# Result

Outcome: completed

Result: Lehmann's explicit decimal prime digit-sum surjectivity
threshold decreases to
`138033986655720044641496559943864`.

For every integer

\[
m\ge138033986655720044641496559943864,
\qquad \gcd(m,9)=1,
\]

there exists a prime

\[
p\le10^{2m/9}
\]

with decimal digit sum \(s_{10}(p)=m\).

The previous exact value in arXiv:2606.04677v1 is
`177843590339381623423137292296355`. The decrease is
`39809603683661578781640732352491`, or approximately
22.3846153846154%.

The proof uses exact rational parameters

```text
eta = 545473/10000000
nu  = 714321/2500000
c43 safety = 1/200
```

and a signed one-sided envelope for the even-floor error in C13.
Every C1--C13 condition and every major, minor, AP, and absorption
threshold is certified permanently from the exact integer digit
length

```text
30674219256826676586999235543080.
```

Python-FLINT/Arb passes at 256, 384, and 512 bits. A separately typed
Julia/Nemo reconstruction passes at 384 bits, and the pinned author
code corroborates a sharper empirical value below the round proof
threshold. The complete wrapper reproduces all retained transcripts
byte for byte.

The focused literature search found no prior explicit threshold at
most the new integer. Novelty is qualified through 2026-07-19.

Two requested delegated hostile reviews failed before execution
because the required agent pool returned HTTP 503. The release audit
records this explicitly; no external-review result is claimed.
