Closing the general vanishing lemma for doubled plaquettes touching a loop edge: a symbolic proof of the helper-promoted order-4 case

In a previous note we proved that a doubled plaquette with a loop-type edge and at least two isolated edges among the remaining three vanishes identically (Theorem 3.2 of that note), and reported, as a computationally verified but symbolically unproven statement (Proposition 4.1 there), that the same vanishing persists whenever at least one of the remaining three edges is non-loop, whether isolated or promoted to multiplicity 4 via an auxiliary doubled "helper" plaquette -- the specific mechanism used throughout that note's 108-case computational verification. We close this gap: we prove an order-4 reduction lemma showing that a helper-promoted edge contributes, after summing out the helper's own internal indices, exactly the same delta pattern and the same scalar weight as an isolated edge, so that, for the purposes of the vanishing mechanism, a helper-promoted edge is indistinguishable from an isolated one. Combined with an elementary connectivity observation (a 4-cycle has no nonempty proper subset of edges closed under adjacency), this gives a symbolic proof of Proposition 4.1 in exactly the scope of its original 108-case verification, for every m≥3 -- not only m∈{3,5,7}. We do not extend the result beyond that scope: an order-4 edge whose two extra occurrences arise from some other mechanism than a doubled helper plaquette with isolated remaining edges is not covered.