Exact Critical Dissipation and Undecidability for a Laminar Passive-Scalar Equation on the Three-Torus

The paper studies the critical vanishing-diffusivity limit of a forced, stationary passive-scalar equation inside a laminar, steady, unforced Navier–Stokes flow on the three-torus. The velocity field is the parallel field u = ∂ₓ on T³ = Tₓ × T²_{y,z}. For κ > 0 and a friction parameter γ ≥ 0, a smooth scalar source Q is advected, diffused, and damped according to γ g_{κ,γ} + ∂ₓ g_{κ,γ} − κ Δ g_{κ,γ} = Q. The associated dissipation functional is D_{κ,γ}(u,Q) = κ ‖∇g_{κ,γ}‖²_{L²}. At the critical scaling γ = √κ the paper proves the exact formula D_{crit}(∂ₓ,Q) := lim_{κ↓0} D_{κ,√κ}(∂ₓ,Q) = ‖∇_{y,z} Q̄‖²_{L²(T²)}, where Q̄(y,z) = ∫_{Tₓ} Q(x,y,z) dx is the x-average of the source. In particular, the critical defect vanishes exactly when Q̄ is constant; if Q has zero mean, this is equivalent to Q being an x-coboundary, i.e. Q = ∂ₓφ for some φ ∈ H¹(T³). The formula is completely elementary in Fourier variables, yet it carries a sharp logical consequence: the paper constructs a uniformly computable family of smooth sources Q_e, indexed by Turing machines e, such that D_{crit}(∂ₓ,Q_e) = 0 ⟺ e does not halt. Hence the zero set of the critical passive-scalar defect, even inside the simplest laminar flow, is Π⁰₁-complete. The paper also proves that no uniform positive gap can hold in this source-encoded family. Finally, it proves the analogous exact formula for steady shear fields u_c = c(y,z)∂ₓ with c ∈ W^{1,∞}(T²) and c ≥ c₀ > 0, giving a fully rigorous operator-theoretic proof that removes a prior gap in the estimate of the transport term. The computational content is placed strictly in the source Q, not in the velocity field u, and the result is presented as a baseline for the broader programme of undecidability in fluid dissipation functionals. The paper begins by explaining that the passive-scalar defect measure is one of the cleanest linear probes of an incompressible fluid. One fixes a velocity field u, advects a scalar with small diffusivity κ, and studies the limiting dissipation κ‖∇g^{κ}‖². In time-dependent settings, weak-* limits of the matrix-valued measure κ ∇g^{κ} ⊗ ∇g^{κ} dx dt can capture anomalous dissipation, Batchelor spectra, and possible singularities of the flow. In stationary settings, with a friction or relaxation parameter γ, the natural object is the critical dissipation D_{κ,γ}(u,Q) = κ ‖∇g_{κ,γ}‖²_{L²}, where g_{κ,γ} solves the stationary forced–diffusive–damped passive-scalar equation γ g_{κ,γ} + u·∇g_{κ,γ} − κΔg_{κ,γ} = Q. The scaling γ = √κ is the stationary analogue of the Batchelor scale: it is the balance at which diffusion and damping can produce a finite, nonzero limit for generic sources. The paper analyses this object in the simplest possible nontrivial fluid: the laminar steady Navier–Stokes flow u = ∂ₓ on the flat three-torus T³ = Tₓ × T²_{y,z}, with Tₓ = R/Z and T²_{y,z} = (R/Z)², equipped with the flat metric. This field is divergence-free, harmonic, and solves the unforced steady Navier–Stokes equations with constant pressure: (u·∇)u = 0 and Δu = 0. Thus there is no turbulence, no vortex stretching, no blow-up, and no forcing of the momentum equation. The only nontrivial datum is the passive-scalar source Q. Even in this completely laminar situation, the paper