Electron mass

SIMPLIFIED DERIVATION & CONCEPTUAL CONNECTION

How could a persistent matter record acquire a rest-energy scale?

A positive transfer operator gives dimensionless excitation gaps. The action scale and completion time convert a gap into energy. A candidate gap becomes an electron mass only after the physical sector and rest-energy relation have been established.

Read the common SMM/DTT foundations and notation →
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The SMM/DTT connection

In the proposed temporal account, a particle is a persistent excitation of the common generative process. Its inertia should reflect the cost of sustaining that excitation across completed records. A transfer description makes this idea calculable: compare the propagation of an excited sector with the vacuum over one completion step.

Choose one positive transfer theory

Let T be a self-adjoint, strictly positive transfer operator with eigenvalues t₀ ≥ t_s > 0. Use a common vacuum reference t₀ and define the dimensionless generator:

Ĥ = −log(T/t₀), ε_s = −log(t_s/t₀).

The ratios remove a common normalization. A gap ε_s ≥ 0 measures relative excitation cost in this model. The transfer operator and its physical sectors must be supplied before naming a particular gap.

Convert a step into physical energy

For a physical completion interval τ* and action scale a, matching T/t₀ = exp(−τ*H/a) gives:

E_s − E₀ = (a/τ*)ε_s.

This is a transfer-to-energy dictionary. It does not identify τ* with an experimentally known clock without a further physical argument.

Identify the rest threshold

If a charged spinor sector has the required low-momentum dispersion E(p)² = m²c⁴ + c²p² + …, its rest threshold is mc². Thus:

mₑ = aεₑ/(c²τ*), [mₑ] = kg.

Charge, spin, statistics, stability and interactions are indispensable. In a theory with massless photons, a charged threshold need not be an isolated one-particle eigenvalue; a controlled finite-volume or infrared prescription is required.

Vacuum and candidate excitation levels with the conversion from transfer ratio to mass — Figure m1. The levels illustrate the dictionary. Their positions are not a predicted electron spectrum. Download SVG

Inspect the cyclic alternative

For the particular difference operator U − I on N anti-periodic ticks, Uᴺ = −I, the smallest singular frequency is:

ε_N = 2 sin[π/(2N)].

This follows from the distance of the nearest allowed phase eigenvalue to 1. It is an exact property of that selected operator. A singular frequency is not automatically the transfer gap of a physical electron; an additional dynamical identification is needed. The workbench exposes that choice as an alternative input rule.

A useful dimensional distinction

For a temporal circle with radius R_t measured in seconds, ω_n = n/R_t gives m_n = an/(R_tc²) after energy matching. For a spatial circle with R_x in metres, k_n = n/R_x; only after supplying ω = ck does m_n = an/(R_xc). The radius units cannot be interchanged.

What would make this a mass prediction?

Derive T from the source update, identify the electron sector independently of its measured mass, compute its gap and establish a common scale through independent physics. Fitting τₑ = aεₑ/(mₑ,_refc²) reproduces the target by construction and introduces a separate electron clock.

Try the workbench

Vary the supplied gap and watch the mass change linearly. Switch to the anti-periodic mode to compare the operator-specific value. Turn on the mass fit to see the circular dependency and the warning that a separate fitted clock breaks a common-scale interpretation.

Source basis: C · §§5, 11, 14; pp. 7–8, 18–19, 22–23. These explanations summarize the stated models; they do not add a new physical selection theorem.

Explore mₑ.