Generalized Harmonic Laws for Prime Digit-Length Transitions in Base Representations: Unique Structure in Base 7

We study the digit-length behavior of prime numbers in integer bases. For any base b ≥ 2, the length of a prime’s representation increases by +1 at thresholds b^k. We prove that the index n_k of the first prime crossing b^k satisfies n_k ≈ b^k / (k·log b), and that log_b(p_nk) ≈ k – log_b(k·log b). This reveals a harmonic law governing digit-length transitions. In base 7, this structure becomes visually regular: digit-lengths form long plateaus, with precise +1 jumps at primes just above 7^k. The jump indices grow quasi-geometrically. Within each plateau, the log-distribution of primes follows a 1/(1+u) bias. These features, unique to base 7, align with previous work on Rational Harmonic Arithmetic, where constants like φ₇ = 8/7 and r ≈ 1.12678 describe similar growth dynamics. Our computations extend to the first 1,000,000 primes and show that this behavior is not captured by classical bounds like those of Bertrand, Schoenfeld, or Dusart.