The exact RoPE line asked: what commutes with the dynamics? The observable quotient line asks a different question: what changes are invisible to the chosen measurement interface?
This is where the word “symmetry” becomes more delicate. It may now mean “behaviorally indistinguishable under the observables we decided to track,” not “exact algebraic commutant of the underlying evolution.”
Lean anchors.
IndistinguishableUnder,
BoundaryWritableRegion
Math statement.
In English. Two states are equivalent when every observable in the chosen family agrees on them. A state lies in the writable region when some nontrivial rewrite moves it without changing anything your chosen interface can see.
Physical intuition. The quotient is defined by what you measure. Change the observable family, and you may change which internal moves count as “free.”
The underlying system has not changed. Only the measuring interface has. But that alone changes which internal rewrites look invisible.
A property of the dynamics itself.
A property of dynamics plus the observables you chose to retain.
An equivalence that vanishes once the observables are refined.
This is the point where the paper needs conceptual care. The same word “symmetry” is being used for several different layers. The page sequence is designed to keep those layers separate.
This gives a principled way to move from exact algebra to empirical model analysis without pretending every invisible rewrite is “fundamental.”