Phase 14 · loop residue, reversibility and hidden debt
If the Route Matters, Something Remembers the Route
AT-0111 → AT-0118. Path-dependent coupling is the smallest change that lets route choice matter at all. This lab asks the immediate follow-up question: what does the model have to remember for that to work, and is the memory a hidden debt?
The two-sided result
Path dependence resolves one blocker and creates another. If the outcome depends on the route, something in the model remembers the route, so the true state is larger than R alone. That extra state must be tracked explicitly or the result is hidden memory, not protection.
Model & audit limits
AT-0115 · same destination, different path
Toy model / not physical transportation
The requested toy vector field. Its loop residue density is constant and nonzero, so equal-endpoint routes accumulate different identity response and closed loops do not return to zero.
Interpolate both coordinates together. · endpoint ΔI 0.000 · path length 1.414
Move the first coordinate fully, then the second. · endpoint ΔI 1.000 · path length 2.000
Move the second coordinate fully, then the first. · endpoint ΔI -1.000 · path length 2.000
All three routes start at A and finish at B. Under a constant coupling their accumulated identity response would be identical. Here it is not, and the difference is the whole point.
If the rule depends only on where you start and finish, a smarter route cannot help. If the rule depends on the route, different paths can have different outcomes — but then we must account for what remembers the route.
AT-0112 · what a closed loop leaves behind
Toy model / not physical transportation
Current H = 2.0000 · residue after one full circuit L = 2.0000. The relational coordinates come back exactly. The accumulator does not, so something in the model remembers the route — and that memory has to be tracked as part of the state.
Running the loop-residue audit…