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Drift Variation Autoencoder: Unifying Generation and Representation Learning through Conditional Posterior Flow Matching

Jiarui Cao

arXiv:2608.25138v1cs.LGcs.AI

TL;DR

Stochastic observations can leave multiple plausible clean completions, making deterministic reconstruction incomplete. Drift Variation autoencoder aligns conditional generation and representation learning through posterior-conditioned Flow Matching, with ideal theory and controlled experiments supporting posterior sufficiency and coherent joint sampling.

  • Problem

    Stochastic observations can admit multiple clean completions, motivating the posterior P(X | C) as a shared target for generation and representation learning.

  • Method

    The method combines conditional KL analysis with clean-prediction Conditional Flow Matching through an observation-conditioned encoder, retaining the complete multimodal tuple as target under every mask.

  • Results

    CrossGeom-4 supports the finite-model mechanism across representation, conditioning, reconstruction, and joint sampling, with joint attention reducing shared-factor disagreement by 90-93% relative to independent decoders.

  • Takeaways & Limitations

    Under ideal affine-Gaussian conditions, generation and representation learning share a posterior-sufficient zero set and endpoint, rather than identical numerical objectives.

  • Takeaways & Limitations

    CrossGeom-4 is a deliberately low-dimensional controlled system, so the results do not establish transfer to natural images, text, or audio.

Abstract

from arXiv · show

Stochastic masking, cropping, or modality removal makes deterministic reconstruction an incomplete target: one observation can admit many clean completions. This work takes the corresponding posterior $P(X\mid C)$ as the common statistical object for conditional generation and generatively sufficient representation learning. Drift Variation autoencoder trains a masked encoder $Z=E(C)$ and a conditional flow decoder with one clean-prediction Flow Matching loss. The analysis first decomposes the ideal conditional KL into generator approximation and the representation deficiency $I(X;C\mid Z)$. It then derives orthogonal risk decompositions for conditional Flow Matching. For an affine Gaussian path, the clean-prediction representation gap is zero if and only if $P(X\mid Z)=P(X\mid C)$. Thus the encoder-dependent excess clean-prediction risk induced by Flow Matching and the profiled ideal conditional KL have the same posterior-sufficient zero set, without being numerically equal objectives. An exact conditional field with a zero-noise endpoint then generates $P(X\mid Z)$ and hence $P(X\mid C)$ at a joint ideal optimum. The result extends to continuous multimodal product spaces when the complete modality tuple remains the Flow target for every observation mask. On CrossGeom-4, an 18-run controlled benchmark, observable factors have linear-probe $R^2$ of $0.9990$-$0.9992$, shuffling the joint model's encoder condition increases conditional error by $13.5\times$-$15.7\times$, and joint target attention reduces disagreement on an unobserved factor shared by two outputs by $90.1$-$92.8\%$ relative to independent target decoders. Visible modalities are also generated and reconstructed, directly validating the full-tuple objective. Unconditional mode balance remains imperfect, delimiting the empirical claim to a controlled multimodal proof of concept.

1 Introduction

Drift Variation autoencoder treats the posterior P(X | C) as the shared target for conditional generation and generative representation learning. Its conditional Flow Matching objective makes posterior sufficiency an encoder criterion and extends to joint multimodal completion.

  • Motivation: Information-removing observations require modeling the posterior P(X | C), because one observation can admit multiple plausible clean completions.This posterior provides a common statistical target for generation and representation learning.
  • Posterior principle: The ideal conditional KL decomposes into representation deficiency I(X; C | Z) and generator approximation after compression.Representation correctness is posterior sufficiency: P(X | Z) = P(X | C).
  • Flow objective: Conditional Flow Matching jointly trains a masked encoder and clean-prediction decoder on an affine path with an analytic sample target.The decoder receives only (Xt, t, Z), without teacher, contrastive target, or generated reference set.
  • Flow risk and sufficiency: rep(E) = 0 if and only if P(X | Z) = P(X | C) if and only if I(X; C | Z) = 0.This establishes matching ideal encoder zero sets, not numerical equality between Flow Matching and KL objectives or their gradients.
  • Endpoint and multimodal guarantees: For complete continuous multimodal tuples, every observation mask conditions the encoder while the joint Flow target remains the full tuple.An exact representation-conditioned field generates P(X | Z), and posterior sufficiency closes the connection to P(X | C).

