---
title: "NS-DRC Cycle III Handoff"
version: "v1.0"
date: "2026-08-15"
status: "Cycle-III closure and Cycle-IV launch state"
---

# NS-DRC Cycle III Handoff v1.0

## Completed cycle

Cycle III contains:

$$
\boxed{
\text{DRC-01 through DRC-07}.
}
$$

## Cycle-III result

At the Type-I reservoir-mechanism classification layer:

$$
\boxed{
\text{no unexplained DRC reservoir class remains}.
}
$$

This is not a regularity theorem.

## Proven Type-I state skeleton

For a Type-I singular point:

1. Barker--Prange give absolute parabolic enstrophy concentration.
2. Low-frequency exclusion gives absolute local UV stock with:
   $$
   2^J R\asymp1.
   $$
3. Backward concentration gives a same-center nested parabolic core chain.
4. Final Type-I singular-center branching is finite.
5. The local UV core implies a nontrivial global high-pass Hdot1 strain state.

## Proven dynamic source reductions

High-frequency state cannot remain genealogically source-free.

Source renewal is reduced through:

- strain--vorticity residual forcing;
- vorticity-parent forcing;
- low-mode dissipation-wavenumber driver action;
- finite state-backed parent clusters;
- backward state re-rooting.

Old-stock survival, source amplification, gross cancellation and many-parent multiplicity no longer form independent unexplained reservoir classes.

## Exact remaining Type-I gap

The global source ancestor of a high-pass state has not been proved to belong to the same nested singular-core branch.

Define:

$$
\boxed{
\text{SCPB — Source--Core Provenance Bridge}.
}
$$

Also, selected parent states have not been proved recursively compatible as nodes of one source tree.

Define:

$$
\boxed{
\text{REC — Recursive Edge Compatibility}.
}
$$

These are Chain-Necessity problems, not reservoir problems.

## Non-Type-I gap

The Type-I core architecture assumes:

$$
\sup_{t<T_\ast}
\|u(t)\|_{L^{3,\infty}}
<
\infty.
$$

The branch:

$$
\sup_{t<T_\ast}
\|u(t)\|_{L^{3,\infty}}
=
\infty
$$

requires a different ancestry entry theorem.

## Cycle-IV launch

$$
\boxed{
\textbf{Navier--Stokes Ancestry Necessity Program (NS-ANP)}
}
$$

### ANP-01

Source--Core Provenance.

### ANP-02

Recursive Edge Compatibility.

### ANP-03

Non-Type-I critical-Lorentz ancestry entry.

### ANP-04

Arbitrary-depth compatible finite source ancestry and infinite-path extraction.

### ANP-05

Finite Obstruction retest.

## Hard status

$$
\boxed{
\text{Reservoir classification closure}
\neq
\text{Chain Necessity}
}
$$

$$
\boxed{
\text{Chain Necessity}
\neq
\text{Finite Obstruction}
}
$$

$$
\boxed{
\text{Navier--Stokes regularity remains open}.
}
$$