---
title: "Navier–Stokes Minimal Obstruction Rigidity Program 02：Native Defect Extraction、Defect-Completed Compactness、Profile Splitting、Harmonic-Pressure Quotients 與 Minimal-Profile Existence"
short_title: "NS-MORP 02"
series: "Navier–Stokes Minimal Obstruction Rigidity Program"
cycle: "VII"
version: "v0.1"
date: "2026-08-16"
author: "Neo.K / EveMissLab"
language: "zh-TW"
status: "Native extraction / compactness architecture / conditional minimal-profile existence"
epistemic_status: "Builds an explicit defect-completed compactness topology for normalized Navier-Stokes obstruction packages. Under standard uniform local suitable-weak bounds, proves strong local L3 compactness of velocity and strong L3/2 compactness of the active Calderon-Zygmund pressure, while retaining the spatially harmonic pressure as a separate weak quotient/tail coordinate. Extracts a nonnegative dissipation defect measure from weak H1 convergence. Proves a Time-Slice Extraction Barrier showing that terminal-time danger alone may vanish in spacetime compactness, and a Thickened State/Defect Carrier theorem showing that any uniformly time-thickened dissipation packet survives as either state dissipation or a genuine defect measure. Proves a Band-Limited Tight Trace Compactness theorem and introduces a one-point compactification of relative-frequency shell distributions; terminal trace mass therefore splits into a compact finite-relative-frequency carrier or a scale-escape defect. Defines a defect-completed package topology and proves sequential compactness under explicit local bounds, pressure gauge normalization, trace/scale tightness or defect completion, and weak compactness of residual coordinates. Gives a conditional minimal-profile existence theorem by the direct method. The paper does not prove the missing universal thickening/native-separation theorem, does not rule out diffuse profile splitting, and does not promote quotient/profile compactness to actual original-solution realization. Minimal NS obstruction existence remains conditional, and no Forest Coercive Budget, Finite Forest Obstruction, atomic CN3, or Navier-Stokes regularity is proved."
canonical_source: "UTF-8 Markdown"
---

# Navier–Stokes Minimal Obstruction Rigidity Program 02

# Native Defect Extraction、Defect-Completed Compactness、Profile Splitting、Harmonic-Pressure Quotients 與 Minimal-Profile Existence

## 0. 本文定位

MORP-01 reduced minimal-obstruction construction to:

$$
\boxed{
M\mbox{-}XTR,
\quad
M\mbox{-}COM,
\quad
M\mbox{-}TR,
\quad
M\mbox{-}RIG.
}
$$

The present paper attacks the first two modules.

The central problem is subtle:

> a dangerous terminal state may be quantitatively nonzero at one pre-singularity time, while the ordinary spacetime compactness topology forgets it completely.

Therefore MORP needs a topology that is:

1. generated by native Navier--Stokes quantities;
2. compact enough to attain a minimizer;
3. strong enough not to erase the extracted obstruction;
4. flexible enough to retain defect-only limits.

---

# 1. External compactness calibration

Several primary Navier--Stokes results calibrate the strategy.

Rusin--Šverák, Jia--Šverák, Kenig--Koch, and Gallagher--Koch--Planchon show that minimal singular data can be extracted in concrete critical topologies once suitable compactness/stability or profile decomposition is available.

Jia--Šverák develop local-in-space energy and smoothing estimates.

Recent harmonic-pressure finite-scale work formulates pressure compactness modulo spatially harmonic functions because the harmonic pressure may fail strong time compactness even when the active pressure is stable.

Recent finite-window obstruction work similarly treats compact pressure images and harmonic tails as explicit structural hypotheses rather than automatic consequences of a bounded critical norm.

These results motivate the native package topology below.

---

# 2. Normalized cylinder

Work on the fixed normalized cylinder:

$$
\boxed{
Q_2
=
B_2\times(-4,0).
}
$$

Let:

$$
(u_n,p_n)
$$

be smooth pre-singularity rescalings or suitable weak packages on:

$$
Q_2.
$$

Assume the uniform local suitable-weak bound:

$$
\boxed{
\sup_n
\left[
\|u_n\|_{L_t^\infty L_x^2(Q_2)}
+
\|\nabla u_n\|_{L^2(Q_2)}
+
\|p_n-(p_n)_{B_2}(t)\|_{L^{3/2}(Q_2)}
\right]
\le
M.
}
$$

The pressure mean is only one convenient gauge.

