Emergent hypernetworks in weakly coupled oscillators

The paper proves, under network non-resonance, that two near-identity transformations eliminate all O(α) pairwise terms and expose α^2 triadic interactions with explicit replacement rule denominators (d1−1)γk + d2 γ̄k + d3 γℓ + d4 γ̄ℓ, leading to the hypernetwork normal form uk̇ = fk(uk) − α^2 Σℓ,p[AkℓAkp 1Gℓp_k − AkℓAℓp 2Gℓp_k] plus higher-order remainders; see the replacement rule and main normal form statement (Eq. (11) and (12)) and the definitions of 1G and 2G via derivatives of h̃ contracted with h (Eq. (27)) . The homological equation is solved by Pk = Σℓ Akℓ h̃kℓ, and, after the first transformation, the only α^2 terms up to total degree 4 come from −[Pk||H5] (cf. S110–S116 and Lemma S111) . The candidate solution follows exactly this two-step normal-form strategy, uses the same denominators, correctly identifies −[H5,P] as the leading α^2 source, matches the degree bookkeeping (1G/2G start at degree ≥3), and provides a remainder bound consistent with the paper’s expansions. Hence both are correct and essentially the same proof.