The Compton-Counting Ontology and the Bruhat-Tits Tree

In the paper's words

The Compton frequency of a particle, normalized by the Planck frequency, yields a dimensionless rational number --- the Compton count. We argue that this count subsumes every other physical quantity as a rational function, that the physical number system is therefore [\Q\]{.math .inline} rather than the full Archimedean continuum [\R\]{.math .inline}, and that the natural coordinate system for the non-Archimedean completions of [\Q\]{.math .inline} is the Bruhat--Tits tree --- a regular [\(p+1)\]{.math .inline}-valent combinatorial tree with no privileged origin, axes, or absolute scale. The chain --- Compton count [\\→\]{.math .inline} rational number [\\→\]{.math .inline} prime factorization [\\→\]{.math .inline} [\p\]{.math .inline}-adic valuation [\\→\]{.math .inline} Bruhat--Tits tree depth [\\→\]{.math .inline} non-anthropocentric coordinate system --- appears in no external paper across six litera...

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

Numbers it reports

1. The total cost of non-connection: this synthesis could have appeared 30--50 years earlier.

   from 4. The Century-Long Silo Failure

2. We recommend: 40% Compton ontology extension (immediate, interpretive), 25% place-democracy formalization, 20% BT-coordinate development, 10% $\mathbb{Q}$-over-$\mathbb{R}$ foundations, and 5% hedge allocation for unknown candidates.

   from 5.3 Research Effort Allocation

3. These are rational numbers (or, more precisely, computable reals with rational definitions --- we address the Archimedean projection in §2.3). [established — standard quantum mechanics with Planck units] The key ontological claim: the Compton count is the ONLY physical quantity that is not a ratio of anything more basic.

   from 2.1 The Compton Count as Fundamental Integer

4. The primes $\{2,3,5\}$ define the Pythagorean semigroup $\mathcal{P} = \{2^a \cdot 3^b \cdot 5^c\}$ (5-smooth numbers), which has special status in the adelic factorization of physical mass ratios. [speculative — active QNFO research program; see ACRP papers for 5-smooth mass-ratio analysis]

   from 3.1 From Compton Count to p-Adic Valuation

How it is done

1. Keywords: Compton frequency, Bruhat--Tits tree, $p$-adic physics, dimensionless reformulation, place-democracy, Ostrowski's theorem, Zitterbewegung, combinatorial coordinates

   from Abstract

2. We assess four paradigm-shift candidates through the structured forecast protocol (see companion artifact deep-research.md for the full 11-stage protocol).

   from 5. Structured Forecast

3. Under pessimistic perturbations (all judgments at lower bounds), the ranking shifts to B \> D \> C \> A --- the foundational $\mathbb{Q}$-over-$\mathbb{R}$ claim survives even if the specific Compton-ontology and BT-coordinate claims fail.

   from 5.1 Sensitivity Analysis

4. The AI conducted the external literature search (six APIs), performed the cross-domain consilience gate analysis, executed the structured forecast protocol, and assisted with manuscript formatting.

   from Declarations

What would falsify it

1. Candidate A (BT tree as physical coordinates) is the highest-impact but longest shot --- it requires a testable BT-scattering prediction that current Lorentz-invariance constraints (~$10^{-20}$ precision) do not exclude.

   from 5. Structured Forecast

2. Candidate B ($\mathbb{Q}$ over $\mathbb{R}$) is the most provocative but hardest to test; it is a foundational claim about what numbers exist in physics, not a prediction about what numbers will be measured.

   from 5. Structured Forecast

3. The Bruhat--Tits tree as physical coordinates (Candidate A) is the most ambitious and fragile candidate, requiring a testable BT-scattering prediction that survives Lorentz-invariance constraints.

   from 6. Conclusion

Where the claim stops

1. Recognizing this structural isomorphism is not only a matter of intellectual housekeeping; it opens the door to transferring decades of results from each silo into the others --- and to reformulating physics in a coordinate system that does not privilege the Archimedean completion.

   from 6. Conclusion

2. The dimensionless reformulation achieves place-democracy: the formula carries no assumption about which completion is being used. [established — ODR v4.0.4; Ostrowski, 1916] This paper extends the dimensionless program by identifying the ontological substrate beneath the ratios.

   from 1. Introduction

3. The Standard Model's dimensionless parameters become statements about the relative cycle-counts of different field modes. [speculative — the mapping from SM parameters to Compton-count ratios is not yet constructed]

   from 2.2 Ratio Ontology

Where to read

1. Abstract174 words · 1 min
2. 1. Introduction282 words · 1 min
3. 2. The Compton-Counting Ontology452 words · 2 min
4. 3. The Bruhat--Tits Tree as a Non-Anthropocentric Coordinate System460 words · 2 min
5. 4. The Century-Long Silo Failure247 words · 1 min
6. 5. Structured Forecast444 words · 2 min
7. 6. Conclusion232 words · 1 min
8. Declarations177 words · 1 min