The Position-wise Prime Digit Distribution Theorem: A Formal Proof of Position-wise Digit Equidistribution in the Prime Numbers
We state and prove the Theorem: for primes $p < 10^n$ with base-$10$ expansion $p = \sum_{k=0}^{n(p)-1} d_k(p) 10^k$, the positional digit probabilities $P_n(d \mid k)$ satisfy \[ \lim_{n \to \infty} P_n(d \mid k) = \begin{cases} 1/10, & k \ge 1,\ d \in \{0,\dots,9\}, \\[4pt] 1/9, & k = \mathrm{lead},\ d \in \{1,\dots,9\}. \end{cases} \] The limiting behavior splits cleanly into two distinct mechanisms: an arithmetic regime for interior digits and an Archimedean regime for the leading digit. For fixed interior positions ($k \ge 1$), digit extraction modulo $10^{k+1}$ reduces the problem to prime counts in reduced residue classes, where uniform distribution follows from Siegel--Walfisz (with Bombieri--Vinogradov allowing $k$ to grow with $n$). For the leading digit, the $1/9$ limit is not a Benford-type scale invariance, but arises from the local near-constancy of the prime density $1/\log t$ within single decades combined with a Toeplitz-type error averaging. Explicit classical and conditional error bounds are recorded for both regimes.