BKT–39 The Belle/Belle II Null-Anchor Matrix in the LOM–GTSFC–USC–GTCW Framework as a New-Physics Filter

BKT-39 presents the Belle/Belle II null-anchor matrix as a formal new-physics filter within the LOM–GTSFC–USC–GTCW framework. The article does not interpret null results as evidence for USC. Its role is to define a constraint geometry: a model variant is rejected in a given channel if its predicted projection exceeds the corresponding experimental upper bound. The formal core of the article is the orthogonal-residual construction in observable space. The empirical observable vector is denoted by O_exp, the null model by M₀(θ), and the best-fit null parameter by θ̂₀. The raw residual is d = O_exp − M₀(θ̂₀), and the orthogonal residual is R_⊥ = (I − P₀)d. The operator P₀ is the covariance-weighted projection onto the local tangent space of the null model. In the linear approximation, P₀ = J(JᵀΣ⁻¹J)⁺JᵀΣ⁻¹, where J = ∂M₀/∂θ |θ̂₀ is the Jacobian of the null model, Σ is the covariance matrix, and (·)⁺ denotes the Moore–Penrose pseudoinverse. The defining orthogonality condition is ⟨R_⊥, J_a⟩_Σ⁻¹ = 0 for every tangent direction J_a of the null model. This means that the residual component is not locally absorbable by refitting the parameters of the null model. For a USC variant with parameters λ, each experimental channel k defines a projection map Π_k(λ). The anchor condition is |Π_k(λ)| ≤ U_k,exp(CL), where U_k,exp(CL) is the experimental upper bound at the chosen confidence level. The allowed parameter set is therefore Λ_allowed = { λ ∈ Λ : |Π_k(λ)| ≤ U_k,exp(CL) for all k }. The article defines five Belle/Belle II anchor classes: A39-1: missing-energy channels in B decays, including B → hX_inv.A39-2: M1 bottomonium transitions.A39-3: LFV tau channels, including τ → ℓα.A39-4: time-dependent CP observables in B⁰ → K_S⁰π⁰γ.A39-5: meson-production and fragmentation data near √s ≃ 10.58 GeV. The statistical layer of BKT-39 includes profile likelihood, Toy Monte Carlo, CL_s, Fisher information geometry, the Schur complement, Amari information geometry, natural gradients, KFAC and RMHMC. The Fisher metric is denoted by I(θ). After profiling null-model or nuisance parameters, the effective metric on the extension parameters is g_eff = I_λλ − I_λθ I_θθ⁻¹ I_θλ. This metric quantifies how much information about the extension parameter λ remains after the directions degenerate with the null model have been removed. BKT-39 does not claim that Belle/Belle II data confirm the full LOM–GTSFC–USC–GTCW framework. Its function is methodological: it converts null results and constraining measurements into a quantitative selection protocol for new-physics variants. A variant that violates at least one anchor is excluded in that channel. A variant that satisfies all anchors is not thereby validated; it remains compatible with the constraint matrix. BKT-39 is directly connected to BKT-40. BKT-39 defines the formal null-anchor and orthogonal-residual apparatus, whereas BKT-40 applies this apparatus to a concrete numerical benchmark in the B⁺ → K⁺ + inv channel.