Latent Strategy Variable

This note describes the concept of a latent strategy variable and gives reference language for writing about it in the paper. The key convention is that $z$ is always a latent variable in the model, while being strategy-aligned is a desired property of a learned factorization rather than something guaranteed by the notation.

Core Language

Use latent strategy variable at first technical introduction.

Use latent variable $z$ in equations and technical derivations once the intended role has been defined.

Use strategy variable when discussing the intended role of $z$ in prose.

Use latent strategy sparingly as a compact prose shorthand.

Avoid calling $z$ simply the strategy unless the context is clearly about a successful strategy-aligned factorization or a ground-truth benchmark label.

Reference phrasing:

We write $z \in \mathcal Z$ for a latent strategy variable: an unobserved variable intended to capture high-level strategy structure in the response $y$ beyond what is determined by the input $x$.

Strategy

In this paper, a strategy is a high-level, reusable attribute of a response trace that determines the broad procedure used to solve the input, while abstracting away from low-level surface details such as wording, formatting, or incidental token choices.

Reference phrasing:

A strategy is a high-level, reusable attribute of a response trace: it describes the broad procedure used to generate a solution, not the incidental surface form of that solution.

This language avoids overclaiming that a strategy is a literal cognitive plan, intention, or causal mental state inside the model. It names an operational structure in the distribution of response traces.

Latent Variable Versus Recovered Strategy

The notation $z$ should not by itself imply that the latent is semantically meaningful. A model can use $z$ to encode arbitrary predictive information, surface details, input-specific partitions, or noise. The paper's goal is to learn factorizations where $z$ becomes strategy-aligned.

Reference phrasing:

We seek factorizations in which $z$ is not merely an arbitrary latent index, but a latent strategy variable: conditioning on $z$ should select a high-level procedure for generating $y$ from $x$.

Failure-mode phrasing:

A latent can be informative without being strategy-aligned: it may improve reconstruction by encoding surface details, input-specific partitions, or arbitrary predictive features rather than reusable strategy structure.

Success phrasing:

The learned latent is strategy-aligned when samples from $r(z|x)$ separate ground-truth strategies and the generator $g(y|x,z)$ produces strategy-consistent responses.

Strategy Versus Surface Variation

A strategy should be coarser than ordinary sequence variation. It should describe the procedure or high-level structure of the response, not low-level differences inside the same procedure.

Good descriptors:

• high-level
• abstract
• procedural
• reusable
• strategy-level
• response-trace attribute
• shared across inputs
• coarser than surface variation
• not directly observed during training

Reference phrasing:

The latent should capture strategy-level variation rather than low-level variation within a strategy.

Different strategies may solve the same input correctly, while different surface realizations may instantiate the same strategy.

Local And Global Semantics

The paper should distinguish local separation from stronger cross-input semantics.

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

Strategy consistency means that conditioning on the same latent value or region repeatedly yields outputs with the same strategy.

Cross-input semantic stability means that the same latent value or region tends to correspond to the same strategy across related inputs.

Global strategy alignment means that the latent has a consistent strategy meaning across the task distribution.

Strong recovery is theory-facing language for recovery up to a single global permutation of strategy labels. Use it mainly when the paper is explicitly discussing theoretical recovery definitions.

Reference phrasing:

A useful latent should do more than separate outputs for a single input. Ideally, its meaning should be stable across inputs: the same latent region should correspond to the same high-level strategy on related problems.

Benchmark Operationalization

In the controlled synthetic tasks, strategies are operationalized by known labels attached to solution traces. These labels are used only for evaluation.

Reference phrasing:

In our controlled tasks, strategies are operationalized by ground-truth labels $\mathrm{strat}(y)$ attached to solution traces. These labels are used only for evaluation; training observes $(x,y)$ pairs but not strategy labels.

This phrasing makes clear that the benchmark supplies an observable proxy for strategy alignment, while the training procedure remains unsupervised with respect to strategy labels.

Relationship To Modes

Use mode mainly in the introduction and motivation, where distributional-mode language helps readers understand why multiple strategies can coexist under $p_\theta(y|x)$.

In technical sections, prefer strategy, latent strategy variable, and strategy structure. The term mode is overloaded: it can mean a distributional mode, a behavioral mode, an algorithmic strategy, or a model operating mode.

Reference phrasing:

Intuitively, the model's response distribution can place mass on several high-level modes of reasoning. We formalize the object of interest as a latent strategy variable in a router-generator factorization.

Terms That Need Care

• plan: can be too anthropomorphic and too deterministic.
• intent or intention: implies agency and should be avoided.
• thought: overclaims internal cognitive status.
• reasoning trace inside the model: risks confusing generated response traces with internal model states.
• cluster: useful for visualizations but too geometric and post-hoc as a main concept.
• strategy code: sounds representation-learning-centric rather than probabilistic.
• mode: useful in motivation but overloaded in technical prose.

Usage Across The Paper

In the abstract and introduction, use strategy and occasional mode language to motivate the problem.

In the problem statement, define $z$ as a latent strategy variable and immediately clarify that strategy alignment is a desideratum, not an assumption.

In methodology, mostly use latent variable $z$ and strategy-conditioned generator after the initial definition.

In experiments, use strategy label, strategy decodability, strategy-consistent generation, within-input strategy separation, and cross-input semantic stability.

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