The Standard Model contains nineteen free parameters: quark masses, lepton masses, mixing angles, coupling constants. These numbers are extracted from experiment and inserted into the theory; nothing in the theory itself predicts them. They are not derived from any deeper principle, and their specific values carry no apparent geometric or algebraic meaning. This has always been a source of dissatisfaction. A theory of everything should be able to tell you why the electron mass is what it is.
Decoding Reality addresses this directly. Its central claim is that the dimensionless ratios among Standard Model constants are the special values of automorphic L-functions evaluated at arithmetic points determined by the representation theory of the Standard Model gauge group. These L-functions are the same family of objects that appears in the Birch–Swinnerton-Dyer conjecture and the Langlands programme. The constants of physics are, in this sense, numbers that the structure of arithmetic forces.
Quark Masses from Casimir Invariants
The quark masses are the most numerous of the free parameters and the least understood. In the framework, they arise from the Kac–Moody Casimir invariants of the boundary current algebra. The Sugawara formula gives the conformal weight hR = C2(R)/(k + h∨) for a representation R of the gauge group at level k. Under the mass conversion MR = 2hRΛQCD, each quark mass corresponds to a specific representation of SU(3) × SU(2) × U(1) at the Kac–Moody level appropriate to that sector.
The predictions are verified to fifteen significant digits in the companion numerical verification paper. This is not fitting: the representations are determined by the gauge group structure, the level by the coupling constants at the unification scale, and the output is the mass ratios. If the prediction fails a precision measurement, the framework is falsified in a specific and testable way.
The Cabibbo Angle
The Cabibbo angle θC ≈ 13.04° governs quark mixing between the first and second generations. It appears in the CKM matrix, the matrix that rotates quark mass eigenstates into weak interaction eigenstates. In the Standard Model, its value is measured and inserted; nothing predicts it.
In the framework, the Cabibbo angle is the Fubini–Study metric angle between two specific lines in ℂP2 — the complex projective space that parametrizes the two-quark mixing sector of the CKM matrix. The geometry of complex projective space is fixed by the inner product on the underlying complex vector space, which is in turn fixed by the Haar measure on the gauge group. The Cabibbo angle is not a parameter; it is a geometric invariant of the mixing sector.
The Koide Locus
In 1982, Yoshio Koide observed that the three charged lepton masses me, mμ, mτ satisfy the relation (me + mμ + mτ) = ⅔(√me + √mμ + √mτ)2 to experimental precision better than one part in ten thousand. This is a non-trivial relation among measured masses, not a tautology, and its origin has been mysterious for four decades.
The framework derives it from the geometry of the representation ring of SU(2). The three lepton masses correspond to the three spin-j representations of SU(2) at j = 1/2, 1, 3/2 (in a specific parametrization), and the Koide relation is the condition that these three representations lie on a specific geodesic in the representation variety. For quarks, the colour Casimir CF = 4/3 shifts the representations off this geodesic in a calculable way, giving specific predictions for the quark geometry that differ from the lepton geometry by a computable colour correction.
The constants of physics are not arbitrary numbers written into the universe's source code. They are the values that the representation theory of the gauge group forces upon the arithmetic invariants of the boundary conformal field theory.