The cheap embedding principle: Dynamical upper bounds for homology growth

The paper establishes the dynamical upper bounds b̂_n(Γ, Γ_*; Z) ≤ medim^Z_n(Γ ↷ Γ̂_*) and t̂_n(Γ, Γ_*) ≤ mevol_n(Γ ↷ Γ̂_*) via a precise three-step mechanism: (i) quantitative strictification to obtain Γ_*-adapted α-embeddings over cylinder sets (Theorem 5.10 specialized as Theorem 8.3), (ii) passage to finite index subgroups that yields homology retracts (Theorem 7.6 restated as Theorem 8.4), and (iii) a torsion estimate controlled by the logarithmic norm (Theorem 7.7 restated as Theorem 8.5), combined in the proof of Theorem 8.1 (Theorem 1.2) . The candidate’s outline captures the right high-level idea (approximate by finite factors and compare via a corner/Morita argument), but it omits two essential ingredients that the paper treats carefully: (A) strictification of almost chain complexes/maps to preserve exact chain relations after approximation to finite σ-algebras (Sections 4–5; Theorem 5.10) and (B) the retraction step that ensures H_n(Γ_i; Z) is a Z-retract of the finite-level homology, which is crucial for the rank inequality (Theorem 7.6/8.4). The model incorrectly asserts that H_n(Γ_i; Z) is a subquotient of E_n ⊗ Z from a mere chain map C_* → E_*; without the retraction this conclusion does not follow. The paper’s proof also controls torsion via the lognorm invariant and its finite-level compatibility (Theorem 7.7/8.5), which the model replaces with a Hadamard-style bound but again without establishing the retraction framework. Thus the paper’s argument is complete and correct, while the model’s proof sketch has substantive logical gaps in the approximation and retraction steps.