---
title: "NS-FCBP Cycle VI Handoff"
version: "v1.0"
date: "2026-08-16"
status: "Cycle-VI closure and minimal-obstruction-rigidity frontier"
---

# NS-FCBP Cycle VI Handoff v1.0

## Core outcome

Cycle VI does not produce an unconditional Forest Coercive Budget.

It does produce a complete conditional critical-lift compiler and identifies the exact PDE obligations still missing.

## Solved structural issues

- generic one-derivative forcing gap identified;
- filtered near-field derivative loss structurally recovered;
- non-summable pressure--flux schedule constructed;
- moving filters made compatible with exact/coercive ledgers;
- horizon time-thickness threshold sharpened to the half-exponent;
- paid-side absorption criterion isolated.

## CAR decomposition

$$
CAR0:
\text{NS audit coordinate realization}
=
\mathrm{EXTERNAL/CONSTRUCTED}.
$$

$$
CAR1:
\text{dangerous causal mark}\to\text{native audit separation}
=
\mathrm{OPEN}.
$$

$$
CAR2:
\text{fixed-window anti-phantom transfer}
=
\mathrm{EXTERNAL/CONDITIONAL}.
$$

$$
CAR3:
\text{moving-window uniformity}
=
\mathrm{OPEN}.
$$

## Anti-cheating rule

A copied dangerous gate is not a coercive extraction theorem.

A valid obstruction must remain nonzero in a native audit geometry that does not contain a disguised copy of the dangerous norm.

## Invisible cascade frontier

The strongest remaining combined-invisible object is NS-realizable and simultaneously invisible to pressure, flux, energy, and selected-time adjoint trace, while retaining a nontrivial residual pairing.

Mechanism channels such as model-cone excess and filtered critical increment defects can shrink this class in particular sustaining regimes.

They do not yet eliminate the full class.

## Paid side

A sufficient closure would be:

$$
\sum w_kW_k^-
\le
\gamma
\sum w_kW_k^+
+
C,
\qquad
\gamma<1,
$$

plus caloric leakage absorption.

Neither is universally proved.

## Final Cycle-VI obligations

$$
\boxed{
XTR
\vee
UNI
\vee
RIG
\vee
SIGN.
}
$$

These are theorem-construction obligations:

- non-tautological extraction;
- moving-window uniformity;
- invisible-cascade rigidity;
- paid-side sign/leakage coercivity.

## Next program

$$
\boxed{
\textbf{
NS-MORP —
Navier--Stokes Minimal Obstruction Rigidity Program
}
}
$$

The next goal is not another ledger.

It is to extract a canonical minimal nonzero obstruction and force it into a rigid dynamical normal form that can be classified or excluded.