We develop an arithmetic framework for studying the Riemann Hypothesis through the summatory Liouville function L (x) = Σ_ (n ≤ x) λ (n), where λ (n) = (-1) ^ (Ω (n) ) is the Liouville completely multiplicative function. Our contributions are organized in three parts. Part I: An arithmetic heuristic. We present a heuristic argument toward the bound |L (x) | = O (x^ (2/3+ε) ) built from the Square Root Identity, the Dirichlet hyperbola method, and a Sobolev interpolation lemma applied to the Matomäki–Radziwiłł mean value theorem. We emphasize that this bound is not established here: it is a power saving, stronger than the classical Vinogradov–Korobov estimate and currently unknown unconditionally. The value of the argument is that it is entirely arithmetic—it bypasses the Riemann zeta function and its zeros—and that it isolates the precise estimate that would be required: an averaged square-root-cancellation input at every dyadic scale (Hypothesis (SR), Section 3), strictly beyond the Matomäki–Radziwi () theorem and belonging to the same difficulty class as the Chowla-type inputs needed elsewhere in the paper. Given (SR), the cross-arc terms are controlled by Cauchy–Schwarz, and signed cross-arc cancellation is needed only to lower the exponent further; an earlier draft, working from a square-root-cancellation misreading of Matomäki–Radziwi (), located the obstruction in the cross-arc terms, and that assessment is corrected here. Part II: Exact identities and the energy framework. We derive four exact identities from the complete multiplicativity of λ: the Dyadic B-Chain, linking the additive interaction term B (x) at consecutive dyadic scales; the Energy Identity, connecting a weighted variance functional V (X) = Σ_ (n ≤ X) L (n) ²/n (n+1) to Chowla-type correlations; the Prime Modulator Decomposition, expressing B (x) as a Leibniz alternating series at each prime; and the Squarefree Representation. We establish the energy equivalence in the direction V (X) = O (log X) ⇒ RH unconditionally; the converse RH ⇒ V (X) = O (log X), together with the sharp constant V (X) CVlog X, is established conditionally on a Gonek-type second-moment hypothesis for Σ_ (n≤ X) L (n) ² (which is stronger than RH alone), and we are explicit that RH by itself does not yield it. Here CV = ζ (1/2) ^ (-2) + 2Σ_ (γ>0) |ζ (2ρ) / (ρ ζ' (ρ) ) |^ (2) ≈ 0. 5408 is an explicit sum over the nontrivial zeros; this corrects the squarefree-density value 6/π² ≈ 0. 608 used in an earlier draft, and the corrected constant is confirmed by the numerics at the percent level (Section 14. 8). The variance recursion establishes the exact weighted decomposition V (2X) = (1/4) V (⌊ X/2⌋) + 2C_ (×) (X) + N (X) + E (X), with an explicit boundary term |E (X) | ≤ 8 (L (⌊ X/2⌋) ²+1) /X², in which the contraction factor 1/4 is unconditional and exact; an unweighted form of this recursion asserted in an earlier draft is false and is retracted with a numerical refutation (Remark 9. 1). The Quarter-Scale Contraction identifies 1/n (n+1) as the unique weight producing exact algebraic telescoping, and the Weight Duality Theorem shows this weight simultaneously enables the contraction and interfaces with Tao's logarithmic Chowla results; an exact Beta-integral representation V (X) = ₀¹ (1-t) Σ_ (n ≤ X) L (n) ²t^ (n-1) dt further identifies the weight as the Beta moment B (n, 2) and recasts the energy condition in Tauberian terms. We identify the bounded odd-increment condition (BOI), verified to 2^ (29) ≈ 5. 4 × 10⁸, as an unconditionally sufficient condition for RH (via V (X) = O (log X) ; Theorem 10. 12) ; its necessity (that RH implies BOI) we establish only conditionally on the same second-moment hypothesis, and we prove BOI cannot be derived from the Matomäki–Radziwiłł theorem alone. We introduce the prime factorization tower, a novel decomposition that iteratively removes prime oscillations from λ, expressing cross-interval correlations as infinite products of prime-specific transmission factors. Three additions sharpen Part II into an exact dictionary. A growth-exponent theorem proves, unconditionally, limsupX log V (X) /log X = 2Θ - 1 where Θ is the supremum of the real parts of the zeros; hence RH is equivalent, with no auxiliary hypothesis, to V (X) = X^ (o (1) ), and each fixed exponent < 1 for V is a strictly weaker, currently open, zero-free half-plane target. An Abel–Karamata bridge shows (using only L (n) ² ≥ 0) that the leading term of the second-moment hypothesis is exactly equivalent to the boundary limit (1-t) ²Σₙ L (n) ² t^ (n-1) → CV, measured here as 0. 5406 at t = 1 - 10^ (-6). And the anti-channel constants acquire closed forms: the measured per-octave slope -0. 874 and restoring conspiracy -0. 167 are identified, at the same conditional level as the CV computation, as the zero-side averages ⟨cos (γlog 2) ⟩w = -0. 8766 and ⟨ 2^ (-1/2) +cos (γlog 2) ⟩w = -0. 1695, dominated by the phase of the first zero, with the sign decided by the first 120 zeros. Part III: A conditional framework. We construct a conditional reduction of the Riemann Hypothesis to a single analytical condition—the extension of the Chowla correlation Euler product from logarithmic to Cesàro averages (Condition C). This reduction proceeds through a Multi-Prime Induction utilizing the variance recursion. A second, heuristic path via a Prime Factorization Tower with Descending Prime Induction is presented as motivation only: it would require transmission-factor positivity across all primes, which we do not establish (the relevant infinite product vanishes; see Remark 12. 5) and which remains conjectural (Conjecture 1. 13). We introduce the Sieve Variance Inequality, which proves (conditionally on C) that cross-terms in the prime sieve decomposition are non-positive, using a Pair Density Amplification mechanism rooted in the Euler product structure. The prime sign contraction, verified by exhaustive enumeration at small scales, provides a combinatorial perspective on why forcing λ (p) =-1 at successive primes contracts partial sums. We identify the Cesàro-level Chowla autocorrelation as the single irreducible barrier and present extensive computational verification, re-run and extended at 10^ (8) in this version, including a residual-energy ladder showing that after subtracting K zeros the walk's remaining variance equals the explicit-formula tail at every K (terminal floor 0. 00108 against predicted 0. 00106). One previously claimed structural regularity is corrected: half-scale sign coherence to 10⁹ fails at full integer resolution precisely inside the known Pólya window near 9. 06×10^ (8), and is restated as a density phenomenon. — Version 2 (July 2026). This version supersedes Version 1 (31 pp. , May 2026, https: //doi. org/10. 5281/zenodo. 20370176). It corrects several substantive errors of v1 — including the unweighted variance recursion (replaced by an exact weighted contraction with explicit boundary term), the treatment of the Matomäki–Radziwiłł theorem (now routed through the explicitly named open Hypothesis (SR) ), the sharp constant of the energy equivalence (6/π² corrected to CV ≈ 0. 5408, with the converse now conditional on a second-moment hypothesis), the CRT descent factor, and the Weight Optimality theorem — and roughly doubles the paper to 61 pages. Every key claim is now proved, conditionalized on a named hypothesis, or explicitly retracted with a numerical refutation; all numerics are recomputed exactly (sieve to 2²9, 120-zero constants). A complete changelog is included in this record.
Matteas Ellis (Wed,) studied this question.