Coulomb and fine-structure calculations supply quantitative baselines. A source-derived correction must enter through compatible level operators, with fixed coefficients and a correlated pattern across reserved transitions.
Begin with one common set of levels
Label each stationary state a by its quantum numbers and assign an energy Ea. The signed transition frequency is νab=(Eb−Ea)/h. This definition immediately imposes a consistency law: νac=νab+νbc. Any proposed temporal correction to stationary spectroscopy must obey the same law.
SMM’s common-source ambition makes this constraint especially useful. Inner histories may differ between states, but their outer completion must produce one compatible energy structure. Independent offsets assigned to each spectral line can destroy that structure even when individual numbers appear plausible.
Recover the leading spectrum first
For the nonrelativistic Coulomb operator, En=−B/n², with B=μc²α²/2 after physical matching. The energies approach the zero continuum threshold. Adjacent spacing is B(2n+1)/[n²(n+1)²], approaching 2B/n³ at large n. These are exact consequences of the supplied scalar model.
Open the full-size diagram →
For a worked scale-free check, the 1→2 interval is 3B/4 and the 2→3 interval is 5B/36, giving ratio 5/27. A source construction that recovers this ratio has reproduced part of the Coulomb operator’s structure; it has not thereby fixed B or derived radiative corrections.
Fine structure links spin and motion
In the point-nucleus, infinite-nuclear-mass Dirac–Coulomb benchmark, the leading correction is δEfs(n,j)=−[mc²α⁴/(2n⁴)][n/(j+1/2)−3/4]. The α⁴ scale follows because typical bound momentum is of order mcα, so the relativistic kinetic correction p⁴/(m³c²) has that energy order.
At n=2, the bracket is 5/4 for j=1/2 and 1/4 for j=3/2. Their energy difference at this order is mc²α⁴/32. Correct spin-orbit and contact coefficients are needed to obtain the combined pattern. Reduced mass alone does not supply every recoil correction, and nuclear, radiative, and external-field effects enter later.
An explicit way inner modes can shift levels
Let P select retained atomic states and Q=I−P select auxiliary states of one self-adjoint Hamiltonian. Eliminating the Q component of the stationary equation gives Heff(E)=PHP+PHQ(E−QHQ)−1QHP whenever the resolvent exists. This displays how unresolved generative structure can leave an energy-dependent response in completed records.
For one retained level E0, one auxiliary level E0+Δ, and coupling v, the lower exact energy is E0+Δ/2−√(Δ²/4+v²). At |v|≪Δ its shift is −v²/Δ+O(v⁴/Δ³). This is a calculable auxiliary response. A radiative field loop additionally needs its field states, symmetry constraints, regulator, and matching prescription.
Worked example: one operator, several lines
As a first-order effective insertion, take δH=λδ³(r). It shifts nS states by K/n³, where K=λ/(πaB³), while states with nonzero orbital angular momentum have zero contact matrix element at this order. The delta is used here as a perturbative insertion, not an unregularized standalone three-dimensional Hamiltonian.
The ratio is fixed before λ is chosen. It is a useful cross-state consequence, although ordinary nuclear short-distance effects can share contact-like behavior. Other angular or isotope channels are needed to identify an origin.
Keep the baseline and new operator consistent
A Lamb-shift comparison requires a specified reference containing the relevant self-energy, vacuum-polarization, recoil, nuclear, and higher-order terms. A missing conventional contribution is not a new DTT effect. Likewise, a holonomy phase needs an energy map: a dimensionless cycle fraction alone cannot be subtracted from a measured frequency.
Calculate any proposed correction from a common operator δH=ΣckOk. For nondegenerate levels its first-order shifts are the corresponding expectation values; within a degenerate subspace, diagonalize the perturbation first. This procedure enforces level compatibility and shows which coefficients are independently fixed, fitted, or still unknown.
For a simple holonomy-based example, assign three dimensionless state fractions η1=0, η2=ε, and η3=0 with one shared energy map δEa=Kηa. The first transition shifts by Kε/h, the next by −Kε/h, and the direct 1→3 shift vanishes. The cycle closes automatically. Separate transition-dependent multipliers would generally lose this constraint.
Freeze shared inputs and use complementary channels
Define the calibration set before selecting reserved predictions. A transition already used indirectly to determine a mass, radius, or frequency scale cannot simultaneously count as independent confirmation. Shared uncertainties produce covariance: two lines involving the same level or parameter are generally correlated. A transition network with q connected levels contains only q−1 independent stationary energy differences.
The theoretical priority is to derive an operator relation from the six nested inner roles and one completion law. Hydrogen, isotope variation, and high-angular-momentum states can then test distinct consequences of that relation. A successful result would organize many levels with fewer independent coefficients, making the common source explanatory rather than a separate label attached to each spectral correction.
Continue with the articles
- Hydrogen fine structure and its temporal interpretation
Examine how relativistic, spin, and angular-momentum structure split the Coulomb spectrum. The standard baseline gives precise requirements for a temporal source model and shows which additional quantities could become independent predictions.
- The Lamb shift as a demanding benchmark
A proposed guide to using the Lamb shift as a precision benchmark for the program. Explain why naming a temporal-loop mechanism is weaker than reproducing a quantitative shift.
- Radiative inner-time loops: defining the proposal
Develop the skeleton draft’s temporal-loop language into a precise construction. Specify the states, interactions, and regulator before claiming a connection with radiative corrections.
- Freeze parameters before predicting spectroscopy
Explain the post-freeze strategy proposed for atomic observables. A spectroscopy comparison becomes informative when its coefficients are fixed without using the same transitions as targets.
- Recovering a QED baseline before adding residuals
A proposed methodology article for the fine-structure program. Explain how agreement with a standard baseline must be established before a residual can be attributed to new temporal dynamics.
- What would count as a DTT spectroscopic residual?
Outline criteria for a meaningful residual: a specified model, fixed coefficients, uncertainty control, and a pattern across observables. The skeleton draft motivates this task but does not complete it.
- State-dependent record cycles in hydrogen
Introduce the skeleton draft’s proposal to extend a ground-state holonomy description to different atomic states. Focus on the definitions needed before state-dependent predictions can be evaluated.
- Differential spectroscopy as a test strategy
Explain why differences between transitions can test a shared model more strongly than a single fitted line. Develop the proposed holonomy comparison as an overconstrained protocol.
- Hydrogen and deuterium as complementary probes
A proposed isotope-comparison article within the differential-test program. Identify which effects are shared and which require separate mass and nuclear inputs before interpreting a residual.
- Rydberg states and long record cycles
Outline how highly excited states could extend the proposed state-dependent test program. Explain which scaling law the temporal model must provide before measurements can discriminate it.
- Designing an overconstrained atomic test
Turn the skeleton draft’s research direction into a practical analysis plan. Require more independent observables than fitted parameters and predeclare the cross-state prediction rules.