What the experiment establishes
The positive contribution is a reproducible cross-operator comparison of finite structural signatures, with distinct results for recurrence, robustness and conditional discrimination. Primary transformed hydrogen and five image graphs give the strongest recurring shape. The embryo exception and moderate Kerr associations locate its heterogeneity. Kerr's U/P results exceed the specified uniform-group and coarse-bin/phase references. Shell-block stability demonstrates robustness to changes in weight placement that preserve detailed block multisets. The failed joint specificity criterion leaves those findings intact.
The signatures are functionals of spectral multisets. Isospectral operators necessarily give identical signatures. Fifteen windows cannot restore information already absent from the spectrum. Log autocorrelation also loses translation and phase information. A signature can register geometry carried through the spectrum without identifying a unique geometry or eigenbasis. The measured coefficients establish no common physical carrier, physical realization law, universal sign inversion between coherence and dissipation, or uniquely selected invariant. In particular, preserving a weight histogram need not destroy relevant physical structure: shell multisets retain detailed radial information.
The contribution can develop in three directions: cross-operator application, a compact density-sensitive descriptor alongside gap statistics, or reproducible residuals beyond a separately specified density-and-pair model. General classification and added value over density, quantile, moment or spacing summaries would require an explicit task and independent objects. A different observable by itself does not establish novelty. An ordinary density mechanism would not erase its possible diagnostic value. The two-paper comparison is not an exhaustive priority search.
A bounded next calculation would fit a fixed low-complexity density model using the saved hydrogen shell spectra and test it against the retained tables. Declare the endpoint split, propagate fit uncertainty, preserve the minimum anchor, and keep all fifteen slots and all three widths. The already seen endpoints make that an exploratory calibration. Separate adjacent-gap calculations could ask whether local correlations change while the all-pair profile survives; declare coordinate choice, sectors and degeneracy rules in advance. Neither extension was executed. For independent explanation, derive density from operator parameters or separate objects and validate on new objects. Broader universality and physical interpretation remain questions for those tests.
The result in full scope
Finite recurrence is strong in several specified constructions, heterogeneous, and generally shared by alternatives. Kerr distinguishes two nominated references in development and holdout, with weaker holdout separation from the overtone/Gram-preserving law. Hydrogen profile shape survives shell-block permutation, but aggregate abundance depends on the readout and the short fiber pilot is uninformative. The frozen joint library test fails at 0.2339.
The log-autocorrelation representation identifies precisely what is being measured and what an explanatory density model must predict. It does not turn the comparison into a common physical mechanism. Related random-matrix results do not already explain these operators quantitatively. Independent prediction, independent-object testing and utility comparisons are the direct next steps.
Data and code availability
The public verification supplement supplies retained inputs, library definitions, compact controls, result tables, executable verification and regeneration code, plot scripts, dependency pins and checksums. Its README separates quick checks on stored results from full numerical regeneration and reports actual executions. Billions of repeated derived matrix cells are reconstructed or recomputed in batches rather than stored. The original publication preparation did not run a clean full numerical reproduction or start a new scientific campaign.
Spatial numerical replay begins with the retained centroids. The supplement excludes photographs, thumbnails and overlays. It retains exact hashes, known acquisition identifiers and extraction rules. Six exact mappings and all seven photographic permission records are unresolved. Fully independent acquisition and pixel replay are therefore not represented as public or complete.
The supplied manuscript labels 10.5281/zenodo.23888292 a reserved publication identifier and does not assert public upload at the time of preparation. The author's research platform is everythingequation.com. The new author-created text, figures and tables are licensed CC BY 4.0; the adaptation retains the provenance and limits of the source.
Author statement
Jeremy Rodgers developed the research programme and core manuscript. AI tools supported additional numerical work, cross-checking, literature reconciliation and editorial revision. The author retains responsibility for the interpretation and final manuscript.
Published source: Cross-Domain Spectral Ratio Structure: Refined Recurrence, Robustness and Diagnostic Limits. Read the original PDF alongside the web treatment.