Toward a Spectral-Geometric Mass Framework

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Paper V — Extended Spectral-Geometric Mass Framework (PDF) Paper IV — Primary Start with Paper IV Repository GitHub — vfd-crystallisation

Epistemic Status

This paper is exploratory, not the primary entry point. It contains four distinct levels of claim, listed from strongest to weakest:

1. Exact computations — graph-theoretic theorems on the 600-cell (strongest)
2. Numerically established correspondences — mass predictions, α expression
3. Derived within model — consequences of framework postulates
4. Framework postulates — structural assumptions (weakest)

This paper does not claim first-principles derivation, a complete dynamical theory, or absolute mass scale. For the foundation, start with Paper IV.

What the 600-Cell Provides

The mathematical foundation rests on six exact graph theorems — computations on a finite graph, not approximations:

9 eigenvalues in Q(√5) with n² multiplicities
600-cell uniqueness for φ-eigenvalues among regular 4-polytopes
Dual populations with intersection numbers from the association scheme
RD(4) = 15 — the Rogers bound on 4-dimensional kissing number
Hopf fiber eigenvalues in Q(φ)
Spectral gap structure constraining admissible standing-wave modes

These are exact computations on a finite graph. They do not depend on any physical assumption and would hold in any framework that uses the 600-cell as a combinatorial object.

The Mass Formula

The mass of each particle is determined by three ingredients, each with a different epistemic status:

Closure invariant ΔC — from the graph spectrum of the 600-cell
Standing-wave self-consistency correction — from WKB round-trip phase on the polytope
Winding contribution — from the Hopf fibration of the 600-cell

m = m_e × φ^E(θ)

Mass as the electron mass scaled by a φ-power whose exponent E(θ) is determined by the spectral geometry.
Zero fitted continuous parameters. All exponents from discrete graph data.

The Fine-Structure Constant

The four integer eigenvalues of the 600-cell adjacency matrix — {9, 12, 14, 15} — yield an expression for the inverse fine-structure constant:

α⁻¹ = 87 + 50 + π/87 = 137.036

From the integer eigenvalues {9, 12, 14, 15} of the 600-cell.
Status: numerically established (0.81 ppm), not analytically proven.

Status

The agreement is 0.81 parts per million. This is a numerically established correspondence — the expression reproduces the observed value to high precision, but the derivation connecting integer eigenvalues to the electromagnetic coupling has not been proven from first principles.

The Particle Table

13 non-reference particle masses predicted from the 600-cell spectral geometry:

Particle	Status	Error
Electron	Reference	—
Up	Predicted	0.002%
Down	Predicted	0.030%
Muon	Predicted	0.010%
Strange	Predicted	0.047%
Proton	Predicted	0.014%
Neutron	Predicted	<0.001%
Charm	Predicted	0.016%
Tau	Predicted	0.009%
Bottom	Predicted	0.007%
Top	Predicted	0.028%
W	Predicted	<0.001%
Z	Predicted	0.009%
Higgs	Predicted	0.005%

Average Error

Average error across 13 non-reference predictions: 0.014%. All masses from the spectral geometry of a single polytope with zero fitted continuous parameters.

What This Does Not Claim

A first-principles derivation — the framework postulates are assumptions, not proven axioms
A complete dynamical theory — no equations of motion, no S-matrix, no scattering amplitudes
Absolute mass scale — the electron mass is a reference input, not derived
Neutrino masses — not included in the current framework
Running masses — scale dependence of quark masses is not addressed
That chain ratios are algebraically proven — they are numerically established correspondences

The spectral geometry of a single polytope may organise the particle mass spectrum.

Paper open-access. Part of the crystallisation programme at github.com/vfd-org/vfd-crystallisation.