We resolve the status of the three open problems identified in the Spectral Program for the Riemann Hypothesis. Open Problem 1 (the uniform lower bound θₑff (k) ≥ c > 0) is shown to be false. Open Problem 3 (μ (log 2/log 3) ≤ 2) remains a wall. Open Problem 2 (restricted Weil positivity on T_√2) is the sole surviving path, supported by strong numerical evidence. The negative result for OP1 is proved via a tube variational argument. The alignment operator Hₖ = −X² + Vₖ on Tᵏ has a rank-one kinetic structure: the directional derivative X = Σ ωⱼ ∂/∂θⱼ penalizes variation only along the 1D Kronecker flow direction ω = (log 2,. . . , log pₖ). Variation in the (k−1) -dimensional transverse subspace costs zero kinetic energy. A trial function that is Gaussian along the flow (width L) and delta-localized transversally gives inf σ (Hₖ) ≤ 1/ (4L²) + Σ aⱼ (1 − exp (−ωⱼ² L²/2) ). Optimizing over L yields inf σ (Hₖ) ≤ √ (Σωⱼ³/2), hence θₑff = O (√ (log k) /√k) → 0 by PNT. The "plateau" θₑff ≈ 0. 47 observed in earlier Fourier-truncated computations was an artifact: truncation at |m| ≤ M limits transverse resolution, creating an artificial energy floor that disappears as M → ∞. Numerical verification confirms the tube bound falls below truncated eigenvalues for k ≥ 3. This closes the Bridge Theorem path to RH (Path 27 in the cartography of failed approaches). Open Problem 3 (μ (log 2/log 3) ≤ 2) is surveyed with reference to 17+ documented dead ends. Baker gives exponential bounds, Schmidt cascade gives μ ≤ 3, no technique bridges the gap to μ = 2. This is Path 28 — not a dead end but a bypass, since OP2 reaches RH without it. The positive result for OP2: the Weil functional Wₖ (η) is strictly positive for all k = 1 to 20 and all η tested, including the regime η = 1/ (k log k) required by density. The off-diagonal to diagonal ratio |O/D| decreases monotonically from 3. 3 (k=2) to 0. 03 (k=20). The mechanism is diagonal dominance via the Fundamental Theorem of Arithmetic: off-diagonal terms contain |m log p − log pⱼ| > 0 (since pᵐ ≠ pⱼ for distinct prime powers), and the Gaussian suppression exp (−Δ²/ (2η²) ) kills them. The integer gap lemma gives |m log p − log pⱼ| ≥ 1/ (2 max (pᵐ, pⱼ) ), providing explicit suppression bounds. Two technical gaps remain for a complete proof via OP2: (1) formal proof that |Oₖ (η) | ≤ (1−ε) Dₖ for ε > 0 independent of k, requiring explicit summation of the off-diagonal tail; (2) verification that the η₀ (k) required for diagonal dominance is compatible with the η required for density of T_√2 in the Schwartz topology. An energy decomposition diagnostic is provided, showing the ground state splits between bulk modes (high |m·ω|, growing fraction 31%→67%) and resonant modes (low |m·ω|, shrinking fraction), with a stable "resolution tax" kin/|Vₒsc| ≈ 0. 27. Reproducible Python verification code is included in the appendix. This paper serves as Paths 27–28 in the cartography of failed approaches to RH initiated in "Twenty-Five Ways Not to Prove the Riemann Hypothesis, " continuing with the Battery bound diagnostic (Path 26) and the DA Spectral Attack (Path 28 technical detail).
Thierry Marechal (Wed,) studied this question.