obtains two results of different characters. First, it proves an exact formula: D_{crit}(∂ₓ,Q) = ‖∇_{y,z} Q̄‖²_{L²(T²)}, with Q̄(y,z) = ∫_{Tₓ} Q(x,y,z) dx. The formula is elementary; it follows from Fourier expansion in the x-variable and the transverse variables y,z. It shows that the critical defect sees only the transverse variation of the x-average of the source. It vanishes precisely when the source is, up to an additive constant, an x-coboundary. Second, the paper proves that the map Q ↦ D_{crit}(∂ₓ,Q) has an undecidable zero set on the class of computable smooth sources. More precisely, there exists a uniformly computable family Q_e such that D_{crit}(∂ₓ,Q_e) = 0 ⟺ the Turing machine e never halts. Consequently the set {e : D_{crit}(∂ₓ,Q_e) = 0} is Π⁰₁-complete. The paper emphasizes what this result does not claim. It is not a claim that the fluid velocity computes, nor does it imply that the full Navier–Stokes equations are undecidable. The velocity is fixed and trivial; the computational content is placed entirely in the scalar source Q. The paper should be viewed as the closed laminar layer of a larger programme. The next layer asks whether the same phenomenon can be moved inside the velocity field itself, using steady Euler or Navier–Stokes plugs. That stronger question is delicate and is not solved here; it is addressed in companion work under explicit local-embedding hypotheses. Finally, the paper also proves a rigorous extension to steady shear fields u_c = c(y,z)∂ₓ with c ∈ W^{1,∞}(T²) and c ≥ c₀ > 0, correcting a prior estimate in the literature that overlooked the contribution of the high x-frequency modes to the full gradient. The corrected proof confirms the same exact formula in this larger class. The structure of the paper is as follows. Section 2 sets up the stationary forced–diffusive–damped passive-scalar equation and states the main theorems. Section 3 gives the Fourier proof of the exact laminar formula (Theorem A). Section 4 analyses the critical scaling γ = √κ. Section 5 introduces the computability framework and proves Π⁰₁-completeness of the zero set (Theorem B). Section 6 proves that no uniform gap is available in the source-encoded family (Theorem C). Section 7 gives the corrected and fully rigorous proof of the shear-flow generalization. Section 8 discusses regularity, the Batchelor analogy, and the limitations of the result. Section 9 gives examples and consistency checks. Section 10 collects concluding remarks. In the setup, let M = T³ = Tₓ × T²_{y,z} with coordinates (x,y,z) ∈ (R/Z) × (R/Z)² and the flat metric. Throughout, the measure on Tₓ and on T²_{y,z} is normalized to have total mass one, so that the volume of T³ is one. This normalization is used consistently in the definition of the x-average and in the Fourier coefficients below. Let u = ∂ₓ = (1,0,0). Then ∇·u = 0, (u·∇)u = 0, and Δu = 0, so u is a steady unforced Navier–Stokes solution with constant pressure, for every value of the viscosity. Let Q ∈ H¹(T³;R) be a passive-scalar source. For κ > 0 and γ > 0, define g_{κ,γ} as the unique L² solution of (2.1) γ g_{κ,γ} + ∂ₓ g_{κ,γ} − κΔg_{κ,γ} = Q. Here Δ = ∂ₓ² + Δ_{y,z}, where Δ_{y,z} is the Laplacian on the transverse torus. When γ > 0, the operator γ + ∂ₓ − κΔ has Fourier symbol γ + 2πin + 4π²κ(n² + ‖ℓ‖²), whose real part γ + 4π²κ(n² + ‖ℓ‖²) is