The Character Structure of the Collision Fluctuation

Universal across bases The Mertens growth law holds in every base tested:

The Collision Fluctuation Sum

The fluctuation sum The collision count at each prime fluctuates around its

Silent Primes and the Variance of the Collision Count

Silent primes Fix a rotation angle (lag) and ask: which primes have

Bin Derangements and the Gate Width Theorem

The ring Every fraction of a prime is the same ring of

Phase-Filtered Ramanujan Sums and the Spectral Gate

The Ramanujan connection The classical Ramanujan sum c_p(ℓ) = −1 for

The Autocorrelation Formula

The missing formula Paper 7 proved that eigenvalues are the DFT of

The Spectral Power of the Digit Function

The bridge The previous papers left a gap: the autocorrelation R(ℓ)

The Spectral Structure of Fractional Fields

From matrices to eigenvalues The cross-alignment matrix A(p) encodes the full

The Cross-Alignment Matrix

The complete object The previous papers measured the fractional field {k/n}

The Coherence Decomposition

Beyond alignment The previous papers measured one thing: how coherently the fractional

The Three-Tier Theorem

The conjecture The first paper in this series proved the alignment formula

The Alignment Limit for All Primes

The gap The first paper proved that the alignment formula α = (2m