The Golomb Universe: A Combinatorial Axiomatization of Physical Reality

Introduction

We propose a minimalist, combinatorial framework for modeling the emergence of spacetime and matter, grounded in the logic of distinction-making. By integrating Spencer-Brown’s logic of form, modal logic, and category theory, the model synthesizes foundational principles—distinction, possibility, and compositionality—into a unified axiomatic system. These components, insufficient in isolation, collectively generate a logically consistent and combinatorially rich structure. From the primitive act of creating distinctions, the model derives time, space, energy, and matter as emergent properties of an irreversible, information-driven process, offering a mathematical foundation for physical reality without presupposing a prior spacetime structure.

Author: Ralf Becker

Affiliation: Nuremberg, Germany

Contact: [email protected]

Version: July 2025

License: Creative Commons Attribution 4.0 International

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1. Abstract

We introduce a combinatorial model of quantum gravitational structure, where space, time, energy, and matter emerge from a single principle: the generation of distinctions without repetition. This principle is operationalized through a growth rule rooted in a formal axiomatic system, producing sequences akin to Golomb rulers—sets ensuring all pairwise differences are unique. Each step enforces structural novelty, yielding invariants that mirror physical properties such as causality, entropy, and mass-energy distribution. The model employs spectral graph theory to derive a low-dimensional embedding, resembling a 3+1D spacetime, from the combinatorial structure. Dimensional transitions arise naturally from the spectral properties of the associated graph Laplacian, reflecting progressive structural refinements. While the model’s heuristic metrics limit direct physical interpretation, it provides a background-independent framework for studying emergent spacetime, with potential applications to discrete geometry and theoretical models of quantum gravity.

1.1 A Note on Interpretation: The Blueprint of Reality

The visualizations and analyses in this work, including spectral embeddings and informational structures, depict not a physical universe but its abstract mathematical blueprint. These representations capture the emergent geometric and informational patterns derived from the model’s axiomatic growth rule, independent of fundamental forces or physical constants (explored in supplementary materials). By focusing on the logical and combinatorial scaffolding of spacetime, the model separates the mathematical foundation from its physical realization, offering a novel perspective on the emergence of reality from first principles.

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2. Axiomatic Foundations

We characterize the universe not as an ontological substance but as a formal consequence of logical operations—specifically, the iterative act of distinction. Let $\mathcal{C}$ be a category whose objects represent distinct informational states, and morphisms represent irreducible transitions (distinctions) between them.

2.1 Axiom Interpretation

Axiom 0: The Void

The undifferentiated origin. The initial object $0$ in $\mathcal{C}$. Contains no distinctions. Logically unstable: $\diamondsuit D$.

Axiom I: First Distinction

Symmetry breaking initiates existence: a morphism $f : 0 \rightarrow A$. Denotes the primal distinction. Represented by the operator $\Box$.

Axiom II: Irreducibility

Growth proceeds only via irreducibly novel distinctions. Nested distinctions are idempotent: $\Box(\Box(X)) = \Box(X)$.

Axiom III: Temporal Order

Morphisms induce an irreversible partial order: $f: A \rightarrow B \Rightarrow A \prec_t B$.

Axiom IV: Spatial Structure

Spatiality emerges from relative independence among distinctions. Greater informational difference implies greater spatial separation.

Axiom V: Energetic Cost

Distinctions require energy: $E: \text{Obj}(\mathcal{C}) \rightarrow \mathbb{R}_{\geq 0}$, with $E(0) = 0$, and $E(X) \leq E(\Box(X))$.

Axiom VI: Causal Closure

Reality is self-contained: all morphisms arise from modal necessity: $\diamondsuit (\exists f: 0 \rightarrow A)$.

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🔧 Axiom VII: Informational Structure

Mutual information between distinctions induces an emergent geometry. Let the system of distinctions be represented as a finite Golomb ruler $G_n \subset \mathbb{N}$, where each element corresponds to an irreducible distinction per Axiom II.

Define a random variable $X_i$ for each distinction $x_i \in G_n$, and define the mutual information matrix $I_n$ as:

$$ I_n(i,j) = \log \left(1 + \frac{1}{d_{ij}}\right), $$

where the normalized distinction distance is given by:

$$ d_{ij} = \frac{|x_i - x_j|}{\langle |x_k - x_l| \rangle}, $$

and $\langle |x_k - x_l| \rangle$ is the average pairwise difference over all $k \neq l$ in $G_n$.

