Summary

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

$$C_{21} \le 0.2$$

([PT2021] applied to the Polymath15 criterion [P2019], the table's current
record) to

$$C_{21} ;\le; 0.1965 \quad\left(= \tfrac{1965}{10000}\ \text{exactly}\right),$$

via the Polymath15 criterion ([P2019], Theorem 1.2) instantiated at the barrier
site $X = 6000000185827$ with parameters $t_0 = 177/1000$,
$y_0 = \sqrt{39/1000}$, $N_0 = 690988$, so that
$C_{21} \le t_0 + y_0^2/2 = 1965/10000$ exactly. The only height input anywhere
in the chain is [PT2021] ($X/2 = 3000000092913.5 \le T = 3000175332800$, exact
slack $350479773/2$). The same certificate chain certifies two intermediate
rungs of the ladder at the same site: the re-tuned record row
$t_0 = 71/400$, giving $C_{21} \le 0.197$ exactly (secondary exhibit below),
and the historical parameter row $t_0 = y_0 = 0.1809$, giving
$C_{21} \le 0.197262405$ (exact terminating decimal) — see the prior-art
section below, which is essential reading for this submission.

Framed against the public record: $C_{21} \le 0.197$ is a certified
improvement below the recorded $C_{21} \le 0.20$ consequence of [PT2021]
(noted by Tao in the Polymath15 thread, April 2020 — the current table row);
$C_{21} \le 0.1965$ improves further and lies below the earlier uncertified
0.1972624050 sighting (Rudolph, May 2020) at the same site. The history of
the bound at this RH height is: 0.22 ([P2019]) → 0.20 ([PT2021] + Tao's
comment) → uncertified 0.1972624050 (Rudolph's comment, never verified) →
certified 0.197 and 0.1965 (this submission).

Prior art (please read first)

This bound must not be read as a new numerical discovery. An essentially
0.1973-class computation at the same barrier site was publicly reported in
the comment thread of the Polymath15 announcement post on T. Tao's blog:

• T. Tao, comment of 22 Apr 2020
  (#comment-554457):
  the [PT2021] verification "immediately improves the upper bound Λ ≤ 0.22 in
  the Polymath15 paper to Λ ≤ 0.20" — the step that became the current table
  row.
• Rudolph [Dwars], comment of 1 May 2020
  (#comment-556524):
  pursuing the paper's "extra decimals could be wrought out … We have not
  pursued this" remark, derives $N = 690988$ from the new height, reports that
  the mollified bound "becomes and stays larger than 0.03 for the choice
  $t_0 = y_0 = 0.1809$", concludes Λ ≤ 0.1972624050 ("to be on the safe
  side let's take 0.1973"), and reports a winding number of 0 at the
  barrier location $6000000185827$, linking a raw output file in
  km-git-acc/dbn_upper_bound as evidence.

That comment is parameter-exact prior art for the value class of this PR. It
was, however, never certified, written up, refereed, or independently
verified (no in-thread reply exists). The new content of this submission is
the certification: a machine-checkable certificate chain with two
mechanically independent verification lines for every ingredient, which
(a) certifies the comment's exact row ($C_{21} \le 0.197262405$), and
(b) sharpens it to $C_{21} \le 0.197$ exactly by re-tuning $(t_0, y_0)$ inside
the same certified slab. Full credit for identifying the site, the parameter
class, and the reachability of 0.197-class values from the [PT2021] height
belongs to that comment.

The headline bound: Λ ≤ 0.1965

$$C_{21} ;\le; 0.1965 \quad\left(= \tfrac{1965}{10000}\ \text{exactly}\right),$$

at the barrier site $X = 6000000185827$, parameters $t_0 = 177/1000$,
$y_0 = \sqrt{39/1000}$, $N_0 = 690988$, so that
$C_{21} \le t_0 + y_0^2/2 = 1965/10000$ exactly. Same height input
([PT2021] only, exact slack $350479773/2$); same certified barrier slab
(containment exact: $39/1000 \ge 0.1809^2$, $0.1770 \le 0.1809$). This
value lies strictly beyond the 2020 comment's value class (0.1973) and —
as certified — beyond the reach of the Euler-2-class mollifier its
calculation used (certified grid-sharp Euler-2 maximum $0.0201508110 <
0.03$ at this row); the route runs through a certified Euler-3 mollifier
bound ($0.1848895863 &gt; 0.03$ at the row point). The prior-art section
above is unaffected by this headline.

