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A New Idea for Physics Based on Tree-Like Geometry

A plain-language version of “Unity of Ultrametric Physics”

3 min readNo jargon, no mathematicsPhysics

The short version

This paper says the universe might be built on a branching, tree-like geometry instead of smooth space. It lays out the full math from the start, then shows how this idea could explain several puzzles and how to test it.

Why ordinary space might not be the base

Most physics assumes space is smooth and continuous. This paper asks what if space is actually built like a tree, with branches inside branches. In this kind of geometry, the closest path between two points is not a straight line. Instead, you go up the tree to a common branch, then down. The paper calls this "ultrametric" geometry. It is the geometry of family trees and of some complex systems in nature. The paper says this tree-like structure might be the true base of reality. Smooth space would then just be what we see when we look at large scales.

Building the math from nothing

The paper starts with the simplest ideas: what a set is, what a function is, what distance means. It then introduces a stricter rule for distance. In ordinary space, the direct path is never longer than going through a third point. In the tree-like geometry, the direct path is never longer than the longer of the two legs. This small change makes triangles always have two equal longest sides. It makes every point inside a ball the center of that ball. It means there are no smooth paths at all. The paper then builds a whole new kind of number system based on prime numbers. These "p-adic" numbers have the strange property that a number is small when it can be divided many times by a chosen prime. The paper shows how to do calculus with these numbers. It then unites all these number systems into one big structure called the adele ring. This structure holds both the usual real numbers and all the p-adic numbers together.

How this could explain physics puzzles

The paper says many hard problems in physics become simpler in this tree picture. The huge gap between the strength of everyday forces and gravity is just a counting fact about tree levels. The fact that quantum gravity calculations blow up in ordinary physics is fixed because the tree has a finite number of levels. A small angle in the strong force that seems fine-tuned to zero is exactly zero at the base of the tree. The energy of empty space, which is off by a huge factor in ordinary physics, cancels out when all the number systems are combined. Some small differences between measured particle properties and standard predictions, like the muon's magnetic behavior and the W boson's mass, get natural corrections from tree loops. Patterns in rare particle decays match the tree structure.

Ways to test the idea

The paper lists eighteen ways to check if this is true. Particle colliders might see sudden jumps in how often particles scatter, at energies tied to different primes. Dark matter detectors might see signals at depths set by tree levels, or might see nothing if the levels are too deep. The early universe might have left ripples in space with a step-like pattern. Quantum computers built on tree-shaped hardware would have built-in error protection. The paper gives specific numbers for when effects should appear and what current experiments already rule out.

What this paper does not show

The paper does not claim any of these tests have already worked. It says the idea needs to be checked against future data. It does not prove that ordinary space must emerge from the tree picture; it argues this is plausible. It leaves open whether all the prime-based number systems matter equally or some dominate. It does not resolve whether the tree has a finite or infinite depth. The paper presents the full framework but says the work of matching every detail to known physics remains ongoing.

Words used here

ultrametric
a way to measure distance where the direct path is never longer than the longer of two legs, like moving through a family tree
p-adic
a number system where size depends on how often a chosen prime divides a number, opposite to usual size
adele
a combined structure holding all number systems at once, both usual and p-adic

Learn more in plain language

This plain-language version was written by an AI model from the text of the paper and checked against plain-language rules: everyday words, no mathematics, no unexplained names. It leaves details out and can still be wrong; the paper is the source of record. To report a problem, open the paper, select the passage and choose Report.

For specialists: the technical digest, quoted from the paper

In the paper's words

This document develops a unified framework for physics grounded in ultrametric geometry. No prior mathematical knowledge is assumed. Every concept — from the notion of a set to the adelic field equations — is defined in place, every operation explained, and key properties proved.

