Terminology Decision Log

This note tracks paper-terminology decisions for the strategy-factorization paper. It is a working decision log, not the final glossary. The compact final terminology should eventually live in Notation and terminology, while this note preserves alternatives, reasoning, rejected terms, and section-specific usage guidance.

Working Goal

Develop a stable language system for the paper: terminology, notation, claim phrasing, figure language, and short reusable prose patterns that make the methodology clear without overclaiming. The terminology should foreground that the paper studies how to factorize an already-capable conditional language model into a latent-variable router-generator model whose latent variable captures reusable strategy structure.

Source Context

Primary references for the current terminology pass:

• Notation and terminology
• Contributions
• Claims
• Problem framing
• Story boarding
• Operational definition of a strategy variable
• CVAE post-training methodology
• CVAE objective and loss terms
• Reconstruction weighting and normalization
• Formalizing strategy identification
• path:paper/storyboard
• path:paper/conference-paper

Current Research Message

The paper studies a conditional language model $p_\theta(y|x)$ that already defines a distribution over response sequences. For reasoning and algorithmic tasks, this distribution can mix several high-level strategies. The goal is to learn a latent-variable router-generator factorization

$$p_\theta(y|x) \approx \int r(z|x) g(y|x,z)\,d\mu(z)$$

where $r(z|x)$ chooses or represents a latent strategy and $g(y|x,z)$ generates responses conditioned on that strategy.

The central difficulty is that the base generator can already model $p_\theta(y|x)$ without using $z$. Standard variational training can therefore preserve the observable conditional distribution while leaving the latent variable inert. The methodology introduces objective terms and training choices that prefer factorizations where $z$ explains information about $y$ beyond what $x$ already predicts, while preserving distributional fidelity to the base model.

The empirical story should distinguish at least three outcomes:

• strategy information is present in the base model but not exposed as a generative variable;
• the proposed objective can recover a strategy-aligned router-generator factorization in controlled settings;
• stronger evidence asks whether the recovered latent has cross-input semantic stability rather than only input-local separation.

Decision Workflow

Work through terminology clusters in dependency order. For each cluster, record:

1. Concept being named.
2. Mathematical or empirical referent.
3. Candidate terms.
4. Advantages.
5. Risks and reader-confusion modes.
6. Recommended usage by section.
7. Final decision.
8. Follow-up propagation targets.

Do not make a term final until its neighboring terms are checked for consistency. For example, the objective name should be decided together with the names for the base model, z-free surprisal, reconstruction metric, and figure axes.

Terminology Clusters

1. Main Object Of Study

Concept: the object the paper is trying to learn from $p_\theta(y|x)$.

Current and candidate terms:

• latent-variable factorization
• router-generator factorization
• latent-variable router-generator factorization
• probabilistic decomposition
• latent-variable decomposition
• strategy decomposition
• disentangled factorization
• strategy-structure recovery
• strategy-aligned factorization

Open issue: decide whether the primary noun is factorization, with recovery/disentanglement used as properties and outcomes, or whether strategy recovery should be co-primary in the paper framing.

Decision: use latent-variable router-generator factorization as the formal term, with router-generator factorization as the short form. Use strategy-structure recovery for the success claim and strategy-aligned factorization or disentangled factorization for the desired property. See Router-generator factorization terminology.

2. Latent Variable And Strategy Semantics

Concept: what $z$ represents, and what kind of variation it should and should not encode.

Current and candidate terms:

• latent strategy variable
• strategy variable
• latent variable
• strategy
• reasoning strategy
• abstract strategy
• high-level strategy
• mode
• reasoning mode
• distributional mode
• strategy code
• latent embedding

Open issue: decide how much the paper should rely on the intuitive word "strategy" before giving the benchmark-specific operational definition.

Decision: use latent strategy variable at first technical introduction, latent variable $z$ in equations after definition, and strategy variable when discussing the intended role of $z$. Treat strategy alignment as a desired property of the learned factorization, not as guaranteed by the notation. Use mode mainly in intro/motivation. See Latent strategy variable.

3. Router, Generator, And Posterior Components

Concept: named model components in the factorization and training procedure.

