Research · Papers · The quadratic hull and its defects · MF-025
The false path 8 -> 0 -> 0 in the natural five-code Ghost-P5 quadratic hull
No subset of H achieves soundness via repetition alone (two-column sequential composition admits exact false path 8 -> 0 -> 0)
Published 2026-08-29
For everyone
Plain summary
MF-025 shows that repeating a local rule across two steps won't fix its blind spots. The natural five-code Ghost-P5 construction produces 66 local quadratic equations. Applied across two successive columns, this entire equation set accepts the false path 8 -> 0 -> 0. Because this bogus path passes every check, repeating the checks cannot make any subset of these local equations sound. This finding doesn't apply to equations that couple columns directly, nor does it cover the separate Cartesian period-two p14 relation. The register records no prior art.
Result
Let H denote the complete 66-equation natural five-code Ghost-P5 quadratic hull. Its two-column sequential composition admits the exact false path 8 -> 0 -> 0. Therefore, no subset of H achieves soundness via repetition alone. This excludes cross-column factors and the Cartesian period-two p14 relation.
Setting and definitions
The local system is the natural five-code Ghost-P5 quadratic hull, composed sequentially across two columns. Soundness requires that every accepted path satisfies the underlying relation. Cross-column factors and the Cartesian period-two p14 relation are distinct register items outside this scope.
Method
Verified via the global quadratic hull composition certificate, replay receipt, and reconciliation receipt in this paper's downloadable evidence pack.
Discussion
MF-025 rules out repetition as a soundness repair for the natural five-code Ghost-P5 local family. It does not generalize to arbitrary quadratic systems or constructions with direct inter-column constraints. No corrections or prior-art positions are recorded in the register.
For everyone — the takeaway
What this means
If a local check lets a bad state slip through, running it twice in a row won't clean it up. In this Ghost-P5 setup, an invalid state travels 8 -> 0 -> 0 across two columns without triggering any alarms. Rerunning the checks or keeping only a subset of them cannot guarantee soundness. Designers must either add constraints that link columns directly or turn to the separate two-period p14 relation.
Register references
MF-025
Receipts: Package-3 global_quadratic_hull_composition_certificate.json; CONT package3_replay_receipt.json; package3_e_vs_period_two_p14_reconciliation.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 3 of 3 receipt files bundled (6 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.