Strategy Alignment And Recovery

This note describes the language for successful router-generator factorizations. It separates the model object, the desired property, and the empirical or theoretical claim.

Language Hierarchy

Use three layers consistently:

1. Object: the learned object is a latent-variable router-generator factorization.
2. Property: a useful factorization is strategy-aligned when its latent variable tracks strategy-level variation.
3. Outcome or claim: an experiment or theorem shows strategy-structure recovery when the learned latent recovers the strategy structure of the response distribution.

Reference phrasing:

We learn a latent-variable router-generator factorization. A successful factorization is strategy-aligned: its latent variable separates and controls strategy-level variation in responses. In the controlled benchmark, we evaluate strategy-structure recovery by comparing the learned latent to ground-truth strategy labels.

Strategy-Aligned

Use strategy-aligned as the default descriptor for a successful factorization. A strategy-aligned factorization is one whose latent variable corresponds to the intended strategy structure rather than arbitrary predictive information, surface form, or input-specific partitions.

Useful forms:

• strategy-aligned factorization
• strategy-aligned latent
• strategy alignment
• aligns with ground-truth strategy labels

Reference phrasing:

Not every factorization is useful: a pretrained model admits trivial factorizations in which $z$ is ignored. We seek strategy-aligned factorizations, where variation in $z$ corresponds to strategy-level variation in $y$.

Strategy-Structure Recovery

Use strategy-structure recovery for empirical or theoretical success claims. This term names the outcome: the learned latent recovers the strategy structure of the response distribution.

In controlled synthetic tasks, recovery is appropriate because the task has known strategy labels. In broader language-model motivation, avoid overclaiming recovery of a unique true latent strategy; use softer language such as surfacing or exposing latent strategy structure.

Reference phrasing:

We evaluate strategy-structure recovery by measuring whether router samples separate ground-truth strategy labels and whether the generator produces strategy-consistent responses when conditioned on $z$.

For paper-facing metric and frontier language around strategy alignment and strategy decodability, see Metrics and frontier language.

Strategy-Disentangled

Use strategy-disentangled as a reader-facing synonym for a strategy-aligned, strategy-separating factorization, after the intended meaning has been defined.

Use disentangled factorization or strategy-disentangled factorization when the paper needs a compact phrase for the qualitative goal. Avoid making disentanglement the standalone name of the problem, because it invites comparison to beta-VAE and FactorVAE-style unsupervised disentanglement.

Reference phrasing:

We use "strategy-disentangled" to mean that the learned latent separates strategy-level variation in responses, rather than arbitrary predictive or surface-level variation.

Local Properties

Within-input strategy separation means that, for a fixed input $x$, different latent values correspond to different strategies for solving that same input.

Strategy-separating is useful for latent-space plots and diagnostics, but it is not by itself the full success condition: a latent can separate arbitrary properties.

Reference phrasing:

Within-input strategy separation asks whether different latent values correspond to different strategies for the same input.

Generator Behavior

Strategy-consistent generation describes the behavior of $g(y|x,z)$: conditioning on a fixed latent value or region should produce responses with the corresponding strategy.

This phrase is useful for qualitative examples, controlled-latent generation, and captions where latent geometry alone is insufficient.

Reference phrasing:

The objective is designed to select factorizations in which $z$ carries information about $y$ beyond $x$ in a way that supports strategy-consistent generation.

Stronger Semantic Property

Cross-input semantic consistency is the stronger property that the same latent value or region tends to correspond to the same strategy across related inputs. See Cross-input semantic consistency.

This is stronger than within-input strategy separation. It supports claims about reusable strategy representations and analogy-based transfer.

Reference phrasing:

Beyond within-input strategy separation, we ask whether the latent has cross-input semantic consistency: whether the same latent region corresponds to the same strategy across related inputs.

Related Language

Decomposition is an intuitive prose synonym for factorization. It can be useful when explaining the idea informally, but it is less mathematically precise because it does not imply marginal preservation under the latent variable.

Reference phrasing:

Intuitively, the factorization decomposes the response distribution into strategy-conditioned components.

Unmixing is vivid but narrow. It can imply that the mixture components are already well-defined and simply need to be recovered. Use it sparingly, mainly for intuition in the introduction or discussion.

Separation is concrete and useful for diagnostics, but incomplete as a success criterion. A latent may separate arbitrary features without being strategy-aligned.

Surfacing and exposing are useful motivation terms when arguing that strategy information is already present in the base model.

Reference phrasing:

The method surfaces strategy structure already implicit in the base model by exposing it as a controllable latent variable.

Usage Across The Paper

In the problem statement, introduce strategy-aligned factorization as the target property after defining the exact latent-variable factorization.

In methodology, describe objective terms as selecting or encouraging factorizations in which $z$ supports strategy-consistent generation.

In experiments, use strategy-structure recovery, within-input strategy separation, strategy-consistent generation, and cross-input semantic consistency.

In theory, use strategy-structure recovery, weak recovery, strong recovery, and global label alignment when formal definitions require them.

In titles, abstracts, and figure captions, use strategy-aligned or strategy-disentangled depending on the intended level of rhetorical force.

Terms That Need Care

• disentanglement: useful but loaded; define the intended strategy-specific meaning.
• unmixing: vivid but can overstate identifiability.
• decomposition: intuitive but less formal than factorization.
• separation: useful local property, not the full success condition.
• recovery: appropriate in synthetic tasks and theory, but should be softened for open-ended natural-language settings.