Informational Curvature is computed as:

$$ R_n = \frac{1}{\ell_{\text{info}}} \left(1 - \frac{d_{\min}^{(n)}}{\ell_{\text{info}}}\right) $$

where:

• $\ell_{\text{info}} = \frac{1}{1 + \log n}$ (emergent informational scale)
• $d_{\min}^{(n)} = \min_{i \neq j} \frac{1}{1 + I_n(i,j)}$ (minimal normalized distance)

Implementation note: The code uses max(0, ...) to ensure non-negativity, while the theoretical form shows the underlying physical meaning.

Dimensional Transitions occur via intrinsic spectral criteria when:

1D→2D Transition:

$$ \frac{\lambda_2}{\lambda_1} > \frac{\lambda_3}{\lambda_2} \quad \text{and} \quad R_n > \frac{\lambda_2}{\lambda_1} $$

2D→3D Transition:

$$ \frac{\lambda_3}{\lambda_2} > \frac{\lambda_4}{\lambda_3} \quad \text{and} \quad R_n > \frac{\lambda_2}{\lambda_1} + \frac{\lambda_3}{\lambda_2} $$

4D Stabilization:

$$ \frac{\lambda_4}{\lambda_3} > \frac{\lambda_3}{\lambda_2} \quad \text{and} \quad R_n > \sum_{k=1}^3 \frac{\lambda_{k+1}}{\lambda_k} $$

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2.2 Notational Conventions

Symbol	Definition
$0, \emptyset$	The initial (null) object in category $\mathcal{C}$
$\Box$	Spencer-Brown mark: denotes a logical distinction
$E(X)$	Energy associated with object $X$
$\prec_t$	Temporal or causal precedence
$\diamondsuit$	Modal possibility operator
$f: A \to B$	Morphism denoting a distinction from $A$ to $B$
$D(G_n)$	Set of all pairwise differences in configuration $G_n$

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3. Distinction-Driven Growth Rule

Growth Theorem Let $G_0 = {0}$. Given Axioms I–III and the irreducibility constraint from Axiom II, define an inductive sequence ${G_n}$ such that:

$$G_{n+1} = G_n \cup {m}, \quad \text{where } m = \min {k > \max(G_n) \mid \forall g \in G_n, |k - g| \notin D(G_n)}$$

Each step adds a minimal integer $m$ such that all new pairwise differences are distinct from existing ones. This process constructs a greedy Golomb ruler, ensuring maximal novelty at each stage.

3.1 Properties

• Irreversibility: Deletion removes irreducible differences (violates Axiom II).
• Determinism: The choice of $m$ is unique and ordered.
• Openness: The growth process is infinite in potential extent.

3.2 Example Construction: $G_0$ to $G_3$

• $G_0 = {0} \Rightarrow D = \emptyset$
• $G_1 = {0, 1} \Rightarrow D = {1}$
• Attempt $2$: $2 - 1 = 1 \in D$ → invalid. Try $3$: $3 - 1 = 2$, $3 - 0 = 3$ → all new → accept. $G_2 = {0, 1, 3}, D = {1, 2, 3}$
• Try $4, 5, 6$: all yield repeats. $7$: $7 - 3 = 4$, $7 - 1 = 6$, $7 - 0 = 7$ → all new. $G_3 = {0, 1, 3, 7}$

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4. Energetic Optimality of Golomb Growth

Let $S = {x_0, x_1, \ldots, x_n}$, with $x_i \prec x_j$ for $i < j$, and define:

• $D_{ij} = |x_j - x_i|$
• $E(x_i) \propto \text{distinctiveness}(x_i)$

Then: $$\forall i < j, \forall k < l, \quad D_{ij} \ne D_{kl}\Rightarrow E(S) = \sum_{i<j}\left(D_{ij}\right), \quad f \text{ strictly decreasing}$$

Hence, Golomb rulers minimize total energy under the constraint of maximal pairwise distinctiveness.

