Improve upper bound for the de Bruijn–Newman constant to 0.1875 (certified record package)

[463464q435q43]

Summary

This updates the best known unconditional upper bound for the de Bruijn–Newman
constant from

$$C_{21} \le 0.1965$$

(our previous certified record, PR #101 / [MI2026]) to

$$C_{21} ;\le; \tfrac{3}{16} ;=; 0.1875 \quad\text{exactly},$$

via the Polymath15 criterion ([P2019], Theorem 1.2) instantiated at the same
barrier site $X = 6000000185827$ with parameters $t_0 = 1680/10^4$,
$y_0 = \sqrt{39/1000}$, $N_0 = 690988$, so that
$C_{21} \le t_0 + y_0^2/2 = 1875/10^4 = 3/16$ exactly. The only height input
anywhere in the chain remains [PT2021] ($X/2 = 3000000092913.5 \le
T = 3000175332800$, exact slack $350479773/2$).

The route differs from the 0.1965 chain: instead of a single mollified sweep
at one $(t_0, y_0)$ row, the hypothesis-(ii) region is closed by a certified
window tiling — the integer $N$-range $[N_0, \infty)$ is split at
$N_{\mathrm{mid}} = 2745000$ (the split point covered by both adjacent legs),
with the finite window closed by a certified mollified window certificate and
the tail closed by a certified cutoff-descent tail certificate, and every
joint (containment, tiling identity, split-point overlap, margin) gated in
exact rational / outward-rounded interval arithmetic by a standalone assembly
verifier (58 checks, exit 0, re-run twice from a clean copy). The same
assembly ladder certifies the intermediate rung $C_{21} \le 0.1891$ at the
same site (shipped in the bundle).

Prior art (please read first)

The prior-art framing of PR #101 / [MI2026] carries over unchanged and remains
essential reading: the barrier site, the parameter class, and the reachability
of sub-0.20 values from the [PT2021] height were identified in the Polymath15
blog thread (Tao, 22 Apr 2020, #comment-554457; Rudolph [Dwars], 1 May 2020,
#comment-556524, uncertified $C_{21} \le 0.1972624050$). The values below
0.1965 in this submission have, to our knowledge, no prior published or
informally reported counterpart at any site; the contribution remains the
certified, machine-checkable chain with independent verification lines.

The certificate chain

All three hypotheses of [P2019] Theorem 1.2 are discharged by certified
artifacts. Every decimal digit string below is a machine-derived
floor/ceiling truncation of a certified interval endpoint or an exact
rational — never nearest-rounded.

