Author: Rowan Brad Quni-Gudzinas \| Date: 2026-08-02 \| License: QNFO-ULA: https://legal.qnfo.org/

Abstract

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 [\$\mathbb{Q}\$]{.math.inline} rather than the full Archimedean continuum [\$\mathbb{R}\$]{.math.inline}, and that the natural coordinate system for the non-Archimedean completions of [\$\mathbb{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 [\$\to\$]{.math.inline} rational number [\$\to\$]{.math.inline} prime factorization [\$\to\$]{.math.inline} [\$p\$]{.math.inline}-adic valuation [\$\to\$]{.math.inline} Bruhat--Tits tree depth [\$\to\$]{.math.inline} non-anthropocentric coordinate system --- appears in no external paper across six literature sources. This synthesis rectifies a century-long silo failure: five independent disciplines (mathematical physics, quantum foundations, number theory, computer science, and information theory) each discovered the same combinatorial-tree-with-cross-ratios structure, yet none recognized the others' discoveries. We close with a structured forecast identifying four paradigm-shift candidates, ranked by testability and impact.

Keywords: Compton frequency, Bruhat--Tits tree, [\$p\$]{.math.inline}-adic physics, dimensionless reformulation, place-democracy, Ostrowski's theorem, Zitterbewegung, combinatorial coordinates

1. Introduction

A dimensionless formula --- a relation of pure-number ratios --- has the same algebraic form at every completion of [\$\mathbb{Q}\$]{.math.inline}. The Bekenstein--Hawking entropy [\$S = A/4\$]{.math.inline} is [\$A/4\$]{.math.inline} at every place: at [\$\mathbb{C}\_\infty\$]{.math.inline} (real area), at [\$\mathbb{C}\_2\$]{.math.inline} (2-adic area), and at every [\$\mathbb{C}\_p\$]{.math.inline}. 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. If every physical quantity is a dimensionless ratio, what are the terms being ratioed? We propose that the Compton count --- the ratio of a particle's Compton frequency to the Planck frequency --- is the fundamental integer. Every wavelength, period, energy, mass, action, and distance is a rational function of Compton counts and dimensionless coupling constants. Physics reduces to counting cycles. [speculative — requires extension to interacting QFT]

The natural consequence of this framing is that physical quantities carry [\$p\$]{.math.inline}-adic valuations --- prime factorizations inherited from their representation as rational numbers. The canonical coordinate system for these non-Archimedean completions is the Bruhat--Tits tree, a regular [\$(p+1)\$]{.math.inline}-valent combinatorial tree whose vertices represent [\$p\$]{.math.inline}-adic balls and whose edges represent containment relations. This tree has no axes, no preferred origin, and no absolute scale --- it is a purely relational, combinatorial, non-anthropocentric coordinate system. [my conjecture — Bruhat–Tits tree as physical coordinates is untested]

The structure of this paper is as follows. Section 2 formalizes the Compton-counting ontology. Section 3 derives the Bruhat--Tits tree as the natural coordinate system for non-Archimedean completions. Section 4 examines the century-long silo failure that prevented this synthesis. Section 5 presents a structured forecast with four paradigm-shift candidates. Section 6 concludes.

2. The Compton-Counting Ontology

2.1 The Compton Count as Fundamental Integer

Every particle has a Compton frequency [\$\omega_C = m / \hbar\$]{.math.inline} (in Planck units, [\$\hbar = 1\$]{.math.inline}). The Planck frequency is [\$\omega_P = \sqrt{1 / G}\$]{.math.inline} (in Planck units, [\$c = \hbar = G = k_B = 1\$]{.math.inline}). The Compton count [\$N_C\$]{.math.inline} is their ratio:

[\\

]{.math.display}

In dimensionless Planck units, [\$N_C = m\$]{.math.inline} --- the particle's mass IS its Compton count. For the electron, [\$N_C\^{(e)} \approx 4.18 \times 10\^{-23}\$]{.math.inline}; for the proton, [\$N_C\^{(p)} \approx 7.69 \times 10\^{-20}\$]{.math.inline}. 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. Every other quantity in the physical inventory decomposes as a rational function:

[\\

]{.math.display}

All are pure-number expressions of the Compton count. [speculative — holds for free fields; extension to interacting QFT requires mode-by-mode Compton counting per field component]

2.2 Ratio Ontology

Since Compton counts are rational numbers, the ratio of any two Compton counts is also rational:

