interactionDefect_RCLCombiner
Why this theorem is linked from SOCKET: SOft Collision Kernel EsTimator for Sparse Attention unclear
Pith linked this Lean declaration because the review connected a specific passage in the paper to this theorem. The relation tag says how strong that connection is; it is not a generic placeholder.
Lemma 5: Var(X) ≤ μ(1−μ) with equality iff X ∈ {0,1}
Relation between the paper passage and the cited Recognition theorem.
plain-language theorem explainer
The interaction defect of the RCL combiner with parameter c equals c u v at every pair. Branch selection arguments cite the identity to convert the RCL parameter into a coupling witness. The proof is a direct algebraic reduction that substitutes the two definitions and simplifies.
Claim. Let $P(u,v)=2u+2v+cuv$. The interaction defect satisfies $P(u,v)-P(u,0)-P(0,v)+P(0,0)=cuv$.
background
The module formalizes the Recognition Composition Law family $F(xy)+F(x/y)=2F(x)+2F(y)+c F(x)F(y)$ whose associated combiner is the polynomial $P(u,v)=2u+2v+cuv$. The interaction defect of any combiner $P$ is the bilinear form $P(u,v)-P(u,0)-P(0,v)+P(0,0)$, which vanishes if and only if $P$ is separately additive. RCLCombiner attaches the real parameter $c$ to this polynomial.
proof idea
One-line wrapper that unfolds interactionDefect and RCLCombiner then applies the ring tactic.
why it matters
The identity is invoked by RCLCombiner_isCoupling_iff and RCLCombiner_nonzero_couples. It supplies the explicit coupling witness required by the strengthened (L4*) composition consistency in RS_Branch_Selection.tex, thereby excluding the additive branch and isolating the bilinear representative J (T5).
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papers checked against this theorem (showing 5 of 5)
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MSE vector field in multiuser detection is conservative
"g / (g^T v + sigma^2) = nabla_v log(sigma^2 + g^T v) ... R_sum = log(1 + sum P_k |h_k|^2 / sigma^2) which is independent of the path"
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Log-likelihood strongly concave near true tree parameters
"Z_u = q(θ̂_v Z_v, θ̂_w Z_w) with q(x,y)=(x+y)/(1+xy); Hessian off-diagonals decay as (C9 δ)^⌊(dist(e,f)−1)/4⌋ (Prop. 3.2, Thm. 2.2)"
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Nonlinear heads escape collapse by generating negative curvature
"the pullback metric G(z) = Jh(z)⊤Jh(z) is singular ... v⊤G(z)v = 0 for v in Vaug"
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Semantic prior lets one model detect fakes across four domains
"we compute the projection of the update matrix ΔW onto the two subspaces ... outside energy ratio ... cosine similarity"
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Soft LSH collisions score tokens for sparse attention
"Lemma 5: Var(X) ≤ μ(1−μ) with equality iff X ∈ {0,1}"