Research · Papers · Direct sums, wedges and the p14 frontier · ML-075
The Mirwald–Schnorr 3-form transfer gap and n = 6 enumeration barrier
MC(W) = qMC(W) for dim W = 3 is proved only for n ≤ 5; closing it at n = 6 is an open bottleneck for packing filtration bounds at j ≥ 2
Published 2026-09-04
For everyone
Plain summary
Multiplicative complexity measures the minimum number of AND gates needed to compute a set of Boolean functions using only XOR and AND operations. A central question is whether feeding intermediate products into later multiplication gates helps compute systems of quadratic functions, or whether multiplying linear inputs directly is always optimal.
For systems of up to two quadratic functions, general multiplications offer no advantage over linear multiplications. For three quadratic functions, this equivalence is proven only up to five input variables. At six variables, testing every possible circuit exceeds standard computing limits. Because an inductive shortcut across all input sizes failed, lower-bound methods cannot yet reach three or more outputs in the general circuit model. Resolving the six-variable case would settle the exact multiplicative complexity of finite field multiplication and establish baseline complexity floors for small cryptographic components.
Result
Multiplicative complexity equivalence between the unrestricted model and the quadratic restriction for quadratic form subspaces, MC(W) = qMC(W) with dim W = 3, holds only for n ≤ 5 variables (MF-165). For dim W ≤ 2, equivalence transfers via LEMMA B, leaving THEOREM A′ (MF-144) unrestricted only at index j ≤ 1.
At j ≥ 2, THEOREM A′ bounds qMC alone. The general-model plane-parity analogue fails (MF-174 ii), and the gate-local induction candidate LEMMA X for dim W = 3 across all n is refuted (MF-174 iii). The Mirwald–Schnorr 3-form transfer gap at n = 6 is therefore the sole obstacle preventing the packing filtration bounds from operating in the unrestricted model at j ≥ 2.
Setting and definitions
Let W be a linear subspace of quadratic forms over GF(2) in n variables.
- MC(W) denotes the multiplicative complexity of W in standard XAGs over GF(2).
- qMC(W) denotes the quadratic multiplicative complexity of W, where each multiplication gate computes the product of two affine forms in the primary inputs.
- THEOREM A′ (MF-144) denotes the packing filtration bound yielding complexity floors indexed by step parameter j.
- LEMMA B transfers qMC bounds to MC bounds when dim W ≤ 2.
- LEMMA X refers to the proposed gate-local induction for dim W = 3 across arbitrary n, refuted in MF-174 (iii).
- The general-model plane-parity analogue refers to the affine-plane parity property used in quadratic lower bounds, refuted for the unrestricted model in MF-174 (ii).
Method
The n ≤ 5 boundary was settled by canonical state space enumeration over GL(n, 2) orbits (MF-165). At n = 5 and multiplicative depth k = 4, the verification evaluated 3.8e8 gate options across 11,830 canonical states in roughly 1 hour.
For n = 6, the first-gate universe contains (63·62)/6 = 651 GL(6, 2) orbit classes. The corresponding 4-AND layer expands roughly two orders of magnitude relative to n = 5. As estimated in wave1-ms3/REPORT.md §4 MS-08, closing n = 6 exceeds local search and requires a dedicated compute job (such as an AX162 run) incorporating GL(6, 2) orbit reduction. The planned exploration avenue is a box design targeting n = 6, k ≤ 4.
Discussion
The transfer gap for dim W = 3 at n = 6 restricts THEOREM A′ to j ≤ 1 in the unrestricted model. Because the general plane-parity analogue fails (MF-174 ii) and gate-local induction via LEMMA X is refuted (MF-174 iii), no theoretical bridge connects qMC and MC for 3-forms at arbitrary n.
Closing MC(W) = qMC(W) at n = 6 for dim W = 3 would:
- Extend THEOREM A′ to the unrestricted model at j ≥ 2.
- Settle the multiplicative complexity of GF(16) multiplication at MC(GF(16) mult) = 9 (ML-076).
- Unlock exact lower bounds for three-output S-box designs.
Without an n = 6 computation or a non-local proof technique, the general-model packing filtration remains bounded at j = 1.
For everyone — the takeaway
What this means
We do not yet know whether intermediate multiplications can save gates when computing three quadratic equations on six variables. The equivalence holds for up to five variables, but checking six variables exhausts standard computing setups.
Because an inductive proof across all variable counts failed, orbit-reduced search is the remaining open path. Settling this six-variable case will pin down the exact circuit cost of GF(16) multiplication and establish concrete complexity floors for small cryptographic components.
Register references
- ML-075
- MF-144 (THEOREM A′)
- MF-165
- MF-174 (ii, iii)
- ML-076
- geometry/REPORT.md §6
- wave1-ms3/REPORT.md §4 MS-08
- PROGRESS.log 86–87
Every artifact named above is bundled in, or hashed by, this paper's evidence pack below.
Evidence pack
Everything needed to check this entry against its receipts: the register text, a manifest with a SHA-256 hash for every named receipt, and 2 of 2 receipt files bundled (12 KB). Anything not bundled is still hashed in the manifest and lives in the compute-box working trees.
Changelog
Last reviewed 2026-09-04
- 2026-09-04Published on this site.