Andrea Cavazzini
This paper proves unconditional global regularity with quantitative exponential decay for the three-dimensional incompressible Navier–Stokes equations on the periodic box, restricted to velocity fields invariant under the Kida–Pelz symmetry group of order 48, for viscosities above an explicit threshold. The entire proof reduces, through a chain of six independently verifiable steps, to a single integer arithmetic fact: 20,625 < 31,104. This is, to the author's knowledge, the first time a Navier–Stokes regularity result has been distilled to a verifiable inequality between two five-digit integers, with every constant computed exactly and no numerical approximation entering the argument at any stage. The Kida–Pelz flow and why it matters. The Kida–Pelz initial datum, introduced by Kida (1985) and studied extensively by Pelz (2001), has occupied a special place in the blow-up literature for decades. It was originally proposed as a candidate for finite-time singularity formation precisely because its high octahedral symmetry concentrates vortex stretching into a small number of interacting structures, producing some of the most intense enstrophy growth observed in direct numerical simulations. The fact that the same symmetry that was expected to promote blow-up turns out to prevent it is itself a significant finding: it demonstrates that vortex stretching intensity and blow-up potential are fundamentally different quantities, a distinction that is often blurred in heuristic discussions of turbulence. The proof architecture. The argument has a deliberately transparent two-layer structure separating analysis from arithmetic, so that each layer can be checked independently by specialists in different fields. The analytic layer establishes three quantitative inputs. First, a spectral gap: representation-theoretic analysis of the octahedral group acting on Fourier space shows that the first two shells of the Laplacian spectrum are entirely killed by symmetry, tripling the effective Poincaré constant from 1 to at least 3. This means the KP symmetry forces vorticity to reside at higher wavenumbers where viscous dissipation is three times stronger than for generic flows. Second, a triadic density bound: the GKP equivariance constrains the Fourier support so severely that the number of resonant triads contributing to the nonlinear stretching term is reduced by a factor involving the group order, yielding a geometric density bound of at most 2. Third, an exact initial enstrophy: the KP datum is monochromatic, with all Fourier modes sitting at a single shell of squared wavenumber 11, giving the exact rational value 33/4 for the initial enstrophy. No floating-point computation, truncation, or discretisation enters this calculation. These three inputs feed into a Bernoulli differential inequality for the enstrophy whose separatrix is computed in closed form. The arithmetic layer then verifies that the initial enstrophy lies below this separatrix, which reduces to the integer comparison 33 times 625 equals 20,625, which is less than 31,104 equals 4 times 7,776. The safety margin is 50.8 percent, meaning the result would survive even if the analytic constants were degraded by up to 20 percent. Bounded enstrophy then gives global existence via the standard H1-continuation criterion, and exponential decay in all Sobolev norms follows by a Gronwall bootstrap. Beyond the core result. The paper establishes several extensions that go beyond mere regularity. Exponential decay is proved not only for the enstrophy but for all Sobolev norms simultaneously, with explicit prefactors and rates. The decay is shown to hold in all Lebesgue spaces from L2 to L-infinity and for all derivative orders, meaning that every physically measurable quantity associated with the flow decays exponentially. The pressure decays at double the velocity rate, a consequence of the quadratic structure of the pressure Poisson equation. Time analyticity is established for all positive times, meaning the solution extends to a holomorphic function in a strip around the real time axis. A shell-by-shell energy spectrum analysis shows that higher Fourier shells decay faster, with rates proportional to the squared wavenumber — a quantitative version of the physical intuition that small-scale structures are dissipated more rapidly. A Reynolds number characterisation shows that the Bernoulli closure holds if and only if the KP Reynolds number is below approximately 235, giving a concrete, physically interpretable criterion. The stability result deserves particular emphasis: global regularity is shown to persist under small perturbations that need not respect the KP symmetry. This means the result is not a fragile artifact of exact symmetry but a robust property of a neighbourhood in function space around the KP datum. The self-frustration connection. This paper is designed as a companion to the author's monograph "Self-Frustration of Vortex Stretching and