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.
Some readers have voiced the following concerns, which seem correct to me.
It seems that the text was entirely generated by LLMs without much human effort.
In particular, it is extremely hard to read. For example, even the abstract is full of non-standard jargon
("full saturation of the universal cap", "witness choice, not analysis slack, is what previously kept this regime open").
The ECCC board is determined not to allow this in the future,
and a statement reflecting this was added to the CFP.
As you noted, and as I tried to make very clear in the paper, I did this work with the assistance of Fable. The work was all done interactively, but the paper was largely written by AI. My goal was just to get the work in front of the authors of the referenced papers, but eccc was the wrong forum ... I've now just tried to contact them directly, which was the right approach. I apologize for the trouble.
