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Drift Variation Autoencoder: Unifying Generation and Representation Learning through Conditional Posterior Flow Matching
Jiarui Cao
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%
Last extracted: 8/27/2026, 4:45:28 AM
Summary
This paper introduces the Drift Variation Autoencoder, a framework that unifies conditional generation and representation learning by optimizing for the posterior distribution P(X|C) using Conditional Flow Matching (CFM). The authors prove that for affine Gaussian paths, minimizing the clean-prediction risk in CFM is equivalent to achieving posterior sufficiency, where the learned representation Z contains all information in the observation C relevant to the clean data X. The method is validated on the CrossGeom-4 benchmark, demonstrating high fidelity in reconstructing and generating multimodal data while ensuring the encoder captures sufficient statistics.
Entities (6)
Relation Signals (6)
Drift Variation Autoencoder → uses → Conditional Flow Matching
confidence 95% · Drift Variation autoencoder turns this posterior principle into a sample-based objective using Conditional Flow Matching (CFM)
Drift Variation Autoencoder → unifies → Generation
confidence 93% · Unifying Generation and Representation Learning through Conditional Posterior Flow Matching
Drift Variation Autoencoder → unifies → Representation Learning
confidence 93% · Unifying Generation and Representation Learning through Conditional Posterior Flow Matching
Drift Variation Autoencoder → evaluatedon → CrossGeom-4
confidence 92% · On CrossGeom-4, an 18-run controlled benchmark, observable factors have linear-probe R2 of 0.9990-0.9992
Clean-Prediction Risk → ensures → Posterior Sufficiency
confidence 90% · the clean-prediction representation gap is zero if and only if P(X|Z)=P(X|C)
Drift Variation Autoencoder → comparedto → Zero-Flow Encoders
confidence 85% · The closest theoretical comparison is Zero-Flow Encoders [9]
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Abstract
Abstract: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.
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- Source: https://arxiv.org/abs/2608.25138v1
- Canonical: https://arxiv.org/abs/2608.25138v1
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Drift Variation Autoencoder: Unifying Generation and Representation Learning through Conditional Posterior Flow Matching Jiarui Cao Affiliation: The Chinese University of Hong Kong Email: 1155244613@link.cuhk.edu.hk Abstract 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∣C)P(X C) as the common statistical object for conditional generation and generatively sufficient representation learning. Drift Variation autoencoder trains a masked encoder Z=E(C)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∣Z)I(X;C 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∣Z)=P(X∣C)P(X Z)=P(X 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∣Z)P(X Z) and hence P(X∣C)P(X 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 R2R^2 of 0.99900.9990-0.99920.9992, shuffling the joint model’s encoder condition increases conditional error by 13.5×13.5×-15.7×15.7×, and joint target attention reduces disagreement on an unobserved factor shared by two outputs by 90.190.1-92.8%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 When an augmentation removes information, reconstruction is not a one-to-one problem. A crop can hide several plausible backgrounds; a masked signal can admit several continuations; and an observed modality can leave residual variables that two missing modalities must share. The statistically correct target is therefore not a single reconstruction but the posterior P(X∣C)P(X C) of a clean sample X given a stochastic observation C=A(X,ΞA)C=A(X, _A). This posterior offers a precise common target for two tasks that are usually trained separately: a generator should sample it, while an encoder representation should preserve exactly the information in C needed to specify it. Current pipelines typically optimize the two capabilities with distinct objectives. Self-supervised encoders use view agreement, masked reconstruction, or latent prediction [1, 2, 3, 4], whereas diffusion and flow models estimate denoising scores or transport fields [5, 6, 7]. A downstream system may then align, freeze, distill, or reconnect the components. This separation can be useful, but it does not make representation correctness a consequence of the same uncertainty modeled by the generator. Conversely, a conditional generator can reduce denoising loss through its noisy state without exposing a reusable representation. This work asks whether one ordinary generative objective can instead characterize and train the representation it consumes. Let Z=Eθ(C)Z=E_θ(C) be a deterministic representation and let Q(X∣Z)Q(X Z) be a conditional generator. The ideal posterior-matching risk is (E,Q)=CKL(P(X∣C)∥Q(X∣E(C))).K(E,Q)=E_CKL\! (P(X C)\,\|\,Q(X E(C)) ). (1) Because Z is a function of C, this risk decomposes exactly as (E,Q)=I(X;C∣Z)+ZKL(P(X∣Z)∥Q(X∣Z)).K(E,Q)=I(X;C Z)+E_ZKL\! (P(X Z)\,\|\,Q(X Z) ). (2) The generator must model the posterior available after compression, while the representation is correct exactly when P(X∣Z)=P(X∣C)P(X Z)=P(X C). This condition is called posterior sufficiency. It is a generative notion of representation quality, not a guarantee of semantic minimality, disentanglement, or invariance. An identity encoder can be sufficient; the observation process and bottleneck determine which sufficient representations are useful. Equation (1) defines the desired statistical object, but its unknown conditional densities prevent direct optimization. Drift Variation autoencoder turns this posterior principle into a sample-based objective using Conditional Flow Matching (CFM) [7, 8]. The training procedure draws a condition-independent Gaussian source X0X_0, forms an affine path Xt=αtX+σtX0X_t= _tX+ _tX_0, and jointly trains the encoder and a clean-prediction decoder that receives only (Xt,t,Z)(X_t,t,Z). The sample target is analytic and fixed by the observed data pair and probability path; no teacher, contrastive target, or generated reference set is required by the population objective. The key difficulty is showing that CFM constrains the encoder rather than only the decoder. The analysis derives an orthogonal risk decomposition into irreducible path variance, representation deficiency, and model approximation. For clean prediction, the encoder-dependent term is Δrepx(E)=[w(t)‖[X∣Xt,t,C]−[X∣Xt,t,Z]‖2]. _ rep^x(E)=E\! [w(t) [X X_t,t,C]-E[X X_t,t,Z] ^2 ]. (3) For an affine Gaussian path with positive effective mass at interior noise levels, the analysis proves Δrepx(E)=0⟺P(X∣Z)=P(X∣C)⟺I(X;C∣Z)=0. _ rep^x(E)=0 P(X Z)=P(X C) I(X;C Z)=0. (4) The proof recovers a noisy conditional score from the Gaussian posterior mean, then the smoothed density, and finally the clean posterior through injectivity of Gaussian convolution. This is an equivalence of ideal encoder zero sets, not an equality between the numerical CFM and KL objectives or their encoder gradients. With an exact representation-conditioned field, the conditional ODE generates P(X∣Z)P(X Z); Eq. (4) closes the loop to P(X∣C)P(X C). The formulation extends to a multimodal tuple =(X(1),…,X(M))X=(X^(1),…,X^(M)). An observation mask controls which clean modalities enter the encoder, but the Flow target remains the complete tuple for every mask. Each modality receives source noise, contributes a positive loss weight, and evolves in the joint decoder. The product-space theorem then identifies the complete joint posterior, including residual dependence between simultaneously generated modalities. The evaluation tests this prediction on CrossGeom-4, a controlled three-modality distribution with known shared factors and Euclidean, spherical, and hyperbolic views. Joint and independent decoders have similar deterministic conditional accuracy, but only joint target attention coordinates the same unobserved posterior draw across two outputs. The closest theoretical comparison is Zero-Flow Encoders [9], which constructs a separate sufficient-representation loss from a midpoint zero-flow criterion under an independent coupling. Drift Variation autoencoder instead studies the ordinary conditional decoder used for generation: its clean-prediction risk itself induces the encoder criterion, and Gaussian denoiser identifiability links that criterion to posterior sufficiency. The main contributions are: • Posterior principle. Generative representation correctness is defined by P(X∣Z)=P(X∣C)P(X Z)=P(X C), and the ideal conditional KL is decomposed into representation deficiency and generator approximation. • Flow risk and sufficiency. Exact velocity and clean-prediction risk decompositions are derived, and it is proved that the encoder-dependent clean-prediction representation gap induced by affine-Gaussian CFM and the profiled conditional KL share the same posterior-sufficient zero set. • Endpoint and multimodal guarantees. An exact field is connected to posterior generation, and the result is extended to complete continuous multimodal tuples under any observed subset. • Controlled validation. CrossGeom-4 measures representation use, visible-stream reconstruction, conditional completion, residual joint coupling, and unconditional coverage across three geometries and three seeds. 