What this programme is about
Digital circuits compute over bits, but system designers can assign those bits to internal machine states in thousands of ways. This programme asks how state representations and coordinate symmetries control the exact nonlinear gate cost of finite-state controllers, carry chains, and logic allocators. Choosing an encoding acts as a search gauge: an arbitrary choice that leaves external behavior intact while reshaping the difficulty of circuit optimization.
Results here give circuit designers and solver developers concrete bounds on the gains available from state recoding. They prove which state transformations preserve nonlinear gate counts, how to prune symmetric branches during SAT-based logic synthesis without creating false proofs, and where arithmetic carry representations resist state reduction. That spares engineers from chasing impossible state minimizations and speeds up automated circuit synthesis.
What has been settled
Invertible affine maps over F₂ preserve XOR–AND count and depth MF-092. For an eight-state controller, 40,320 3-bit encodings collapse into 30 affine orbits with exact minima of 2, 3, 4, or 5 AND gates. A minimum-rank filter reduces those 40,320 encodings to two affine orbits MF-008: the natural encoding and orbit29 = [0,1,2,3,4,7,6,5], which previously had no certificate. Finite-horizon Nerode minimization of the 22 natural SHA carry states yields 22 classes through bit 29, 16 at bit 30, four at bit 31, and one terminal class across all 64 round constants MF-050, ML-033. This leaves only 128 terminal occurrences mergeable, closing full-alphabet recoding as a large-saving route.
Structural complexity bounds form the second theme. Multiplicative complexity equals shortest-path distance on the complete semantic gate-state graph, with an exact linear programming dual as a 1-Lipschitz potential problem MF-113, where quotient certificates require complete edge sets ML-055. An exhaustive census of the two-row cyclic model proves 575,968 systems realize only 32,768 Boolean functions, all with degree at most 3 MF-004. For the two-column C7 transducer with dim V = 4, algebraic normal form computation confirms 12 nonzero cosets have degree ≥ 3 and 3 cosets are quadratic MF-175.
Symmetry reduction governs the third theme. A sparse exact CEGIS loop with 120× row-order symmetry reduction synthesizes a 69,862-variable allocator in 12 rounds taking 26 seconds, whereas monolithic search timed out MF-017. Sorting the three nonzero vectors u, v, and u+v of an independent two-plane in S/<1> yields an exact GL(2,2) symmetry quotient for catalyst parameterization MF-042. For subset-UNSAT certificates, soundness requires ∀g ∈ G, g(R) = R MF-018.
What is still open
The exact multiplicative cost of degree-22 symmetric functions remains unresolved under ML-079. The open question asks whether these functions require 21 or 22 multiplications. Existing evidence shows that the specific five-step two-phase construction route at t = 5 yields 22 multiplications, ruling out that single path to 21. Alternative two-phase parameterizations at t = 6 and t = 7 remain untested and open. The register records no prediction for whether a higher-step construction will close the gap or establish 22 as the true lower bound.
How to read the evidence
Exhaustive checks and execution receipts dominate this programme, backed by machine-certified proofs. Exhaustive checks cover every valid state assignment across finite controller spaces and carry pipelines, leaving no room for sampling bias. Receipts document verifiable solver runtimes and concrete algebraic normal form expansions. Certified proofs verify the underlying graph dualities. Readers can treat the settled orbit counts and complexity floors as definitive within their stated finite interfaces, while keeping open parameterizations strictly uncommitted.