---

# 3. Energy interpolation

The local energy bounds imply:

$$
\boxed{
u_n
\text{ bounded in }
L^{10/3}(Q_2).
}
$$

Indeed the standard three-dimensional interpolation gives:

$$
\|u_n\|_{L^{10/3}(Q_2)}
\le
C_M.
$$

Hence:

$$
u_n\otimes u_n
$$

is uniformly bounded in:

$$
L^{5/3}(Q_2).
$$

---

# 4. Local time-derivative bound

Inside:

$$
Q_{3/2},
$$

the Navier--Stokes equation gives:

$$
\partial_tu_n
=
\nu\Delta u_n
-
\nabla\cdot
(
u_n\otimes u_n
)
-
\nabla p_n.
$$

The three terms are uniformly bounded in a common negative Sobolev space, for example:

$$
\boxed{
\partial_tu_n
\text{ bounded in }
L^{3/2}
\left(
-9/4,0;
W^{-1,3/2}(B_{3/2})
\right).
}
$$

No global pressure formula is needed for this local compactness step.

---

# 5. CIV/VII-2.1 — Local State Compactness

## Theorem 5.1

Under Sections 2--4, after passing to a subsequence:

$$
\boxed{
u_n
\to
u_\ast
\quad
\text{strongly in }
L^2(Q_{4/3}).
}
$$

Moreover:

$$
\boxed{
u_n
\to
u_\ast
\quad
\text{strongly in }
L^3(Q_{4/3}).
}
$$

Also:

$$
\boxed{
\nabla u_n
\rightharpoonup
\nabla u_\ast
\quad
\text{weakly in }
L^2(Q_{4/3}).
}
$$

### Proof

Apply Aubin--Lions using:

$$
H^1(B_{3/2})
\Subset
L^2(B_{3/2})
\hookrightarrow
W^{-1,3/2}(B_{3/2}).
$$

This gives strong:

$$
L^2
$$

compactness on an interior cylinder.

Interpolate the strong:

$$
L^2
$$

convergence against the uniform:

$$
L^{10/3}
$$

bound to obtain strong:

$$
L^3.
$$

$\square$

---

# 6. Suitable-weak stability

Under the standard local pressure and local-energy bounds, the limit:

$$
(u_\ast,p_\ast)
$$

can be taken to be a suitable weak solution on the interior cylinder.

### Status

This is CLASSICAL/EXTERNAL compactness infrastructure.

The present paper uses it only as a compactness carrier.

---

# 7. Active pressure decomposition

Choose:

$$
\eta
\in
C_c^\infty(B_{3/2}),
$$

with:

$$
\eta\equiv1
$$

on:

$$
B_{5/4}.
$$

Define the active pressure:

$$
\boxed{
p_n^{act}
=
\mathcal R_i\mathcal R_j
(
\eta u_{n,i}u_{n,j}
).
}
$$

Set:

$$
\boxed{
h_n
=
p_n-p_n^{act}.
}
$$

Then for almost every time:

$$
\boxed{
\Delta h_n
=
0
\quad
\text{in }
B_{5/4}.
}
$$

---

# 8. CIV/VII-2.2 — Active-Pressure Strong Compactness

## Theorem 8.1

Under Theorem 5.1:

$$
\boxed{
p_n^{act}
\to
p_\ast^{act}
\quad
\text{strongly in }
L^{3/2}(Q_{5/4}).
}
$$

### Proof

Strong:

$$
u_n\to u_\ast
\quad
\text{in }L^3
$$

implies:

$$
\eta u_n\otimes u_n
\to
\eta u_\ast\otimes u_\ast
\quad
\text{in }L^{3/2}.
$$

The Riesz transforms are bounded on:

$$
L^{3/2}.
$$

$\square$

---

# 9. Harmonic pressure class

The full local pressure may fail strong compactness because:

$$
h_n
$$

is only spatially harmonic; its time dependence is not controlled by the same Calderón--Zygmund formula.

Define the harmonic quotient:

$$
\boxed{
[p]_{\mathcal H}
=
p
\mod
\{
h:
\Delta_xh=0
\}.
}
$$

On:

$$
Q_{5/4},
$$

the class:

$$
[p_n]_{\mathcal H}
$$

is represented by:

$$
p_n^{act}.
$$

---

# 10. CIV/VII-2.3 — Harmonic-Quotient Pressure Compactness

## Theorem 10.1

The pressure classes satisfy:

$$
\boxed{
[p_n]_{\mathcal H}
\to
[p_\ast]_{\mathcal H}
\quad
\text{strongly in }
L^{3/2}_{loc}.
}
$$

Thus harmonic pressure tails are not a compactness obstruction for the quotient pressure variable.