bounded below by γ > 0, so the operator is coercive on L² and invertible. Moreover, the operator maps H^{s+2} → H^s for every s ≥ 0, and for Q ∈ H¹ the solution g_{κ,γ} lies in H³(T³). In particular ∇g_{κ,γ} ∈ H²(T³) ⊂ L²(T³), so the dissipation functional defined below is finite. The dissipation functional is (2.2) D_{κ,γ}(u,Q) = κ ‖∇g_{κ,γ}‖²_{L²(T³)}. The critical scaling is (2.3) γ = √κ. The critical defect is defined by (2.4) D_{crit}(u,Q) = lim_{κ↓0} D_{κ,√κ}(u,Q), whenever the limit exists. In the laminar case treated here, the limit always exists and is given by an explicit formula. For f ∈ L¹(T³), define the x-average (2.5) f̄(y,z) = ∫_{Tₓ} f(x,y,z) dx. Since the measure on Tₓ is normalized, this is the average of f over x. The main theorem is Theorem A (Exact laminar defect): For every Q ∈ H¹(T³), (2.6) D_{crit}(∂ₓ,Q) = ‖∇_{y,z} Q̄‖²_{L²(T²)}. The limit (2.4) exists and equals the right-hand side. In particular, (2.7) D_{crit}(∂ₓ,Q) = 0 ⟺ Q̄ is constant. Equivalently, if ∫_{T³} Q = 0, then (2.8) D_{crit}(∂ₓ,Q) = 0 ⟺ Q = ∂ₓφ for some φ ∈ H¹(T³). Theorem B (Undecidability): There exists a uniformly computable family Q_e ∈ C^∞(T³), indexed by e ∈ N, such that (2.9) D_{crit}(∂ₓ,Q_e) = 0 ⟺ e does not halt. Consequently the set {e : D_{crit}(∂ₓ,Q_e) = 0} is Π⁰₁-complete. Theorem C (No uniform gap): For the family in Theorem B, (2.10) inf{D_{crit}(∂ₓ,Q_e) : e halts} = 0. Hence no alternative of the form D_{crit} = 0 or D_{crit} ≥ c₀ > 0 can hold uniformly in this laminar source-encoded class. Theorem D (Shear flow generalization): Let c ∈ W^{1,∞}(T²) satisfy c ≥ c₀ > 0. For the steady Euler field u_c = c(y,z)∂ₓ and every Q ∈ H¹(T³), the critical passive-scalar defect at γ = √κ satisfies (2.11) D_{crit}(u_c,Q) = ‖∇_{y,z} Q̄‖²_{L²(T²)}. The proofs of Theorems A, B, C, and D occupy Sections 3, 5, 6, and 7 respectively. The proof of the exact laminar formula (Theorem A) uses Fourier series. For (n,ℓ) ∈ Z × Z², write e_{n,ℓ}(x,y,z) = exp(2πi(nx + ℓ·(y,z))). For f ∈ L²(T³), let f̂_{n,ℓ} = ∫_{T³} f(x,y,z) e_{−n,−ℓ}(x,y,z) dx dy dz be the Fourier coefficient. Then ∂ₓ f ↔ 2πin f̂_{n,ℓ}, and −Δf ↔ 4π²(n² + ‖ℓ‖²) f̂_{n,ℓ}. Equation (2.1) becomes, mode by mode, (3.1) (√κ + 2πin + 4π²κ(n² + ‖ℓ‖²)) ĝ_{n,ℓ} = Q̂_{n,ℓ}. Therefore ĝ_{n,ℓ} = Q̂_{n,ℓ} / (√κ + 2πin + 4π²κ(n² + ‖ℓ‖²)), and the dissipation is (3.2) D_{κ,√κ} = κ ∑_{n∈Z} ∑_{ℓ∈Z²} 4π²(n² + ‖ℓ‖²) |ĝ_{n,ℓ}|². The sum is split into n = 0 and n ≠ 0. For the zero x-frequency modes (n = 0), if n = 0, then ĝ_{0,ℓ} = Q̂_{0,ℓ} / (√κ + 4π²κ‖ℓ‖²). The contribution to D_{κ,√κ} is (3.3) κ · 4π²‖ℓ‖² |Q̂_{0,ℓ}|² / (√κ + 4π²κ‖ℓ‖²)². For ℓ = 0 this is zero. For ℓ ≠ 0, factor √κ from the denominator: (√κ + 4π²κ‖ℓ‖²)² = κ(1 + 4π²√κ‖ℓ‖²)². The numerator κ · 4π²‖ℓ‖² then cancels the factor κ, so (3.3) equals (3.4) 4π²‖ℓ‖² |Q̂_{0,ℓ}|² / (1 + 4π²√κ‖ℓ‖²)². As κ ↓ 0, the denominator tends to 1, so the pointwise limit is 4π²‖ℓ‖² |Q̂_{0,ℓ}|². Since Q ∈ H¹(T³), the coefficients Q̂_{0,ℓ} are exactly the Fourier coefficients of Q̄ in the transverse variables, and ∑_{ℓ∈Z²} 4π²‖ℓ‖² |Q̂_{0,ℓ}|² = ‖∇_{y,z} Q̄‖²_{L²(T²)} < ∞. Moreover, for every κ > 0, 4π²‖ℓ‖² |Q̂_{0,ℓ}|² / (1 + 4π²√κ‖ℓ‖²)² ≤ 4π²‖ℓ‖² |Q̂_{0,ℓ}|², and the majorant is summable in ℓ because Q̄ ∈ H¹(T²). By dominated convergence, (3.5) lim_{κ↓0} ∑_{ℓ} contribution(n = 0) = ‖∇_{y,z} Q̄‖²_{L²(T²)}. For the nonzero x-frequency m