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Curvature Domain Ownership

Status: Analysis-only. No renderer changes. All domain boundaries are inferred from transport-metric structure, not from scene labels.

This note synthesises the regime-clustering, anomaly-detection, and radial-structure analyses into a coherent account of how transport domains emerge from the wormhole observer ladder data and why domain boundaries — not sampling artifacts — are the primary explanation for visible banding.

Source: papers/paper_001_causal_observer_ladders/clustering_summary.md, papers/paper_001_causal_observer_ladders/analysis/bridge_anomaly_scoring.md, papers/paper_001_causal_observer_ladders/analysis/regime_clustering.md

The Domain Decomposition

Transport domains in xPRIMEray are not assigned — they emerge. Clustering of the five-feature stored-hit metric table (OPL mean, OPL max, portal-hit density, throat-event density, crossings per pixel, segments per crossing) produces a k = 3 partition that recovers the following structure with ARI = 0.5946, silhouette = 0.5547:

Domain Checkpoints Transport signature
Near-side mouth, mouth-to-throat, throat Dense interaction; smooth cost decrease; radial-dominant
Bridge post-throat backstep Sparse; maximum cost; minimum OPL; multi-metric outlier
Far-side post-throat exit-approach, exit-lookback Re-densification; tangential shift; maximum portal density

Both agglomerative and k-means clustering produce identical partitions at k = 3. At k = 2, 4, or 5, alignment with the manual regime labels degrades significantly (ARI drops to 0.14 or below). k = 3 is the natural dimensionality of the observer ladder's transport-phase space.

Why the Bridge Is the Anomaly, Not the Throat

A common assumption is that the wormhole throat is the primary discontinuity — the place where transport behaviour changes most sharply. The data contradict this.

The throat clusters with the near-side family, not with the bridge. It extends the near-side trend (increasing portal density, decreasing segments-per-crossing, decreasing OPL mean) rather than breaking it. The sharp discontinuity occurs at the bridge.

Feature Near-side trend at throat Bridge departure
portal-hit density 0.147 → 0.163 → 0.175 (increasing) 0.096 (collapse)
crossings/pixel 0.650 → 0.699 → 0.748 (increasing) 0.210 (collapse)
segments/crossing 153 → 140 → 128 (decreasing) 366 (spike)
OPL mean 9.96 → 9.73 → 9.51 (decreasing) 7.59 (minimum)

The bridge is not a slightly more extreme version of the throat. It is a structurally distinct transport regime defined by simultaneous collapse in density metrics and spike in per-crossing cost. This is the multi-metric character that makes it a genuine outlier.

Multi-Metric Anomaly Quantification

Three independent anomaly measures all rank post_throat_backstep_01 (the bridge) as the top anomaly:

Measure Bridge value Next ranked Method
Euclidean z-score 4.40 exit-lookback: 2.84 Standardised feature-space distance from centroid
Isolation forest 0.616 exit-lookback: 0.554 Expected path length for isolation
LOF 1.35 exit-lookback: 1.24 Local density ratio

The bridge anomaly score (overall: 2.195, bridge-signature: 1.953) is more than 3× larger than the next-highest bridge-signature checkpoint (mouth: 0.325). No other checkpoint is an outlier in all three measures simultaneously.

Spectral and Periodicity Evidence

FFT analysis of the six-checkpoint ladder sequences rules out an oscillatory interpretation:

  • OPL mean, throat-event density, and crossings-per-pixel: dominant frequency = 1/6 cycles per checkpoint (slowest possible mode — regime-scale drift, not oscillation).
  • Segments-per-crossing: dominant frequency = 1/3 cycles per checkpoint, with a large bridge residual that cannot be explained by any simple periodic model.

The bridge excursion in segments-per-crossing cannot be fit by a smooth oscillatory model. It is a singular, localised transport anomaly — a transition state — rather than one phase of a repeating pattern.

Radial Structure

Image-derived radial intensity profiles were reconstructed for each checkpoint. Key finding: a consistent global feature radius does not persist across the full ladder.

  • Near-side checkpoints remain close to the marked aperture radius.
  • The throat shows the strongest local slope magnitude at the apparent aperture radius (0.025), exceeding all other checkpoints. The throat is the sharpest horizon-like state in the ladder, not the bridge.
  • Post-throat checkpoints progressively shift away from a single-ring model. The bridge is where the visible radial structure becomes harder to represent with a single near-aperture feature radius — but this fragmentation is consistent with the broader morphological disorder of the bridge (190 connected components, 214 Hough lines).

Implication: Curvature Domain Boundaries as Band Cause

If banding were caused by insufficient sampling or numerical precision, the band locations would not correlate with transport-regime boundaries. But the evidence shows:

  • Phase coherence is lower inside visible bands than outside (gap 0.162 at mouth).
  • Neighbour-normal-delta discontinuities align with visible bands.
  • First-hit divergence originates before stored-hit refinement (ruling out post-integration artifact injection).
  • Band-boundary proximity and domain-boundary proximity are correlated.

This supports the hypothesis that visible bands mark locations where the transport field transitions between distinct solution families — i.e., domain-boundary transitions — rather than where sampling collapses due to renderer limitations.

Caution: This is a hypothesis grounded in correlations, not an experiment with a controlled intervention. A definitive test would require varying the domain-boundary location (e.g., by modifying the field profile) and checking whether band locations follow.

Connection to Render Guidance

The domain decomposition has immediate practical implications:

Domain Interpolation Tile recommendation Coherence expectation
Near-side Valid Adaptive square (direction-preserving) High (coherence gap > 0.16 at bands)
Bridge Not valid for world-space interpolation Multi-centre polar probes (no stable global attractor) Low (diffuse disorder, small gap)
Far-side Valid within domain Adaptive square, reoriented for tangential features To be measured

Cross-References