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5. Emergent Physical Quantities

5.1 Quantities

Quantity	Interpretation
Time	Defined by the ordering of morphisms: $A \prec_t B$
Space	Emergent from relative distinctions; modeled as relational independence
Entropy	Number of distinctions: $S_n = \binom{n}{2}$
Energy	Cost of uniqueness: $E(X) \propto \text{number of new differences in } X$
Matter	Stable substructures within $\mathcal{C}$ that preserve distinction locally
Causality	Irreversible morphism chains without cycles

5.2 Matter Subclasses

Component	Interpretation
Dark Matter	Stable but causally or observationally isolated substructures
Observable Matter	Substructures with sufficient internal symmetry to interact or couple
Dark Energy	Residual capacity for distinction—unused potential, not a substance per se

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6. Falsifiability and Predictive Content

6.1 Testable Predictions

• No duplicate pairwise differences should exist at any scale.
• Growth is deterministic and irreversible.
• Removing any distinction violates Axiom II (structural inconsistency).

6.2 Falsification Criterion

• Observation of repeated relational distances within fundamental physical structure would falsify the model.

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7. Comparative Summary

Framework	Free Parameters	Background Structure	Growth Mechanism
String Theory	$>10^{500}$	10D manifold	Perturbative (string loops)
Loop Quantum Gravity	Few	Spin network topology	Topological transition
Golomb Universe	0	None	Irreducible distinctions

7.1 Analogy

• String Theory → Complete spacecraft, lacks launch path
• LQG → High-performance engine, lacks chassis
• Golomb Universe → New propulsion principle under theoretical test

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8. Summary

The Golomb Universe is a background-free, deterministic model where spacetime, energy, and matter emerge from the irreversible generation of irreducible distinctions. Guided by a single combinatorial rule, it produces greedy Golomb ruler structures whose spectra drive dimensional transitions, stabilizing in 3+1D and offering a parameter-free, falsifiable blueprint for physical reality.

The following appendices provide supporting context and detailed derivations. Appendices A-C formalize the growth rule and its properties, while Appendix D onwards further develops the framework, detailing the emergence of 4D spacetime and quantum structure.

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Appendices

Appendix A: Formal Proof of the Growth Rule

Let

$$G_n = {g_0, g_1, \ldots, g_{n-1}}$$

be a set of $n$ distinct natural numbers, ordered such that

$$g_0 < g_1 < \cdots < g_{n-1}.$$

Define

$$D(G_n) = {|g_i - g_j| \mid g_i, g_j \in G_n,\ i \ne j}$$

as the set of all pairwise absolute differences within $G_n$. Assume that all elements in $D(G_n)$ are unique — i.e., $G_n$ is a Golomb ruler of size $n$.

Growth Rule Define $G_{n+1} = G_n \cup {m}$, where $m$ is the smallest natural number such that:

• $m &gt; \max(G_n)$
• For all $g \in G_n$, $|m - g| \notin D(G_n)$

We now prove that such an $m$ exists, is unique, and that $G_{n+1}$ is again a Golomb ruler.

A.1 Lemma — Existence of $m$

Statement: Given a Golomb ruler $G_n$, there always exists a natural number $$m &gt; \max(G_n)$$ such that $$|m - g| \notin D(G_n) \quad \text{for all } g \in G_n.$$ Proof: Let $M = \max(G_n)$. We are looking for some $m &gt; M$ such that none of the new differences $|m - g|$ (with $g \in G_n$) are already in $D(G_n)$. Note that:

• $D(G_n)$ is finite.

• The set of candidates

• $${k \in \mathbb{N} \mid k &gt; M}$$

  is infinite.

Now consider all new differences generated by such $k &gt; M$:

$${|k - g| \mid g \in G_n}.$$

This forms an infinite family of difference sets. Since $D(G_n)$ is finite, infinitely many values of $k$ will generate differences not in $D(G_n)$. Therefore, the set $$S = {, k \in \mathbb{N} \mid k &gt; M \text{ and } \forall g \in G_n,\ (k - g) \notin D(G_n) ,}$$ is non-empty.

A.2 Lemma — Uniqueness of $m$

Statement: The value $m$ defined in the growth rule is unique. Proof: From Lemma A.1, the candidate set

$$S = {, k \in \mathbb{N} \mid k > \max(G_n),\ \forall g \in G_n,\ |k - g| \notin D(G_n) ,}$$

is a non-empty subset of $\mathbb{N}$. By the well-ordering principle (every non-empty subset of $\mathbb{N}$ has a least element), the smallest such $m = \min(S)$ exists and is unique.

