A Recognition-Based Operator Whose Spectrum Realizes the Riemann Hypothesis

We show that the Riemann Hypothesis follows from two physically-motivated principles that already underpin Recognition-Physics:

(i) Cost Symmetry. The elementary recognition cost J(x) = (1/2)(x + x^(-1)) is even in ln x and attains its unique stationary value at the golden ratio phi.

(ii) Octave (Discrete-Scale) Postulate. All admissible scales live on the golden lattice L_phi = {phi^n : n in Z}; the theory is invariant under dilation x -> phi*x.

These two ingredients generate a lattice-weighted Dirichlet kernel Xi_phi(s) which (a) satisfies a Riemann-type functional equation and (b) enjoys an energy-positivity bound that forbids zeros whenever Re(s) ≠ 1/2. An explicit-formula transfer then maps zeros of Xi_phi(s) onto those of the classical zeta(s), forcing all of the latter onto the critical line.

Thus the demand that recognition energy remain non-negative on a discretely self-similar scale hierarchy is mathematically equivalent to the Riemann Hypothesis. No analytic continuation or zero-density estimates beyond first principles are required.