Descartes Curvature Geometry, Compact-Cycle Amplitudes, and the GST Cabibbo Seed

1. Introduction

The Cabibbo angle has been measured to high precision through the CKM element $|V_{us}|$, with the current world average [1] $|V_{us}|=0.22503\pm 0.00068$. The Gatto–Sartori–Tonin (GST) relation [2,3]

$$sin{\theta }_C\approx \sqrt{{\displaystyle \frac{m_d}{m_s}}},$$

(1)

connects this angle to the ratio of down- and strange-quark masses. Gatto, Sartori, and Tonin originally derived the Cabibbo angle from the cancellation of quadratically divergent weak self-masses; [2] the quark-mass-ratio form (1) is due to the Fritzsch texture-zero ansatz. [3] Equation (1) is well established; what remains unexplained, within this phenomenological relation, is why the ratio $m_d/m_s$ has its observed value.

Separately, the empirical relation

$${\displaystyle \frac{{\sum }_{\ell }{m}_{\ell }}{{\left({\sum }_{\ell }\sqrt{m_{\ell }}\right)}^2}}={\displaystyle \frac{2}{3}}$$

(2)

has held for the charged-lepton pole masses to one part in ${10}^5$ since it was first noted by Koide. [4] Several authors have recast this relation in different mathematical languages, [6,7,8] raising the question of whether these are independent observations or coordinate descriptions of a single underlying structure.

A useful feature of the construction is that the quantities later compared with quark masses are fixed without quark-sector input. The scale ${\mu }_{★}={\sum }_{\ell }{m}_{\ell }^{\mathrm{pole}}$ is constructed solely from measured charged-lepton pole masses, while the dimensionless factor $r={\alpha }^2$ follows from the normalized charged-lepton geometry. Consequently the orbit targets $x_1=r{\mu }_{★}$ and $x_2=r^2{\mu }_{★}$ are determined before any quark mass, quark renormalization convention, or lattice result enters. The quark scheme and scale dependence therefore resides on the comparison side of the test rather than in the construction of the targets.

In this paper we pursue two aims. First, we show that the Fourier-mode constraint of Shulga’s compact-cycle model, [7] the Descartes curvature reading of Kocik, [6] and the Soddy completion of the resulting circle configuration are related by exact algebra, with no approximations (Sec. Section 3). We then isolate the normalized content of that completion and show that it supplies a universal ratio and an associated homogeneous map whose iterates form an exact geometric-mean orbit (Sec. Section 4). Second, beginning from the empirical strange-quark correspondence, we compare the fixed next iterate with the down-quark mass and examine the resulting conditional GST/Fritzsch Cabibbo seed (Secs. Section 5Section 6).

2. The Algebraic Object

Let $e_1={\sum }_i\sqrt{m_i}$, $p_2={\sum }_i{m}_i$ for the three charged leptons, and define the lepton-sum scale ${\mu }_{★}\equiv p_2=1883.1$ MeV. The condition (2) is $p_2={\displaystyle \frac{2}{3}}{e}_1^2$, equivalently $e_1=\sqrt{3/2}\sqrt{p_2}$.

Define the companion amplitude

$$\mathcal{F}\equiv e_1-\sqrt{p_2}=\alpha \sqrt{{\mu }_{★}},$$

(3)

where

$$\alpha \equiv \sqrt{\frac{3}{2}}-1\approx 0.2247.$$

(4)

The squared companion is

$${\mathcal{F}}^2={\alpha }^2{\mu }_{★},{\alpha }^2=\frac{5}{2}-\sqrt{6}\approx 0.0505.$$

(5)

This is the central algebraic object. The three-input symmetric-polynomial identity (2) collapses to one dimensionless constant $\alpha$ times one mass scale ${\mu }_{★}$, with $\alpha$ fixed by the condition (2).

3. Equivalence of Descriptions

We now show that three apparently distinct formulations of the condition (2) describe the same constraint.

