Zero simplicity — that each Riemann zero is a simple zero — is a distinct condition from zero location, and classical analytic approaches cannot reach it. This paper proves why: every classical method sees only the invariant subspace of the relevant operator and cannot access the anti-invariant subspace where the arithmetic content lives. The paper constructs the correct mathematical object — arithmetic cohomology H_arith for Spec(ℤ) with Hodge structure from Cl(1,1) — and names two programs for closing the condition: a Beurling-type sampling theorem (Program A) or a Hodge Index analog for H_arith (Program B). The Fundamental Theorem of Arithmetic is the non-degeneracy theorem both programs need to instantiate. The route is cleared; the passage is open. Abstract We identify the precise mechanism by which the simplicity of the zeros of the Riemann zeta function ζ(s) is enforced by the fundamental theorem of arithmetic. The condition C1.1-rev — Z″(t₀) ≠ 0 at all simple on-line zeros, equivalently Im(B/A) ≠ 0 — resists every classical analytic approach. We prove this resistance is categorical: a complete route taxonomy shows every known approach fails for the same structural reason, that analytic tools see only the invariant subspace (Re[H_Λ] = 0, a tautology) and cannot reach the anti-invariant subspace (Im[H_Λ]) where the arithmetic content lives. We then construct the correct mathematical object for the problem: an explicit arithmetic cohomology H*_arith for Spec(ℤ) carrying a complete Hodge structure, built from the proved Cl(1,1) structure on L²(ℝ). Within this object, C1.1-rev is precisely equivalent to the non-degeneracy of the second-order Hecke operator Π = Σ_p (log p)² T_p on H¹_arith. The algebraic skeleton of the function-field Hodge Index Theorem is reproduced — the same objects and pairings — but as an imposed construction, not as the hard theorem that, in the function-field case, produces the non-degeneracy. Reproducing the skeleton is not yet reproducing its force; that force is exactly Program B's open step. We identify the correct proof type as spectral rigidity from primal independence: the explicit formula makes the zeros the Fourier dual of the primes, and the linear independence of {log p} over ℚ — the fundamental theorem — forces a non-degenerate dual spectrum. Two programs are named: Program A (Beurling-type sampling theorem for the von Mangoldt-weighted comb) and Program B (Hodge Index analog for H*_arith). Subject to either program, unique factorization is the non-degeneracy theorem for the zeros. Keywords: Riemann zeta function, zero simplicity, arithmetic independence, fundamental theorem of arithmetic, explicit formula, prime-zero duality, arithmetic cohomology, Hodge structure, Hecke operators, Beurling-Kadec sampling, codimension-2 obstruction, C1.1-rev, scale separation, Cl(1,1) MSC2020: 11M06; 11M26; 42C15; 46C05; 11M41; 46E22 This paper is part of a single continuous derivation beginning from the axiom 'orientation capacity actualizes.' The full stack derives physics, consciousness, and organizational structure from the iterative operator z² + c — one axiom, one operator, seventeen papers. Each paper stands alone. Together they are one argument. The complete framework is at diasdimensions.org and the full stack is collected in the Dias Dimensions Research community on Zenodo.