arxiv: 2605.09651 · v1 · submitted 2026-05-10 · ✦ hep-ph

Recognition: 2 theorem links

· Lean Theorem

A minimization theorem for the Koide ratio and its Standard Model calibration

K. H\"ubner

Authors on Pith no claims yet

Pith reviewed 2026-05-12 03:07 UTC · model grok-4.3

classification ✦ hep-ph

keywords Koide ratiominimization theoremone-particle extensioncharged leptonscharm quarkeffective participantsStandard Model flavorkinematic benchmark

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The pith

Any positive mass set with Koide ratio Q0 reaches a unique minimum of Q0/(1+Q0) when extended by one optimally chosen mass.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a minimization theorem for the Koide ratio of any collection of positive real masses. Adding one new mass x to an existing set produces an extended ratio that always attains a global minimum value of exactly Q0 divided by one plus Q0. This minimum is achieved when the new mass equals the square of the ratio of the sum of the original masses to the sum of their square roots. The result supplies an exact benchmark that can be applied to Standard Model flavor data, such as the charged leptons where the predicted mass is 1.255 GeV and the ratio sits 6 ppm above the ideal 2/5 limit. The authors then scan many combinations of known particles to rank how closely they approach this kinematic minimum.

Core claim

For any positive mass set with Koide ratio Q0, the one-particle extension Q(m1,…,mN,x) has a unique global minimum Qmin=Q0/(1+Q0) at m*=[(∑mi)/(∑√mi)]2. For the measured charged leptons this yields m*ℓ=1.25534(16) GeV and Q4,min exp=0.3999978(43); in the ideal Koide limit QℓK=2/3 the minimum is exactly 2/5. In effective-participant language Neff≡1/Q the optimal one-particle extension increases Neff by one. The one-particle Neff profile is exactly Lorentzian in a dimensionless share-mismatch coordinate u.

What carries the argument

The one-particle extension Q(m1,…,mN,x) of the Koide ratio Q=(∑√mi)2/∑mi, which reaches its unique global minimum when the added mass equals [(sum of original masses) divided by (sum of their square roots)] squared.

Load-bearing premise

The Koide ratio is defined in the standard way using sums of masses and sums of square roots of masses, all masses are positive real numbers, and the extension is formed by adding exactly one new mass x.

What would settle it

A direct computation showing that the extended Koide ratio does not attain its claimed minimum value at m* or that the minimum is not unique when all masses remain positive.

Figures

Figures reproduced from arXiv: 2605.09651 by K. H\"ubner.

**Figure 2.** Figure 2: FIG. 2. Distribution of the relative miss ∆ [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗

read the original abstract

The charged-lepton Koide relation remains a striking empirical regularity in Standard-Model flavor data. We prove that for any positive mass set with Koide ratio $Q_0$, the one-particle extension $Q(m_1,\ldots,m_N,x)$ has a unique global minimum $Q_\text{min}=Q_0/(1+Q_0)$ at $m^*=\bigl[(\sum_i m_i)/(\sum_i \sqrt{m_i})\bigr]^2$. This exact kinematic result defines a unique extension benchmark. For the measured charged leptons it gives $m_*^\ell = 1.255\,34(16)\,\text{GeV}$ and $Q_{4,\min}^{\mathrm{exp}} = 0.399\,997\,8(43)$; in the ideal Koide limit $Q_\ell^{\mathrm{K}}=2/3$, the corresponding minimum is exactly $2/5$. In the effective-participant language $N_{\mathrm{eff}}\equiv 1/Q$, the optimal one-particle extension increases $N_{\mathrm{eff}}$ by one, while the equal-$k$ multiplet extension increases it by $k$. The one-particle $N_{\mathrm{eff}}$ profile is exactly Lorentzian in a dimensionless share-mismatch coordinate $u$, which we interpret kinematically rather than dynamically. Using charged-lepton pole masses with the PDG~2024 own-scale $\overline{\text{MS}}$ charm mass gives $Q(e,\mu,\tau,c)=0.400\,002\,5(64)$, i.e. $11.7\,\text{ppm}$ above the measured-input benchmark and $6.2\,\text{ppm}$ above $2/5$. This intentionally mixed-definition comparison is treated only as a phenomenological coincidence. To calibrate it within a stated benchmark class, we perform an exhaustive common-scale scan over non-neutrino Standard Model 2-body and 3-body seeds with one added mass. The charged-lepton-plus-charm continuation ranks $33/12{,}720$ in the raw trial set, $24/2{,}640$ after collapsing repeated scale realizations, and $6/756$ within the fermion-only collapsed subset. We present the charm case as an empirically calibrated example of the theorem, not as a dynamical flavor model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a clean new minimization theorem for the Koide ratio when adding one mass, plus a numerical check that leptons plus charm sit close to the minimum as a labeled coincidence.

read the letter

The main takeaway is a direct calculus result: for any positive masses with Koide ratio Q0, the one-particle extension reaches a unique global minimum of Q0/(1+Q0) at a specific added mass m* = [(sum mi)/(sum sqrt(mi))]^2. This is new and stated without prior results or fits. The paper then applies it to the charged leptons, getting m* around 1.255 GeV and Qmin near 0.4, and notes that adding the charm mass produces a four-particle Q only a few ppm away from both the benchmark and the ideal 2/5 value. They back this with an exhaustive scan over SM 2- and 3-body seeds, where the lepton-plus-charm case ranks reasonably high but not uniquely after collapsing scales. The theorem itself is internally consistent and requires only positivity of the masses, with limits at zero and infinity confirming the minimum. The SM part is framed explicitly as a phenomenological coincidence rather than a prediction or model, which keeps the claims proportionate. The scan details are summarized rather than fully reproduced, and the ppm-level closeness depends on the chosen PDG inputs and scale conventions, so small shifts in masses could move the ranking. No dynamical explanation is offered for why any particular set should approach the minimum. This work is mainly for people already tracking Koide-type relations or numerical patterns in flavor masses; it supplies a reproducible benchmark they can check against their own extensions. A broader audience will find little new physics. I would send it to peer review because the central mathematical statement is self-contained and falsifiable, and the numerical exercise is transparent enough to be cited as a reference point even if the interpretation stays modest.

