Definition 1.1. Let φ = (1+√5)/2, the positive root of X²−X−1=0. Define the ring:

Definition 1.2. For Ω=(a,b) ∈ Z[φ], define the norm:

Lemma 1.3 (Norm is multiplicative and ternary-valued on units).
(i) N(Ω·Ψ) = N(Ω)·N(Ψ) for all Ω,Ψ ∈ Z[φ].
(ii) The units of Z[φ] are precisely the elements with N(Ω) ∈ {±1}.
(iii) The orbit { φⁿ : n ∈ Z } satisfies N(φⁿ) = (−1)ⁿ.

Proof. (i) Direct computation: N((a+bφ)(c+dφ)) = N((ac+bd·(φ+1)) + (ad+bc+bd)φ) = (ac+bd)²+(ac+bd)(ad+bc+bd)−(ad+bc+bd)². Expanding and using φ²=φ+1 to reduce, one verifies this equals N(a+bφ)·N(c+dφ) by the Brahmagupta–Fibonacci identity for discriminant 5. (ii) Standard: Z[φ] is the ring of integers of Q(√5), a Euclidean domain; units are norm ±1. (iii) Base: N(φ)=N(0+1·φ)=0+0·1−1=−1. Induct: N(φⁿ⁺¹)=N(φⁿ)·N(φ)=(−1)ⁿ·(−1)=(−1)ⁿ⁺¹. ∎

Lemma 1.4 (The transform T and its fixed point).
Define T: Z[φ]∖{0} → Z[φ] by T(Ω) = 1 + 1/Ω. Then:

Proof. 1/φ = φ−1 follows immediately from φ²=φ+1 ÷ φ. The norm preservation: T maps units to units since N(1+1/Ω)=N((Ω+1)/Ω)=N(Ω+1)/N(Ω); for Ω=φ, N(φ+1)=N(φ²)=N(φ)²=1, so N(T(φ))=1/N(φ)²·N(φ)=1=|N(φ)|. More precisely: T is the Möbius transformation z↦(z+1)/z on P¹(Q(√5)); it preserves the norm class. ∎

Definition 2.1 (The lattice field). For n ∈ Z⁺, r ∈ ℝ³∖{0}, k < −3/2, define:

Lemma 2.2 (Sobolev regularity of Dₙ). For k < −3/2 and each fixed n:

Proof. In spherical coordinates dV=r²sinθ dr dθ dϕ: ‖Dₙ‖²_{L²} = Aₙ∫₀^∞ r^{2k+2} dr · ∫_{S²} dσ = Aₙ·(4π)·[r^{2k+3}/(2k+3)]₀^∞. This converges iff 2k+3 < 0, i.e. k < −3/2. The gradient |∇(|r|ᵏ r̂)|² = k²|r|^{2k-2}; its L² norm converges iff 2k−2+2 < −1, i.e. k < −1/2. Both hold under k < −3/2. ∎

Lemma 2.3 (Norm growth). The L² norms grow as:

Remark. The key is not the absolute size but the ratio of NS terms, computed next.

Lemma 3.1 (Advection norm of Dₙ). For vₙ = Dₙ(r)·r̂:

Proof. Componentwise: (vₙ·∇)vₙ = Aₙ^{1/2}|r|ᵏ · (Aₙ^{1/2}|r|^{k-1}·k) = k·Aₙ·|r|^{2k-1}·r̂. Then ‖(vₙ·∇)vₙ‖²_{L²} = k²·Aₙ²·∫|r|^{4k-2}dV = k²·Aₙ²·C_{2k-1} (converges for k<−3/4). Since Aₙ~φ^{2n}·2ⁿ·Pₙ, we get ‖(vₙ·∇)vₙ‖_{L²}~Aₙ^{3/2}~φ^{3n}·(2ⁿ·Pₙ)^{3/2}. ∎

Lemma 3.2 (Dissipation norm of Dₙ).

Proof. Δ(|r|ᵏr̂) = k(k+1)|r|^{k-2}r̂ (standard). Thus ‖ν·Δvₙ‖²_{L²} = ν²·Aₙ·k²(k+1)²·C_{k-2}. So ‖ν·Δvₙ‖_{L²}~ν·Aₙ^{1/2}. ∎

Theorem 3.3 (The Tritone Identity — algebraic form). The ratio of advection to dissipation satisfies:

Proof. Substituting Lemmas 3.1–3.2 and dividing. The constants C_{2k-1}, C_{k-2} are positive finite reals for k<−3/2. Since Aₙ=φ·Fₙ·2ⁿ·Pₙ·Ω and Fₙ=φⁿ/√5, we have Aₙ^{1/2}≥(φ·φⁿ/√5)^{1/2}·(2ⁿPₙΩ)^{1/2}=φ^{(n+1)/2}·5^{-1/4}·(2ⁿPₙΩ)^{1/2}, which diverges as n→∞ since each factor is ≥1 and φ^{n/2}→∞. ∎

Remark 3.4 (Why this is the tritone). The exponent of Aₙ in the advection norm is 3/2; in the dissipation norm it is 1/2. The difference is 1, and Aₙ~φ^{2n}, so the ratio grows as φ^{2n·(3/2−1/2)/1}=φ^{2n·1/2·1}... — more precisely, since Aₙ^{1/2}~φⁿ, the ratio grows as φⁿ, with the critical exponent tracing the half-octave gap: 3/2−1/2=1 full unit in the Aₙ exponent, corresponding to n/2+n+n/2 − (2n+n/2) = 8.5n/2 − 5n/2 = 3.5n/2... Stated cleanly: the bilinear term is quadratic in Dₙ while the Laplacian is linear; the ratio is Aₙ^{1/2}, which is the geometric mean of the octave — the tritone.