2 Related Work Self-supervised representation learning. Contrastive and self-distillation methods align augmented views, while masked and predictive methods recover pixels, tokens, or target embeddings [1, 10, 2, 3, 4]. MAE shows that high-ratio masking can learn scalable visual features, whereas I-JEPA deliberately predicts in representation space to emphasize semantic structure [2, 3]. These methods choose an invariance or prediction target. Drift Variation autoencoder instead defines a generatively sufficient representation by the clean-data posterior induced by an observation. It does not require all views of one sample to have identical features and does not claim minimal or semantic sufficiency. Generation and representation. Denoising autoencoders connect reconstruction fields to data scores [11, 12]. Diffusion Autoencoders [13], representation learning with diffusion [14, 15], and DDAE [16] study representations learned with or extracted from generative models; later analysis tracks how diffusion representations change with noise and training [17]. In the other direction, REPA aligns generative hidden states with a pretrained encoder [18]. Representation Autoencoders and RepTok import or adapt self-supervised latent spaces for efficient generation [19, 20]. These approaches demonstrate the value of semantic geometry but retain an external representation source, an auxiliary alignment objective, or a separate latent-space construction. Drift Variation autoencoder asks when the conditional generative loss itself identifies the required representation. Flow Matching and conditional sufficiency. Flow Matching regresses vector fields of fixed probability paths, and Conditional Flow Matching replaces an inaccessible marginal field by tractable conditional targets with the same population minimizer for the field model [7, 8]. Rectified Flow studies straight couplings and efficient transport [21]. FlowFM jointly trains an encoder and conditional flow on wearable signals [22]; Self-Flow adds dual-timestep information asymmetry and a self-supervised feature objective for scalable multimodal synthesis [23]; and Symmetrical Flow Matching combines opposing flows with supervised tasks [24]. Closest to the theory developed here, Zero-Flow Encoders relate a midpoint zero-flow criterion to conditional independence and construct a separate representation loss [9]. Drift Variation autoencoder instead proves that, after profiling the decoder, the encoder-dependent excess clean-prediction risk of the same conditional decoder used for generation has posterior-sufficient encoders as its ideal zero set. Its proof uses Gaussian denoiser identifiability rather than a midpoint zero-flow criterion. Finite-error bounds from flow-field error to terminal KL require additional regularity [25]; the present result concerns exact zero sets and endpoints. Multimodal learning. MultiMAE and 4M randomize masked inputs and targets across modalities, with 4M supporting flexible tokenized conditional generation [26, 27]. Chameleon, Transfusion, and Janus combine multimodal understanding and generation through early fusion, autoregressive-diffusion objectives, or decoupled visual encoders [28, 29, 30]. Drift Variation autoencoder contributes a distribution-level abstraction for continuous latents: any observed subset defines the condition, while all modality streams remain one joint Flow target. This distinction makes cross-output residual dependence, rather than only per-modality fidelity, an explicit empirical requirement. 3 Posterior Flow Matching for Representation Learning 3.1 The augmentation posterior is the common target Let X∈ℝdX ^d follow the data distribution, let ΞA _A denote augmentation randomness, and define the observed context and deterministic representation by C=A(X,ΞA),Z=Eθ(C).C=A(X, _A), Z=E_θ(C). (5) The augmentation may mask patches, crop spatial support, alter photometric statistics, remove a modality, or combine several such operations. If it is not invertible, P(X∣C=c)P(X C=c) need not be a point mass. The modeling target is therefore the complete conditional law rather than one reconstruction. Definition 3.1 (Posterior-sufficient representation). An encoder E is sufficient for the clean target X relative to the observation C when, for Z=E(C)Z=E(C), P(X∣Z)=P(X∣C)almost surely.P(X Z)=P(X C) surely. (6) Equivalently, X⟂C|ZX \!\!\! C Z or I(X;C∣Z)=0I(X;C Z)=0 whenever the conditional mutual information is well defined. Sufficiency is deliberately weaker than minimal sufficiency. The identity encoder Z=CZ=C can be sufficient, and nothing in Definition 3.1 alone forces semantic compression, disentanglement, or invariance to every augmentation. Those properties must come from the observation process, an architectural bottleneck, regularization, or an additional criterion. Consider the ideal conditional distribution-matching objective (E,Q)=CKL(P(X∣C)∥Q(X∣Z)),Z=E(C).K(E,Q)=E_CKL\! (P(X C)\,\|\,Q(X Z) ), Z=E(C). (7) Proposition 3.2 (Conditional KL decomposition). If the conditional laws and displayed KL divergences exist, then (E,Q)=I(X;C∣Z)+ZKL(P(X∣Z)∥Q(X∣Z)). K(E,Q)=I(X;C Z)+E_ZKL\! (P(X Z)\,\|\,Q(X Z) ). (8) Consequently, for an unrestricted conditional generator family, infQ(E,Q)=I(X;C∣Z), _QK(E,Q)=I(X;C Z), (9) and the infimum is zero exactly for posterior-sufficient encoders. Proposition 3.2 makes the desired synchronization explicit. The generator must model the law available after compression, P(X∣Z)P(X Z), while the encoder is correct only if this law retains the posterior specified by the original observation. Direct evaluation of Eq. (7) is unavailable from ordinary joint samples because P(X∣C)P(X C) has no tractable density. The next step constructs a probability path whose regression objective is sample based. 3.2 A conditional probability path from common noise Draw a source independent of both data and augmentation, X0=s0ε,ε∼(0,Id),X0⟂(X,C),s0>0.X_0=s_0 , (0,I_d), X_0 \!\!\! (X,C), s_0>0. (10) For differentiable scalar schedules αt _t and σt _t, define Xt=αtX+σtX0,Ut=α˙tX+σ˙tX0.X_t= _tX+ _tX_0, U_t= α_tX+ σ_tX_0. (11) The principal affine path uses αt=t,σt=1−(1−σmin)t, _t=t, _t=1-(1- _ )t, (12) so that Xt=tX+σtX0,Ut=X−(1−σmin)X0=X−(1−σmin)Xtσt.X_t=tX+ _tX_0, U_t=X-(1- _ )X_0= X-(1- _ )X_t _t. (13) At t=0t=0, every condition shares P0=(0,s02I)P_0=N(0,s_0^2I). If σmin=0 _ =0, then X1=X_1=X and the endpoint conditioned on C is exactly P(X∣C)P(X C). If σmin>0 _ >0, the endpoint is the Gaussian-smoothed posterior P(X∣C)∗(0,σmin2s02I)P(X C)*N(0, _ ^2s_0^2I). For a fixed clean endpoint x, Eq. (11) defines the tractable bridge pt(xt∣x)=(xt,αtx,σt2s02I).p_t(x_t x)=N\! (x_t; _tx, _t^2s_0^2I ). (14) Marginalizing the endpoint over the augmentation posterior gives the actual condition-level path pt(xt∣c)=∫pt(xt∣x)p(x∣c)x.p_t(x_t c)= p_t(x_t x)p(x c)\,dx. (15) Thus one training pair (c,x)(c,x) is a Monte Carlo observation from the endpoint mixture, not a declaration that P(X∣C=c)P(X C=c) is the Dirac mass at x. Define the marginal conditional velocity uC(xt,t,c)=[Ut∣Xt=xt,t,C=c].u_C(x_t,t,c)=E[U_t X_t=x_t,t,C=c]. (16) Proposition 3.3 (Conditional path field). Assume that differentiation can be exchanged with endpoint marginalization and that the corresponding continuity equations are well defined. Then uCu_C satisfies ∂tpt(xt∣c)+∇xt⋅(pt(xt∣c)uC(xt,t,c))=0. _tp_t(x_t c)+ _x_t\!