RCLCombiner_isCoupling_iff
plain-language theorem explainer
The equivalence shows that the RCL polynomial combiner with parameter c is a coupling combiner precisely when c is nonzero. Branch-selection arguments in the Recognition framework cite it to exclude the additive branch of the composition-law family. The proof rewrites the coupling predicate through the interaction-defect characterization and dispatches both directions by substitution plus a test-point evaluation.
Claim. Let $P_c(u,v) := 2u + 2v + c uv$. A combiner $P$ is coupling when it is not separately additive. Then $P_c$ is coupling if and only if $c ≠ 0$.
background
The Recognition Composition Law family is $F(xy) + F(x/y) = 2F(x) + 2F(y) + c F(x)F(y)$, realized by the polynomial combiner $P(u,v) = 2u + 2v + c uv$. The module distinguishes the bilinear branch ($c ≠ 0$, representative $J(x) = ½(x + x^{-1}) - 1$) from the additive branch ($c = 0$, representative ½(ln x)²). The strengthened (L4*) requires the combiner to be coupling rather than separately additive, which forces the bilinear branch.
proof idea
The proof rewrites the statement with isCouplingCombiner_iff_interactionDefect_nonzero. The forward direction assumes a pair with nonzero defect, supposes c = 0 for contradiction, substitutes the defect formula interactionDefect_RCLCombiner and obtains zero by ring. The reverse direction applies RCLCombiner_nonzero_couples at the test point (1,1) to exhibit a nonzero defect.
why it matters
This supplies the central equivalence for the downstream branch_selection theorem, which is the Lean rendering of the branch-selection result in RS_Branch_Selection.tex. It isolates the bilinear representative J of the RCL family, aligning with the J-uniqueness step (T5) of the forcing chain and the Recognition Composition Law. Residual α-coordinate freedom is deferred to separate calibration conditions outside this operator-level statement.
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papers checked against this theorem (showing 30 of 58)
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Generalized means create infinite thermodynamic speed limits
"The TSLs with the arithmetic mean... and the one with the logarithmic mean... are the same as the ones in the previous studies [21,22]."
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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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Fused geometry and appearance metric predicts synthetic data transfer
"We define SADGE as a bilinear interaction because it is the simplest function that captures complementarity while staying monotone and low-capacity. ... SADGE = a Ĝ + b  + c Ĝ  where a, b, c ≥ 0"
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Importance sampling misallocates token credit in LLM RL
"This mismatch ... over-amplifying already high-probability tokens ... rich-get-richer dynamics"
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Free energy minimization yields convergent policy composition
"The softmax gating rule emerges exactly from the proximal operator of the entropic barrier; as ε→0 the rule recovers argmax (sparse) or Gumbel-softmax (dense) gating."
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Perturbing latents reawakens erased concepts in diffusion models
"Lorth = E [sum I[v_m^e ≠ v_d] |<v_m^e , v_d>| / (||v_m^e|| ||v_d||) + ξ]"
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Reciprocal cost yields rank-one Hessian in log coordinates
"Theorem 2.1. Let F satisfy composition law F(xy)+F(x/y)=2F(x)F(y)+2F(x)+2F(y) and unit log-curvature lim 2F(e^t)/t²=1. Then F(x)=J(x)."
