Research · Papers · Direct sums, wedges and the p14 frontier · MF-073
Exact catalyst dimension in a rank-tight XAG
For a p-gate acyclic XAG with W ⊆ G, dim G = p, dim W = r: catalyst dimension = p − dim W = p − r
Published 2026-08-29
For everyone
Plain summary
This note gives an exact count of the temporary degrees of freedom in certain small logic circuits. Consider an acyclic circuit of XOR and AND gates containing p gate functions. Suppose each gate adds an independent function even after ignoring constants and XOR combinations (the affine part). If the desired target outputs span r independent non-affine directions, the remaining catalyst dimension is exactly p−r. A catalyst direction is a gate function available inside the circuit that sits outside the target output space. This formula holds whenever the gate functions are independent. It does not prove that every circuit satisfies this independence condition or establish a general lower bound on circuit size. The register classifies the identity as definition-level linear algebra and uses it to classify nearby frontier cases as zero-, one-, or two-catalyst circuits.
Result
Let a p-gate acyclic XAG compute a target quotient W of rank r. Let G be the span of the p gate functions modulo affine functions, satisfying
W ⊆ G and dim G = p.
Because the gate functions are linearly independent modulo affine functions,
catalyst dimension = p − dim W = p − r.
The complementary gate-function directions in G outside W span a subspace of dimension exactly p − r.
Setting and definitions
Work in the vector space of Boolean functions modulo affine functions. Let G be the span of the equivalence classes of the p gate functions. The circuit is rank-tight when dim G = p, meaning the p gate functions remain linearly independent modulo affine functions. Let W ⊆ G denote the target quotient, with rank(W) = dim W = r.
The catalyst space is any subspace in G complementary to W. Because W ⊆ G, its dimension is dim G − dim W = p − r.
Method
The derivation is a direct linear-algebra proof. It evaluates the catalyst dimension from W ⊆ G and substitutes dim G = p via rank-tightness. The entry uses no SAT solvers or exhaustive replays.
Discussion
The identity holds whenever an acyclic XAG has gate functions that are linearly independent modulo affine functions and contain the target quotient. It is a dimension identity on vector spaces; it does not establish that arbitrary circuits are rank-tight, construct circuits for a given target, or certify that p is minimal.
The register treats this result as definition-level linear algebra used to classify MF-070–072 as zero-, one-, or two-catalyst instances. No prior-art works or corrections are recorded for this entry.
For everyone — the takeaway
What this means
If a circuit has p independent gate directions and its output target uses r of them, exactly p−r directions remain. That difference is the circuit's catalyst dimension. The formula turns the idea of "extra workspace" into a strict count. It applies when the gate outputs remain independent after filtering out XORs and constants, and when the target sits entirely within their span. It does not synthesize circuits or prove absolute lower bounds, but it provides the exact accounting needed to classify rank-tight circuits across the catalog.
Register references
- Entry: MF-073.
- Receipt: CONT
lens_r2c_frontier_verification_report.md. - Prior-art works: the register does not record this.
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Changelog
Last reviewed 2026-08-29
- 2026-08-29Published on this site.