Exact multiplicative complexity of a Fano 3-space at n = 8 and floor census at n = 4

Result

Let W <= Λ^2(F2^8) be the 3-dimensional subspace spanned by vectors sum_{i ∉ L} d_i, where L ranges over the lines of a Fano configuration P_1, ..., P_7 of 2-planes in the dual of F2^8, and each d_i is the corresponding decomposable class. Then:

1. dim W = 3, and every nonzero element in W has rank 8 with mc = 4.
2. The coordinate code associated with W is the [7,3,4] simplex code.
3. The 2-multiplication complexity satisfies m_2(W) = 6.
4. qMC(W) = MC(W) = 7 = dim W + 4.

For the census across all 2,824 linear subspaces of Λ^2(F2^4):

1. The self-contained floor ρ* matches qMC on 2,439 subspaces and underestimates qMC on 385 subspaces (210 of 651 in dimension 2, and 175 of 1,395 in dimension 3).
2. On the 1,395 subspaces of dimension 3, the joint distribution of pairs (LB, qMC) is:

• (3, 3): 1,220 subspaces
• (3, 4): 105 subspaces (LB underestimates by 1)
• (4, 4): 70 subspaces

Setting and definitions

Let GF(2) denote the two-element field, and let Λ^2(F2^n) denote the space of alternating 2-forms in n variables. The multiplicative complexity MC(W) of a subspace W <= Λ^2(F2^n) is the minimum number of multiplication gates in an XAG over GF(2) evaluating a basis of W. The quadratic multiplicative complexity qMC(W) restricts multiplications to bilinear products of linear forms.

For f ∈ Λ^2(F2^n), mc(f) denotes its rank-based multiplicative complexity, equal to half its algebraic rank over GF(2). The quantity m_2(W) denotes the 2-multiplication complexity across 2-dimensional projections of W.

A Fano configuration consists of seven 2-planes P_1, ..., P_7 in the dual of F2^8 whose incidence structure is PG(2, 2) (7 points and 7 lines). Each 2-plane determines a decomposable rank-2 class d_i ∈ Λ^2(F2^8). For each of the 7 lines L in PG(2, 2), its complement contains 4 points; the sum of the corresponding d_i forms an element of W.

The floors ρ* and LB are theoretical lower bounds on qMC derived from rank distributions and combinatorial dimension constraints.

Method

Upper and lower bounds match tightly on W <= Λ^2(F2^8):

1. Upper bound: W <= span(d_1, ..., d_7) with each d_i decomposable (computable in 1 multiplication), so qMC(W) <= 7 and MC(W) <= 7.
2. Quadratic lower bound: All 7 nonzero classes in W have rank 8 (mc = 4). LEMMA B₃ yields qMC(W) >= 28 / 4 = 7.
3. General lower bound: Every 2-plane in W contains three classes of mc = 4. LEMMA B combined with Mirwald–Schnorr rank conditions establishes m_2(W) >= 6. Applying THEOREM A′ at j = 1 yields MC(W) >= 3 - 1 + 6 - 1 = 7. Thus qMC(W) = MC(W) = 7.

Search scripts identified 384 Fano configurations at n = 8. One explicit representative configuration was fully expanded, written to disk, and replayed.

For the census at n = 4, all 2,824 subspaces in Λ^2(F2^4) were generated and classified, computing exact qMC, rank distributions, ρ*, and LB.

Artifact receipts:

• wave1-ms3/u3_fano_floor.py: Fano configuration construction and floor evaluation.
• u3_fano_floor.json: coordinates of the 2-planes P_1, ..., P_7 in the dual of F2^8.
• fano.py: configuration search and verification script.
• fano_result.json: output metrics for the 384 identified configurations.
• verify_fano.py: certificate replay and rank validation.
• n4_full.json: census data for all 2,824 subspaces of Λ^2(F2^4).
• n4_dim3.json: joint floor distribution on the 1,395 dimension-3 subspaces.
• PROGRESS.log: execution logs across lines 1–61 and 101–112, detailing the 105 shortfall spaces.

Recorded evidence tiers: FR and P for the Fano space; FC for the n = 4 census.

Discussion

The construction confirms an excess of qMC(W) - dim W = 4 on a 3-dimensional space at n = 8 without 3-form transfer theorems, matching the largest transfer-proved excess in this dimension class. Rank uniformity of the [7,3,4] simplex coordinate code ensures every nonzero element has maximal rank 8.

Novelty is categorized as PARTIAL / novelty unverified. Simplex-code packing is standard in coding theory in spirit, and using its incidence structure to select linear combinations of decomposable 2-forms follows established algebraic principles.

The n = 4 census delineates the limits of standard floor functions. The self-contained floor ρ* is sharp on 2,439 of 2,824 subspaces, failing on 385 subspaces confined to dimensions 2 (210 of 651) and 3 (175 of 1,395). Upgrading to LB on the 1,395 dimension-3 subspaces matches qMC on 1,220 spaces at complexity 3 and 70 at complexity 4, and underestimates by 1 on 105 spaces.

Attribution and prior art

Prior art: This result is partial, and its novelty remains unverified because simplex-code packing is conceptually a standard technique in coding theory.

Register references

• Register Entry: MF-166
• Receipt Artifacts: wave1-ms3/u3_fano_floor.py, u3_fano_floor.json, fano.py, fano_result.json, verify_fano.py, n4_full.json, n4_dim3.json, PROGRESS.log 1–61, 101–112
• Prior Art: Simplex-code packing (standard coding theory in spirit; partial / novelty unverified)

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Evidence pack

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