A Minimal Closure-Active SU(2)L Algebra in ECSM: Branch-Conversion Generators for Neutral-Charged Weak Pairs

This paper presents a notebook-backed algebraic test of the ECSM weak-sector bridge. Recent ECSM matter-sector work proposed that saturated localised excitations may require triadic internal branch structure: Q_loc → (Q_+, Q_0, Q_-)_loc where Q_+ and Q_- provide polarity or mirror imbalance, while Q_0 provides neutral closure. Subsequent ECSM work proposed that the left-active neutral-charged pair Ψ_L^ECSM = (Q_0, Q_ch)_L may provide a structural route toward weak SU(2)_L-like behaviour, with Q_0 acting as a neutral closure branch and Q_ch as a charged branch. The present paper performs the first minimal algebraic test of this proposal. The closure-active pair is represented as a two-state vector space, with generators defined by T_i = σ_i / 2 where σ_i are the Pauli matrices. The notebook calculation verifies exact closure of the SU(2) commutator algebra, [T_i, T_j] = i ε_ijk T_k with maximum numerical error 0.0. The raising and lowering operators T_+ = T_1 + iT_2 T_- = T_1 - iT_2 act as branch-conversion operators: T_+ |Q_ch⟩ = |Q_0⟩ T_- |Q_0⟩ = |Q_ch⟩ The diagonal generator T_3 assigns opposite weak-isospin-like labels: Q_0 : +1/2 Q_ch : -1/2 A finite-response deformation test replaces the coherent generators with T_i(χ) = χT_i + (1-χ)Δ_i For generic deformation matrices Δ_i, the algebra fails to close away from the coherent limit, while closure is recovered exactly at χ = 1. This supports the ECSM interpretation that weak SU(2)_L-like structure is a coherent-limit closure algebra of a left-active neutral-charged branch pair. The paper does not claim to derive the full electroweak theory, W/Z masses, the Weinberg angle, weak coupling constants, Yukawa couplings, neutrino oscillations, beta-decay spectra, or precision electroweak observables. Its purpose is narrower: to establish that the proposed ECSM neutral-charged closure-active doublet satisfies the minimum algebraic condition required for an SU(2)_L-like weak-sector bridge.