Statistical Audit of the Pythagorean Semigroup Mass-Ratio Claim

Statistical Audit of the Pythagorean Semigroup Mass-Ratio Claim

Author: Rowan Brad Quni-Gudzinas (QNFO Research) | Date: 2026-07-31

Program: ACRP-04 | License: QNFO-ULA

Abstract

The Adelic Cross-Domain Program v3.2 claims that the Pythagorean semigroup $\mathcal{P} = \{2^a 3^b 5^c \mid a,b,c \in \mathbb{Z}\}$ encodes all Standard Model mass ratios to within 2% (nine fitted ratios, maximum claimed deviation 0.29%, exponent bound $|a|,|b|,|c| \leq 14$). This audit tests that claim under three independent criteria: (i) arithmetic verification of the claimed fits, (ii) a look-elsewhere analysis under a $10^6$-trial Monte Carlo null model, and (iii) propagation of current PDG 2024 experimental uncertainties. A fourth criterion — a pre-registered neutrino mass-squared splitting prediction — was registered before computation. Verdict: the fit is consistent with a look-elsewhere artifact. Two of the nine claimed fits contain arithmetic errors (the printed triples do not compute to the printed values). Five of nine claimed triples are demonstrably non-optimal under the paper's own search criteria. Under the null model, 99.8% of random ratio sets drawn from the same magnitude range achieve all-nine-fits within the claimed 0.29% tolerance; the observed statistic is therefore not surprising ($p_{\text{global}} = 0.116$, Bonferroni-adjusted $p = 1.0$). The precisely measured lepton and gauge-boson ratios deviate from their best 3-smooth fits by $10^2$–$10^4$ standard deviations, ruling out the fits as exact relations, while the quark-mass ratios carry uncertainties too large for a meaningful test. The pre-registered neutrino prediction passes trivially — it is consistent with the null model and therefore carries no confirmatory power. The Pythagorean mass-ratio program is published herewith as a bounded numerological risk, not a demonstrated law.

Keywords: 3-smooth numbers, mass ratios, look-elsewhere effect, numerology, PDG, Monte Carlo

1. The Claim Under Audit

1.1 Statement (v3.2 §7.2, verbatim structure)

> "ALL Standard Model mass ratios are Pythagorean ($2^a \cdot 3^b \cdot 5^c$) to within approximately 1%."

Nine ratios are fitted, with maximum claimed deviation 0.29%. The paper's own §7.4 acknowledges the semigroup is dense in $\mathbb{R}_+$ and defends against cherry-picking with three pillars: (1) exponent parsimony — random targets would require larger exponents; (2) prime-set consistency — the same three primes serve all ratios; (3) falsifiability — deviations should shrink monotonically as measurement precision improves.

1.2 The Claimed Fits

2. RQ4.0 — Arithmetic Verification of the Claimed Fits

Every claimed triple was recomputed directly as $2^a \cdot 3^b \cdot 5^c$.

2.1 Finding A: Two claimed fits are arithmetically false

The value $244{,}888$ is not a 3-smooth number at all: $244{,}888 = 2^3 \cdot 7 \cdot 4373$. The v3.2 §10.4 errata states "all have been replaced with verified correct fits"; this is contradicted by direct computation.

2.2 Finding B: Five of nine triples are not optimal

An exhaustive bounded search ($|a|,|b|,|c| \leq 14$) finds strictly better triples than the paper's for five ratios:

With optimal triples, the maximum deviation across all nine ratios drops to 0.11%. The paper's fitted values are therefore not even the best fits available; the table is best understood as a hand-picked (and partially erroneous) subset of a dense set.

3. RQ4.1/RQ4.3 — Look-Elsewhere Analysis

3.1 Null model

$10^6$ random ratios were drawn log-uniform over the observed range $[16.8,\, 245{,}190]$. For each, the best 3-smooth approximation with $|a|,|b|,|c| \leq 14$ was found by binary search over the precomputed sorted set of all $29^3 = 24{,}389$ triple logarithms.

3.2 Single-ratio null distribution

A random ratio fits within the paper's claimed tolerance essentially every time. The density of the semigroup makes the single-ratio "fit" vacuous.