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

Numbers it reports

  1. The Einstein–Hilbert action emerges from the entanglement entropy of the tree boundary: $$S_{\text{EH}} = \frac{1}{16\pi G_N} \int R \sqrt{-g} \, d^4x = \frac{k_B}{4\ell_P^2} \int_{\partial\mathcal{T}_p} S_{\text{EE}}(A) \, dA$$ where $S_{\text{EE}}(A)$ is given by the Ryu–Takayanagi formula on the tree (Chapter 12c).

    from Derivation of the

  2. In $\mathbb{Q}_5$, solving $3x = 1$ digit by digit yields $\frac{1}{3} = 2 + 3 \cdot 5 + 1 \cdot 5^2 + 3 \cdot 5^3 + \cdots = 2.\overline{31}_5$.

    from Chapter 6: The

  3. Searches for log-periodic oscillations in Planck data have set upper limits $A_2 \lt 0.01$ at 95% CL.

    from Chapter

  4. Quantitative predictions for xenon-based experiments: Tree depth $d_2$ $\sigma_{\text{SI}}$ ($\mathrm{cm}^{2}$) Experimental status 40 $2 \times 10^{-46}$ Excluded by XENONnT/LZ 43 $1 \times 10^{-47}$ Near current 90% CL limit 45 $2 \times 10^{-48}$ Accessible with $\times 10$ exposure 50 $2 \times 10^{-50}$ Below neutrino fog Prediction.

    from Chapter 17b: Dark

  5. Theorem 11.3 (Gauge coupling running with p-adic steps). $$\alpha_i^{-1}(E) = \alpha_i^{-1}(E_0) + \frac{b_i}{2\pi} \log\frac{E}{E_0} + \sum_p \frac{b_i^{(p)}}{2\pi} \left\lfloor \log_p \frac{E}{E_0} \right\rfloor$$ The floor function produces step-like contributions at $E = p^n E_0$, rather than the smooth logarithmic running of standard renormalization group equations.

    from Chapter

  6. For a gauge theory with coupling $g$: $$\beta(g) = \beta_{\text{SM}}(g) + \sum_p \beta_p(g), \quad \beta_p(g) = -\frac{g^3}{16\pi^2} \cdot b_p \cdot \Theta(E - \Lambda_p)$$ where $\Theta$ is a step function (smeared by threshold effects) and $\Lambda_p = p^{n_p} \Lambda_0$ are the characteristic p-adic scales.

    from 11.5

How it is done

  1. The standard absolute value $|\cdot|_\infty: \mathbb{Q} \to \mathbb{R}_{\geq 0}$ is: $$|x|_\infty = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x \lt 0 \end{cases}$$ Theorem 2.2 (Properties).

    from 2.1 From Naturals to Rationals

  2. Factorization and the p-adic Valuation Theorem 2.3 (Fundamental Theorem of Arithmetic).

    from 2.2 Prime

  3. Theorem 3.7 (Metric completion).

    from Chapter 3: Distance and

  4. Theorem 4.2 (Isosceles triangles).

    from Chapter 4: The Ultrametric

What would falsify it

  1. It culminates in concrete computational architectures and eighteen falsifiable experimental protocols spanning collider physics, cosmology, dark matter detection, and tabletop quantum simulation.

    from A

  2. The W-boson mass anomaly — the CDF measurement $M_W = 80\,433.5 \pm 9.4$ MeV exceeds the Standard Model prediction by $\sim 7\sigma$; ultrametric tree corrections shift $M_W$ upward (Chapter 17a).

    from Compelling Clues

  3. The Unity framework predicts normal ordering with $m_1 \lt m_2 \lt m_3$, consistent with current global fits ($\Delta\chi^2 \approx 10$ in favor of normal ordering).

    from 13.2 The Neutrino Sector

Where the claim stops

  1. No prior mathematical knowledge is assumed.

    from A

  2. Continuous, Archimedean spacetime — the spacetime of general relativity and quantum field theory — is an emergent, large-scale approximation.

    from Prologue: Why Ultrametric?

  3. This satisfies $(a, b) = (c, d)$ if and only if $a = c$ and $b = d$.

    from Chapter 1: Sets,

Where to read

  1. Part I: Mathematical1,425 words · 6 min
  2. Part II: The p-Adic Universe749 words · 3 min
  3. Part III: Physics on1,629 words · 7 min
  4. Part IV: The Unity2,731 words · 12 min
  5. Part V: Implementations1,402 words · 6 min
  6. Part VI: Experimental1,223 words · 5 min
  7. Part VII: Appendices375 words · 2 min
  8. |2,155 words · 9 min

Read from the paper itself: 115 sections, 12,839 words. Nothing in this digest is written by a model; a line that is not a sentence of the paper is not shown. Have a specific question? Ask this paper.

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