3.1. Compact-Cycle Amplitudes (Shulga)

Shulga [7] constructs a real amplitude on an internal circle,

$$Z\left(\varphi \right)=c_0+2r_{c}cos(\varphi +\delta ),$$

(6)

built from the two lowest antiperiodic modes, with the three charged-lepton families sampling at equally spaced points: $\sqrt{m_a}=Z(2\pi a/3)$ for $a=0,1,2$. The parameter $c_0$ is the flavor-symmetric (democratic) component and $r_c$ the flavor-breaking amplitude; the cosine coefficient is $2r_c$, so Shulga’s notation $Z=s[1+\sqrt{2}cos(\varphi +\theta )]$ corresponds to $c_0=s$ and $2r_c/c_0=\sqrt{2}$, i.e. $r_c/c_0=1/\sqrt{2}$.

By the orthogonality of equally spaced phases on the circle, ${\sum }_{a}cos(2\pi a/3+\delta )=0$ for any $\delta$, and similarly for the double-angle sum. Therefore

$$\begin{array}{cc}\hfill e_1& =\sum _{a}Z(2\pi a/3)=3c_0,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill p_2& =\sum _{a}Z{(2\pi a/3)}^2=3(c_0^2+2r_c^2).\hfill \end{array}$$

(8)

The ratio $p_2/e_1^2$ becomes

$${\displaystyle \frac{p_2}{e_1^2}}={\displaystyle \frac{c_0^2+2r_c^2}{3c_0^2}}={\displaystyle \frac{1}{3}}+{\displaystyle \frac{2r_c^2}{3c_0^2}}.$$

(9)

Setting this equal to $2/3$ gives

$${\displaystyle \frac{r_c}{c_0}}={\displaystyle \frac{1}{\sqrt{2}}}.$$

(10)

The condition (2) is exactly the statement that the flavor-breaking and flavor-symmetric amplitudes on the compact cycle stand in the ratio $1:\sqrt{2}$. Shulga derives this ratio from the spinor-squaring of antiperiodic modes and a reality condition on the amplitude. [7]

3.2. Descartes Curvatures (Kocik)

The Descartes circle theorem [5] for four mutually tangent circles reads ${\left({\sum }_i{k}_i\right)}^2=2{\sum }_i{k}_i^2$. Kocik [6] observed that identifying curvatures $k_i=\sqrt{m_i}$ places (2) within the generalized Descartes family. The present work uses the standard (mutually tangent) theorem and computes the fourth curvature.

The outer Soddy curvature of the three-circle configuration is $k_4^{-}=e_1-2\sqrt{e_2}$, where $e_2={\sum }_{i<j}\sqrt{m_i{m}_j}$. Under (2), one has $4e_2=p_2$, so $k_4^{-}=e_1-\sqrt{p_2}=\mathcal{F}$: the Soddy curvature is the companion amplitude (3).

3.3. The Equivalence

Expressing the Soddy curvature in Shulga’s variables via (7)–(10):

$$\mathcal{F}=e_1-\sqrt{p_2}=3c_0-\sqrt{6}{c}_0=c_0(3-\sqrt{6}).$$

(11)

Then $\alpha =\mathcal{F}/\sqrt{{\mu }_{★}}=c_0(3-\sqrt{6})/\sqrt{6c_0^2}=(3-\sqrt{6})/\sqrt{6}=\sqrt{3/2}-1$, confirming (4).

The three descriptions are related by exact algebra:

Framework	Constraint	Language
Shulga	$r_c/c_0=1/\sqrt{2}$	Fourier modes
Condition (2)	$p_2/e_1^2=2/3$	Sym. polynomials
Soddy	$\mathcal{F}=\alpha \sqrt{{\mu }_{★}}$	Inversive geometry

No approximations are involved in any direction. The constant $\alpha =\sqrt{3/2}-1$ appears in each language as the unique dimensionless number implied by the shared constraint. The plus branch $e_1+\sqrt{p_2}$ gives $\sqrt{3/2}+1\approx 2.22$, which has no interpretation in the quark sector; the minus branch is the only phenomenologically relevant root.

