# Sources and dependency boundary

## Imported theorem framework

This result specializes the parameter-uniform proof in:

Jens Lehmann, *An Explicit Surjectivity Threshold for Digit Sums of
Primes*, arXiv:2606.04677v1.

The imported analytic statements are exactly Lehmann's:

- explicit moment comparison, Lemma 28;
- characteristic-function replacement, Proposition 2;
- explicit major-arc approximation, Proposition 3;
- minor-arc prime-sum bound, Lemma 25;
- Fourier positivity criterion, Lemma 36; and
- the integer-length conversion in Section 6.

No constant from Drmota--Mauduit--Rivat, Mauduit--Rivat, or
Iwaniec--Kowalski is changed here. Their role remains inside the
hash-pinned manuscript's proof.

## New layer

The new contribution is a rigorous numerical specialization at

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

and a sharper permanence proof for the second C13 inequality. The
source paper bounds the even-floor error by an absolute-value
remainder. Here the exact sign is retained:

```text
D = 2 floor(theta L^nu/log L)
Dtilde - 2 < D <= Dtilde
```

The C13 exponent is decreasing as a function of `D` on this interval,
so evaluating it at `Dtilde - 2` gives a one-sided continuous upper
envelope. This is the only changed analytic estimate.

## Corroboration boundary

`author_replay.py` runs the tagged author's exploratory threshold
code at the new rational decimal parameters. Its bisection values are
corroborative only. The theorem uses the outward-directed Arb and
Nemo certificates, not those floating-point roots.

## Classical prior work

- M. Drmota, C. Mauduit, and J. Rivat, *Primes with an average sum of
  digits*, Compositio Mathematica 145 (2009), 271--292.
- C. Mauduit and J. Rivat, *Sur un probleme de Gelfond: la somme des
  chiffres des nombres premiers*, Annals of Mathematics 171 (2010),
  1591--1646.
- G. Harman, *Counting Primes whose Sum of Digits is Prime*, Journal
  of Integer Sequences 15 (2012), Article 12.2.2.

These sources establish the qualitative context, not the new explicit
integer threshold.
