A certified de Bruijn–Newman upper bound of 0.1729

The statement

Let $H_t$ and $\Lambda$ have the normalization of the pinned Polymath15 source. Set $X=6000000185827$, $t_0=3377/20000$, and $y_0=9/100$. Polymath15 Theorem 1.2 converts the three certified zero-free regions into the displayed bound.

Relative to the established Polymath15 theorem $\Lambda\le 11/50$, the exact decrease is $471/10000=0.0471$.

The three zero-free regions

The initial-time strip uses the published S-0001 theorem only for critical-line location through height $3000175332800$. The required heights $X/2$ and $(X+1)/2$ leave reserves $175239886.5$ and $175239886$; no zero-simplicity input is used.

At $t=t_0$, 1,649 low-$y$ cells and 107 high-$y$ cells cover the finite ray. Four Euler primes are used through $N=770000$, three through $N=2000000$, and two thereafter, followed by complementary analytic tails.

The intermediate barrier is covered by a pinned V3 stored-sum matrix and 844 adaptive argument-principle rectangles. The winding enclosure contains only the integer zero, and the exact-model boundary margin exceeds $0.95116$.

The cutoff repair

The sparse checker evaluates the literal state at the initial cutoff and after every reachable divisor threshold before taking termwise maxima. This repairs the terminal-cutoff substitution defect identified by O-0177.

The package-native regression enumerates all 451 reachable zero-through-four-prime state patterns, including all 353 four-prime patterns. On an actual theorem cell it rejects an omitted-initial-state counterfeit that passes the older bounded regression, closing the fail-open boundary identified by O-0184.

Independent finite evidence

Local arm64 and decorrelated x86 FLINT 3.6.0 evidence cover all 1,756 finite cells in canonical order. The weakest low-$y$ and high-$y$ endpoints are $0.0016881143766\ldots$ and $0.0241907009721\ldots$, both above the raw threshold $0.00114$.

The complete effective-model transfer cost is at most $0.00113438368318\ldots$, leaving strict positive reserve. The default verifier authenticates the package, reruns every analytic seam and state-pattern gate, checks both architecture evidence layers, and fails closed on source, runtime, grid, matrix, or transcript drift.