Compton Frequency Cross-Ratios on Bruhat-Tits Trees: A Pre-Registered Search for Adelic Structure in the Standard Model Mass Spectrum (Version 2.3)

In plain language

The authors set out to see whether the masses of known particles line up with special mathematical numbers that appear in a theory called an “Adelic Theory of Everything.” They found that three particle masses (the tau lepton, the proton, and the bottom quark) match those special numbers within about 3.5 percent, but a later, broader test showed the overall fit was actually worse than random, with an improvement factor of 0.54× and an empirical p‑value of 0.973. In a separate check they identified 156 groups of four particle masses (31.5 % of all possible groups) whose ratios almost equal simple fractions with denominators ≤ 5000 and errors < 10⁻⁴. Among all 12‑choose‑4 possible groups, the combination 976/919 gave the smallest denominator and matched a pattern expected from the theory’s four‑step process, a coincidence the authors say is unlikely to happen by chance. Of the five pre‑registered ratio tests at the primes 2, 3, and 5, two ratios (976/919 and 430/419) came close to simple fractions, just barely beating a standard mathematical guarantee. The work matters because it shows that, despite some intriguing near‑matches, the proposed link between particle masses and these exotic numbers does not hold up under rigorous testing, suggesting that the “Adelic” idea is not a useful guide for understanding the Standard Model.

Restated for a non-expert reader; the original wording follows. Still uses some technical language.

In the paper's words

Version 1.0 of this programme [1] established that physical quantities can be formulated without anthropocentric conventions. Its extension into an "Adelic Theory of Everything" [2] attempted to match Standard Model particle masses to CM j-invariants via decimal PDG data — an approach we now recognize as suffering from multiple-comparisons inflation, absence of a physical mechanism, and the methodological self-contradiction of using base-10 percent errors to critique anthropocentric number syste

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

Numbers it reports

1. Version 1.0's "Adelic Theory of Everything" extension [2] claimed to find three significant alignments between Standard Model particle masses and class number 1 CM j-invariants: the tau lepton mass with $j(D=-7)$, the proton mass with $j(D=-4)$, and the bottom quark mass with $j(D=-8)$ — all within 3.5% in decimal ratio.

   from 1.1 What was claimed

2. An initial unconstrained grid search found exactly this degenerate solution and mistakenly reported it as a "4.8× improvement over random." That result is retracted.

   from 2.3 Anharmonic Oscillator Hypothesis (Added 2026-07-22, corrected after methodological failure)

3. An initial unconstrained grid search over $(\omega_0, \alpha)$ in the anharmonic oscillator model of §2.3 reported a "4.8× improvement over random" ($p\lt 0.001$).

   from 5.6 Harmonic Analysis Results — Retracted and Corrected

4. We compared this fit against 1000 surrogate datasets of log-uniform random masses spanning the identical range as the real Standard Model spectrum: Metric Value Real particle data RMS 107,500 Surrogate median RMS 57,985 Improvement factor 0.54× (real data fits WORSE than random) Empirical $p$-value 0.973 The anharmonic-oscillator hypothesis for Standard Model particle masses is decisively disconfirmed.

   from 5.6 Harmonic Analysis Results — Retracted and Corrected

5. 156 quadruples (31.5%) yield cross-ratios that approximate rational numbers with denominator $q \leq 5000$ and relative error $\lt 10^{-4}$.

   from 5.4 CODATA 2022 / PDG 2025 Replication

6. Its extension into an "Adelic Theory of Everything" [2] attempted to match Standard Model particle masses to CM j-invariants via decimal PDG data — an approach we now recognize as suffering from multiple-comparisons inflation, absence of a physical mechanism, and the methodological self-contradiction of using base-10 percent errors to critique anthropocentric number systems.

   from Abstract

How it is done

1. Its extension into an "Adelic Theory of Everything" [2] attempted to match Standard Model particle masses to CM j-invariants via decimal PDG data — an approach we now recognize as suffering from multiple-comparisons inflation, absence of a physical mechanism, and the methodological self-contradiction of using base-10 percent errors to critique anthropocentric number systems.

   from Abstract

2. Under this pre-registered constraint, we test whether Standard Model particle masses fit the anharmonic ladder better than random surrogate data spanning the same mass range.

   from 2.3 Anharmonic Oscillator Hypothesis (Added 2026-07-22, corrected after methodological failure)

3. The Dirichlet approximation theorem guarantees that for ANY real number $x$ and any integer $Q$, there exists a rational $p/q$ with $q \leq Q$ such that $|x - p/q| \lt 1/(Q(Q+1))$.

   from 5.5 Null-model analysis

4. An initial unconstrained grid search over $(\omega_0, \alpha)$ in the anharmonic oscillator model of §2.3 reported a "4.8× improvement over random" ($p\lt 0.001$).

   from 5.6 Harmonic Analysis Results — Retracted and Corrected

What would falsify it

1. Falsified if an independent precision measurement of the muon, tau, or Higgs mass shifts the cross-ratio outside $[976/919 - 10^{-5}, 976/919 + 10^{-5}]$. [F2] $\text{CR}(u,d,s,c) = 430/419$ is exactly correct.

   from 6. Falsifiability Conditions

2. Falsified if lattice QCD determinations of quark masses shift the cross-ratio outside the error envelope. [F3] The overall null result for the adelic cross-ratio hypothesis can be overturned if: (a) at least 3 of 5 pre-registered cross-ratios show $\text{ord}_p(\text{CR}) \neq 0$ at any prime after mass uncertainties are reduced by a factor of 10, or (b) a mechanism connecting the Bruhat-Tits tree to the Standard Model Lagrangian is discovered.

   from 6. Falsifiability Conditions

3. We report a null result with one weak hint warranting further investigation with expanded pre-registration and independent data.

   from Abstract

Where the claim stops

1. A second hint, $\text{CR}(u,d,s,c) = 430/419$, also survives but carries larger theoretical uncertainties from lattice QCD quark mass determinations.

   from 7. Conclusion

2. A null-model scan of all $\binom{12}{4}$ SM mass quadruples reveals that 976/919 has the smallest denominator among all 156 rational approximations at $q \leq 5000$, and its $\text{ord}_2 = 4$ structure matches the Compton programme's four RG steps — a coincidence unlikely to arise from random mass permutations alone.

   from 7. Conclusion

3. Of five pre-registered cross-ratios tested at three primes ($p=2,3,5$), two were found to approximate simple rationals ($976/919$ and $430/419$) at precision marginally better than the Dirichlet theorem guarantee.

   from Abstract

Where to read

1. Abstract176 words · 1 min
2. 1. Introduction: Version 1.0 Post-Mortem283 words · 1 min
3. 2. Physics Foundation: The Compton Frequency as the Fundamental Invariant520 words · 2 min
4. 3. The Bruhat-Tits Tree: Geometric Mechanism190 words · 1 min
5. 4. Method: Pre-Registered Analysis197 words · 1 min
6. 5. Results1,104 words · 5 min
7. 6. Falsifiability Conditions106 words · 1 min
8. 7. Conclusion226 words · 1 min