branch_selection
Why this theorem is linked from Cross Modality Image Translation In Medical Imaging Using Generative Frameworks 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.
The results show that GANs outperform latent generative models across all tasks, with SRGAN achieving statistically significant superiority.
Relation between the paper passage and the cited Recognition theorem.
plain-language theorem explainer
The theorem asserts that if the RCL combiner with parameter c satisfies the coupling condition, then c must be nonzero. Researchers formalizing the Recognition Science composition law cite it to exclude the additive branch in favor of the bilinear representative. The proof is a direct one-line application of the forward direction of the RCL coupling equivalence.
Claim. If the combiner $P(u,v)=2u+2v+cuv$ is not separately additive, then $c≠0$.
background
The module formalizes branch selection under the strengthened composition consistency (L4*). A combiner is coupling when it is not separately additive, so that the interaction defect is nonzero for some pair of arguments. The RCL combiner is the polynomial $P(u,v)=2u+2v+cuv$ attached to the Recognition Composition Law family with free real parameter c. The upstream theorem RCLCombiner_isCoupling_iff states that this combiner is coupling if and only if c≠0.
proof idea
The proof is a one-line wrapper that applies the forward implication of RCLCombiner_isCoupling_iff to the supplied coupling hypothesis.
why it matters
This result forces the bilinear branch of the RCL family once the coupling requirement of the strengthened (L4*) is imposed. It is invoked by the contrapositive additive_branch_not_coupling and by the certificate branchSelectionCert. In the framework it completes the operator-level selection between the two branches of the RCL family, leaving only residual α-coordinate freedom for later calibration steps.
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papers checked against this theorem (showing 30 of 1147)
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Cold-start recommendation recast as zero-shot learning
"the reconstruction part is effective in mitigating the domain shift problem... the demand for more truthful reconstruction from the attributes to behavior is generalizable across warm and cold domains"
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Re-weighted method learns intrinsic view weights for clustering
"Lemma 1 and Theorem 1 (monotonic decrease via h(t)=t^p - (p/2)t^2 + (p/2)-1)"
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Double-phase growth controls existence in nonlinear eigenvalue problems
"double-phase energy functional Pp,q(u) := ∫(|∇u|^p + a(x)|∇u|^q) dx"
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DP utility gap shrinks as 1 over n² and ε²
"Assumption 2. g1 and g2 are convex functions of θ … f is also a convex function"
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Minimal interaction ranks hemoglobin cooperativity differently than Hill slope
"minimize ∏ max(w_I, w_I^{-1}) subject to Φ(W)=P ... lifted to linear objective with monomial constraints after s_I = ∏_{I'⊆I} w_{I'} change of variables"
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Bandit pure exploration gets finite guarantees by solving a game
"The Pure Exploration Game... value of the game is Dμ := sup_w inf_λ ∑ w_k d(μ_k, λ_k)"
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Volterra equations yield conditions for linear-quadratic games
"state equation y(t)=y0 + ∫[A(t,s)y(s)+B(t,s)u(s)+C(t,s)v(s)]ds... performance functional quadratic in y,u,v"
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Gradient bisector recovers full-resolution interferogram phase
"û1 = ∇O* Cerr / ||∇O* Cerr||2 , û2 = ∇O* CTV / ||∇O* CTV||2 , u = (û1 + û2)/2"
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Costs turn sequent proofs into budget-constrained game strategies
"We generalize the concept of costs in proofs by using a semiring structure"
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Multi-task learning lifts anomaly detection in driving data
"the network is trained by minimizing the difference, |x−a|² ... overall loss LO = wA LA + wB LB + wR LR"
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Variance reduction works for TFW random lattice energies
"Lemma 7 / Theorem 3: variance reduction by conditioning on statistical functionals F with multilevel local dependence"
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Safe optimization tunes room PID to cut costs 32%
"Safe Contextual GP-LCB... Sn ← {a ∈ A | Pr(Ji(ã) ≤ c_i) ≥ 1−ϵ}"
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RaRecognize flags emerging rare subclasses while ignoring majority noise
"RaRecognize … estimates a general decision boundary … by construction it is dissimilar to the specialized learners"
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Uniform corrections turn GGA spectra into experimental matches for solar materials
"ΔEg = Eg,HSE – Eg,PBE … ε2,PHS(E) = [E/(E+ΔEg)] ε2,PBE(E−ΔEg)"
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Entropy minimization recovers phase-space density from Gaia streams
"Measurement uncertainties ... act to effectively increase the entropy ... decrease the phase-space density"
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Weinberg operator favored for neutrino masses by economy
"The Majorana mass term (38) is the most economical mass term. The flavor neutrino fields νlL enter into the interaction Lagrangian and into the neutrino mass term (the number of neutrino dof is minimal)."
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Convex order ranks two failure probability estimators
"SUR criterion JRn(x) = En[Var[Rn+1] | Xn+1=x] derived from the same convex-order inequality"
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Copying top-consuming neighbors splits economy into rich and poor
"critical transition at tau_c where population becomes strongly bimodal... endogenous aperiodic oscillation"
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Association relation proves type soundness in multiparty sessions
"Theorem 3 (Safety, Deadlock-Freedom, and Liveness by Association)"
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ML correction cuts wind farm fault errors by 76%
"Feature selection... Mutual Information (MI) and Pearson correlation analysis... final feature set that is highly informative, non-redundant"
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Dual-branch fusion reduces error for cross-domain graph prompting
"Assumption 3.2 (Second-order error model ... σ_g² + σ_a² − 2ρ > 0, ρ < min{σ_g², σ_a²})"
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Continued fraction blocks match transformers at half the size
"depth d and number of ladders L ... parameter savings ... no expansion (α=1)"
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Entropy-aware KL switch lifts math scores in model distillation
"LEOPD_t(θ;ct)=LOPD_t(θ;ct)+I[Hte_t>τ]LFKL_t(θ;ct)"
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Linear drift yields nonlinear Fokker-Planck via q-geometry
"By identifying the generalized chemical potential as the natural dynamical ansatz, we construct a general thermodynamic framework where the drift term remains linear in the probability density"
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Marker calibration shortens reasoning paths
"PATHCAL uses marker probabilities to detect local reasoning-mode competition and softly rebalances marker logits when competing-branch evidence becomes excessive"
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Kuramoto flow fully described by Morse theory
"The flow of ψ in Q_I is topologically equivalent to the flow of the Perfect Morse potential M on T^d (Theorem 10.1)"
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Convex factors let energy models scale to larger reasoning tasks
"nonnegative factor summation preserves convexity of the relaxed global objective... convex composition rules out spurious relaxed local minima"
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Multi-gate residuals stabilize deep nets without extra comms cost
"gated mechanism replaces the rigid unitary gain with a learnable dissipative coefficient (1−gl) via a convex combination"
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Compact coordinator expands diffusion models to larger domains
"ViT-based coordinator ... overlap averaging ... MultiDiffusion-like updates"
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Sparse physics model enables lighter robust MPC for quadrotors
"tube radius adapted online from PIML residual error"