2 Related Work

Prior work learns representations through invariance, prediction, reconstruction, or generative modeling, while Drift Variation autoencoder defines generative sufficiency through the clean-data posterior induced by an observation. It also builds on conditional Flow Matching and multimodal masking, extending these ideas toward distribution-level conditional generation for continuous latents.

  • Self-supervised representation learning: Contrastive, self-distillation, masked, and predictive methods align augmented views or recover pixels, tokens, or target embeddings.MAE demonstrates scalable visual features through high-ratio masking, while I-JEPA predicts in representation space to emphasize semantic structure.
  • Self-supervised representation learning: Drift Variation autoencoder defines generative sufficiency by the clean-data posterior induced by an observation, without requiring identical features, minimal sufficiency, or semantic sufficiency.Its representation target differs from methods that choose invariance or prediction targets.
  • Flow Matching and conditional sufficiency: Flow Matching regresses vector fields along fixed probability paths, while Conditional Flow Matching uses tractable conditional targets with the same population minimizer for the field model.Related directions include straight-coupling transport in Rectified Flow and encoder–conditional-flow training in FlowFM.
  • Multimodal learning: Multimodal methods randomize masked inputs and targets or combine understanding and generation through early fusion, autoregressive-diffusion objectives, or decoupled visual encoders.Drift Variation autoencoder instead offers a distribution-level abstraction for continuous latents in which any observed subset defines the condition.

3 Posterior Flow Matching for Representation Learning

The section formalizes posterior sufficiency as the shared target of conditional generation and representation learning. Conditional Flow Matching yields this target through sample-based risk decompositions, exact Gaussian zero-set equivalence, and joint posterior recovery at a zero-noise endpoint.

  • Posterior sufficiency: Posterior sufficiency requires P(X | Z) = P(X | C), equivalently X ⊥⊥C | Z or I(X; C | Z) = 0.This condition does not itself imply semantic compression, disentanglement, or augmentation invariance.
  • Conditional KL: The conditional KL decomposes into representation deficiency I(X; C | Z) and generator mismatch EZ KL(P(X | Z) ∥Q(X | Z)).With an unrestricted conditional generator, the infimum is zero exactly for posterior-sufficient encoders.
  • Conditional Flow Matching: Conditional Flow Matching replaces inaccessible posterior-density evaluation with a sample-based regression objective whose population minimizer is the marginal conditional velocity.Its orthogonal risk decomposition separates predictor error from irreducible path variance.
  • Gaussian representation sufficiency: For the affine Gaussian path, clean-prediction representation error is zero if and only if P(X | Z) = P(X | C) almost surely.Thus the Flow Matching representation gap and conditional KL share the same posterior-sufficient zero set, without being numerically identical objectives.
  • Endpoint correctness: An exact conditional field with a zero-noise endpoint generates P(X | Z), which equals P(X | C) when the representation is posterior-sufficient.Decoder field fitting and encoder sufficiency appear as distinct responsibilities in the population risk.
  • Joint multimodal sufficiency: For continuous multimodal product spaces, the complete joint tuple must remain the Flow target under every observation mask, enabling reconstruction, completion, and any-subset generation.Visible modalities are generated and reconstructed, while missing modalities are completed; raw discrete text requires a separate construction and proof.

4 Controlled Validation

Controlled CrossGeom-4 validation supports the model’s representation-use, complete-tuple generation, and conditional posterior-coupling predictions. Joint target attention substantially improves cross-output consistency, while imperfect unconditional mode balance limits the claim to a controlled multimodal proof of concept.

  • Validation design: The benchmark tests representation use, marginal completion, and coherent joint posterior sampling against known conditional factors and an exact sampling oracle.The evaluation is designed to distinguish these predictions without relying on a learned evaluator.
  • Representation use: Mean observed-factor probe R2 is 0.9990-0.9992 across all levels and decoder types, while shuffling valid joint-model conditions increases conditional error by 13.5×-15.7×.The unavailable uBC factor has R2 between −0.023 and −0.016 under an A-only condition, consistent with no leakage.
  • Complete-tuple generation: Visible-stream factor MAE is 0.0912-0.1041 across nonempty masks, and all-visible ABC reconstruction reaches MAE 0.0817, 0.0818, and 0.0944 on L1-L3.These results show that the full-tuple objective generates and reconstructs visible modalities rather than only predicting complements.
  • Joint posterior coupling: Joint attention reduces disagreement on the shared unknown factor by 90.1-92.8% and its W1 error by 61.2-76.9% versus matched independent decoders.Known-factor accuracy remains nearly identical, so the improvement concerns posterior coupling rather than easier conditional prediction.
  • Unconditional limitation: Unconditional TV-to-uniform remains 0.086-0.108 versus a finite-sample reference of about 0.034, despite joint decoding reducing unconditional disagreement by 86.6-89.4%.Both decoder types reach all 16 sign modes, but the independent baseline trades lower mode-frequency TV for failed joint factor consistency.