the Architecture of the Navier–Stokes Blow-Up Barrier" (Cavazzini, 2026), which identifies a twelve-link chain of structural obstructions to finite-time blow-up for general three-dimensional Navier–Stokes. Three of those twelve links have concrete, quantitative realisations in the Kida–Pelz class. The enhanced spectral gap is a realisation of Link 5 (the spectral gap threshold that governs alignment stability). The triadic density reduction is a realisation of Link 3 (the oscillation bound that controls the pressure Hessian for tube-like vorticity). The identically vanishing helicity — proved here as a consequence of the parity inversion in the octahedral group — is a realisation of Link 6 (the gap–alignment complementarity), because it eliminates the eigenframe injection mechanism entirely: with zero helicity budget, the pressure Hessian cannot rotate the strain eigenframe to sustain the dangerous compressive component identified in the companion paper as the sole variable separating regularity-compatible from blow-up-compatible configurations. When all three mechanisms act simultaneously, as enforced by the octahedral symmetry, the self-frustration chain that remains open for general flows closes completely and unconditionally. The arithmetic inequality 20,625 < 31,104 is the quantitative expression of this closure. This provides the first concrete validation of the self-frustration framework as a genuine regularity tool rather than merely a classification scheme: the structural architecture described in the companion monograph is not an abstract taxonomy but a machinery that produces theorems when supplied with sufficient quantitative input. The minimal symmetry result strengthens this connection further: the octahedral group of order 48 is proved to be the smallest finite subgroup of O(3) for which the Bernoulli method closes. This characterises the precise boundary between symmetry groups where the self-frustration mechanisms are strong enough to guarantee regularity and those where they are not, providing a sharp answer to the question of how much geometric structure is needed to resolve the regularity problem within this framework. Context within the broader landscape. The Navier–Stokes regularity problem has a long history of partial results exploiting symmetry, from the classical two-dimensional theory (where regularity is known unconditionally due to the absence of vortex stretching) to various axisymmetric and helical reductions. The present work differs from these in a fundamental respect: the Kida–Pelz flow is fully three-dimensional with active, sustained vortex stretching — the mechanism responsible for the supercritical character of the equations is present and operative, not eliminated by dimensional reduction. What the symmetry does is not remove the stretching but quantitatively constrain it, tilting the balance between stretching and dissipation in favour of dissipation by a computable margin. This is a qualitatively different use of symmetry from the classical approach, and it suggests that the boundary between regularity and potential blow-up may be more accessible than previously thought — not through eliminating the dangerous mechanism, but through measuring and constraining it. The paper also contributes to the broader programme of understanding which structural properties of the Navier–Stokes equations are responsible for regularity. The identification of three independent mechanisms (enhanced dissipation, triadic depletion, topological obstruction) that close the regularity chain when acting together, combined with the companion monograph's demonstration that these same mechanisms are present but quantitatively insufficient for general flows, suggests a precise research programme: strengthen the quantitative estimates on any one of the three mechanisms sufficiently to close the chain without symmetry. The open problems listed in the paper — removal of the viscosity threshold, full Gevrey bootstrap, exact spectral gap computation — are formulated with this programme in mind. Methodological note. Every result in the paper carries an explicit epistemic label. All constants are computed exactly as rational numbers or algebraic expressions. The paper makes no claim regarding the Clay Millennium Prize and explicitly discusses the four gaps separating the present result from the Prize requirements: symmetry restriction, periodic domain, viscosity threshold, and partial Gevrey bootstrap. The distance from each gap to a resolution is assessed individually, with the viscosity threshold identified as an artifact of the Bernoulli method rather than a physical phase transition. MSC 2020 Classification: 35Q30 (primary — Navier–Stokes equations); 76D03 (existence, uniqueness, and regularity for incompressible viscous fluids); 42B25 (maximal functions and Littlewood–Paley theory); 20C15 (ordinary representations and characters of finite groups); 35B65 (smoothness and regularity of solutions to PDE