The certified legs of the 0.1965 chain (each with a standalone exit-0
verifier; line counts as in the paper's status table):

1. Hypothesis (i) — unchanged ([PT2021] only).
2. Hypothesis (ii) — the same certified 444-rectangle slab; the row's
   barrier region is strictly contained (exact-rational containments).
3. Hypothesis (iii) — gap-free: the Euler-3 mollified bound is
   ball-certified $&gt; 0.03$ at every integer $N \in [690988, 5000000]$
   (4,309,013 contiguous points; min $0.184889586327$ at $N_0$), plus a
   $t$-box-uniform certificate over $[0.1770, 0.1775]$ and an independent
   zero-shared-code block-uniform certificate over the same box (margin
   $\ge 6.066621\cdot 10^{-3}$ at $y_0$). A three-stage certified
   conversion chain (exponent alignment, cost $\le 1.8\cdot 10^{-14}$;
   normalization/leak accounting, certified conservative; $y$-transfer
   routing through the reduction theorem, composite $y$-uniform floor
   $\ge 0.0060665140$ for all $y \in [y_0, \sqrt{0.646}]$) carries the
   sweeps into the row-parameterized binding theorem (error budget
   $\le 1.06981\cdot 10^{-7}$, binding margin $\ge 0.0201721712$ at the
   0.03 threshold). The box-uniform tail covers all $N &gt; 5\cdot 10^6$
   ($|H_t/B_t| \ge 0.0833079015$ uniformly over a $(t,y)$-box containing
   the row's full hypothesis range).
4. Bookkeeping — the site-uniform window-gluing lemma covers every
   $t_0 \le 0.1809$ at this site; row margins $\ge 5377393.0179$ /
   $\ge 11989041.1446$; exact identities $t_0 + y_0^2/2 = 1965/10000$,
   $y_0^2 + 2t_0 = 393/1000 \le 1$.

Assembly certificate. The combined assembly manifest (bundle directory
certified1965/assembly_1965/) binds the legs: a citation-arithmetic
package whose standalone verifier re-derives the exact assembly
$C_{21} \le t_0 + y_0^2/2 = 177/1000 + 39/2000 = 393/2000 = 0.1965$
exactly and gates every joint (containments, identities, gap-free window
tiling of $[X,\infty)$ with certified splice overlap $\ge 125663718.7099$,
margins, composite finite-leg floor $\ge 0.0060665140$ floor-truncated) in
exact rational / outward-rounded interval arithmetic — 52 checks,
exit 0, re-run from a clean copy. An independent second-line sign-off
on the assembled statement (bundle directory
certified1965/assembly_secondline/; zero shared code, 42 gates,
exit 0) re-establishes the exact record arithmetic and every joint,
machine-reproduces the certified anchor values in two parameter regimes,
and adds a falsification probe at an in-window point; no discrepancy
found. Chain completed 2026-06-12; sign-off same day.

Secondary exhibit: the intermediate record Λ ≤ 0.197

The same certificate chain certifies the re-tuned record row
$t_0 = 71/400$, $y_0 = \sqrt{39/1000}$ at the same site, giving
$C_{21} \le 197/1000$ exactly — the intermediate rung of the ladder, and
the chain that the 0.1965 assembly builds on. All three hypotheses of
[P2019] Theorem 1.2 are discharged by certified
artifacts; every leg carries two independent verification lines with zero
shared code (C/Arb vs. from-scratch Python interval arithmetic / exact
rational arithmetic, formulas independently transcribed from the paper).
Every decimal digit string below is a machine-derived floor truncation
(toward $-\infty$) of a certified interval endpoint or exact rational — never
nearest-rounded.

1. Hypothesis (i) — RH height. Supplied by [PT2021] only:
   $X/2 = 3000000092913.5 \le 3000175332800$, checked in exact rationals on
   both lines. ($X$ sits within $5.85\cdot 10^{-5}$ relative of the
   $X = 2T$ ceiling — the last site this height can serve.)
2. Hypothesis (ii) — barrier winding number 0. Full slab
   $x \in [X, X+1]$, $y \in [0.1809, 1]$, $t \in [0, 0.1809]$: 444 rectangles,
   every winding field $0.0000000000$, overall winding 0, minimum mesh modulus
   $1.4850313937$ (five orders above the certified error budget
   $e_A+e_B+e_{C_0} \le 1.0466\cdot 10^{-7}$); reproduces the 2018 committed
   artifact 444/444 to ≥ 20 common digits — an independent re-validation of
   the file linked in the 2020 comment. Second line: a zero-shared-code
   interval recomputation of hash-chosen rectangles (selection seeded by the
   sha256 of the first line's own output) plus a hash-seeded corner audit.
   The stored-sums input matrix matches the sha256-pinned committed artifact
   on all 7692 fields by exact string equality, and an independent 170-bit
   line re-marched all 7688 components at 20 floor-truncated digits, 0 fail.
3. Hypothesis (iii) — Dirichlet lower bound. Gap-free: the Euler-2
   mollified selection bound is ball-certified $&gt; 0.03$ at every integer
   $N \in [690988, 5000000]$ (grid min $0.031340098433$ at $N_0$), an analytic
   tail certificate covers all real $N \ge 5\cdot 10^6$, and a mollified tail
   package gives $|f_{t_0}| \ge 0.2477133542$ uniformly beyond. A certified
   $y$-monotonicity bridge transfers the $y = y_0$ sweep to the full
   hypothesis range, and a record-binding lemma closes the assembly with
   certified margin $m_{\min} - E_{\max} \ge 0.0231662557 &gt; 0$. Second line:
   an independent block-uniform interval certificate covers the entire range
   with no gap, and an independent re-derivation of the binding lemma agrees
   on the final budgets.
4. Bookkeeping. $N(x,t) = 690988$ everywhere on the slab and the sweep
   window covers $[X, X+1]$ with exact margins ($\ge 5377392.8789$ /
   $\ge 11989041.1415$, uniformly over every $t_0 \le 0.1809$) — second line
   in pure integer arithmetic (no interval library; $\pi$ enclosed by two
   intersecting Machin-type brackets).