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Trajectory selection gives 10x faster training and better out-of-domain web agents
"D(i, j) = max(δ(si, sj), δ(yi, yj)) with δ = 1 − BERTScore"
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Sign-aware aggregation sustains unlearning across sequential VLM requests
"sign conflict occurs at parameter dimension i when ˆτ(a)_i · ˆτ(b)_i < 0"
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Convex potential targets population-level balance in MoE
"RCLCombiner_isCoupling_iff ... branch_selection (c ≠ 0 forces bilinear branch)"
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RI rankings compete with lasso for variable selection
"CRI.Z replaces reallocation term with identity, yielding w²_G = (VrU⊤_r y)⊙(VrU⊤_r y)"
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Homotopy training lifts unrolled imaging PSNR by 2.5 dB
"g_α(x)=(1−α)½∥y−Hx∥²₂ + α½∥y_t−H_t x∥²₂ ... α:1→0 during training"
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Adaptive gate skips retrieval for 70-90 percent of RAG queries
"Lemma 1 (order-equivalence). For any strictly decreasing ϕ, thresholding U_mar at τ is equivalent to thresholding the mean gap"
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Single polynomial parametrizes all Riccati hierarchy solutions
"gi = xi xm for i < m, gm = x_m²"
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Local positivity yields turnpike in mean field games
"the symmetry property(ii) is a symmetry property that is clear within the quadratic Hamiltonian framework by identity D¯u=-D¯m/¯m... (BB*)A*=A(BB*)"
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Affordance cues and progress signals lift robot success to 91.8%
"fine-grained affordance queries comprise four subqueries <Global>, <Local>, <Spatial>, and <Dynamic>"
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f-divergence MI matrices are PSD for nonnegative power-series generators
"replica embedding turns monomial (t-1)^m into Gram matrices; nonnegative mixtures preserve PSD"
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Degree-10 polynomial divides no Newman polynomial
"solving the system of linear inequations ... 0 ≤ c_k ≤ 1 ... using the mixed-integer linear programming package Gurobi"
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Accelerated Sinkhorn reaches O(n^{7/3}ε^{-5/3}) for partial transport
"Lemma 1. Define the Lipschitz constant L = ||A||_{1→2}^2 / μ_f where μ_f = γ / (||r||_1 + ||c||_1 − s)."
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LLM posteriors form curved manifolds in representation space
"parameter posteriors are encoded as curved manifolds in representation space... Standard linear steering moves representations off-manifold, inducing unintended, coupled changes, whereas geometry-aware methods preserve the target belief family"
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Bonding descriptors raise ML accuracy for materials properties
"bonding heterogeneity (skew, kurtosis of BWDF) and effective interaction number (EIN ICOHP) reduce lattice thermal conductivity; significant MAE gains for max pfc, klat, moduli"
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Necessity-operator closures factorize fuzzy formal contexts
"we will consider the frame ([0,1],≤,&G) … ⊤-normalized context … g↑N ≤ g↑π … intervals of concepts"
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Dynamic inhibition sets neural assembly sizes
"inhibition process based on the ratio between excitatory and inhibitory neurons ... pi=0.2"
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Probability gradients plus asymmetric decay stabilize soft clipping
"Left Boundary: A positive integer power function of πθ... Right Boundary: A reciprocal radical power function of πθ... Cleft = (1−εlow)−n π−(n+1)θold ... Cright = (1+εhigh)1/m π1/m−1θold"
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Alphabet growth boosts shuffled privacy only on collapse to delta_1
"Diluting/persistent dichotomy: family diluting iff worst ν_adbd,d ⇒ δ1 (equivalently I*→0); persistent otherwise, no extra d-gain"
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Stationary velocity field makes flow matching adaptive for robots
"g⋆(a,ε,γ)=(ε−a)c(γ) with c(1)=0... transforming ground-truth action sequences into natural stationary equilibrium points"
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Bandit-guided distributed design cuts regret in black-box experiments
"Kriging Believer variance deflation σ^(j)²(x) = σ^(0)²(x) − Σ k(x,x^(l))² / (σ²(x^(l)) + σ_n²)"
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Wrapper makes any network normalization-equivariant
"WNE is parameter-free; it exactly parameterizes the NE class without constraining internal layers."
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Quadratic cost correction lifts VLA success 28.8% in dynamic scenes
"From a single quadratic cost, joint minimization yields a unified solution that decomposes orthogonally into two distinct channels. The pace channel compresses execution along the planned direction, while the path channel applies an orthogonal spatial offset"
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Web video pretraining yields strong zero-shot monocular 3D
"Multi-View Signal Proxy (MVS) ... Pt,t+1 = rH / rF ... MVS(v) = average Pt,t+1"