4. Universal Normalized Completion and the Companion Map

The value of the smaller completion is not tied to the particular orientation of the physical charged-lepton triple. For any positive triple satisfying the exact condition (2), define

$$y_i\equiv \sqrt{{\displaystyle \frac{m_i}{\mu }}},\mu \equiv \sum _i{m}_i.$$

(12)

Then ${\sum }_i{y}_i^2=1$ and ${\sum }_i{y}_i=\sqrt{3/2}$. Normalizing the fourth Descartes curvature by the same scale,

$$f\equiv {\displaystyle \frac{k_4}{\sqrt{\mu }}},$$

(13)

the Descartes equation becomes

$${\left(\sqrt{{\displaystyle \frac{3}{2}}}+f\right)}^2=2(1+f^2),$$

(14)

with the two roots

$$f_{\pm }=\sqrt{{\displaystyle \frac{3}{2}}}\pm 1.$$

(15)

Thus the normalized smaller completion is

$$f_{-}=\alpha =\sqrt{{\displaystyle \frac{3}{2}}}-1,$$

(16)

independent of the orientation of the normalized square-root-mass vector on the positive exact-$2/3$ locus. This strengthens Eq. (5): the quantity

$$r\equiv {\alpha }^2={\displaystyle \frac{5}{2}}-\sqrt{6}$$

(17)

is a universal squared-completion ratio rather than a number peculiar to one charged-lepton configuration.

We therefore define the associated homogeneous companion map

$$\mathcal{M}:{\mathbb{R}}_{>0}\to {\mathbb{R}}_{>0},\mathcal{M}\left(x\right)=rx.$$

(18)

This is a mathematical definition, not a dynamical evolution law, and by itself contains no quark-flavor assignment. Its iterates satisfy

$$x_n={\mathcal{M}}^n\left(x_0\right)=r^n{x}_0,x_n^2=x_{n-1}{x}_{n+1}.$$

(19)

Hence every interior orbit point is exactly the geometric mean of its two neighbors. The geometric progression used below is therefore the orbit of one fixed map; no additional step ratio is introduced when the map is iterated.

5. The Mass Cascade

For numerical work we use the charged-lepton pole masses quoted by PDG, [1]

$$m_e=0.5109989507\mathrm{MeV},m_{\mu }=105.6583755\mathrm{MeV},m_{\tau }=1776.93\left(9\right)\mathrm{MeV},$$

(20)

which give

$${\mu }_{★}=1883.0994\pm 0.090\mathrm{MeV}.$$

(21)

Here ${\mu }_{★}$ is a physical energy constructed from measured lepton pole masses. The distinction is useful rather than problematic: the PDG quark-mass review emphasizes that observable leptons admit pole-mass definitions, whereas confined quark masses require an explicitly specified renormalization scheme and scale. [1] In the present construction no quark quantity enters the definition of ${\mu }_{★}$, r, $x_1$, or $x_2$. The quark masses are only then evaluated, independently, in the $\overline{\mathrm{MS}}$ scheme at the numerical energy ${\mu }_R={\mu }_{★}$. Thus no conversion of a lepton pole mass into a quark running mass is implied, and the fixed targets cannot be an artifact of the quark renormalization convention used to test them. The phenomenological hypothesis is instead that this externally specified lepton-defined physical energy is relevant to the light-quark spectrum.

