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
A common constants program must connect dimensionless source outputs to response, matter, and geometry through one physical dictionary. A dependency map separates those deductions from supplied scales and empirical calibration.
- Identifiability: can observations determine the model?
Four dimensionless outputs leave two exact parameter directions unresolved in the displayed constants dictionary. The calibration fiber shows which additional diagnostics could distinguish otherwise equivalent models.
- A common dimensional dictionary
A shared dictionary converts source outputs into calibrated actions, lengths, times, masses, and curvature coefficients. Dimensional checks expose missing scales and distinguish coefficient matching from an independent derivation.
- An exact singlet-loop response
A mobile carrier, stationary dual, and supplied singlet-binding Hamiltonian reduce to one exact resolvent equation. Its unique physical root determines the holonomy-dependent ground response and a conditional completion-cost candidate.
- A protected gap in the finite singlet model
The finite singlet-loop Hamiltonian has a unique ground state and a uniform lower bound on its excitation gap. The derivation identifies the supplied penalty, carrier rank and limits that separate this result from a continuum gauge-theory mass gap.
- A normalizable hydrogen-like ground state
Square completion gives the exact normalized ground state of the supplied nonrelativistic Coulomb Hamiltonian. Its spectrum and relative root-mean-square speed provide precise atomic benchmarks without selecting the coupling itself.
- Completion as ground-state selection
A normalized two-outcome energy filter approaches the ground state along accepted sequences when the initial overlap is nonzero. Its fidelity bound is distinct from preparation yield, radiative relaxation and an autonomous completion mechanism.