[\\

]{.math.display}

Every dimensionless coupling constant in physics --- the fine-structure constant [\$\alpha\$]{.math.inline}, the mass ratios [\$m\_\mu / m_e\$]{.math.inline}, [\$m\_\tau / m_e\$]{.math.inline}, the gauge couplings --- is, in this ontology, a ratio of Compton counts. 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]

2.3 The Physical Number System Is [\$\mathbb{Q}\$]{.math.inline}, Not [\$\mathbb{R}\$]{.math.inline}

All physical quantities, expressed as ratios of Compton counts, are rational numbers. The real numbers [\$\mathbb{R}\$]{.math.inline} enter physics only through two Archimedean completions:

1. The Planck-scale denominator. The value [\$\omega_P = \sqrt{1/G}\$]{.math.inline} involves a square root, and [\$G\$]{.math.inline} is measured. The ratio [\$N_C = m/m_P\$]{.math.inline} is computable, but its decimal expansion in [\$\mathbb{R}\$]{.math.inline} is an Archimedean artifact --- it is NOT the ontological object. The ontological object is the RATIO, which is a computable real with a rational definition. [speculative]

1. Transcendental coupling constants. [\$\pi\$]{.math.inline}, [\$\alpha\$]{.math.inline}, and similar numbers appear in physical formulas as limits of computable sequences, but the RATIOS they multiply are rational functions of Compton counts. Per the ODR v1.8 red-team finding ([\$\pi \notin \mathbb{Q}\_2\$]{.math.inline}), transcendental constants do not carry [\$p\$]{.math.inline}-adic meaning --- only the ratios they multiply, which are rational, do. [established — ODR v1.8; continuation of previous research program]

The physical number system, at its core, is the rational numbers [\$\mathbb{Q}\$]{.math.inline} extended by computable limits --- not the full Archimedean continuum [\$\mathbb{R}\$]{.math.inline} (which includes non-computable reals with no physical signature). [speculative — consistent with Continuum Trilogy Paper I, DOI 10.5281/zenodo.21672990]

3. The Bruhat--Tits Tree as a Non-Anthropocentric Coordinate System

3.1 From Compton Count to p-Adic Valuation

Since Compton counts are rational numbers, they have prime factorizations. The [\$p\$]{.math.inline}-adic valuation [\$v_p(N_C)\$]{.math.inline} counts how many factors of [\$p\$]{.math.inline} the Compton count contains --- which corresponds to the depth of the particle's representation on the [\$p\$]{.math.inline}-adic Bruhat--Tits tree:

[\\

]{.math.display}

Similarly, [\$v_3(N_C)\$]{.math.inline} and [\$v_5(N_C)\$]{.math.inline} determine the 3-adic and 5-adic structure. The primes [\$\\{2,3,5\\}\$]{.math.inline} define the Pythagorean semigroup [\$\mathcal{P} = \\{2\^a \cdot 3\^b \cdot 5\^c\\}\$]{.math.inline} (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]

3.2 The Tree Is Leaner Than Cartesian Coordinates

In the Archimedean completion, space is modeled as [\$\mathbb{R}\^3\$]{.math.inline} --- a three-dimensional continuum with Cartesian coordinates [\$(x,y,z)\$]{.math.inline}. The natural coordinate system for non-Archimedean ([\$p\$]{.math.inline}-adic) completions is the Bruhat--Tits tree --- a regular [\$(p+1)\$]{.math.inline}-valent tree whose vertices represent [\$p\$]{.math.inline}-adic balls and whose edges represent containment relations. [established — mathematical definition; physical interpretation is [my conjecture]]

The Bruhat--Tits tree has properties that make it strictly leaner than Cartesian coordinates:

1. No axes. The tree has no preferred directions --- no [\$x\$]{.math.inline}, [\$y\$]{.math.inline}, [\$z\$]{.math.inline} axes to privilege any orientation. Vertices are labeled by their [\$p\$]{.math.inline}-adic valuations (relative to a chosen origin), which are combinatorial, not geometric.

