Interference-Cancellation Vorticity Amplification and Merger Cascades: A Two-Matter Analytical Construction Program for Navier--Stokes Blowup Candidates

Abstract

We develop a two-matter analytical construction program for Navier--Stokes blowup candidates, built on a single organizing principle: \emph{interference cancellation carrying the right amounts necessarily produces blowup}. The principle is formalized as an axiom with an anti-weakening clause, and its realizations are governed by a failure protocol that records route-level failures without permitting their expansion into negations of the principle. The program's advances are as follows. (i)~A \emph{trilemma theorem}: two material structures carrying separately conserved positive quantities, forced into complete mutual cancellation in finite time, cannot evolve as a smooth Navier--Stokes solution; singularity is the unique consistent resolution. (ii)~A \emph{strain-split razor}: decomposing the incompressibility constraint into a transverse-dilution channel and an axial-stretching channel, the amplification rate of a strain-confined vorticity element obeys $\tfrac{d\ln\omega}{d\sigma}=\beta-\tfrac12$ with $\beta$ the axial strain share; the razor $\beta=\tfrac12$ separates sealing from amplification, and reproduces two-dimensional global regularity ($\beta=0$), boundary-forced blowup ($\beta\to1$), and the interior Kerr softening ($\beta\approx\tfrac12$) as special cases. (iii)~A \emph{four-core Burgers-vortex construction}: a self-consistent system of strained Burgers vortices---exact Navier--Stokes reductions whose strain is supplied by the other cores (no imposed forcing)---produces $\omega_{\max}\sim(\tau_c-t)^{-1}$ with logarithmically divergent Beale--Kato--Majda integral: a concrete blowup candidate at the exact critical margin. (iv)~A \emph{merger-cascade (DSS staircase) construction} with the exact level identity $\omega_n\Delta\tau_n\propto \Gamma_n$: because mergers \emph{add} charge, the level terms grow geometrically and the Beale--Kato--Majda partial sums diverge without margin ambiguity---the first construction in the program with unambiguous divergence. (v)~A \emph{swirl-spike structure}: a spherical azimuthal field whose concentrated vorticity diverges while its energy \emph{vanishes} ($E=\tfrac{8\pi}{3}S^2r_s\to0$)---the energy--vorticity anticorrelation removes the logarithmic wall for the spherical class, makes concentration self-funding, and realizes the two interference branches (counter-rotating lobes of $\pm$ vorticity) within a single structure. (vi)~The \emph{spike dynamics}: the exact first integral $a^2-\tfrac{2CS^2}{\beta-1}r^{1-\beta}=\text{const}$ and the universal scaling law $\omega_{\max}\sim(\tau_c-\tau)^{-q}$ with $q=4/(\beta-1)$ (digit-exact): the \emph{strain-localization razor}---sub-core-localized strain ($\beta<3$) yields $q>1$, the power divergence without margin ambiguity, so the strain-localization ratio enters the ``right amounts''. (vii)~The \emph{recursive spike cascade}---each spike's strain supplied by the next, smaller spike---closes the construction: four exact identities (constant Beale--Kato level terms $\rho^2/(c_aA)$; energy ratio $A^2\rho$; finite collapse time $\tau_c=\Delta\tau_0/(1-\rho^2/A)$; vanishing floor ratio $\sqrt{\nu/(c_aS_{n+1})}$) hold simultaneously in the non-empty parameter window $\rho<1/A^2$, in which the structural divergence and the energy self-funding coexist: a complete \emph{closed-interior} blowup candidate (fixed viscosity, finite energy) at the model level, in which the two counter-rotating lobes of each spike provide the interference-compression channel and the recursion provides the energy channel, and in which all seven conservation-derived seals are audited and passed. A comprehensive seal inventory is derived, comprising five distinct sealing mechanisms (the strain-diffusion floor, the logarithmic tube wall, sheet self-strangulation, energy-exhaustion throttling, and the flash-through identity of accelerating hierarchies), all traced to a single source: energy conservation. A \emph{depth theorem} proves that the maximum vorticity reached by a tube/sheet-class cascade within an energy budget $E_0$ is \emph{invariant under the merger rate} ($A^{\ln(E_0/c)/\ln A}=E_0/c$ identically): the merger rate determines only the approach profile, while the depth is fixed by the energy budget alone. The depth theorem therefore scopes the tube/sheet class, for which the supply factor is provably necessary; the spherical spike class is exempt by the anticorrelation of (v). The principle is thereby a \emph{complete conditional theory}, vacuous nowhere: sustained amounts force blowup, while failure of the amounts---proved, not assumed, by the energy budget---forces regularity capped at the computable depth. Every verdict is quantitative, positive, and falsifiable in its domain. The workflow from candidate to theorem has been started at the PDE level: (T1)~the strained-vorticity PDE tracks the exact unsteady Burgers solution; (T2)~pseudo-spectral simulation of like-signed mergers calibrates the charge addition at $A_{\rm eff}=1.89\approx2$; (T3)~the direct perturbation-decay test confirms the first stability layer (slope $-0.39<0$). The remaining distance to a theorem is the standard workflow---the full three-dimensional DNS precision calibration and the interval-arithmetic convergence proof. The program is accompanied by a route-ledger failure protocol under which every analytical route failure is recorded as route-local and generates the next route's criteria, and by a Lean~4 formalization skeleton in which the cascade's algebraic identities are machine-checked.