Linear Propagation Assumption (LPA)

• LPA is a hypothesis asserting that local gradient updates propagate coherently across related facts through linear feature mappings.
• Its formalization utilizes relation algebra and tensor factorizations, exposing challenges with logical operations like negation, converse, and composition.
• Empirical and theoretical insights show failures in knowledge editing and multi-hop reasoning, prompting the need for higher-order and memory-based strategies.

The Linear Propagation Assumption (LPA) is a hypothesis central to understanding the limits of logical coherence in neural networks under first-order, gradient-based parameter updates. It posits that a local update intended to alter the score of a specific fact will automatically propagate coherently to all logically related facts, without requiring the resolution of a separate constraint-satisfaction problem. Recent formalization and analysis of LPA using relation algebra have revealed deep structural obstacles to its general validity in neural models, especially when implementing logical operations such as negation, converse, and composition of relations.

1. Formalization of the Linear Propagation Assumption

The LPA asserts that a small, first-order parameter change $\Delta\theta$ designed to increase the model’s score $s_\theta(q)$ for a query $q$ should systematically induce predictable changes to related queries, specifically negations and converses. The formal requirements are: $$\Delta s(\neg q) = -\Delta s(q), \qquad \Delta s(\mathrm{rev}(q)) = \Delta s(q)$$ with

$$\Delta s(q) \approx \langle \phi_q, \Delta\theta \rangle$$

where $\phi_q$ is the gradient feature for $q$. The gradient-level equivariance necessary for LPA is expressed as

$$\phi_{\neg q} = -\phi_q, \qquad \phi_{\mathrm{rev}(q)} = \phi_q$$

Maintaining linearly independent feature vectors $\{\phi_q\}$ for different logical orbits is also required to permit independent editability of unrelated facts (Chang et al., 29 Jan 2026).

2. Representation Theory and Tensor Factorization

For the algebraic operations of negation and converse, LPA’s structural requirements forcibly induce a tensor factorization of the query-feature space. Specifically, representation theory for the group $G_E \times \mathbb{Z}_2$ (permutations of entities and negation) and Maschke’s theorem imply that the feature space $W$ decomposes into direct sums of tensor products: $W \cong \bigoplus_i (C_i \otimes R_i)$ The feature map further factorizes as

$$\phi(h,r,t) = \bigoplus_i \sum_{k=1}^{m_i} [u_{i,k}(h,t) \otimes v_{i,k}(r)]$$

with $u_{i,k}(h,t)$ as entity-pair context and $v_{i,k}(r)$ as relation content. Negation acts to flip the sign of the relation content: $v_{i,k}(\neg r) = -v_{i,k}(r)$ Converse equivariance ($\phi(t,r^\smile,h) = \phi(h,r,t)$) dictates a further split into symmetric and antisymmetric blocks: $\phi_i(h,r,t) = \phi^+_i(h,r,t) + \phi^-_i(h,r,t)$ where swapping $h$ and $t$ swaps sign in the antisymmetric component (Chang et al., 29 Jan 2026).

3. Relational Composition and Bilinearity

Relational composition, in the “unique-witness” case, reduces to logical conjunction: $$(r;s)(h,t) \Longleftrightarrow \exists!\,b:\,r(h,b)\wedge s(b,t)$$ LPA here requires a feature-level binary operator $F$ such that

$\phi_{p \wedge q} = F(\phi_p, \phi_q)$

Imposing symmetry ($F(u,v) = F(v,u)$), idempotence ($F(u,u) = u$), and respect for linear dependencies, one establishes that $F$ must be a symmetric bilinear map: $$F(u,v) = \tilde F(u,v), \quad \tilde F: W^\wedge \times W^\wedge \to W^\wedge$$ This property follows from the sentential logic structure and the need for linear propagation (Chang et al., 29 Jan 2026).

4. Incompatibility of Bilinearity and Negation—Feature Collapse

A central negative result arises when combining bilinearity and the negation-equivariance demanded by LPA. Idempotence for conjunction gives

$\tilde F(u, u) = u, \quad \tilde F(-u, -u) = -u$

but bilinearity insists

$\tilde F(-u, -u) = (-1)^2 \tilde F(u, u) = u$

yielding $u = -u$ and thus $u = 0$. Consequently, all atomic feature vectors necessarily collapse to zero, making conjunction (and therefore relational composition) impossible under strict linear propagation. This collapse demonstrates the structural impossibility of achieving fully systematic logical propagation via first-order gradient updates for binary relations (Chang et al., 29 Jan 2026).

5. Empirical and Theoretical Consequences

Several observed failures in neural knowledge manipulation are explained by the impossibility results associated with LPA:

Failure Mode	Underlying Mechanism	LPA Constraint
Knowledge Editing	Feature geometry collapse	Anti-alignment under negation, bilinear edits
Reversal Curse	Converse not enforced	No systematic reverse propagation
Multi-Hop Reasoning	Composition collapse	Bilinearity incompatible with negation

Local editing approaches (e.g., ROME, MEMIT) operate in the linearized tangent space, and LPA would require impossible geometric relations among feature gradients. The reversal curse refers to the inability of updates to $p(h,t)$ to propagate systematically to $p^\smile(t,h)$. Multi-hop reasoning failures stem from the collapse of conjunction under LPA for composed relations (Chang et al., 29 Jan 2026).

6. Mitigation Strategies and Future Directions

To address the limitations inherent in LPA, several approaches have been suggested:

• Higher-order or nonlinear editing: Moves beyond tangent-space geometry, aiming for more expressive updates.
• Memory-based edits: Methods such as SERAC and MeLLo maintain corrected facts externally, routing queries to ensure coherence.
• Architectures with explicit tensor-product or binding modules: These can enforce logical symmetries by design rather than by propagation.
• Regularization of gradient features: Partially aligning $\phi_{\neg q}$ with $-\phi_q$ or $\phi_{\mathrm{rev}(q)}$ with $\phi_q$ during training can ameliorate some constraints.

A plausible implication is that models seeking true logical coherence under parameter updates must abandon the pure first-order regime, incorporate architectural inductive biases, or utilize multi-stage memory augmentation techniques (Chang et al., 29 Jan 2026).

7. Summary and Structural Insights

The Linear Propagation Assumption, while conceptually attractive for scalable knowledge editing and logical consistency, is structurally incompatible with binary relational logic when implemented using first-order gradient updates. Systematic propagation of edits through negation, converse, and composition requires tensor factorizations and bilinear mappings that are mutually incompatible under the constraints of LPA, leading to the collapse of feature maps. These findings elucidate the need for alternative strategies in neural network design for relational knowledge, indicating that progress in systematic knowledge editing and logical coherence will rely on mechanisms transcending simple linear propagation and local gradient methods (Chang et al., 29 Jan 2026).