Proof Sketches
Proof Sketches
This file should accumulate theorem sketches and impossibility arguments once the definitions stabilize.
Sketch 1: Local usefulness does not imply global consistency
Candidate statement:
Even if a latent factorization is perfectly predictive for each individual task, there may exist a family of equivalent factorizations that differ by task-dependent label permutations. In that case, task-wise identifiability can hold while globally consistent identifiability fails.
Proof idea:
- build two tasks with the same strategy set,
- define two latent-to-strategy maps that are each valid on their own task,
- then swap the latent labels on one task only,
- showing that prediction is unchanged but global semantics are not.
Sketch 2: Permutation-invariant recovery is weaker than semantic recovery
Candidate statement:
A permutation-invariant alignment score can certify that the model recovers the correct number of strategy groups, but it does not by itself prove that the same latent index means the same strategy across tasks.
Proof idea:
- a single best matching score ignores task-dependent relabelings,
- so high score can coexist with inconsistent latent semantics.
Sketch 3: Teacher forcing can limit what is identifiable
Candidate statement:
If only a short prefix is strategy-sensitive, then identifiability may only be possible for the prefix-level decision boundary and not for the full strategy semantics unless extra assumptions are added.
Proof idea:
- reduce the sequence to an informative prefix plus a deterministic suffix,
- show that the suffix contributes no identifying information,
- and argue that many latent assignments remain observationally equivalent.
Sketch 4: What extra assumptions might restore global identifiability?
Possible assumptions:
- strategy persistence across examples,
- task family structure with shared latent semantics,
- intervention-stability of forced
z, - or explicit supervision / anchor examples.
The next step is to identify which assumptions are strong enough to make the desired notion of identifiability mathematically meaningful but still realistic for the synthetic setting.
