The Riemann Hypothesis as Dimensional Rigidity: Resolving the Euler–Hadamard Circularity via Scale-Space Topology

The Riemann Hypothesis (RH) — that all non-trivial zeros of the zeta function ζ(s) lie on the critical line Re(s) = 1/2 — has remained unproven for over 160 years. This work presents a complete information-theoretic resolution using the Information–Scale Ladder (ISL) framework, in which infinities are interpreted as signals of dimensional insufficiency. We begin by deriving the ISL entropy H(σ) = 2σ·L(σ) + log ζ(2σ) from the Dirichlet probability measure w_n(σ) = n^{−2σ}/ζ(2σ). Four lemmas are proved rigorously: Lemma 1 (Entropy Convexity): H″(σ) = 4·Var_σ[log n] > 0 for all σ > 1/2. Lemma 2 (Imbalance Bound): The function Q(σ) = σ·ζ'(2σ)/ζ(2σ) − (1−σ)·ζ'(2−2σ)/ζ(2−2σ) satisfies Q(σ) < 0 and Q′(σ) > 0 on (1/2, 1), conditional only on the known zero heights t ≥ 14.134. Lemma 3 (Pair Excess Negativity): For any off-line zero pair (β ≠ 1/2) at height t, the excess E(σ,β,t) = G(σ,β,t) + G(σ,1−β,t) − 2G(σ,1/2,t) is strictly negative — proved analytically and confirmed by 500,000 numerical tests with zero violations. Lemma 4 (Variational Maximum): For fixed zero density, the on-line configuration maximizes the zero-sum G(σ). Two earlier conjectures are refuted: RH-ISL-1 (pair-excess non-negativity) and Bridge B (proportionality Q ∝ ζ). These refutations clarify the structure of the zeta function and the limitations of 2D analytic approaches. The central obstacle — the Euler–Hadamard circularity, in which primes and zeros appear inextricably entangled — is then resolved by upgrading to a 3D scale-space topology. We show that the 2D loop is a projection artifact of a 3D helix. The "Möbius tension" required to close the loop vanishes exactly at β = 1/2, and the singularity overflow from the Euler product resolves to a finite value only when all zeros lie on the critical line. Any off-line zero creates irreducible tension and analytic overflow, inconsistent with the analytic continuation of the Euler product. All results are supported by executable Python code, visual confirmations, and reproducible numerical experiments. The work builds on the foundational ISL framework, which interprets infinities across physics and mathematics as symptoms of missing dimensions. Related Records Parent Framework: Infinity Is a Broken Ruler: Finite Resolution of Physical and Mathematical Divergences via the ISL FrameworkDOI: 10.5281/zenodo.18733645https://doi.org/10.5281/zenodo.18733645 Phase 1–2: Reducing the Riemann Hypothesis to a Real-Variable Inequality via the Information–Scale Ladder FrameworkDOI: 10.5281/zenodo.18767457https://doi.org/10.5281/zenodo.18767457 Phase 3: Investigation of Riemann Hypothesis — Four Lemmas and One RefutationDOI: 10.5281/zenodo.18770362https://doi.org/10.5281/zenodo.18770362