Cross-Input Semantic Consistency

This note records terminology and reference language for describing when the learned latent variable has a stable strategy meaning across different inputs. The central property is cross-input semantic consistency: a latent value should represent the same kind of strategy across different problems, not merely separate outputs within one input-specific response set. When the paper needs alignment-forward language, cross-input latent strategy alignment is a strong name for the property being measured, though it is likely too long for compact figure legends.

Reference phrasing:

Strategy separation within a single input is not enough: the latent could simply learn an input-specific partition of possible outputs. We therefore evaluate cross-input semantic consistency, the extent to which a latent value carries the same strategy meaning across different inputs. Operationally, we test this through analogical transfer: we infer or sample a latent on one problem, reuse it on another problem, and measure whether the generated solution follows the corresponding strategy. The resulting Analogical Consistency score measures whether sharing $z$ induces matching strategy labels across inputs.

Terminology Hierarchy

Use cross-input semantic consistency for the property.

Use cross-input latent strategy alignment when connecting the property directly to the paper's broader goal of strategy alignment. This phrase is precise but verbose.

Use analogical transfer for the operational test or application of the property: reuse a latent value inferred or sampled for one input on another input, then evaluate whether the corresponding strategy transfers.

Use Analogical Consistency for the metric: an empirical score measuring whether generated outputs produced with the same latent value have matching strategy labels across different inputs.

Use Strategy Alignment as a compact figure-facing label when the figure reports the paper-level property and the caption defines it as cross-input latent strategy alignment measured by Analogical Consistency. Use Analogical Consistency when the figure is specifically naming the metric.

Use within-input strategy separation for the weaker local property: different latent values produce different strategies for the same input. This property is necessary but not sufficient for a reusable strategy variable.

Use strong recovery when a compact theory-facing phrase is needed for the combination of within-input strategy separation and cross-input semantic consistency. Avoid making strong recovery the main empirical phrase unless it has been formally defined in the surrounding text.

Property

Cross-input semantic consistency asks whether the meaning of $z$ is stable across inputs.

The failure mode is an input-local partition: for each input $x$, different latent values may separate different possible outputs, but the same latent value may correspond to unrelated strategies for different inputs. Such a factorization can look locally disentangled while failing to expose a reusable strategy variable.

Reference phrasing:

A factorization can separate strategies within each input while still failing to assign a stable meaning to $z$ across inputs. Cross-input semantic consistency rules out this input-local partitioning failure: the same latent value should induce the same strategy type across different problems.

Operational Test

Analogical transfer tests cross-input semantic consistency by transferring a latent value across problems.

One operational pattern is:

1. Sample a latent value $z$ from the router $r_\phi(\cdot|x)$ on a source input $x$.
2. Generate a source response $y$ from $g_\phi(\cdot|x,z)$.
3. Reuse the same latent value $z$ on a different target input $x'$.
4. Generate a target response $y'$ from $g_\phi(\cdot|x',z)$.
5. Check whether $y'$ follows the same strategy as $y$.

Reference phrasing:

In an analogical transfer test, a latent value selected on one problem is reused on another problem. If $z$ has cross-input semantic consistency, the target response should follow the strategy associated with the source response.

Another compact reference sentence:

Strategy Alignment is measured as cross-input latent strategy alignment, using the Analogical Consistency metric under an analogical transfer intervention.

Metric

Use Analogical Consistency for the empirical metric associated with analogical transfer.

A compact form is:

$$\operatorname{AnalogicalConsistency} = \Pr[ \operatorname{strat}(Y)=\operatorname{strat}(Y') ],$$

where $Z$ is sampled from the router on a source input and then shared across two inputs from the same task family:

$$Z\sim r_\phi(\cdot|X), \qquad Y\sim g_\phi(\cdot|X,Z), \qquad X'\sim \mathcal D_X(\cdot|\operatorname{task}(X)), \qquad Y'\sim g_\phi(\cdot|X',Z).$$

The agreement target is the strategy label of the generated source response, $\operatorname{strat}(Y)$. Equivalently, the metric can be written as:

$$\Pr[ \operatorname{strat}(Y')=\operatorname{strat}(Y) ].$$

The exact estimator can depend on the experiment design. The important naming convention is that Analogical Consistency names the score, while cross-input semantic consistency or cross-input latent strategy alignment names the property being measured.

Reader-facing metric question:

This metric answers: does the same latent $Z$ induce the same strategy when transferred from $X$ to a related $X'$?

Relationship To Other Recovery Terms

Within-input strategy separation means that, for a fixed input $x$, different latent values induce different strategy-conditioned outputs. This is a local property.

Cross-input semantic consistency means that the latent value has a stable strategy meaning across different inputs. This is a reuse property.

Strong recovery can be used when both properties are present: the factorization separates strategies within an input and aligns the meaning of latent values across inputs.

Reference phrasing:

Within-input strategy separation shows that the latent variable affects generation for a fixed problem. Cross-input semantic consistency shows that the same latent value has reusable strategy meaning across problems. Strong recovery requires both.

Usage Across The Paper

In methodology or evaluation descriptions, introduce cross-input semantic consistency or cross-input latent strategy alignment as the property and analogical transfer as the operational test.

In metric tables and captions, use Analogical Consistency for the scalar score when technical precision matters. In compact main-figure labels, Strategy Alignment is acceptable when the caption defines it as cross-input latent strategy alignment measured by Analogical Consistency.

In theory-facing discussion, use strong recovery only when the intended condition combines local separation with cross-input semantic consistency.

Avoid using global semantic consistency as the main term. It is intuitive, but cross-input semantic consistency is more concrete and makes the evaluated axis explicit.

Terms That Need Care

Global semantic consistency is a useful synonym in prose, but can sound stronger than the evidence supports.

Analogical transfer should name the test or application, not the abstract property.

Analogical agreement is a descriptive phrase for the estimator's equality check, but Analogical Consistency is the preferred paper-facing metric name.

Weak recovery and strong recovery are useful in formal contexts, but they should be defined before use.