Depth gauge is the local cross-layer analogue of a symmetry-compatible rewrite: absorb part of one layer into its neighbor, compensate there, and keep the chosen boundary observables unchanged.
This is useful because many mechanistic stories are too literal about which exact layer “contains” a computation. The theorem family says some of that localization is gauge.
Lean anchors.
DepthGaugeOn.transport_equivalent_at,
neighbor_layer_absorption_preserves_observable_at,
ApproximatelyDepthGaugeOn
Math statement.
In English. If your chosen adjacent-layer observable family cannot see a push/pull rewrite across a layer boundary, then the rewritten stack is observationally equivalent to the original one at that window.
Physical intuition. Some “where in depth is the computation?” questions are partly gauge questions. You can slide representation mass across nearby layers while preserving the boundary-facing story.
The point is not that every cross-layer rewrite is harmless. The point is that some are harmless relative to a chosen observable window, and those should be treated as gauge-like.
If two neighboring layer descriptions differ only by a depth gauge, they may be the same computation in different layer charts.
The approximate theorem turns this into a controlled error statement instead of an all-or-nothing idealization.
This is the cleanest non-RoPE example of exact/approximate observable-relative symmetry inside the current queue.
Feynman reading: if the detector only sees the combined local pattern, then shifting some amplitude from one layer to its neighbor may be a coordinate change, not a new mechanism.