Discrete Dirac Reduction of Implicit Lagrangian Systems with Abelian Symmetry Groups

The paper’s Theorem 4.1 states that for a (+)-discrete Lagrange–Dirac system with a G-invariant discrete Lagrangian, the reduced triple ([X̂d], [D̂+Ld], [D̂d+]) satisfies the reduced inclusion, and gives the local Discrete Lagrange–Poincaré–Dirac equations (25) in trivialized coordinates; see Theorem 4.1 and its displayed system (25) . The candidate solution reproduces the same structure: (i) uses the G-invariance of the (+)-discrete Dirac structure and Tulczyjew map to descend dynamics, matching Proposition 3.2’s invariance and the construction of [D̂d+] ; (ii) employs the right trivializations λd and λ̂d (with λd(q0,q1)=(q0,x1,g1−hd(q0,x1))) to express Ld in terms of the reduced ld, exactly as in (11)–(12) and the definition of ld via L̂d ○ λd ; (iii) writes the local Lagrange–Dirac equations qk^+=qk+1, pk+1=D2Ld, pk=−D1Ld (Def. 2.2) and right-trivializes momenta, consistent with the paper’s local equations and the λ̂d/Λd inverses showing p = (w − h*_{d,Σ}(µ), µ) ; (iv) applies the chain rule with the discrete connection derivatives D1hd(q0,x1)(q)=hd(q,0), D2hd(q0,x1)(x)=hd(0,x) (equation (6)) to obtain the listed partial derivative identities of Ld in terms of ld (Lemma 4.2) ; and (v) combines Lemma 3.1 for [Ω̂♭d+] and Proposition 4.1 for [D̂+Ld] to produce exactly the six-line reduced system (25) and the kinematic identities xk^+=xk+1 and gk^+ + hd,Q(xk,0)=gk+1 − hd,Σ(xk+1) . No substantive gaps or contradictions are found; the candidate’s derivation follows the same scheme as the paper with equivalent steps and formulas.