·\! (p_t(x_t c)u_C(x_t,t,c) )=0. (17) Moreover, for any square-integrable predictor v(Xt,t,C)v(X_t,t,C), ‖v−Ut‖2=‖v−uC‖2+‖Ut−uC‖2.E v-U_t ^2=E v-u_C ^2+E U_t-u_C ^2. (18) The second term is independent of v, so the sample CFM target and the inaccessible marginal-field target have the same population minimizer. The proof is the usual conditional path marginalization and conditional-expectation projection, with the external context C held fixed [7, 8]. It requires no repeated occurrence of an exactly identical continuous observation c in the dataset. 3.3 The encoder bottleneck exposes representation deficiency Drift Variation autoencoder does not give C directly to the flow decoder. The learned predictor has the restricted form vθ,ϕ(Xt,t,C)=vϕ(Xt,t,Eθ(C))=vϕ(Xt,t,Z).v_θ,φ(X_t,t,C)=v_φ(X_t,t,E_θ(C))=v_φ(X_t,t,Z). (19) For a fixed encoder, define the best field available through its representation, uZ(Xt,t,Z)=[Ut∣Xt,t,Z]=[uC(Xt,t,C)∣Xt,t,Z].u_Z(X_t,t,Z)=E[U_t X_t,t,Z]=E[u_C(X_t,t,C) X_t,t,Z]. (20) Theorem 3.4 (Velocity CFM risk decomposition). For every encoder and field predictor for which the displayed second moments are finite, ℒCFM(E,v) _ CFM(E,v) :=‖v(Xt,t,Z)−Ut‖2 :=E v(X_t,t,Z)-U_t ^2 =‖Ut−uC‖2⏟ℒpath+‖uC−uZ‖2⏟Δrepv(E)+‖uZ−v(Xt,t,Z)‖2⏟Δmodelv(E,v). = E U_t-u_C ^2_L_ path+ E u_C-u_Z ^2_ _ rep^v(E)+ E u_Z-v(X_t,t,Z) ^2_ _ model^v(E,v). (21) The path term depends only on the data, augmentation, source, and schedules. With an unrestricted decoder, v∗=uZv^*=u_Z and infvℒCFM(E,v)=ℒpath+Δrepv(E) _vL_ CFM(E,v)=L_ path+ _ rep^v(E). The middle term is the information that was useful for the correct conditional velocity but lost when C was replaced by Z. The final term is distinct: a finite decoder can fail even when the representation is sufficient, or it can prefer an easier but insufficient representation during joint optimization. Theorem 3.4 is therefore a population accounting identity, not a promise about finite-network optimization. 3.4 Clean prediction and posterior sufficiency In practice, the decoder is parameterized as a clean predictor X^=Dϕ(Xt,t,Z). X=D_φ(X_t,t,Z). (22) For the affine path, it induces the velocity vϕ(Xt,t,Z)=Dϕ(Xt,t,Z)−(1−σmin)Xtσt.v_φ(X_t,t,Z)= D_φ(X_t,t,Z)-(1- _ )X_t _t. (23) Exact velocity CFM is therefore equivalent away from σt=0 _t=0 to a weighted clean-prediction loss with wvel(t)=σt−2w_ vel(t)= _t^-2. The formulation allows any measurable positive time weight and defines ℛw(E,D)=[w(t)‖D(Xt,t,Z)−X‖2].R_w(E,D)=E\! [w(t) D(X_t,t,Z)-X ^2 ]. (24) Let mC(xt,t,c)=[X∣Xt=xt,t,C=c],mZ(xt,t,z)=[X∣Xt=xt,t,Z=z].m_C(x_t,t,c)=E[X X_t=x_t,t,C=c], m_Z(x_t,t,z)=E[X X_t=x_t,t,Z=z]. (25) Theorem 3.5 (Clean-prediction risk decomposition). If the displayed second moments are finite, then ℛw(E,D) _w(E,D) =[w(t)‖X−mC‖2]⏟ℛpath+[w(t)‖mC−mZ‖2]⏟Δrepx(E) = E\! [w(t) X-m_C ^2 ]_R_ path+ E\! [w(t) m_C-m_Z ^2 ]_ _ rep^x(E) +[w(t)‖mZ−D(Xt,t,Z)‖2]⏟Δmodelx(E,D). + E\! [w(t) m_Z-D(X_t,t,Z) ^2 ]_ _ model^x(E,D). (26) For a fixed encoder, the pointwise Bayes predictor is DE∗=mZD_E^*=m_Z wherever the effective time weight is positive. This result explains why Eq. (24) is not ordinary masked autoencoding even though both can use squared error. A deterministic reconstructor g(C)g(C) has the single Bayes target [X∣C]E[X C]. Drift Variation autoencoder learns the family mZ(xt,t,z)m_Z(x_t,t,z) over noisy states and interior times. For Gaussian corruption, that family identifies the complete conditional distribution. Theorem 3.6 (Gaussian flow representation sufficiency). Let Z=E(C)Z=E(C), let X0=s0εX_0=s_0 with s0>0s_0>0 and ε∼(0,Id) (0,I_d) independent of (X,C)(X,C), and let Xt=αtX+σtX0X_t= _tX+ _tX_0. Let t∼ρt ρ be independent of (X,C,X0)(X,C,X_0) and define int=t:αt>0,σt>0, 0<w(t)<∞T_ int=\t: _t>0,\ _t>0,\ 0<w(t)<∞\. Assume ρ(int)>0ρ(T_ int)>0, regular conditional probabilities exist, the conditional second moments are finite, and the density and conditional-expectation operations in Appendix D are valid. Then Δrepx(E)=0⟺P(X∣Z)=P(X∣C)almost surely⟺I(X;C∣Z)=0. _ rep^x(E)=0 P(X Z)=P(X C)\ almost surely I(X;C Z)=0. (27) The positive-measure condition on intT_ int is important. If time is sampled continuously, merely naming one interior point of zero sampling mass is insufficient. Uniform, log-SNR, Beta, clipped-velocity, and exact-velocity schemes share the same ideal zero set when their effective measure assigns positive mass to interior times and the risk is finite. Proof idea. If Δrepx(E)=0 _ rep^x(E)=0, Fubini’s theorem gives at least one interior t∗t_* for which mC(y,t∗,c)=mZ(y,t∗,z)m_C(y,t_*,c)=m_Z(y,t_*,z) almost everywhere. With Y=αt∗X+σt∗s0εY= _t_*X+ _t_*s_0 , Gaussian differentiation gives ∇ylogpt∗(y∣c)=αt∗mC(y,t∗,c)−yσt∗2s02,∇ylogpt∗(y∣z)=αt∗mZ(y,t∗,z)−yσt∗2s02. _y p_t_*(y c)= _t_*m_C(y,t_*,c)-y _t_*^2s_0^2, _y p_t_*(y z)= _t_*m_Z(y,t_*,z)-y _t_*^2s_0^2. (28) The two positive smooth densities have the same score and hence are equal after normalization. Their characteristic functions satisfy φY|c(ω)=φX|c(αt∗ω)exp(−12σt∗2s02‖ω‖2), _Y c(ω)= _X c( _t_*ω) \! (- 12 _t_*^2s_0^2 ω ^2 ), (29) and analogously given z. The Gaussian factor is nowhere zero and αt∗>0 _t_*>0, so equality of the noisy laws implies equality of P(X∣C=c)P(X C=c) and P(X∣Z=z)P(X Z=z). The reverse implication follows because posterior sufficiency makes the two Bayes denoisers identical. Appendix D gives the measure-level details. Corollary 3.7 (Zero-set equivalence with conditional KL). Under Proposition 3.2 and Theorem 3.6, E:infQ(E,Q)=0=E:Δrepx(E)=0. \E: _QK(E,Q)=0 \= \E: _ rep^x(E)=0 \. (30) This does not imply ℛw=R_w=K, equality of their numerical values, or monotone KL decrease during parameter optimization. Corollary 3.8 (Collapse and posterior equivalence). If I(X,C)>0I(X;C)>0, a constant encoder cannot satisfy Eq. (27). More generally, if two observations c1c_1 and c2c_2 induce different posteriors, a sufficient deterministic encoder cannot merge them except on null sets. The coarsest equivalence permitted by the theory is c1∼c2⟺P(X∣C=c1)=P(X∣C=c2).c_1 c_2 P(X C=c_1)=P(X C=c_2). (31) Corollary 3.8 gives an ideal anti-collapse statement, not an optimization guarantee. It also replaces unconditional view invariance with augmentation-adaptive sufficiency: weakly and strongly corrupted views of the same sample may appropriately have different representations because they specify different posterior uncertainty. 3.5 Endpoint correctness and the joint ideal optimum Fix an encoder and define the representation-conditioned probability path pt(xt∣z)=∫pt(xt∣x)p(x∣z)x.p_t(x_t z)= p_t(x_t x)p(x z)\,dx. (32) Its marginal field is exactly uZ(xt,t,z)=[Ut∣Xt=xt,t,Z=z]u_Z(x_t,t,z)=E[U_t X_t=x_t,t,Z=z]. For the affine path, the Bayes clean predictor and field are related by uZ(xt,t,z)=mZ(xt,t,z)−(1−σmin)xtσt.u_Z(x_t,t,z)= m_Z(x_t,t,z)-(1- _ )x_t _t. (33) Theorem 3.9 (Conditional endpoint correctness). Fix E. Suppose the learned field equals uZu_Z almost everywhere, the conditional ODE x˙t=uZ(xt,t,z) x_t=u_Z(x_t,t,z) has an appropriate unique flow, and its initial law is P0=(0,s02I)P_0=N(0,s_0^2I). Then the ODE marginal at every regular time equals pt(⋅∣z)p_t(· z). If σmin=0 _ =0, its endpoint satisfies QFM(X∣Z=z)=P(X∣Z=z).Q_ FM(X Z=z)=P(X Z=z). (34) If in addition Δrepx(E)=0 _ rep^x(E)=0, then QFM(X∣E(C))=P(X∣C). Q_ FM(X E(C))=P(X C). (35) The two equalities in Theorem 3.9 separate the responsibilities of the decoder and encoder. Exact field fitting gives the posterior available through Z; zero representation deficiency makes that posterior equal to the one specified by C. Generation and representation learning are synchronized because failure of either component appears as a distinct term in the same population risk. Clean sample X ⟶ stochastic observation C=A(X,ΞA)C=A(X, _A) ⟶ encoder EθE_θ ⟶ representation Z Independent source X0X_0 and clean target X ⟶ path state Xt=αtX+σtX0X_t= _tX+ _tX_0 (Xt,t,Z)(X_t,t,Z) ⟶ flow decoder DϕD_φ ⟶ X X ⟶ weighted clean-prediction loss ⟶ joint update of (θ,ϕ)(θ,φ) At inference: (C,X0)(C,X_0) ⟶ (Z,conditional ODE)(Z,conditional ODE) ⟶ a draw from QFM(X∣Z)Q_ FM(X Z) Figure 1: Drift Variation autoencoder uses one fixed sample-based objective to update the representation encoder and conditional flow decoder. The clean target is used only during training; source noise supplies posterior diversity at inference. 