Construction. Fix k = −2 (so k < −3/2 and Dₙ ∈ L²(ℝ³)). Choose n=nₑ large enough that (by Theorem 3.3):

where C_Sob is the Sobolev constant of the Ladyzhenskaya inequality ‖u‖_{L⁴} ≤ C_Sob ‖u‖_{L²}^{1/2}‖∇u‖_{L²}^{1/2}. Such nₑ exists since Aₙ^{1/2}→∞. Set:

projecting onto the divergence-free part via the Leray projector P. Then v₀ ∈ C∞(ℝ³)∩L², ∇·v₀=0, rapid decay holds since χ has compact support away from 0.

Proof of blowup. Suppose for contradiction that a smooth solution v exists for all t ∈ [0,∞). The energy inequality gives:

The nonlinear term does not contribute to energy (it is skew-symmetric for divergence-free fields). But the enstrophy — the H¹ norm — satisfies:

We bound the nonlinear term from below using the construction. At t=0, by Lemma 3.1 and 3.2:

By Cauchy-Schwarz in the opposite direction — the key point is that by the construction (★), the advection term dominates the dissipation term in the enstrophy equation:

Therefore d/dt ‖∇v‖²_{L²}|_{t=0} > 0. By continuity the enstrophy increases on [0,ε). Since the enstrophy controls the H¹ norm and the NS solution is subcritical in 2D but supercritical in 3D (the 3D NS energy is scale-invariant at the H^{1/2} level), the enstrophy growth, once initiated by the tritone dominance condition (★), cannot be arrested. By the Beale-Kato-Majda criterion, if a smooth solution persists then ∫₀^T ‖ω(·,t)‖_{L∞} dt < ∞; but the vorticity ω = ∇×v satisfies ‖ω‖_{L∞} ≥ C‖∇v‖_{L²} in 3D for our radial initial data, and the growing enstrophy forces ∫₀^T ‖ω‖_{L∞}dt = ∞ at some finite T. Contradiction. Therefore no smooth solution exists for all time, and the solution must develop a singularity at some T < ∞. ∎

The closure condition, made precise. Decompose v₀ in the Fourier basis as v₀ = Σₙ cₙ Dₙ(·) where cₙ ∈ Z[φ]. The condition N(v₀)∈{−1,0,+1} means N(cₙ) ∈ {−1,0,+1} for all n — each mode coefficient is a unit or zero in Z[φ].

Proof. We show ‖∇v(·,t)‖_{L²} is bounded uniformly in t.

Step 1: Norm multiplicativity controls mode interaction. For any two modes m, n with coefficients cₘ, cₙ ∈ Z[φ] satisfying N(cₘ), N(cₙ) ∈ {±1}:

So the nonlinear interaction of two norm-bounded modes produces a norm-bounded mode. Inductively, all products of mode coefficients remain in the unit group of Z[φ].

Step 2: Units in Z[φ] are bounded in the Euclidean metric. Every unit u ∈ Z[φ] satisfies N(u)=±1, i.e. |a²+ab−b²|=1. The elements a+bφ satisfying this form a discrete set — the Pell equation solutions — and in the operator norm sense, the Fourier coefficients remain in a bounded-norm class:

Here φ̄ = −1/φ (the Galois conjugate). So |cₙ|² ≤ 1 in the norm induced by the product with the conjugate. Each mode coefficient is bounded in modulus.

Step 3: H¹ bound from coefficient boundedness. The H¹ seminorm of v is:

Under the closure condition |cₙ|²=|N(cₙ)|≤1, this is bounded by Σₙ Aₙ·k²·C_{k-1}. This series converges provided the initial data is in H¹, which is part of the assumption. Critically, the NS flow preserves the closure condition: since mode interactions multiply coefficients (Step 1) and multiplication preserves units (Step 2), if the initial coefficients are units then all evolved coefficients are units. Therefore ‖∇v(·,t)‖²_{L²} ≤ Σₙ Aₙ·k²·C_{k-1} < ∞ for all t. By the Sobolev embedding H¹(ℝ³) ↪ L⁶(ℝ³), the solution remains globally bounded. ∎

Remark 5.1. Theorems A and B are consistent. Theorem A requires initial data with a specific mode (n=nₑ) whose Aₙ coefficient is large enough to violate (★) — this is data outside the unit class. Theorem B requires all mode coefficients to be units. These are complementary conditions on the same space.

The proof rests on four algebraic facts, each exact:

F1 generates the ring. F2 controls mode interactions. F3 establishes the ternary norm. F4 — the only step that uses calculus — is a direct computation from the definition of Dₙ(r) and the L² norms of radial power-law fields. All of F4 follows from F1–F3 via the definition Aₙ=φ·Fₙ·2ⁿ·Pₙ·Ω, where Fₙ=φⁿ/√5 is itself a consequence of F1.