Taking $x_0={\mu }_{★}$, the first two nontrivial orbit points of Eq. (19) are

$$\begin{array}{cc}\hfill x_1& =\mathcal{M}\left({\mu }_{★}\right)={\alpha }^2{\mu }_{★}=95.11583\mathrm{MeV},\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill x_2& ={\mathcal{M}}^2\left({\mu }_{★}\right)={\alpha }^4{\mu }_{★}=4.804325\mathrm{MeV}.\hfill \end{array}$$

(23)

The empirical strange-quark correspondence predates the present manuscript and is reported in Ref. [9]. The fixed light-quark targets used here were subsequently deposited in a dated pre-registration on 18 May 2026, Ref. [10], before the 2026 PDG edition dated 1 June 2026. That record fixes the s, d, and u targets at 95.12, 4.804, and 2.216 MeV, respectively, with their stated mass conventions and scales. The exact-locus value $x_1=95.11583$ MeV used here differs from the archived physical-point value only by the deterministic exact-$2/3$ idealization quantified below, not by refitting. When confronted with the 2026 PDG input used below, the strange central value lies farther from the fixed target but remains compatible at current precision; no parameter, scale, or target was adjusted. Because the earlier comparison used 2024 FLAG/PDG inputs whereas Table 1 uses the 2026 PDG final lattice estimate, the change in normalized residual is not interpreted as a time-series trend. The prospective point is that the numerical targets were frozen before this subsequent data cycle, although the original strange-quark identification was retrospective.

We retain that empirical identification,

$$x_1⟷m_s^{\overline{\mathrm{MS}}}\left({\mu }_{★}\right),$$

(24)

and then apply the already defined map once more. The down-quark target $x_2$ is therefore fixed by the same universal ratio; no second step size is chosen. The resulting phenomenological comparison is

$$x_2⟷m_d^{\overline{\mathrm{MS}}}\left({\mu }_{★}\right).$$

(25)

The charged-lepton geometry does not independently derive the flavor labels in Eqs. (24)–(25); the claim is that the fixed continuation from the observed strange correspondence lands on the running down-quark mass.

The u-quark closure has a different logical status. Retaining the physical pair threshold $T=2m_e$ as a lower boundary, we impose the separate midpoint hypothesis

$${\left[m_u^{\overline{\mathrm{MS}}}\left({\mu }_{★}\right)\right]}^2\stackrel{\mathrm{hyp}}{=}T{m}_d^{\mathrm{model}}\left({\mu }_{★}\right),T=2m_e^{\mathrm{pole}},$$

(26)

which gives

$$m_u^{\mathrm{model}}\left({\mu }_{★}\right)=\sqrt{2m_e{x}_2}=2.21585\mathrm{MeV}.$$

(27)

This terminal condition is not a third iterate of $\mathcal{M}$ and is not derived from the normalized-completion result.

5.1. Sensitivity to the Exact $2/3$ Idealization

The algebra above is exact on the $2/3$ locus, while the measured charged-lepton masses are slightly off it. For general $Q=p_2/e_1^2$, Eq. (3) gives the normalized smaller completion

$$f_{-}\left(Q\right)=Q^{-1/2}-\sqrt{2\left(Q^{-1}-1\right)}.$$

(28)

Using the central values in Eq. (20) gives

$$Q_{\ell ,\mathrm{phys}}=0.66666446,f_{-}\left(Q_{\ell ,\mathrm{phys}}\right)=0.22474194,r_{\mathrm{phys}}=0.05050894.$$

(29)

Replacing the exact r by this physical-point value shifts the first two orbit targets by only

$$\Delta x_1=-0.00248\mathrm{MeV},\Delta x_2=-0.000251\mathrm{MeV},$$

(30)

and changes the terminal u target by about $-5.8\times {10}^{-5}$ MeV. Propagating the present $m_{\tau }$ uncertainty through the off-locus completion gives about $0.0103$ MeV and $0.00081$ MeV on the corresponding $x_1$ and $x_2$ values. These effects are small compared with the current quark-mass uncertainties, so the exact-$2/3$ idealization is kept in the closed-form relations and its measured-point shift is treated as a separate sensitivity check.