### Safety

The physical harmonic representative:

$$
h_n
$$

is not discarded.

If a detector or work ledger depends on harmonic pressure, its class must be retained separately as a weak/tail coordinate.

$\square$

---

# 11. Weak harmonic-tail compactness

After fixing a spatial harmonic gauge, for example a mean or finite jet normalization, interior harmonic estimates give:

$$
\boxed{
h_n
\text{ bounded in }
L_t^{3/2}
W_x^{m,\infty}(Q_1)
}
$$

for each fixed:

$$
m,
$$

provided the local:

$$
L^{3/2}
$$

pressure bounds are uniform.

Therefore one may extract:

$$
\boxed{
h_n
\rightharpoonup
h_\ast
}
$$

weakly in the corresponding time-space topology.

This is weaker than strong time compactness and is intentionally retained as a separate carrier.

---

# 12. Dissipation measure

Define:

$$
\boxed{
\mu_n^{diss}
=
|\nabla u_n|^2
dxdt.
}
$$

The uniform local energy bound gives uniformly bounded total mass on compact interior cylinders.

After a subsequence:

$$
\boxed{
\mu_n^{diss}
\stackrel{\ast}{\rightharpoonup}
\mu_\ast^{diss}
}
$$

as Radon measures.

---

# 13. CIV/VII-2.4 — Dissipation Defect Measure

## Theorem 13.1

There exists a nonnegative Radon measure:

$$
\boxed{
\nu_{\rm diss}\ge0
}
$$

such that:

$$
\boxed{
\mu_\ast^{diss}
=
|\nabla u_\ast|^2dxdt
+
\nu_{\rm diss}.
}
$$

### Proof

Weak convergence:

$$
\nabla u_n\rightharpoonup\nabla u_\ast
$$

and convex lower semicontinuity give, for every nonnegative compactly supported:

$$
\phi,
$$

$$
\int
\phi
|\nabla u_\ast|^2
\le
\liminf_n
\int
\phi
|\nabla u_n|^2.
$$

Identify the weak-star measure limit and subtract the absolutely continuous state part.

$\square$

---

# 14. Defect-only limits are legitimate

Strong:

$$
u_n\to u_\ast
$$

does not imply:

$$
\nabla u_n\to\nabla u_\ast
$$

strongly.

Thus:

$$
\boxed{
\nu_{\rm diss}\neq0
}
$$

may survive while the velocity state itself is strongly compact.

This is one precise realization of a MORP defect-only obstruction carrier.

---

# 15. Terminal-time danger

Many ANP dangerous certificates are selected-time statements.

Schematic form:

$$
\boxed{
\mathfrak d_n(0)
\ge
c_0>0.
}
$$

A spacetime topology does not automatically preserve this lower bound.

---

# 16. CIV/VII-2.5 — Time-Slice Extraction Barrier

## Theorem 16.1

There exist smooth scalar functions:

$$
g_n
$$

such that:

$$
\boxed{
g_n(0)=1
}
$$

for every:

$$
n,
$$

but:

$$
\boxed{
\int_{-1}^{0}
|g_n(t)|^2dt
\to0.
}
$$

### Proof

Take a smooth compactly supported:

$$
\phi
$$

with:

$$
\phi(0)=1,
$$

and set:

$$
g_n(t)=\phi(nt).
$$

Then the integral is:

$$
O(n^{-1}).
$$

$\square$

---

# 17. Consequence for M-XTR

A terminal-time dangerous mark does not, by itself, produce a nonzero spacetime defect measure.

Therefore M-XTR must supply at least one of:

$$
\boxed{
\textbf{TRACE-CARRIER}
}
$$

or:

$$
\boxed{
\textbf{THICKENED-DEFECT-CARRIER}.
}
$$

This is a genuine extraction obligation.

