Let f be a product of D generic linear forms in m variables over a field of characteristic 0, and for integers k, l >= 0 let Gamma_{k,l}(f) = dim S_l * partial^k f be its shifted partial derivative measure, the complexity measure behind the known lower bounds for homogeneous depth-four algebraic circuits. Two universal upper bounds hold for every homogeneous f of degree D: Gamma_{k,l}(f) = 0. For every fixed m >= 3 we prove that for all (k,l) and all D >= D_0(m,k,l) = poly(k,l), a generic product of D linear forms satisfies Gamma_{k,l}(f) = min(N_k N_l, N_{D-k+l}) -- full saturation of the universal cap. For derivative spaces (l = 0) we prove exact equality dim partial^k f = min(N_k, N_{D-k}) for all k = D - D/m, and (once D >= 2m^2) within a factor (2/e)^{m-1}/(2e^2 m) of the cap at every k, via a standalone combinatorial comparison lemma for capped compositions (a Polya-urn coupling plus log-concavity). We also compute exactly, by a filtration calculus, the measure of products with disjoint-pair block structure, and show these are genuinely deficient in the row-dominated regime -- for even m >= 6 and k = l >= m^2, by a factor at least (k/32m)^{(m-4)/2} -- witness choice, not analysis slack, is what previously kept this regime open.

The complexity-theoretic reading: against a single product gate of generic linear forms in any fixed number of variables, shifted partial derivatives certify nothing beyond a polynomial degree threshold. This is an exact, per-gate form of the saturation phenomenon underlying the rank-measure barriers of Efremenko-Landsberg-Schenck-Weyman, Efremenko-Garg-Oliveira-Wigderson, and Bhargav-Dutta-Saxena, here established with exact constants in the few-variable, high-degree regime relevant to algebraic hardness-randomness bootstrapping. Characteristic 0 is essential: over small finite fields the evaluation variant of the measure (Armand-Behera-Tavenas, 2026) reverses the polarity.

The commutative-algebra reading: we determine the Hilbert function of the ideal generated by partial^k f in each degree k+l -- for f a generic hyperplane multi-arrangement form -- extending the study of apolar algebras of products of linear forms initiated by DiPasquale-Flores-Peterson. The proofs are elementary throughout (no recourse to Froberg-type conjectures or Alexander-Hirschowitz): the main theorem reduces, by an exact "master reduction," to the rank of an explicit Laurent-polynomial family, which is resolved by a zero-multiplicity bound for exponential polynomials, layered confluent and twist Vandermonde arguments, and a residual core lemma valid over any field. Every machine-checkable step of the derivation has been verified exactly (integer or modular arithmetic, multiple primes and seeds), including the full generic-m code path at m = 4, ..., 9.