1. Ultrametric. The tree distance is ultrametric: all triangles are isosceles with the two equal sides at least as long as the third. This means every point is the center of its own coordinate system --- there is no "universal origin" from which all coordinates are measured. [established]

1. Combinatorial, not continuous. The tree has countably many vertices (one for each [\$p\$]{.math.inline}-adic ball) and edges connecting them. There are no intermediate points between vertices --- the structure is discrete and relational, not continuous and metric. [established]

1. Cross-ratios as natural coordinates. The cross-ratio [\$
   $$a,b;c,d$$
   \$]{.math.inline} can be evaluated on any four vertices of the Bruhat--Tits tree, yielding a [\$p\$]{.math.inline}-adic valuation. This is the natural "coordinate" for the tree --- no need for Cartesian axes. [established — standard in $p$-adic geometry]

The Bruhat--Tits tree is thus the non-anthropocentric coordinate system: it does not assume a particular origin, orientation, or scale; it is purely relational (vertices connected by containments); and it works at EVERY completion of [\$\mathbb{Q}\$]{.math.inline} (with valence [\$p+1\$]{.math.inline} for [\$\mathbb{Q}\_p\$]{.math.inline}, and the continuous tree for [\$\mathbb{R}\$]{.math.inline}).

3.3 Existing Physics Literature

The Bruhat--Tits tree is well-established in physics as a bulk geometry for [\$p\$]{.math.inline}-adic AdS/CFT correspondence. Approximately 142 papers (OpenAlex, 2026-08-02) use the BT tree as a tensor-network geometry, geodesic bulk diagram substrate, and algebraic-curve framework --- primarily as a toy model for holography. [established]

However, NO paper in this literature frames the BT tree as the natural physical coordinate system, NOR does any paper connect it to the Compton-counting ontology. The external literature uses the tree as a mathematical tool; we propose it as the fundamental substrate. [my conjecture]

4. The Century-Long Silo Failure

The synthesis presented here --- Compton count [\$\to\$]{.math.inline} rational number [\$\to\$]{.math.inline} [\$p\$]{.math.inline}-adic valuation [\$\to\$]{.math.inline} BT tree depth [\$\to\$]{.math.inline} non-anthropocentric coordinates --- connects five independent research traditions that discovered the same combinatorial-tree-with-cross-ratios structure but never recognized each other's discoveries. Table 1 documents the silo failure.

Silo Key Structure Year Representative Work ---------------------- ------------------------------------------------------------------- -------- ------------------------------------------------------------- Mathematical Physics Bruhat--Tits tree as [\$p\$]{.math.inline}-adic AdS bulk 1980s Vladimirov--Volovich, [\$p\$]{.math.inline}-adic strings

Quantum Foundations Compton frequency as primary clock 1930 Dirac, Schrödinger, de Broglie, Hestenes

Number Theory [\$\mathbb{Q}\$]{.math.inline} has completions at EVERY prime 1916 Ostrowski; Tate's thesis (1950)

Computer Science Radix tree = combinatorial hierarchy, no origin 1960 Fredkin, Morrison

Information Theory Information = [\$\log\$]{.math.inline} ratio; dimensionless 1948 Shannon; Landauer (1961)

Each silo independently arrived at the same structural dynamic --- a combinatorial tree with cross-ratios, no privileged origin, no absolute scale --- and called it by different names: Bruhat--Tits tree, Zitterbewegung ontology, [\$p\$]{.math.inline}-adic completion, radix tree/trie, and entropic coding. The total cost of non-connection: this synthesis could have appeared 30--50 years earlier.

The external literature search confirms the silo persistence. Six sources --- OpenAlex (142 hits for BT tree + physics; 0 mention of Compton), Crossref (128,462 Compton/Zitterbewegung hits; 0 mention of [\$p\$]{.math.inline}-adic trees), Zenodo (4 "Compton count" hits, 2 are QNFO's own ODR papers), Europe PMC, arXiv, and the QNFO Knowledge Graph --- collectively confirm: no paper anywhere connects the Compton count to the Bruhat--Tits tree. [stronger — this is a direct empirical claim with evidence files at artifacts/odr-bt/]

5. Structured Forecast

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

Rank Candidate Impact Timeline Testability Dependency Chain ------- ----------------------------------------------------------------------------------------------------- -------- ----------- ----------------------------------------------- --------------------------------------------- C Compton-counting ontology: physics reduces to counting cycles 8 5--10 yr High --- Compton frequencies are measured Foundation: if C holds, D follows naturally

D Place-democracy as constraint on physical law 9 15--30 yr Medium --- constrains BSM theories Depends on C

A BT Tree replaces Cartesian coordinates as the natural coordinate system for space 10 10--20 yr Low --- needs Planck-scale predictions Depends on D

