A Descartes Companion Identity at Q = 2/3 and a Geometric Light-Fermion Mass Cascade

1. Introduction

Three coherent beams separated by ${120}^{\circ }$ produce hexagonal intensity patterns: a standard result in interference lithography [1,2]. The lattice constant is $a=2\lambda /(3sin\theta )$, reaching a minimum of $a=2\lambda /3$ at grazing incidence. The mathematical toolkit underlying this result ($C_3$ phase cancellation, geometric means of transport coefficients, Berry phases on compact cycles) is native to condensed matter [3,4,5,6]; cross-domain applications of these tools to particle physics have been explored at book length by Volovik [7].

The same $C_3$ phase-cancellation identities that govern three-beam interference also enforce $Q=2/3$ in the charged-lepton parametrization. When evaluated on lepton masses, the resulting algebraic carrier yields closed-form candidate expressions for light-quark masses. This paper develops the interference geometry first, on its own condensed-matter terms, and then notes that the resulting mass relations match particle data at the sub-percent level.

The approach is deliberately narrow: if the geometric relations survive further tests, they could serve as analytic anchors for cross-domain work. If they do not, the interference geometry stands on its own as a $C_3\times Z_2$ structure with integer resonances. The Koide relation has been addressed via family gauge symmetry [8] and compact-cycle Berry phase [9]; the present approach uses neither, relying instead on interference geometry. We make no claim to a fundamental theory of fermion masses; we observe that the mathematical structures organizing these relations are familiar from condensed matter and report the numerical correspondences.

2. $C_3$ Phase Cancellation

Place $N=3$ identical sources at ${120}^{\circ }$ intervals on a circle. Three coherent beams at these angles produce an intensity pattern whose structure follows from two trigonometric identities that hold for any offset $\delta$:

$$\begin{array}{cc}\hfill {\displaystyle \sum _{k=0}^2}cos{\theta }_k& =0,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill {\displaystyle \sum _{k=0}^2}{cos}^2{\theta }_k& =\frac{3}{2},\hfill \end{array}$$

(2)

where ${\theta }_k=\delta +2\pi k/3$. The first identity encodes the $C_3$ phase cancellation underlying the hexagonal pattern; the second fixes the quadratic normalization.

Writing the intensity at each sampling point as [19,27]

$$I_k={\displaystyle \frac{A}{6}}{\left(1+\sqrt{2}cos{\theta }_k\right)}^2,$$

(3)

the total intensity is $\sum I_k=A$ (by Eqs. 1–), and the ratio

$$Q\equiv {\displaystyle \frac{\sum I_k}{{(\sum \sqrt{I_k})}^2}}={\displaystyle \frac{2}{3}}$$

(4)

is forced by the $C_3$ identities alone. The coefficient $\sqrt{2}$ is simultaneously fixed: requiring $\sum {(1+ccos{\theta }_k)}^2=6$ gives $c^2=2$. Neither Q nor $\sqrt{2}$ is fitted. The parametrization (3) can be viewed as an algebraic realization of the $C_3$ identities; whether it corresponds to a physical interference pattern in an internal space is an open question. Throughout, $\sqrt{I_k}$ denotes the positive branch; the $C_3$ identity is exact for signed roots, and the charged-lepton sector uses the branch where $1+\sqrt{2}cos{\theta }_k\ge 0$.

A $Z_2$ factor at each position distinguishes the two signs of $\sqrt{I_k}$, giving a sixfold $C_3\times Z_2$ structure. Rivero [10] independently showed that the eigenvalue condition $x_i=x_0+z_i$ with $\sum z_i=0$ and $\mathrm{avg}\left(z_i^2\right)=x_0^2$ forces $Q=2/3$.

3. Resonance Conditions

The hexagonal lattice constant from three-beam interference is [1]

$$a\left(\theta \right)={\displaystyle \frac{2\lambda }{3sin\theta }}.$$

(5)

Integer-wavelength resonances, $a=N\lambda$, occur at $sin\theta =2/\left(3N\right)$:

N	$sin\theta$	$a/\lambda$	Interpretation
1	$2/3$	1	$Q=2/3$ (intensity ratio)
2	$1/3$	2	$Z_2$ sublattice doubling
3	$2/9$	3	Three wavelengths per cell
6	$1/9$	6	$C_3\times Z_2$ fully resolved

The rational number $2/9$ appears in three distinct roles: as $sin{\theta }_3$ in the resonance table, as the Shulga orientation parameter $\delta$ in the charged-lepton spectrum, and as the compact-cycle Berry phase. The proposed connection is an equality of dimensionless rational values, not an identification of the physical incidence angle with the Koide phase.