5 Limitations

The paper’s equivalence is limited to population and realizability settings, with additional assumptions on sufficiency, corruption, endpoint integration, conditioning, and the controlled benchmark. It does not establish optimization, finite-risk information bounds, broad modality transfer, or practical endpoint correctness.

  • The equivalence does not show optimization reaches a global optimum, finite models realize Bayes denoisers, or low empirical loss quantitatively bounds I(X; C | Z).With finite capacity, representation deficiency and decoder approximation interact, and the decoder’s easiest representation may not be sufficient.
  • Posterior sufficiency does not imply minimal sufficiency or human semantic quality; the code may retain identity, texture, or any detail affecting P(X | C).Observation design and bottleneck structure determine which information is emphasized.
  • CrossGeom-4 demonstrates factor accessibility and joint sampling only in a deliberately low-dimensional system, not transfer to natural images, text, or audio.Broader multimodal claims require modality-specific tokenizers, larger models, semantic and dense probes, and standard generative metrics.
  • The identifiability proof relies on additive full-rank Gaussian corruption, a positively weighted interior time, and injective Gaussian convolution.Singular paths, non-Gaussian sources, discrete tokens, and manifold-valued flows require separate arguments; CrossGeom-4 L3 uses Euclidean Flow Matching with endpoint projection.
  • Endpoint correctness assumes an exact field, a regular conditional ODE, and controlled integration, while practical solvers, codecs, clipping, and projection add untheorized errors.The work provides no general finite-risk conditional-KL or mutual-information bound.
  • The sufficiency claim requires a condition-independent source and genuine encoder bottleneck; informative decoder side channels instead become part of the combined condition.In the full-tuple implementation, noisy Flow states are provided for every modality, while the visibility mask affects only the encoder.

6 Conclusion · A Notation, Assumptions, and Scope

The paper treats P(X | C) as the common object for conditional generation and representation learning, with conditional KL and Flow Matching yielding complementary decompositions. Its guarantees rely on explicit probabilistic, Gaussian-path, regularity, realizability, and multimodal-target assumptions, while CrossGeom-4 provides controlled empirical support.

  • 6 Conclusion: The conditional posterior P(X | C) unifies generation and representation learning, while encoder-dependent Flow Matching risk isolates representation deficiency from path variance and model approximation.For an affine Gaussian path, the representation term vanishes exactly when P(X | Z) = P(X | C).
  • 6 Conclusion: 90-93% relative to independent decoders: joint target attention reduces disagreement on an unobserved shared factor in CrossGeom-4.The benchmark also reports observable-factor exposure, order-of-magnitude degradation after condition shuffling, and reconstruction of visible modalities as active targets.
  • A Notation, Assumptions, and Scope: The multimodal extension requires the complete tuple to remain the Flow target under every mask, and deterministic decoder side channels must be absorbed into R = (Z, B).Claims about Z alone require that B add no posterior information or that the bypass be removed.
  • A Notation, Assumptions, and Scope: Because Z = E(C) is deterministic, σ(Z) ⊆ σ(C) and X → C → Z forms a Markov chain under common-probability-space and regular-conditional-law conventions.Conditional-law and conditional-expectation equalities are understood up to corresponding null sets.
  • A Notation, Assumptions, and Scope: The analysis assumes standard Borel spaces, suitable conditional moments, an independent Gaussian source X0 = s0ε, differentiable schedules, and exact posterior endpoint conditions when required.The endpoint conditions are α0 = 0, σ0 = 1, α1 = 1, and σ1 = 0.
  • A Notation, Assumptions, and Scope: Continuous-time results require positive-measure interior support plus valid differentiation, Fubini, tower-property, score-identification, moment, and domination conditions.The interior set requires αt > 0, σt > 0, and positive finite effective loss weight.
  • A Notation, Assumptions, and Scope: Inferring equality of ODE marginals requires continuity-equation and conditional-ODE existence, uniqueness, boundary, and integrability conditions; exact predictors or endpoints additionally require decoder realizability.The risk decompositions themselves do not require realizability.
  • A Notation, Assumptions, and Scope: The KL decomposition applies to probability kernels under a valid chain rule, with I(X; C | Z) = E_C KL(P(X | C) ∥ P(X | Z)) for deterministic Z = E(C).The sample CFM target is encoder-independent, whereas the representation-conditioned Bayes field depends on Z, so encoder training is not direct KL gradient descent.