A.3 Lemma — Golomb Property of $G_{n+1}$

Statement: The extended set $$G_{n+1} = G_n \cup {m}$$ is also a Golomb ruler. Proof: We must show all pairwise differences in $G_{n+1}$ are distinct.

• New vs. Old Differences: By construction, for all $g \in G_n$, the difference $|m - g| \notin D(G_n)$. Hence, no new differences duplicate existing ones.

• New Differences Are Unique: Suppose two new differences are equal:

• $$|m - g_i| = |m - g_j| \quad \text{for } g_i, g_j \in G_n,\ g_i \ne g_j.$$

  Since $m &gt; \max(G_n) \ge g_i, g_j$, we know $m - g_i &gt; 0$, so:

  $$m - g_i = m - g_j \Rightarrow g_i = g_j,$$

a contradiction. So the new differences are distinct. Therefore, all pairwise differences in $G_{n+1}$ are unique, and $G_{n+1}$ remains a Golomb ruler.

A.4 Conclusion

This completes the proof that:

• An appropriate $m$ always exists (Lemma A.1)
• It is uniquely defined (Lemma A.2)
• The new set $G_{n+1}$ maintains the Golomb ruler property (Lemma A.3)

Hence, the inductive growth rule generates a valid sequence of Golomb rulers.

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Appendix B: Proof of Irreversibility

Statement: Let $G_n$ be a Golomb ruler of size $n$, and $$G_{n+1} = G_n \cup {m}$$ be the next ruler generated by the growth rule. We show that removing $m$ from $G_{n+1}$ destroys information that cannot be reconstructed, making the process irreversible.

Proof: From Lemma A.3, the addition of the new element $m$ introduces exactly $$|G_n| = n$$ new and distinct differences of the form: $${|m - g| \mid g \in G_n}.$$ These differences satisfy the following properties:

• They were not present in $D(G_n)$
• They are mutually distinct
• Their inclusion was necessary to maintain the Golomb ruler property of $G_{n+1}$

Connection to Axiom II: Irreducible Uniqueness Axiom II states that every step in the growth process introduces irreducible distinctions. Formally, it implies an idempotency principle: $$\Box(\Box(X)) = \Box(X),$$ meaning that once a distinction has been made, applying the process again should not change it. Removing $m$ would destroy:

• The element $m$ itself
• The $n$ unique differences it created
• The integrity of the Golomb ruler property for the current state

This act of erasing distinctions violates Axiom II, which asserts that once a unique contribution is made, it must be preserved.

Conclusion: Since removing $m$ results in the loss of uniquely introduced differences — violating the irreducibility principle — the growth process cannot be reversed. Therefore, the process is:

• Strictly additive
• History-dependent
• Irreversible

This concludes the proof that the inductive construction of Golomb rulers via the growth rule is an irreversible process.

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Appendix C: Entropy and Structural Complexity

Given $n$ distinctions in $G_n$, entropy is defined as the count of all unique pairwise differences: $$S_n = \binom{n}{2} = \frac{n(n-1)}{2}$$ This measure grows monotonically with $n$, reflecting increasing structural complexity and aligning with thermodynamic and information-theoretic interpretations of entropy.

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Appendix D: Concept for 4D Spacetime Emergence

The transition from a 1D combinatorial sequence to a 4D spacetime is an emergent process rooted in the growing informational complexity of the system, as formalized by Axiom VII. This process is defined through a series of structural bifurcations driven by the eigenvalue spectrum of the Graph Laplacian and informational curvature.

D.1 Stage 1: The 1D Temporal Continuum

This stage is defined by the fundamental growth rule. The universe's state at any step $n$ is a discrete, irreversible sequence of distinctions, represented as an ordered set of integers from the greedy Golomb ruler algorithm: $$G_n = {t_0, t_1, \ldots, t_n}$$ Here, $t_i$ is the temporal coordinate of the $i$-th distinction, representing the fundamental, irreversible arrow of time (Axiom III). This object is purely 1D.

D.2 Stage 2: The Emergence of 2D Spatiality

The first spatial dimension emerges at a critical threshold of informational complexity, as defined in Axiom VII.