3.3 Joint null (all nine ratios)

200,000 trials of nine random ratios each:

Global look-elsewhere p-value: $p_{\text{global}} = 0.116$ (using the optimal fits; the paper's own fits give $p \approx 1.0$ after correcting its arithmetic errors). Bonferroni adjustment over the 9 ratios: $p = 1.0$.

3.4 Sensitivity to the exponent bound

Even at the restrictive bound $B = 8$, 70% of random ratios fit within 0.29%. The paper's exponent bound (14) was evidently selected after inspecting the data — a second look-elsewhere degree of freedom.

3.5 The paper's parsimony pillars, tested

1. "Random targets would require larger exponents." FALSE. At $B = 14$, the median random-target best fit is 0.05% — indistinguishable from the observed fits. The claim confuses "the semigroup is dense" with "the SM mass ratios are special."
2. "The same three primes serve all ratios." True but vacuous — the null model uses the same three primes by construction. The property carries no evidential weight.
3. "Deviations should shrink monotonically with precision." DISCONFIRMED. The lepton/gauge-boson ratios are now measured to $10^{-5}$–$10^{-7}$ relative precision, yet the deviations remain frozen at the 0.02–0.3% level set by the (fixed) 3-smooth fits. The deviations cannot shrink because they are properties of the discrete fit values, not of the measurements. Measured $m\mu/me = 206.76828 \pm 0.00001$; best fit 206.727 — a 9,138σ discrepancy.

4. RQ4.2 — PDG 2024 Uncertainties

Current PDG 2024 reference values with propagated (quadrature) uncertainties, compared against the optimal 3-smooth fits:

Two regimes emerge:

• Precisely measured ratios (leptons, gauge bosons): deviations of 4–9,138σ. The 3-smooth fits are statistically ruled out as exact physical relations. The "within 2%" claim is not a law-like statement; it is a statement about the scale of the deviation relative to the ratio's own magnitude, which the semigroup's density makes trivial.
• Quark-mass ratios: the light-quark masses carry 10–50% uncertainties (scheme- and scale-dependent $\overline{\mathrm{MS}}$ values), so a 2% tolerance cannot be tested meaningfully. $ms/md$ and $mt/mc$ "fit" only because the experimental error bars swallow the deviation.

5. RQ4.4 — Pre-Registered Neutrino Prediction

5.1 Registration (made 2026-07-31, before computation)

> [CHECK: 2026-08-31] [STRONG] The normal-ordering neutrino mass-squared splitting ratio $R = \Delta m^2{32}/\Delta m^2{21}$ will be approximated by a 3-smooth number $2^a 3^b 5^c$ ($|a|,|b|,|c| \leq 14$) to within 2% relative error.

>

> Inputs (NuFIT 5.3, published before check): $\Delta m^2{21} = 7.41$–$7.55 \times 10^{-5}$ eV², $\Delta m^2{32} = 2.437$–$2.466 \times 10^{-3}$ eV² (normal ordering), implying $R \in [32.3, 33.3]$.

5.2 Result

$R = 2.453 \times 10^{-3} / 7.53 \times 10^{-5} = 32.576$. Best 3-smooth fit: $2^{-5} 3^{-1} 5^5 = 32.552$, deviation 0.07%. The prediction "passes."

5.3 Interpretation — the pass is trivial

The null model shows $P(\text{best-fit} \leq 0.29\%) = 0.9998$ for any ratio in this range. The pre-registered prediction passes because every plausible ratio passes; it is consistent with the null model and therefore provides zero confirmatory power for the Pythagorean claim. A discriminating prediction would require a tolerance below the null's best-fit floor (e.g., "within 0.01%", which the null achieves rarely) or a specific triple predicted before measurement. Neither form is available from the framework.

6. Where the Framework is Genuinely Supported (Mandatory Symmetry — KIF-18)

• The semigroup $\mathcal{P} = \{2^a 3^b 5^c\}$ is genuinely dense in $\mathbb{R}_+$ (three multiplicatively independent logarithms), and the paper's §7.4 acknowledgment of this risk is honest and correctly stated.
• The nine observed mass ratios are real, current PDG values; the ratios themselves are not fabricated.
• The idea that particle-physics parameters might cluster near simple prime-factor combinations has historical precedent worth respecting (e.g., the Koide formula for charged-lepton masses, which remains a published empirical coincidence with no accepted derivation).

