Statistical Audit of the 5-Smooth Semigroup Mass-Ratio Claim

In the paper's words

ACRP-04 statistical audit. Verdict: CONSISTENT WITH LOOK-ELSEWHERE ARTIFACT. p_global=0.116; 2 claimed-fit arithmetic errors; 5 non-optimal triples; precise ratios deviate 4-9138 sigma; pre-registered neutrino prediction trivially passes.

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

Numbers it reports

1. 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 $|a|,|b|,|c| \leq 14$).

   from Abstract

2. 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 $p = 1.0$).

   from Abstract

3. Ratio Observed (v3.2) Claimed triple $(a,b,c)$ Claimed value Claimed deviation $m_\mu/m_e$ 206.77 $(6,4,-2)$ 207.36 0.29% $m_\tau/m_e$ 3477.2 $(-14,6,7)$ 3476.14 0.03% $m_\tau/m_\mu$ 16.82 $(-3,8,-5)$ 16.80 0.14% $m_t/m_c$ 136.6 $(11,-1,-1)$ 136.53 0.05% $m_s/m_d$ 20.0 $(2,0,1)$ 20 exact $m_b/m_s$ 45.3 $(-6,-3,7)$ 45.21 0.20% $m_W/m_e$ 157356 $(3,9,0)$ 157464 0.07% $m_Z/m_e$ 178450 $(-6,6,6)$ 177978.5 0.26% $m_h/m_e$ 245190 $(-5,14,-4)$ 244888.0 0.12%

   from 1.2 The Claimed Fits

4. Ratio Printed triple Printed value Actual value Actual deviation Claimed deviation $m_\tau/m_\mu$ $(-3,8,-5)$ 16.80 0.2624 98.4% 0.14% $m_h/m_e$ $(-5,14,-4)$ 244888.0 239.15 99.9% 0.12% The value $244{,}888$ is not a 3-smooth number at all: $244{,}888 = 2^3 \cdot 7 \cdot 4373$.

   from 2.1 Finding A: Two claimed fits are arithmetically false

5. An exhaustive bounded search ($|a|,|b|,|c| \leq 14$) finds strictly better triples than the paper's for five ratios: Ratio Paper's triple Paper's dev Optimal triple Optimal dev $m_\mu/m_e$ $(6,4,-2)$ 0.29% $(-2,-10,11)$ 0.02% $m_\tau/m_\mu$ $(-3,8,-5)$ (98.4%) $(-11,-11,14)$ 0.02% $m_b/m_s$ $(-6,-3,7)$ 0.20% $(2,11,-6)$ 0.11% $m_Z/m_e$ $(-6,6,6)$ 0.26% $(3,-7,11)$ 0.09% $m_h/m_e$ $(-5,14,-4)$ (99.9%) $(6,-13,14)$ 0.07% With optimal triples, the maximum deviation across all nine ratios drops to 0.11%.

   from 2.2 Finding B: Five of nine triples are not optimal

6. Statistic Value Median best-fit error 0.05% 90th percentile 0.14% 99th percentile 0.18% $P(\text{best-fit} \leq 0.29\%)$ 0.9998 $P(\text{best-fit} \leq 1.0\%)$ 1.0000 A random ratio fits within the paper's claimed tolerance essentially every time.

   from 3.2 Single-ratio null distribution

How it is done

1. 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 $|a|,|b|,|c| \leq 14$).

   from Abstract

2. $10^6$ random ratios were drawn log-uniform over the observed range $[16.8,\, 245{,}190]$.

   from 3.1 Null model

3. Bound $B$ Triples Median null error $P(\text{fit} \leq 0.29\%)$ 8 4,913 0.19% 0.70 12 15,625 0.08% 0.98 14 24,389 0.05% 1.000 20 68,921 0.03% 1.000 Even at the restrictive bound $B = 8$, 70% of random ratios fit within 0.29%.

   from 3.4 Sensitivity to the exponent bound

What would falsify it

1. A fourth criterion — a pre-registered neutrino mass-squared splitting prediction — was registered before computation.

   from Abstract

2. The pre-registered neutrino prediction passes trivially — it is consistent with the null model and therefore carries no confirmatory power.

   from Abstract

3. 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.

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

Where the claim stops

1. 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.

   from 8. Conclusion

2. 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.

   from 8. Conclusion

3. 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.

   from 8. Conclusion

Where to read

1. Abstract252 words · 1 min
2. 1. The Claim Under Audit130 words · 1 min
3. 2. RQ4.0 — Arithmetic Verification of the Claimed Fits163 words · 1 min
4. 3. RQ4.1/RQ4.3 — Look-Elsewhere Analysis335 words · 1 min
5. 4. RQ4.2 — PDG 2024 Uncertainties158 words · 1 min
6. 5. RQ4.4 — Pre-Registered Neutrino Prediction159 words · 1 min
7. 6. Where the Framework is Genuinely Supported (Mandatory Symmetry — KIF-18)78 words · 1 min
8. 7. Where the Framework is Constrained or Contradicted (Mandatory Symmetry — KIF-18)127 words · 1 min
9. 8. Conclusion140 words · 1 min
10. 9. Calibration Register91 words · 1 min
11. 10. Methodology and Reproducibility75 words · 1 min
12. 11. Declarations59 words · 1 min