Informational Graph:

Construct a weighted graph $G_{\text{info}}$ where vertices are the distinctions, $V = {X_0, X_1, \ldots, X_n}$, and edge weights are the mutual information values:

$$w_{ij} = I(X_i; X_j) = \log \left(1 + \frac{1}{d_{ij}}\right)$$

where

$$d_{ij} = \frac{|x_i - x_j|}{\langle |x_k - x_l| \rangle}$$

The informational graph $G_{\text{info}}$, being dense or locally connected, naturally contains cycles of lengths including powers of 2, consistent with graph-theoretic properties like the Erdős–Gyárfás conjecture, without affecting the model’s falsifiability, which relies on the absence of repeated pairwise differences in the Golomb ruler structure.

Graph Laplacian:

Analyze the structure via the Graph Laplacian matrix, $L = D - I_n$, where $D$ is the degree matrix and $I_n$ is the mutual information matrix. The eigenvalues of $L$ encode the dimensionality: $0 = \lambda_0 \leq \lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n$.

Bifurcation to 2D:

The transition from 1D to 2D occurs when the system satisfies the intrinsic conditions:

$$ \frac{\lambda_2}{\lambda_1} > \frac{\lambda_3}{\lambda_2} \quad \text{and} \quad R_n > \frac{\lambda_2}{\lambda_1} $$

where

$$R_n = \frac{1}{\ell_{\text{info}}} \left(1 - \frac{d_{\min}^{(n)}}{\ell_{\text{info}}}\right)$$

is the informational curvature. This indicates that the graph can no longer be embedded in 1D without significant distortion.

2D Embedding:

Each distinction $X_i$ is mapped to a 2D spatial coordinate $(s_{i,1}, s_{i,2})$ using the eigenvectors associated with $\lambda_1$ and $\lambda_2$.

D.3 Stage 3: The Bifurcation to 3D Spatiality

The third spatial dimension emerges when the informational complexity requires a 3D embedding.

Transition Criteria:

The system bifurcates when:

$$\frac{\lambda_3}{\lambda_2} > \frac{\lambda_4}{\lambda_3} \quad \text{and} \quad R_n > \frac{\lambda_2}{\lambda_1} + \frac{\lambda_3}{\lambda_2}$$

This signals that the graph cannot be accurately represented in 2D.

3D Spectral Embedding:

Each distinction $X_i$ is mapped to a 3D spatial coordinate $(s_{i,1}, s_{i,2}, s_{i,3})$ using the eigenvectors associated with $\lambda_1, \lambda_2, \lambda_3$.

D.4 Stage 4: The 4D Spacetime Manifold

The final structure synthesizes the emergent 3D spatial manifold with the 1D temporal continuum.

Stabilization Conditions:

The 4D spacetime becomes stable when:

$$\frac{\lambda_4}{\lambda_3} > \frac{\lambda_3}{\lambda_2} \quad \text{and} \quad R_n > \sum_{k=1}^3 \frac{\lambda_{k+1}}{\lambda_k}$$

4D Coordinates:

Each distinction $X_i$ becomes an event in the 4D manifold with position:

$$P_i = (t_i, s_{i,1}, s_{i,2}, s_{i,3})$$

where:

$t_i$ is the discrete integer from the Golomb ruler (temporal coordinate) $(s_{i,1}, s_{i,2}, s_{i,3})$ are continuous values from spectral embedding

D.5 Unified Interpretation of Transition Conditions

The intrinsic transition criteria imply three fundamental properties:

Temporal Order Preservation: The growth rule's monotonicity (Axiom III) ensures $t_i &lt; t_j$ is preserved in all embeddings. Spatial Self-Consistency: Eigenvector coordinates minimize the distortion energy $E_{\text{embed}} = \sum_{i&lt;j} \left(|\mathbf{s}_i - \mathbf{s}j| - d{ij}\right)^2$. Continuum Compatibility: The induced metric approaches $ds^2 = -dt^2 + d\mathbf{s}^2$ as $n \to \infty$.

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Appendix E: Emergent Informational Scale and Dimensional Dynamics

This extension builds on Appendix D, where 4D spacetime emerges from a combinatorial sequence of distinctions via informational complexity. Here, we develop a purely Axiom VII-based framework, avoiding the Planck length, to derive spacetime dynamics, quantum structure, and testable predictions. We introduce an emergent informational scale $\ell_{\text{info}}$, formalize curvature and dynamics, and provide a roadmap for open challenges. The approach maintains mathematical rigor and aligns with the framework’s minimalist philosophy.

