Research · Papers · Adders, counters and the heap law · ML-062
Refutation of the universal carry-bond tensor lower bound
Universal carry-bond floor fails: literal full-carry sector injection fails at J₂, degree-4 indicators ∉ span of 2 gates.
Published 2026-09-04
For everyone
Plain summary
A proposed universal floor for digital addition circuits, the carry-bond tensor lower bound, fails on small cases. The method aimed to show that computing carry bits across digital addition always requires a set minimum number of non-linear logic gates. Direct evaluation on a two-stage junction shows that degree-four indicator polynomials fall outside the reach of two gates, breaking the theoretical argument.
The refutation kills the universal bounding technique, but it leaves the exact gate cost of addition circuits open and leaves existing upper bounds intact.
Result
The universal carry-bond tensor lower bound floor is false. Literal full-carry sector injection fails at J₂, where degree-4 indicators fall outside the linear span of two multiplicative gates.
Setting and definitions
Let J₂ denote the 2-stage carry junction. Multiplicative complexity bounds over GF(2) evaluated via tensor rank and sector injection assign lower bounds by embedding indicator functions of the carry sector into the linear span generated by intermediate non-linear gate outputs.
Method
Automated verification across evidence tiers P and FR established the refutation. Testing full-carry sector injection at J₂ explicitly computed the algebraic degrees of the resulting indicators, which fall outside the span of the two available gate products.
Verification receipts and scripts:
- Reports: zkgolf-decomp/reports/VERIFY-FEYNMAN.md, zkgolf-decomp/reports/CERT-TENSOR.md, and zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md.
- Result dataset: zkgolf-decomp/verify-feynman-scratch/bond-results.json (SHA-256 5103dbbacd613a110945546000f64ed543b1ea361e638367380470bb76219c49).
- Certificate verifier: zkgolf-decomp/verify-feynman-scratch/verify_bond.py (SHA-256 c05660e5454507676dc420378ee6ae976b677837df8a5b38e79a2ae9e836aea7).
Discussion
The counterexample at J₂ eliminates the proposed universal carry-bond floor as a general lower-bound mechanism. Its scope is strictly bounded:
- The projected-heap construction count remains a valid upper bound.
- The addition-specific multiplicative lower-bound problem remains OPEN.
- Related register reference: MF-121.
For everyone — the takeaway
What this means
A proposed universal method for proving minimum gate costs across addition circuits fails on simple two-step components. The breakdown eliminates this entire family of universal lower-bound arguments. It does not determine whether addition circuits are inherently cheap or expensive to build; finding the exact minimum gate count for binary addition remains open and will require different methods.
Register references
- ML-062
- MF-121
- zkgolf-decomp/reports/VERIFY-FEYNMAN.md
- zkgolf-decomp/reports/CERT-TENSOR.md
- zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md
- zkgolf-decomp/verify-feynman-scratch/bond-results.json (SHA-256 5103dbbacd613a110945546000f64ed543b1ea361e638367380470bb76219c49)
- zkgolf-decomp/verify-feynman-scratch/verify_bond.py (SHA-256 c05660e5454507676dc420378ee6ae976b677837df8a5b38e79a2ae9e836aea7)
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.