ON THE HOFER-ZEHNDER CONJECTURE ON WEIGHTED PROJECTIVE SPACES

The paper proves a weighted-orbifold version of Shelukhin’s Hofer–Zehnder-type result for CP(q) via generating-function barcodes and a Smith-inequality argument. The main statement (Theorem 1.2) exactly asserts that if N(ϕ; F) > |q| (char F = 0 or prime to the weights), then ϕ has infinitely many periodic points; moreover, in char 0 there is a p-periodic point for all large primes p, and in char p there are infinitely many p^k-periodic orbits, matching the headline conclusion the model aims for . However, the model’s proof outline contains decisive errors: (i) it misidentifies the Z/p-fixed locus of the p-iterated generating function as “the diagonal,” whereas the paper shows it is a disjoint union of p projective subspaces P_r with r ∈ Z/pZ, and the restrictions carry action shifts by r/p (Lemma 4.1 and Proposition 4.2) ; (ii) it asserts a φ- and p-independent “total rank = |q|” for the global generating-function homology and links it to Chen–Ruan cohomology. The paper does not make this identification and, in fact, structures its argument around barcode periodicity and a universal bound on bar lengths, not a global rank bound (Theorem 3.4, Proposition 3.8, Theorem 3.9) ; (iii) it relies on degree-separation by mean-index iteration to inject a direct sum of local summands into Z/p-invariants, which the paper neither needs nor justifies in this weighted-orbifold generating-function setting. The paper’s route—Smith inequality for the p-iterate, the decomposition of the fixed locus, an integral formula for β_tot, and a universal length bound—yields the correct conclusion with precise hypotheses (Proposition 4.3, Proposition 4.4, Corollary 4.5, Proposition 4.6) . Hence, the paper is correct; the model’s proof is flawed in key steps.