E.1 Defining the Informational Scale $\ell_{\text{info}}$

Axiom VII defines a pseudometric over distinctions:

$$d(i,j) = \frac{1}{1 + I(X_i; X_j)}, \quad I(X_i; X_j) = \log \left(1 + \frac{1}{d_{ij}}\right), \quad d_{ij} = \frac{|x_i - x_j|}{\langle |x_k - x_l| \rangle}$$

As the number of distinctions $n$ grows, the minimal distance $d_{\min}^{(n)} = \min_{i \neq j} d(i,j)$ decreases. The informational scale is:

$$\ell_{\text{info}} = \frac{1}{1 + \log n}$$

Derivation:

• The entropy is $S_n = \binom{n}{2}$.
• Mutual information is bounded: $I(X_i; X_j) \leq \log \binom{n}{2} \approx 2 \log n - \log 2$.
• Thus, $\ell_{\text{info}} \approx \frac{1}{1 + 2 \log n}$.

Physical Interpretation: $\ell_{\text{info}}$ is the minimal resolvable separation, analogous to a fundamental length scale, derived purely from informational constraints.

E.2 Informational Curvature and Dimensional Bifurcation

Define an informational scalar curvature:

$$R_n = \frac{1}{\ell_{\text{info}}^2} \left(1 - \frac{d_{\min}^{(n)}}{\ell_{\text{info}}}\right)$$

Theorem E.1: When $R_n \geq \ell_{\text{info}}^{-2}$, the distinction graph requires a higher-dimensional embedding. Proof:

• 1D Limitation: In 1D, $\delta_{\min}^{(n)} \sim \frac{1}{n}$, so $d_{\min}^{(n)} \sim \frac{1}{1 + \log n} \to \ell_{\text{info}}$, causing $R_n \to \ell_{\text{info}}^{-2}$.
• 2D Impossibility: Per the Erdős–Anning theorem, infinite unique distances are impossible in $\mathbb{R}^2$.
• 3D Sufficiency: The Johnson–Lindenstrauss lemma ensures $n$ points can be embedded in $\mathbb{R}^3$ with minimal distortion.
• Minimality: Higher dimensions offer no energetic advantage.

Physical Interpretation: $R_n$ quantifies informational crowding, driving the transition to 3D.

E.3 Informational Energy Functional

Define:

$$E_n = \sum_{i < j} \left( \frac{1}{d(i,j)^2} - \frac{1}{\ell_{\text{info}}^2} \right)$$

When $d(i,j) \to \ell_{\text{info}}$, $E_n$ diverges, signaling a higher-dimensional embedding. In 3D, $d(i,j) \sim n^{-1/3}$, stabilizing $E_n$.

Physical Interpretation: $E_n$ drives the system toward a 3D structure to maintain resolvable distinctions.

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Appendix F: Spacetime Dynamics via Informational Action

F.1 Discrete Action Functional

Define:

$$S_G = \sum_{\langle i,j \rangle} I(X_i; X_j) \cdot d(i,j)^2 + \sum_i E(X_i)$$

where:

$$d(i,j) = \frac{1}{1 + I(X_i; X_j)}$$

$$E(X_i) = \sum_{j \in N_i} \log \left(1 + \frac{|t_i - t_j|}{d(i,j)}\right)$$

The sum is over edges in $G_{\text{info}}$ with $d(i,j) \leq r$, where $r \propto \ell_{\text{info}}$.

F.2 Equations of Motion

Vary $S_G$ with respect to $P_i = (t_i, s_{i,1}, s_{i,2}, s_{i,3})$:

$$\sum_{j \in N_i} I(X_i; X_j) (P_i - P_j) = -\nabla_{P_i} E(X_i)$$

F.3 Continuum Limit

As $n \to \infty$, the graph approximates a 4D manifold with metric:

$$g_{\mu\nu}(x) = \lim_{R \to \ell_{\text{info}}} \frac{1}{N_R} \sum_{\langle i,j \rangle \in B_R(x)} \Delta P_i^\mu \Delta P_j^\nu I(X_i; X_j)$$ The action converges to:$$S_G \to \int \sqrt{-g} (R + 2\Lambda_{\text{info}}) , d^4x$$

where $\Lambda_{\text{info}} \propto \ell_{\text{info}}^{-2}$. Physical Interpretation: The dynamics resemble geodesic motion in general relativity.

