Frequencies of Events in the Actual World: A Reconciled Cross-Domain Framework with Explicit Arithmetic

In the paper's words

Every empirical discipline counts events, but the practice of measuring event frequencies is fragmented across fields that rarely compare methods. This paper develops a cross-domain framework for event-frequency observation in the actual world, organized around three questions: what counts as an event, what sampling process generates the observed counts, and what a claimed frequency structure requires as evidence. We formalize event observation as a detected point process, distinguish Archimedean (freely placed) from radix‑locked (p‑adic) frequency parameterizations, and impose a single statistical discipline: Poisson count modeling with explicit sample‑size requirements.

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

Numbers it reports

1. A burst at 132 MHz exists as an “event” only relative to an instrument with a specific band, sensitivity, and trigger logic; a “real‑world event” in social media exists only relative to a keyword lexicon and a platform’s data access.

   from 1. Introduction

2. The paper's contributions are: (1) a unified event‑frequency formalism applicable to linguistic, social, acoustic, simulated, and physical event streams (Section 3); (2) explicit arithmetic for the radix‑locked angular frequencies, their ratios, Poisson probabilities, zero‑event upper limits, and the sample‑size requirement $N \gtrsim 1/\epsilon^{2}$ (Section 4); (3) a worked radix‑locking test on the four quiet‑corona radio‑impulsive frequencies recorded by the Murchison Widefield Array at 98, 120, 132, and 160 MHz [6], showing that the channel set is not radix‑locked (Section 4); and (4) a discussion of failure modes and falsification conditions (Section 6).

   from 1. Introduction

3. In its evaluation, [1] reports that more than 80% of the learned relations are assessed as causal, which grounds the linguistic event ontology: an event in text is individuated by causal role, and frequency counts over such corpora therefore count causal units.

   from 2. Background and Related Work

4. The $p$‑adic CMB bispectrum search [10] supplies the quantitative anchor: radix‑locked oscillations with angular frequency $\omega_p = 2\pi/\ln(p)$ for $p \in \{2,3,5,7\}$, upper bounds $\epsilon_p \lt 2.5$ at 95 % confidence, no detection, within a single‑modulation model.

   from 2. Background and Related Work

5. The zero‑event 95 % upper limit is $\lambda_D^{\text{UL}} = 2.996/T_{\text{obs}}$, and the sample‑size requirement for resolving a fractional modulation of amplitude $\epsilon$ is $N \gtrsim 1/\epsilon^{2}$.

   from 5. Results

6. Because $\omega_p = 2\pi/\ln(p)$, ratios of $\omega$’s are inverse log ratios: $$ \frac{\omega_2}{\omega_3} = \frac{\ln 3}{\ln 2} = \frac{1.098612}{0.693147} = 1.584962 = \log_2 3, $$ $$ \frac{\omega_2}{\omega_5} = \frac{1.609438}{0.693147} = 2.321929 = \log_2 5, \qquad \frac{\omega_3}{\omega_5} = \frac{1.609438}{1.098612} = 1.464974 = \log_3 5. $$ These irrational logarithm‑ratio values are the signature of radix‑locking: a radix‑locked spectrum is not harmonically spaced, since $\omega_2/\omega_3 = \log_2 3 \approx 1.584962$ is not a ratio of small integers.

   from 4.1 Radix‑locked angular frequencies

How it is done

1. As worked examples we compute the radix‑locked angular frequencies $\omega_p = 2\pi/\ln(p)$ for $p \in \{2,3,5,7\}$, obtaining $\omega_2 = 9.064722$ through $\omega_7 = 3.228919$; we convert the impulsive radio‑event frequencies $\{98, 120, 132, 160\}$ MHz recorded in the quiet solar corona to angular frequencies, averaging $\overline{\Omega} = 8.01\times10^{8}\ \mathrm{rad\,s^{-1}}$; we show by two complementary tests that this frequency set carries no radix‑locked structure; and we derive Poisson zero‑event upper limits and the sample‑size rule $N \gtrsim 1/\epsilon^{2}$.

   from Abstract

2. But the frequency of an event is not a property of the world alone; it is a property of the joint system of world, detector, sampling protocol, and counting convention.

   from 1. Introduction

3. Our radix‑locking tests are a narrower, parametric cousin: where [5] compares whole frequency distributions against a universal prior, we test a specific one‑parameter family of log‑periodic structures.

   from 2. Background and Related Work

4. We apply two complementary tests.

   from 3.2 Archimedean versus radix‑locked parameterizations

What would falsify it

1. The framework converts an open question about event frequencies in the actual world into falsifiable, arithmetically explicit tests, and its contribution is methodological rather than empirical.

   from Abstract

2. The paper's contributions are: (1) a unified event‑frequency formalism applicable to linguistic, social, acoustic, simulated, and physical event streams (Section 3); (2) explicit arithmetic for the radix‑locked angular frequencies, their ratios, Poisson probabilities, zero‑event upper limits, and the sample‑size requirement $N \gtrsim 1/\epsilon^{2}$ (Section 4); (3) a worked radix‑locking test on the four quiet‑corona radio‑impulsive frequencies recorded by the Murchison Widefield Array at 98, 120, 132, and 160 MHz [6], showing that the channel set is not radix‑locked (Section 4); and (4) a discussion of failure modes and falsification conditions (Section 6).

   from 1. Introduction

3. This does not refute the AQN interpretation of the events themselves [6]; it establishes only that the observing frequencies do not carry $p$‑adic log‑periodic structure.

   from 4.3 Radix‑locking tests on the MWA frequency set

Where the claim stops

1. (ii) With only four frequencies, the tests have limited power, and this paper derives no power analysis: a radix‑locked set with missing members could fail the consecutive‑spacing test while passing the pairwise test, and vice versa.

   from 6. Discussion

2. As a rough order‑of‑magnitude false‑positive estimate for Test 1, if residuals under the Archimedean null are taken as approximately uniform on $[0, 0.5]$, the probability that a single residual falls below $\epsilon = 0.15$ is about $0.15/0.5 = 0.3$, so the probability that all six pairwise residuals fall below tolerance is roughly $0.3^{6} \approx 7.3\times10^{-4}$.

   from 6. Discussion

3. This estimate rests on the uniformity assumption and is not a derived power analysis, which remains an open question.

   from 6. Discussion

Where to read

1. Abstract192 words · 1 min
2. 1. Introduction392 words · 2 min
3. 2. Background and Related Work1,055 words · 5 min
4. 3. Methods508 words · 2 min
5. 4. Analysis700 words · 3 min
6. 5. Results235 words · 1 min
7. 6. Discussion370 words · 2 min
8. 7. Practical Implications and Engineering Pathways333 words · 1 min
9. 8. Conclusion71 words · 1 min
10. Provenance and AI Involvement225 words · 1 min
11. Appendix A. Divergence report71 words · 1 min
12. Appendix B. Claim attribution144 words · 1 min