Research · Papers · Direct sums, wedges and the p14 frontier · MF-178
Domination of tensor slice rank floors for systems of quadratic forms
srank(T) ≤ m for m-form tensors T, so ⌈srank(T)/3⌉ ≤ m ≤ MF-137 lower bound for all non-affine targets over F_2^n.
Published 2026-09-04
For everyone
Plain summary
Multiplicative complexity measures the minimum number of multiplications required to compute a system of Boolean functions. We tested whether tensor slice rank (or partition rank) could establish stronger lower bounds on the multiplication count for systems of quadratic equations.
It cannot. Slicing the 3D tensor of an m-equation quadratic system along one axis caps its slice rank at m. Because slicing across three modes introduces a 1/3 penalty, the resulting lower-bound floor cannot exceed ceil(m / 3). Standard linear packing bounds already guarantee at least m multiplications for every non-affine system. Slice rank is therefore a dead end for lower-bounding quadratic systems.
Result
For any system of m quadratic forms over F_2^n represented by the trilinear tensor T(x, y, z) = sum_{k=0}^{m-1} z_k (x^T A_k y):
- srank(T) <= m = dim(V).
- The tensor slice rank lower-bound floor satisfies ceil(srank(T) / 3) <= ceil(m / 3) <= m.
- For every non-affine target, the MF-137 packing lemma gives MC >= m + mu - 1 >= m, since mu(V) >= 1. Hence ceil(srank(T) / 3) <= MF-137 holds on 100% of non-affine targets.
- For cubic scalar targets (m = 1), srank(T) <= n, yielding ceil(srank(T) / 3) <= ceil(n / 3), which is dominated by Schnorr's degree floor (deg - 1 = 2), Walsh spectral floors, and restriction floors.
- In an exhaustive evaluation across all 89 corridor units in the geometry sweep, the slice rank floor provides 0 tightenings out of 89 units (0.0%).
Setting and definitions
Let V be an m-dimensional subspace of quadratic forms over F_2^n, represented by the trilinear tensor T(x, y, z) = sum_{k=0}^{m-1} z_k (x^T A_k y) with A_k in F_2^{n x n}.
Tao's slice rank srank(T) is the minimal number of 1-slice tensors summing to T, where a 1-slice tensor has the form u(x) f(y, z), v(y) g(x, z), or w(z) h(x, y). The associated multiplicative complexity floor is MC >= ceil(srank(T) / 3). Naslund's partition rank generalizes this decomposition. The baseline linear packing bound MF-137 provides MC >= m + mu - 1, where mu = mu(V) is the linear packing deficiency of V.
Method
- Obstruction proof: Decomposing T along the z-basis coordinates yields T(x, y, z) = sum_{k=0}^{m-1} z_k (x^T A_k y). Each term z_k (x^T A_k y) is a 1-slice tensor of mode z. This explicit length-m decomposition proves srank(T) <= m unconditionally.
- Algebraic comparison: Because non-affine targets have mu(V) >= 1, MF-137 sets MC >= m. The mode-slicing loss factor reduces the slice rank bound to ceil(srank(T) / 3) <= ceil(m / 3) <= m, strictly weaker than the dimension bound.
- Empirical sweep: The Gemini-map hypothesis from scout units 003, 088, and 091 (HANDOFF-20260902 Wave 2 SR) was tested across all 89 geometry corridor units via wave2-slice-rank/test_all_cells_slice_rank.py. Audited in wave2-slice-rank/slice_rank_audit.json, yielding 0 tightenings across all 89 cells.
Discussion
The 1/3 mode-slicing loss factor prevents tensor slice rank and partition rank from producing non-trivial multiplicative complexity bounds on quadratic systems. For vector targets, the output dimension m directly caps slice rank, bounding ceil(srank(T) / 3) below the linear packing floor MF-137.
For scalar cubic targets (m = 1), slicing along x or y gives srank(T) <= n. The floor ceil(n / 3) remains dominated by Schnorr's degree bound (deg - 1 = 2), Walsh spectral bounds, and restriction bounds. Slice rank and partition rank yield no tightenings across the test corridor.
For everyone — the takeaway
What this means
Slice rank is useful in combinatorics, but it fails to yield meaningful circuit lower bounds for quadratic systems. Decomposing a 3D tensor across three directions incurs an unavoidable 3x penalty. This loss makes the resulting bound weaker than simply counting the system's independent output equations. Lower-bounding multiplicative complexity for quadratic circuits over bits requires different tools.
Register references
- MF-178
- HANDOFF-20260902 Wave 2 SR
- Scout units 003, 088, 091
- wave2-slice-rank/slice_rank_audit.json
- wave2-slice-rank/test_all_cells_slice_rank.py
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Changelog
Last reviewed 2026-09-04
- 2026-09-04Published on this site.