The Full Sample of 117 Galaxies — Blind Application

1. The result first

3. Graph 1 — Error distribution histogram

The distribution of signed prediction errors across the 117 galaxies, stacked by calibration group:

4. Graph 2 — Cumulative accuracy curve

The fraction of galaxies within a given absolute error threshold:

5. Graph 3 — Error vs disk scale

The error for each of the 117 galaxies, plotted against its disk scale $R_d$, coloured by Hubble type, and shaped by calibration group (circles for CALIB and MW, squares for BLIND):

7. What this means

7.1 The model captures real signal

The blind sample reaches $20.6\%$ median accuracy with parameters frozen from a $23$-galaxy calibration. A theory that was simply overfitting the training set would degrade by a factor of two or more on a $94$-galaxy blind set. Here, the degradation is from $18\%$ (CALIB) to $21\%$ (BLIND) — three percentage points. This is the expected behaviour of a model that captures genuine physics.

7.2 The remaining error structure is identifiable

The $+18\%$ positive bias and the correlation with $R_d$ are not random; they reflect the assumption of universal $(\ell_0, \lambda)$. The pattern visible in Graph 3 — large $R_d$ galaxies over-predicted, small $R_d$ galaxies under-predicted — directly indicates the form of the next refinement: the coherence length must depend on local baryonic density. This was already the recommendation of Notes XI and XXV; the $117$-galaxy sample confirms it on a much larger statistical base.

7.3 The MW is an anomaly that points the same direction

The Milky Way at $+78\%$ is the most over-predicted single galaxy. Its $\Sigma_d \sim 600\,M_\odot/\text{pc}^2$ (with $\Upsilon_\star = 0.5$, the equivalent for the SPARC scale) is in the highest decile of the sample. A density-dependent $\ell_0$ would naturally suppress the wave field in such a high-density disk, bringing the MW error toward zero. The fact that the MW alone (Note XXIV) fitted with $\ell_0 = 0.51$ kpc, $\lambda = 1.02$ — a $40\%$ longer coherence length and $50\%$ smaller coupling than the global fit — is consistent with this interpretation.

8. Summary

1. The BeeTheory framework with the corrected kernel and parameters $\ell_0 = 0.31$ kpc, $\lambda = 1.95$ (frozen from Note XXV) is applied without any further fitting to 117 galaxies.

2. Of these, 94 are blind: they were never used in any calibration step.

3. Global performance: median $|\text{err}| = 20.4\%$, $50\%$ within $20\%$, $68\%$ within $30\%$, $85\%$ within $50\%$.

4. Blind sample (94 galaxies): median $|\text{err}| = 20.6\%$, mean signed $+12\%$ — essentially the same accuracy as the calibration set ($18.1\%$ median). The model generalises.

5. The Milky Way is the most over-predicted single galaxy ($+78\%$), consistent with its anomalously high surface density.

6. The residual error structure correlates with $R_d$ and indirectly with $\Sigma_d$, confirming on a $117$-galaxy statistical base what Note XI identified on the smaller CALIB sample.

7. The clear next step is to introduce a density-dependent coherence length $\ell_0(\Sigma_d)$ — the simplest physical modification capable of removing the residual structure visible in Graph 3.