Post-Positional Numeracy: Finite-Adele Encoding and Product-Formula-Verified Exact Rational Arithmetic

In the paper's words

Exact computation on the rationals today inhabits a single completion. Digital arithmetic runs in the real numbers and accepts rounding; exact alternatives run in a single p-adic completion through Hensel codes and reconstruct through the Chinese remainder theorem with Farey bounds. This paper develops the multi-place realization that joins the two.

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

Numbers it reports

1. Product-formula checks: the golden values $x \in \{6, 2/3, 12\}$ over $S = \{2, 3\}$; the boundary value $x = 5/2$, where the truncated product is $5 = 1/|5/2|_5$; $10^4$ seeded random $S$-smooth trials over $S = \{2, 3, 5, 7\}$; and $10^4$ seeded trials of the general identity with an outside prime factor.

   from 5.1 Method

2. Ostrowski checks: the strong triangle inequality on $2 \times 10^4$ random integer pairs, the bound $|n|_2 \le 1$ for integers, and the full product formula for four values.

   from 5.1 Method

How it is done

1. Ostrowski's theorem states that the choice is not a free one — every nontrivial absolute value on $\mathbb{Q}$ is equivalent to the real absolute value or to a $p$-adic one — but the theorem also implies that no single completion is privileged.

   from 1. Introduction

2. Ostrowski's theorem states that every nontrivial absolute value on $\mathbb{Q}$ is equivalent to one of these; the completions are $\mathbb{R}$ and the fields $\mathbb{Q}_p$.

   from 2.1 Absolute values and Ostrowski's theorem

3. Given a residue $r$ modulo a composite $M$ together with a bound $B$ with $2B^2 \lt M$, the rational $a/b$ with $\gcd(b, M) = 1$, $|a| \le B$, $|b| \le B$, and $a b^{-1} \equiv r \pmod M$ is unique, and a two-step Euclidean algorithm recovers it.

   from 2.2 Hensel codes and rational reconstruction

4. That literature is unrelated to Ostrowski's theorem on absolute values, which is the theorem used here.

   from 2.4 A remark on terminology

What would falsify it

The paper's text states no explicit falsification criterion; the limits below are what it says about where the claim stops.

Where the claim stops

1. Lemma 1 quantifies the map's faithfulness — on the window $W$ the map is injective, and Remark 1 shows the faithfulness fails if the window is stated one-sided.

   from 9. Discussion

2. An open question the verification suggests is the cost of exactness: what a multi-place round-trip costs against a single-place one in cycles, memory, and energy, and where the crossover lies for practical workloads.

   from 9. Discussion

3. The measurement is straightforward to set up and is left for future work.

   from 9. Discussion

Where to read

1. 1. Introduction375 words · 2 min
2. 2. Preliminaries295 words · 1 min
3. 3. The finite-adele encoding313 words · 1 min
4. 4. The product formula as a verification invariant215 words · 1 min
5. 5. Computational verification349 words · 2 min
6. 6. Crosswalk142 words · 1 min
7. 7. Practitioner relevance120 words · 1 min
8. 8. Related work127 words · 1 min
9. 9. Discussion120 words · 1 min
10. 10. Declarations81 words · 1 min