Irrelevant Operators, Anisotropy Bounds, and Operator Insertions in Balaban's RG for 4d SU(N) Lattice Yang-Mills: Symanzik Classification and Quantitative Irrelevance of O(4)-Breaking Operators

We classify gauge-invariant local lattice operators of classical dimension 6 on the four-dimensional hypercubic lattice into O(4)-invariant, hypercubic-invariant but O(4)-breaking (anisotropic), and on-shell-redundant components, following the Symanzik improvement programme and the on-shell technique of Luscher-Weisz. The anisotropic sector is one-dimensional (Theorem 3.6, Proposition 3.7: uniqueness of the hypercubic harmonic) - a purely representation-theoretic fact, machine-verified in the companion suite together with the classical Symanzik a^2/12 anisotropic term of the Wilson plaquette itself. Inside Balaban's renormalization group framework (small-field regime, g_k <= gamma_0, k <= k* - the window bookkeeping of the audited chain), we extract the anisotropic projection of the effective action via local Taylor (jet) expansion of polymer activities and prove the quadratic bound |c^(k)_{6,aniso}| <= C a_k^2, uniformly in lattice spacing eta, physical volume, and RG step k within the window - conditionally on the structural package, whose audited status is now attached (traceable per 2602.0069 v2; beta_LF dichotomy for the large-field remainder). We further prove an insertion integrability estimate for connected correlators with one anisotropic insertion; Version 3 corrects its status: v1-v2 called it unconditional, resting on an imported clustering/mass-gap bound (Theorem 4.5, from the unaudited companion [1]) claimed uniform in eta and L_phys - per the audited program (2602.0053/0054 v2) such a gap is conditional on (H-DOB-blk)+(H-P0) and windowed, so Theorem 6.6 is conditional on that input. Combined with the rotational Ward identity of the companion [2] (unaudited, identifier pending), the O(4)-breaking distribution tested against Schwartz functions is O(eta^2 |log((Lambda_YM eta)^{-1})|) and vanishes as eta -> 0 - under the same conditional load. This paper thereby supplies, conditionally, the operator-classification input required by 2602.0063 v3 (Proposition 5.3/Remark 5.4) for continuum SO(4) restoration. All verifiable content (representation counts, the unique hypercubic harmonic, the plaquette's own a^2/12 anisotropy, Cauchy-jet mechanics, the a_k^2/log bookkeeping, and the clustering-to-integrability conversion) is adjudicated in a companion suite.