The ℚ-vs-ℝ Question: Why Physical Law Requires Only the Rational Numbers

In plain language

The paper set out to show why the rational numbers (fractions of whole numbers) are enough to describe physical law, even though many equations use the real numbers (all possible decimal values). It argued that measurements can only store a finite amount of information, because the Bekenstein bound limits the entropy of any finite system, and the Holevo bound limits how much quantum information we can extract. It pointed out that the measured fine‑structure constant is about 1/137.035999084 and the electron g‑factor is about 1.00115965218059, both of which can be expressed as rational numbers to the precision of the experiments. They noted that the QNFO physics programme, which includes the Adelic Physics Program (7 papers), the Ostrowski Programme (30+ artifacts), Hensel Codes (3 versions), and the ultrametric foundation, rests on this rational‑number assumption. This matters because it means we can build physical theories using only fractions, simplifying calculations and avoiding unnecessary reliance on the infinite precision of real numbers.

Restated for a non-expert reader; the original wording follows. 1 statement(s) without support in the text were removed. Still uses some technical language.

In the paper's words

The QNFO physics programme — spanning the Adelic Physics Program (7 papers), the Ostrowski Programme (30+ artifacts), Hensel Codes (3 versions), and the ultrametric foundation — rests on an assumption that has never been systematically defended: that the rational numbers (ℚ) are the physically correct base field, and that the real numbers (ℝ) are a mathematical convenience rather than a physical necessity. This memo provides that defense. We establish four sub-claims — Measurement Finitude, Observational Equivalence, Ostrowski Theorem Applicability, and Cardinality Non-Constraint — and defend each against the strongest known objections.

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

Numbers it reports

1. Adversary 1's objection (Ostrowski Programme red-team v2, 2026-07-28) puts the challenge directly: "Measurements yield real numbers.

   from 1.1 The Question

2. The Bekenstein bound [established] (Bekenstein 1981) states that the maximum entropy S of a physical system confined to a finite region of space with finite energy E and characteristic radius R is: $$S \leq \frac{2\pi k R E}{\hbar c}$$ From this, the maximum information I (in bits) that can be stored in the system is: $$I = \frac{S}{k \ln 2} \leq \frac{2\pi R E}{\hbar c \ln 2}$$ For a system of finite spatial extent and finite energy — which describes every physically realizable experimental apparatus — the information content is finite .

   from 2.2 The Physical Argument: Bekenstein Bound

3. The Holevo bound [established] (Holevo 1973) states that the accessible information extractable from a quantum system prepared in an ensemble of states is bounded by the von Neumann entropy of the ensemble.

   from 2.3 The Quantum Argument: Holevo Bound

4. The measured value of the fine-structure constant $\alpha \approx 1/137.035999084$ is a rational number (specifically, the ratio of two integers to the precision of the measurement).

   from 2.4 The Practical Argument: Every Measurement Ever Made

5. The electron g-factor $g/2 \approx 1.00115965218059$ is a rational number.

   from 2.4 The Practical Argument: Every Measurement Ever Made

How it is done

1. We establish four sub-claims — Measurement Finitude, Observational Equivalence, Ostrowski Theorem Applicability, and Cardinality Non-Constraint — and defend each against the strongest known objections.

   from Abstract

2. Ostrowski Theorem Applicability (OTA — §4): Because $\mathbb{Q}$ is physically accessible, Ostrowski's theorem applies to the physical base field.

   from 1.3 Structure

3. No experiment has ever produced — or can produce — an irrational number with infinite precision. [established]

   from 2.1 The Claim

4. The Bekenstein bound [established] (Bekenstein 1981) states that the maximum entropy S of a physical system confined to a finite region of space with finite energy E and characteristic radius R is: $$S \leq \frac{2\pi k R E}{\hbar c}$$ From this, the maximum information I (in bits) that can be stored in the system is: $$I = \frac{S}{k \ln 2} \leq \frac{2\pi R E}{\hbar c \ln 2}$$ For a system of finite spatial extent and finite energy — which describes every physically realizable experimental apparatus — the information content is finite .

   from 2.2 The Physical Argument: Bekenstein Bound

What would falsify it

1. To meet the QNFO falsifiability requirement, we specify four concrete conditions that would disconfirm the $\mathbb{Q}$-as-base claim.

   from 6. Falsifiability Conditions

2. We provide four concrete falsification conditions that would disconfirm the $\mathbb{Q}$-as-base position, meeting the QNFO falsifiability requirement.

   from Abstract

3. Keywords: rational numbers, real numbers, base field, measurement theory, Ostrowski's theorem, adelic physics, Bekenstein bound, constructive mathematics, falsifiability

   from Abstract

Where the claim stops

1. Cardinality does not constrain the $\mathbb{Q}$-as-base claim.

   from 5.6 Conclusion of §5

2. The QNFO physics programme — spanning the Adelic Physics Program (7 papers), the Ostrowski Programme (30+ artifacts), Hensel Codes (3 versions), and the ultrametric foundation — rests on an assumption that has never been systematically defended: that the rational numbers ($\mathbb{Q}$) are the physically correct base field, and that the real numbers ($\mathbb{R}$) are a mathematical convenience rather than a physical necessity.

   from Abstract

3. We directly address Adversary 1's objection from the Ostrowski Programme red-team v2 (CONDITIONAL FAIL): "Measurements yield real numbers.

   from Abstract

Where to read

1. Abstract179 words · 1 min
2. 1. Introduction408 words · 2 min
3. 2. Measurement Finitude (MF)577 words · 3 min
4. 3. Observational Equivalence (OE)833 words · 4 min
5. 4. Ostrowski Theorem Applicability (OTA)737 words · 3 min
6. 5. Cardinality Non-Constraint (CNC)441 words · 2 min
7. 6. Falsifiability Conditions379 words · 2 min
8. 7. Response to Adversary 1 (Red-Team v2)513 words · 2 min
9. 8. Conclusion352 words · 2 min
10. Declarations94 words · 1 min