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Fixing Physics by Changing Its Basic Building Blocks

A plain-language version of “REIFICATION AND NON-ARCHIMEDEAN FOUNDATIONS IN THEORETICAL PHYSICS”

4 min readNo jargon, no mathematicsPhysics

The short version

This paper says modern physics has grown too complicated. It argues that replacing the smooth, endlessly divisible space we use with a lumpy, branching, tree-like space would remove many invented entities. Four new frameworks try this, but none yet computes a known physical number from scratch.

Why physics looks too complicated

The paper starts with a lesson from history. Long ago, people thought Earth sat at the center of everything. Planets seemed to loop backward in the sky. To explain this, thinkers added circle upon circle to their model. The system grew to about forty circles for one planet. Then the viewpoint shifted. The Sun became the center. Orbits became oval. All the extra circles vanished. They were never real things. They were ghosts of a bad choice of viewpoint.

The paper says physics today makes the same mistake. It keeps adding unseen things to fix problems. These modern extras include virtual particles, dark matter, a field that drove early cosmic expansion, and others. The paper calls these "epicycles" after the old circles. The real problem is not nature. The problem is the framework we use to describe it.

The hidden assumption causing trouble

All these troubles share one root. Physics assumes space and numbers are smooth and can be split forever without end. The paper calls this the Archimedean assumption. This assumption forces physicists to invent fixes. Particles appear out of nowhere in calculations. Matter we cannot see must fill galaxies. A special field must explain why the early universe looks so uniform. Space itself must contain points with no size.

The paper proposes a different kind of space. In this space, distances do not add the normal way. Instead they branch like a tree. Regions are either separate or one sits inside the other. There is no gradual blending. This tree-like space is called non-Archimedean.

Four new frameworks

The paper looks at four attempts to build physics without the smooth continuum.

The first is holography. It says a theory of gravity in a higher space matches a simpler theory on the boundary of that space. The extra dimension emerges from connections between parts of the boundary. A network pattern makes the tree shape visible. But this only works for one special kind of space. It fails for the space we think we live in.

The second is causal set theory. It replaces smooth spacetime with a set of discrete events linked by before-and-after relations. Space and time emerge from counting these events. This theory predicts tiny random wiggles in the expansion of space. But it has not yet been fully connected to known physics.

The third comes from algebraic quantum field theory. It replaces point-like fields with networks of operator algebras. A construction called the crossed product fixes an infinite entropy problem in one type of space. This is a real result within existing quantum field theory. But it awaits further work to connect to other areas.

The fourth is the Syntactic Token Calculus. It is the most stripped-down. It starts from one single act: drawing a boundary or mark. Everything in physics must come from this alone. The marks nest like a tree. A mapping connects this tree to our usual smooth numbers. The framework already produces unchanging patterns for particles. But it has not yet computed any known physical number starting from nothing.

How to judge which framework wins

The paper offers a test. A better framework must make calculations simpler. It must compute known numbers using fewer free choices. This is called the Keplerian criterion. The Syntactic Token Calculus faces a barrier. It produces pure patterns. But turning these into human-style numbers requires an extra step. The paper calls this the "pentadactility problem." It means our base-ten number system comes from having five fingers. That choice is not fundamental. It is a human accident. The challenge is to bridge this gap.

What remains unknown

The paper is honest about limits. None of the four frameworks has yet computed a known constant from first principles with fewer free choices. The Syntactic Token Calculus makes predictions that could be tested. These include special patterns in the oldest light in the universe, extra bumps in the Higgs particle, and tree-like clustering in brain data. But these tests lie ahead. The crossed product construction needs more development too. The paper ends with a warning. The line between our map of reality and reality itself is itself a mark we draw. Drawing that line is the only thing that is not a human artifact.

Words used here

Archimedean
the property of a space where any size can be exceeded by adding enough of a smaller size, like how enough small steps cover any distance
Crossed product
a mathematical construction that combines an algebra with a group action to fix infinite quantities
Epicycle
a circle added to a model to fix a problem, later understood as a ghost of a bad viewpoint
Non-Archimedean
a space where distances branch like a tree rather than blending smoothly
Reification
treating a feature of our description as if it were a real thing in nature

Learn more in plain language

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For specialists: the technical digest, quoted from the paper

