Exact renormalisation for patch frequencies in inflation systems

The paper proves (i) an exact renormalisation formula for m-point patch frequencies in self-similar inflation systems and (ii) a finite, closed “self‑consistent” subsystem with bound |x_i| ≤ (λ−1)^{-1} max{|r−r'|} together with uniqueness of the solution, via a block-matrix normal form and a coupling argument that forces a one-dimensional λ-eigenspace. Theorem 2 states the precise renormalisation ν_{a_1…a_m}(x_1,…,x_{m−1}) = λ^{-1} Σ να_1…α_m((x_1+r(1)−r(2))/λ,…), derived using local recognisability and rescaling of the averaging radius, and the self-consistent bound is given explicitly in Eq. (3); these appear verbatim in the paper’s Sections 3–4. The model reproduces the same renormalisation step-by-step (expand by the inflation, translate, factor out λ), identifies the same finite self-consistent box with B = Δ/(λ−1), and proves uniqueness by Perron–Frobenius theory on the induced finite “collared” substitution matrix M(m), whose spectral radius is λ. The only minor imprecision in the model is claiming that the difference sets Λ_{a_{i+1}}−Λ_{a_i} are uniformly discrete; the paper uses the weaker and correct locally finite/FLC property, which is sufficient for finiteness in bounded windows. Otherwise, the arguments agree on substance; the uniqueness proofs use different but compatible routes (block normal form and coupling in the paper; primitivity plus PF in the model). See Theorem 2 (exact renormalisation), the bound (3) defining the closed finite subsystem, and the discussion around (5)–(6) for the 1-dimensional λ-eigenspace via coupling in the paper’s proof (, , , ).