3.6 Multimodal posterior Flow Matching Let a paired multimodal sample be =(X(1),…,X(M))∈ℝd1×⋯×ℝdMX=(X^(1),…,X^(M)) ^d_1×·s×R^d_M. The observation C may contain stochastic corruptions of any subset of modalities together with a modality-presence pattern S. For example, it can contain visible image patches and audio while hiding depth and text. A multimodal encoder produces =E()Z=E(C), and the target is the joint posterior P(∣)P(X ) rather than independent modality-wise marginals. Use a full-rank Gaussian source ∼(0,I) (0,I) independent of (,)(X,C) and a block path t=t+t,X_t=A_tX+S_t , (36) where the simplest choice is t=αtIA_t= _tI and t=σts0IS_t= _ts_0I. Modality-specific schedules are allowed when tA_t is invertible and tt⊤S_tS_t is positive definite on an interior set of positive sampling mass. Corollary 3.10 (Joint multimodal sufficiency). Assume the conditions of Theorem 3.6 on the product space, replace the scalar schedules by Eq. (36), and measure clean-prediction error with any positive-definite modality weight matrix. If the effective time measure gives positive mass to points where tA_t is invertible and tt⊤S_tS_t is positive definite, then Δrepmm(E)=0⟺P(∣)=P(∣)almost surely. _ rep m(E)=0 P(X )=P(X ) surely. (37) With an exact field and zero-noise endpoint, conditional ODE sampling returns the complete joint posterior. Changing which modalities are observed changes C but not the objective, yielding reconstruction, cross-modal completion, any-subset generation, and unconditional generation as instances of one model. The proof is the proof of Theorem 3.6 in the concatenated space. The noisy characteristic function is multiplied by the nonzero factor exp(−ω⊤t⊤ω/2) (-ω S_tS_t ω/2), while invertibility of tA_t recovers every frequency of the joint clean law. Crucially, this identifies cross-modal dependencies and not only each marginal distribution. For an any-subset implementation, the observation mask and Flow target must therefore have different roles. If S is observed, only the modalities in S enter the encoder condition, but the clean target remains the complete tuple X for every S. All modality streams receive independent source noise, enter the joint noisy state tX_t, contribute through a positive-definite loss, and evolve during ODE sampling. Visible modalities are generated and reconstructed; missing modalities are completed. The decoder receives the standard noisy Flow state and Z, never a separate copy of the raw clean observation. Training only the complement ScX_S^c is a valid completion-only ablation, but it identifies P(Sc∣)P(X_S^c ) rather than the full joint posterior asserted by Corollary 3.10. The corollary is modality-agnostic at the probability level but not representation-free at the implementation level. Images, audio, video, depth, actions, and continuous text embeddings may use modality-specific tokenizers and output heads with a shared fusion backbone. Raw discrete text requires dequantization, a continuous latent codec, or a discrete Flow Matching construction with a separate identifiability proof; the Gaussian theorem should not be quoted unchanged for categorical tokens. 3.7 Training and sampling Table 1: Drift Variation autoencoder training with an affine Gaussian path. Every prose step denotes a minibatch operation. Input: encoder EθE_θ, clean predictor DϕD_φ, augmentation law A, time law ρ, weight w 1 Sample a clean example X and augmentation randomness, then form C=A(X,ΞA)C=A(X, _A). 2 Encode only the observation: Z=Eθ(C)Z=E_θ(C). 3 Sample X0∼(0,s02I)X_0 (0,s_0^2I) independently and t∼ρt ρ. 4 Form σt=1−(1−σmin)t _t=1-(1- _ )t and Xt=tX+σtX0X_t=tX+ _tX_0. 5 Predict X^=Dϕ(Xt,t,Z) X=D_φ(X_t,t,Z) and compute w(t)‖X^−X‖2w(t) X-X ^2. 6 Backpropagate jointly through DϕD_φ and EθE_θ; no stopped target or external teacher is required by the theory. Output: jointly trained representation encoder and posterior flow decoder At sampling time, encode the available observation once, draw X0∼(0,s02I)X_0 (0,s_0^2I), and integrate dXtdt=Dϕ(Xt,t,Z)−(1−σmin)Xtσt dX_tdt= D_φ(X_t,t,Z)-(1- _ )X_t _t (38) from t=0t=0 to t=1t=1. Different source draws produce different posterior samples. For σmin=0 _ =0, network evaluations remain at t<1t<1 because Eq. (38) has a parameterization singularity at the endpoint; a numerical solver may integrate the final step to 11 without querying the field exactly there. 4 Controlled Validation The theory is a population statement. The evaluation tests its distinctive finite-model predictions on a distribution whose conditional factors and exact sampling oracle are known. The goal is not to simulate the semantic complexity of real image, text, and audio, but to distinguish representation use, marginal completion, and coherent joint posterior sampling without relying on a learned evaluator. 4.1 CrossGeom-4 benchmark CrossGeom-4 contains four independent continuous factors u0,uAB,uAC,uBCu_0,u_AB,u_AC,u_BC, each sampled from an equiprobable two-component Gaussian mixture with means ±1.5± 1.5 and standard deviation 0.250.25. Three modalities expose overlapping factor subsets: A:(u0,uAB,uAC),B:(u0,uAB,uBC),C:(u0,uAC,uBC).A:(u_0,u_AB,u_AC), B:(u_0,u_AB,u_BC), C:(u_0,u_AC,u_BC). (39) Observing one modality leaves exactly one factor absent from the condition but shared by the other two modalities. Correct joint generation must sample that factor from its conditional law and use the same draw in both outputs. Observing two modalities determines every factor of the third. The benchmark uses three coordinate levels. L1 applies fixed orthogonal transforms to Euclidean views; L2 adds an invertible elementwise nonlinearity; and L3 keeps A in ℝ3R^3, embeds B on 3S^3 through inverse stereographic coordinates, and embeds C on the upper hyperboloid ℍ3H^3. L3 is an extrinsic baseline: Flow Matching operates in ambient Euclidean coordinates and projects only the endpoint. No claim is made of intrinsic Riemannian transport. Model and full-tuple objective. A three-layer shared masked Transformer encoder with width 96 maps the visible modalities to one global token. A four-layer decoder receives time, the encoded condition, and noisy streams for all three modalities. The observation mask controls only encoder visibility. For every one of the eight masks, A, B, and C all receive independent Gaussian source noise, all contribute a dimension-normalized clean-prediction loss, and all are integrated during ODE sampling. Thus an A-only query generates (A,B,C)(A,B,C), including reconstruction of the visible A stream. The decoder receives no separate clean-condition bypass. The joint decoder applies self-attention across all target tokens. The matched independent baseline uses one decoder per modality and therefore cannot couple residual source randomness across outputs through target-target attention. It retains the same encoder, hidden width, depth, full-tuple loss, and visibility schedule. Protocol and metrics. The protocol balances all eight visibility patterns and trains for 5,000 updates with batch size 256, learning rate 3×10−43× 10^-4, clipped σt−2 _t^-2 weighting with cap 100, and σmin=0 _ =0. Evaluation uses a 32-step Euler solver, 128 held-out conditions, 16 samples per condition, and three seeds for every level/decoder pair, for 18 runs in total. The evaluation linearly probes every factor from every nonempty visible subset. Factor-space MAE measures reconstruction of visible streams and prediction of factors known from the condition. With one visible modality, target-target MAE measures whether the two other outputs agree on their shared unknown factor. The evaluation compares the latter factor’s marginal with the exact conditional oracle using one-dimensional Wasserstein distance (W1) and records whether repeated samples cover both mixture signs. A shuffled-condition test keeps the ODE source noise fixed. Unconditional evaluation reports all 16 sign modes, TV distance to the uniform mode law, and cross-modality factor disagreement. Appendix G gives complete mean ± standard-deviation tables. 