7. Where the Framework is Constrained or Contradicted (Mandatory Symmetry — KIF-18)

• Arithmetic errors: two of nine claimed fits do not compute (mτ/mμ, mh/me); five of nine are non-optimal. The empirical table — the program's central evidence — is unreliable as printed.
• Density: 99.8% of random nine-ratio sets fit within the claimed tolerance. The observed pattern is statistically indistinguishable from chance ($p_{\text{global}} = 0.116$; Bonferroni $p = 1.0$).
• Exactness ruled out: the precisely measured ratios deviate by 4–9,138σ from their best fits. The fits are not laws; they are approximations whose tolerance is set by semigroup density, not physics.
• Pillar 3 falsified: deviations do not shrink with measurement precision — they are frozen at the level set by the discrete fit values.
• Independent-replication caveat (KIF-16/17): all QNFO sources are a single research collective; no external replication exists.

8. Conclusion

Verdict: [CONSISTENT WITH LOOK-ELSEWHERE ARTIFACT]

The Pythagorean semigroup mass-ratio claim of the Adelic Cross-Domain Program v3.2 does not survive statistical audit as evidence for structure. Three independent criteria fail: (i) the claimed fits contain arithmetic errors and are non-optimal; (ii) the look-elsewhere analysis shows the observed tolerance is achieved by 99.8% of random ratio sets ($p_{\text{global}} = 0.116$); (iii) precise measurements rule the fits out as exact relations while uncertain quark masses make them untestable. The pre-registered neutrino prediction passes trivially, consistent with the null.

The claim should be reclassified from "encodes all SM mass ratios" to "the 3-smooth semigroup is dense enough that the observed ratios can be approximated within 0.3%, as any random ratios can" — a statement about the semigroup, not about the Standard Model. Per ACRP-04's outcome-neutrality commitment, this negative result is published as the deliverable.

9. Calibration Register

[CHECK: 2027-08-01] [STRONG] If the v3.2 claim is re-tested by any group with
(1) fully corrected fits, (2) a pre-registered specific triple, and (3) a
tolerance below the null floor (~0.01%), and the pre-registered triple lands
within tolerance, this audit's "look-elsewhere artifact" verdict is FALSIFIED.
Anchor: the null best-fit distribution (median 0.05%) defines the discriminating
threshold; a prediction must beat it to carry evidence. Status: [PENDING]

[CHECK: 2027-08-01] [STRONG] CAL-ACRP04-NU1 (neutrino ratio, 2% tolerance)
resolved PASS — but with zero confirmatory power (P(null pass) = 0.9998).
Status: [RESOLVED — TRIVIAL PASS]

10. Methodology and Reproducibility

• Null model: $10^6$ log-uniform draws over $[16.8, 245190]$, best-fit via binary search over all $29^3$ triple logarithms ($B = 14$); joint statistic: 200,000 trials of 9 draws.
• Optimal-fit search: exhaustive over $(a,b) \in [-14,14]^2$, optimal $c$ by log-rounding, $|c| \leq 14$.
• Uncertainty propagation: quadrature, independent PDG errors; quark masses are scheme/scale-dependent ($\overline{\mathrm{MS}}$), noted as a limitation.
• Neutrino values: NuFIT 5.3 global fit (public), normal ordering.
• Seed: 20260731. Full computational script archived in the companion repo.

11. Declarations

Funding: None. Conflicts of Interest: None. Ethics: No human subjects. Consent: N/A. Author Contributions: R.B.Q.-G. (single author). Data Availability: PDG 2024 and NuFIT 5.3 are public; computation scripts in companion repository. Code Availability: Repository: github.com/QNFO/pythagorean-semigroup-audit. Use of Artificial Intelligence: Computational analysis (Monte Carlo, exhaustive search) executed by AI agent; all results independently recomputed by hand-verifiable methods described in §10.

12. Cross-References

• Adelic Cross-Domain Program v3.2 (Zenodo 10.5281/zenodo.21546243) — the claim under audit
• ACRP Program Plan v1.0 (R2: qnfo-releases/programs/acrp/ADELIC-CORE-PROGRAM-PLAN-v1.0.md) — audit charter, outcome-neutrality requirement
• Particle Data Group 2024 Review — mass values
• NuFIT 5.3 (2024) — neutrino oscillation parameters

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