---
id: C-0092
type: claim
title: "A smaller explicit surjectivity threshold for decimal digit sums of primes"
status: proved-unconditional
family: adjacent-unconditional
interestingness: 3
statement: >
  For every integer
  m>=138033986655720044641496559943864 with gcd(m,9)=1,
  there exists a prime p<=10^(2m/9) whose decimal digit sum is m.
  This lowers Lehmann's arXiv:2606.04677v1 exact explicit threshold
  177843590339381623423137292296355 by 22.3846153846154%.
verifier: canon/witnesses/C-0092/verify_all.py
depends: []
supersedes: []
sources:
  - literature/2026-07-19-lehmann-prime-digit-sum-surjectivity-threshold-search.md
  - canon/witnesses/C-0092/PROOF.md
  - canon/witnesses/C-0092/PIN.md
  - canon/witnesses/C-0092/RELEASE-AUDIT.md
  - canon/witnesses/C-0092/source/lehmann-2606.04677v1-source.tar
  - scratch/adjacent-unconditional--lehmann-prime-digit-sum-surjectivity-threshold/RESULT.md
tags: [prime-digit-sum, prescribed-digit-sum, explicit-threshold, circle-method, lehmann, drmota-mauduit-rivat, interval-arithmetic, arb, nemo]
caveats: >
  The result imports the analytic lemmas proved in Lehmann
  arXiv:2606.04677v1 and recertifies their parameter specialization
  and complete threshold closure; it does not rederive the full
  DMR/MR analytic chain. The source remained an arXiv v1 preprint on
  2026-07-19. No optimality or least-threshold claim is made. Novelty
  is qualified by the documented search through 2026-07-19. Two
  requested delegated hostile reviews failed before execution
  because their account pool returned HTTP 503; the independent Nemo
  implementation is a separate computational reconstruction, not a
  third-party review.
---

## Statement

Let \(s_{10}(n)\) be the sum of the decimal digits of \(n\). For
every integer

\[
m\ge
138033986655720044641496559943864
\]

with \(\gcd(m,9)=1\), there exists a prime

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

such that \(s_{10}(p)=m\).

Lehmann's arXiv:2606.04677v1 proves the same theorem with

\[
177843590339381623423137292296355.
\]

The exact decrease is

\[
39809603683661578781640732352491,
\]

or \(22.3846153846153846\ldots\%\).

## Mechanism

Use Lehmann's parameter-uniform major/minor-arc theorem with

\[
\eta=\frac{545473}{10000000},
\qquad
\nu=\frac{714321}{2500000},
\]

and the source's \(1/200\) safety factor for \(c_{43}\). Put

\[
Y_\dagger=7.063\times10^{31},
\qquad
L_\dagger
=30674219256826676586999235543080.
\]

Directed Arb arithmetic proves

\[
L_\dagger
=\left\lceil\frac{Y_\dagger}{\log10}\right\rceil
\]

and

\[
\left\lceil
\frac92\frac{Y_\dagger}{\log10}+\frac92
\right\rceil
=138033986655720044641496559943864.
\]

For every integer digit length \(L\ge L_\dagger\), the witness proves
all moment-comparison conditions C1--C8 and characteristic-function
conditions C9--C12 permanently.

For C13, write

\[
\widetilde D=\frac{2\theta L^\nu}{\log L},
\qquad
D=2\left\lfloor\frac{\widetilde D}{2}\right\rfloor,
\qquad
W=\widetilde D-2.
\]

The exact C13b exponent is

\[
h(D)=(D-1)T+\frac D2-\frac D2\log D,
\qquad
T=\eta\log L+\log C.
\]

On the whole floor interval \(W<D\le\widetilde D\),

\[
\frac{\partial h}{\partial D}=T-\frac12\log D<0.
\]

Hence \(h(D)\le h(W)\). The resulting continuous upper envelope has
endpoint margin

\[
14.59162285254167\ldots
\]

and a permanently negative derivative; after division by \(L^\nu\),
the directed derivative reserve is

\[
2.906191324110626\ldots\times10^{-9}.
\]

This signed floor treatment replaces the paper's coarser
absolute-value remainder envelope.

At \(Y_\dagger\), the major-arc inequality has directed margin

\[
2.84673134769017\ldots\times10^{-6},
\]

all its exponents are negative, and the minor-arc logarithmic
inequality has margin

\[
0.004475302156707718\ldots
\]

with permanently decreasing left side. The witness separately
checks \(Y_{41}\), \(Y_{42}\), the AP threshold, the integer
\(Y_{47}\) ceiling, and the C1--C8 integer threshold.

Thus \(m\) above the displayed threshold gives

\[
L=\left\lfloor\frac{2m}{9}\right\rfloor\ge L_\dagger.
\]

Every source hypothesis holds at \(x=10^L\), so Lehmann's positivity
argument gives a prime

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

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

```text
exact_map: PASS -- hash-pinned Lehmann v1 parameter-uniform theorem -> exact rational eta,nu,c43 -> permanent directed C1-C13 and major/minor/AP/absorption certificates -> exact integer floor/ceiling handoff -> prescribed decimal digit sum of a prime
counterfeit: PASS -- the tagged author's mpmath bisection, pointwise passage of the exact predicates, or a rounded real threshold without permanent monotonicity and integer-ceiling control does not prove the theorem
counterfeit_failure: PASS -- three Arb precisions and a separately typed Nemo implementation prove positive endpoint reserves, permanence, source hashes, and the exact integer handoff; the author-code roots are retained only as corroboration
success_delta: PASS -- the explicit sufficient threshold on the identical theorem domain decreases by 39809603683661578781640732352491, or 22.3846153846154%
failure_delta: PASS -- any failed condition identifies the exact C1-C13, major, minor, AP, absorption, source-pin, or integer-conversion branch and leaves Lehmann's published threshold unchanged
stop_condition: PASS -- stop on a prior explicit threshold at most the candidate, source/hash drift, any nonpositive directed reserve, an unproved permanence step, an omitted source branch, or inward rounding in the integer conversion
```

## What survives

The result is unconditional relative to the published analytic lemmas
it imports and does not assume RH. It changes only the explicit
numerical threshold, not the congruence obstruction
\(\gcd(m,9)=1\), the exponent \(2m/9\) in the prime upper bound, or
the underlying circle-method architecture.

The certificate does not show that the new threshold is least,
globally optimal over admissible parameters, or close to the true
onset. Smaller rational specializations, sharper C13 envelopes, or
improved analytic constants remain open.

## Provenance

Primary source: Jens Lehmann, *An Explicit Surjectivity Threshold for
Digit Sums of Primes*, arXiv:2606.04677v1, submitted 2026-06-03.
The accompanying repository is pinned at tag
`v1.0-mcom-submission`, commit
`9f9903a862b83ad3b1c2fbb961af1ecf7857d214`.

Parameter search, directed certification, independent reconstruction,
and literature search were completed 2026-07-19.