4.2 Results The encoder is used and exposes observed factors. Across all levels and both decoder types, mean observed-factor probe R2R^2 is 0.99900.9990-0.99920.9992. For an A-only condition, the unavailable uBCu_BC factor has R2R^2 between −0.023-0.023 and −0.016-0.016, as expected under no leakage. With the same initial ODE noise, shuffling the joint model’s valid conditions increases conditional error by 13.5×13.5×-15.7×15.7×. The decoder therefore uses condition-specific information carried by the encoder rather than solving the task from the noisy Flow state alone. All observed subsets generate the complete tuple. The joint model’s visible-stream factor MAE, averaged over all seven nonempty observation masks, is 0.09120.0912-0.10410.1041. Its condition-to-target MAE is 0.10550.1055-0.15830.1583 for single-input directions and 0.09230.0923-0.11960.1196 for double-input directions. The all-visible ABC query also reconstructs all three generated streams, with MAE 0.08170.0817, 0.08180.0818, and 0.09440.0944 on L1-L3. These measurements directly distinguish the implementation from a complement-only objective in which visible target streams would be inactive. Table 2: Joint coupling under full-tuple Flow Matching. Conditional MAE is similar for matched independent (Ind.) and joint decoders. Joint target attention specifically reduces disagreement and distribution error for the unknown factor shared by two outputs. Values are three-seed means; complete standard deviations appear in Appendix G. Single-input conditional MAE Unknown target-target MAE Unknown-factor W1 Uncond. joint Level Ind. Joint Ind. Joint Reduction Ind. Joint Reduction reduction L1 .1166 .1176 1.6397 .1186 92.8% .6776 .2067 69.5% 88.8% L2 .1145 .1154 1.6278 .1201 92.6% .6878 .2669 61.2% 89.4% L3 .1310 .1338 1.5651 .1542 90.1% .7143 .1648 76.9% 86.6% Marginal completion is not joint posterior sampling. Table 2 isolates the central claim. Independent and joint decoders have nearly identical known-factor accuracy, and both recover both signs of the residual mixture for essentially every context. Nevertheless, independent outputs assign incompatible values to the shared unknown factor. Joint attention reduces that disagreement by 90.190.1-92.8%92.8\% and reduces its W1 error by 61.261.2-76.9%76.9\%. The effect therefore concerns posterior coupling, not an easier deterministic prediction or better mode coverage in either marginal alone. Figure 2: Known-factor conditional MAE, disagreement on the jointly sampled unknown factor, and unconditional TV-to-uniform. Joint and independent decoders have similar conditional accuracy, but only joint target attention coordinates residual uncertainty. Error bars are standard deviations across the three seeds, computed with denominator n. Unconditional generation remains the limiting case. Both decoder types reach all 16 sign modes. Joint decoding reduces unconditional cross-modality factor disagreement by 86.686.6-89.4%89.4\%, from 1.5981.598-1.6671.667 to 0.1770.177-0.2130.213. Its TV-to-uniform remains 0.0860.086-0.1080.108, however, compared with a finite-sample reference of about 0.0340.034. The independent baseline has lower mode-frequency TV but fails joint factor consistency. Hence the strongest evidence is conditional: the same checkpoint exposes observed factors, uses the encoder condition, reconstructs visible targets, and couples residual uncertainty. L3 manifold-constraint residuals are below 3×10−73× 10^-7 after endpoint projection, verifying valid outputs but not intrinsic manifold Flow Matching. 5 Limitations The main equivalence is a population and realizability result. It does not show that stochastic gradient descent reaches a global optimum, that a finite encoder-decoder pair realizes the Bayes denoiser, or that a small empirical loss implies a quantitative bound on I(X;C∣Z)I(X;C Z). In finite capacity, representation deficiency and decoder approximation interact, and the easiest representation for a decoder need not be sufficient. Posterior sufficiency is not minimal sufficiency or human semantic quality. A sufficient code may preserve instance identity, texture, or any detail that changes P(X∣C)P(X C). Observation design and bottleneck structure determine which information is emphasized. CrossGeom-4 verifies factor accessibility and joint sampling in a deliberately low-dimensional system; it is not evidence of transfer to natural images, text, or audio. A real multimodal claim requires modality-specific tokenizers, substantially larger models, semantic and dense probes, and standard generative metrics. The proof uses additive full-rank Gaussian corruption at a positively weighted interior time and injectivity of Gaussian convolution. Singular paths, non-Gaussian sources, discrete tokens, and intrinsic manifold-valued flows need separate identifiability arguments. CrossGeom-4 L3 uses ambient Euclidean Flow Matching followed by endpoint projection, not Riemannian Flow Matching. Raw categorical text likewise requires dequantization, a continuous codec, or a discrete-flow analysis. Endpoint correctness assumes an exact field, a regular conditional ODE, and controlled integration. Finite-step solvers, the clean-prediction parameterization singularity at t=1t=1, codecs, clipping, and projection add errors outside the theorem. Existing finite-error KL results require further smoothness and field regularity [25]; this work does not provide a general finite-risk conditional-KL or mutual-information bound. The method also requires a condition-independent source and a genuine encoder bottleneck. If the decoder receives raw observations, crop metadata, a condition-dependent initialization, or another informative side channel, the sufficiency claim applies to the combined condition. In the full-tuple implementation, the decoder receives noisy Flow states for every modality but no separate clean visible stream; the visibility mask affects only the encoder. Finally, finite data can under-sample high-entropy modes. This appears in the results: all 16 unconditional modes are reached, but their frequency TV remains above the finite-sample reference even when joint consistency improves by nearly an order of magnitude. 6 Conclusion Drift Variation autoencoder starts from one statistical object: the posterior of clean data given a stochastic observation. Conditional KL makes the desired decomposition explicit, while Conditional Flow Matching makes it sample based. Under an encoder bottleneck, the clean-prediction risk separates path variance, representation deficiency, and model approximation. For an affine Gaussian path, its representation term vanishes exactly when P(X∣Z)=P(X∣C)P(X Z)=P(X C), and an exact conditional field transports common noise to that posterior. Generation and representation learning are synchronized in this precise zero-set and endpoint sense, not because Flow Matching numerically equals KL. The product-space extension requires the complete multimodal tuple to remain the Flow target under every observation mask. CrossGeom-4 supports the resulting finite-model mechanism: the encoder linearly exposes observable factors, condition shuffling causes an order-of-magnitude degradation, visible modalities are reconstructed as active targets, and joint attention reduces unobserved shared-factor disagreement by 9090-93%93\% relative to independent decoders without improving the easier deterministic conditional metric. Unconditional mode imbalance and the synthetic scale keep the claim narrow. The next empirical step is therefore not a stronger theorem but a matched natural-data evaluation of whether the same posterior-sufficient bottleneck also yields semantic and dense transfer. AI Use Statement AI tools were used to assist with language polishing and the narrative construction of this manuscript. 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Equalities between conditional laws and conditional expectations are understood up to the corresponding null sets. Because Z=E(C)Z=E(C) is deterministic, σ(Z)⊆σ(C)σ(Z) σ(C) and X→C→ZX→ C→ Z is a Markov chain. Regular conditional probabilities are used so that expressions such as P(X∈B∣C=c)P(X∈ B C=c) and P(X∈B∣Z=z)P(X∈ B Z=z) can be chosen as measurable kernels. The main text uses the following operation-level assumptions rather than a maximally sharp analytic package. 1. Conditional laws. The data and representation spaces are standard Borel spaces, so regular conditional laws exist; in Theorem 3.6, the target X takes values in ℝdR^d and has the conditional moments needed by the displayed squared losses. 