The complete argument

Strong recurrence, limited library specificity

Five primary image graphs track the hydrogen profile, while the embryo and paired-library controls show exactly where that agreement stops.

Jeremy Rodgers · Independent Researcher · 2026-11-02
doi:10.5281/zenodo.23888292

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Native profile recurrence

The common shape and the embryo exception

The observed comparison contains 17 views: Kerr, two hydrogen readouts and 14 spatial variants. The eight-view primary descriptive core contains Kerr, hydrogen log1p and six dark families. All identities, fifteen slots and three tolerances remain in the comparison. Figure 1 and Appendix D show every core pair, including image–image comparisons. The public supplement includes all 17 views.

Table 2. Primary hydrogen log1p versus six dark image signatures. $F_{G,s}$ counts the fraction of all 10,000 paired-library coefficients at least the target, including target and ties.

Source	Spearman 3%	Spearman 1%	Spearman 0.5%	Pearson 3%	3% Spearman reference fraction
Embryo	0.23900	0.26856	0.04482	-0.25984	0.54060
Galaxy	0.92063	0.92328	0.96373	0.98928	0.97900
Pollen-like	0.88641	0.88641	0.88518	0.95464	0.86000
Leaf	0.99553	0.99463	0.99642	0.98673	0.16900
Shroud	0.94868	0.94492	0.94765	0.97981	0.71350
Snowflake	0.98743	0.98652	0.98921	0.99637	0.44180

At 3%, five hydrogen–image Spearman coefficients span 0.886 to 0.996. They remain between 0.885 and 0.996 at narrower widths. Embryo dark is the clear exception, at 0.239, 0.269 and 0.045, with negative Pearson coefficients at all widths. Its bright version differs strongly but remains a sensitivity view. Pairwise agreement among the five other primary images is also high: the 3% Spearman range is 0.851–0.988. The recurrence is broad and heterogeneous.

Kerr is more moderately related. Its 3% Spearman coefficients are 0.52143 with hydrogen, 0.66848 with embryo, 0.65708 with galaxy, 0.38277 with pollen-like microscopy, 0.52189 with leaf, 0.38399 with Shroud and 0.53335 with snowflake. The full comparisons retain each of these pairs and their narrower-width results.

Six heat maps compare all eight primary views at three tolerances. Five image families agree strongly with hydrogen; dark embryo is the exception and Kerr agreement is more moderate. — Figure 1. Complete primary-core profile comparisons at all three tolerances. Spearman is primary descriptive and Pearson secondary. The fifteen incidence slots retain ties and algebraic dependence. These are value-ratio profile correlations, not adjacent-gap statistics, null-excess correlations or repeats of earlier full-histogram correlations. The exact coefficients are in Table 6.

Similar profiles also occur under alternative libraries

For hydrogen versus the six dark families, the 36 coefficient/tolerance comparisons give conservative fractions at least the target from 0.0598 to 0.9790. Strong recurrence in native profiles does not establish that the target library has an exceptional correlation in this bank.

Kerr–embryo at 3% is an exception to that general pattern. Its Spearman coefficient is 0.66848 with reference fraction 0.0086; Pearson is 0.54929 with fraction 0.0190. The distinction weakens at narrower widths. This is descriptive among the many retained pairs, not a replacement primary test. Both conservative fractions and symmetric midpositions are supplied because exact ties can be substantial. No new selected significance threshold is imposed.

Color-coded matrix of the fractions of paired library correlations that equal or exceed the target for each core pair and window width. — Figure 2. Every primary-core target coefficient's position in the complete paired alternative-library population. Fractions include the target and ties. They describe retained-bank specificity, not physical discovery probabilities or independent image replications. All 28 pairs, three widths and both coefficients are shown; Table 6 gives the exact values.

Published source: Cross-Domain Spectral Ratio Structure: Refined Recurrence, Robustness and Diagnostic Limits. Read the original PDF alongside the web treatment.