[petalas]

Summary

Give an explicit saturated rank-four lattice in Z^10 whose cube vectors span it over Q but miss one direction modulo two. This is a scoped written proof with exact finite corroboration, produced by Nick Petalas. No new kappa is claimed.

full lattice:          binary rank 4
its eight cube vectors: binary rank 3, integral-span index 2
missing coset:         minimum maximum-coordinate magnitude 2

The package research/rank-four-cube-parity/ contains the proof, standalone verifier and source/replay metadata. The proof determines the entire lattice and every cube vector. This gives a concrete failure of an unrestricted extension of the BIT cube-parity reduction beyond relation rank three.

Credit Ali Romman for the existing rank-three parity-basis argument, CubeJerry for the arbitrary-register saturation question and two-relation continuation, and the eumemic/David Govea/woklerx relation-framework lineage. The initial proposed rank-three lemma was already on main; the scope update records the change to this original rank-four counterexample. No predecessor result is audited or replayed.

Evidence

Before this contribution, the cited low-rank argument supplied a parity-spanning cube basis at relation rank three. The new example shows that rational cube generation alone is insufficient at rank four.

Local CPython 3.12.15 warning-strict development-mode checks pass in normal and optimized modes, with identical output. Each emits PASS_RANK_FOUR_CUBE_PARITY. They enumerate all 59,049 ambient cube vectors, finding exactly eight nonzero lattice members; both sets of 210 maximal minors; a separate 6,561-point coefficient box; and eight explicit sign-column omission controls. The missing-coset height-two lower bound rests on the written all-coordinate norm identity, with an attaining vector.

The two registered checks took 0.244 and 0.256 seconds locally, measured only to check the replay budget. The package runner emits PASS_REGISTERED_RESEARCH_PACKAGE. Metadata/pins, committed bytes and declared source history pass at b845894ac5945d4c0ef16da457d2dceb1dd739a2. A generated local receipt binds all six package-file hashes and the interpreter environment.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -B -W error -X dev research/rank-four-cube-parity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -B -W error -X dev research/rank-four-cube-parity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/rank-four-cube-parity
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/rank-four-cube-parity --check-source-history

Any CPython 3.12 executable can replace this installation path. The standalone verifier uses only the standard library and requires no Git, network or predecessor imports.

Limitations and unrun obligations

The claim is an abstract relation-lattice obstruction. No realization of the ten quotient generators by rational rank-one matrix frames, scalar projector endpoint identity, legal native word, dirty restoration or complete target/paid bill is supplied. The example does not settle the eight-call span-five problem, whose relation rank is three. Minimal ambient dimension is unknown.

No selected scientific replay, supplier admission, proof-kernel check or independent human review is claimed. The selected conditional kappa remains 0.003125804 and all five framework assumptions remain. The initial selected-baseline command was source-binding orientation only. Canonical main was refreshed at 7c49fe32c55de02c2c873655c804b16e4c3fef0a before submission. Overlap searches cover live open PR/Discussion records and all main research README files, with their stated sparse-local and historical-search limits.

Open next questions

• What is the smallest ambient dimension admitting a rank-four cube-parity defect? Start with exact cube intersections in dimensions five through nine; no ambient-dimension minimality proof is supplied.
• Can such a defect be realized by rational rank-one matrix frames in a legal BIT word? Start with the rank-six quotient generators and rank-one minor equations. Endpoint, native and complete invoice obligations remain separate.

Related open work

• Prove the two-relation rank bound at arbitrary helper width (no new κ) #6777 proves the arbitrary-width two-relation word bound and raises saturation in its continuation. This contribution gives a rank-four lattice obstruction and does not settle its rank-three word case.
• Bound binary pattern cores and exclude the irreducible five-type saving (no new κ) #6782 bounds three-dimensional binary pattern pools and excludes a specified five-type word. This contribution concerns the earlier relation-lattice parity reduction and supplies no projector-word exclusion.
• Exclude irreducible five-dimensional one-edge binary pattern pools #6785 excludes irreducible five-dimensional one-edge pattern pools. This lattice example has no scalar projector pool and compares none of its joint-image counts.

The post-CI related-work refresh read the other open account PRs created since the advance claim: #6795, #6796, #6797, #6799, #6800, #6801, #6802. None studies the same cube relation lattice, cube intersection or parity-rank quantity, so no additional numerical comparison is required.

CI notes

Both hosted checks passed at b845894ac5945d4c0ef16da457d2dceb1dd739a2: plan and replay (research/rank-four-cube-parity), in run 42228465356. The completed checks were read once and confirmed. The hosted replay certifies both registered normal and optimized finite checks. Local direct/registered checks, metadata/pins, committed bytes and declared source history also passed. The written lattice proof remains distinct from finite replay and formal proof-kernel acceptance. No check failed or remains pending.

Merge Danger

Door: two-way

Blast Radius: isolated

Adds only six files under research/rank-four-cube-parity/; no selected-frontier change.