Research · Papers · Direct sums, wedges and the p14 frontier · MF-014
Direct-sum additivity barrier for extension-field batching over F₂ᵐ
Conjecture MF-014: batching over F₂ᵐ via Karatsuba or CRT cannot beat direct-sum additivity under the register's multiplicative cost model
Published 2026-08-29
For everyone
Plain summary
MF-014 asks whether packing several binary computations into an extension field F₂ᵐ can lower the multiplication bill below the direct sum of their individual costs. The conjecture asserts that neither Karatsuba nor Chinese Remainder Theorem (CRT) batching can beat direct-sum additivity in the register's cost model. The claimed proof remains unverified, so the entry stands as an open conjecture rather than an established lower bound. The register records no prior-art attribution; this page makes no novelty claim.
Result
Conjecture MF-014 states that under the register's multiplicative cost model, batching over F₂ᵐ via Karatsuba or CRT cannot beat direct-sum additivity. The recorded proof is unverified, leaving the lower bound open.
Setting and definitions
The base field is F₂, with extension field F₂ᵐ. Batching packs multiple component computations into F₂ᵐ. The evaluated multiplication arrangements are Karatsuba and CRT. The target metric is multiplicative cost, evaluated against direct-sum additivity (the sum of independent component costs).
The register omits the explicit specification of the cost model, the batched input class, and any auxiliary admissible operations.
Method
MF-014 originates in BRIEF §B.5, probe 04, as an unverified claimed proof. The register records no solver output, replay script, certificate, or independent audit. Provenance is limited to the probe citation and its unverified status.
Discussion
The conjecture is restricted to Karatsuba and CRT batching over F₂ᵐ under the register's cost model against direct-sum additivity. It asserts neither a universal lower bound across arbitrary batching techniques nor bounds under alternative cost metrics.
Index curation notes list corrections as NONE. With no recorded prior art or verified proof, the entry serves as an open target without a novelty claim.
For everyone — the takeaway
What this means
If you pack several binary jobs into a larger field F₂ᵐ to save on multiplications, MF-014 predicts you will not beat doing them separately, at least when using Karatsuba or CRT under this cost model. Because the proof is unverified, the question remains open. A concrete construction beating direct-sum additivity would refute it; a formal proof would set a firm barrier for these batching techniques.
Register references
MF-014 Receipt: BRIEF §B.5 (probe 04) 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 0 of 0 receipt files bundled (1 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.