Mass and dispersion from a recurrent carrier

A mass is more than an internal frequency

DTT motivates the possibility that matter corresponds to persistent internal recurrence. To turn this into inertial mass, the model must explain how a moving excitation carries energy and momentum. A frequency label by itself supplies neither an energy unit nor the coefficient controlling the response to momentum. The desired result is a dispersion relation in which the rest frequency and propagation share the same physical clock.

The familiar positive-energy relation E²=c²p²+m²c⁴ then gives E(0)=mc². Its small-momentum expansion gives the kinetic term p²/(2m). Recovering both from one operator is more informative than naming a recurrence frequency a mass.

An explicit two-component update

Take a one-dimensional lattice with spacing ℓ and a two-component complex carrier at each site. One update first rotates the components through C=exp(−iθσ_x) and then shifts the two components in opposite directions by one site. In a momentum sector with dimensionless wave number k, choose the shift matrix S(k)=exp(−ikσ_z). The update is U(k)=S(k)C.

The Pauli matrices satisfy σ_x²=σ_z²=I and anticommute. Both factors are unitary with determinant one. This is a supplied toy carrier and spatial lattice; the construction illustrates what a later source-generated carrier would need to reproduce.

Derive the exact dispersion

A two-by-two unitary matrix with determinant one has conjugate eigenvalues e^±iω. Its trace is therefore 2cosω. Expanding the two factors of U and taking the trace removes every Pauli term. The remaining scalar term gives an exact dispersion relation without a continuum approximation.

For small k and θ, expand each cosine to second order. The constant terms cancel and give ω²≈k²+θ². This is the form of a massive relativistic dispersion near the long-wavelength, small-angle regime. At finite lattice scale the exact cosine law has additional corrections and is not exactly Lorentz invariant.

Connect the same phase to physical units

Let each update correspond to a measured duration τ and use one action scale ħ. Set E=ħω/τ, p=ħk/ℓ, and c_*=ℓ/τ. In the stated limit, the dispersion becomes E²≈c_*²p²+(ħθ/τ)². Defining m_*=ħθ/(τc_*²) identifies its rest-energy scale and low-momentum kinetic coefficient consistently.

This dictionary does not determine ℓ, τ, ħ, or θ numerically. Moreover, identifying c_* with the light speed measured by another sector requires a common-source matching result. Equality cannot be established by giving both speeds the same symbol.

Worked example and the controlled limit

Choose θ=0.02 and k=0.01. The continuum estimate is ω≈√0.0005≈0.02236 radians per update. The exact value is obtained from arccos(cos0.02 cos0.01), allowing the lattice error to be checked directly. For k much smaller than θ, expanding the square root gives ω≈θ+k²/(2θ), and thus the expected nonrelativistic kinetic term after applying the same dictionary.

There is a further useful audit. At fixed nonzero θ, the exact small-k coefficient is cotθ/2, not 1/(2θ). The two agree as θ approaches zero. This identifies the precise approximation needed when the rest-frequency mass is also called the inertial mass.

A discrete update also defines eigenphases modulo 2π. Selecting a low-energy branch is therefore part of the continuum description, not an absolute energy assignment at every phase. A source theory must explain which band is physically retained and why its clock calibration continues consistently across that band.

Propagation and the limit of a mode

Differentiating the exact dispersion gives dω/dk=cosθ sin k/sinω. Since sin²ω−cos²θ sin²k=sin²θ≥0, its magnitude is at most one where the branch derivative is defined. The group speed therefore satisfies |v_g|≤ℓ/τ. Independently, the update itself moves information at most one lattice site per step, giving an exact discrete propagation bound.

A group-velocity bound and a relativistic invariant metric are different statements. The first follows here from the specific unitary walk. Exact Lorentz symmetry is a property of an appropriate continuum limit, and a full three-dimensional theory would need compatible carriers, locality, and interactions.

What SMM/DTT should select next

The example makes the temporal interpretation productive: internal mixing supplies a rest phase, while spatial transfer supplies momentum dependence. The six nested inner roles would need to generate these operations and their retained memory together, with outer completion defining τ operationally. They should not merely program unrelated values of θ and the shift rule.

The substantive research question is whether one source law fixes the allowed mixing phases and forces all physical sectors to share the same clock and propagation conversion. A successful answer could connect a discrete recurrence spectrum to mass ratios. Until that selection is derived, the walk is a transparent reconstruction example with an adjustable mass parameter.

Continue with the articles

• The Mass-Energy Equivalence

  Mass–energy equivalence provides a focused test of a temporal interpretation of familiar physics. The relativistic derivation identifies which energy, clock and propagation relations a deeper SMM/DTT construction must explain.

• The Properties of Matter

  Material properties are identified through invariants and reproducible responses. Examples involving mass, weight and collective behaviour show the distinct measurement and dynamical requirements for an SMM/DTT account of matter.

• A charged threshold is not a full mass gap

  A positive charged cycle threshold leaves neutral centralizer modes in the spectrum. The exact finite-operator calculation and projected inequality identify why compact recurrence alone does not establish a full physical mass gap.

• The remaining gap-proof obligation

  Finite-volume gaps, curvature masses and screening estimates do not yet establish a volume-uniform quantum gap. This audit identifies the connected-correlation estimate and the additional gauge and continuum matching steps left open in the cited manuscript.