Localized Symbolic Activation in the Cathedral Window: The Paper 20 Valley as a Slow-Orbit Diversity Minimum

Paper 22 in the "Geometry of the Critical Line" programme.

This paper shows that the global symbolic α-invariance established in Paper 21 masks a strong localized transition. Restricting the analysis to the pullback of the Paper 20 Cathedral window reveals that both symbolic and metric complexity grow substantially with α.

Within the slow-convergence subset (top 10% of orbits by iteration count), reduced-word entropy attains a robust local minimum in the α ≈ 0.47–0.483 window, providing the first fiber-framework observable to recover the location of the Paper 20 entropy valley. At this parameter value, the hesitation orbits follow maximally concentrated relay-dominated symbolic itineraries. As α increases past 0.5, the slow orbits diversify into increasingly complex relay-bounce patterns.

The minimum survives four independent robustness tests: slow-orbit threshold variation (5%–30%), relay cutoff variation (R₀ = 1.0–2.0), alternative symbolic coding (corridor vs Voronoi), and iteration cap analysis. The observed minimum survives four independent robustness tests and is therefore best interpreted as a genuine localized dynamical signal rather than a simple artifact of thresholding, partition choice, or iteration truncation.

The global symbolic scaffold is thus approximately α-invariant as a bulk property, but locally activatable: the α-dependence lives not in the set of allowed symbolic paths but in the probability distribution over those paths, restricted to the signal-bearing region and concentrated in the hesitation orbits.

This is a working paper, part of the Geometry of the Critical Line programme. The correspondence between the slow-orbit word-entropy minimum and the Paper 20 basin-entropy valley is a location match, not a full metric identity; a precise alignment audit is noted as future work.