Research · Papers · Direct sums, wedges and the p14 frontier · MF-047
A lower bound on chained products for the period-two p14 target
rank(T) = 10, rank(T≤2) = 4, dim(T ∩ W) = 4 ⇒ every p14 construction requires ≥ 6 chained or non-input-only product functions
Published 2026-08-29
For everyone
Plain summary
This paper studies the period-two p14 target. The question is how much of the target circuit can come from input-only products—AND operations built directly from original inputs rather than intermediate signals.
After ignoring affine operations (XOR additions and constants), the target requires ten independent directions. Standalone input-only products can supply at most four. The remaining six directions must come from chained products that reuse earlier computed values. Because the p14 budget allows only fourteen products, any topology with nine or more input-only products leaves at most five products for the remaining directions and cannot succeed. This eliminates a broad class of layouts while leaving a smaller candidate frontier open for search.
Result
Modulo affine functions, the target quotient decomposes as
rank(T) = 10, rank(T≤2) = 4, rank(T>2) = 6.
Every standalone input-only product lies in T≤2. Consequently, every p14 construction requires at least six chained or non-input-only product functions. Any topology with nine or more input-only products is impossible.
The certified obstruction is
dim(T ∩ W) = 4,
where W is the span of input-only products. The fifteen rank-tight eight-wedge pairs constitute the dead zero-slack layer. The live frontier consists of six oriented seven-wedge seeds (three under copy swap), with four pre-catalyst subspaces per seed.
Setting and definitions
All vector spaces are defined modulo affine functions. Let T denote the target space in this quotient, partitioned into degree-at-most-two (T≤2) and degree-above-two (T>2) subspaces. Let W denote the span of standalone input-only products, which take only primary inputs as operands. Chained products take at least one earlier product output as an operand. The terms eight-wedge pair, seven-wedge seed, zero-slack layer, and pre-catalyst subspace denote register candidate-prefix classes.
Method
The bound was established through two exact enumerations: target capture and rank-tight prefix enumeration. These censuses evaluate the four-dimensional input-only intersection and the six-dimensional high-degree component in the quotient, separating the zero-slack layer from the live search frontier. An independent audit verified the obstruction certificate without altering the live frontier. Full raw outputs and verification artifacts are available in this paper's downloadable evidence pack.
Discussion
This quotient lower bound constrains product topologies for the period-two p14 target. It rules out configurations with nine or more input-only products because fourteen total multiplications cannot house nine base products alongside the six required chained products.
The obstruction stems from product degree: every standalone input-only product lands in the four-dimensional subspace T≤2, so no combination of input-only products can capture more than four target dimensions regardless of factor selection. The result eliminates the fifteen rank-tight eight-wedge pairs while preserving six oriented seven-wedge seeds across three copy-swap classes (each with four pre-catalyst subspaces). It prunes the search space without resolving whether the remaining frontier contains a valid p14 implementation.
For everyone — the takeaway
What this means
A fourteen-product circuit cannot spend nine products on fresh input pairs. Those nine products supply only four independent target pieces once XORs and constants are discounted, leaving ten minus four, or six pieces, to be built by the remaining five multiplications. That arithmetic does not work. Designers can immediately discard layouts with nine or more input-only AND gates and focus on the remaining six seed designs that use chained intermediate values.
Register references
Entry: MF-047.
Receipts: Fleet B p14_target_capture.json; Fleet B p14_rank_tight_prefix_census.json; CONT return7_fleet_b_independent_audit.json (SHA-256 1bfcaa7be4ddd023e928667e2ca11b06004b83b745a578752895a2524ed6399c).
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 1 of 3 receipt files bundled (3 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.