Research · Papers · Direct sums, wedges and the p14 frontier · ML-026
On the multiplicative complexity of the 26-input Cartesian period-two law
Two-step full interface has exact MC=4; exposed-sum deterministic shared-operand family has MC=2L for every L
Published 2026-08-29
For everyone
Plain summary
Searching for a circuit that handles several carries together has ruled out many specific shortcuts while leaving the main fourteen-operation target open. The label p14 denotes a circuit using fourteen product operations, where a product operation is an AND-like gate counted by the project's circuit-size metric. UNKNOWN means the runs found neither a valid circuit nor a proof that none exists. An exact two-step version requires four operations. The (4,7) route with one shared helper has no compatible instance, and a family with one exposed input shared across copies needs 2L operations for L copies. A newly built eight-operation component passes all 8,192 test inputs, but running two separate copies gives a sixteen-operation baseline rather than the fourteen-operation goal. The main route covers every combination of two block situations and survives the obstruction screens. Twelve named subroutes remain UNKNOWN. Later addenda rechecked the evidence and left those twelve subroutes UNKNOWN. The register records no prior art.
Result
Let L be the number of copies in the exposed-sum family.
The two-step full interface has exact MC=4, proved by a p3 UNSAT projection and a replayed p4. For k=4, the (4,7) one-shared-flat-catalyst intersection is empty. The exposed-sum deterministic shared-operand family has MC=2L for every L.
A 13→7 component p8 replays on all 8,192 inputs. Two disjoint copies yield the existing p16 period baseline instead of p14. The live integration census shows a structural obstruction: the 672 natural interiors form a single directed chain where all 225,456 unordered pairs are comparable and zero are incomparable. The 336 natural adjacent pairs are 23-input sequential specializations rather than independent Cartesian pairs.
The 26-input Cartesian period-two p14 law passes the H2 and defect screens with delta_flip=4, nonlinear-output rank 10, and a verified 12-wedge cover. Circuit existence remains UNKNOWN.
The certified wedge span intersects the nonlinear target quotient in dimension 4. Every p14 construction therefore requires at least six gate functions outside that span; every topology materializing at least nine wedge-span functions is PROVED dead. Fleet B independently establishes that all input-only products fall inside the four-dimensional low-degree target part. Every p14 thus needs at least six chained or non-input-only functions, ruling out any topology with nine or more input-only products. This subsumes all twelve eleven-wedge variants across four independent derivations.
Exactly fifteen eight-wedge pairs are rank-tight with six gates remaining. The separated-product theorem and exact component shells eliminate those fifteen pairs alongside all 1,080 rank-tight named prefixes, including the 130 zero-slack seven-wedge seeds.
The maximal named frontier consists of six ordered seven-wedge one-slack pairs, forming the three copy-swap orbits (7,15), (7,23), and (7,39). Each orbit splits into four pre-catalyst subspaces, giving twelve in total. The unique slack catalyst is PROVED to appear within the first three remaining gates and is genuinely mixed. All twelve maximal named one-catalyst subspace frontiers remain UNKNOWN.
Setting and definitions
MC denotes multiplicative complexity, and pn denotes a target circuit using n product operations. The p14 target is the 26-input Cartesian period-two law, where Cartesian indicates that all combinations of the two block situations are evaluated. The p16 baseline comes from two disjoint p8 components.
The p14 screens evaluate wedge functions, their certified span, the nonlinear target quotient, and rank-tight prefixes. A one-slack pair is a named seven-wedge pair with one remaining gate of slack in the target count. A catalyst is a shared intermediate function used across multiple subcircuits. The sequential substitution sets r=rank(I+A) for the three-state substitution.
Method
The two-step proof combines a p3 UNSAT projection with a replayed p4. The 13→7 component p8 was verified across all 8,192 inputs. The natural-interior census and the classification of the 336 adjacent pairs yielded the integration obstruction.
For the Cartesian p14 object, exact H2 and defect screens established delta_flip=4, nonlinear-output rank 10, and a certified 12-wedge cover. The wedge-span intersection dimension is 4. The Fleet B audit provided the independent restriction on input-only products. The separated-product theorem, exact component shells, rank-tight prefix screens, and direct-fusion scans were evaluated across the named families.
