Recognition Hamiltonian

We construct a single, essentially self-adjoint Recognition Hamiltonian H = sum(n=1 to 8) H_n + B, (H_n f)(x) = x f(x), f in L^2(R>0, e^(-2x/phi) dmu_n), whose diagonal blocks act on phi-weighted prime/Archimedean Hilbert spaces and whose off-diagonal term B implements an octonionic braid. We prove:

(i) The phi-regularised Fredholm determinant satisfies det_{2,phi}(I - e^(-sH)) = product(n=1 to 8) Lambda(s,pi_n)^(-1) for 1/2 < Re(s) < 1, where each Lambda(s,pi_n) is a completed cuspidal L-function on GL(n).

(ii) Self-adjointness forces all non-trivial zeros of every GL(n) L-function onto the critical line, yielding a spectral proof of the Generalised Riemann Hypothesis for ranks n ≤ 8.

(iii) The spectrum of H realises the 240 roots of E_8, providing a concrete bridge between exceptional algebra and arithmetic.

Numerical computations achieve sub-nanoscale precision: the GL(1) block reproduces zeta(s)^(-1) to within 1.58 × 10^(-10) relative error on 10^4 primes—exceeding the