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Localized Defect Glide Method

Method — instantiates Mobile-Defect Reconfiguration

Advances a single bounded defect through a coupled structure one legal local step at a time, so a large rearrangement happens as a chain of small moves instead of one global cutover.

This is the core motion primitive of the archetype: take one tightly bounded unit of mismatch — the mobile defect — and walk it forward one permitted local step at a time, so the whole structure reconfigures through the passage of the defect rather than through change everywhere at once. Its defining discipline is a single-locus invariant: at any instant only one local relation is out of place, and the defect carries that lone mismatch onward, leaving satisfied relations behind it and ahead of it. The name borrows from dislocation glide in a crystal, where a metal deforms not by shearing a whole plane of bonds simultaneously but by letting a line defect slip through one row of bonds at a time — cheap because only the defect's immediate neighbourhood is ever disturbed.

Example

The plainest first encounter is a sliding-tile puzzle. Fourteen numbered tiles and one empty square sit in a 4×4 frame, and the empty square is the mobile defect. You never lift a tile; you slide a neighbour into the hole, which moves the hole one cell — one legal local step. To carry a tile from a corner to the centre, you walk the empty square around it, shuffling it into place slide by slide. Everything on the board changes only when the hole passes through it; the single travelling gap is the only thing that ever "moves freely." Run enough legal glides and any reachable arrangement can be assembled — a full global rearrangement achieved without ever having more than one cell out of place at a time.

How it works

  • One active defect. Hold the structure to exactly one travelling mismatch; reasoning stays local because nothing else is mid-change.
  • Per-step legality. Every step must leave all other local relations satisfied — the move is admissible only if it disturbs nothing but the defect's own neighbourhood.
  • Corridor-bound travel. The defect may only advance where the structure permits it (its glide path); the route is chosen, not free.
  • Path matters. Because each move resets the context for the next, the sequence of steps — not just the endpoint — determines cost and what gets disturbed en route.

It is emphatically not pairwise change scattered across the structure; it is one moving locus of mismatch chained into a large effect.

Tuning parameters

  • Step granularity — how much the defect advances per move (one bond vs. a small cluster). Bigger steps finish faster but widen the momentary mismatch.
  • Defect bounding — how sharply the mismatch is confined. A tighter defect disturbs less per step but usually needs more steps.
  • Route choice — shortest path vs. least-disruptive path through the corridor; the two rarely coincide.
  • Reversibility per step — whether each individual glide is independently undoable, which trades speed for the option to back up one move.

When it helps, and when it misleads

Its strength is turning an impossible simultaneous cutover into a tractable series of small, individually checkable moves — only ever one thing half-changed, so any single step can be validated or reversed. Its central limitation is path-dependence: a locally legal sequence can wander into a dead end, and some target configurations are simply unreachable by legal local moves at all. The sliding puzzle makes this concrete — exactly half of all tile arrangements can never be reached from a given start, because a parity invariant is conserved by every legal slide.[1] The classic misuse is forcing an illegal move to "make progress" — shoving a tile through the frame — which corrupts the structure the method was meant to preserve. The discipline that guards against both is to check the target's reachability against the conserved invariants before starting, and to verify each step's legality rather than trusting the chain.

How it implements the components

  • mobile_defect_definition — specifies the bounded unit of mismatch that travels: its extent, and what state it carries as it moves.
  • local_move_legality_rule — defines which single steps are admissible, i.e. leave every other local relation satisfied.
  • propagation_corridor_map — the permitted glide path(s) along which the defect may advance.

It does not nucleate the defect in the first place (defect_introduction_gate — Canary Handoff Sequence), clear the obstacles that pin it (pinning_and_barrier_map, mobility_enabler — Pinning-Site Removal Procedure), or absorb it at journey's end (terminal_sink_or_absorber — Terminal-Sink Handoff).

Notes

The nearest sibling, Adjacent-Swap Sequence, is a discrete cousin: it exchanges pairs of neighbours, whereas this method moves a single mobile locus (a gap or dislocation) whose motion reconfigures everything it passes. Choosing between them is choosing whether your "defect" is a swap of two elements or one travelling vacancy — a distinction that changes which invariants are conserved and therefore which targets are reachable.

References

[1] A sliding-tile puzzle conserves the parity of the tile permutation combined with the blank's position under every legal slide, so only half of all arrangements are reachable from any given board — the classic solvability result for the 15-puzzle. It is the standard illustration that legal local moves preserve an invariant, and that any target violating it cannot be reached however cleverly the defect is routed. ↩