Why Quantum Physics Does Not Need the Real Numbers

In the paper's words

We propose and formalize a principle of completion-invariance for physical ontology: since all recorded measurement outcomes are rational numbers, the archimedean completion R and the non-archimedean completions Q_p of the rational base field Q are equally instrumental constructs, and any claim that the continuum is physically real must, to be consistent, extend equal reality-status to every completion of the same base field. We develop a finite-precision (rational) formulation of quantum mechanics in which amplitudes are restricted to rationals of bounded denominator and prove an explicit error bound: for a d-dimensional system with amplitudes approximated at n-bit precision, the deviation of every Born-rule probability from its standard value is at most (2+δ)√d δ with δ = 2^-n-1, which for d=4 and n=64 is ≈ 1.08× 10^-19 — far below any operational discrimination. [...]

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

Numbers it reports

1. We develop a finite-precision (rational) formulation of quantum mechanics in which amplitudes are restricted to rationals of bounded denominator and prove an explicit error bound: for a $d$-dimensional system with amplitudes approximated at $n$-bit precision, the deviation of every Born-rule probability from its standard value is at most $(2+\delta)\sqrt{d}\,\delta$ with $\delta = 2^{-n-1}$, which for $d=4$ and $n=64$ is $\approx 1.08\times 10^{-19}$ — far below any operational discrimination.

   from Abstract

2. We derive the bit budget required to push this bound below any target tolerance ($n=41$ bits for $d=4$ at tolerance $10^{-12}$; $n=21$ bits at tolerance $10^{-6}$), identify scale-invariant and ultrametric regimes as the only plausible loci of genuine completion-divergence, and formalize the symmetry argument that any realist case for $\mathbb{R}$-reality transfers mutatis mutandis to $\mathbb{Q}_p$-reality.

   from Abstract

3. Then $\delta = 2^{-65}$ and $$\epsilon_P \le (2 + 2^{-65}) \cdot \sqrt{4} \cdot 2^{-65} = (2 + 2^{-65}) \cdot 2 \cdot 2^{-65} = (4 + 2^{-64})\, 2^{-65}.$$ Numerically: $2^{-64} = 1/2^{64} = 1/1.8446744\times 10^{19} \approx 5.421011\times 10^{-20}$, so $2^{-65} \approx 2.710505\times 10^{-20}$, and $$\epsilon_P \le 4 \times 2.710505431\times 10^{-20} + 2^{-64}\cdot 2^{-65} \approx 1.084202172\times 10^{-19} + 1.47\times 10^{-39} \approx 1.084202173\times 10^{-19}.$$ The second term is negligible; the bound is $\epsilon_P \approx 1.084202\times 10^{-19}$, i.e. $\approx 1.08\times 10^{-19}$ to two significant figures.

   from 4.2 Numerical evaluation for a two-qubit system

4. Using the leading-order bound $\sqrt{d}\,2^{-n} \le \epsilon_{\mathrm{target}}$ (a conservative simplification: the true bound $(2+\delta)\sqrt{d}\,2^{-n-1}$ is smaller, so the derived $n$ is sufficient), $$n \ge \log_2\!\left(\frac{\sqrt{d}}{\epsilon_{\mathrm{target}}}\right) = \frac{1}{2}\log_2 d + \log_2\frac{1}{\epsilon_{\mathrm{target}}}.$$ Example 1. $d = 4$, $\epsilon_{\mathrm{target}} = 10^{-12}$: $\frac{1}{2}\log_2 4 = 1$, and $\log_2(10^{12}) = 12 \log_2 10 = 12 \times 3.321928 = 39.86314$.

   from 4.3 Bit budget for a target tolerance

5. This bound is far weaker than the bit-truncation bound of Section 4.1 (it fixes the precision at the apparatus rather than at the state description) and is included only to show that even the crudest rational model already sits below typical probability-level uncertainties; the main results use the bit-truncation convention.

   from 4.5 Grid-rounding convention (alternative bound)

6. The bound of Section 4.1 covers finite-dimensional systems with rational observables.

   from 4.6 Where completions can genuinely diverge

How it is done

1. We develop a finite-precision (rational) formulation of quantum mechanics in which amplitudes are restricted to rationals of bounded denominator and prove an explicit error bound: for a $d$-dimensional system with amplitudes approximated at $n$-bit precision, the deviation of every Born-rule probability from its standard value is at most $(2+\delta)\sqrt{d}\,\delta$ with $\delta = 2^{-n-1}$, which for $d=4$ and $n=64$ is $\approx 1.08\times 10^{-19}$ — far below any operational discrimination.

   from Abstract

2. Section 4 carries out the explicit derivations, including the probability-error bound and the bit-budget calculation.

   from 1. Introduction

3. The supplied bibliography contains twelve entries; several are topically distant from quantum foundations, and we use each only for what its own summary states.

   from 2. Background and Related Work

4. The proof is immediate: the three cited properties are shared by every completion of $\mathbb{Q}$ by construction, so no argument appealing to them alone can discriminate.

   from 3.1 The rational base field and its completions

What would falsify it

1. If the answer is no for all operationally accessible observables, then the choice of completion is a calculational convention, and adelic methods (the joint use of real and $p$-adic techniques) are legitimate tools rather than ontological commitments.

   from 1. Introduction

2. If the answer is yes in some regime, completion-invariance tells us exactly where to look for physically observable completion-divergence.

   from 1. Introduction

3. Section 6 discusses limitations, failure modes, and what would falsify the claims.

   from 1. Introduction

Where the claim stops

1. (ii) The derivation assumes exact rational arithmetic; floating-point implementations introduce their own rounding, which our $n$ counts do not model.

   from 6. Discussion

2. (iv) The benchmark precision levels in R3 and R6 are assumed, not measured; the qualitative conclusions survive several orders of variation in the assumed floor, but a reader requiring a specific experimental claim must supply their own precision figure.

   from 6. Discussion

3. The principle is falsified if a single, reproducible operational procedure is exhibited whose outcomes cannot be reproduced by any rational-valued reformulation of quantum mechanics at any feasible bit budget $n$ — that is, if the answer to the central question of Section 1 (do standard quantum predictions depend on the archimedean completion at all?) is demonstrably yes in an operationally accessible regime.

   from 6. Discussion

Where to read

1. Abstract218 words · 1 min
2. 1. Introduction505 words · 2 min
3. 2. Background and Related Work1,115 words · 5 min
4. 3. Methods478 words · 2 min
5. 4. Analysis802 words · 3 min
6. 5. Results247 words · 1 min
7. 6. Discussion441 words · 2 min
8. 7. Conclusion199 words · 1 min