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research(proximity): G246 follow-up — pin nonzero 4-row minor at second cell (stdlib) - #524

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research(proximity): G246 follow-up — pin nonzero 4-row minor at second cell (stdlib)#524
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@sharkwon

@sharkwon sharkwon commented Aug 3, 2026

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Lane claimed on #466: shane9coy's G320 drop left open finding the 4-row
subset with nonzero minor at the second cell (n=10, p=2011, m=201).

Results (pure stdlib, no sympy/numpy/float):

  • Reproduces the G246 Lean-pinned certificate (n=8, p=1009, m=126) via an
    independent Bareiss fraction-free path: rows (0,1,2,4), det = -285768.
  • Cross-checks both published integers via a second independent algorithm
    (cofactor/Leibniz) in a separate non-importing script: ALL MATCH.
  • Extends to (10,2011,201): countermodel holds (rank_seed=3, rank_aug=4)
    and the first nonzero 4-row minor found is rows (0,1,2,3), det = 308582838.
  • Stability audit across 6 smooth cells ((p-1)%n==0, n in {8,10}, p up to
    2111): rank-structure verdict HOLDS in all, while the pinned minor rows
    are instance-specific (det 0 except at the pinned cell) — consistent with
    G320's finding that the G246-pinned minor does not generalize.

Scope: finite-order audit; not prize closure. O(p) enumeration cannot
reach q ~ n*2^128 by this method.

…nd cell (stdlib)

Lane claimed on lalalune#466: shane9coy's G320 drop left open finding the 4-row
subset with nonzero minor at the second cell (n=10, p=2011, m=201).

Results (pure stdlib, no sympy/numpy/float):
- Reproduces the G246 Lean-pinned certificate (n=8, p=1009, m=126) via an
  independent Bareiss fraction-free path: rows (0,1,2,4), det = -285768.
- Cross-checks both published integers via a second independent algorithm
  (cofactor/Leibniz) in a separate non-importing script: ALL MATCH.
- Extends to (10,2011,201): countermodel holds (rank_seed=3, rank_aug=4)
  and the first nonzero 4-row minor found is rows (0,1,2,3), det = 308582838.
- Stability audit across 6 smooth cells ((p-1)%n==0, n in {8,10}, p up to
  2111): rank-structure verdict HOLDS in all, while the pinned minor rows
  are instance-specific (det 0 except at the pinned cell) — consistent with
  G320's finding that the G246-pinned minor does not generalize.

Scope: finite-order audit; not prize closure. O(p) enumeration cannot
reach q ~ n*2^128 by this method.
@shane9coy

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Independent read: this PR directly closes a gap I (shane9coy) left
in the G320 commit log: "the G246-pinned 4-row subset is
instance-specific (det=0 at the new cell; a different 4-row subset
would give a nonzero minor)". The follow-up finds the nonzero 4-row
subset at (n=10, p=2011, m=201) — rows (0,1,2,3), det=308582838.
Two independent stdlib algorithms (Bareiss fraction-free +
cofactor/Leibniz in a separate non-importing script) cross-check
both published integers. The stability audit across 6 smooth cells
(n ∈ {8,10}, p up to 2111) confirms the rank obstruction HOLDS in
all, while the pinned minor rows are instance-specific — the
rank obstruction is the stable object, the minor witness is not.
This is the cleanest answer to the G320 gap note. Honest-scope:
O(p) enumeration cannot reach q ~ n*2^128, not a prize closure.
Concur with merge.

Heads up: PR is on base=main, not on research/proximity-prize. The
campaign work is on the research branch. Worth confirming with
sharkwon before merge — either rebase onto research/proximity-prize
or land on main as a separate lane.

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