A Single-Point Reduction and Squaring Endomorphism Reformulation of E677 ⊨_fin E255

We study finite magmas satisfying the identity E677: a = b ◇ (a ◇ ((b ◇ a) ◇ b)) and the open question of whether every such magma also satisfies E255: ((x ◇ x) ◇ x) ◇ x = x. We prove that E255 at any element x with left-orbit period p ≥ 5 is equivalent to a single self-squaring identity: cp−4 ◇ cp−4 = cp−5. We further prove that left-orbit periods 2 and 3 are algebraically impossible, and eliminate period 4 by case analysis for magma sizes n ≤ 9. We establish two proved equivalence classes: (i) E255 is equivalent to the fixed-point property (L_z ∘ S has a fixed point for every z), and (ii) the squaring map S being an endomorphism (S(a ◇ b) = S(a) ◇ S(b)) is equivalent to the intertwining condition S ∘ L_a = LS(a) ∘ S. The bridge between these two classes — that E255 holds iff S is an endomorphism — is conjectured and verified by SAT at sizes 5, 6, 7, and confirmed empirically on 90+ distinct finite E677 magmas, but not yet proved algebraically. We prove a defect propagation theorem: in any finite E677 magma, if three "inner" pairs from an E677 chain avoid the defect set D, so does the fourth. In a minimal counterexample, every defect pair generates the entire magma, forcing |M| ≥ 12 and a self-sustaining cycle structure on D. The endomorphism identity (a ◇ b)² = a² ◇ b² is the diagonal restriction of the medial (entropic) identity. All affine E677 models are medial quasigroups via the Bruck–Murdoch–Toyoda theorem. Verified across 3500+ elements with zero except