Figure 1. (a) Steady-state intensity $\langle {\psi }^2\rangle$ for three cylindrical-wave sources at ${120}^{\circ }$ on a ring of radius $R/\lambda =0.27$, showing the $C_3$ interference pattern with sixfold visual symmetry. (b) At $R/\lambda =1.10$, additional fringes resolve between the primary arms, consistent with the $N=6$ row of the resonance table ($1/9$ regime). Peak counts follow the sequence $3,6,9,12,15,18,\dots$ as $R/\lambda$ increases, adding three peaks per step in the numerical examples studied here. Each resolved interference ring contributes one peak per $C_3$ source. (c) The geometric-mean cascade from ${\mu }_{★}$ to $2m_e$, with ring closure.

Figure 1. (a) Steady-state intensity $\langle {\psi }^2\rangle$ for three cylindrical-wave sources at ${120}^{\circ }$ on a ring of radius $R/\lambda =0.27$, showing the $C_3$ interference pattern with sixfold visual symmetry. (b) At $R/\lambda =1.10$, additional fringes resolve between the primary arms, consistent with the $N=6$ row of the resonance table ($1/9$ regime). Peak counts follow the sequence $3,6,9,12,15,18,\dots$ as $R/\lambda$ increases, adding three peaks per step in the numerical examples studied here. Each resolved interference ring contributes one peak per $C_3$ source. (c) The geometric-mean cascade from ${\mu }_{★}$ to $2m_e$, with ring closure.

At $N=3$, three standing-wave antinodes fit inside one lattice cell. Numerical simulation of three cylindrical-wave point sources (2D Helmholtz equation, equal amplitude $A=2/3$, wavelength $\lambda =82$ grid units, exponential damping $e^{-\gamma r}$ with $\gamma =0.004$) on a ring of radius R illustrates this: at $R/\lambda \approx 0.27$, six intensity peaks appear at $2\pi /6$ spacing. Doubling R resolves the sublattice structure, doubling the peak count and halving the sector width to $1/9$. Simulation code is available at https://github.com/AndBrilliant/quarkscrew.

Shulga [9] independently derived $\delta =2/9$ as a Berry phase on a compact family cycle, from integrating out higher Fourier harmonics on an internal circle. The resonance interpretation proposed here reads the same rational value as a boundary-condition analogue: the cycle accommodates three wavelengths, suggesting three selected modes. Shulga’s compact-cycle calculation remains the derivation of $\delta =2/9$; the resonance picture is a geometric correspondence. All prior Berry-phase-fixes-an-observable examples in condensed matter (TKNN, Zak, Jauregui; see Xiao et al. [11] for a review) give integer or $\pi$-rational values; a non-integer rational like $2/9$ is, to our knowledge, without CM precedent outside Shulga’s construction. Until Shulga’s result is independently verified, the lepton-mass expressions should be regarded as a one-parameter fit to $\delta$.

3.1. Quark-Sector Offsets

Żenczykowski [12] observed that quark-sector offsets are fractions of $1/9=\delta /2$:

$${\delta }_U={\displaystyle \frac{Q}{9}}={\displaystyle \frac{2}{27}},{\delta }_D={\displaystyle \frac{2Q}{9}}={\displaystyle \frac{4}{27}},$$

(6)

with the charge $Q=2/3$ as the selection factor. In the interference picture, $1/9$ corresponds naturally to the $N=6$ row, where the full $C_3\times Z_2$ structure is resolved.

4. Effective Medium: Geometric Means

Dykhne [4] proved that for a two-dimensional isotropic two-phase system with equal area fractions, the effective transport coefficient is exactly the geometric mean:

$${\sigma }_{\mathrm{eff}}=\sqrt{{\sigma }_1{\sigma }_2}.$$

(7)

This is a theorem, not an approximation; variational bounds by Hashin and Shtrikman [13] show that the geometric mean is the exact attainable optimum for such systems.

