Characterizations of P-like continua that do not have the fixed point property

The paper’s Theorem 3.2 proves the equivalence (T1) ⇔ (T2) ⇔ (T3) via carefully constructed taut P-covers and a “strong” fixed-point-free pattern, including (D2′), with the key buffer property (Pn): B(f(Cl(V)), mesh(U_{n+1})) ⊆ φ_n(V) and the Lebesgue-number control ensuring strong refinements; this yields (D1),(D2),(D2′),(D3) and finishes both directions (T1 ⇒ T2,T3 and T2,T3 ⇒ T1) correctly . The candidate solution omits necessary mesh/Lebesgue-number bounds to guarantee strong refinement (chooses mesh(U_{n+1}) < min{δ_n/2, ρ/2^{n+1}} but not < λ_n), proposes an invalid shortcut to obtain (D2′) by setting φ_{n+1}(U)=φ_n(V(U)) without a monotonicity guarantee, and uses incorrect set-distance/“composition preserves intersections” arguments in the constructions for (T2) ⇒ (T1) and (T3) ⇒ (T1). These gaps conflict with the paper’s rigorous proofs of (D2′), continuity, and fixed-point-freeness of the induced maps .