Numeral Trees as the Native Ultrametric Geometry of Cyclic Time

In the paper's words

Cyclic timestamps are stored linearly but intrinsically form mixed‑radix positional strings that nest hour, day, month, and finer cycles. Existing pipelines treat timestamps as points on the real line, ignoring this hierarchy and causing mis‑grouping across cycle boundaries. The gap is a lack of a mathematically grounded distance that respects the hierarchical nesting of calendar cycles.

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

Numbers it reports

1. To incorporate mixed radices (e.g., hour = 24, day = 30, month = 12), we treat each level $k$ of the tree as having its own branching factor $b_k$.

   from 3. Methods

2. Hour branching factor: $b_{\text{hour}}=24$.

   from 4.1. Input numbers

3. Day branching factor: $b_{\text{day}}=30$.

   from 4.1. Input numbers

4. Month branching factor: $b_{\text{month}}=12$.

   from 4.1. Input numbers

5. According to the mixed‑radix ultrametric, the distance is the reciprocal of the hour branching factor: $$ d_{\text{mix}}(t_1,t_2)=b_{\text{hour}}^{-1}=24^{-1}= \frac{1}{24}=0.041666\ldots $$ Rounded to seven significant digits: $0.0416667$.

   from 4.2. Derivation steps

6. The day branching factor $30$ is an approximation; real months vary between $28$ and $31$ days.

   from 6.1. Limitations

How it is done

1. Proof that the derived distance satisfies the strong triangle inequality (Section 4).

   from 1. Introduction

2. Theorem 8.9 shows that the quotient map $\kappa\colon T\to B(T,v)$ induces an isomorphism of groupoids, establishing a rigorous bridge between tree combinatorics and ultrametric geometry.

   from 2. Background and Related Work

3. We model a numeral in base $b\ge2$ as a finite path from the root of a rooted $b$‑ary tree to a leaf.

   from 3. Methods

4. We derive explicit distances for a concrete mixed‑radix calendar: hour (base $24$), day (base $30$), month (base $12$).

   from 4. Analysis

What would falsify it

1. If empirical clustering of timestamped events does not improve over Euclidean distance, the ultrametric model would be falsified.

   from 6.2. Failure modes

2. The experiment falsifies the result if the measured latency exceeds that of a conventional linear subtraction by more than a factor of $2$.

   from 7.3. Smallest next experiment

Where the claim stops

1. The day branching factor $30$ is an approximation; real months vary between $28$ and $31$ days.

   from 6.1. Limitations

2. The model assumes a single global time zone and ignores daylight‑saving adjustments, which would introduce additional branching at the hour level.

   from 6.1. Limitations

3. Future work will empirically test the cognitive hypothesis and prototype the hardware counter described in Section 7.3.

   from 8. Conclusion

Where to read

1. Abstract156 words · 1 min
2. 1. Introduction229 words · 1 min
3. 2. Background and Related Work863 words · 4 min
4. 3. Methods176 words · 1 min
5. 4. Analysis163 words · 1 min
6. 5. Results90 words · 1 min
7. 6. Discussion182 words · 1 min
8. 7. Practical Implications and Engineering Pathways272 words · 1 min
9. 8. Conclusion91 words · 1 min
10. Provenance and AI Involvement225 words · 1 min
11. Appendix A. Divergence report119 words · 1 min
12. Appendix B. Claim attribution37 words · 1 min