Cohomological Undecidability in Steady Forced Navier–Stokes Flows via a Hyperbolic Periodic Orbit, and the Vanishing of the Critical Defect for a Single Orbit

The paper studies the stationary passive-scalar resolvent on the flat three-torus T³ = T_L × D, where T_L = ℝ/Lℤ is the clock circle and D ⊂ ℝ² is a small open disk containing the origin with coordinates (s, y, z). The central equation is √κ g_κ + X·∇g_κ − κΔg_κ = Q, where X is a smooth divergence-free velocity field, Q is a smooth scalar source, and κ > 0 is a small diffusivity. The Batchelor scaling is γ_κ = √κ, the dissipative observable is D_κ(X,Q) = κ‖∇g_κ‖²_{L²(M)}, and the critical defect is D_crit(X,Q) = liminf_{κ↓0} D_κ(X,Q), with liminf used so that lower bounds are unconditional. The motivating question is how little geometry is needed for a steady flow to force a nonzero periodic-orbit obstruction from a fixed smooth source, and whether this obstruction can encode the halting problem. The answer is that a single hyperbolic periodic orbit suffices for the cohomological undecidability mechanism, provided the source has a nonzero weighted average along that orbit. This is a cohomological, not a pointwise, obstruction: a coboundary Q = X·∇φ has zero integral along every closed orbit, so a hyperbolic orbit with nonzero period integral cannot be a coboundary. At the same time, the paper separates sharply the logical obstruction from the analytic dissipation observable: a single hyperbolic periodic orbit carries a Livšic obstruction but has zero H¹-capacity in dimension three, and D_crit is an H¹-type functional that does not see obstructions supported on zero-capacity sets. For a closed smooth three-manifold M with Riemannian metric g, Laplace–Beltrami operator Δ, and smooth divergence-free X with div_μ X = 0, Proposition 2.1 states that for every κ > 0 and every Q ∈ L²(M), the equation √κ g_κ + X·∇g_κ − κΔg_κ = Q has a unique solution g_κ ∈ H²(M); if Q ∈ C^∞(M) then g_κ ∈ C^∞(M); and the energy identity √κ‖g_κ‖²_{L²} + κ‖∇g_κ‖²_{L²} = ⟨Q, g_κ⟩_{L²} holds. The proof tests against g_κ and integrates over M: the transport term vanishes because ∫_M (X·∇g_κ)g_κ dμ = (1/2)∫_M X·∇(g_κ²) dμ = −(1/2)∫_M g_κ² div_μ X dμ = 0, using ∂M = ∅ and div_μ X = 0; the diffusion term gives −∫_M (Δg_κ)g_κ dμ = ∫_M ‖∇g_κ‖² dμ. The operator A_κ = √κ + X·∇ − κΔ has positive real part √κ on L²(M,μ), hence is coercive and boundedly invertible by the Lax–Milgram theorem; smoothness follows from elliptic regularity since X and Q are smooth and M is closed. The