Research · Papers · What rank-one constraints can express · MF-061
Classification of the identity-C fixed-point consumer on toy Boolean graphs
1*(D⊕r)=r forces D=0; with m multiplication pins, affine parity compiles in m+1 rows (AND3 in 3 rows, AND4 in 4 rows)
Published 2026-08-29
For everyone
Plain summary
MF-061 analyzes an algebraic gadget for checking Boolean logic. One step (the creator) builds a temporary value q=AB, and a later step (the consumer) reads it. Testing all 16 base cases of the identity-C fixed-point consumer shows that the combined system projects to a three-input product relation. On the three-input AND graph y=x0 x1 x2, this rejects the invalid point (111,0) where all inputs are 1 but the output is 0.
A companion sentinel check forces discrepancy D=0 while preserving an even number of internal states. Combined with m multiplication pins, it compiles an affine parity check in m+1 rows: 3 rows for a 3-input AND graph and 4 rows for a 4-input AND graph. Whether this mechanism extends to full SHA circuits remains UNKNOWN.
Result
Within the strict canonical identity-C class, the identity-C fixed-point consumer is completely classified across all 16 base cases. A creator row AB=q paired with a consumer reading q projects to a principal cubic relation. On y=x0 x1 x2, this system rejects the H2 point (111,0). In this class, a single creator and consumer pair does not define the full cubic graph.
The sentinel constraint 1*(D⊕r)=r forces D=0 across a nontrivial even fibre. Together with m multiplication pins, it compiles affine parity in m+1 rows, producing the canonical AND3 compiler result in 3 rows and AND4 in 4 rows. The SHA instance status is UNKNOWN.
Setting and definitions
Variables evaluate over GF(2); ⊕ denotes addition modulo 2 and juxtaposition denotes multiplication. A creator is a row defining an intermediate AB=q. A consumer is a row reading q. The identity-C fixed-point consumer designates the specific form classified across 16 base cases in the register.
The toy graph is y=x0 x1 x2, and its H2 point is (111,0). The principal cubic is the projected Boolean relation generated by the creator-consumer pair on this graph. The sentinel equation is 1*(D⊕r)=r, where D is the discrepancy, r is the sentinel variable, and the satisfying assignments over internal variables form a nontrivial even fibre. Affine parity denotes the affine row system's XOR constraint, and m is the count of multiplication pins.
Method
The classification evaluates all 16 base cases of the identity-C fixed-point consumer exhaustively. Projecting the creator-consumer system onto y=x0 x1 x2 identifies the rejection of (111,0). Combining the sentinel constraint with m multiplication pins yields the row-count bounds for AND3 (3 rows) and AND4 (4 rows).
Verification relies on the catalyst verification report alongside four package certificates (covering the canonical fixed-point normal form, the cubic hull escape on the toy graph, and the canonical compiler results for AND3 and AND4), with raw certificates provided in this paper's downloadable evidence pack.
Discussion
The single-creator, single-consumer construction projects to a principal cubic and eliminates (111,0) on the toy graph, but cannot isolate the full cubic graph within the canonical class. The sentinel mechanism achieves the m+1 row parity compilation bound for AND3 and AND4.
The register contains no JSC separator, replay script, recount, or Lean proof for full hash graphs; the SHA instance remains UNKNOWN. Curation corrections: NONE. Prior-art position: the register does not record this.
For everyone — the takeaway
What this means
This entry verifies a small algebraic gadget on toy Boolean networks. It links a temporary multiplication step to a downstream reader, ruling out the specific bad assignment (111,0) on a three-input AND gate. A separate parity sentinel zeroes out errors while keeping an even balance of hidden states, compiling 3-input and 4-input AND gates in 3 and 4 rows. These certificates prove the mechanism works on small graphs, but they don't prove it scales to SHA.
Register references
Entry: MF-061.
Receipts: CONT lens_r2a_catalyst_verification_report.md; package certificates/canonical_fixed_point_normal_form.json, certificates/cubic_hull_escape_toy.json, certificates/canonical_and3_graph_compiler_result.json, certificates/canonical_and4_graph_compiler_result.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 5 of 5 receipt files bundled (9 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.