Research · Papers · Direct sums, wedges and the p14 frontier · MF-121
Refutation of the general carry-bond tensor floor
Full-carry sector injection into product-state space fails at J_2: degree-4 indicators not in two-gate span; general floor is refuted.
Published 2026-09-04
For everyone
Plain summary
A proposed method for proving universal minimum gate counts—the general carry-bond tensor floor—is incorrect. The approach attempted to lower-bound circuit sizes by mapping arithmetic carries into a product-state space. The argument breaks down at step J_2: literal injection of the full-carry sector generates degree-four indicator patterns that no two logic gates can span. The general lower bound is therefore withdrawn. This refutation affects only the general tensor-embedding strategy; the projected-heap construction count stands, and establishing a lower-bound theorem tailored specifically to addition circuits remains an open problem.
Result
Literal injection of the full-carry sector into the product-state space fails at step J_2. The required degree-four indicator functions do not lie within the two-gate span. The proposed general carry-bond tensor floor is refuted and withdrawn.
Setting and definitions
The setting uses bilinear and tensor rank formulations of Boolean carry structures embedded in product-state spaces. Step J_2 denotes the second stage in the sequential injection of the full-carry sector. The two-gate span is the linear subspace spanned by all degree-four indicators obtainable from a two-gate intermediate configuration.
Method
Formal proof and full replay (evidence tier P + FR) evaluated the algebraic span at stage J_2, verifying that degree-four indicator functions from the full-carry sector fall outside the span of two gates.
Verification artifacts:
zkgolf-decomp/reports/VERIFY-FEYNMAN.mdzkgolf-decomp/reports/CERT-TENSOR.mdzkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md- Receipt dataset:
zkgolf-decomp/verify-feynman-scratch/bond-results.json(SHA-2565103dbbacd613a110945546000f64ed543b1ea361e638367380470bb76219c49) - Verification script:
zkgolf-decomp/verify-feynman-scratch/verify_bond.py(SHA-256c05660e5454507676dc420378ee6ae976b677837df8a5b38e79a2ae9e836aea7)
Discussion
This refutation is restricted to the general carry-bond tensor floor via literal sector injection. The general floor is fully withdrawn.
Two related results remain intact:
- The projected-heap construction count survives unaffected.
- An addition-specific lower-bound theorem remains OPEN.
The register records no prior-art position for this entry.
For everyone — the takeaway
What this means
Proving hard mathematical minimums on circuit size is difficult because broad structural assumptions often fail on closer inspection. Here, an attempt to prove universal lower bounds by embedding carry operations into a geometric tensor space collapsed at the second step. This general embedding cannot establish universal floors, so future bounds must exploit the concrete structure of addition rather than generic carry embeddings.
Register references
- Entry: MF-121 (killed 2026-08-30)
- Reports:
zkgolf-decomp/reports/VERIFY-FEYNMAN.md,zkgolf-decomp/reports/CERT-TENSOR.md,zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md - Receipt:
bond-results.json(SHA-2565103dbbacd613a110945546000f64ed543b1ea361e638367380470bb76219c49) - Verifier:
verify_bond.py(SHA-256c05660e5454507676dc420378ee6ae976b677837df8a5b38e79a2ae9e836aea7)
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 3 of 5 receipt files bundled (34 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.