Theory Outline

This note briefly ingests the current draft in path:paper/conference-paper/sections/05-theory.tex. The theory section already has a working draft, so this note is intentionally compact: it records the section role, main formal objects, result message, and source-level issues to remember during later revision.

Related notes:

• Strategy recovery theory language
• Strategy-structure recovery problem
• Cross-input semantic consistency
• Strategy alignment and recovery

Current Section Role

The section is titled Toward a Theory of Strategy-Structure Recovery. Its role is to connect the empirical objective-selection story to a formal question: when can a variational approach identify a latent variable $Z$ aligned with a reference strategy structure $S^\star$ implicit in a fixed conditional distribution $p(y|x)$?

The section should be read as a high-level theory preview for the main paper, with precise statements deferred to supplementary material. It does not need to fully rederive the methodology objective. It should clarify that fitting $p(y|x)$ is not enough; recovery depends on objective selection among possible latent-variable factorizations.

Current Formal Setup

The draft considers finite sets $\mathcal X$ and $\mathcal Y$, an input law $\mu$, and a fixed conditional distribution $p(y|x)$. It introduces:

• encoder $q(z|x,y)$;
• router $r(z|x)$;
• decoder/generator $g(y|x,z)$;
• joint law $Q$ induced by $X\sim\mu$, $Y|X\sim p(\cdot|X)$, and $Z|X,Y\sim q(\cdot|X,Y)$;
• population variational loss $\mathcal L_\beta(q,r,g)$;
• profiled cost $\mathcal A_{\beta,\lambda}(q)$ after optimizing over $r,g$ and adding a theoretical complexity penalty $C(r,g)$;
• surrogate/profied objective $\mathcal J_{\beta,\lambda}(q)$;
• target strategy $S^\star=f(X,Y)$ and ideal strategy encoder $q_{S^\star}(z|x,y)=\mathbf 1\{z=S^\star\}$;
• discrepancy $d(q,q_{S^\star})$ measuring alignment with $S^\star$;
• recovery gap $\Delta(\delta)$ separating the target strategy encoder from encoders outside a $\delta$-neighborhood.

Main Result Message

The high-level result says that a positive recovery gap is sufficient for strategy-structure recovery. If $\Delta(\delta)>0$, then every population maximizer of the profiled objective lies within $\delta$ of the reference strategy encoder. A uniform-convergence condition at the scale of $(1-\beta)\Delta(\delta)$ transfers the population selection result to empirical minimizers, with a finite-class sample complexity of order

$$n=\mathcal O\left(\frac{\log|\mathcal Q|+\log(1/\eta)}{(1-\beta)^2\Delta(\delta)^2}\right).$$

Paper-facing interpretation: the theory formalizes objective selection. Strategy recovery is possible when the intended strategy encoder has a positive profiled margin over competing encoders after accounting for latent information, router-generator fit, and complexity/realizability bias.

Connection To Experiments

The experiments show that standard variational objectives can fit observable behavior without selecting a strategy-aligned latent, while model-directed reconstruction pressure empirically improves selection. The theory section should connect to this by saying that latent-variable expressivity is not enough: the objective and inductive biases must prefer the strategy-compatible factorization over other marginally adequate factorizations.

Use empirical language such as within-input strategy separation, cross-input semantic consistency, strategy alignment, and Analogical Consistency outside the theory. Reserve weak/strong recovery terminology for formal discussion if the section defines the corresponding distances.

Source-Level Revision Notes

The current source contains inline author notes that should be resolved later:

• decide whether to use decoder or generator in the theory section; main-paper terminology favors generator, with decoder only as a variational-model bridge;
• clarify the role of the complexity penalty $C(r,g)$ and why it is introduced;
• decide whether $\Delta(\delta)$ should be defined inside or before the result box;
• explain why positivity of the gap matters before stating the result;
• decide whether the feature box should be labeled as a theorem, proposition, or informal result.

The section is otherwise ready to leave aside while outlining the remaining paper sections.