The Distinction-Based Ultrametric: A Hierarchy Distance Without Primes, and the Statistical Test of Arithmetic Structure in Physical Spectra

In the paper's words

Quantum mechanics is written in one completion of the rational numbers, the real line, though Ostrowski's theorem gives one completion for every prime. A line of work reads physical structure through those other places, but its empirical tests of prime-specific geometry in physical systems have repeatedly returned null results. This paper separates what survives from what does not.

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

Numbers it reports

1. A generic clock-rest coupling violates Parisi ultrametricity in 29-35% of simulated instances.

   from 3. What has been tested, and what has not

2. The ultrametric-QEC independent-error threshold is $2.0\times10^{-4}$, roughly fifty-five times below the surface-code threshold.

   from 3. What has been tested, and what has not

How it is done

1. The complex Hilbert space of the standard formalism is built on the real line, and the real line is one completion among all of them: Ostrowski's theorem classifies the completions of the rational numbers as the real line and one p-adic field per prime.

   from 1. The Archimedean shadow

2. A hierarchy-detection toolbox — the distinction-based distance plus an ultrametricity test — classifies a dataset as hierarchical or not; the auditable-attention proof of concept is the same distance applied to a transformer.

   from 7. What a practitioner can do

What would falsify it

1. The tests that have not been run are statistical — whether the distribution of a large spectrum carries arithmetic information beyond what universal random-matrix theory predicts.

   from 1. The Archimedean shadow

2. The formula requires the tree, and a deposited computation exhibits the counterexample.

   from 2. The distinction-based ultrametric formula

3. An anharmonic mass ladder was falsified by its pre-registered null.

   from 3. What has been tested, and what has not

Where the claim stops

1. The GUE null is realized by a GOE ensemble, whose bulk pair correlation is identical to GUE; the ensemble difference enters only at the arithmetic-correction order, which this test does not probe.

   from 10. Limitations

2. The Fenna-Matthews-Olson complex has seven sites; seven levels cannot support a correlation function.

   from 3. What has been tested, and what has not

3. If a large-N physical spectrum shows pure GUE statistics with no arithmetic corrections, the claim of a physics-relevant arithmetic substrate is falsified at the distribution level, not merely at the level of finite geometric approximation.

   from 4. The surviving empirical claim

Where to read

1. 1. The Archimedean shadow281 words · 1 min
2. 2. The distinction-based ultrametric formula288 words · 1 min
3. 3. What has been tested, and what has not260 words · 1 min
4. 4. The surviving empirical claim206 words · 1 min
5. 5. Computational verification230 words · 1 min
6. 6. Real data196 words · 1 min
7. 7. What a practitioner can do116 words · 1 min
8. 8. Where the premises end85 words · 1 min
9. 9. Related work107 words · 1 min
10. 10. Limitations119 words · 1 min
11. 11. Conclusion106 words · 1 min