Tight Size Characterization and Exponential Product-Depth Separations for Non-Commutative Circuits via Normal-Form-Free Split-Rank

Over the free algebra F X, we develop a normal-form-free split-rank calculus for unbounded-fan-in product-depth non-commutative circuits and prove matching two-sided size characterizations. Every syntactically homogeneous size-s, product-depth-Δ circuit satisfies rk*(f)≤ sΔ. An explicit nested-split family SN,Δ of middle split-rank NΔ admits a product-depth-Δ circuit of size at most 4NΔ, yielding N≤ s*(SN,Δ;Δ)≤ 4NΔ and proving that the exponent α(Δ)=Δ is tight. For the full-split polynomial Pn,m=∑i₁,,iₘyi₁⋯ yiₘzi₁⋯ ziₘ the same calculus gives the exponential product-depth-$1$ characterization nᵐ≤ s*(Pn,m;1)≤ 4m\,nᵐ. The induction closes directly on DAGs with sharing---no multiplication-by-affine or skew normal form---and degree may grow independently of product-depth. We do not claim Limaye--Srinivasan--Tavenas strength for IMM at large Δ; the contribution is a tight split-rank instrument with proved matching bounds, complementary to that line. A degree-truncation lemma transfers homogeneous bounds to inhomogeneous circuits with explicit overhead. All arguments are measured in circuit size, product-depth, and degree.