The Brindel Transformation: Arithmetic Isomorphism, Coaxial Circles, and Dynamical Fixed Points

Description We introduce and develop the geometric and arithmetic properties of the Brindel transformation, defined by the fixed-point equation n^ (sₙ) = n for integers n ≥ 2. The map φ (n) = ln (n) /2π is shown to be a monoid isomorphism from (N*, ×) to an additive subgroup of (R, +), in which prime numbers are exactly the generators. A rotation identity for the Riemann zeta function is established as a direct consequence. Extending the transformation to complex base z, we characterize the level sets Re (sᵦ) = σ as a family of coaxial circles in the (ln|z|, arg z) plane, with the critical circle C₁/₂ of radius 2π passing through the origin. The dynamical fixed points of the map z → sᵦ form an infinite family zₖ* satisfying the exact symmetry σₖ + σ-₊-₁ = 2 and the asymptotic expansion zₖ* = (k+1) /k + i/ (2π k³) + o (k⁻³). For roots of unity ω₍, ₊ = exp (2πik/n), we prove Re (s⏨_₍, ₊) = 1 + n/k. All results are fully proved and numerically verified.