Ahlfors-regular conformal dimension and energies of graph maps

The paper proves Theorem A with a complete modulus-based argument: (i) it sets up visual metrics and snapshot covers, (ii) relates combinatorial modulus on J to weighted multi-curve energies on Gn, and (iii) derives the two inequalities and the critical-equality Eq=1 at q=ARCdim via Propositions 6.5 and 6.8 together with Theorem 5.11 and Corollary 5.12 . By contrast, the model’s writeup asserts key steps without justification: (a) it replaces the paper’s star-covers Vn by edge-preimages Ke and assumes admissibility of test weights ωn(e)=diam(K_e) for the q-energy, and (b) it builds a path metric dq and a Carathéodory measure μq from extremal weights ρn and claims Ahlfors q-regularity and visuality. These require nontrivial connectivity/overlap and diameter–length comparability results that the paper avoids precisely by working with stars and combinatorial modulus; they are not supplied by the model and are generally false without additional hypotheses. Hence, as a proof, the model’s solution is incomplete.