Koide tuples
The original Koide formula
K = (m₁+m₂+m₃) / (√m₁+√m₂+√m₃)² = 2/3 for charged leptons
inspires a whole family of triplet sum rules. Some require taking the
negative square root for one of the masses (those entries are
marked − below). The down-quark inverse tuple uses
K⁻¹ = (1/m₁+1/m₂+1/m₃)/(1/√m₁+1/√m₂+1/√m₃)² = 2/3.
Source: Rivero, Phys.
Lett. B 877 (2026) 140510.
| tuple | type | predicted K | K from PDG 2025 | deviation | postdicted | note (hover for citation) |
|---|
Square-root sum rule (Eq. 10 of the paper):
√m_b = √3·√m_s + √m_c → predicts
m_b = 4184.5 MeV (PDG 2025: 4183 ± 7 MeV, match
to 0.04%). Drawn as a pink dashed line on the b-quark mass card on the
main page.
The down-quark inverse Koide tuple hits the
fixed point K⁻¹ = 2/3 exactly at
Q ≈ 280 TeV under SM RG running (Martin-Robertson
SMDR, 5-loop). At Q = M_Z the value is K⁻¹ = 0.66750 (+0.13% from 2/3);
at Planck scale, 0.66539 (−0.19%).
K(year) per tuple
Computed live from the PDG mass history. Pink dashed
line is the prediction K = 2/3 = 0.66667.
Poincaré–Casimir quartic spectrum
Identifying the four electroweak masses with a signed
quartet of Poincaré representations indexed by spin s and
parity, with mass-squared values r² M_Z². The four predicted
ratios r come out of a single dimensionless construction and depend on no
free parameter beyond M_Z as the input scale.
Source:
Rivero, Casimir-quartic / Poincaré manuscript.
| slot | identified with | r = ratio to M_Z | tree prediction (GeV) | with 3/8 correction (GeV) | PDG 2025 (GeV) |
|---|
The fourth column (3/8 correction) applies
ε = (C_F/C_A)(M_Z² − M_W²) = (3/8)·(91.188² − 80.369²) ≈ 696 GeV²
to the negative-sector slots with the spin-grading sign factor
(−1)^(2s+1) (Eq. 33 of the manuscript). The positive slots are
unchanged. With this single correction the s = ½ negative slot lands at
125.20 GeV, the s = 1 negative slot at
174.17 GeV, and the s = 3/2 positive slot stays at
96.54 GeV. All four electroweak masses are then reproduced
within experimental error.
Equivalent on-shell weak mixing angle:
s²_dV = 1 − (β_{1/2}/β_1)² = 0.22310132… (the pink dashed
line on the on-shell sin²θ_W card on the main page).
Same construction, read as a one-parameter
over-determined fit, gives a Poincaré rest mass
m ≈ 106.57 GeV:
m_W = M_W/√((√57−3)/8) = 106.5705 GeV,
m_Z = M_Z/√(√3−1) = 106.5779 GeV
(two agreeing-determinations to a part in 10⁴, reproducing the
m ≈ 106.5 GeV Rivero noticed in his 2004 posting).
The construction also has a positive s = 3/2 slot at 96.54 GeV, coincident with the LEP/CMS 95-96 GeV anomaly. No Standard Model particle is currently identified at this mass.
How to read the deviations
For the W mass slot the +0.4σ agreement is striking because the construction has only one input (M_Z), so the W mass is a pure prediction with no fitted parameter — competitive with the global SM electroweak fit's ±5 MeV prediction (80.359 GeV).
The Higgs and the Fermi VEV deviate at the percent level. The paper interprets these as the tree-level signal to be perturbed by O(α) corrections; the splitting structure of the signed negative-sector eigenvalues then naturally produces values close to the observed m_H and v/√2.