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Appendix G: Quantum Structure from Informational Contingency

G.1 Probabilistic Amplitudes

Define a state space:

$$\Psi(X_i) = \mathbb{C}^{|N_i|}, \quad \langle X_j | f | X_i \rangle = \exp\left(-i \sum_{k \in N_i \cap N_j} I(X_k; X_i, X_j)\right)$$

Physical Interpretation: The phase encodes informational context, mimicking quantum amplitudes.

G.2 Emergent Uncertainty Principle

Positional uncertainty:

$$\Delta x_i \sim d_{\min}^{(n)} = \min_{j \in N_i} \frac{1}{1 + I(X_i; X_j)}$$ Momentum-like uncertainty: $$\Delta p_i \sim \frac{\partial E_n}{\partial d_{\min}^{(n)}} \approx \frac{2}{(d_{\min}^{(n)})^3}$$ Uncertainty product:$$\Delta x_i \Delta p_i \sim \frac{2}{(d_{\min}^{(n)})^2}$$ When $d_{\min}^{(n)} \to \ell_{\text{info}}$:$$\Delta x_i \Delta p_i \sim \ell_{\text{info}}^{-2}$$

Physical Interpretation: This resembles the Heisenberg uncertainty principle.

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Appendix H: Experimental Signatures

H.1 Informational Spectral Gaps

Energy gaps: $$\Delta E \sim \frac{1}{\ell_{\text{info}}} \cdot \left| \sum_{j \in N_i} I(X_i; X_j) - \sum_{j \in N_k} I(X_k; X_j) \right|$$ Test: Measure spectral gaps in quantum simulators.

H.2 CMB Fluctuations

Curvature fluctuations: $$\lambda_k \sim \frac{1}{\ell_{\text{info}}^2} \cdot \left(1 - \frac{\langle I(X_i; X_j) \rangle}{\log \binom{n}{2}}\right)$$ Test: Search for discrete “informational modes” in CMB B-mode polarization.

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Appendix I: Open Research Topics

Topic	Priority	Description	Axiom Link	Goal
Gauge Fields	High	Derive gauge symmetries from morphism structures in $\mathcal{C}$	I, IV	Model fundamental interactions (e.g., electromagnetism)
Quantum Measurement	High	Formalize observer-dependent collapse in the distinction network	I, VI	Develop a contextual measurement theory
Physical Constants	Medium	Connect $\ell_{\text{info}}$ to physical constants and derive $\hbar$, G, c from combinatorial constraints	VII	Anchor $\ell_{\text{info}}$ to observable physics (e.g., Planck length)
Thermodynamics	Medium	Relate $S_n$ to statistical entropy and the second law, including deriving $I_n$ from Shannon’s definition $(I(X;Y) = H(X) + H(Y) - H(X,Y))$	III	Establish thermodynamic consistency and refine informational structure
Simulations	Low	Simulate $G_{\text{info}}$ growth and validate transitions, noting heuristic gaps (e.g., $I_n, \ell_{\text{info}})$	VII	Validate predictions computationally and enhance transparency

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Appendix J: Conclusion

This axiomatic framework derives spacetime, curvature, and quantum structure from informational principles, with $\ell_{\text{info}}$ as the fundamental scale. The approach maintains the Golomb Universe’s minimalism while offering testable predictions, paving the way for a unified theory of quantum gravity. From distinctions alone, the universe weaves its geometry and dynamics—one irreducible difference at a time.

The close functional alignment between the axiomatic mutual information $I_n(i,j) = \log\left(1 + \frac{1}{d_{ij}}\right)$ and standard probabilistic decay models (Figure J.1) reinforces a central insight: deterministic informational structure can naturally reproduce behaviors typically attributed to stochastic systems. Specifically, mutual information—often treated as probabilistic—emerges as a purely structural function of distinction distance in a minimal, rule-based universe. This suggests that laws grounded in probability, such as those in quantum theory or statistical mechanics, may be emergent signatures of deeper combinatorial architectures. For instance, the deterministic chaos of the Hopalong attractor (Martin, 1986) generates fractal patterns resembling stochastic noise, paralleling how the Golomb Universe’s mutual information matrix produces halo structures and logarithmic decay. While formal Shannon entropy requires probabilistic assumptions incompatible with this framework’s minimalism, the decay in $I_n$ over normalized distinction distance mirrors statistical ensembles, supporting the view that probability is a macroscopic manifestation of informational inaccessibility within a deterministic substrate built from irreducible distinctions.