The Dykhne theorem applies to transport coefficients in two-phase media; it does not, in itself, apply to fermion masses. We borrow only the geometric-mean functional form as an ansatz for interpolation between adjacent resonance scales:

$$\begin{array}{cc}\hfill m_s^2& ={\mu }_{★}·m_d,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill m_u^2& =m_d·2m_e,\hfill \end{array}$$

(9)

treating adjacent scales as if they were the two self-dual boundary values. Whether a valid physical mapping exists between the effective-medium setting and the fermion mass hierarchy is an open question; the ansatz is adopted here because it is the simplest interpolation consistent with the $Z_2$ structure and because it reproduces the data. The pair-production threshold $2m_e$ enters as the $Z_2$ base unit: the minimum energy at which mass is created from the electromagnetic field.

5. A Companion Identity

The $C_3$ phase cancellation and the Dykhne interpolation suggest a natural companion quantity. Given any three intensities $I_1,I_2,I_3$, define

$$F\equiv \sum _k\sqrt{I_k}-\sqrt{{\sum }_k{I}_k}.$$

(10)

The companion $F^2$ uses only addition, square roots, and subtraction.

The Descartes circle theorem [14,15] gives a companion via curvatures ${\kappa }_k=\sqrt{I_k}$: ${\kappa }_{\mathrm{Soddy}}=\sum {\kappa }_k-2\sqrt{{\sum }_{i<j}{\kappa }_i{\kappa }_j}$. Writing $E=\sum {\kappa }_k$, $P=\sum {\kappa }_k^2$, $C={\sum }_{i<j}{\kappa }_i{\kappa }_j$, the Soddy formula subtracts $2\sqrt{C}$; the F-identity subtracts $\sqrt{P}$. They agree when $4C=P$:

$$4C=P\iff Q={\displaystyle \frac{P}{E^2}}={\displaystyle \frac{2}{3}}.$$

(11)

This is proved in Sec. Section 2: the $C_3$ cancellation forces $Q=2/3$ identically. The companion identity therefore reduces to the Descartes circle theorem as a consequence of the interference geometry.

The identification $\kappa =\sqrt{m}$ as a curvature variable was noted by Kocik [16]. In condensed matter, $m^{*}\propto {({\partial }^{2}E/\partial k^2)}^{-1}$ is standard [3]; Satija [17] showed that the integer Apollonian gasket maps onto the Hofstadter butterfly energy spectrum, with curvatures encoding quantum Hall conductivities; this establishes a precedent for Descartes-type circle geometry appearing as a physical energy spectrum.

From $Q=2/3$:

$$F^2=\left(\frac{5}{2}-\sqrt{6}\right)A={\alpha }_K^{2}A,$$

(12)

where ${\alpha }_K\equiv \sqrt{3/2}-1$. The cascade ratio

$$R_K\equiv {\displaystyle \frac{1}{{\alpha }_K^2}}=19.798$$

(13)

is unitless and independent of $\delta$.

6. Correspondence with Charged-Lepton Masses

The Koide relation [18,19] for charged-lepton pole masses,

$$Q={\displaystyle \frac{m_e+m_{\mu }+m_{\tau }}{{(\sqrt{m_e}+\sqrt{m_{\mu }}+\sqrt{m_{\tau }})}^2}}={\displaystyle \frac{2}{3}},$$

(14)

has held to one part in ${10}^5$ for forty-four years and is approximately RG-invariant [20]. Setting $A={\mu }_{★}\equiv m_e+m_{\mu }+m_{\tau }=1883.1$ MeV and $\delta =2/9$ in Eq. 3 reproduces $m_{\tau }$, $m_{\mu }$, and $m_e$ to four significant figures.

The companion evaluates to

$$F^2(e,\mu ,\tau )=95.12\mathrm{MeV}.$$

(15)

All comparisons use PDG 2024 / FLAG 2024 values as a frozen reference convention. Comparison with PDG 2024 [21], run to ${\mu }_{★}$ via four-loop $\overline{\mathrm{MS}}$ (RunDec 3 [22], ${\alpha }_s\left(M_Z\right)=0.1180$, $N_f=3$):

$$m_s^{\overline{\mathrm{MS}}}\left({\mu }_{★}\right)=95.0\pm 0.8\mathrm{MeV},\Delta =+0.12\sigma .$$

(16)

The companion of the charged-lepton intensity pattern lies on the strange-quark $\overline{\mathrm{MS}}$ mass at ${\mu }_{★}$ within current uncertainties. This coincidence is empirical; the interference model does not predict which particle the companion should match. Among QCD mass scales below 1 GeV, only $m_s$ falls within the $1\sigma$ band of $F^2$; the pion decay constant $f_{\pi }=92.1$ MeV lies $3.2\%$ below, and no meson, baryon, or other quark mass falls within $5\%$.

