lin-0118
9.1 Quotient before measuring
A learning system is usually presented with redundancy. Neural parameters have permutations and rescalings; low-rank adapters have factorization gauge; reward models have additive baselines; local trivializations have gauge transformations; causal fields depend on coordinates even when their distribution does not.
Let \(P\) be a presentation space and let \(\pi :P\to M=P/{\sim }\) identify representatives with the same declared model behavior. A diagnostic \(e:P\to Z\) descends to the quotient when there is \(\bar e:M\to Z\) such that
If this condition fails, the diagnostic can change while the declared model does not. It is then a presentation artifact, not an obstruction on the model space.
A field \(V_P:P\to TP\) is projectable through \(\pi \) when there is a field \(V_M:M\to TM\) satisfying
Alternatively, the realization may declare a gauge fixing or an equivariant horizontal lift.
Measure and repair on the semantic quotient whenever possible. If computation must use representatives, declare the gauge and certify that the resulting signal descends.