Two reporting notes for the maintainers, found while certifying against the
committed artifacts (details + certified replacement data in the artifact
bundle; neither affects any published row): the committed 0.21-row winding
log prints $y_0 = 0.15429$ in its header vs. $0.15492$ in its filename/Table 1
(favorable containment direction — row unaffected), and the committed
0.19/0.18-row stored-sums artifacts contain 10 entries (all im components)
that are nearest-rounded rather than floor-truncated at 20 digits (proven from
exact endpoints; certified corrected dumps supplied).

Verification

Paper and certificate bundle: https://doi.org/10.5281/zenodo.20724170 (digest
table in the paper's §Artifacts; MANIFEST.sha256 root-of-trust). The bundle is
self-contained; checkers are standalone (Python 3 + mpmath, or C sources
compiled against FLINT/Arb by the bundled script) and share no code with the
producers. Start with the bundle's front-page THEOREM_MAP.md (machine-readable
THEOREM_MAP.json): one row per certificate leg of both chains — exact
statement proved, bundle path, manifest digests, toolchain pins, frozen
expected output, run command. Expected replay, from the bundle root:

# One-shot self-check: bundle-wide MANIFEST.sha256 + all pure-Python
# verifiers (Python 3 + mpmath); the FLINT/Arb + Lean packages auto-skip when
# the toolchain is absent and print the exact command to run them elsewhere.
bash check_bundle.sh

# Representative individual replays, from the bundle root:
# -- pure-Python checkers (seconds each) --
python3 certificates/certified1965/assembly_1965/verify_assembly_1965.py    # 0.1965 assembly manifest, 52 checks, exit 0 (~2 s)
bash    certificates/certified1965/assembly_secondline/verify.sh            # independent 0.1965 sign-off, 42 gates, exit 0 (~3 s)
python3 certificates/record/binding/verify_record_binding.py                # record-binding lemma (line 1), exit 0
bash    certificates/record/binding_secondline/verify.sh                    # record-binding independent second line, exit 0
bash    certificates/record/winding_rects_secondline/verify.sh             # zero-shared-code winding rectangles (line 2), exit 0
bash    certificates/record/sweep_secondline/verify.sh                      # gap-free Dirichlet sweep second line, exit 0
bash    certificates/record/tail_secondline/verify.sh                       # tail second line (N0 row), exit 0
# -- FLINT/Arb packages (gcc + FLINT/Arb, Linux; re-run on Linux 2026-06-13, exit 0) --
bash    certificates/record/record_package_197/verify.sh                    # full 0.197 record manifest (winding/stored-sums/sweep/tail/y-bridge legs + assembly), exit 0
bash    certificates/certified1965/grid_full/verify.sh                      # Euler-3 full-grid sweeps (t = 0.1770 / 0.1775), exit 0
bash    certificates/certified1965/grid_tbox/verify.sh                      # t-box-uniform sweep over [0.1770, 0.1775], exit 0

Re-verification is cheap by design (the certificates function as the
"checksums" suggested in the same 2020 thread,
#comment-554639):
the heaviest single replay (full independent stored-sums re-march) is ~45
core-minutes; the barrier leg re-runs in ~256 cpu-s; everything else is
seconds-to-minutes.

Proposed page edit

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

| 0.1965 | [MI2026] | certified record package (Euler-3 mollifier route); same barrier site; see PR |

References — append:

- [MI2026] Mosaic Intelligence. "A certified unconditional upper bound Λ ≤ 0.1965
  for the de Bruijn–Newman constant." 2026. Paper and certificate bundle:
  https://doi.org/10.5281/zenodo.20724170

Attribution

Criterion, approximation machinery, and production code: the Polymath15
project ([P2019], km-git-acc/dbn_upper_bound). Height: [PT2021]. Value-class
prior art: Rudolph [Dwars]'s 1 May 2020 blog comment (cited above). New-content
tag: [MI2026], Mosaic Intelligence (@111111) — same
attribution as our submissions #92/#93/#94/#95.

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.