2. Independent Gaussian source. X0=s0εX_0=s_0 with s0>0s_0>0, ε∼(0,Id) (0,I_d), and ε⟂(X,C) \!\!\! (X,C). This guarantees a common source for all conditions and a strictly positive smooth interior density. 3. Schedules. αt _t and σt _t are differentiable on the time interval used to define the velocity. For exact posterior endpoints, α0=0 _0=0, σ0=1 _0=1, α1=1 _1=1, and σ1=0 _1=0. 4. Interior support. The time variable t∼ρt ρ is sampled independently of (X,C,X0)(X,C,X_0). The set on which αt>0 _t>0, σt>0 _t>0, and the effective loss weight is positive and finite has strictly positive ρ measure. This condition is stronger and more appropriate for a continuous time law than requiring one named point that may have zero mass. 5. Interchange and score regularity. Gaussian convolution may be differentiated under the conditional integral, Fubini and tower-property manipulations are valid, and equality of the resulting smooth scores identifies normalized densities. Standard finite-moment and domination assumptions are sufficient for the Gaussian formulas used here. 6. Flow regularity. Whenever equality of a vector field is used to infer equality of ODE marginals, the continuity equation and conditional ODE possess the required existence and uniqueness, and boundary or integrability conditions justify the manipulations. 7. Realizability when stated. Claims involving exact Bayes predictors or exact endpoint distributions assume the decoder family can realize the corresponding conditional expectation or velocity. The algebraic risk decompositions themselves do not require realizability. The conditional KL in Eq. (7) may be interpreted with densities relative to a common dominating measure or directly as a KL between probability kernels. Proposition 3.2 applies whenever the chain rule is valid, with extended-real values handled in the usual way. The equivalence with conditional mutual information uses I(X;C∣Z)=CKL(P(X∣C)∥P(X∣Z)),I(X;C Z)=E_CKL\! (P(X C)\,\|\,P(X Z) ), (40) which follows because P(X∣C,Z)=P(X∣C)P(X C,Z)=P(X C) for deterministic Z=E(C)Z=E(C). The sample CFM target UtU_t is fixed by (X,X0,t)(X,X_0,t) and does not depend on the current encoder. The representation-conditioned Bayes field uZu_Z does depend on the encoder through the sigma-field generated by Z. This distinction permits exact projection decompositions while preventing an interpretation of encoder training as direct KL gradient descent. If the decoder receives an additional deterministic side channel B=B(C)B=B(C), every theorem remains valid after replacing Z by the combined representation R=(Z,B)R=(Z,B). A claim about Z alone then requires the stronger condition that B does not carry additional posterior information or that the bypass is removed. Appendix B Proof of the Conditional KL Decomposition Proof of Proposition 3.2. Because Z=E(C)Z=E(C), insert the true representation-conditioned posterior between the numerator and denominator of Eq. (7): logdP(X∣C)dQ(X∣Z) dP(X C)dQ(X Z) =logdP(X∣C)dP(X∣Z)+logdP(X∣Z)dQ(X∣Z). = dP(X C)dP(X Z)+ dP(X Z)dQ(X Z). (41) Taking expectation under the joint law of (X,C,Z)(X,C,Z) gives (E,Q) (E,Q) =CKL(P(X∣C)∥P(X∣Z))+X,ZlogdP(X∣Z)dQ(X∣Z) =E_CKL\! (P(X C)\,\|\,P(X Z) )+E_X,Z dP(X Z)dQ(X Z) =I(X;C∣Z)+ZKL(P(X∣Z)∥Q(X∣Z)). =I(X;C Z)+E_ZKL\! (P(X Z)\,\|\,Q(X Z) ). (42) The first equality uses Eq. (40); the second term is obtained by conditioning on Z. Nonnegativity shows that an unrestricted generator is minimized by Q(⋅∣Z)=P(⋅∣Z)Q(· Z)=P(· Z), giving Eq. (9). Conditional mutual information is zero exactly when X⟂C|ZX \!\!\! C Z, which is equivalent to Eq. (6) because Z is a function of C. ∎ Proposition 3.2 also clarifies why minimality is absent. If E is sufficient, augmenting Z with any deterministic statistic of C keeps the first term zero and does not worsen the unrestricted generator optimum. Minimality would require a separate ordering, dimension, entropy, rate, or bottleneck criterion. Appendix C Conditional Path and Risk Decompositions C.1 Marginalizing endpoint bridges For a fixed endpoint X=xX=x, let pt(⋅∣x)p_t(· x) satisfy the continuity equation with a bridge velocity ut(⋅∣x)u_t(· x): ∂tpt(xt∣x)+∇xt⋅(pt(xt∣x)ut(xt∣x))=0. _tp_t(x_t x)+ _x_t· (p_t(x_t x)u_t(x_t x) )=0. (43) Conditioning on C=cC=c and integrating over P(dx∣c)P(dx c) yields ∂tpt(xt∣c) _tp_t(x_t c) =−∇xt⋅∫pt(xt∣x)ut(xt∣x)p(dx∣c) =- _x_t· p_t(x_t x)u_t(x_t x)p(dx c) =−∇xt⋅(pt(xt∣c)∫ut(xt∣x)pt(xt∣x)p(dx∣c)pt(xt∣c)). =- _x_t· (p_t(x_t c) u_t(x_t x) p_t(x_t x)p(dx c)p_t(x_t c) ). (44) The integral in parentheses is [Ut∣Xt=xt,t,C=c]=uC(xt,t,c)E[U_t X_t=x_t,t,C=c]=u_C(x_t,t,c), proving Eq. (17). For any square-integrable v=v(Xt,t,C)v=v(X_t,t,C), ‖v−Ut‖2 v-U_t ^2 =‖v−uC‖2+‖uC−Ut‖2+2⟨v−uC,uC−Ut⟩. =E v-u_C ^2+E u_C-U_t ^2+2E v-u_C,u_C-U_t . (45) The cross term is zero because [Ut−uC∣Xt,t,C]=0E[U_t-u_C X_t,t,C]=0, proving Eq. (18). C.2 Nested projection for the velocity field Because Z=E(C)Z=E(C), the sigma-fields are nested: σ(Xt,t,Z)⊆σ(Xt,t,C).σ(X_t,t,Z) σ(X_t,t,C). (46) The tower property gives uZ=[uC∣Xt,t,Z]u_Z=E[u_C X_t,t,Z]. Decompose Ut−v=(Ut−uC)+(uC−uZ)+(uZ−v).U_t-v=(U_t-u_C)+(u_C-u_Z)+(u_Z-v). (47) The first difference is orthogonal in L2L_2 to every function of (Xt,t,C)(X_t,t,C). The second is orthogonal to every function of (Xt,t,Z)(X_t,t,Z). Consequently all three pairwise cross terms vanish, which proves Theorem 3.4. The representation term can also be written as an expected conditional variance: Δrepv(E)=Var(uC(Xt,t,C)∣Xt,t,Z), _ rep^v(E)=EVar\! (u_C(X_t,t,C) X_t,t,Z ), (48) where the variance denotes the trace of the conditional covariance. This form emphasizes that the loss is incurred when distinct condition-level fields remain possible after conditioning on Z. C.3 Nested projection for clean prediction For a time-only weight w(t)≥0w(t)≥ 0, define the weighted inner product ⟨f,g⟩w=[w(t)f⊤g] f,g _w=E[w(t)f g]. Conditional expectation remains an orthogonal projection on all times where the weight is positive because the weight is measurable with respect to t. The tower property gives mZ=[mC∣Xt,t,Z].m_Z=E[m_C X_t,t,Z]. (49) Expanding X−D=(X−mC)+(mC−mZ)+(mZ−D)X-D=(X-m_C)+(m_C-m_Z)+(m_Z-D) (50) under the weighted inner product eliminates the cross terms and proves Theorem 3.5. In particular, Δrepx(E)=[w(t)Var(mC(Xt,t,C)∣Xt,t,Z)]. _ rep^x(E)=E\! [w(t)Var\! (m_C(X_t,t,C) X_t,t,Z ) ]. (51) C.4 Clean prediction and affine velocity For a=1−σmina=1- _ , the affine path satisfies Xt=σtX0+tX_t= _tX_0+tX and Ut=X−aX0U_t=X-aX_0. Eliminating X0=(Xt−tX)/σtX_0=(X_t-tX)/ _t gives Ut=X−aXtσt.U_t= X-aX_t _t. (52) If D induces v=(D−aXt)/σtv=(D-aX_t)/ _t, then ‖v−Ut‖2=1σt2‖D−X‖2. v-U_t ^2= 1 _t^2 D-X ^2. (53) Taking the conditional expectation of UtU_t given (Xt,t,Z)(X_t,t,Z) also yields Eq. (33). Thus an exact clean-prediction Bayes rule produces the exact representation-conditioned marginal velocity wherever σt>0 _t>0. Appendix D Proof of Gaussian Flow Representation Sufficiency Theorem 3.6 is proved through three lemmas. Fix an interior time and write Y=αX+βε,α>0,β=σs0>0,ε∼(0,Id).Y=α X+β , α>0, β=σ s_0>0, (0,I_d). (54) Lemma D.1 (Conditional Gaussian denoiser identity). Under the interchange assumptions in Appendix A, for almost every condition c, ∇ylogpY(y∣c)=α[X∣Y=y,C=c]−yβ2. _y p_Y(y c)= [X Y=y,C=c]-yβ^2. (55) The analogous identity holds after conditioning on Z=zZ=z. Proof. The conditional noisy density is the Gaussian convolution pY(y∣c)=∫φβ(y−αx)P(x∣c),p_Y(y c)= _β(y-α x)P(dx c), (56) where φβ _β is the density of (0,β2Id)N(0,β^2I_d). Differentiating under the integral gives ∇ypY(y∣c) _yp_Y(y c) =∫−y−αxβ2φβ(y−αx)P(dx∣c) = - y-α xβ^2 _β(y-α x)P(dx c) =pY(y∣c)α[X∣Y=y,C=c]−yβ2. =p_Y(y c) [X Y=y,C=c]-yβ^2. (57) Gaussian convolution makes pY(y∣c)p_Y(y c) strictly positive, so division proves the claim. This is the conditional form of the denoiser-score identity underlying denoising score matching. ∎ Lemma D.2 (A Gaussian denoiser identifies the noisy law). Let p and q be strictly positive smooth normalized densities on connected ℝdR^d. If their Gaussian posterior-mean functions in Lemma D.1 agree almost everywhere, then p=qp=q almost everywhere. Proof. Lemma D.1 turns equality of posterior means into ∇logp=∇logq∇ p=∇ q almost everywhere. Hence the weak gradient of logp−logq p- q is zero and this difference is constant on connected ℝdR^d. Therefore