B Proof of the Conditional KL Decomposition · C Conditional Path and Risk Decompositions · C.1 Marginalizing endpoint bridges

The conditional KL decomposes into representation deficiency and generator approximation, while conditional path risks decompose orthogonally around the marginalized conditional field. These results characterize sufficiency and clarify that minimality requires an additional criterion.

  • B Proof of the Conditional KL Decomposition: The conditional KL equals I(X; C | Z) plus the expected KL from P(X | Z) to Q(X | Z).An unrestricted generator is minimized by Q(· | Z) = P(· | Z).
  • B Proof of the Conditional KL Decomposition: Sufficiency is equivalent to I(X; C | Z) = 0, or equivalently X ⊥⊥ C | Z when Z is a function of C.This is also equivalent to equality of the relevant posteriors in the stated decomposition.
  • B Proof of the Conditional KL Decomposition: If E is sufficient, augmenting Z with any deterministic statistic of C preserves zero representation deficiency and does not worsen the unrestricted generator optimum.Thus the decomposition alone does not enforce minimality.
  • B Proof of the Conditional KL Decomposition: Minimality requires a separate ordering, dimension, entropy, rate, or bottleneck criterion beyond sufficiency.The proof explicitly distinguishes representation sufficiency from minimality.
  • C.1 Marginalizing endpoint bridges: For fixed endpoint bridges, marginalizing over P(dx | c) yields the conditional field uC(xt, t, c) through the continuity equation.The conditional field is the conditional expectation of the bridge velocity given Xt, t, and C.
  • C.1 Marginalizing endpoint bridges: The cross term is zero because E[Ut − uC | Xt, t, C] = 0, establishing the orthogonal risk decomposition.This conditional-mean property removes the interaction between predictor error and bridge variability.
  • C Conditional Path and Risk Decompositions: For any square-integrable v, the conditional path risk separates into prediction risk around uC, intrinsic bridge variance, and a cross term.The decomposition is stated in Eq. (45).

C.2 Nested projection for the velocity field · C.3 Nested projection for clean prediction · C.4 Clean prediction and affine velocity

Nested conditional expectations yield orthogonal risk decompositions for both velocity fields and clean prediction, with representation error equal to residual conditional variance. Under the affine path, an exact clean-prediction Bayes rule recovers the exact representation-conditioned marginal velocity wherever σ_t > 0.

  • C.2 Nested projection for the velocity field: Because Z = E(C), conditioning on (X_t, t, Z) is nested within conditioning on (X_t, t, C), enabling tower-property projections.This nesting gives u_Z = E[u_C | X_t, t, Z].
  • C.2 Nested projection for the velocity field: The velocity residual decomposes into conditional, representation, and decoder terms whose pairwise cross terms vanish by L2 orthogonality.The decomposition is U_t − v = (U_t − u_C) + (u_C − u_Z) + (u_Z − v).
  • C.2 Nested projection for the velocity field: The velocity representation loss equals E Var(u_C(X_t, t, C) | X_t, t, Z), measuring unresolved variation among condition-level fields.The conditional variance is the trace of the conditional covariance.
  • C.3 Nested projection for clean prediction: Under a nonnegative time-only weight, conditional expectation remains an orthogonal projection on times where the weight is positive.The weight is measurable with respect to t, preserving the projection argument.
  • C.3 Nested projection for clean prediction: The weighted clean-prediction residual decomposes into conditional, representation, and decoder terms, with cross terms eliminated by the weighted inner product.This proves the clean-prediction risk decomposition.
  • C.3 Nested projection for clean prediction: The clean-prediction representation loss equals E[w(t) Var(m_C(X_t, t, C) | X_t, t, Z)].It quantifies conditional variance remaining after replacing C with Z.
  • C.4 Clean prediction and affine velocity: For the affine path with a = 1 − σ_min, conditioning the affine velocity on (X_t, t, Z) yields the clean-prediction relation in Eq. (33).Therefore, an exact clean-prediction Bayes rule produces the exact representation-conditioned marginal velocity wherever σ_t > 0.

