Spherical Causal Discretization: A Combinatorial Origin of Relativity, Quantum Behaviour, Thermodynamics, and Energy–Mass Equivalence
Description
This paper introduces a minimal discrete model in which spacetime geometry, quantum behaviour, thermodynamics, cosmological scaling, and the mass–energy relation emerge from a single underlying mechanism: spherical causal discretization. Assuming only a fundamental spatial increment ℓ₀, a fundamental temporal increment τ₀, and the causal rule that influence advances one radial shell per tick at c = ℓ₀/τ₀, the model shows that the structures of general relativity and quantum mechanics arise from the combinatorics of expanding spherical shells.
Curvature appears as population deficits δN in the ideal area law N(r) = 4πr²/ℓ₀². Quantum-like propagation follows from path-count multiplicities across shells. Thermodynamic irreversibility results from the monotonic growth of the causal record, S(t) ∼ ln N(t), and the cosmological scaling relation β = 2α emerges directly from spherical growth. Most notably, maintaining a stable deficit of m relations requires preventing m outward updates per tick, yielding the mass–energy relation E = mc² as an inevitable bookkeeping identity of the discrete causal dynamics.
The spherical model provides the conceptual core of the Discrete Gravitational Ontology (DGO) programme, revealing how relativity, quantum mechanics, thermodynamics, and cosmology can arise from the same simple causal principle.
Rev B - Appendix A added derivation of the Einstein–Hilbert action from the spherical sector of the Discrete Gravitational Ontology (DGO), and minor editorial polish.
Rev C - Appendix B added complementary geometric interpretation of mass and E = mc² in a null-adapted “c-space” frame. Fixed lost section 3 that was missing in Rev B.
Rev D - Section 4.7 added. Emergent $\pi$.
Abstract (English)
This paper develops a minimal discrete model in which spacetime, quantum behaviour,
thermodynamics, curvature, and the mass--energy relation all emerge from a single principle:
spherical causal discretization. Assuming only (i) a discrete spatial increment
$\ell_0$, (ii) a discrete temporal increment $\tau_0$, and (iii) radial causal propagation
at the invariant rate $c=\ell_0/\tau_0$, we show that continuum physics arises from the
combinatorics of an expanding family of relational shells.
Each shell at radius $r_n = n\ell_0$ contains $N_0(n)=4\pi n^2$ relational elements, and
the causal frontier grows as $R(t)=ct$. From this structure, quantum-like propagation
follows from path-count multiplicities; thermodynamic irreversibility from the monotonic
growth of the realized record $S\sim\ln N(t)$; and curvature from population deficits
$\delta N(n)$ relative to the ideal quadratic scaling. In the weak-field limit the universal
small-sphere relation $\delta N/N_0 \approx -Rr^2/12$ identifies these deficits with the
scalar curvature. Most strikingly, a persistent deficit requires preventing $m=-\delta N$
outward causal updates per tick, and the update energy consumed by this resistance is
inevitably $E=mc^2$.
Spherical causal discretization therefore provides a unified combinatorial mechanism from
which the structural laws of general relativity, quantum mechanics, and thermodynamics
arise without continuum assumptions. Relativity appears as spherical causal bookkeeping;
quantum behaviour as path combinatorics; entropy and the arrow of time as record
accumulation; and mass as a stable radial deficit whose maintenance costs $mc^2$ units of
causal update energy.
In this formulation, the entire structure of general relativity, quantum mechanics, and
thermodynamics emerges from exactly three physical primitives: a discrete spatial grain
$\ell_0$, a discrete temporal grain $\tau_0$, and the rule that causal influence advances
one radial shell per tick at the invariant rate $c=\ell_0/\tau_0$.
Notes (English)
Author’s Note
The Discrete Gravitational Ontology did not begin as research. It began as a quiet, private refusal to accept that general relativity and quantum mechanics could remain forever incompatible.
For nearly ten years I carried that tension like a splinter in the mind. I read widely but unsystematically — popular articles, Einstein’s own papers, forum debates, anything that touched the rift — and every explanation left me more convinced that the incompatibility was illusory. There had to be a deeper, simpler coherence.
The decisive spark came from a single diagram in the Swedish popular-science magazine Illustrerad Vetenskap illustrating Jonathan Oppenheim’s post-quantum gravity ideas. One glance at that discrete-spacetime sketch triggered the question that changed everything: What if, gravity itself is the discrete relational fabric — not something that lives in space, but the very thing space is made of?
From that moment the pieces fell together with surprising speed.
There is no graviton because gravity is not particle-like; it is the web that constitutes spacetime itself. The character and the drawn line in the 1970s Italian cartoon La Linea became the perfect image: the “world” and the “drawing act” are one and the same. Quantum behaviour emerged naturally as probabilistic updating on that discrete web, and the arrow of time as the irreversible act of realization — the universe continuously measuring itself by choosing what to keep.
Modern AI tools (first ChatGPT, later Grok by xAI) became indispensable collaborators. They served as patient mathematical interpreters, consistency checkers, and merciless critics, translating raw visual and philosophical intuitions into precise equations and derivations. Every major insight — dual discreteness (“pixels and frames”), the identification of mass as causal drag, the spherical shell combinatorics, the derivation of E = mc² from update resistance — was stress-tested and refined in hours and hours of dialogue.
Yet the conceptual direction always remained mine, driven by the same stubborn minimalist question:
If we truly strip physics to its absolute bones, what must remain?
The answer turned out to be astonishingly little:
- a discrete spatial grain ℓ₀,
- a discrete temporal tick τ₀,
- and the single rule that causal influence expands exactly one radial shell per tick at speed c = ℓ₀/τ₀.
From these three primitives alone — with no manifold, no fields, no Hilbert space, and no action principle postulated — general relativity emerges in the weak-field spherical limit with the exact Einstein–Hilbert action (including its 1/(16πG) prefactor), a Schrödinger-like dynamics arises from path combinatorics in the non-relativistic regime, and rest mass appears as the precise energy cost of resisting causal expansion.
This paper is the current, most stripped-down expression of that decade-long journey. It is offered not as the final word, but as evidence that causality, discreteness, and irreversible record-keeping are not just enough — and perhaps exactly enough — to build a universe.
— Björn Asplind
November 18, 2025
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Additional details
Related works
- Is supplement to
- Preprint: 10.5281/zenodo.17602735 (DOI)
- Preprint: 10.5281/zenodo.17398767 (DOI)
Dates
- Issued
-
2025-11-17Rev A
- Updated
-
2025-11-18Rev B
- Updated
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2025-11-19Rev C
- Updated
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2025-11-30Rev D
References
- 10.5281/zenodo.17398767
- 10.5281/zenodo.17602735