-dimensional and higher beings.

Thus here, when we modify a novel, it is three dimensions modifying two dimensions, not three dimensions becoming four dimensions. Dimension is not spatial superposition, but a hierarchy of modification authority.

Core Propositions and Corollaries

1. Causality precedes Temporal Sequence

Cause-strings and effect-strings appear in pairs. Causality itself is not defined by temporal sequence. Temporal sequence, rather, appears only after the intersection of time-strings and space-strings. Temporal sequence is the projection of causal structure after the intersection of spacetime strings, not a premise of causality itself.

2. Black Holes

Black holes still contain rule-strings and large causal strings, but lack small causal strings. Within the proposed rule-string model, we represent a black hole as having only space-strings and no time-strings; therefore time-strings and space-strings cannot intersect normally, and no "present" is produced. After an object is drawn into a black hole, it loses the "present" and remains only as a theoretical existence on space-strings. Here, we assume that rule-strings corresponding to underlying conservation rules (for example, energy conservation) remain applicable, so the object cannot vanish into nothing.

3. White Holes

White holes likewise contain large causal strings but lack small causal strings. A white hole "spits out" an object not as a process of spatial movement, but as a direct occurrence of rule interference: when the rule changes, the object directly appears outside.

4. Rule-String Repair and Modifiers

When lower-dimensional beings attempt to break rules, rule-strings repair themselves. What is repaired are those who lack the ability but wish to break through; those who have the ability and successfully break through are retained and become modifiers and alterers of rule-strings.

5. Higher and Lower Dimensions

Higher and lower dimensions are not a simple relation of superposition, but a relation of inside and outside. Higher-dimensional beings exist between rules, that is, between strings. They can read all strings, but cannot interfere with rule-strings higher than themselves. Higher-dimensional beings can modify lower-dimensional worlds, but cannot directly observe lower-dimensional worlds as we do. This is analogous to reading a novel: we can modify the plot, but cannot enter the two-dimensional world and directly observe it in that world's manner.

6. Blank Strings

When a being transcends time and space to influence the past, new strings are generated. In the newly generated strings, the original beings do not exist. Because the original causal relations change, a number of new strings equal to the original causal number are generated. In these newly generated strings, there are blank strings, which contain only rules. The being is then drawn into these strings and becomes a modifier and alterer of the rule-strings below the fundamental rules.

7. The Replay of the Universe

When a modification affects subsequent development, the modifier may be modified by a

higher-dimensional being. This also triggers a consequence: the spacetime that has been modified must replay from the Big Bang or from the beginning. If a higher-level rule-string is added, a change will occur.

Relation to Existing Theories

1. Relativity

This article does not replace relativity. Special and general relativity can be regarded as local rules within lower-dimensional strings. We use two conclusions of special relativity as evidence: time and space are unified into four-dimensional spacetime; and no object with mass can exceed the speed of light. The former is consistent with "time-strings and space-strings must intersect"; and the latter with "an object cannot escape a black hole by speed."

2. Quantum Mechanics

Small causal strings are produced by the choices of a subject, which has an analogical relation to the quantum measurement problem. But this article does not provide a formalization of quantum mechanics, nor does it claim to solve the measurement problem.

3. Causal Set Theory

Causal set theory holds that causality precedes spacetime. This article is consistent with that direction, but adds the structure of "rule-string hierarchy."

4. String Theory

The "string" in this article is not the one-dimensional physical object of string theory, but a rule unit. The names are the same, but the meanings differ, and this must be clearly distinguished.

Conclusion

This article proposes a conceptual model centered on the hierarchy of rule-strings. Its core propositions are: causality precedes temporal sequence; dimension is not the number of spatial axes, but a hierarchy of modification authority; black holes and white holes are extreme cases in which time-strings and space-strings cannot intersect normally. The model remains conceptual, lacking mathematical form and testable predictions. But it offers a dimension approach different from "adding one dimension": higher and lower dimensions are not simple superposition, but inside and outside, modifier and modified.

References

[1] Einstein, A. (1905). “Zur Elektrodynamik bewegter Körper.” Annalen der Physik, 17, 891–921. https://doi.org/10.1002/andp.19053221004

[2] Einstein, A. (1923). “On the Electrodynamics of Moving Bodies.” Translated by W. Perrett and G. B. Jeffery. In The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity, pp. 35–65. London: Methuen & Co.

[3] Dai, Y., & Dai, L. (2014). “Universal Mass-Velocity Relation and Special Relativity.” Acta Scientiarum Naturalium Universitatis Pekinensis, 50(3), 403–410.

[4] Mayer, J. R. (1842). “Bemerkungen über die Kräfte der unbelebten Natur.” Annalen der Chemie und Pharmacie, 42(2), 233–240. https://doi.org/10.1002/jlac.18420420212

[5] Maron, S. H., & Lando, J. B. (1974). Fundamentals of Physical Chemistry. New York: Macmillan.

[6] Kittel, C., Knight, W. D., & Ruderman, M. A. (1973). Mechanics. Berkeley Physics Course, Vol. 1, 2nd ed. New York: McGraw-Hill.

[7] Surya, S. (2019). “The Causal Set Approach to Quantum Gravity.” Living Reviews in Relativity, 22, Article 5. https://doi.org/10.1007/s41114-019-0023-1

---

Author Biographies

David Yan is an eighth-grade student at Chongqing No.1 Experimental Middle School. He is interested in theoretical physics and philosophy of science, and enjoys exploring questions about dimension, causality, and spacetime. This article is his first attempt to propose a conceptual framework that connects these ideas.

Zonghan Wen is an eighth-grade student at Chongqing No.1 Experimental Middle School. He is interested in physics and often discusses thought experiments with his co-author. He helped develop the model and write this article.