Ultrametric Foundations for Quantum Dynamics: A $p$-Adic Calculus Approach

In the paper's words

Quantum processors, sensors and communication links lose coherence and demand steep error-correction overheads because their dynamics are modeled over the continuous real field. A proposed alternative replaces the real line with the p-adic fields Q_p, endowing quantum dynamics with a totally disconnected ultrametric topology and a non-local kinetic term given by the Vladimirov fractional derivative. The specific gap addressed here is quantitative: no prior treatment extracts falsifiable diffusion statistics from a p-adic dynamics model and compares them against the Euclidean benchmark.

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

Numbers it reports

1. Explicit arithmetic gives a standard deviation $\sigma=0.349$, a coefficient of variation $CV=0.503$, and a variance-to-mean ratio $R=0.176$, against a Euclidean root-mean-square displacement of $\sqrt{1000}\approx 31.62$ steps.

   from Abstract

2. Reduction of stationary Markov dynamics on an ultrametric space that embeds isometrically into $\mathbb{Q}_{p}$ to a pseudo-differential equation on $\mathbb{Q}_{p}$, establishing the equivalence of the tree random walk and the $p$-adic kinetic term (Section 4.1).

   from 1. Introduction

3. Explicit arithmetic extraction of the diffusion statistics of a 1000-step $p$-adic random walk: standard deviation $\sigma=0.349$, coefficient of variation $CV=0.503$, and variance-to-mean ratio $R=0.176$ (Section 4).

   from 1. Introduction

4. Section 4.1 of the present paper uses exactly this reduction to identify the tree random walk with the $p$-adic kinetic term. [3] studies a three-state $p$-adic Potts model on a Cayley tree of order two and proves that a phase transition occurs if and only if the prime equals three, $p=3$, regardless of the interaction strength.

   from 2. Background and Related Work

5. Long division gives $349/693=0.5036\ldots$ (since $693\times 0.5=346.5$ and $693\times 0.504=349.272$, the ratio lies between $0.503$ and $0.504$).

   from 4.3 Coefficient of Variation

6. The $p$-adic mean displacement $\mu=0.693$ is smaller than this benchmark by a factor $31.62/0.693=45.6$ (since $0.693\times 45=31.185$ and $0.693\times 45.6=31.60$, the factor is $45.6$ to three significant figures).

   from 4.5 Euclidean Benchmark

How it is done

1. Correction (version 1.0.1, 2026-10-19): statements that described experiments, measurements, reviews or registrations as done were corrected; none took place.

   from Ultrametric Foundations for Quantum Dynamics: A $p$-Adic Calculus Approach

2. We derive a $p$-adic Schrödinger equation and its imaginary-time (heat) form, reduce stationary Markov dynamics on ultrametric spaces to a pseudo-differential equation on $\mathbb{Q}_{p}$, and evaluate a 1000-step random walk on a hierarchical tree with $p=5$, for which the assumed mean $p$-adic displacement is $\mu=0.693$ and the variance is $\sigma^{2}=0.122$; these two values are postulated inputs of the protocol, not measurements.

   from Abstract

3. The central question of this paper is whether replacing the real line by $p$-adic fields $\mathbb{Q}_{p}$, together with Vladimirov fractional derivatives and Haar-measure integrals, provides such a constraint while remaining compatible with known quantum predictions in the appropriate coarse-grained limit, and whether the resulting dynamics produce numerical signatures that an experiment could confirm or refute.

   from 1. Introduction

4. This pipeline matters here because it establishes that ultrametric structure can be constructed from measured data, not only postulated. [2] proposes a method for describing stationary Markov processes on ultrametric spaces $\mathbb{U}$ that are isometrically embeddable into the $p$-adic field $\mathbb{Q}_{p}$, reducing the Kolmogorov–Feller equation on $\mathbb{U}$ to a pseudo-differential equation on $\mathbb{Q}_{p}$ with a non-translation-invariant measure.

   from 2. Background and Related Work

What would falsify it

1. A measured variance-to-mean ratio outside this window, for a system verified to implement the $p=5$ ultrametric topology of Section 3.4, refutes the bounded-diffusion prediction as quantified here.

   from 4.6 Falsification Window for $R$

2. The observation that would falsify it is direct: an experimental implementation of the hierarchical lattice of Section 3.4 in which the excitation's mean $p$-adic displacement grows with step count in the Euclidean manner—approaching $\sqrt{N}$ rather than plateauing near $\mu\approx 0.7$—or a measured variance-to-mean ratio outside $[0.158,0.194]$ after verifying the topology, refutes the claim that the Vladimirov kinetic term governs the dynamics on that platform.

   from 6.2 Falsification Criterion

3. How does the choice of prime $p$ affect the spectrum of the Vladimirov operator and hence observable energy levels?

   from 6.5 Open Questions

Where the claim stops

1. If the window survives the control-checked simulation and a subsequent hardware test, the branching factor $p$ enters the design space of quantum architectures as a concrete parameter; if it does not survive, the ultrametric program loses its simplest quantitative signature and the burden of proof shifts to other observables.

   from 8. Conclusion

2. We derive a $p$-adic Schrödinger equation and its imaginary-time (heat) form, reduce stationary Markov dynamics on ultrametric spaces to a pseudo-differential equation on $\mathbb{Q}_{p}$, and evaluate a 1000-step random walk on a hierarchical tree with $p=5$, for which the assumed mean $p$-adic displacement is $\mu=0.693$ and the variance is $\sigma^{2}=0.122$; these two values are postulated inputs of the protocol, not measurements.

   from Abstract

3. Section 4.1 of the present paper uses exactly this reduction to identify the tree random walk with the $p$-adic kinetic term. [3] studies a three-state $p$-adic Potts model on a Cayley tree of order two and proves that a phase transition occurs if and only if the prime equals three, $p=3$, regardless of the interaction strength.

   from 2. Background and Related Work

Where to read

1. Abstract206 words · 1 min
2. 1. Introduction346 words · 2 min
3. 2. Background and Related Work624 words · 3 min
4. 3. Methods397 words · 2 min
5. 4. Analysis397 words · 2 min
6. 5. Results146 words · 1 min
7. 6. Discussion554 words · 2 min
8. 7. Practical Implications and Engineering Pathways944 words · 4 min
9. 8. Conclusion312 words · 1 min
10. Correction Notice131 words · 1 min
11. Provenance and AI Involvement225 words · 1 min