7. Light-Quark Cascade

The geometric-mean cascade motivated by the Dykhne construction (Eqs. 8–) gives:

$$\begin{array}{cc}\hfill m_d& ={\alpha }_K^4{\mu }_{★}=4.804\mathrm{MeV},\hfill \\ \hfill m_u& ={\alpha }_K^2\sqrt{2m_e{\mu }_{★}}=2.216\mathrm{MeV}.\hfill \end{array}$$

(17)

PDG 2024 gives $m_u\left({\mu }_{★}\right)=2.20\pm 0.07$ MeV ($+0.23\sigma$). QED corrections for charge-$2/3$ quarks reach $\sim 1\%$ (Appendix A), giving $\pm 0.02$ MeV systematic. The relation selects $m_u=2.216$ MeV. The quoted $\pm 0.02$ MeV scale reflects the estimated size of the missing QED correction (QED is not applied; see Appendix A), not an experimental uncertainty. The $+0.23\sigma$ deviation is computed in a QCD-only scheme against PDG values.

The prediction chain forms a loop:

$${\mu }_{★}\stackrel{{\alpha }_K^2}{\to }{m}_s\stackrel{÷R_K}{\to }{m}_d\stackrel{\sqrt{·2m_e}}{\to }{m}_u\to 2m_e\stackrel{Z_2}{\to }e\stackrel{\mathrm{ring}}{\to }{\mu }_{★}.$$

(18)

The light-quark sum, pair-normalized, is empirically close to $100=5\times 20$, with the light-quark hierarchy 20 approximated by the Koide–Soddy cascade ratio $R_K=19.798$:

$${\displaystyle \frac{m_u+m_d+m_s}{2m_e}}\approx 100.$$

(19)

FLAG 2024 [24] values run to ${\mu }_{★}$ give $99.8\pm 0.6$. This empirical closure provides a fourth equation for three unknowns; equations 8– plus the closure alone give $m_s=95.17$ MeV versus $F^2=95.12$ MeV. Agreement: $0.058\%$.

The ratio $m_s/m_d$ is RG-invariant under QCD [23]. FLAG 2024 [24] reports $m_s/m_d=20.1\pm 0.3$, consistent with $R_K=19.80$ at $1.0\sigma$.

8. Cross-Checks

The tau-slice: applying the F-identity to $(m_Z,m_e,m_u)$ at the $\tau$ mass scale gives $F^2(m_Z,m_e,m_u\left(m_{\tau }\right))=m_d\left(m_{\tau }\right)$. Using PDG running masses directly (no cascade input, with $m_u\left(m_{\tau }\right)$ and $m_d\left(m_{\tau }\right)$ taken from external running-mass tables rather than from Eqs. 8–), the agreement is $0.015\%$; using cascade-derived $m_u$ run to $m_{\tau }$, the agreement is $0.49\%$. The first uses no cascade-derived quantities and thus tests the algebraic consistency of the F-identity with externally measured running masses; the second is a self-consistency check of the cascade itself.

A separately preregistered Monte Carlo analysis [25] tests the null hypothesis that a random Koide-compatible lepton spectrum ($Q=2/3$, random $\delta$) produces an $F^2$ residual as small as the observed $|F^2-m_s|=0.12$ MeV. Under four independent priors (log-uniform, log-normal, Yukawa anarchy, and linear uniform on ${\mu }_{★}$), the probability is $p\le 5\times {10}^{-4}$ in all cases. A look-elsewhere correction over 24 natural mass functions (ratios, roots, sums) reduces the effective significance by a factor of $\sim 2$. The analysis has not yet been independently scrutinized and is cited as supporting evidence, not as a standalone result. Pre-registered predictions are archived at [26]. An earlier version of this work appeared as [28].

9. Frozen-Input Outputs and Comparisons

The outputs are chained: $m_s$ determines $m_d$, which determines $m_u$. No free parameters buffer a discrepancy at any link.

