Ultrametric Quantum Gravity and Computation

In the paper's words

This report presents a unified theoretical framework that bridges two seemingly disparate frontiers: ultrametric quantum computation and ultrametric quantum gravity. Both paradigms replace the conventional Archimedean continuum with a discrete, hierarchical, tree‑like geometry described by p‑adic numbers and their associated Bruhat–Tits trees. The framework demonstrates that: The synthesis yields a coherent picture: reality is a timeless, branching fractal; quantum computation is navigation of this fractal; and quantum gravity is the constraint that selects admissible branches.

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

Numbers it reports

1. 3.2 p‑adic Minisuperspace Consider the simplest cosmological model: a Friedmann–Lemaître–Robertson–Walker universe with scale factor $a$ and a scalar field $\phi$.

   from 3. From Minisuperspace to p‑adic Superspace

2. Then $$ -D_p^{\alpha_a}A=\lambda A,\qquad \frac{1}{a_p^{2}}D_p^{\alpha_\phi}\Phi+V(a_p,\phi_p)\Phi=-\lambda\Phi . $$ 4.2 Scale‑Factor Eigenfunctions The first equation is solved by p‑adic plane waves $A_k(a_p)=\chi_p(k a_p)$ with eigenvalue $-|k|_p^{\alpha_a}$.

   from 4. Solutions – Fractal Branches and Hierarchical Cosmologies

How it is done

No section of this paper is shaped like a method, so no method sentence is quoted.

What would falsify it

1. This predicts a minimum length $\ell_{\min}\propto p^{-1/2}$ (in appropriate units) and a holographic scaling of degrees of freedom with area.

   from 7. Physical Predictions and Experimental Signatures

Where the claim stops

1. 1.1 The Continuum Hypothesis and Its Discontents Conventional quantum mechanics and quantum field theory are built on the field of real (or complex) numbers, an Archimedean structure that assumes distances can be subdivided arbitrarily.

   from 1. The Archimedean Limitation and the p‑adic Alternative

2. 1.5 Intrinsic Fault‑Tolerance in Quantum Computation Encoding quantum information on the vertices of the Bruhat–Tits tree exploits the strong triangle inequality: small perturbations cannot accumulate.

   from 1. The Archimedean Limitation and the p‑adic Alternative

3. An error becomes significant only if it is large enough to jump to a different branch.

   from 1. The Archimedean Limitation and the p‑adic Alternative

Where to read

1. Executive Summary148 words · 1 min
2. Part I: Foundations Revisited – Non‑Archimedean Quantum Theory454 words · 2 min
3. Part II: Ultrametric Quantum Gravity – The Wheeler–DeWitt Equation486 words · 2 min
4. Part III: Synthesis – Bridging the Two Frontiers505 words · 2 min
5. Conclusion – The Ultra‑Metric Paradigm147 words · 1 min