In the paper's words

This synthesis examines the persistent difficulties in contemporary theoretical physics through the lens of the Copernican diagnostic: the proliferation of unobservable entities (virtual particles, dark matter, the inflaton, etc.) signals not that nature is complex but that the descriptive framework is suboptimal. The common denominator underlying these “epicycles” is identified as the Archimedean axiom—the assumption of a smooth, infinitely divisible continuum that underpins both general relativity and quantum mechanics. Replacing this continuum with a non‑Archimedean, hierarchical, discrete substrate (exemplified by the Bruhat‑Tits tree) dissolves the epicycles and reveals the invariant structure beneath.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

  1. The Cross‑Ratio: A Clarification 2.7 Comparative Evaluation Matrix PART III: METHODOLOGICAL TENSION AND THE CRITERION OF ADEQUACY 3.1 Foundational Monism vs.

    from Table of Contents

  2. 10 A MERA network is a tensor contraction pattern with the structure of a tree: each layer corresponds to a coarse-graining step, and the branching factor encodes the number of degrees of freedom integrated out at each scale.

    from 2.2 Holography and

  3. Recent work (Chandrasekaran, Longo, Penington, Witten, 2022–2024) demonstrates that the algebra of observables for subregions in quantum gravity is a Type $\mathrm{I}\mathrm{I}\mathrm{I}_{1}$ von Neumann factor .

    from 2.4

  4. Theorem (Takesaki Duality, 1973): The crossed product of a Type $\mathrm{I}\mathrm{I}\mathrm{I}_{1}$ factor by its modular automorphism group is a Type $\mathrm{I}\mathrm{I}_{1}$ factor .

    from 2.4

  5. 17 The Type $\mathrm{I}\mathrm{I}_{1}$ factor possesses a faithful semifinite normal trace $\tau$, allowing the definition of a finite, renormalized entropy.

    from 2.4

  6. Physical result (Chandrasekaran et al., 2023): For the static patch of de Sitter space, the crossed product entropy reproduces the Generalized Entropy : $$S(\rho) = \frac{A}{4G_N} + S_{\text{out}} + \text{constant}$$ where $A$ is the horizon area, $G_N$ is Newton’s constant, and $S_{\text{out}}$ is the entropy of matter fields.

    from 2.4

How it is done

  1. Formally, for an ordered field $F$, the Archimedean property states: $$\forall a, b \in F \text{ with } a \gt 0, \exists n \in \mathbb{N} \text{ such that } n \cdot a \gt b$$ This axiom characterizes the real number continuum $\mathbb{R}$ and, by extension, the smooth Lorentzian manifolds of general relativity and the Hilbert spaces of quantum mechanics.

    from 1.3 The Archimedean Axiom as Unrecognized Coordinate Choice |

  2. The Pusey-Barrett-Rudolph (PBR) theorem (2012) demonstrates that if the wavefunction is merely epistemic (encoding an agent’s information about an underlying real state), a contradiction with quantum predictions arises under the assumption that the underlying state is independent of measurement choice.

    from 1.4 Contemporary Epicycles: A Diagnostic Catalog |

  3. Formally, an ultrametric space is a metric space $(X, d)$ satisfying the strong triangle inequality.

    from 2.1 The Common

  4. Theorem (Takesaki Duality, 1973): The crossed product of a Type $\mathrm{I}\mathrm{I}\mathrm{I}_{1}$ factor by its modular automorphism group is a Type $\mathrm{I}\mathrm{I}_{1}$ factor .

    from 2.4

What would falsify it

  1. 13 The theory predicts a fundamental stochastic fluctuation in the cosmological constant of order $\delta\Lambda/\Lambda \sim 1/\sqrt{N}$, where $n$ is the number of causal set elements in the observable universe.

    from 2.3 Causal Set Theory

  2. The scientific requirement (exemplified by the Crossed Product) holds that a scientific theory must account for existing data and predict new data.

    from 3.1 Foundational

  3. It may use any mathematical tool necessary to achieve this, provided the tool is well-defined and the theory is falsifiable.

    from 3.1 Foundational

Where the claim stops

  1. The common denominator underlying these “epicycles” is identified as the Archimedean axiom—the assumption of a smooth, infinitely divisible continuum that underpins both general relativity and quantum mechanics.

    from Abstract

  2. The STC claims the greatest parsimony—all of physics from a single primitive, the mark—but has not yet computed a known empirical number from first principles.

    from Abstract

  3. 1 The quantitative measure of the simplification is precise: Kepler’s first law reduced the description of Mars’ orbit from approximately forty epicyclic terms to a single ellipse.

    from 1.1 The Copernican Precedent |

Where to read

  1. REIFICATION7,351 words · 32 min
  2. maps to the unit Chu space44 words · 1 min
  3. corresponds to the idempotence of the4,538 words · 20 min

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