1. Hypothesis (i) — RH height. [PT2021] only, checked in exact rationals:
   $X/2 = 3000000092913.5 \le 3000175332800$, exact slack $350479773/2$,
   anchored against the published 0.2-row identities.
2. Hypothesis (iii) — barrier winding. A certified boundary-winding-0
   certificate on the slab $[X, X+1] \times [33/200, 1]$, uniform over all
   real $t \in [0, 0.1809]$ (two verification lines: the producing
   certificate and an independent corner audit), combined with a certified
   $N$-constancy certificate on the slab whose two exactness anchors
   ($\ge 5377392.8789$ / $\ge 11989041.1415$, floor-truncated) are
   re-derived inside the assembly verifier at 220-bit interval precision.
   The slab-to-hypothesis binding is the same one used by the published
   0.1965 assembly, consumed here for a strictly smaller $t$-range; the slab
   joints are gated exactly (slab floor $(33/200)^2 \le y_0^2$ with margin
   $471/40000$; lower-edge monotonicity with $q(0) = 3/8 \le 1$ symbolic;
   $t_0 \le 0.1809$).
3. Hypothesis (ii) — Dirichlet lower bound, window tiling.
   • $N \in [690988, 2745000]$, all real
     $y \in [\sqrt{39/1000}, \sqrt{1-2t}]$: a certified mollified
     finite-window certificate on the closed tile
     $t \in [1680/10^4, 1685/10^4]$ (containing $t_0$ as its left endpoint,
     membership exact; block bottom $N_c = N_0$, so no uncovered sub-block;
     composite floor $\ge 0.0134320455$, floor-truncated, re-gated in the
     assembly).
   • $N \ge 2745000$: a certified tail certificate at cutoff
     $N_1' = 2745000$ on the band $t \in [1680/10^4, 1696/10^4]$, full
     $y$-range, re-derived live inside the assembly verifier: box enclosure
     on the $y$-hull at head depth $M = 50000$, termwise $y$-monotonicity
     above the hull, all 28 monotonicity constants certified negative
     (binding value $-0.033771$, ceiling-truncated), cap side conditions
     re-gated at the cutoff ($D \le 0.9854851520 < 1$; tail slack
     $\ge 0.0091331201 > 0$), the machinery anchored bit-identically against
     its certified reference regime ($N_1 = 5\cdot 10^6$, $M = 20000$:
     $D \le 0.7334562057$, flow $\ge 0.1747906321$).
   • The two legs meet at the single split point $N_{\mathrm{mid}} =
     2745000$, covered by both; coverage of every real $x \ge X$ is gated by
     the exact window-boundary identity and window ordering. Zero-freeness
     is mollifier-independent, so the legs compose across their different
     mollifier classes, exactly as in the published 0.1965 assembly.
4. Assembly certificate. The record package (bundle directory
   certificates/certified1875/assembly_1875/) binds the legs: a standalone
   verifier (Python 3 + mpmath interval arithmetic at 220 bits + sympy exact
   symbolics; reads no files) re-derives the exact record arithmetic
   $C_{21} \le t_0 + y_0^2/2 = 3/16$ and gates every joint — 58 checks,
   exit 0, re-run twice from a clean copy; independently re-run on a
   separate machine before submission. An independent second-line
   sign-off on both assembled statements (bundle directory
   certificates/certified1875/assembly_secondline/; zero shared code,
   enforced by an AST audit that forbids file reads in the certificate
   script) re-establishes the record arithmetic and every joint on its own
   line and re-proves the live tail leg on an independent engine with
   strictly sharper endpoints — 76 checks, exit 0, completed
   2026-07-03; no discrepancy found.

The intermediate rung $C_{21} \le 0.1891$ ($t_0 = 1696/10^4$, same $y_0$;
pure citation-arithmetic over the same certified inventory, 58 checks,
exit 0) ships in the bundle as certificates/certified1875/assembly_1891/.

Verification

Paper and certificate bundle: https://doi.org/10.5281/zenodo.21175533 (digest
table in the paper's §Artifacts; MANIFEST.sha256 root-of-trust). The bundle is
self-contained; checkers are standalone and share no code with the producers.
Start with the bundle's THEOREM_MAP.md. Representative replay, from the
bundle root:

bash check_bundle.sh
python3 certificates/certified1875/assembly_1875/verify_assembly_1875.py   # record assembly, 58 checks, exit 0 (~21 s)
python3 certificates/certified1875/assembly_1891/verify_assembly_1891.py   # intermediate rung, 58 checks, exit 0 (~2 s)

Re-verification is cheap by design; the heaviest single replay completes in
under a minute.

Proposed page edit

constants/21a.md, "Known upper bounds" table — append:

| 0.1875 | [MI2026b] | certified record package (window-tiling assembly route); same barrier site; see PR |

References — append:

- [MI2026b] Mosaic Intelligence. "A certified unconditional upper bound Λ ≤ 0.1875
  for the de Bruijn–Newman constant." 2026. Paper and certificate bundle:
  https://doi.org/10.5281/zenodo.21175533

Attribution

Criterion, approximation machinery, and production code: the Polymath15
project ([P2019], km-git-acc/dbn_upper_bound). Height: [PT2021]. Site and
value-class prior art (0.197 class): Rudolph [Dwars]'s 1 May 2020 blog
comment (as credited in PR #101). New-content tag: [MI2026b], Mosaic
Intelligence — same attribution as PR #101 and submissions #92–#95, #124.

AI assistance disclosure

This is a fully AI-derived result: the construction was found and certified by
Mosaic Intelligence's automated search-and-verification system, and the
submission text was AI-prepared. All numerical results and references were
independently re-run and verified before submission.