Assumptions, results and research status
All residuals here are synthetic. A physical DTT claim requires a source-derived operator, independently fixed coefficients, a conventional baseline, and uncertainty analysis.
What this explorer calculates
This is a synthetic calculation. Its coefficients illustrate declared level-response models and are not fitted to measurements or computed DTT radiative coefficients. The explorer first assigns one correction to each state, then subtracts endpoint energies to form intervals. Every occurrence of a state therefore uses the same correction.
Use the controls to compare two contributions with the same contact-like state dependence, and test how a common offset cancels. A pattern can be restrictive without uniquely identifying its origin: finite nuclear size and a proposed local source response can have the same leading S-state dependence. [Differential programme]
Explore the model
Explore shared level shifts and differential intervals
Synthetic illustration. Assign one correction to each state, then obtain every interval correction by subtracting the two state corrections. This shows why independent per-line adjustments can fail and why a shared response pattern may still be physically ambiguous.
δνa→b = (δEb − δEa)/h.
The contact profile acts on S states (ℓ = 0). P and D states have zero contact weight in this simplified model. A and B are separately adjustable labels for exactly the same profile. C is a common offset. No measured spectral residuals are loaded.
- Total contact coefficient A + B
- +10 kHz
- Signed correction: 2S → 8D
- −1.25 kHz
Synthetic model. Total contact coefficient +10 kHz. The 2S to 8D correction is −1.25 kHz. The shared-level cycle closes.
| Interval a → b | Weight difference | A contribution / kHz | B contribution / kHz | Total / kHz |
|---|---|---|---|---|
| 1S → 2S | -0.875 | -8.75 | -0 | -8.75 |
| 2S → 8S | -0.123046875 | -1.23046875 | -0 | -1.23046875 |
| 1S → 8S | -0.998046875 | -9.98046875 | -0 | -9.98046875 |
| 2S → 8D | -0.125 | -1.25 | -0 | -1.25 |
| 2S → 2P | -0.125 | -1.25 | -0 | -1.25 |
| 2P → 8D | 0 | 0 | 0 | 0 |
State corrections used in every interval
| State | Contact weight | A / kHz | B / kHz | C / kHz | δE/h / kHz |
|---|---|---|---|---|---|
| 1S | 1 | +10 | 0 | 0 | +10 |
| 2S | 0.125 | +1.25 | 0 | 0 | +1.25 |
| 2P | 0 | 0 | 0 | 0 | 0 |
| 3S | 0.037037037 | +0.37037037 | 0 | 0 | +0.37037037 |
| 8S | 0.001953125 | +0.01953125 | 0 | 0 | +0.01953125 |
| 8D | 0 | 0 | 0 | 0 | 0 |
1. Verify the cycle relation
For three levels a, b, and c, a shared energy model gives δνa→c = δνa→b + δνb→c. The cancellation is algebraic, independent of the size of A, B, or C.
-8.75 + (-1.23046875) = -9.98046875 kHz
Cycle residual: 0 kHz (to numerical rounding).
2. Keep the lower S-state correction
For 2S → 8D, the upper D state has zero contact weight. The lower 2S state has weight 1/8, so the interval correction is −(A + B)/8. The contact contribution therefore remains present. Only an interval with equal contact weights cancels this profile.
3. Check whether the mechanism can be identified
Every interval depends on A and B through their sum. Moving 5 kHz from one coefficient to the other leaves all interval predictions unchanged. Even arbitrarily precise measurements of these same intervals could not separate A from B without additional physical information or observables with a different response pattern.
This is an illustration of a possible degeneracy with another contact contribution, such as a leading S-state finite-size term in an appropriate effective model. A real analysis must derive the coefficients, their dependence on mass and nuclear properties, their uncertainties, and the applicable perturbation operators. The equality of the two profiles here is deliberately stipulated to make the identifiability issue visible.
4. Specify the next theoretical calculation
A physical DTT correction needs a defined operator or state-energy map, a fixed normalization, and a coefficient derived from the source construction. A holonomy or structural quantity cannot be converted to kilohertz by selecting an arbitrary scale. Once that calculation is available, the same level-shift machinery can propagate it consistently across many intervals.
This tool contains synthetic coefficients and no measured residual dataset. Background and derivation requirements: differential comparisons, research status, and sources. The common formula is deliberately elementary so that every displayed value can be checked by hand.
A worked check you can reproduce
For a contact example, set δEnS = K/n³ and zero leading shifts for states with nonzero orbital angular momentum. Then the 1S → 2S correction is −7K/(8h), while 2S → nD retains −K/(8h). The upper D state does not remove the shift of the lower S state.
Try changing one shared coefficient and observe which combinations cancel. A transition cycle must close because its intermediate energy cancels algebraically. A dimensionless holonomy would require a separate physical energy map before it could enter this calculation in hertz.
Approximation and interpretation
All residuals here are synthetic. A physical DTT claim requires a source-derived operator, independently fixed coefficients, a conventional baseline, and uncertainty analysis.
Use the reset control to restore the reference example. Exported tables should retain their input values and units. The reproducible-calculations guide explains how to distinguish numerical evaluation, uncertainty, model limitations, and independent prediction.
Continue reading
Sources for this guide
Source manuscripts, version notes, and standard references are listed in the reference library.