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.
An atomic test is overconstrained when the model makes more independent predictions than it has freely fitted quantities. The word “independent” is essential: duplicated lines, exact transition-cycle identities and nearly proportional sensitivities do not create new information. Paper 14 proposes a design matrix for cross-state residuals. This provides a practical way to test SMM/DTT’s central aspiration that many observed quantities arise from one source law. The test should have enough freedom to represent its declared mechanism and little enough freedom that the pattern can genuinely fail.
Choose the effective operator family and compute its level matrix elements. Form a row for each reserved transition and a column for each unknown coefficient. Add nuisance columns for independently constrained conventional quantities such as nuclear parameters or calibration offsets. Label which coefficients are derived, externally measured or fitted within the test. The resulting matrix exposes accidental degeneracies: two physical explanations can be indistinguishable if they produce the same columns over the chosen channels. Improving precision alone cannot separate exactly identical directions.
Imagine four independent residual channels and one proposed source amplitude. If the baseline is fully fixed, the vector r must lie near a one-dimensional line in four-dimensional data space. Three orthogonal combinations test the model without changing its amplitude. If two additional unconstrained nuisance directions are admitted, only one independent orthogonal combination remains. If the source column lies entirely in the nuisance span, the data cannot identify the source amplitude at all. This elementary geometry shows why a count of four lines and one named DTT parameter may overstate the actual test power.
With Gaussian errors and a justified covariance matrix, weighted least squares compares the residual vector to the model subspace using (r−Sc−Nη)ᵀΣ⁻¹(r−Sc−Nη), with suitable treatment of nuisance constraints. Exact singular directions should be removed or handled explicitly rather than inverted numerically without diagnosis. The nominal number of residual degrees of freedom is the data rank minus the rank of the fitted design, not necessarily the raw line count minus the number of parameter names. Strong nonlinearities or uncertain theory-error distributions may require a more careful inference procedure.
State the reserved channels, coefficient bounds, covariance treatment and test statistic before inspecting the final comparison. A failure can mean the operator family is wrong, the upstream normalization is wrong or the baseline is incomplete; the analysis should investigate those possibilities without silently changing the original prediction. If a revised model is introduced, preserve the earlier result and identify new tests. This practice is particularly important for structural-number programs because an apparently successful coefficient can conceal a choice among many grammars, branch conventions or normalization rules made after seeing data.
After weighting by the inverse square root of the covariance, inspect the singular values of the fitted sensitivity matrix. A zero singular value means an exact combination of coefficients is unidentifiable; a very small one means noise strongly amplifies its uncertainty. For example, two shape columns proportional across every chosen transition cannot be separated regardless of how many times those transitions are measured. Adding a channel where the columns have a different ratio can restore rank. This gives a practical way to select complementary measurements before data collection. It also tests whether a proposed source relation genuinely reduces ambiguity or merely renames two existing correlated parameters.
An overconstrained test is most valuable when the source architecture predicts relations among operator coefficients. For example, a single completion law might fix the ratio of a contact-like term and a spin-sensitive term. That ratio should enter the matrix as a derived restriction, leaving fewer free directions. The six nested stages and outer completion must then be explicit enough that another reader can reproduce the restriction. The experimental design becomes an audit of the claimed common origin: do several physically different atomic channels obey the same source-imposed relation? The skeleton outlines this goal; the operator derivation and data analysis remain to be completed.
KEEP THE CONNECTION
Scope of the argument. A concrete analysis design for future work, with no fitted coefficients, significance level or successful result invented.
Draft file: paper14_hydrogen_differential_holonomy_tests.pdf
Working manuscript. Article-level references identify the consulted sections and the scope of the interpretation. No public URL was supplied for this exact manuscript version.
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