The published p8 factorization was tested for direct fusion across three skeletons. Direct assignments numbered 1,920/1,920/38,400, and relaxed candidate factor universes numbered 49/49/60; none reached a seven-gate target span. Plücker two-plane normalization was evaluated as a factor-symmetry quotient: it is sound once concrete products are deduplicated modulo the full current signal space, and it survives the eight-row mixed-cancellation counterexample. Sound unrestricted fixed-seed CNF runs timed out without resolving the orbits.
Sequential integration was measured against r. Whole-circuit totals are 21,738 for r=3, 22,074 for r=2, 22,410 for r=1, and 22,746 for r=0. Only r=3 and r=2 beat the incumbent leader. The p16 positive control passed on all 8,388,608 sequential inputs.
Discussion
The entry status is PROVED DEAD because the named direct routes, wedge-span and input-only families, rank-tight prefix families, and published-p8 mixed-fusion seed class are eliminated in their stated forms. This outcome does not establish a p14 circuit or prove general p14 impossibility. The twelve maximal named one-catalyst subspace frontiers remain UNKNOWN. Outside the closure are the remaining 942 positive-slack named prefixes, circuits with at most six standalone wedges, absorbed or alternative quadratic directions, cyclic rows, and unconstrained p14 constructions.
The mixed-seed addendum rechecked the mixed-seed results (available in the downloadable evidence pack), the direct-fusion scan, the factor-universe scan, and the NumPy seed verification. The decomposable-kernel addendum proves that the two-copy catalyst kernel is:
ker(μ0) ⊕ ker(μ1)
Rank-two-cross planes are residue-injective, and rank-one fibers have size at most 4. The sound quotient leaves 12,297,829,216,765,457,700 distinct pre00 residues and 196,765,269,276,340,245,782 pre11 residues per orbit. The addenda leave the twelve frontiers UNKNOWN.
All unconstrained lanes timed out or hit interruption receipts and are frozen. Future searches must enforce or pre-filter for r=3 (with r=2 secondary) and supply an acyclic XAG or a sound, complete, computable simultaneous witness, a full Cartesian replay, a 23-input sequential replay, a recount, and Lean artifacts. The passing p16 control does not imply p14 existence. The register records no prior art.
For everyone — the takeaway
What this means
The search has mapped out dead branches and isolated the live frontier. Four-operation two-step circuits, several shared-helper setups, the published eight-operation fusion family, and every listed rank-tight named prefix are ruled out. A fourteen-operation circuit remains possible because the main two-block law passed the obstruction tests. That survival simply means the candidate is not yet eliminated; it does not prove a circuit exists. The sequential circuit's gate count depends on the candidate's rank, and only ranks 2 and 3 improve on the current leader. Any future resolution must deliver an explicit circuit or computable witness verified across the required input domains. The register records no prior art.
Register references
ML-026; receipts: CONT joint_carry_e0_export_receipt.json, joint_carry_k4_cit_receipt.json, period_two_p14_hull_wedge_result.json, period_two_p14_prefix_and_rank_screen.json, pro_request3_p14_rank_independent.json, separated_product_theorem_independent.json, pro_request3_task1_replay_receipt.json, p14_one_slack_exact_attempt.json, pro4b_catalog_audit_replay.json, pro4b_return_c_crossmap.md, return6_d_verification_receipt.json, return6_d_component_p8_independent_replay.json, return6_e_replay_receipt.json, return6_e_plucker_soundness.json, return6_e_verification_report.md, p14_live_occurrence_dependency_certificate.json, p14_sequential_substitution_control/p16_control.sequential_substitution_audit.json, RETURN6-P8-MIXED-SEED-RESULT.md, return6_p8_direct_fusion_scan.json, return6_p8_factor_universe_scan.json, p14_sequential_saving_certificate.json, priority0_p14_sequential_saving_audit.json, return7_fleet_b_independent_audit.json; Return B period_two_p14_wedge_span_obstruction_results.json; addenda return6_p8_seed_numpy_recheck.json, return6_p14_decomposable_kernel.json; 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 2026-08-30
- 2026-08-29Published on this site.
- 2026-08-30Superseded (2026-08-30): While the scoped topology and shell closures in ML-026 remain valid, its 12-maximal-cell, 817-name frontier is updated to 221 cells—comprising 68 unresolved one-catalyst and 153 unchanged multi-catalyst cells. Unrestricted p14 remains unknown, and no p15 conclusion follows.