A positive transfer spectrum gives εs=−log(ts/t0), and common time–action matching converts it to ms=aεs/(c²τ*). Electron charge, spin, stability and dispersion must identify which threshold the formula describes.
Persistence as a spectral question
In SMM/DTT, matter is proposed to be a persistent pattern of the generating process rather than an independently supplied substance. To make persistence quantitative, compare the completed propagation of an excited sector with that of a reference vacuum. The difference should be encoded in one source-derived operator.
The six inner levels organize nested work before an outer record is available. They do not, by themselves, specify a mass spectrum. A source model needs an autonomous update, retained memory, admissible charged states and a completion rule. Once those are supplied, its spectral problem can ask which patterns remain stable and what cost is associated with their persistence.
Calculate a dimensionless excitation gap
Let T be a self-adjoint strictly positive transfer operator with a largest reference eigenvalue t0 and an excitation eigenvalue ts≤t0. Define the dimensionless relative generator by taking a logarithm.
After n transfer steps, the excitation-to-reference ratio is exp(−nεs). A common multiplicative normalization of T cancels. Thus the gap measures a relative cost, not an absolute vacuum energy. This positive transfer description is not itself unitary real-time evolution; connecting the two requires an appropriate physical reconstruction and clock dictionary.
Convert the gap using one clock and action
Suppose the normalized transfer operator is physically matched to exp(−τ*H/a), where τ* is a completion duration and a is the action scale. Then the energy difference is Es−E0=aεs/τ*. If the relevant threshold is a rest energy mc², the mass follows.
As a symbolic example, let ts/t0=exp(−δ). Its gap is δ and its conditional mass is aδ/(c²τ*). Halving δ halves that mass with the dictionary fixed. Fitting τ* from the measured electron mass would instead calibrate the conversion and could not independently predict the same mass.
A worked anti-periodic mode
A different illustrative construction takes a unitary shift U on N ticks with UN=−I. Its phases are exp[i(2k+1)π/N]. The singular values of U−I are their distances from one: |exp(iθ)−1|=2|sin(θ/2)|. The nearest permitted phase therefore gives:
At N=3 this value is one; for large N it behaves as π/N. This is an exact property of the selected difference operator and boundary condition. It is not automatically the logarithmic transfer gap defined above. A source theory must derive the operator, justify anti-periodicity and show how its singular frequency enters a physical dispersion before treating it as a particle mass.
What identifies the electron sector?
A label on a spectral level does not determine particle identity. The candidate sector must have the appropriate charge, spin transformation, statistics, stability and interactions. Its low-momentum energy must match a rest threshold and an independently defined propagation law, for example E(p)²=m²c⁴+c²p² plus controlled corrections.
For a charged sector coupled to massless radiation, the threshold need not be an isolated one-particle eigenvalue in an infinite-volume description. A finite-volume or infrared prescription must state what is being calculated and how it is compared across limits. The finite singlet-loop gap discussed in the article collection is a useful protected auxiliary example, but it is not by itself an electron spectrum or a continuum mass theorem.
Temporal and spatial circles give different dictionaries
A temporal circle with radius Rt measured in seconds has angular frequencies ωn=n/Rt. With energy matching E=aω, the mass expression is an/(Rtc²). A spatial circle with radius Rx measured in metres instead gives wave numbers kn=n/Rx. If a linear dispersion ω=ck is additionally supplied, its corresponding mass form is an/(Rxc).
The distinction prevents a dimensional error from masquerading as a mass prediction. Neither compactness nor a discrete mode label chooses the radius. A common source must determine its relation to clock readings, spatial distances and the matter sector without importing the target mass through a recurrence formula.
Calculate a common spectrum before fitting masses
The next constructive target is a source-derived transfer or Hamiltonian problem with explicitly identified charged spinor sectors. Compute its dimensionless threshold, derive the shared action and clock matching, and test its dispersion. The hierarchy between that threshold and geometric or other matter scales is then a physical output rather than an inserted label.
The electron-mass workbench compares a supplied gap with the anti-periodic difference example and exposes the effect of fitting a clock. The mass-ratio guide removes the overall conversion factors altogether. Together they provide two complementary checks: whether the source produces the right relative spectrum, and whether one independently established scale converts that spectrum consistently.
Continue with the articles
- A dependency map for physical constants
Introduce the constants program as a chain from completion dynamics to response and then physical matching. Identify which quantities are structural, calibrated, or still free.
- Identifiability: can observations determine the model?
Explain why a collection of observable constants can leave parameter directions unresolved. Use the draft’s two remaining directions to distinguish a good fit from unique reconstruction.
- A common dimensional dictionary
Introduce the conversion between dimensionless model outputs and quantities expressed in physical units. Explain why unit consistency is necessary but does not determine an absolute action, length, or time scale.
- An exact singlet-loop response
Introduce the four-position mobile carrier and stationary dual used in the solvable model. Explain how the holonomy-dependent resolvent determines the ground-state response under the supplied Hamiltonian.
- A protected gap in the finite singlet model
Explain the model’s unique ground state and gap bound. Distinguish a finite-system spectral statement from a continuum mass gap in a physical gauge theory.
- A normalizable hydrogen-like ground state
Introduce the lower-bound and bound-state results of the effective model. Explain how the supplied kinetic and potential rules produce a meaningful atomic test case.
- Completion as ground-state selection
Explain the selective instrument that converges toward the model’s ground state when initial overlap is nonzero. Keep this construction separate from a universal physical mechanism of atomic formation.