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Research · Papers · Direct sums, wedges and the p14 frontier · MF-124

Reduction of named p14 census residual to 221 cells

596 named p14 cells proved dead (327 s=0, 269 s=1); 68 s=1 timed out and 153 multi-catalyst remain, leaving 221 named surviving cells

MF-124OPENOPEN QUESTIONDirect sums, wedges and the p14 frontier

Published 2026-09-04

For everyone

Plain summary

This work tests whether candidate circuit decomposition templates (called cells) can compute the target transformation p14 with minimal non-linear operations under a GF(2) XOR-free cost metric.

An automated census eliminated 596 candidate cells by proving they cannot produce valid circuits. All 327 rank-tight cells with zero auxiliary slack (s=0) and 269 single-catalyst cells (s=1) are dead. No tested cell yielded a valid circuit model. With 68 single-catalyst cells timing out and 153 multi-catalyst cells remaining untested, 221 named candidate cells survive out of the initial 817.

This result supersedes the earlier snapshot in MF-105. The general existence question for p14 remains open.

Result

In the GF(2) XAG multiplicative complexity model, the candidate census for named p14 decompositions drops from 817 to 221 surviving cells:

  1. All 327 named rank-tight s=0 cells are dead (UNSAT / certified empty).
  2. Of 337 named single-catalyst s=1 cells, 269 are dead, 0 produced a valid model (SAT), and 68 timed out without resolution (N/E).
  3. The 153 named multi-catalyst candidate cells remain unaddressed.
  4. Total eliminated cells: 327 + 269 = 596.
  5. Total surviving residual pool: 68 + 153 = 221 candidate cells.

This constitutes a scoped finite classification over the 596 evaluated cells. General p14 SAT/UNSAT remains open.

Setting and definitions

Synthesis operates over GF(2) under the XOR-free (multiplicative complexity) metric, targeting decompositions of p14 into structured component maps.

Candidate spaces are partitioned into cells parameterized by catalyst slack s:

  • s=0: rank-tight cells introducing no intermediate catalyst variables.
  • s=1: single-catalyst cells.
  • s > 1: multi-catalyst cells.
  • Dead: the constraint system is proven unsatisfiable (no valid circuit model exists).
  • N/E (not evaluated / non-exhaustive): solver execution timed out prior to determination.

Method

Constraint generation and search ran automatically across the enumerated candidate cells. Run records and verification artifacts include:

  • Reports:
  • zkgolf-decomp/reports/EXP-CENSUS-S1.md
  • zkgolf-decomp/reports/SB-P14INJ.md
  • zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md
  • Receipts (AX162 verification artifacts):
  • zkgolf-decomp/exp-census-scratch/aggregate-audit.json (SHA-256 9a578de8058e019351613c5db260d23b433900a8ac5ac28293c5b6e19e3d7c03)
  • zkgolf-decomp/sb-p14inj-scratch/census_crosscheck_receipt.json (SHA-256 a5eab1b5af2f0e77d56141cfaac229e4e661215128a34cdb561886b0aed70883)
  • zkgolf-decomp/EXP-CENSUS-PROGRESS.log (SHA-256 3c1963e46a9b799ca2b43bdcad9f32d245263b2e6f3d4a89990fc2ad9048fb50)
  • zkgolf-decomp/sb-p14inj-scratch/component_p7_probe_receipt.json (SHA-256 776468d26a21dec8cc01fdd452c01b84ff1598f763f0c9628cce2824b194cf74)

Automated solvers proved 596 candidate cells dead. The 68 unresolved s=1 cells reached execution timeouts and are cataloged as N/E.

Discussion

This entry provides a finite classification covering the 596 decided cells, superseding the snapshot in MF-105.

Scope boundaries:

  • It does not prove general SAT or UNSAT for p14.
  • It leaves 68 s=1 cells unresolved due to timeouts and 153 multi-catalyst cells unexamined.
  • It carries no implications for p15 decompositions.

The program status for p14 remains OPEN.

For everyone — the takeaway

What this means

Proving lower bounds on non-linear operations requires checking structural possibilities systematically. Automated solvers can decisively rule out candidate layouts that cannot work.

Here, 596 candidate structures for p14 were proven impossible. This narrows the named search space down to 221 surviving cells (68 unresolved single-catalyst cells and 153 multi-catalyst cells), focusing future compute on the remaining open instances.

Attribution and prior art

Prior art: This result supersedes the 817-cell snapshot from MF-105; the general p14 SAT/UNSAT problem remains open.

Register references

  • Entry: MF-124
  • Prior snapshot superseded: MF-105
  • Reports:
  • zkgolf-decomp/reports/EXP-CENSUS-S1.md
  • zkgolf-decomp/reports/SB-P14INJ.md
  • zkgolf-decomp/reports/STATE-OF-PROGRAM-V2.md
  • Machine receipts:
  • zkgolf-decomp/exp-census-scratch/aggregate-audit.json
  • zkgolf-decomp/sb-p14inj-scratch/census_crosscheck_receipt.json
  • zkgolf-decomp/EXP-CENSUS-PROGRESS.log
  • zkgolf-decomp/sb-p14inj-scratch/component_p7_probe_receipt.json

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 7 receipt files bundled (41 KB). Anything not bundled is still hashed in the manifest and lives in the compute-box working trees.

Download evidence.zip

Changelog

Last reviewed 2026-09-04

  • 2026-09-04Published on this site.

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