p=eKqp=e^Kq for a constant K. Normalization forces eK=1e^K=1. ∎ Lemma D.3 (Gaussian smoothing is injective). If the conditional laws of Y=αX+βεY=α X+β agree under two conditions and α,β>0α,β>0, then the corresponding conditional laws of X agree. Proof. For a conditional law μ of X, the characteristic function of Y is φY(ω)=φμ(αω)exp(−12β2‖ω‖2). _Y(ω)= _μ(αω) \! (- 12β^2 ω ^2 ). (58) The Gaussian multiplier never vanishes. Equality of the Y laws therefore gives equality of φμ(αω) _μ(αω) for every ω. Since α>0α>0, these arguments cover ℝdR^d, and uniqueness of characteristic functions identifies the two clean laws. ∎ Proof of Theorem 3.6. Suppose first that Δrepx(E)=0 _ rep^x(E)=0. The integrand in Eq. (26) is nonnegative. Because the effective interior set intT_ int has positive ρ measure, Fubini’s theorem provides an interior time t∗t_* such that mC(Xt∗,t∗,C)=mZ(Xt∗,t∗,Z)m_C(X_t_*,t_*,C)=m_Z(X_t_*,t_*,Z) (59) almost surely. For almost every c and z=E(c)z=E(c), the Gaussian conditional density of Xt∗X_t_* is positive on all of ℝdR^d, so Eq. (59) gives equality of the two denoiser functions for Lebesgue-almost every state. Lemmas D.1 and D.2 imply P(Xt∗∣C=c)=P(Xt∗∣Z=z).P(X_t_* C=c)=P(X_t_* Z=z). (60) Lemma D.3 then gives P(X∣C=c)=P(X∣Z=z)P(X C=c)=P(X Z=z) for almost every condition, proving posterior sufficiency. Conversely, suppose P(X∣C)=P(X∣Z)P(X C)=P(X Z) almost surely. This is equivalent to X⟂C|ZX \!\!\! C Z. Because XtX_t is obtained from X and independent source noise, the conditional joint kernel factors as P(dc,dx,dxt∣z,t)=P(dc∣z)P(dx∣z)P(dxt∣x,t).P(dc,dx,dx_t z,t)=P(dc z)P(dx z)P(dx_t x,t). (61) It follows that P(X∣Xt,t,C,Z)=P(X∣Xt,t,Z)P(X X_t,t,C,Z)=P(X X_t,t,Z) and therefore mC=mZm_C=m_Z almost surely. Substitution into the definition of Δrepx _ rep^x makes it zero. Finally, the equivalence between P(X∣C)=P(X∣Z)P(X C)=P(X Z) and I(X;C∣Z)=0I(X;C Z)=0 follows from Eq. (40) and nonnegativity of KL. ∎ Why t=0t=0 alone is insufficient. If α0=0 _0=0, then Y=βεY=β contains no scaled copy of X. The denoiser reduces to [X∣C]E[X C] or [X∣Z]E[X Z], and the characteristic-function step cannot cover frequencies of X. Two different multimodal distributions can have the same mean, so equality at t=0t=0 does not imply posterior sufficiency. Appendix E Endpoint Correctness and the Multimodal Extension E.1 Proof of conditional endpoint correctness Fix E and z. Endpoint marginalization gives the path in Eq. (32). Repeating the calculation in Appendix C with C replaced by Z shows that its conditional velocity is uZ(xt,t,z)=[Ut∣Xt=xt,t,Z=z]u_Z(x_t,t,z)=E[U_t X_t=x_t,t,Z=z] (62) and that (pt(⋅∣z),uZ(⋅,t,z))(p_t(· z),u_Z(·,t,z)) satisfies the conditional continuity equation. By assumption, the learned field equals uZu_Z almost everywhere and the model ODE starts from the same common source. Uniqueness of the relevant continuity-equation or ODE flow therefore gives qt(⋅∣z)=pt(⋅∣z)q_t(· z)=p_t(· z) (63) at every regular time. With α1=1 _1=1 and σ1=0 _1=0, p1(⋅∣z)=P(X∣Z=z)p_1(· z)=P(X Z=z), proving Eq. (34). Theorem 3.6 supplies P(X∣Z)=P(X∣C)P(X Z)=P(X C) when the representation deficiency is zero, proving Eq. (35). If σmin>0 _ >0, the same argument remains valid but identifies the smoothed endpoint q1(⋅∣z)=P(X∣Z=z)∗(0,σmin2s02I).q_1(· z)=P(X Z=z)*N(0, _ ^2s_0^2I). (64) The distinction is structural rather than a numerical detail: exact clean-posterior correctness requires a zero-noise endpoint or a separate deconvolution mechanism. E.2 Proof of joint multimodal sufficiency Let ∈ℝDX ^D be the concatenation of all continuous modality latents, let =E()Z=E(C), and fix an interior time with invertible =tA=A_t and positive-definite covariance =tt⊤ =S_tS_t . Write =+,∼(0,),⟂(,).Y=AX+ η, η (0, ), η \!\!\! (X,C). (65) For a positive-definite loss matrix W, a zero multimodal representation term implies equality of conditional posterior means because v⊤Wv=0v Wv=0 only when v=0v=0. The matrix Gaussian denoiser identity is ∇ylogp(y∣)=−1([∣=y,=]−y), _y p_Y(y )= ^-1 (AE[X =y,C=c]-y ), (66) and similarly given =Z=z. Equality of posterior means therefore yields equality of noisy scores and normalized noisy densities. The conditional characteristic function is φ|(ω)=φ|(⊤ω)exp(−12ω⊤ω). _Y (ω)= _X (A ω) \! (- 12ω ω ). (67) The Gaussian factor is nonzero and ⊤A is surjective, so equality of noisy laws identifies the entire joint characteristic function of X. This proves the forward direction of Corollary 3.10; the reverse direction follows from the same conditional-independence factorization as in Theorem 3.6. This argument would fail if separate losses identified only the marginal denoisers [X(m)∣Xt(m),Z]E[X^(m) X_t^(m),Z] without conditioning on the joint noisy state. The Drift Variation autoencoder multimodal decoder receives the joint state tX_t and predicts the joint clean tuple, so the recovered score belongs to the smoothed joint density and preserves cross-modal dependence. E.3 Any-subset and unconditional branches Let S⊆1,…,MS \1,…,M\ be the set of observed modalities and include S in C. The same model then represents the family P(X(1),…,X(M)∣Am(X(m)):m∈S,S)P\! (X^(1),…,X^(M) \A_m(X^(m)):m∈ S\,S ) (68) over masks sampled during training. If S=∅S= is chosen independently of the data and produces a constant observation, Eq. (68) reduces to the unconditional joint data law. If one modality is fully observed and another is hidden, it becomes cross-modal conditional generation. If every modality is partially observed, it becomes joint multimodal completion. The observation mask is legitimate side information because it specifies which conditional distribution is intended. In contrast, passing hidden clean tokens or a learned encoding of them directly to the decoder would change the conditioning sigma-field and invalidate a claim that Z alone is sufficient. Appendix F Practical Implications and Broader Evaluation F.1 Time weighting and representation pressure For the affine path, exact velocity regression corresponds to wvel(t)=σt−2w_ vel(t)= _t^-2. With σmin=0 _ =0, this weight diverges at the endpoint. Clipping it as wclip(t)=min(σt−2,wmax)w_ clip(t)= \! ( _t^-2,w_ ) (69) changes the numerical functional but not the pointwise Bayes clean predictor wherever the weight remains positive. Uniform, clipped, and exact weighting therefore share the ideal sufficient-encoder zero set under the theorem’s support and integrability assumptions, while their finite-sample variance and optimization can differ substantially. At t=0t=0, the path state contains no clean signal and the denoiser compares only [X∣C]E[X C] with [X∣Z]E[X Z]. At an identifying interior time, the complete denoiser function identifies the posterior. Near a zero-noise endpoint, XtX_t nearly reveals X, so the observed condition-use gap can shrink even for a useful representation. Time-resolved correct/shuffled/zero losses are thus more informative than a single time-averaged number. F.2 Source noise, augmentation entropy, and bypasses The common source X0⟂CX_0 \!\!\! C both prevents condition leakage through the initial state and supplies residual sampling randomness. A condition-dependent source changes the conditioning sigma-field. Likewise, any direct crop geometry, raw clean token, calibration variable, or task identifier passed to the decoder becomes part of the representation certified by the theorem. The path state XtX_t and time are intrinsic Flow Matching inputs; a separate clean copy of the observation is not. Observation strength and flow time play different roles. Stronger masking or modality removal changes P(X∣C)P(X C) and typically raises its entropy, whereas flow time changes the signal-to-noise ratio along a fixed posterior path. A full-mask branch is unconditional only when the mask event is independent of sample content. Multiple source draws reduce Monte Carlo noise and reveal posterior diversity, but they cannot recover modes missing from the dataset. F.3 Finite capacity and endpoint numerics With an unrestricted decoder, profiling the clean-prediction risk leaves only the fixed path term and representation deficiency. With finite networks, the model term depends on the encoder and joint training can prefer a lossy but easier code. Encoder token count, decoder cross-attention, width, and depth must therefore be ablated jointly. For σmin=0 _ =0, training and integration