A Discrete Geometric Toy Model with Sub-Extensive Constraints for Scale-Dependent Cosmological Expansion

Authors: Bruce E. Camber and AI-assisted editing-and-review by (alphabetically)
ChatGPT, Claude, DeepSeek, Gemini, Grok, Meta, Mistral, Perplexity (3 April 2026)

Abstract

A formal mathematical treatment, this is the “chatgpt version.”
We introduce a discrete geometric toy model in which large-scale cosmological expansion emerges from iterative binary growth at the Planck scale. The model defines a sequence of configurations generated by a simple doubling rule applied to fundamental units, interpreted as Planck-scale volumetric elements. We construct a mapping between discrete iteration steps and effective cosmological observables, including a scale factor and Hubble-like parameter. While highly simplified, the framework provides a heuristic laboratory for exploring how hierarchical structure and expansion may arise from discrete microphysical rules. Limitations and potential observational connections are discussed.

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1. Introduction

One of the central open problems in theoretical physics is the reconciliation of Planck-scale discreteness with the smooth spacetime description of cosmology. While general relativity successfully describes large-scale dynamics and quantum theory governs microscopic phenomena, a unified description remains elusive.

In this work, we present a toy model designed to explore whether large-scale expansion can emerge from simple discrete geometric rules. The model is not intended as a replacement for established frameworks, but rather as a heuristic tool to investigate scale hierarchies bridging quantum and cosmological domains. No claim is made that the present framework constitutes a complete cosmological theory.

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2. Model Definition

We define a discrete sequence of configurations $S_n$​, where $n\in \mathbb{N}$ represents a discrete iteration step.

Each configuration consists of a collection of identical fundamental units, interpreted as Planck-scale volumetric elements.

2.1 Binary Growth Rule

The system evolves according to a binary rule:

$$N(n)=2^n$$

where $N(n)$ is the number of fundamental units at step $n$.

2.2 Geometric Interpretation

We associate each unit with a characteristic volume $v_P\sim l_P^3$​, where $l_P$​ is the Planck length.

The total volume is then:

$$V(n)=N(n)\cdot v_P$$

For heuristic scaling purposes, we associate the configuration with an effective isotropic radius, and define that radius:

$$R(n)\sim V(n{)}^{1\mathrm{/}3}\sim 2^{n\mathrm{/}3}{l}_P$$

2.3 Extended Growth Dynamics

We generalize the binary growth rule by introducing a correction term:$$N(n+1)=2N(n)-ϵ(n)$$

where $ϵ(n)\ge 0$ϵ(n)≥0 represents geometric or interaction constraints (e.g., packing inefficiencies or boundary effects).

A natural class of corrections is given by sub-extensive scaling:$$ϵ(n)=\kappa N(n{)}^{\alpha },0<\alpha <1$$

where $\kappa$ and α are constants and where $\kappa$ is equal to the strength of the constraint and α is equal to how strongly the constraints scale.

This form preserves monotonic growth while introducing scale-dependent deviations from ideal exponential behavior.

In the generalized model, the expansion rate acquires scale dependence. To leading order:$$H(n)\approx H_0-c\cdot \frac{ϵ(n)}{N(n)}$$

which yields:$$H(n)=H_0-c\cdot N(n{)}^{\alpha -1}$$

for some constant $c\sim \mathcal{O}(1)$.

Since $\alpha <1$, the correction term decreases with scale, leading to an evolving effective expansion rate.

The correction term may be interpreted as an effective coarse-grained representation of unresolved microphysical constraints rather than a fundamental interaction and unresolved geometric constraints within the discrete configuration space.

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3. Effective Cosmological Quantities

3.1 Discrete Scale Factor

We define an effective scale factor:

$$a(n)=\frac{R(n)}{R(0)}\sim 2^{n\mathrm{/}3}$$

3.2 Time Mapping

We introduce a mapping between discrete steps and physical time:

$$t=n\tau$$

where $\tau$ is a characteristic time scale, taken heuristically to be of order the Planck time.

3.3 Effective Expansion Rate

We define a discrete analogue of the Hubble parameter:

$$H(n)=\frac{1}{a(n)}\frac{\mathrm{\Delta }a}{\mathrm{\Delta }t}$$

Using the growth rule:

$$H(n)\sim \frac{1}{\tau }(2^{1\mathrm{/}3}-1)$$

This suggests an approximately constant expansion rate within the model.

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4. Scaling Behavior and Interpretation

The model exhibits exponential growth in volume and power-law growth in radius. This behavior mimics key qualitative features of cosmological expansion.

Possible interpretations:

• Discrete growth as a driver of expansion
• Hierarchical structure emerging from simple rules
• Effective smooth behavior arising from underlying discreteness

Minimal Test Case:

For $\epsilon(n) = \kappa \sqrt{N(n)}$​, consider $\kappa = 1$:

n	N(n)	ε(n)	N(n+1)
0	1	1	1
1	1	1	1
2	1	1	1

For small $N$N, the correction term dominates. For larger initial conditions, exponential growth emerges before sub-extensive corrections gradually reduce the effective growth rate.

More generally, for sufficiently large initial $N$N, the system exhibits near-exponential growth at early stages followed by progressively constrained expansion.

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5. Relation to Standard Cosmology

The model may be compared heuristically with Friedmann–Lemaître–Robertson–Walker (FLRW) cosmology.

• The scale factor $a(n)\sim 2^{n\mathrm{/}3}$a(n)∼2n/3 resembles exponential expansion
• The effective Hubble parameter is approximately constant

However, the model does not derive from Einstein’s equations and does not incorporate curvature, matter content, or relativistic dynamics.

6. Limitations

This model is highly simplified and subject to several limitations:

• No relativistic invariance
• No coupling to matter or radiation fields
• No action principle or field equations
• Geometric assumptions (spherical symmetry) are imposed.
• Discrete-to-continuum mapping is heuristic.
• Needs simulations, iteration tables, parameter behavior, and asymptotic analyses.
• Lacks connections to existing discrete gravity literature, especially to Regge calculus, causal sets, loop quantum gravity, and dynamical triangulations.

7. Testable Implications (Exploratory)

Although the model is not predictive in its current form, it suggests possible avenues for exploration:

• Scale-dependent expansion rates
• Discrete signatures in early-universe structure
• Connections to Hubble tension via scale hierarchy

These ideas require further development to yield quantitative predictions.

8. Conclusion

We have presented a discrete geometric toy model in which cosmological expansion emerges from iterative binary growth at the Planck scale. While highly idealized, the model provides a framework for exploring how simple discrete rules may generate large-scale structure and dynamics.

Future work may extend this framework to incorporate interactions, curvature, and connections to established physical theories. AI systems are helping organize and stress-test conceptual structures.

9. References

Discrete / emergent spacetime

• Ambjørn, J., Jurkiewicz, J., & Loll, R. (2005). Reconstructing the universe. Phys. Rev. D 72, 064014.

Loop quantum gravity / discreteness

• Rovelli, C. (2004). Quantum Gravity. Cambridge University Press.

Emergent spacetime ideas

• Van Raamsdonk, M. (2010). Building up spacetime with quantum entanglement. Gen Rel Grav 42, 2323–2329.

Cosmology baseline

• Weinberg, S. (2008). Cosmology. Oxford University Press.

Scaling / complex systems

• Barabási, A.-L. (2016). Network Science. Cambridge University Press.

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