The claim
Under a product-of-experts factorization, s₁₂ = s₁ + s₂ − s₀, which is what Composable Diffusion exploits (LIT-770). The manuscript adds that this needs conditional independence at the relevant noisy state and does not guarantee correct joint bindings. Schrödinger bridges give a path-space view of conditional transport (LIT-776).
The exact condition. Write s₀ = ∇ₓ log p(x), sᵢ = ∇ₓ log p(x|yᵢ) and s₁₂ = ∇ₓ log p(x|y₁,y₂), all at the same noisy state x. By Bayes, ∇ₓ log p(x|y) = ∇ₓ log p(x) + ∇ₓ log p(y|x), so
s₁₂ − s₁ − s₂ + s₀ = ∇ₓ log [p(y₁,y₂|x) / (p(y₁|x)p(y₂|x))].
The scores add exactly when the right side vanishes, that is, when the ratio does not vary with x on a connected region where the densities are positive and differentiable. Conditional independence given x makes the ratio 1 for every x, so it is sufficient. It is not necessary. Let y₁ and y₂ each take values in {0, 1, 2}, with p(yᵢ|x) = (σ(x)/2, σ(x)/2, 1 − σ(x)) for any smooth σ into (0, 1), and let v = (1, −1, 0), so that Σ_y v(y)p(y|x) = 0 for every x. Then
p(y₁,y₂|x) = (1 + ε v(y₁)v(y₂)) p(y₁|x) p(y₂|x), 0 < ε < 1,
is a conditional distribution with the given marginals for every x. The conditions are dependent given x (the factor is 1 + ε at y₁ = y₂ = 0), and the ratio is 1 + ε v(y₁)v(y₂) for every x, so the scores add exactly. The example is constructed; nothing here says prompts behave this way.
The manuscript’s caution therefore stands with one correction. Conditional independence is the condition under which addition is guaranteed for every pair of condition values, not the only one under which it holds, and the criterion is about one pair (y₁, y₂) at one noise level.
What it does not say
It does not say a generative failure is evidence of sheaf contextuality, and it does not identify denoising time with historical or telephone-game time. The reading of Composable Diffusion (NOTE-597) supports the manuscript’s caution: the paper derives the composition for clean data, applies it at every noise level without discussion, and loses to an energy-based baseline when composing relations.
In C7’s appendix
C7 Appendix D, omitted from the extracted manuscript, states the identity’s conditions: “The product-of-experts conditional score identity follows from the conditional independence of c1 and c2 given x_t and a shared prior p_t(x_t), subject to positivity and differentiability. It is not a generic identity for arbitrary prompt embeddings or finite-step diffusion implementations. When scores are approximate, compositional failures may reflect estimator error, model misspecification, or sampling limitations.” It adds controls: prompts paired by entities, scene type and length, randomized order and seeds, and evaluation blind to condition. The open question behind it is QUESTION-018.
A second caveat on adding scores
Ho and Salimans (LIT-839) show that learned scores are not conservative, so a guided field is in general the score of no density. That is a caveat on s₁₂ = s₁ + s₂ − s₀ beyond the condition on the ratio this claim states. Attend-and-Excite (LIT-806) documents conjunctions failing by omission (“catastrophic neglect”), and in T2I-CompBench++ (LIT-798) Composable Diffusion is again the weakest method.
Note of 2026-10-13: a second line
Under ADR-040 (group C, decided by the owner) this claim also holds
formal-apparatus. It rests on no claim, and the conditions on the score identity are mathematics about conditional generation, not about translation.
Note of 2026-10-15: split under ADR-043
ADR-043’s classification found this claim compound: that adding scores composes conditions only under conditional independence at the noisy state is a derivation (demonstrated), while that each condition can be met while their conjunction or relational binding fails is documented by benchmarks and method papers (empirical). The empirical part is now CLAIM-321, which corroborates this claim. This claim keeps the derivation and its conditions, and stays demonstrated. Its title, which led with the empirical part, was changed to state only the derivation; the first disjunct of its defeat condition, a model whose per-condition satisfaction predicts its joint binding accuracy, is CLAIM-321’s.
Note of 2026-10-25: masked composition of perspectives’ predictions
A45 of the exchange of 2026-10-24, continued, combined perspectives’ conditional noise predictions as ε* = ε_∅ + g·Σwᵢ Mᵢ (εᵢ − ε_∅) / max(1, Σwᵢ Mᵢ), a masked and normalized form of THEORY-405’s composition, with one scheduler step per timestep. For one full-mask perspective of weight at least 1 it is classifier-free guidance; where perspectives overlap with total weight at least 1 it takes the weighted mean of their differences; where the total weight is below 1 it dilutes the guidance. By this claim the combination samples a composition of the perspectives’ conditions only where the ratio condition holds. Elsewhere it is in general the score of no distribution (THEORY-611). The reply cited neither, and no real model was run.
The reading of MultiDiffusion (LIT-1522, NOTE-1158) places it. With unit weights where coverage is at least 1 and a deterministic sampler, A45’s combination is MultiDiffusion’s region-based step: a per-pixel weighted mean of the views’ noise predictions (THEORY-958). The exchange did not cite it. A45 adds the per-perspective weights and the dilution rule below total weight 1. Where the weights are normalized the combination is an average, not a sum, and this claim’s ratio condition is a condition on sums. Under THEORY-405’s idealization an average of k scores is the score of the normalized geometric mean Π pᵢ^{1/k}, not of the product (NOTE-1160). So “only where the ratio condition holds” above applies to the sum, and the average composes a different target. The claim is unchanged.