Paper release: r1578 Released: July 25, 2026

Keywords: Koide relation, circulant matrices, cyclic symmetry, icosahedral symmetry, finite event algebra, GNS representation

The face-corner carrier

Koide’s charged-lepton relation  is usually written in terms of the three positive square-root masses. The construction below identifies the exact condition it imposes on a cyclic Hermitian response.

Let \(G=A_5\) act by proper rotations on the icosahedron. The twenty outward oriented faces form the transitive orbit \(G/C_3\). The stabilizer of one face cyclically permutes its three corners, so every face carries a local copy of the regular \(C_3\) representation. If \(R\) denotes the cyclic shift, then \[ R^3=I,\qquad R^\dagger=R^2. \] The commutant of this regular action is the three-dimensional circulant algebra \[ \{x_0I+x_1R+x_2R^2:x_j\in\mathbb C\}. \] Hermiticity restricts a response to \[ \begin{equation} C=aI+bR+\overline bR^2,\qquad a\in\mathbb R,\quad b\in\mathbb C. \label{eq:circulant} \end{equation} \] Write \(b=\rho e^{i\delta}\), with \(\rho=|b|\geq0\). Fourier diagonalization gives \[ \begin{equation} \lambda_k=a+2\rho\cos\left(\delta+\frac{2\pi k}{3}\right), \qquad k=0,1,2. \label{eq:eigenvalues} \end{equation} \] The unordered spectrum is independent of the chosen face representative. This is a local bundle statement. The sixty face-corner flags form the regular \(A_5\) torsor; the geometry alone supplies no canonical global three-dimensional physical family space.

The positive-chamber identity

Conditional finite tracial balance

The balance condition in Eq. [eq:balance] can arise exactly from a finite event packet. This subsection states the packet as an independent conditional theorem.

Let \[ \mathcal V=\mathbf 1\oplus\chi\oplus\overline\chi, \qquad \mathcal H_{\mathrm{or}}=\mathbb C^2, \qquad \mathcal A=B(\mathcal V\otimes\mathcal H_{\mathrm{or}}) \simeq M_6(\mathbb C). \] Here \(\mathbf 1\) is the neutral cyclic mode, \(\chi\oplus\overline\chi\) is the two-dimensional charged plane, and \(\mathcal H_{\mathrm{or}}\) is a two-state orientation record. Let \(P_0\) and \(P_c\) project onto the neutral line and charged plane. For a rank-one oriented event \(q_+\), define \[ Z_0=P_0\otimes I_{\mathrm{or}}, \qquad Z_c=P_c\otimes q_+, \qquad E_+=Z_0+Z_c. \] Both \(Z_0\) and \(Z_c\) have rank two.

Theorem 4 is finite and exact. Its event algebra, block choice, conditioned trace, and cyclic response map are premises. A physical charged-lepton statement additionally requires a quotient-visible map from the local face bundle to one chiral three-family response, preservation of the relevant block powers, and a mass readout. Those objects are absent from the theorem.

Numerical diagnostic and provenance

Using the Particle Data Group 2026 central masses , \[ (m_e,m_\mu,m_\tau) =(0.51099895069,\ 105.6583755,\ 1776.93)\ {\rm MeV}, \] gives \[ Q_{\mathrm{PDG}}=0.6666644634026367. \] The declared minimal complete public response model gives \[ \begin{aligned} Q_{\mathrm{MCPR}}&=0.6666644634090389,\\ \rho/a&=0.7071044442750720,\\ \frac{\rho/a-1/\sqrt2}{1/\sqrt2} &=-3.3049\times10^{-6}. \end{aligned} \] The response dimensions, path table, amplitude, phase, and determinant exponent were historically selected with knowledge of the charged-lepton target. The executable evaluation reads no charged target at runtime, yet that runtime separation does not make the model blind or source-derived. No significance or prospective evidence claim follows from the proximity of the two displayed \(Q\) values.

Why residual symmetry does not finish the ratios

The remaining family-shape problem can be stated in the real five-dimensional representation \[ W_5\simeq\operatorname{Sym}^2_0(\mathbb R^3), \] the traceless symmetric \(3\times3\) matrices with action \(A\mapsto gAg^T\).

Threefold and fivefold symmetry cannot produce three distinct masses. Twofold symmetry leaves two scale-free coordinates, exactly enough to carry the two mass ratios without selecting them. A numerical spectrum therefore requires a specific invariant potential derived from the screen dynamics. This proposition locates the missing input; it supplies no replacement fit.

Formal proof boundary

The algebraic core has a Lean 4 formalization. It proves the root sum, square sum, quotient formula, reciprocal-square-root identity, and the nonnegative-modulus equivalence \[ \frac13+\frac23r^2=\frac23 \quad\Longleftrightarrow\quad r=\frac1{\sqrt2}. \] The formal module deliberately does not identify signed eigenvalues with physical square roots. Positivity of all three eigenvalues, the phase selection, and the source-to-charged-family attachment remain outside its statement. This division matches Theorem 1: the checked algebra is exact, and the physical reading is a separate premise.

Conclusion

The regular three-corner face carrier turns the Koide relation into one transparent modulus condition. In the positive chamber, \[ Q=\frac23 \quad\Longleftrightarrow\quad \rho/a=\frac1{\sqrt2}. \] The finite tracial event packet supplies this balance conditionally through equal block weights and the canonical square-root representation. The result does not determine the phase or mass ratios and does not attach the local face fiber to physical charged leptons. Its durable content is an exact circulant identity, an exact finite conditional balance theorem, and a sharp statement of the remaining physical work.