Research · Papers · The SHA-256 record and exact synthesis · ML-023
On the natural stationary p7 route targeting 21,738
p7 target 21,738: 2,018 dead_tail / 181,441 blockers across 768 rows, 0 unresolved tails, global closure = UNKNOWN, status = LIVE-PARKED
Published 2026-08-29
For everyone
Plain summary
This paper reports on an unfinished search for a natural stationary seven-product (p7) multiplication route targeting count 21,738, where every row follows the same local rule. A resumed search covered 768 rows and logged 181,441 blockers—cases that halt extension. The audit proved that 2,018 of these are dead tails that cannot lead to a solution, leaving zero unresolved tails in the audited batch. We have proved two abstract lemmas covering state values 0 and 1 and a fixed positive condition. These lemmas do not yet close the entire search space because the remaining prefix lacks a compact, rank-tight coefficient and divisor export to connect them across all cases. The entry is marked LIVE-PARKED: the audited data is preserved while the global question stays open. An earlier count of 1,994 dead tails was corrected to 2,018 without altering the verdict. The register records no prior-art comparison.
Result
For the natural stationary p7 route targeting 21,738, the direct resume audit records 2,018 exact dead_tail entries among 181,441 blockers across 768 rows, with zero unresolved tails in the audited batch. The abstract state-{0,1} lemma and the positive-pin lemma are proved.
The route remains unexhausted. Without a rank-tight natural-p7 coefficient/divisor export, the hypotheses of these lemmas cannot be instantiated across all audited worlds. Global closure remains UNKNOWN, and the register status remains LIVE-PARKED.
Setting and definitions
The route is the natural stationary p7 family: a seven-product construction where every row applies the identical local rule, targeting count 21,738.
A blocker is an audited search state that prevents the current prefix from extending. A dead_tail is a blocker classified as provably non-extendable. An unresolved tail is a branch whose feasibility is not settled by the audit. The coefficient/divisor export is the missing rank-tight data record required to satisfy the hypotheses of the state-{0,1} and positive-pin lemmas. Global closure requires an argument covering the entire natural p7 prefix, beyond the audited dead tails.
Method
The result comes from a direct resume audit of the natural stationary p7 search pipeline. The run logged 181,441 blockers across 768 rows, resolving 2,018 entries as dead_tail with zero unresolved tails in that output. The abstract state-{0,1} and positive-pin lemmas were verified as proved components.
The audit did not generate the rank-tight coefficient/divisor export required for global closure. Verification artifacts and raw logs are available in this paper's downloadable evidence pack under resume SHA-256 4990e43355247aa6059a540e35f08c6481d59dd706a6cd06b78484fc06273dfa.
Discussion
An erratum updates the reported dead_tail count from 1,994 to 2,018 out of 181,441 blockers over 768 rows. The status is unchanged.
The proved abstract lemmas supply structural constraints, but the register lacks the rank-tight coefficient/divisor export required to establish their hypotheses across natural-p7 worlds. The route therefore remains an open conjecture in this class. The audit eliminates only the confirmed dead tails; it establishes neither global impossibility nor global existence, and leaves other p7 presentations unconstrained. The register states no prior-art position.
The zero unresolved tails figure applies strictly to the 181,441 classified blockers. The unexhausted prefix reflects branches that still require either the coefficient/divisor export or an alternative global proof.
For everyone — the takeaway
What this means
We audited 768 rows of the seven-product search and proved that 2,018 dead ends cannot continue. We also proved two helper rules about allowed state values and positive pins. To prove whether the full route works, we still need a compact certificate linking those helper rules to every remaining branch. The search is LIVE-PARKED: the verified dead ends and lemmas are confirmed, but the overall result remains unknown. The register notes no prior art.
Register references
ML-023lens_l3b_verification_report.md- Resume SHA-256
4990e43355247aa6059a540e35f08c6481d59dd706a6cd06b78484fc06273dfa - 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 1 receipt files bundled (4 KB). Anything not bundled is still hashed in the manifest and lives in the compute-box working trees.
Changelog
Last reviewed 2026-08-29
- 2026-08-13An internal audit corrected the earlier count of 1,994 to 2,018 dead_tail entries among 181,441 blockers over 768 rows. The overall verdict is unchanged, and the prefix remains unexhausted.
- 2026-08-29Published on this site.