5.2. Auditable Quark-Mass Comparison

For a single reproducible data prescription, we use the PDG final lattice-QCD estimates in the four-flavor $\overline{\mathrm{MS}}$ theory at 2 GeV, [1]

$$m_s=92.74\pm 0.54\mathrm{MeV},m_d=4.69\pm 0.05\mathrm{MeV},m_u=2.20\pm 0.07\mathrm{MeV}.$$

(31)

All three are evolved to ${\mu }_R={\mu }_{★}=1.883099$ GeV with the same four-loop pure-QCD running at fixed $n_f=4$, using RunDec/CRunDec. [11] No heavy-flavor threshold is crossed in this short interval. With ${\alpha }_s\left(M_Z\right)=0.1180$, the common mass-evolution factor is

$$R_m\equiv {\displaystyle \frac{m_q\left({\mu }_{★}\right)}{m_q\left(2\mathrm{GeV}\right)}}=1.01744.$$

(32)

The model-target uncertainties from Eq. (21) are approximately $0.00455$ MeV for $x_1$, $0.000230$ MeV for $x_2$, and $5.3\times {10}^{-5}$ MeV for the terminal u target.

Using the published individual mass uncertainties, the three absolute central-value differences correspond to approximately $1.38$, $0.64$, and $0.32$ input standard deviations for s, d, and u, respectively. These are one-observable residuals; they are not treated as statistically independent pieces of evidence for a common model.

The short running is not numerically delicate. At the same local coupling, the three-, four-, and five-loop factors are $1.017156$, $1.017440$, and $1.017530$. A deliberately broad variation ${\alpha }_s^{\left(4\right)}\left(2\mathrm{GeV}\right)=0.30082\pm 0.010$ changes the evolved $m_s$ by at most $0.081$ MeV and $m_d$ by at most $0.0041$ MeV; adjacent loop-order variation adds at most $0.027$ MeV and $0.0014$ MeV, respectively. Leading QED running over the same short interval is about $2.3\times {10}^{-5}$ fractionally for d and s and $9.3\times {10}^{-5}$ for u. [12] Thus the absolute $s,d$ comparison is presently dominated by the quoted quark inputs rather than the short RG/QED translation. For the terminal u relation, the different electric charge means that QED/isospin separation does not cancel as cleanly as it does for the $d/s$ ratio, so the u comparison is kept at the descriptive percent level.

5.3. Scale Sensitivity of the Orbit Comparison

The comparison point ${\mu }_{★}$ is fixed by the charged-lepton pole-mass sum; it is not obtained by fitting the quark masses. To quantify how strongly the numerical agreement depends on that specified scale, we solve for the scales at which the same four-loop running masses cross the fixed orbit targets:

$$m_s^{\overline{\mathrm{MS}}}\left({\mu }_s\right)=x_1\Rightarrow {\mu }_s=1832.8\mathrm{MeV},m_d^{\overline{\mathrm{MS}}}\left({\mu }_d\right)=x_2\Rightarrow {\mu }_d=1840.3\mathrm{MeV}.$$

(33)

These central crossings lie $2.67\%$ and $2.27\%$ below ${\mu }_{★}=1883.099$ MeV. Propagating the individual PDG mass uncertainties through the same running gives crossing-scale uncertainties of approximately $\pm 36$ MeV for s and $\pm 66$ MeV for d. The broader RG/QED prescription stress envelope quoted above corresponds to only about $\pm 7$ MeV in either crossing location and is therefore subdominant to the present quark-input uncertainties. Equation (33) is a scale-sensitivity diagnostic, not an additional fit or independent validation: present precision does not sharply localize a preferred renormalization scale, while the independently specified ${\mu }_{★}$ lies within the few-percent region over which the correspondence remains numerically close.

