The Square-Root Horizon: Squares, Reversal and Prime Clocks in the Distribution of the Primes and the Zeros of the Riemann Zeta Function

Source abstract

We read the primes and the zeros of the Riemann zeta function through three exact structures: the residue classes of the divisor in the rebuild law $\sum_{r\le x}M(x/r)=1$, the squares of the primes, and the reversal of the Euler product. Split by $r\bmod3$ the rebuild law is $(1,a(x),-a(x))$. Its zero class reproduces the whole law at $x/3$, so the classes modulo $3^\kappa$ form a self-similar tower whose leaf energy is $\ll x^{1+\varepsilon}$ for every $\varepsilon>0$ if and only if the Riemann Hypothesis (RH) holds; at every zero of $\zeta$ the three classes stand as $(0,a,-a)$. Each prime square enters the logarithm of the Euler product with weight $\frac12$, and at that weight the lean of the prime count vanishes: the weight fitted from the exact counts to $10^{10}$ is $0.4994$. The squares part of $\zeta$ is zero-free for $\sigma>\frac12$ and has a half-order pole at $s=\frac12$, the square-root horizon, matched by a half-order zero of the prime factor, which carries every zero of $\zeta$ in $\sigma>\frac12$. Reversing the sign of every prime turns $\zeta(s)$ into $\zeta(2s)/\zeta(s)$ and every zero into a pole; the Liouville-weighted sawtooths rebuild the square staircase exactly, and if they approximate the odd-square staircase in $L^2(0,\infty)$ then RH holds. On the critical line every forward term of the approximate functional equation is balanced by its reversed partner, and off it no two pairs are balanced at once; the pair truncated without its remainder has zeros off the line, at each of which $\zeta$ itself is far from zero. The Jensen polynomials of $\Xi$, read as Jacobi chains, have every coupling positive to degree $400$, and each prime pulls the zeros toward the reversal of its clock, as Landau's formula prescribes. We state the axiom of harmonic independence: the prime waves are related only through the mechanism of their construction, and a zero off the critical line would be a relation without a mechanism. The Davenport--Heilbronn function, whose zeros leave the line, adds two Euler products with the root number as weight and couples distinct primes in its logarithm, as the axiom predicts.