Optimal Radix q under Resource Constraints: A Reconciled Derivation Framework for Ultrametric Hierarchies

In the paper's words

Hierarchical (ultrametric) representations organize N items in a q-ary tree of depth n = ⌈ln N / ln q⌉, and the choice of branching factor q determines how storage, interconnection, and traversal resources scale. Prior ultrametric modeling has treated the radix as a fixed structural parameter inherited from p-adic conventions, while the resource-constrained design literature treats branching only implicitly. This paper asks: for a fixed number of leaves N and an explicit cost model, what branching factor q minimizes total resource cost?

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

Numbers it reports

1. Exhaustive exact arithmetic for $N = 10^{6}$ with $c_{0}/c_{1} = 10$ yields $q = 4$, total cost 1,957,341.4 cost units, a 22.2% saving over the binary default $q = 2$.

   from Abstract

2. We also analyze, as a clearly labeled alternative convention, a budget-constrained gain model with exponential-in-depth cost, under which the radix choice becomes a second-order effect (optimum $q = 6$ at baseline assumptions, with a 1.9% margin over the next-best radix).

   from Abstract

3. Versus the binary default: $(2,516,581.2 - 1,957,341.4)/2,516,581.2 = 559,239.8/2,516,581.2 = 0.222$, i.e., a 22.2% cost reduction .

   from 4.2 Worked example: $N = 10^{6}, c_{0} = 1, c_{1} = 0.1$

4. The exhaustive search over all $q \geq 2$ therefore gives the optimum $q = 6$, with node efficiency 9331 nodes per $10^{7} J$ (9.33 × $10^{-4}$ nodes/J) versus 8191 for $q = 2$ (ratio $9331/8191 = 1.14\times$); in information terms the advantage over $q = 2$ is $8.959/8.317 = 1.077$, i.e., 7.7%, and the margin over the next-best radix $q = 3$ is $8.959/8.789 = 1.019$, i.e., 1.9%.

   from 4.3 Regime analysis: when does q matter? (alternative convention)

5. The $q = 6$ optimum falls 2.7% short of the exponential-cost bound (8.959 vs 9.210 nats).

   from 4.3 Regime analysis: when does q matter? (alternative convention)

6. Surrogate accuracy: the continuous approximation underestimates the exact cost at $q = 4$ by 4.9% but correctly identifies the optimum.

   from 5. Results

How it is done

1. The mathematical properties of ultrametrics as zero-dimensional analogues of ordinary metrics — including ultrametric versions of the Arens–Eells isometric embedding theorem, the Hausdorff extension theorem, and the Niemytzki–Tychonoff characterization of compactness — are established in [3].

   from 2. Background and Related Work

2. Problem statement (primary convention).

   from 3. Methods

3. Under the exponential cost model $C = c_{0}' q^{\alpha L}$ with budget B, the binding constraint gives $L = \ln(B/c_{0}')/(\alpha \ln q)$ and optimal gain $G = \ln(B/c_{0}')/\alpha$, independent of q.

   from 4.3 Regime analysis: when does q matter? (alternative convention)

What would falsify it

1. We situate the result within ultrametric embedding theory, stochastic resource-spending models, resource-constrained experimental design, and joules-per-solution benchmarking, and we identify falsification conditions and failure modes.

   from Abstract

2. A sensitivity analysis showing the optimum depends only on the cost ratio, not on N, and a discussion of failure modes and falsification conditions.

   from 1. Introduction

3. The surrogate $h(4) = 1.4 \times 4/3 = 1.8667$ predicts E ≈ N·h = 1,866,667 versus exact 1,957,341 — a 4.9% underestimate, arising because $4^{10}$ exceeds N by 4.86% and lower-level nodes add the balance.

   from 4.2 Worked example: $N = 10^{6}, c_{0} = 1, c_{1} = 0.1$

Where the claim stops

1. A quadratic child-cost term $c_{1}q + c_{2}q^{2}$ would change the stationarity condition to a cubic; the qualitative conclusion — an interior optimum depending on a cost ratio — survives, but the specific number 4 does not.

   from 6. Discussion

2. The alternative convention's cost function is likewise assumed, not measured: $c_{0}' = 10^{3}$ J/node is an order-of-magnitude anchor to [12], and if true costs scale as $q^{\alpha L}$ with $\alpha$ near 1, the q-choice becomes nearly irrelevant and the optimization collapses to a rounding exercise.

   from 6. Discussion

3. A level-dependent branching factor — wide at the top, narrow at the leaves — is a genuine generalization our uniform-q model cannot capture.

   from 6. Discussion

Where to read

1. Abstract238 words · 1 min
2. 1. Introduction447 words · 2 min
3. 2. Background and Related Work817 words · 4 min
4. 3. Methods505 words · 2 min
5. 4. Analysis1,088 words · 5 min
6. 5. Results243 words · 1 min
7. 6. Discussion753 words · 3 min
8. 7. Conclusion199 words · 1 min