ON THE GROWTH OF THE FLOER BARCODE

The paper proves the desired upper bound on sequential barcode entropy via a Lagrangian tomograph, a Crofton-type integral inequality, and Yomdin’s volume-growth theorem. Crucially, in the proof of Theorem 2.4 the authors insert an epsilon-dependent small ball B_k in parameter space to compensate for barcode stability (threshold shift), obtaining vol(L_k) >= const * vol(B_k) * b_{ε_k}(L,L_k) with vol(B_k) ~ ε_k^d; taking logs and using that {ε_k} is subexponential yields ~̂(φ;L,L′) ≤ h_top(φ) (see Lemma 4.1, eqs. (4.3)–(4.4), and Theorem 2.4 in the paper ). By contrast, the model’s argument drops the ε-dependent volume factor and asserts a uniform-in-ε inequality b_ε(L,L_k) ≤ C·Vol(Γ_k)/μ(V), which would incorrectly avoid the subexponential requirement and relies on a non-justified monotonicity b_ε(L,L_k) ≤ b_ε(L_v,L_k) for all v. The paper’s precise chain uses the basic intersection lower bound b_ε(L_s,L_k) ≤ |L_s∩L_k| (eq. (2.1)) and a stability shift (eq. (2.3)) to relate L_s back to L_0 inside the small ball B_k, which the model did not account for (cf. (2.1)–(2.3) and Section 4.2 in the paper ). The model also conflates volumes: for the relative setting the Crofton inequality controls ∫|L_s∩L_k| by vol(L_k) (in M), not by the graph volume in M×M; the graph reinterpretation is valid only in the absolute case L=Δ in M×M (cf. Definitions and Theorem 2.4, and Lemma 4.1 in the paper ).