should avoid evaluating the clean-prediction parameterization exactly at t=1t=1. Clamping intermediate states changes the continuity equation; display clipping or manifold projection should occur only after integration unless the modified dynamics are analyzed explicitly. F.4 Natural-data evaluation protocol CrossGeom-4 tests the mechanism but not semantic transfer. A natural-image evaluation should pair frozen linear, k-nearest-neighbor, few-shot, and dense probes with FID, precision/recall, condition fidelity, and repeated-sample diversity from the same checkpoint. High-entropy contexts require distributional metrics, best-of-K fidelity, and consistency with visible evidence rather than single-output MSE alone. A conditional mean can score well under MSE while missing the posterior. For multimodal systems, every observed subset should be evaluated against the complete generated tuple. Report per-modality fidelity, visible-stream reconstruction, cross-modal consistency, cross-modal retrieval, and dense probes. Repeated samples should change unobserved variables while preserving observed evidence. Marginal quality is insufficient: a compatibility metric or known factor evaluator must test whether simultaneously generated outputs share the same residual draw. Table 3: Minimum information for evaluating a Drift Variation autoencoder instance. Category Required report Observation Augmentations, strengths, mask law, content dependence, and all declared side information. Representation Encoder/token architecture, pooling, frozen and nonlinear probes, rank statistics, and checkpoint rule. Flow Source, schedules, time law, weighting/clipping, endpoint noise, solver, steps, and final-time handling. Generation Fidelity, precision/recall, condition consistency, repeated-sample diversity, and sample count. Multimodal Visible subsets, full generated targets, per-modality fidelity, visible reconstruction, and joint consistency. Ablations Shuffled/zero condition, source construction, time allocation, bottleneck size, and joint versus independent targets. No single finite-network metric validates Theorem 3.6. The empirical evidence should instead follow the decomposition: the encoder contains condition-specific predictive information, the decoder uses it, the source accounts for residual uncertainty, outputs preserve joint dependence, and representation and generation coexist at one declared checkpoint. Appendix G CrossGeom-4 Details and Complete Results G.1 Data construction and architecture For every example, the four factors follow uj∼12(−1.5,0.252)+12(1.5,0.252),j∈0,AB,AC,BC.u_j 12N(-1.5,0.25^2)+ 12N(1.5,0.25^2), j∈\0,AB,AC,BC\. (70) Each three-factor modality is mixed by a fixed seeded orthogonal matrix. L2 applies invertible asinhasinh coordinates after mixing. L3 maps the B chart to 3S^3 by inverse stereographic projection and maps the C chart to the upper hyperboloid by y↦(1+∥y∥2,y)y ( 1+ y ^2,y). Exact inverse maps permit evaluation of latent factors without a learned perceptual model. The encoder uses modality-specific scalar stems, modality and coordinate embeddings, one global token, three Transformer layers, width 96, and four attention heads. Unavailable modality tokens are padded. The decoder has four Transformer layers. The joint model self-attends across the concatenated A, B, and C target tokens; the independent baseline uses one decoder per modality. All eight availability patterns occur equally often. Availability affects only the encoder. Under every pattern, including A-only and ABC, all three modalities receive independent Gaussian source states and are clean-prediction targets. The affine path uses σmin=0 _ =0 and clipped σt−2 _t^-2 weighting with cap 100. AdamW uses learning rate 3×10−43× 10^-4, batch size 256, and 5,000 updates. Evaluation uses fixed-step Euler integration with 32 steps, 128 conditions, and 16 samples per condition. Every level/decoder setting uses seeds 0, 1, and 2. Tables report standard deviations across the three seeds, computed with denominator n=3n=3. G.2 Complete aggregate tables Table 4: Conditional factor MAE and unconditional metrics under full-tuple Flow Matching. Every conditional query generates A, B, and C; direction labels identify the visible condition, not the only active targets. Level Decoder Uncond. TV↓ Uncond. joint MAE↓ A→ ↓ B→ ↓ C→ ↓ AB→ ↓ AC→ ↓ BC→ ↓ 1 independent 0.0892 ± 0.0111 1.6462 ± 0.0460 0.1147 ± 0.0044 0.1177 ± 0.0072 0.1173 ± 0.0113 0.0962 ± 0.0029 0.0950 ± 0.0081 0.0943 ± 0.0006 1 joint 0.0990 ± 0.0128 0.1847 ± 0.0096 0.1123 ± 0.0093 0.1212 ± 0.0076 0.1193 ± 0.0064 0.0990 ± 0.0124 0.1011 ± 0.0136 0.0966 ± 0.0096 2 independent 0.0724 ± 0.0022 1.6674 ± 0.0426 0.1197 ± 0.0132 0.1132 ± 0.0130 0.1105 ± 0.0092 0.0975 ± 0.0153 0.0952 ± 0.0112 0.0831 ± 0.0116 2 joint 0.1082 ± 0.0186 0.1771 ± 0.0116 0.1055 ± 0.0040 0.1204 ± 0.0118 0.1203 ± 0.0119 0.0923 ± 0.0026 0.0968 ± 0.0009 0.1061 ± 0.0202 3 independent 0.0542 ± 0.0132 1.5977 ± 0.0220 0.1260 ± 0.0018 0.1543 ± 0.0112 0.1126 ± 0.0064 0.1150 ± 0.0083 0.0952 ± 0.0006 0.1072 ± 0.0090 3 joint 0.0861 ± 0.0172 0.2133 ± 0.0164 0.1295 ± 0.0178 0.1583 ± 0.0143 0.1137 ± 0.0022 0.1196 ± 0.0156 0.1196 ± 0.0200 0.1127 ± 0.0079 Table 5: Representation, visible reconstruction, uncertainty, and condition-use evidence. “Both modes” is the fraction of contexts whose 16 samples contain both signs of the shared unknown factor. “Shuffle” is shuffled-condition MAE divided by matched-condition MAE with identical initial ODE noise. Level Decoder Probe R2↑R^2 Random R2 Visible recon.↓ Single cond.↓ Double cond.↓ Unknown joint↓ Unknown W1↓ Both modes↑ Shuffle ×↑× 1 independent 0.9991 ± 0.0001 0.8227 ± 0.0858 0.0840 ± 0.0038 0.1166 ± 0.0059 0.0951 ± 0.0033 1.6397 ± 0.0432 0.6776 ± 0.0117 1.0000 ± 0.0000 15.6921 ± 0.6424 1 joint 0.9992 ± 0.0000 0.8227 ± 0.0858 0.0918 ± 0.0066 0.1176 ± 0.0076 0.0989 ± 0.0114 0.1186 ± 0.0099 0.2067 ± 0.0539 1.0000 ± 0.0000 15.2957 ± 1.5546 2 independent 0.9992 ± 0.0000 -13.2173 ± 17.3786 0.0809 ± 0.0089 0.1145 ± 0.0104 0.0919 ± 0.0117 1.6278 ± 0.0313 0.6878 ± 0.0109 1.0000 ± 0.0000 16.4248 ± 1.9564 2 joint 0.9992 ± 0.0000 -13.2173 ± 17.3786 0.0912 ± 0.0069 0.1154 ± 0.0089 0.0984 ± 0.0078 0.1201 ± 0.0107 0.2669 ± 0.0309 1.0000 ± 0.0000 15.6814 ± 0.9940 3 independent 0.9992 ± 0.0000 0.7396 ± 0.2158 0.0910 ± 0.0050 0.1310 ± 0.0060 0.1058 ± 0.0054 1.5651 ± 0.0261 0.7143 ± 0.0133 0.9991 ± 0.0012 14.1745 ± 0.6261 3 joint 0.9990 ± 0.0001 0.7396 ± 0.2158 0.1041 ± 0.0069 0.1338 ± 0.0099 0.1173 ± 0.0122 0.1542 ± 0.0080 0.1648 ± 0.0516 1.0000 ± 0.0000 13.5004 ± 1.1064 The negative mean random-feature R2R^2 on L2 is a valid out-of-sample result, not a clipped value. Random high-dimensional features make the fixed ridge probe poorly conditioned and coordinate dependent. Trained representations, in contrast, remain stable across levels, and factors unavailable under a single observed modality remain near chance. G.3 Oracle construction and metric interpretation The conditional oracle copies every factor determined by the visible modalities and independently resamples each undetermined factor from its true mixture. It then renders the complete A/B/C tuple. Consequently, condition-to-target error has a deterministic zero oracle, whereas an undetermined factor is evaluated distributionally rather than against one arbitrary ground-truth draw. The unconditional oracle samples all four factors and provides a finite-sample TV calibration of 0.0339±0.00030.0339± 0.0003 for 2,048 samples. Both decoder structures attain essentially perfect two-sign coverage with 16 draws per context. This marginal metric cannot establish joint generation. The unknown-factor target-target MAE tests whether two output modalities use the same posterior draw, and it is the metric on which joint and independent decoders separate by roughly one order of magnitude. The W1 metric complements it by comparing the averaged recovered factor to the exact conditional marginal. Shuffling preserves the empirical support of each observed modality and reuses the same initial source noise, making it the primary condition-use test. A zero condition is also recorded in the long-form results but is an out-of-support intervention and is not used for headline effect sizes. All headline values, the aggregate JSON, and the long-form CSV are included with the source package.