Numbers Have a Hidden Tree Shape

In the paper's words

Positional numeral systems are taught and implemented as points on the Archimedean real line, yet the digit strings that encode integers form a rooted b‑ary tree. This work identifies the concrete gap that the native metric induced by digit‑string structure has never been formalized, leading to hidden assumptions about distance and clustering. We adopt the valuation framework of non‑Archimedean number theory, define the b‑adic valuation v_b, and prove that the associated distance d_b(x,y)=b^-v_b(x-y) satisfies all ultrametric axioms for every integer base b≥2.

The abstract, verbatim; every line below is quoted from the text.

What it claims

that the native metric of positional notation is the b‑adic ultrametric—would be falsified if a concrete counterexample were found where the strong triangle inequality fails for the distance d_b defined above. Since the proof in Section 4.1 is purely algebraic, any such counterexample would imply an error in the definition of v_b or in the divisibility argument.

Numbers it reports

1. Dividing (4) by the total number of pairs (1) yields the proportion of pairs at distance $b^{-k}$: $$ \begin{aligned} p_{k} &=\frac{N_{k}}{\binom{b^{m}}{2}} =\frac{\displaystyle\frac{b^{\,2m-k-1}(b-1)}{2}} {\displaystyle\frac{b^{m}\bigl(b^{m}-1\bigr)}{2}}\\[2mm] &=\frac{(b-1)\,b^{\,m-k-1}}{b^{m}-1} \xrightarrow[m\to\infty]{}\frac{b-1}{b^{k+1}}. \tag{5} \end{aligned} $$ Thus, for large $m$, the probability that two random $m$‑digit numbers are at ultrametric distance $b^{-k}$ decays geometrically with factor $1/b$ per increase of $k$.

   from 2. Asymptotic distribution

2. For a fixed width $m$, the mean ultrametric distance is $$ \bar d_{b} = \frac{1}{\binom{b^{m}}{2}} \sum_{k=0}^{m-1} N_{k}\,b^{-k}. $$ Substituting (4) gives $$ \begin{aligned} \bar d_{b} &= \frac{2}{b^{m}\bigl(b^{m}-1\bigr)} \sum_{k=0}^{m-1} \frac{b^{\,2m-k-1}(b-1)}{2}\,b^{-k}\\[2mm] &= \frac{(b-1)\,b^{\,2m-1}}{b^{m}\bigl(b^{m}-1\bigr)} \sum_{k=0}^{m-1} b^{-2k}\\[2mm] &= \frac{(b-1)\,b^{\,m-1}}{b^{m}-1} \frac{1-b^{-2m}}{1-b^{-2}}. \tag{7} \end{aligned} $$ For $b=10$, $m=2$ this reduces to $$ \bar d_{10}= \frac{9\cdot10^{1}}{10^{2}-1}\, \frac{1-10^{-4}}{1-10^{-2}} =\frac{90}{99}\cdot\frac{0.9999}{0.99} =\frac{101}{110}\approx0.91818, $$ exactly the value reported in Section 4.4.

   from 6. Worked example: mean $b$‑adic distance for arbitrary base

3. A minimal experiment would construct a depth‑3 radix tree (10 × 10 × 10 nodes) and measure the average lookup time for queries based on trailing‑digit similarity versus a flat hash table of the same size.

   from (c) Smallest Next Experiment

4. Two quantitative illustrations are derived analytically: (1) exact counts of ultrametric distances for all unordered pairs of decimal integers $0\le x\lt y\le999$, yielding $450\,000$ pairs at distance $1$, $45\,000$ at $0.1$, and $4\,500$ at $0.01$; (2) the mean $b$‑adic distance for the interval $0\le x\lt y\le99$ is $\frac{101}{110}\approx0.918$, the diameter‑normalized Euclidean mean is $\frac{101}{297}\approx0.340$, giving a ratio of exactly $2.70$, and all $855$ pairs with Euclidean distance $\le9$ are maximally distant ($d_b=1$).

   from Abstract

5. Analytic computation of the mean $b$‑adic distance for decimal integers $0$–$99$, the normalized Euclidean mean, and the ratio of the two (Section 5).

   from 1. Introduction

6. Normalizing each Euclidean distance by $99$ and averaging yields $$ \bar d_{\text{Euc}}=\frac{166\,650}{4\,950\times99} =\frac{166\,650}{490\,050} =\frac{101}{297}\approx0.34006734. $$ Ratio of means. $$ \frac{\bar d_{10}}{\bar d_{\text{Euc}}} =\frac{101/110}{101/297} =\frac{297}{110}=2.70. $$ Pairs with Euclidean distance $\le9$.

   from 4.4 Mean‑Distance Analysis for Decimal Integers $0$–$99$

How it is done

1. Formal proof that digit‑string comparison in any base $b$ defines a non‑Archimedean valuation and satisfies the ultrametric axioms (Section 4).

   from 1. Introduction

2. We work entirely analytically.

   from 3. Methods

3. By definition $v_b(x-y)\in\mathbb{N}\cup\{\infty\}$, hence $b^{-v_b(x-y)}\ge0$.

   from 4.1 Valuation Axioms

4. We set $b=10$ and $w=2$ (two‑digit width).

   from 4.4 Mean‑Distance Analysis for Decimal Integers $0$–$99$

What would falsify it

1. The main claim—that the native metric of positional notation is the $b$‑adic ultrametric—would be falsified if a concrete counterexample were found where the strong triangle inequality fails for the distance $d_b$ defined above.

   from Falsifiability

2. Since the proof in Section 4.1 is purely algebraic, any such counterexample would imply an error in the definition of $v_b$ or in the divisibility argument.

   from Falsifiability

3. The experiment would falsify the claim if the tree does not outperform the hash table in lookup speed for suffix‑based queries.

   from (c) Smallest Next Experiment

Where the claim stops

1. The proof assumes an integer base; non‑integer or mixed‑radix systems require separate treatment.

   from Limitations

2. Future work should explore algorithmic exploitation of the ultrametric hierarchy and empirical validation in real‑world systems.

   from 8. Conclusion

3. This work identifies the concrete gap that the native metric induced by digit‑string structure has never been formalized, leading to hidden assumptions about distance and clustering.

   from Abstract

Where to read

1. Abstract170 words · 1 min
2. 1. Introduction218 words · 1 min
3. 2. Background and Related Work307 words · 1 min
4. 3. Methods169 words · 1 min
5. 4. Analysis488 words · 2 min
6. 5. Results72 words · 1 min
7. Appendix C. Extended Derivations and Worked Examples735 words · 3 min
8. 6. Discussion227 words · 1 min
9. 7. Practical Implications and Engineering Pathways283 words · 1 min
10. 8. Conclusion95 words · 1 min
11. Provenance and AI Involvement225 words · 1 min