D Proof of Gaussian Flow Representation Sufficiency · E Endpoint Correctness and the Multimodal Extension · E.1 Proof of conditional endpoint correctness

For the affine Gaussian path, vanishing representation gap is equivalent to posterior sufficiency, because Gaussian denoising and smoothing identify the clean conditional law. An exact learned conditional field reaches P(X | Z), and therefore P(X | C) at zero representation deficiency, only with a zero-noise endpoint or deconvolution.

  • D Proof of Gaussian Flow Representation Sufficiency: Gaussian posterior-mean equality identifies the corresponding noisy law under smooth positive densities on connected R^d.The proof converts posterior-mean equality into equal score fields, whose normalized densities must coincide.
  • D Proof of Gaussian Flow Representation Sufficiency: Gaussian smoothing is injective when α, β > 0, so equal laws of Y = αX + βε imply equal conditional laws of X.The nonvanishing Gaussian characteristic-function multiplier and α > 0 allow recovery of all clean-law frequencies.
  • D Proof of Gaussian Flow Representation Sufficiency: The representation gap is zero exactly when P(X | C) = P(X | Z), equivalently I(X; C | Z) = 0.The forward direction uses an interior positive-noise time and the Gaussian denoiser lemmas; the converse follows from conditional independence.
  • D Proof of Gaussian Flow Representation Sufficiency: Using t = 0 alone cannot establish posterior sufficiency when α0 = 0, because distinct multimodal clean distributions may share the same conditional mean.Without a scaled copy of X, the characteristic-function argument cannot cover the frequencies needed to identify P(X | C).
  • E.1 Proof of conditional endpoint correctness: Endpoint correctness follows because the learned field matches the conditional velocity, so flow uniqueness preserves p_t(· | z) from the common source.With α_1 = 1 and σ_1 = 0, the endpoint is P(X | Z = z); posterior sufficiency then gives P(X | C).
  • E.1 Proof of conditional endpoint correctness: If σmin > 0, the same flow argument identifies a smoothed endpoint rather than the clean posterior.Exact clean-posterior correctness therefore requires a zero-noise endpoint or a separate deconvolution mechanism.

E.2 Proof of joint multimodal sufficiency … F.3 Finite capacity and endpoint numerics

The proof establishes that joint clean-tuple denoising recovers the full multimodal posterior, while mask-conditioned branches support unconditional, cross-modal, and completion tasks. Practical analysis distinguishes weighting, source-noise, bypass, capacity, and endpoint effects that shape finite training without changing the ideal sufficiency result under stated assumptions.

  • E.2 Proof of joint multimodal sufficiency: A zero multimodal representation term forces equality of conditional posterior means, which implies equal noisy scores and normalized noisy densities.The argument uses a positive-definite loss matrix and the Gaussian denoiser identity.
  • E.2 Proof of joint multimodal sufficiency: Equality of noisy laws identifies the entire joint characteristic function of X, proving joint posterior sufficiency; the reverse direction follows by conditional-independence factorization.The Gaussian factor is nonzero and A^⊤ is surjective.
  • E.2 Proof of joint multimodal sufficiency: The decoder preserves cross-modal dependence because it conditions on the joint noisy state X_t and predicts the joint clean tuple rather than separate marginal denoisers.Separate marginal losses would identify only modality-specific denoisers and could lose dependence.
  • E.3 Any-subset and unconditional branches: Mask-conditioned training represents unconditional joint generation, cross-modal conditional generation, and joint multimodal completion across observed-modality subsets.The mask is legitimate side information, but hidden clean tokens or their encodings would change the conditioning sigma-field.
  • F.1 Time weighting and representation pressure: Uniform, clipped, and exact time weighting share the ideal sufficient-encoder zero set when weights remain positive, although finite-sample variance and optimization can differ substantially.For σ_min = 0, exact velocity weighting diverges at the endpoint, motivating clipping.
  • F.1 Time weighting and representation pressure: Near a zero-noise endpoint, X_t nearly reveals X, so condition-use gaps can shrink even for useful representations; time-resolved diagnostics are therefore more informative than one average.At t = 0, the denoiser compares E[X | C] with E[X | Z], while an identifying interior time can identify the posterior.
  • F.2 Source noise, augmentation entropy, and bypasses: Source noise must remain independent of C, and direct clean observations, crop geometry, calibration variables, or task identifiers become bypass information certified as part of the representation.The intrinsic Flow Matching inputs are X_t and time; a separate clean observation copy is not.
  • F.3 Finite capacity and endpoint numerics: Finite networks can favor lossy but easier codes, requiring joint ablations of encoder tokens, cross-attention, width, and depth; with σ_min = 0, avoid evaluating the clean-prediction parameterization exactly at t = 1.Stronger masking changes posterior entropy, while flow time changes signal-to-noise along a fixed path; multiple source draws expose diversity but cannot recover absent modes.

