Cantor Sets in Higher Dimensions I: Criterion for Stable Intersections

The paper proves a covering criterion for C^{1+α}-stable intersections via a precise semiconjugacy Φq between affine and full configuration spaces and a strong covering mechanism that yields uniform contraction and bounded word-length renormalizations in Q, see the commutative diagram (5.23) and Theorem 6.6; Theorem C follows as an immediate consequence and is stated already in the introduction (covering condition (1.1)) . By contrast, the model solution incorrectly promotes a general covering by the full renormalization semigroup R to a one-step covering by a finite generating set R∗_1 and then uses this to assert persistence of recurrence under perturbation. That inference is not justified by (1.1) alone and bypasses the strong covering/finite-length contraction that the paper carefully establishes. For intersection (non-stable), the paper provides Lemma 6.2: any bounded set in Q satisfying the covering condition (with respect to R) yields intersection; this can be obtained via Φq from a covering set in QAff, without the model’s extra bounded-distortion argument . Stability—uniform in nearby embeddings and nearby Cantor sets—requires the strong covering neighborhood B_{δ′}(Φq(W)) constructed in the paper, not the model’s appeal to generic structural stability alone . Hence, the paper’s argument is complete and correct, while the model’s argument has a critical gap (it silently assumes a one-step/strong covering property it does not prove).