Deriving the Minimal SU(2)-Like Branch Algebra from ECSM Closure Constraints: Reversible Neutral-Charged Conversion as the Origin of Weak-Like Generators

This paper presents a notebook-backed derivation of the minimal SU(2)-like branch algebra from ECSM closure constraints. Previous ECSM weak-sector bridge papers used the familiar Pauli/SU(2) algebra as a scaffold for interpreting neutral-charged branch pairs, hypercharge-like charge splitting, photon-Z-like neutral mixing, and W-like charged branch conversion. The present work performs the foundational upgrade. Rather than assuming Ti = sigma_i/2, it starts from two ECSM branch states: |Q0> = neutral closure branch, |Qch> = charged branch, together with reversible branch conversion and closure of the finite operator algebra under commutation. The minimal branch-conversion operators are derived as outer products: T+ = |Q0><Qch|, T- = |Qch><Q0|. Their commutator forces the closure-imbalance generator: [T+, T-] = P0 - Pch = 2T3, so that T3 = 1/2(P0 - Pch). Reconstructing T1 = 1/2(T+ + T-), T2 = 1/(2i)(T+ - T-), recovers the Pauli/SU(2)-like generator structure after derivation. The notebook verifies the full commutator algebra [Ti,Tj] = i epsilon_ijk Tk with maximum residual error 0.0. A minimality test shows that diagonal-only branch labels cannot convert the states, one-way conversion is not reversible, and the pair {T+,T-} is not closed without T3. The residual outside the span of {T+,T-} is 1.4142135623730951, confirming that T3 is forced. A finite-response deformation test shows that exact closure is recovered in the coherent limit chi -> 1. Final notebook verdict: PASS_DERIVED_MINIMAL_SU2_BRANCH_ALGEBRA_FROM_ECSM_CLOSURE The paper does not claim to derive the full electroweak theory, coupling constants, measured weak mixing angle, Higgs mechanism, weak decay rates, scattering amplitudes, anomaly cancellation, or precision electroweak observables. Its narrower result is that the SU(2)-like algebra used in the ECSM weak-sector bridge can be recovered as the minimal closed reversible conversion algebra of a two-branch ECSM closure-active system.