What question did this study set out to answer?

To perform independent verification of the face adjacency Laplacian spectrum of the truncated octahedron.

March 21, 2026Open Access

Spectral Verification: Face Adjacency Laplacian of the Truncated Octahedron (UFFT Supplementary)

Key Points

To perform independent verification of the face adjacency Laplacian spectrum of the truncated octahedron.
Numerical verification using numpy with a maximum deviation of 4.44×10⁻¹⁵.
Symbolic verification using sympy to confirm polynomial identity as true.
Deployment of Jupyter notebook and standalone Python script for the calculations.
Verified spectrum includes specific eigenvalues such as 0¹, ((9−√17)/2)³, and 4².
Characteristic polynomial matches expected form: p(λ) = λ(λ²−9λ+16)³(λ−4)²(λ−7)⁴(λ−9).
Confirmed that √17 is exact, impacting UFFT predictions related to neutrino mixing and mass ratios.

Abstract

Independent verification that the face adjacency Laplacian of the truncated octahedron has spectrum Spec (L) = 0¹, ( (9−√17) /2) ³, 4², ( (9+√17) /2) ³, 7⁴, 9¹ with characteristic polynomial p (λ) = λ (λ²−9λ+16) ³ (λ−4) ² (λ−7) ⁴ (λ−9). Verified numerically (numpy, max deviation 4. 44×10⁻¹⁵) and symbolically (sympy, polynomial identity = True). The √17 is confirmed exact. This result underlies all UFFT predictions depending on √17 — solar neutrino mixing, Higgs/Z mass ratio, PMNS angles. Includes Jupyter notebook and standalone Python script. Dependencies: numpy, sympy only. Runtime: < 30 seconds.

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