The claim
Let A, B act on a space V and A′, B′ on a space V′, all linear, and let Φ: V → V′ be linear. Define the intertwining defects E_A = ΦA − A′Φ and E_B = ΦB − B′Φ, and write [X, Y] = XY − YX. Then
Φ[A,B] − [A′,B′]Φ = A′E_B + E_A B − B′E_A − E_B A.
If the spaces are normed and the operators bounded, then
‖Φ[A,B] − [A′,B′]Φ‖ ≤ (‖A‖ + ‖A′‖)‖E_B‖ + (‖B‖ + ‖B′‖)‖E_A‖.
So a transport that nearly intertwines each framing operation carries their commutator, the measure of how much the two interfere, to the commutator of the images, within an error linear in the defects. With E_A = E_B = 0 it is exact: Φ[A,B] = [A′,B′]Φ, which is ARG-006’s case.
Proof
Substitute ΦA = A′Φ + E_A and ΦB = B′Φ + E_B twice:
ΦAB = (A′Φ + E_A)B = A′ΦB + E_A B = A′B′Φ + A′E_B + E_A B,
ΦBA = (B′Φ + E_B)A = B′ΦA + E_B A = B′A′Φ + B′E_A + E_B A.
Subtracting gives the identity. The norm bound follows from the triangle inequality and ‖XY‖ ≤ ‖X‖‖Y‖, grouping the E_B terms (A′E_B and −E_B A) and the E_A terms (E_A B and −B′E_A).
What it does not say
- That the defects can be read off data. Two models with the same observations can carry defects 0 and 1 (CLAIM-253). The bound is in terms of quantities a state model posits.
- What bounds an observed order effect. That is CLAIM-231’s observational 2ε bound, which needs no state model.
- That the commutator structure of the source is preserved. Only its image under Φ is (CLAIM-011).
- That interference between perspectives is part of what a translation preserves. That is CLAIM-063’s thesis.