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The Compton-Counting Ontology and the Bruhat-Tits Tree

DOI: https://doi.org/10.5281/zenodo.21758752
Published: 2026-08-02

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 [\\(\\omegaC = m / \\hbar\\)]{.math .inline} (in Planck units, [\\(\\hbar = 1\\)]{.math .inline}). The Planck frequency is [\\(\\omegaP = \\sqrt{1 / G}\\)]{.math .inline} (in Planck units, [\\(c = \\hbar = G = kB = 1\\)]{.math .inline}). The Compton count [\\(NC\\)]{.math .inline} is their ratio:

[\\\[NC \\equiv \\frac{\\omegaC}{\\omegaP} = \\frac{m}{mP} = \\frac{m}{\\sqrt{\\hbar c / G}} \\in \\mathbb{Q}\^+\\\]]{.math .display}

In dimensionless Planck units, [\\(NC = m\\)]{.math .inline} --- the particle's mass IS its Compton count. For the electron, [\\(NC\^{(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:

[\\\[\\text{wavelength} = \\frac{1}{NC}, \\quad \\text{period} = \\frac{1}{NC}, \\quad \\text{energy} = N_C, \\quad \\text{action} = 1\\\]]{.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:

[\\\[\\frac{NC\^{(X)}}{NC\^{(Y)}} \\in \\mathbb{Q}\^+\\\]]{.math .display}

Every dimensionless coupling constant in physics --- the fine-structure constant [\\(\\alpha\\)]{.math .inline}, the mass ratios [\\(m\\\mu / me\\)]{.math .inline}, [\\(m\\\tau / me\\)]{.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 [\\(\\omegaP = \\sqrt{1/G}\\)]{.math .inline} involves a square root, and [\\(G\\)]{.math .inline} is measured. The ratio [\\(NC = 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 [\\(vp(NC)\\)]{.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:

[\\\[v2(NC\^{(e)}) = v2(4.18 \\times 10\^{-23}) = v2\\left(\\frac{me}{mP}\\right)\\\]]{.math .display}

Similarly, [\\(v3(NC)\\)]{.math .inline} and [\\(v5(NC)\\)]{.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 \[CONDITIONAL\] --- 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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