F.4 Natural-data evaluation protocol · G CrossGeom-4 Details and Complete Results

The evaluation protocol requires distributional, multimodal, and representation-sensitive tests rather than reconstruction error alone. No single finite-network metric validates the theorem, so evidence should follow the proposed encoder–decoder–source–joint-dependence decomposition at one checkpoint.

  • F.4 Natural-data evaluation protocol: Natural-image evaluation should combine frozen representation probes with FID, precision/recall, condition fidelity, and repeated-sample diversity.High-entropy contexts additionally require distributional metrics, best-of-K fidelity, and consistency with visible evidence.
  • F.4 Natural-data evaluation protocol: Single-output MSE can reward a conditional mean while missing the posterior, motivating distributional evaluation for high-entropy contexts.
  • F.4 Natural-data evaluation protocol: Every observed modality subset should be evaluated against the complete generated tuple using per-modality fidelity, visible-stream reconstruction, and cross-modal consistency.Cross-modal retrieval and dense probes should also be reported.
  • F.4 Natural-data evaluation protocol: Repeated samples should vary unobserved variables while preserving observed evidence, with compatibility metrics testing shared residual draws across simultaneous outputs.Marginal quality alone is insufficient for multimodal evaluation.
  • F.4 Natural-data evaluation protocol: No single finite-network metric validates Theorem 3.6; empirical evidence should instead test predictive encoder information, decoder use, and source-accounted residual uncertainty.
  • F.4 Natural-data evaluation protocol: The evaluation should verify preserved joint dependence and the coexistence of representation and generation at one declared checkpoint.

G.1 Data construction and architecture · G.2 Complete aggregate tables · G.3 Oracle construction and metric interpretation

The benchmark constructs geometrically invertible multimodal data and evaluates full-tuple conditional Flow Matching with controlled architectures, oracle references, and condition-use diagnostics. Its metrics distinguish representation quality, distributional completion, and whether multiple outputs share a posterior draw.

  • G.1 Data construction and architecture: The data pipeline mixes three-factor modalities with fixed orthogonal transforms and applies invertible asinh, spherical, and hyperboloid coordinate maps for exact factor evaluation.Exact inverse maps avoid reliance on a learned perceptual model.
  • G.1 Data construction and architecture: The encoder uses modality-specific stems, embeddings, a global token, three Transformer layers, width 96, and four heads, while joint decoding self-attends across all target modalities.Unavailable modality tokens are padded; the independent baseline uses one decoder per modality.
  • G.1 Data construction and architecture: All eight availability patterns are balanced, yet every pattern trains independent Gaussian source states for all three modalities against complete A/B/C targets.Availability affects only the encoder, including for A-only and ABC observations.
  • G.2 Complete aggregate tables: Trained representations remain stable across levels, whereas factors unavailable from a single observed modality remain near chance and random-feature ridge probes can yield negative out-of-sample R2.The negative random-feature result reflects poor conditioning and coordinate dependence, not clipping.
  • G.3 Oracle construction and metric interpretation: The conditional oracle copies visible factors, independently resamples undetermined factors from their true mixtures, and renders the complete modality tuple.This makes condition-to-target error deterministically zero for the oracle.
  • G.3 Oracle construction and metric interpretation: Both decoder structures achieve essentially perfect two-sign coverage with 16 draws per context, but this marginal statistic cannot establish joint generation.Unknown-factor target-target MAE tests whether output modalities share one posterior draw, while W1 compares averaged recovery with the exact conditional marginal.
  • G.3 Oracle construction and metric interpretation: Shuffling is the primary condition-use test because it preserves observed-modality support and reuses identical initial source noise; zero-condition results are out-of-support and excluded from headline effects.Headline values, aggregate JSON, and long-form CSV are included with the source package.
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