The Positive-Chamber Koide Identity for Icosahedral Face Circulants

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

The face-corner carrier

Koide’s charged-lepton relation [source] 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 \(G=A_5\), the alternating group on five letters, 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.

The conditional tau-mass window

Fixing \(Q=2/3\) in Eq. eq:koideformula and inserting the measured electron and muon masses turns the Koide coordinate into one quadratic in \(\sqrt{m_\tau}\). The mass-ordering premise \(m_\tau>m_\mu\) selects the physical root; the excluded second root sits at \(3.317\) MeV, below the muon mass. Propagating the one-sigma corners of the measured \((m_e,m_\mu)\) inputs [source] with outward rounding encloses the selected root in \[\begin{equation} m_\tau\in[1776.968991,\ 1776.969063]\ \mathrm{MeV}, \label{eq:tauwindow} \end{equation}\] a 72-eV window with center \(1776.969027\) MeV. The center sits \(0.43\) standard uncertainties from the measured \(1776.93\pm0.09\) MeV, and the window is three orders of magnitude narrower than that measurement uncertainty.

The comparison is governed by a registered conditional test whose target definition and decision rule were fixed in a frozen prediction record [source]: a world-average or dedicated-measurement central value more than three standard uncertainties from \(1776.969027\) MeV refutes the balanced-circulant premise; compatibility requires agreement within two standard uncertainties at a measurement uncertainty of at most \(0.045\) MeV; every other outcome carries no verdict. The premise ancestry is declared: the balance condition was first abstracted from the measured charged-lepton triple, so the window is a target-informed conditional postdiction whose evidential weight lies in the kill direction. Improving tau-mass averages test the balanced-circulant premise directly.

Numerical diagnostic and provenance

Using the Particle Data Group 2026 central masses [source], \[(m_e,m_\mu,m_\tau) =(0.51099895069,\ 105.6583755,\ 1776.93)\ {\rm MeV},\] 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 are target-informed inputs. The numerical response is therefore neither blind nor source-derived. The proximity of the two displayed \(Q\) values carries no significance or predictive weight.

Why residual symmetry does not finish the ratios

The 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 are 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. Under the balance and mass-ordering premises the measured electron and muon masses fix the tau mass inside the 72-eV window of Eq. eq:tauwindow, and a registered decision rule fixes what refutes the balanced-circulant premise. 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, a conditional tau-mass window with a registered decision rule, and a sharp boundary between the checked algebra and the physical attachment premises.

AI Assistance Disclosure

This research project used research-grade commercial models, including Anthropic’s Fable and OpenAI’s GPT-5.6-Sol, for research support, software development, editing, and synthesis. The authors are responsible for the paper’s claims, methods, and final text.