A Candidate Proof of the Riemann Hypothesis via the Local Band–Variation Theorem (Full Proof)

We develop a detailed annular harmonic–analytic framework around the completed
zeta function ξ(s) and a hierarchy of orthogonal differences on dyadic bands
that power orthogonalized Fejér energies. We then prove a fully detailed version of
the Local Band–Variation Theorem (LBVT)—a local Carleson frame estimate for the
atomic measure generated by the zeros of ξ on the critical line. The proof follows a
four–step scheme (smooth approximation → uniform Bessel bound → limit passage
→ discretization) and is written in a self–contained way with explicit constants.
Combining LBVT with global and localized “bridge” inequalities, per–tent lower
bounds, per–zero upper bounds, and an off–diagonal almost–orthogonality lemma,
we execute a Carleson absorption that yields depth control of lifts and excludes
off–line zeros at all heights in the presence of the local variation bound. We also
present an annulus/Klein–surface model illuminating the symmetry f(w) = f(1/w)
induced by the functional equation for ξ, and include several figures and tables
(parameter ledgers for κ, λ0, Ξ and working ranges of M, c, ϑ).
Key technical corrections. In Step 3 we use the standard identities (P\r ∗ g)(n) =
r|n|bg(n) and d∂θf(n) = in b f(n) and pass to μon via a finite–band continuity argument;
in Step 4 we implement an explicit discretization using (17) with aj = 1−e−iΘj and
the bound |1 − e−iΘj |2 ≤ 4 to obtain the band–variation inequality.