The Transverse Localization Obstruction and the Restricted Weil Path: Why θeff (k) ≥ c > 0 Fails, and What Replaces It in the Spectral Program for the Riemann Hypothesis

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 θ_eff(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 H_k = −X² + V_k on T^k has a rank-one kinetic structure: the directional derivative X = Σ ω_j ∂/∂θ_j penalizes variation only along the 1D Kronecker flow direction ω = (log 2, ..., log p_k). 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(H_k) ≤ 1/(4L²) + Σ a_j(1 − exp(−ω_j² L²/2)). Optimizing over L yields inf σ(H_k) ≤ √(Σω_j³/2), hence θ_eff = O(√(log k)/√k) → 0 by PNT. The "plateau" θ_eff ≈ 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 W_k(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 p_j| > 0 (since p^m ≠ p_j for distinct prime powers), and the Gaussian suppression exp(−Δ²/(2η²)) kills them. The integer gap lemma gives |m log p − log p_j| ≥ 1/(2 max(p^m, p_j)), providing explicit suppression bounds. Two technical gaps remain for a complete proof via OP2: (1) formal proof that |O_k(η)| ≤ (1−ε)D_k 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/|V_osc| ≈ 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).