Frequency as Valuation Theory: The Rational Ratio at the Heart of Physical Law

In the paper's words

Frequency in dimensionless Planck units is a rational ratio a/b in Q. The Compton frequency of a fundamental particle is its invariant identity — its prime spectrum.

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

Numbers it reports

1. This framework would be DISCONFIRMED if: Any fundamental particle mass in Planck units were experimentally shown to be an irrational number with high confidence (requires precision $\sim 10^{-20}$ or better) The continued fraction expansion of a fundamental dimensionless constant ($R\infty /E_{P}, \alpha , m_{e}/m_{\mu }$) fails to terminate after extending beyond measurement noise The prime spectrum approach fails to reproduce the Koide formula to within 0.1% when a concrete derivation is attempted The Bekenstein bound is VIOLATED for a discrete quantum system (not approximated — genuinely violated)

   from 8.2. Disconfirmation Conditions

2. The conventional representation as a real number (e.g., the hydrogen 21-cm line at 1420.405751768 MHz) is the Archimedean completion's projection of this rational at ONE place (the infinite place).

   from 1. Introduction

3. If these are rational, each mass encodes a prime spectrum: Particle Mass ($\mathrm{MeV}/c^{2}$) $m/m_{P}$ (approx) Rational candidate Electron 0.511 ~4.19 × $10^{-23}$ $\omega _{e} = p_{e} / q_{e}$ (unknown) Muon 105.66 ~8.66 × $10^{-21}$ $\omega _{\mu } = p_{\mu } / q_{\mu }$ (unknown) Tau 1776.86 ~1.46 × $10^{-19}$ $\omega _{\tau } = p_{\tau } / q_{\tau }$ (unknown) Up quark 2.16 ~1.77 × $10^{-22}$ ...

   from 4.4. Particle Spectrum as Rational Number Table

4. If a concrete derivation is attempted and successfully reproduces the Koide formula to within 0.1%, this would constitute strong evidence for the framework.

   from 8.1. Falsifiable Predictions

5. Second, the bound contains $\pi$ — a provably irrational number (Lambert, 1761).

   from 7.3. Connection to the Bekenstein Bound

6. Claim Status Frequency = rational ratio in Planck units [speculative — assumes discrete time] Compton frequency = particle's prime spectrum [not yet falsifiable — requires $\sim 10^{-20}$ mass precision] Product formula = conservation law [speculative — no dynamical mechanism specified] Koide formula derivable from valuations [my conjecture — derivation to be attempted] Bekenstein bound as exact integer inequality [speculative — testable with near-term quantum simulation]

   from 8.3. Current Status

How it is done

1. The ODR thesis series has established that (1) all physical quantities become dimensionless pure numbers in Planck units [1], (2) Ostrowski's theorem classifies all completions of the rational numbers Q on equal footing — real and p-adic [2], and (3) the Bruhat-Tits tree provides the natural coordinate system for dimensionless ratios carrying p-adic valuation structure [3,4].

   from 1. Introduction

2. The Archimedean real numbers we measure are approximations of these rationals at finite experimental precision.

   from 1.1. Analog of the Rydberg Constant

3. Section 2 reviews the mathematical foundation: Ostrowski's theorem, p-adic valuations, and the Bruhat-Tits tree.

   from 1.2. Structure of the Paper

4. Theorem (Ostrowski, 1916 [2]): Every nontrivial absolute value on the rational numbers Q is equivalent to either the real absolute value $|\cdot |\infty$ or a p-adic absolute value |·|p for some prime p.

   from 2.1. Ostrowski's Theorem

What would falsify it

1. The derivation has not yet been performed — this is a conjectured consequence of the framework, not a verified prediction.

   from 8.1. Falsifiable Predictions

2. If attempted and it fails, this would disconfirm the conjecture.

   from 8.1. Falsifiable Predictions

3. The distribution of mass ratios should show statistically significant peaks at simple rationals (2/1, 3/2, 4/3,...) beyond what random real numbers would produce. [This is testable with current PDG data — see below.] P4 — Rydberg-Ritz Precision Prediction: The Rydberg constant, when expressed as a dimensionless rational in Planck units, should converge to a simple rational number as measurement precision improves.

   from 8.1. Falsifiable Predictions

Where the claim stops

1. The Archimedean real numbers we measure are approximations of these rationals at finite experimental precision.

   from 1.1. Analog of the Rydberg Constant

2. It is included here as a direction for future work.

   from 4.3. Gauge Charges as Valuation Structure

3. The challenge: these are known only to finite precision (typically $10^{-6}$ relative uncertainty for charged leptons, worse for quarks).

   from 4.4. Particle Spectrum as Rational Number Table

Where to read

1. 1. Introduction454 words · 2 min
2. 2. Mathematical Preliminaries282 words · 1 min
3. 3. Frequency as Valuation Theory264 words · 1 min
4. 4. Compton Frequency as Prime Spectrum360 words · 2 min
5. 5. Fundamental Particles vs. Quasiparticles387 words · 2 min
6. 6. The Rydberg Spectrum as Rational Function273 words · 1 min
7. 7. The Product Formula as Conservation Law487 words · 2 min
8. 8. Predictions and Falsifiability419 words · 2 min
9. 9. Conclusion206 words · 1 min
10. Declarations66 words · 1 min