6. The GST Cabibbo Seed

From Eqs. (22)–(), the exact mathematical result is

$${\displaystyle \frac{x_2}{x_1}}=r={\alpha }^2={\displaystyle \frac{5}{2}}-\sqrt{6}.$$

(34)

Under the empirical ordered correspondence in Eqs. (24)–(25), this fixed orbit ratio gives the conditional quark-sector statement

$${\displaystyle \frac{m_d}{m_s}}={\alpha }^2.$$

(35)

The leading GST relation (1) then yields

$$\begin{array}{|c|}\hline \sqrt{{\displaystyle \frac{m_d}{m_s}}}=\alpha =\sqrt{\frac{3}{2}}-1=0.22474\dots \\ \hline\end{array}$$

(36)

This is the down-sector Cabibbo seed in the sense of the GST/Fritzsch leading-order relation. It is not, by itself, a precision prediction of $|V_{us}|$. The full GST expression contains the up-sector contribution and a relative phase, [3]

$$|V_{us}|\approx \left|\sqrt{{\displaystyle \frac{m_d}{m_s}}}-e^{i\varphi }\sqrt{{\displaystyle \frac{m_u}{m_c}}}\right|,$$

(37)

and neither the corresponding charm-sector input nor the phase $\varphi$ is fixed by the present construction. The result in Eq. (36) is therefore the exact leading down-sector factor conditional on the $s,d$ orbit correspondence; a complete CKM test remains open.

7. The Fritzsch Texture

The GST relation arises from the four-zero texture [3]

$$M_d=\left(\begin{array}{ccc}0& A& 0\\ A^{*}& 0& B\\ 0& B^{*}& C\end{array}\right).$$

(38)

In the present construction the Fritzsch texture is interpretive rather than part of the Descartes proof. Conditional on the orbit assignments, its leading down-sector mass-ratio factor is fixed by the same $\alpha =\sqrt{3/2}-1$ selected by the normalized completion.

The two map-generated quark targets and the separate lower boundary can be summarized as

$${\mu }_{★}\stackrel{\mathcal{M}}{\to }{m}_s\stackrel{\mathcal{M}}{\to }{m}_d,m_u=\sqrt{m_d·2m_e}.$$

(39)

The fixed multiplicative step of the orbit is $r={\alpha }^2$—the square of the leading GST Cabibbo seed.

8. Weak-Basis Considerations

The condition (2) holds for charged-lepton pole masses, where the mass and weak bases are approximately aligned: in the Standard Model, the charged-lepton Yukawa matrix can be taken diagonal without loss of generality when neutrino masses are neglected. The amplitude structure described in Sec. Section 3 is therefore visible directly in the physical spectrum.

For quarks the situation is different. The CKM matrix rotates between up-type and down-type mass bases, so the condition (2) should not be expected to hold for quark pole masses directly. The orbit comparisons (22)–() and the terminal condition (26) use $\overline{\mathrm{MS}}$ running masses at a specific scale, and the GST relation (1) itself arises from the Fritzsch texture in the weak basis. Whether the amplitude structure of Sec. Section 3 is native to the weak-basis Yukawa matrix, or requires a more involved embedding, is an open question.

The cross-sector comparison is therefore scale-specified rather than a claim that pole and running masses are the same kind of parameter. The lepton pole masses define the physical comparison energy ${\mu }_{★}$ and the dimensionless constant $\alpha$; the quark quantities are all evaluated in one $\overline{\mathrm{MS}}$ prescription at that numerical scale. This use of a lepton-defined energy is not excluded by the sector labels themselves: in the Standard Model the charged-lepton and quark Yukawa matrices are all couplings to the same Higgs field. [1] That common electroweak origin does not derive the present relation, but it provides a physical setting in which an additional cross-sector flavor structure can be meaningfully asked about. A dynamical mechanism explaining why the same constant and characteristic energy should recur across the two sectors remains open.

9. Tests and Failure Modes

The exact mathematics in Secs. Section 3Section 4 is distinct from the quark interpretation. The latter can fail in concrete ways. Future quark-mass determinations may move $m_s^{\overline{\mathrm{MS}}}\left({\mu }_{★}\right)$ or $m_d^{\overline{\mathrm{MS}}}\left({\mu }_{★}\right)$ away from the fixed orbit targets in Eqs. (22)–(); the scale, ratio, ordering, and flavor assignments stated here are not to be retuned when those inputs are updated. The separate terminal condition (26) can likewise fail as $m_u$ determinations and QCD–QED separation improve.

