Research · Papers · What rank-one constraints can express · MF-064
An exact conservation law for gauges in GF(2) forest systems
For a forest with N vertices and E edges over GF(2), gauge space dimension = apparent vertex-row savings = created gauges = N-E
Published 2026-08-29
For everyone
Plain summary
This entry proves an exact conservation rule for edge systems where honest and candidate equations share their right-hand side. A gauge is an unconstrained degree of freedom that leaves every edge equation satisfied. Subtracting the honest equation from a candidate equation gives g_u+g_v=0 on each edge in binary arithmetic (GF(2)). Because each connected cluster forces all its vertices to shift together, every cluster contributes one independent gauge. In a forest of N vertices and E edges, the apparent reduction of N-E vertex rows is matched one-for-one by N-E created gauges. Tested examples yield 1,024, 128, and 32 on both sides. This identity requires shared right-hand sides and does not cover cases where the intermediate value U varies freely.
Result
For an edge system where honest and candidate equations share right-hand sides, subtracting equations over GF(2) yields
g_u+g_v=0
for each edge (u, v). The gauge space is the kernel of the GF(2) incidence matrix, with dimension equal to the number of connected components.
In a forest with N vertices and E edges, the component count is N-E. The N-E apparent vertex-row savings therefore equal the N-E created gauges. Evaluated on 32 disjoint round pairs, four 16-round paths, and one 64-round path, both sides evaluate to 1,024 / 128 / 32, respectively. Labelled-graph checks through n=6 match. This identity applies to honest-RHS incidence systems, not free-U composition.
Setting and definitions
Let G=(V,E) be the labelled graph of the edge system, where N=|V| and E=|E|. Each edge carries an honest equation and a candidate equation sharing a right-hand side. The term g_u denotes the GF(2) discrepancy between candidate and honest values at vertex u.
The GF(2) incidence matrix maps edges to vertex pairs. Its kernel defines the gauge space. A connected component is a maximal path-connected vertex set; a forest is an acyclic graph. The free-U setting, where intermediate variable U varies independently, is out of scope.
Method
Subtracting honest from candidate edge equations cancels identical right-hand sides, yielding g_u+g_v=0. The solution space is the GF(2) incidence kernel, whose dimension equals the component count. For forests, this count is N-E, equating apparent row savings to gauge count.
Numerical checks evaluate 32 disjoint round pairs, four 16-round paths, and one 64-round path, returning 1,024, 128, and 32 on both sides. Small labelled graphs through n=6 confirm the count. Verification artifacts are available in this paper's downloadable evidence pack.
Discussion
The theorem establishes an exact conservation law: merging vertices across an acyclic edge system reduces the row ledger by N-E but introduces N-E independent gauges. The equality is structural and verified by graph enumeration.
The result requires shared right-hand sides across honest and candidate edge equations. It governs incidence constraints and does not extend to free-U compositions or edge systems with modified right-hand sides or composition rules. Curation corrections: NONE. Prior-art comparison or novelty claim: the register does not record this.
For everyone — the takeaway
What this means
Imagine points connected by lines, where each line tests whether two points match a fixed target. When candidate and honest equations share that target, their difference forces both endpoints to change together in binary arithmetic. Every connected group of points gets exactly one independent bit-flip.
In a network with no loops (a forest), every removed row creates exactly one hidden freedom. Removing N-E rows creates N-E hidden freedoms, matching 1,024, 128, and 32 across the tested cases. Apparent row savings in constraint systems can be entirely offset by unconstrained gauge variables. This identity holds only for fixed shared targets, not when intermediate values U can float.
Register references
Entry: MF-064.
Receipts: CONT lens_r2b_independent_audit.json; package certificates/maj_edge_forest_rank.json.
Prior art: the register does not record this.
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 (3 KB). Anything not bundled is still hashed in the manifest and lives in the compute-box working trees.
Changelog
Last reviewed 2026-08-29
- 2026-08-29Published on this site.