energy norm is ‖f‖²_{E_κ} := √κ‖f‖²_{L²} + κ‖∇f‖²_{L²}, so that the energy identity reads ‖g_κ‖²_{E_κ} = ⟨Q, g_κ⟩_{L²}; coercivity gives ‖g_κ‖_{E_κ} ≤ ‖Q‖_{E_κ*}, where E_κ* is the dual of E_κ. Constants are invisible to the gradient: if Q = c is constant, then g_κ = c/√κ is constant and D_κ(X,c) = 0, so adding a constant to Q does not change D_crit and one may normalize ∫_M Q dμ = 0. Proposition 2.2 (coboundary nullspace) states that if Q = X·∇φ + c for some φ ∈ C^∞(M) and constant c, then D_crit(X,Q) = 0. The proof normalizes c = 0 and ∫_M Q dμ = 0, takes φ with zero mean, lets g_κ solve the resolvent, and sets h_κ = g_κ − φ. A direct computation using the resolvent gives √κ h_κ + X·∇h_κ − κΔh_κ = −√κ φ + κΔφ. From coercivity, ‖h_κ‖_{E_κ} ≤ ‖−√κ φ + κΔφ‖_{E_κ*}. For the first term, for f ∈ E_κ, |⟨√κ φ, f⟩| ≤ √κ‖φ‖_{L²}‖f‖_{L²} ≤ √κ‖φ‖_{L²} κ^{−1/4}‖f‖_{E_κ} = κ^{1/4}‖φ‖_{L²}‖f‖_{E_κ}, using ‖f‖_{L²} ≤ κ^{−1/4}‖f‖_{E_κ}, hence ‖√κ φ‖_{E_κ*} ≤ κ^{1/4}‖φ‖_{L²} → 0. For the second term, |⟨κΔφ, f⟩| = |−κ⟨∇φ, ∇f⟩| ≤ κ‖∇φ‖_{L²}‖∇f‖_{L²} ≤ κ‖∇φ‖_{L²} κ^{−1/2}‖f‖_{E_κ} = κ^{1/2}‖∇φ‖_{L²}‖f‖_{E_κ}, using ‖∇f‖_{L²} ≤ κ^{−1/2}‖f‖_{E_κ}, hence ‖κΔφ‖_{E_κ*} ≤ κ^{1/2}‖∇φ‖_{L²} → 0. Combining, ‖h_κ‖_{E_κ} ≤ κ^{1/4}‖φ‖_{L²} + κ^{1/2}‖∇φ‖_{L²} → 0. Since ∇g_κ = ∇φ + ∇h_κ, one gets D_κ(X,Q) = κ‖∇g_κ‖²_{L²} ≤ 2κ‖∇φ‖²_{L²} + 2κ‖∇h_κ‖²_{L²} ≤ 2κ‖∇φ‖²_{L²} + 2‖h_κ‖²_{E_κ} → 0, hence D_crit(X,Q) = liminf_{κ↓0} D_κ(X,Q) = 0. Remark 2.3 notes the converse is false in general: there exist divergence-free X and smooth Q with D_crit(X,Q) = 0 but Q ∉ Cob(X) modulo constants. A classical example is the non-resonant linear flow X_α = ∂_x + α∂_y on T² with α ∉ ℚ; for such X_α, Fourier analysis gives D_crit(X_α,Q) = 0 for every smooth zero-mean Q, but for Diophantine α small-divisor phenomena obstruct smooth solutions of X_α·∇φ = Q for certain smooth Q. Hence D_crit(X_α,Q) = 0 does not imply Q ∈ Cob(X_α) in general. This observation is not used in the proofs below; it delimits the scope of Proposition 2.2. Theorem 3.1 (laminar exact formula): let M = S¹_L × Σ with the product metric, X = ∂_s the parallel field in the s-direction, and for Q ∈ C^∞(M) define the s-average Q̄(y) = (1/L)∫₀^L Q(s,y) ds for y ∈ Σ. Then D_crit(∂_s, Q) = ‖∇_Σ Q̄‖²_{L²(Σ)}. The proof decomposes in Fourier modes in s: Q(s,y) = Σ_{m∈ℤ} Q_m(y) e^{2πims/L} and g_κ(s,y) = Σ_{m∈ℤ} g_{κ,m}(y) e^{2πims/L}. Equation (2.1) becomes for each m: (√κ + 2πim/L)g_{κ,m} − κΔ_Σ g_{κ,m} = Q_m. Diagonalizing −Δ_Σ by an orthonormal basis (e_j) with eigenvalues λ_j ≥ 0, writing Q_m = Σ_j q_{m,j} e_j and g_{κ,m} = Σ_j g_{κ,m,j} e_j, gives g_{κ,m,j} = q_{m,j}/(√κ + 2πim/L + κλ_j). The defect is D_κ = κ Σ_{m,j} λ_j|g_{κ,m,j}|² = Σ_{m,j} κλ_j|q_{m,j}|²/|√κ + 2πim/L + κλ_j|². For m ≠ 0, since |√κ + 2πim/L + κλ_j|² ≥ (2πm/L)², one has Σ_{m≠0,j} κλ_j|q_{m,j}|²/(2πm/L)² ≤ Cκ Σ_{m≠0} ‖∇_Σ Q_m‖²/m² ≤ Cκ‖Q‖²_{H¹} → 0, using Σ_j λ_j|q_{m,j}|² = ‖∇_Σ Q_m‖²_{L²} and rapid decay of Fourier coefficients of smooth functions. For m = 0, κλ_j|q_{0,j}|²/(√κ + κλ_j)² = λ_j|q_{0,j}|²/(1 + √κλ_j)², and since 0 ≤ (1 + √κλ_j)^{−2} ≤ 1 and Σ_j λ_j|q_{0,j}|² = ‖∇_Σ Q̄‖²_{L²} < ∞ for smooth Q, dominated convergence gives lim_{κ→0} Σ_j λ_j|q_{0,j}|²/(1 + √κλ_j)² = Σ_j λ_j|q_{0,j}|² = ‖∇_Σ Q̄‖²_{L²(Σ)}. Combining the two cases gives the formula. A remark on monotonicity notes that the m = 0 contribution is decreasing in κ, the m ≠ 0 contribution is not monotone in general, and no monotonicity of the full defect is asserted; the only consequence used below is the equivalence D_crit(∂_s, Q) = 0 ⟺ Q̄ is constant, which follows from the formula and Poincaré on Σ. Section 4 works on M = T³ = T_L(s) × T²(y,z) with the flat metric, so that Σ = T² is closed, and γ₀(s) = (s,0,0) is the central vertical orbit. Lemma 4.1 (existence of log-bumps): with ρ = √(y² + z²) and small R > 0 such that B(0,R) ⊂ T², for 0 < δ ≪ R define η_δ(ρ) = 1 for ρ ≤ δ, = log(R/ρ)/log(R/δ) for δ ≤ ρ ≤ R, = 0 for ρ ≥ R, extended smoothly to T². Then η_δ is continuous, can be smoothed to C^∞ without changing the estimates, and satisfies η_δ(0) = 1, η_δ ≥ 0, and ‖∇η_δ‖²_{L²(T²)} ≤ C/|log δ| for some constant C depending only on R. The proof computes the gradient in polar coordinates: ∂_ρ η_δ = −1/(ρ log(R/δ)) for δ < ρ < R and 0 otherwise, hence ‖∇η_δ‖²_{L²} = ∫_δ^R (1/(ρ log(R/δ)))² 2πρ dρ = (2π/log²(R/δ))∫_δ^R dρ/ρ = 2π/log(R/δ) ≤ C/|log δ| for δ small; smoothing at the two junctions ρ = δ and ρ = R with a mollifier of width δ/4 changes the L² norm of the gradient by at most a factor 2. Theorem 4.2 (pointwise bound fails): let Q_δ = η_δ − c_δ where c_δ = (1/|T²|)∫_{T²} η_δ. Then (i) Q_δ ∈ C^∞(T³) after smoothing, has zero spatial mean, and is independent of s; (ii) Per_{γ₀}(Q_δ) = L(1 − c_δ) ≥ L/2 for δ small; (iii) D_crit(∂_s, Q_δ) = ‖∇η_δ‖²_{L²(T²)} → 0 as δ → 0. Consequently, for any c > 0 there exists δ > 0 with D_crit(∂_s, Q_δ) < c|Per_{γ₀}(Q_δ)|². The proof is: (i) is immediate; (ii) since c_δ ≤ C/|log δ| → 0, Per_{γ₀}(Q_δ) = L(1 − c_δ) ≥ L/2 for δ small; (iii) by Theorem 3.1 applied to Σ = T², D_crit(∂_s, Q_δ) = ‖∇_{T²} Q̄_δ‖² = ‖∇η_δ‖² (since Q_δ is independent of s, Q̄_δ = Q_δ, and the constant c_δ does not affect the gradient), which by the log-bump estimate is 2π/log(R/δ) → 0. Remark 4.3 explains that the pointwise bound D_crit ≥ c|Per_γ(Q)|² fails because Q_δ concentrates in a thin tube around γ₀: the periodic-orbit functional Per_γ evaluates Q at the orbit itself, while D_crit sees the H¹-energy of Q, which is small for thin tubes because a curve has zero H¹-capacity in transverse dimension two. This is made completely explicit in Section 9: the hyperbolic-orbit mechanism itself produces no positive critical defect. Section 5 constructs the stream-function plug on T³ = T_L × D. The data are smooth functions a ∈ C^∞(T_L) with a ≥ c > 0 and ∫₀^L a(s) ds = 1; b ∈ C^∞(T_L) with ∫₀^L b(s) ds = 0; q ∈ C^∞(T_L) with A₁ := ∫₀^L b(s)q(s) ds ≠ 0; a cutoff χ ∈ C_c^∞(D) with χ ≡ 1 near 0 and χ even in y and z; and a fixed constant α ≠ 0. B is the periodic primitive of b with B′ = b and B L-periodic. The revised fixed source is ψ(y,z) := yχ(y,z)(1 + αyz) and Q₀(s,y,z) := b(s)ψ(y,z); since ∫₀^L b ds = 0, Q₀ has zero spatial mean. For Section 9, χ is chosen with a larger plateau: nested disks D₀ ⊂ D₁ ⊂ D₂ ⊂ D with χ ≡ 1 on D₁, supp χ ⊂ D₂, and supp ψ ⊂ D₀. The silent field is X₀ = ∂_s and φ₀(s,y,z) := B(s)ψ(y,z), so X₀·∇φ₀ = ∂_s φ₀ = B′(s)ψ(y,z) = b(s)ψ(y,z) = Q₀; hence Q₀ is an exact coboundary for X₀ and D_crit(X₀,Q₀) = 0. This is the silent state. An explicit choice on T_{2π} is b(s) = sin s, q(s) = sin s, a(s) = 1/(2π), B(s) = −cos s + c₀ with c₀ ≠ 0 small (|c₀| ≤ 1/2), giving A₁ = ∫₀^{2π} sin² s ds = π ≠ 0; α = 1; and a ≡ 1/(2π) > 0 satisfies the lower bound in (5.1) with c = 1/(2π). The stream function on T_L × D is H_ε(s,y,z) = χ(y,z)[εa(s)yz + (εq′(s) − ε²a(s)q(s))z], extended by zero outside supp χ, and the field is X_ε = ∂_s + (∂_z H_ε)∂_y − (∂_y H_ε)∂_z. Theorem 5.3 (properties of the local plug): for every ε > 0, (i) div X_ε = 0 exactly on T³; (ii) X_ε = ∂_s outside the fixed compact set T_L × supp χ; (iii) ‖X_ε − ∂_s‖_{C^k} ≤ C_k ε for all k ≥ 0, where C_k depends only on (a,b,q,χ); (iv) the curve γ_ε(s) = (s, εq(s), 0), s ∈ T_L, is a periodic orbit of X_ε; (v) γ_ε is hyperbolic for every ε > 0 with transverse Floquet multipliers e^ε and e^{−ε}. The proof of (i) writes X_ε = ∂_s + V_y∂_y + V_z∂_z with V_y = ∂_z H_ε and V_z = −∂_y H_ε, so div X_ε = ∂_y∂_z H_ε − ∂_z∂_y H_ε = 0. Property (ii) is immediate from compact support of χ. Property (iii) uses H_ε = O(ε) in C^{k+1} for every k, since a,