Current and candidate terms:

• router
• strategy router
• generator
• strategy-conditioned generator
• decoder
• expert
• inference model
• posterior
• recognition model
• training-time posterior

Open issue: decide whether to ever use "decoder" or "expert" in the main paper, or reserve them only for related-work comparisons.

Decision: use router for $r(z|x)$ and generator for $g(y|x,z)$, with strategy-conditioned generator at first use. Use conditional prior and decoder only as bridges for variational-model readers. Avoid expert unless discussing related architectures. See Router-generator factorization terminology.

4. Base Model And Adaptation Setting

Concept: the pretrained or reference model being factorized, and the regime in which the factorization is trained.

Current and candidate terms:

• base model
• pretrained model
• reference model
• frozen reference model
• z-free model
• latent-free model
• entangled model
• source model
• adapted-from-base
• factorization of a pretrained model
• adaptation
• post-training
• distillation

Open issue: fix a convention for whether "base model" always means the frozen reference $p_\theta(y|x)$, or whether it can also refer to a z-free path through an adapted model.

Decision: use $p_\theta(y|x)$ as the primary notation for the base model, with $\theta$ denoting fixed base parameters. Use $\phi$ for the learned router-generator parameterization and $\xi$ for the training-time posterior. Use frozen reference copy of $p_\theta$ for nonmoving scoring and weighting quantities, and z-free base-model surprisal for $-\log p_\theta(y|x)$. Avoid $p_{\mathrm{base}}$ as the main notation. See Base model and reference notation.

5. Factorization Properties

Concept: properties of a good factorization.

Current and candidate terms:

• disentangled
• strategy-aligned
• separated
• un-mixed
• decomposed
• semantically meaningful
• controllable
• reusable
• distributionally faithful
• marginal-preserving

Open issue: decide how prominently to use "disentanglement" given its baggage from the beta-VAE and FactorVAE literature.

Decision: use strategy-aligned factorization as the default success descriptor, strategy-structure recovery as the empirical or theoretical claim term, and strategy-disentangled as a carefully defined reader-facing synonym. Keep disentanglement from becoming the standalone main object name. See Strategy alignment and recovery.

6. Objective Family

Concept: the methodology's reconstruction and selection pressure relative to the base model.

Current and candidate terms:

• base-model-relative reconstruction
• base-model-relative excess reconstruction
• excess reconstruction
• residual-surprisal reconstruction
• base-normalized reconstruction
• baseline-normalized reconstruction
• $p(y|x)$-normalized reconstruction
• latent-free surprise excess reconstruction
• z-free-relative reconstruction
• reconstruction pressure
• selection pressure

Open issue: decide whether the umbrella term should emphasize the base-model setting ("base-model-relative") or the information-theoretic quantity ("residual-surprisal").

Decision: use $p_\theta$-directed reconstruction objective as the formal term for the objective modification, with base-directed reconstruction as the short prose term and base-directed reconstruction pressure when emphasizing the optimization role. Write the main objective as $\mathcal J_\theta(x,y;\phi,\xi)=\mathcal R_\theta(x,y;\phi,\xi)+\beta\,\mathrm{KL}(q_\xi(\cdot|x,y)\Vert r_\phi(\cdot|x))$, where $\theta$ subscripts indicate fixed base-model reference quantities and semicolon arguments indicate optimized parameters. See Base-directed reconstruction objective.

7. Objective Mechanism And Interpretation

Concept: how the objective changes optimization and how to explain it.

Current and candidate terms:

• weighting
• normalization
• reweighting
• global rescaling
• example-level weighting
• token-level weighting
• surprisal-share weighting
• inverse-surprisal weighting
• absolute code-length reduction
• absolute information gain
• fractional information gain
• leftover fraction of baseline surprisal
• residual surprise explained by $z$
• equalization force
• heterogeneous beta-VAE
• example-dependent latent budget
• rate-distortion rescaling

Open issue: keep the optimization mechanism distinct from the information-theoretic gloss. The ratio changes gradients by reweighting; the code-length difference is the cleaner diagnostic interpretation.

8. ELBO Failure And Non-Identifiability

Concept: why standard variational training does not solve the problem.