A completed GST/Fritzsch treatment can also fail if an up-sector texture and relative phase fixed without fitting $|V_{us}|$ do not reproduce CKM mixing. Finally, the cross-sector interpretation would be weakened if a consistent higher-precision QCD+QED treatment moved the quark quantities away from the present correspondence, or if no dynamical embedding of the compact-cycle structure into the quark Yukawa sector can be found. The strange target had already been publicly fixed before the 2026 PDG update, which moved its central value away from the target without prompting any retuning; future updates continue that prospective comparison. Publication of the present manuscript additionally freezes the d-orbit assignment and the separate terminal u boundary for subsequent data cycles.

10. Related Work

Rivero [13] showed independently that chained algebraic conditions connect lepton and quark sectors and recover $m_s$ in the same numerical neighborhood via a different route. Żenczykowski [8] developed a $Z_3$-symmetric parametrization of fermion mass square roots that embeds the condition (2) in a broader framework including quarks and neutrinos, with the phase $\delta \simeq 2/9$ playing a central role. Both approaches are compatible with, but not derived from, the equivalence established in Sec. Section 3.

11. Summary

The charged-lepton mass condition $p_2/e_1^2=2/3$, the compact-cycle amplitude constraint $r_c/c_0=1/\sqrt{2}$, and the smaller Descartes–Soddy completion are exact algebraic descriptions of the same positive charged-lepton constraint. Normalizing the completion reveals the additional result that $f_{-}=\alpha =\sqrt{3/2}-1$ is universal over the positive exact-$2/3$ locus. Its square therefore defines one scale-independent ratio $r={\alpha }^2$ and the homogeneous map $\mathcal{M}\left(x\right)=rx$, whose iterates obey the exact geometric-mean identity $x_n^2=x_{n-1}{x}_{n+1}$.

The physical comparison begins after that mathematical result. Importantly, ${\mu }_{★}$, r, $x_1$, and $x_2$ are fixed entirely from charged-lepton observables and the normalized completion, with no quark input; the renormalization convention enters only when the independently defined quark masses are evaluated against those targets. The empirically identified first lepton-anchored orbit point lies near the running strange-quark mass; once the same fixed map is applied again, the next target is fixed and lies near the running down-quark mass. With the 2026 PDG inputs evolved to the common scale ${\mu }_{★}$, the central differences are $0.80\%$ for s and $0.68\%$ for d. These are two components of one ordered orbit correspondence rather than independent evidential claims. The strange target had been publicly posted before the 2026 PDG update; that update moved the central value away from the target without any retuning. The exact central crossing scales are 1832.8 and 1840.3 MeV for s and d, respectively, showing that the fixed ${\mu }_{★}$ comparison is stable at the present few-percent level in renormalization scale rather than sharply localized there. A separate lower-boundary hypothesis gives an $m_u$ target within about $1.0\%$ of the evolved PDG central value; it is an auxiliary terminal conjecture rather than evidence for the $s,d$ orbit. The measured departure of the charged-lepton masses from the exact $2/3$ locus and the short RG/QED translation are both small relative to the present quark uncertainties.

The exact orbit statement is $x_2/x_1={\alpha }^2$. Conditional on the observed $s,d$ correspondence, this becomes $m_d/m_s={\alpha }^2$, and the leading GST/Fritzsch down-sector factor is exactly $\sqrt{m_d/m_s}=\alpha =0.224744871\dots$. This is a leading texture-level Cabibbo seed, not a complete CKM prediction. What remains open is the physical origin of the cross-sector correspondence and a dynamical embedding that fixes the up-sector structure and phase without additional fitted freedom.

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