Mapping and fixed point property theorems for inverse limits with set-valued bonding functions

The paper’s Theorem 5.4 already uses diagonal-avoidance in clause (2c): the symbol “<” is consistently used in the paper to mean exclusion/membership-negation (e.g., fk(x) < Hk(x) and (x1,…,xk+1) < Hk(x1,…,xk+2)), which is exactly the fixed–point–free condition the model calls the “corrected (2c−)” . Thus there is no sign error to correct. Aside from that misreading, the model’s construction gives a valid alternate proof of the same equivalence: (1)⇒(2) is obtained by coding a fixed-point-free map into coherent upper-semicontinuous set-valued maps on cylinders with shrinking diameters, and (2)⇒(1) reconstructs a continuous f via nested intersections and shows diagonal avoidance yields no fixed points. The paper proves the equivalence via its T_{n,m}-framework and reduction to the single-valued case (Theorem 4.10), together with the homeomorphism to a Mahavier product (Theorem 5.2) and the same diagonal-avoidance at some level k . In short: the paper is correct as stated; the model’s proof is also correct as an alternative derivation, but its opening claim about a sign issue is unfounded in the provided text.