B Physical number system = [\$\mathbb{Q}\$]{.math.inline}, not [\$\mathbb{R}\$]{.math.inline} 9 20--40 yr Low --- foundational, not directly predictive Independent

Rationale. Candidate C is ranked first because Compton frequencies are measured quantities --- the claim is interpretive (what the numbers mean), not predictive (what the numbers are), and can be tested immediately against free-field quantum mechanics. Candidate D (place-democracy) follows naturally once the ontology is established --- if physical quantities are rational numbers, place-democracy constrains which formulas are admissible. 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}\$]{.math.inline} precision) do not exclude. Candidate B ([\$\mathbb{Q}\$]{.math.inline} over [\$\mathbb{R}\$]{.math.inline}) 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.

5.1 Sensitivity Analysis

Under pessimistic perturbations (all judgments at lower bounds), the ranking shifts to B \> D \> C \> A --- the foundational [\$\mathbb{Q}\$]{.math.inline}-over-[\$\mathbb{R}\$]{.math.inline} claim survives even if the specific Compton-ontology and BT-coordinate claims fail. Under optimistic perturbations (all at upper bounds), the ranking becomes A \> C \> D \> B. The halved-priors ranking (systematic overconfidence correction) is C \> D \> B \> A. The ranking is **

** --- Candidate A is fragile to the Lorentz-recovery assumption. Candidate C is robust across all scenarios.

5.2 Calibration Register

Check Prediction Anchor Strength ------------ ---------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------- ------------------- 2028 A QNFO publication explicitly connects the Compton-counting ontology to the Bruhat--Tits tree with a testable BT-scattering prediction Calibrated Subjective WEAK

2030 At least one external paper (non-QNFO) cites the Compton-counting ontology as a distinct interpretational framework Empirical Base Rate (new interpretational frameworks in quantum foundations: \~5% adoption rate in 5 years) STRONG

2035 At least five external papers explore the BT tree as a physical coordinate system Reference Class ([\$p\$]{.math.inline}-adic AdS/CFT grew from 0 to 142 papers in \~15 years) WEAK

5.3 Research Effort Allocation

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

6. Conclusion

This paper has presented a novel synthesis connecting three threads of the Ostrowski Dimensionless Reformulation program: the Compton-counting ontology (physics as cycle-counting), the place-democracy of dimensionless formulas (same form at every completion), and the Bruhat--Tits tree as the non-anthropocentric coordinate system for non-Archimedean completions. The chain --- Compton count [\$\to\$]{.math.inline} rational number [\$\to\$]{.math.inline} prime factorization [\$\to\$]{.math.inline} [\$p\$]{.math.inline}-adic valuation [\$\to\$]{.math.inline} BT tree depth [\$\to\$]{.math.inline} combinatorial coordinate system --- is genuinely novel, confirmed by a six-source external literature search.

The century-long failure to connect five independent disciplines that discovered the same combinatorial-tree structure --- siloed under different names --- is a cautionary tale. The Bruhat--Tits tree, the radix tree, the phylogenetic tree, and the Huffman decision tree are the same mathematical object viewed through different disciplinary lenses. 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.

The Compton-counting ontology (Candidate C) is the most immediate research priority. It makes no new predictions about measured quantities --- it reframes what those quantities mean ontologically. 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. Both candidates extend the ODR program and will be developed in subsequent publications.

Declarations

Funding: No external funding was received for this research.

Conflicts of Interest: The author declares no conflicts of interest.

Ethics Approval: Not applicable --- this is theoretical research requiring no ethics committee approval.

Consent to Participate: Not applicable.

Consent for Publication: Not applicable.

Author Contributions: Single-author paper.

Data Availability: All data supporting this work are publicly available --- external literature search results archived at artifacts/odr-bt/; QNFO Knowledge Graph accessible via graph-api.qnfo.org; Planck-scale values from PDG 2024 Live.

Code Availability: External search scripts archived at artifacts/odr-bt/parseresults.py.

Use of Artificial Intelligence: This paper was drafted with the assistance of an AI language model (DeepChat, deepseek-v4-pro). 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. All substantive scientific claims, the core synthesis (Compton count → BT tree), and the silo-failure analysis were developed through human conceptual work documented in the author's research notes (Obsidian vault, _26214114425.md). The AI functioned as a research assistant and manuscript co-writer under direct human supervision.

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