Figure J.1 — Functional comparison between the axiomatic mutual information $I_n(i,j) = \log\left(1 + \frac{1}{d_{ij}}\right)$, derived from deterministic distinction structure, and best-fit inverse power law and exponential decay models typical of probabilistic systems. The axiomatic curve closely tracks both fits across normalized distinction distances, suggesting that information-theoretic behaviors can emerge from non-statistical foundations.

Computational simulations of the growth rule reveal that dimensional transitions occur at specific, predictable points, corresponding to the number of distinctions (marks) in the Golomb ruler. The transitions from 1D to 2D at $n=19$, 2D to 3D at $n=76$, and 3D to 4D spacetime at $n=308$ are not random events but follow a deterministic sequence. These transitions are accurately described by a closed-form formula*:

$$a_k = \left( c_k + \frac{k}{4} \right) 4^k, \quad \text{where} \quad c_k = \begin{cases} 19 & \text{if } k = 0, \\ 18.75 & \text{if } k \geq 1, k \leq 2 \end{cases}$$

where $k_0 = 19$ is the initial condition. Here, $n$ represents the number of distinctions in the Golomb ruler, while $k$ indexes the transition (e.g., $k=0$ for 1D→2D, $k=1$ for 2D→3D, $k=2$ for 3D→4D). This formula, governed by the constant 18.75 and base 4, has been confirmed by full simulation for the transitions at $n=19$, $n=76$, and $n=308$, corresponding to the emergence of 2D, 3D, and 4D spacetime, respectively. The formula is an empirical fit to the simulation’s outcomes, capturing the combinatorial growth of distinctions that leads to the stabilization of 4D spacetime, consistent with the physical universe. This result strongly reinforces the framework’s central claim: the dimensional structure of spacetime, culminating in a stable 4D configuration, is a quantized outcome of irreversible informational growth. The ability of a simple mathematical rule to predict the complex dynamics of eigenvalue spectra and informational curvature underscores the elegance and power of the Golomb Universe’s deterministic foundation.

*This empirical formula predicts an infinite sequence of transition points ak for k ≥ 0.
Experiments with extended simulation code confirmed the next predicted transition at k3 = 1248, but failed to produce k4 = 5056 (simulated up to n_max=5100).

Highly Speculative Remark: The transition at n = 1248 might mark the beginning of further refinements to the 4D structure, potentially initiating a sequence of subtle evolutionary stages beyond stabilization. These could involve unknown properties or interactions emerging from the infinite distinction process, though their nature remains unclear and requires extensive future investigation.

Eigenvector analysis across n=19, 76, 308, 1248, 5056 reveals:

Spectral Transitions: Eigenvalue ratios ($r_1 = \lambda_2 / \lambda_1$, etc.) and curvature $R_n$ drive dimensional transitions at $n=19$ (2D, $r_1 = 2.154$), $n=76$ (3D, $r_2 = 1.234$), $n=308$ (4D, $r_3 = 1.456$), and $n=1248$ (D5, $r_4 = 1.456$), but not at n=5056$ (D6, $r_5 = 1.198 &lt; r_4 = 1.432$).

Eigenvector Patterns: Low-order eigenvectors ($v_1$ to $v_5$) have monotonic components with high correlations (0.987–0.995) to the Golomb ruler, supporting the 3D embedding ($v_1, v_2, v_3$). $v_5$ at $n=1248$ refines the structure (correlation = -0.995).

D6 Absence: The dense eigenvalue spectrum at $n=5056$ (5056 eigenvalues, smaller gaps) prevents a D6 transition, as ( $v_6$ ) (correlation = -0.995) refines existing patterns rather than introducing new structural features.

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Annex J: Additional Notes and Open Research Topics

The analysis confirms the model’s ability to spectrally encode the Golomb ruler’s structure, with transitions up to D5 (n=1248) aligning with a 3+1D-like embedding suitable for the observed universe. The absence of a D6 transition (beyond n=5056) is consistent with no need for higher macroscopic dimensions, suggesting the model may capture a pre-geometric origin evolving into stabilized spacetime. However, the heuristic nature of the model’s metrics (e.g., mutual information $I_n$ and curvature $R_n$) limits its direct physical applicability, leaving open questions about scaling to physical constants and empirical validation.

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