Status: WITHDRAWN / FALSIFIED UNDER CURRENT FORMULATION Date withdrawn: 2026-08-15
Claim
The constructive block-positivity residue transplants to the pinned upstream zero side, giving an unconditional lower bound of 0.672529 on the proportion of simple zeros of ζ(s) on the critical line.
Failure Route 1: White-Box Kill
The earlier direct analysis identified the load-bearing structural failure.
The mechanism: The construction assumed that off-line pair blocks interact non-negatively with the on-line part. This fails because the pinned upstream zero side uses the un-conjugated transpose u u^T, not u u*. For an off-line root γ = α + iβ, the vector u = φ̂(γ − τ) is complex (u = x + iy). The pairing of such a root with its conjugate yields:
u u^T + ū ū^T = 2(xx^T − yy^T)This is a hyperbolic block. A matrix of this form has a negative eigenvalue (unless y = 0, which holds only for on-line roots). The off-line block therefore interacts negatively with the on-line part.
Exact witness: u_x = 1, u_z = i, u_conj(z) = −i gives tr(P₁Q′) = −2. With five unit on-line labels the final inequality reads 9 ≥ 13, a contradiction.
Artifacts:
- Report:
hunts/frontier_math/CLEAN-KILL-REPORT.md - Script:
hunts/frontier_math/clean_kill.py - Lean obstruction:
lean/ZetaLean/FrontierMathObstruction.lean - Regression test:
tests/test_frontier_math_clean_kill.py
Failure Route 2: Blind Attack (Independent Confirmation)
A separately executed blind attack was run against blockpos-0.672529 without prior knowledge of the white-box failure route.
Branch: research/blind-attack-blockpos Research commit: b5788357bdd52e137fdf273c5fcd4815dafea535 Canonical PR: #39 Canonical merge: 78656b0230f92de5636c1d2aa604de3ccffa42de
The question posed: What modification of the world would preserve the appearance of this result while making its interpretation false?
Finding: If an implementation evaluates u u* instead of u u^T, it constructs a Gram matrix that is positive semi-definite by definition. This perfectly preserves the appearance of block positivity, but makes the interpretation false. The actual u u^T construction for an off-line root yields a hyperbolic block. A dedicated scan found the cross-block interaction to be approximately −0.000435, strictly negative.
Artifacts:
- Report:
hunts/frontier_math/BLIND-ATTACK-REPORT.md - Script:
hunts/frontier_math/blind_attack.py - Ledger entry:
harness/departments/review_ledger.py(AttackOutcome, role="blind", claim_withdrawn=True) - Test:
tests/test_review.py - Integration commit:
78656b0230f92de5636c1d2aa604de3ccffa42de(PR #39)
The blind attack independently reached the same structural diagnosis as the white-box kill. Both failure routes are preserved separately. They are not collapsed into one event.
What This Does Not Establish
- This does not establish a new zero-density or zero-proportion theorem.
- The number 0.672529 is not a surviving result.
- The siblings 0.6725124 and 0.6725318 are also withdrawn.
Residual Speculative Thread
The possibility that some contour or integration reformulation could replace u u^T with a legitimate u u*-type structure is recorded as:
Status: PARKED / SPECULATIVE
It does not keep blockpos open. It should not receive immediate research priority.
Research Record Note
A candidate improvement was developed far enough to expose a specific structural claim, attacked independently under a blind protocol, falsified, and withdrawn with the failure evidence preserved. The adversarial record is retained in full in the graveyard and review ledgers.
Relation to Pub 1
This result postdates the Pub 1 scientific snapshot (24fce230968e8d78f6d50ccba39f6ee72da926ee). It does not modify Pub 1's claims, evidence snapshot, or frozen SHA.
Next Frontier Item
R-FB9C81 / urms2-0.51: no recorded blind attack yet; review is not yet standing.