Referee Report

1 major / 3 minor

Summary. The paper proves a minimization theorem for the Koide ratio: for any set of positive masses with ratio Q0, the one-particle extension Q(m1,…,mN,x) has a unique global minimum Qmin=Q0/(1+Q0) at m*=[(∑mi)/(∑√mi)]². It applies this to the charged leptons (yielding m*ℓ≈1.25534(16) GeV and Q4,min^exp≈0.3999978(43)), notes proximity to the charm mass in a mixed-definition comparison, and supports the calibration via an exhaustive scan over non-neutrino SM 2-body/3-body seeds with one added mass, where the lepton+charm case ranks 33/12720 (raw), 24/2640 (collapsed), and 6/756 (fermion-only).

Significance. If the theorem holds, it supplies an exact, parameter-free kinematic property of the Koide ratio under one-particle extension, serving as a benchmark rather than a dynamical model. The SM application is explicitly labeled phenomenological coincidence, with the scan providing empirical context; strengths include the direct calculus derivation, clear N_eff≡1/Q interpretation, and Lorentzian profile in the share-mismatch coordinate.

major comments (1)

[SM calibration section] § on SM calibration and exhaustive scan: the selection criteria for the 12,720-trial set of 'non-neutrino Standard Model 2-body and 3-body seeds', the definition of common-scale realizations, and the exact collapsing procedure to 2,640 and 756 subsets are only summarized. This renders the reported rankings (33/12,720 → 24/2,640 → 6/756) non-reproducible and weakens the claim that the charged-lepton-plus-charm continuation constitutes a calibrated example.

minor comments (3)

[Abstract] Abstract: the numerical inputs for m*ℓ and Q4,min^exp cite PDG 2024 pole masses and own-scale MSbar charm mass, but the precise error propagation (yielding the (16) and (43) uncertainties) should be stated explicitly.
[Theorem statement] The one-particle extension Q(m1,…,mN,x) and the effective-participant language N_eff≡1/Q are introduced without an early equation reference; adding a numbered definition early would improve readability.
[N_eff profile paragraph] The Lorentzian form of the N_eff profile in the dimensionless share-mismatch coordinate u is stated but the explicit functional form of u is not highlighted in the text or figure caption.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive overall assessment and for the constructive comment on the SM calibration section. We agree that additional detail is needed to ensure full reproducibility of the scan and rankings, and we will incorporate this in the revised manuscript.

read point-by-point responses

Referee: [SM calibration section] § on SM calibration and exhaustive scan: the selection criteria for the 12,720-trial set of 'non-neutrino Standard Model 2-body and 3-body seeds', the definition of common-scale realizations, and the exact collapsing procedure to 2,640 and 756 subsets are only summarized. This renders the reported rankings (33/12,720 → 24/2,640 → 6/756) non-reproducible and weakens the claim that the charged-lepton-plus-charm continuation constitutes a calibrated example.

Authors: We agree that the manuscript currently summarizes rather than fully specifies the scan procedure, which limits independent verification of the trial counts and rankings. In the revised version we will expand the relevant section (and, if needed, add a short appendix) to provide: (i) the explicit list of non-neutrino SM 2-body and 3-body seeds together with the precise selection rules that generate the raw set of 12,720 trials; (ii) the operational definition of a “common-scale realization,” including how reference masses are chosen and how the added mass is normalized; and (iii) the exact collapsing algorithm, including the duplicate-detection criterion and the resulting reduction steps to 2,640 and 756 entries. With these additions the reported rankings (33/12,720, 24/2,640, 6/756) will be directly reproducible. We retain the phenomenological framing of the calibration and do not claim it constitutes a dynamical model. revision: yes

Circularity Check

0 steps flagged

No significant circularity: theorem is direct calculus from definition

full rationale

The paper's central minimization theorem follows immediately from the standard definition of the Koide ratio Q = (sum sqrt(m_i))^2 / sum m_i by elementary calculus on the one-particle extension Q(x). The critical point and global minimum Q_min = Q0/(1+Q0) are obtained without fitted parameters, self-citations, or prior results. The Standard Model calibration is explicitly presented as a phenomenological coincidence rather than a derived prediction, with no load-bearing steps that reduce to the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The theorem rests solely on the standard definition of the Koide ratio and the assumption that masses are positive; no free parameters are fitted in the proof itself.

axioms (2)

domain assumption The Koide ratio for a set of masses is defined as Q = (sum sqrt(m_i))^2 / sum m_i
This is the conventional definition used to construct the one-particle extension Q(m1..mN,x).
domain assumption All masses are positive real numbers
Positivity is required for the global minimum to exist and be attained at the stated m*.

pith-pipeline@v0.9.0 · 5731 in / 1383 out tokens · 80049 ms · 2026-05-12T03:07:59.511629+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Theorem 1. ... Q_N+1 has a unique global minimum at r* = R2/R1 ... The minimum value is Q_min = Q0/(1+Q0).
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

N_eff ≡ 1/Q ... N_eff,max = N_eff,0 + 1 ... exact Lorentzian coordinate u ... Q(u) = Q_min(1+u^2)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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