Predictions were archived at Zenodo [26] prior to PDG 2026; any drift relative to updated PDG/FLAG values will be reported without retrofitting. The current agreements are suggestive but not yet decisive: the u and d quark masses carry large uncertainties (PDG quotes $\sim 3\%$ for $m_u$, $\sim 4\%$ for $m_d$), and the cascade values fall well within these bands. The true test is convergence: as lattice QCD+QED determinations sharpen over the next several years, the cascade outputs are fixed point values that will either be confirmed or excluded. Only that convergence, not the present-day overlap, can discriminate the construction from coincidence.

Table 1. Six masses from closed-form expressions. Input: ${\mu }_{★}=m_e+m_{\mu }+m_{\tau }$. The lepton masses ($\tau ,\mu ,e$) are reproduced by construction via ${\mu }_{★}$ and $\delta$; they are inputs, not predictions. The quark masses ($s,d,u$) are the testable outputs.

	Closed-form	Pred.	PDG/FLAG	Dev.
$\tau$	$\frac{{\mu }_{★}}{6}{(1+\sqrt{2}cos\frac{2}{9})}^2$	1776.93	1776.93	$+0.00\%$
$\mu$	$\frac{{\mu }_{★}}{6}{(1+\sqrt{2}cos(\frac{2}{9}+\frac{4\pi }{3}))}^2$	105.66	105.66	$+0.00\%$
e	$\frac{{\mu }_{★}}{6}{(1+\sqrt{2}cos(\frac{2}{9}+\frac{2\pi }{3}))}^2$	0.5110	0.5110	$-0.00\%$
s	${\alpha }_K^2{\mu }_{★}$	95.12	$95.0\pm 0.8$	$+0.12\sigma$
d	${\alpha }_K^4{\mu }_{★}$	4.804	$4.{75}_{-0.17}^{+0.49}$	$+0.10\sigma$
u	${\alpha }_K^2\sqrt{2m_e{\mu }_{★}}$	2.216	$2.{20}_{-0.05}^{+0.07}$	$+0.23\sigma$

Table 1. Six masses from closed-form expressions. Input: ${\mu }_{★}=m_e+m_{\mu }+m_{\tau }$. The lepton masses ($\tau ,\mu ,e$) are reproduced by construction via ${\mu }_{★}$ and $\delta$; they are inputs, not predictions. The quark masses ($s,d,u$) are the testable outputs.

	Closed-form	Pred.	PDG/FLAG	Dev.
$\tau$	$\frac{{\mu }_{★}}{6}{(1+\sqrt{2}cos\frac{2}{9})}^2$	1776.93	1776.93	$+0.00\%$
$\mu$	$\frac{{\mu }_{★}}{6}{(1+\sqrt{2}cos(\frac{2}{9}+\frac{4\pi }{3}))}^2$	105.66	105.66	$+0.00\%$
e	$\frac{{\mu }_{★}}{6}{(1+\sqrt{2}cos(\frac{2}{9}+\frac{2\pi }{3}))}^2$	0.5110	0.5110	$-0.00\%$
s	${\alpha }_K^2{\mu }_{★}$	95.12	$95.0\pm 0.8$	$+0.12\sigma$
d	${\alpha }_K^4{\mu }_{★}$	4.804	$4.{75}_{-0.17}^{+0.49}$	$+0.10\sigma$
u	${\alpha }_K^2\sqrt{2m_e{\mu }_{★}}$	2.216	$2.{20}_{-0.05}^{+0.07}$	$+0.23\sigma$

10. Discussion

The central observation is that $C_3$ phase cancellation, standing-wave resonance, and Dykhne effective-medium interpolation, all standard condensed-matter constructions, organize a set of algebraic identities whose numerical values match the light-fermion mass spectrum. The F-identity companion at $Q=2/3$ reduces to the Descartes circle theorem; the $N=3$ resonance condition gives a geometric counterpart to $\delta =2/9$; the Dykhne geometric mean motivates the quark cascade.

This construction does not provide a Lagrangian or dynamical mechanism, nor does it conflict with one. The Standard Model accommodates fermion masses through Yukawa couplings that are free parameters; the present work observes that six of those parameters are matched by closed-form geometric relations motivated by interference geometry. The construction constrains the Yukawa sector rather than competing with it.

Whether the fermion mass spectrum ultimately reflects an interference pattern on an internal compact space, or whether the numerical correspondences are structural parallels between $C_3\times Z_2$ systems with no dynamical connection, remains open. We propose it merits investigation.