Research · Papers · Direct sums, wedges and the p14 frontier · MF-085
The mixed-image tax and rank-tight copy-locality
dim(residual separated target quotient) = r with r new product gates ⟹ every independent target-bearing gate is copy-local
Published 2026-08-29
For everyone
Plain summary
When computing two separate tasks together, sharing multiplication steps between them wastes capacity. Every shared direction destroys at least one direction that could have carried separate task information. If a target needs \(r\) separate directions and the circuit has an exact budget of \(r\) multiplication steps, every step that carries output information must stay entirely within one task or the other. This exact-budget rule breaks down as soon as a circuit has even one extra multiplication step. MF-032 provides the register's named one-slack counterexample showing this boundary. No external prior art is recorded.
Result
A \(q\)-dimensional mixed span costs at least \(q\) potential separated directions. Consequently, \(r\) products with a nonzero mixed span expose at most \(r-1\) independent new separated directions.
Rank-tight copy-locality corollary: if a residual separated target quotient has dimension \(r\) and is computed with exactly \(r\) new product gates, every independent target-bearing gate is copy-local. The gate allocation splits between the components.
This normal-form statement holds strictly on the rank-tight stratum. It asserts nothing about global additivity once a circuit exceeds the separated target dimension.
Setting and definitions
In the transfer-algebra setting, the residual separated target quotient is the remaining separated target after prior reductions, with dimension \(r\). A product gate is new if counted toward that residual target, and target-bearing if it carries an independent direction of the target.
The mixed span is the linear span of product directions involving both components; its dimension \(q\) counts independent mixed directions. A separated direction lies in the separated target space. A gate is copy-local when its target-bearing contribution is local to one component. The rank-tight stratum denotes the exact budget where the number of new product gates equals \(r\).
Global additivity denotes the unrestricted claim that total combined complexity equals the sum of component complexities. MF-085 restricts all claims to the rank-tight stratum.
Method
The result is established via Theorem 3 (the mixed-image tax) and Corollary 4 (the rank-tight copy-locality statement), with the full derivation available in this paper's downloadable evidence pack.
Discussion
The mixed-image tax provides the structural mechanism for copy locality at exact rank: consuming potential separated directions in a nonzero mixed span leaves fewer than \(r\) independent separated directions for the \(r\)-dimensional target. Every independent target-bearing gate must therefore restrict to one component, splitting the gate allocation at the rank-tight boundary.
This scope is strict. MF-085 establishes a normal form solely on the rank-tight stratum, leaving global additivity open once slack exists. The one-slack counterexample in MF-032 confirms that the boundary cannot be extended. No corrections are recorded for this entry. The register records no external prior-art work.
For everyone — the takeaway
What this means
When a computation has an exact budget of multiplication steps to produce two separate targets, mixing the tasks wastes capacity. Every step that matters to the final output must focus entirely on one task. This gives a crisp structural rule for exact-budget designs. The rule ends there: with even one spare multiplication step, mixing might save operations, as demonstrated by the one-slack boundary example in MF-032.
Register references
Entry: MF-085.
Receipt artifacts: 02-direct-sum-tensor/REPORT.md §3 (Theorem 3, Corollary 4).
Prior art: MF-032, named by the register for the one-slack boundary counterexample; the register does not record an external prior-art work.
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 1 of 1 receipt files bundled (10 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.