UNCONDITIONAL BKM FOR IPQ FLOWS VIA AXIS-ADAPTED CONIC CALCULUS AND RSCC

This manuscript addresses the continuation problem for three-dimensional incompressible viscous transport–diffusion systems by revisiting a classical vorticity-based continuation mechanism in the spirit of Constantin–Fefferman and Beale–Kato–Majda. Rather than proposing a new criterion or modifying an existing one, the paper demonstrates that the hypothesis of a classical continuation criterion is automatically satisfied within a precisely defined solution class, using only established analytic tools. The central idea is to replace pointwise or a priori assumptions on the vorticity or velocity gradient with a scale-resolved, geometric control mechanism that propagates in time. The argument is organized around a dyadic decomposition of vorticity combined with axis-adapted conic projections, which isolate the geometrically coherent component of the flow at each scale. The remaining “off-cone” component is treated as a controlled error, quantified using restricted square–Carleson packing estimates that measure the size and persistence of geometrically unfavorable regions. At the analytic level, the paper introduces a dissipative “ledger” quantity: a weighted sum of dyadic vorticity blocks that tracks the accumulation and leakage of vorticity across scales. The evolution of this ledger is governed by a differential inequality whose structure mirrors that of classical continuation arguments, but with all potentially singular terms expressed in ledger variables rather than independent L∞L^\inftyL∞ norms. Aperture-uniform Calderón–Zygmund bounds and commutator estimates ensure that all nonlinear interactions respect the geometric splitting, while the Carleson packing bounds guarantee summability of the error terms across scales. A key technical step is the elimination of any circular dependence on a priori L∞L^\inftyL∞ control. The paper shows that occurrences of ∥ω∥L∞\|\omega\|_{L^\infty}∥ω∥L∞ or ∥∇u∥L∞\|\nabla u\|_{L^\infty}∥∇u∥L∞ in the ledger inequality can be reconstructed from the same dyadic components that define the ledger itself, up to summable geometric leakage. Absolute continuity in time and standard energy inequalities are then used to close the resulting Grönwall-type argument without assuming the continuation hypothesis in advance. The final outcome is an unconditional continuation result within the stated class: once a solution enters the class, no finite-time breakdown consistent with the classical criterion can occur. The role of the Constantin–Fefferman / Beale–Kato–Majda condition is purely instrumental—it serves as a known analytical gateway whose hypothesis is shown to be internally forced by the structure of the flow, rather than imposed externally. The manuscript is deliberately conservative in scope. It does not introduce new equations, alter classical criteria, or claim results beyond the stated framework. All arguments rely on standard harmonic analysis, geometric measure estimates, and well-known properties of incompressible viscous flows. Complete proofs are provided, including appendices detailing the geometric packing bounds, summability arguments, and closure of the ledger inequality. In summary, the paper reframes a classical continuation mechanism as a consequence of scale-resolved geometric control, thereby converting a conditional regularity statement into an unconditional one within a well-defined analytic setting, without changing the underlying criterion itself.