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Map Fold and Collision Scan

Structural scan — instantiates Neighborhood-Preserving Substrate Mapping

Sweeps a finished map for the two categorical ways locality breaks — folds where the ordering reverses and collisions where distinct source items share one cell — and logs each to a register.

A neighborhood-preserving map can fail locality in two structurally distinct, categorical ways, and this scan hunts for exactly those two. A fold is a crease: the map doubles back so that source order reverses across a line — regions far apart in the source become substrate-adjacent, or the sheet overlaps itself. A collision is a coincidence: two genuinely distinct source items are assigned the same substrate cell because capacity ran out there. Map Fold and Collision Scan sweeps the built map for both and writes every occurrence — location, type, severity, the source items involved — into the distortion register. Its defining idea is that it targets discrete topology defects, not the smooth, everywhere-present distortion that graded metrics average: a fold is a break, not a blur, and the scan's job is to find and locate the breaks.

Example

A cartography team flattens a global ocean-temperature dataset onto a 2D web-map tile grid. Most of the sheet is faithful, but near the poles the projection folds — longitudes are squeezed so hard that a tile's "east" and "west" neighbours reverse — and at the deepest zoom several distinct 1-km source cells are crushed onto one tile. The scan walks the tiling. It flags the polar crease as a fold (a run of tiles whose source-adjacency order is inverted) and the over-squeezed tiles as collisions (many source cells → one substrate cell), each written to the register with coordinates and a severity. The deliverable is not "the map has ≈4% distortion" but "these twelve tiles are folded and these thirty collide" — the precise places where local reasoning on the map is unsafe.

How it works

The scan checks two specific predicates and does nothing else:

  • Fold test — walk source-adjacent pairs and flag any whose substrate images reverse local orientation or cross an already-placed neighbour (a sign flip in the local mapping). Runs of such flips are grouped into one fold.
  • Collision test — bucket source items by substrate cell; any cell holding more than one distinct item — beyond the designed redundancy band — is a collision.

Every hit becomes a register entry tagged with type, location, severity, and the source items implicated. The scan locates and classifies; it neither scores smooth distortion nor decides the repair.

Tuning parameters

  • Fold sensitivity — the orientation-reversal threshold. Lower catches subtle creases but raises false alarms in noisy regions.
  • Collision granularity — the cell size at which two items "share" a location. Coarser cells hide collisions; finer cells surface more.
  • Redundancy tolerance — how much intended overlap to exclude before a coincidence counts as a collision. Set it too tight and you flag designed redundancy.
  • Severity weighting — rank folds by extent, or collisions by how far apart the collided source items truly are (a collision of already-near items is milder).
  • Scan scope — a full sweep versus a targeted re-scan of only the regions changed since the last map update.

When it helps, and when it misleads

Its strength is turning a vague "the map looks wrong near the edges" into an enumerable, located defect list that repair can act on cell by cell — and catching the categorical breaks that averaged distortion hides, since a map can post a low mean distortion and still carry one catastrophic fold.

Its failure mode is being over-read as a certificate: a clean scan only rules out the two defects it tests, so a map with zero folds and zero collisions can still be smoothly, globally misleading. The classic misuse is coarsening the collision cell size until nothing registers, then declaring the map clean. The discipline is to pair the scan with a graded fidelity metric and to report the scan's resolution next to its result — an all-clear at 10-km cells is not an all-clear. It answers a different question than a continuous distortion portrait such as a Tissot indicatrix does.[1]

How it implements the components

  • distortion_fold_and_collision_register — its primary output: the populated log of every fold and collision, each typed and located.
  • overlap_and_redundancy_band — it operates against the designed redundancy band to tell intended overlap from a true collision.

It does not score continuous neighbourhood fidelity — that belongs to Topographic Error Measure and Neighborhood Trustworthiness and Continuity Metric — and it does not repair or reallocate a folded region (Adaptive Remeshing and Reallocation).

Notes

A fold and a collision can look alike in a heatmap but need opposite fixes: a fold is a topology fault (the sheet is oriented wrong — reorganize, or cut a seam), while a collision is a capacity fault (too little substrate there — magnify, or add cells). Keeping the register's type field honest is what lets downstream repair pick the right tool instead of smoothing a crease that needed cutting.

References

[1] Tissot's indicatrix visualizes how a map projection distorts area and angle continuously across the sheet. It complements this scan rather than duplicating it: the indicatrix portrays smooth, everywhere-present distortion, whereas the scan isolates the discrete folds and collisions a continuous portrait can wash out. ↩