Current and candidate terms:

• standard ELBO
• conditional ELBO
• naive ELBO
• ELBO baseline
• posterior collapse
• latent collapse
• trivial factorization
• z-ignoring factorization
• absorbing generator
• expressive-generator non-identifiability
• non-identifying objective
• selection problem

Open issue: distinguish objective non-identifiability from the optimization pathology caused by initialization at the base model.

Decision: use standard negative conditional ELBO or standard variational objective for the baseline. Diagnose two distinct failures: objective-level under-specification, where marginal fidelity to $p_\theta(y|x)$ does not identify a strategy-aligned factorization, and adapted-from-base optimization pathology, where the generator is initialized near an absorbing z-ignoring factorization. Use posterior collapse for the observed optimization outcome, but do not let it replace the formal non-identifiability argument. See ELBO under-specification and the z-ignoring solution.

9. Recovery And Semantic Stability

Concept: what kind of success is shown by the latent variable.

Current and candidate terms:

• strategy recovery
• strategy-structure recovery
• local recovery
• within-input strategy separation
• input-local partition
• weak recovery
• strong recovery
• global semantic consistency
• cross-input semantic stability
• analogical consistency
• analogical agreement (older descriptive phrase)
• analogical transfer
• reusable strategy representation
• global label alignment

Resolved issue: Analogical Consistency is the reader-facing metric name under the broader concepts cross-input semantic consistency and cross-input latent strategy alignment. Analogical agreement remains a descriptive phrase for the equality check used by the estimator, but it should not be the primary metric name.

Decision: use cross-input semantic consistency as the property: the same latent value carries stable strategy meaning across different inputs. Use analogical transfer for the operational test or application of this property, and Analogical Consistency for the metric that measures whether sharing $z$ induces matching strategy labels across inputs. Use within-input strategy separation for the weaker local property, and reserve strong recovery for contexts where both within-input separation and cross-input consistency are intended. See Cross-input semantic consistency.

Refinement: when the paper needs alignment-forward language, cross-input latent strategy alignment is a strong name for the property measured by Analogical Consistency, though it is too long for compact figure legends. In main metric-bar figures, use Strategy Alignment as the compact label when the caption defines it as cross-input latent strategy alignment measured by Analogical Consistency.

10. Metrics, Figures, And Claim Language

Concept: language used in captions, plots, and empirical claims.

Current and candidate terms:

• strategy decodability
• router probe accuracy
• posterior probe accuracy
• strategy-match rate
• sampled behavior
• controlled-latent generation
• distributional fidelity
• marginal fidelity
• base-normalized reconstruction ratio
• reconstruction ratio
• $\rho_{\mathrm{recon}}$
• collapsed corner
• operating regime
• recovery frontier
• Pareto-dominant variants

Open issue: figure language should be rechecked after claims and figure organization are updated.

Decision: use strategy alignment as the property and strategy decodability as the metric family, with router strategy decodability and posterior strategy decodability as precise variants. Use base-relative reconstruction as the default term for $\mathcal R_\theta$ and avoid $\rho$ notation. Interpret $\mathcal R_\theta$ as the fraction of baseline surprisal remaining and $1-\mathcal R_\theta$ as residual surprise explained by $z$. Use alignment-reconstruction frontier, strategy-alignment frontier, and collapsed corner for figure and claim language. See Metrics and frontier language.

Refinement: use Router Strategy Decodability for the router-side linear-probe metric. This metric answers: can a simple probe recover strategy information from $Z \sim r_\phi(\cdot|X)$? Avoid labeling this metric simply as Strategy Alignment in figures that also report a stronger cross-input metric, because router decodability can be high even when Analogical Consistency is low.

First Focused Decision

Start with Cluster 1: Main Object Of Study. This decision constrains title language, abstract wording, the problem-statement section title, and the rest of the terminology hierarchy.

Candidate starting question:

Should the paper's primary noun be factorization, with "strategy recovery" and "disentanglement" used as properties/outcomes, or should strategy-structure recovery be the co-primary problem name?

Propagation Targets

Once decisions are made, update in this order:

1. Notation and terminology
2. Contributions
3. Claims
4. Figures
5. path:paper/storyboard
6. path:paper/conference-paper