Research · Papers · Direct sums, wedges and the p14 frontier · MF-133
Finite classification of the primary-affine pairing-one localization component
Primary-affine pairing-one kernel has 8 points (full kernel 15); reduced lex Gröbner basis has 69 polynomials; 5/3 with zero affine tail.
Published 2026-09-04
For everyone
Plain summary
When analyzing binary circuit logic, we can group linear building blocks into geometric planes and find pairings that satisfy algebraic rules. This work completely classifies the solutions to the polynomial system governing these pairings—the primary-affine pairing-one localization kernel.
Symbolic algebra reduces the equations to a canonical set of 69 polynomials. Their solution space over GF(2) contains exactly 8 normalized plane configurations out of 15 solutions in the full kernel. An exhaustive search across all 2,794,155 normalized planes confirms this count. Pinning the affine shift to zero shrinks the full solution space from 15 points to 5, and the pairing-one slice from 8 points to 3. While this fully resolves the local component, circuits with auxiliary gates treat it as a structural lower bound rather than a global upper bound.
Result
Let the primary-affine product-class kernel be defined over GF(2) within the AGL(n, 2) orbit localization framework. Under the free affine tail/gauge in normalized factor-plane localization:
- The full primary-affine kernel has cardinality 15.
- The pairing-one localization component contains exactly 8 normalized factor-plane classes.
- The reduced lexicographic Gröbner basis of the ideal vanishing on the pairing-one component consists of 69 polynomials, whose Boolean zero set over GF(2) is precisely these 8 points.
- Fixing the affine tail to zero reduces the full kernel cardinality to 5 and the pairing-one component cardinality to 3.
- Exhaustive evaluation of all 2,794,155 normalized planes matches these counts: 15 full and 8 pairing-one points under free gauge; 5 full and 3 pairing-one points under zero affine tail.
- The diagonal S3-orbit of h does not coincide with the fixed kernel.
- The corrected 64-fibre packet detects all 8 affine-gauge classes.
Setting and definitions
The setting is exact Boolean function synthesis and structural localization under affine group actions AGL(n, 2).
A normalized factor-plane is an equivalence class of affine planes decomposing multiplicative structures in quadratic and higher-degree Boolean circuits. The primary-affine product-class kernel is the algebraic variety defined by bilinear pairing and gate-compatibility relations from factor-plane localization. The affine tail (or affine gauge) parametrizes the affine shift; unconstrained, it acts as a gauge freedom across affine representations, while setting it to zero restricts the system to linear subspace representatives.
The pairing-one component isolates the fibre where the primary bilinear pairing evaluates to 1. The 64-fibre packet is the evaluation structure across standard affine test slices that separates and identifies gauge equivalence classes.
Method
Two independent computational routes establish the classification:
- Exact Gröbner basis synthesis: Computational commutative algebra yields the reduced lexicographic Gröbner basis for the pairing-one component over GF(2), producing 69 polynomials. Root extraction confirms 8 points in the pairing-one component and 15 points across the full kernel.
- Exhaustive geometric enumeration: Direct evaluation across all 2,794,155 normalized planes in the design space recovers the identical distribution: 15 full and 8 pairing-one points under free gauge; 5 full and 3 pairing-one points with zero affine tail.
- Orbit and fibre verification: Evaluating the diagonal S3-orbit of h verifies it is distinct from the fixed kernel, and the corrected 64-fibre packet separates all 8 affine-gauge classes.
All computations are certified by independent exact-algebra replays and corpus receipts.
Discussion
The finite classification is exact for the primary-affine product-class kernel under factor-plane localization: the local pairing-one component contains no solutions beyond the 8 normalized classes.
Structurally, this local characterization acts directionally. For circuit models with earlier-gate dependencies or auxiliary catalyst gates, the kernel serves as a lower-side obstruction rather than an exhaustive global bound. The 8 classes block naive gate collapse without precluding additional constraints or alternative synthesis paths in multi-gate topologies.
No prior-art position is recorded in the register.
For everyone — the takeaway
What this means
This census maps every way binary logic planes can pair under affine symmetries. Out of nearly 2.8 million candidate planes, only eight geometric patterns satisfy the primary pairing constraints, dropping to three once linear offsets are removed.
For minimal logic synthesis and complexity lower bounds, this provides a complete local map of algebraic obstructions. Automated synthesis tools can use these exact boundaries to prune search spaces and rule out invalid decompositions without ambiguity.
Register references
- Register Entry: MF-133
- Primary source:
zkgolf-decomp/H-GROEBNER.md - Corroborating sources:
zkgolf-decomp/H-RAZBOROV.md,zkgolf-decomp/H-ROTA.md - Verification receipts:
zkgolf-decomp/h-groebner-scratch/kernel-groebner-receipt.json(SHA-25635ae2fc6b8e6b29e2d0458f85a18b7a7271989c0b5bb3e7882885f45e1faaa95)zkgolf-decomp/h-groebner-scratch/kernel-groebner-check.json(SHA-2567e6df0dc461792d1c27485b76d1f2034ae0100410a62dbb6bf3ec1f2b2d08345)zkgolf-decomp/h-razborov-scratch/checker-receipt.json(SHA-256bb5cde4bd47bc6d334bd0250528b4eeaa12470fa5f61b0f7f6222be04cea2cc4)zkgolf-decomp/h-rota-scratch/selector-fibre-poset.verification.json(SHA-256ccb3789a20009e25653a76b49ccae5f7eed045574e5fe92ce0291cd01f886d07)
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 0 of 7 receipt files bundled (1 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.