---

# 18. Thickened packet

Let:

$$
\chi
\in
C_c(Q_1),
\qquad
\chi\ge0.
$$

Suppose an extracted dangerous package satisfies:

$$
\boxed{
\int_{Q_1}
\chi
|\nabla u_n|^2
dxdt
\ge
\eta
>
0
}
$$

uniformly in:

$$
n.
$$

---

# 19. CIV/VII-2.6 — Thickened State/Defect Carrier Theorem

## Theorem 19.1

Under Theorem 13.1 and Section 18:

$$
\boxed{
\int
\chi
|\nabla u_\ast|^2dxdt
\ge
\frac{\eta}{2}
}
$$

or:

$$
\boxed{
\int
\chi
d\nu_{\rm diss}
\ge
\frac{\eta}{2}.
}
$$

### Proof

Measure convergence gives:

$$
\int
\chi
d\mu_\ast^{diss}
\ge
\eta.
$$

But:

$$
\mu_\ast^{diss}
=
|\nabla u_\ast|^2dxdt
+
\nu_{\rm diss}.
$$

At least one nonnegative summand carries half the mass.

$\square$

---

# 20. Meaning

Once a dangerous packet has uniform positive spacetime thickness, compactness cannot erase it.

It survives as:

$$
\boxed{
\textbf{STATE-VISIBLE}
\vee
\textbf{DEFECT-VISIBLE}.
}
$$

The unresolved theorem is the universal thickening step from the existing terminal certificates.

---

# 21. Selected-time trace

Suppose the package retains a selected-time diagnostic trace:

$$
f_n.
$$

Weak:

$$
L^2
$$

compactness alone does not preserve a norm lower bound.

An orthonormal sequence may converge weakly to zero.

Additional spatial/frequency tightness is needed.

---

# 22. Band-limited trace class

Assume:

$$
\boxed{
\operatorname{supp}
\widehat f_n
\subset
\{
\xi:
c\le|\xi|\le C
\}
}
$$

with fixed:

$$
0<c<C<\infty,
$$

and:

$$
\boxed{
\sup_n
\|f_n\|_2
\le
M.
}
$$

Assume spatial tightness:

$$
\boxed{
\lim_{R\to\infty}
\sup_n
\int_{|x|>R}
|f_n(x)|^2dx
=
0.
}
$$

---

# 23. CIV/VII-2.7 — Band-Limited Tight Trace Compactness

## Theorem 23.1

Under Section 22:

$$
\boxed{
\{f_n\}
\text{ is precompact in }
L^2(\mathbb R^3).
}
$$

If in addition:

$$
\boxed{
\|f_n\|_{L^2(B_A)}
\ge
\eta>0,
}
$$

then every strongly convergent subsequence has a nonzero limit:

$$
\boxed{
f_\ast\neq0.
}
$$

### Proof

Fixed Fourier support and the uniform:

$$
L^2
$$

bound give a uniform:

$$
H^1
$$

bound.

Rellich gives compactness on every fixed ball.

Spatial tightness upgrades local compactness to global:

$$
L^2
$$

compactness.

The local lower bound passes to the strong limit.

$\square$

---

# 24. Relative-frequency shells

Let:

$$
J_n
$$

be the normalized terminal reference shell.

For:

$$
m\ge0,
$$

define a nonnegative selected-time shell carrier:

$$
\boxed{
a_{n,m}
=
\int
\chi(x)
|
\omega_{J_n+m,n}(x,0)
|^2dx.
}
$$

Let:

$$
\boxed{
A_n
=
\sum_{m\ge0}
a_{n,m}.
}
$$

Assume:

$$
A_n>0.
$$

Define:

$$
\boxed{
\rho_{n,m}
=
a_{n,m}/A_n.
}
$$

---

# 25. Relative-scale probability measure

Let:

$$
\overline{\mathbb N}_0
=
\mathbb N_0
\cup
\{
\infty
\}
$$

be the one-point compactification.

Define:

$$
\boxed{
\sigma_n^{sc}
=
\sum_{m\ge0}
\rho_{n,m}
\delta_m.
}
$$

This is a probability measure on a compact space.

---

# 26. CIV/VII-2.8 — Relative-Frequency Carrier Compactification

## Theorem 26.1

After passing to a subsequence:

$$
\boxed{
\sigma_n^{sc}
\stackrel{\ast}{\rightharpoonup}
\sigma_\ast^{sc}.
}
$$

Exactly one of the following broad cases occurs.

### Finite-relative-frequency carrier

$$
\boxed{
\sigma_\ast^{sc}(\mathbb N_0)>0.
}
$$

A positive fraction of terminal carrier mass remains at finite relative shell distance.

### Relative-frequency escape

$$
\boxed{
\sigma_\ast^{sc}(\{\infty\})>0.
}
$$

A positive fraction of carrier mass escapes to unbounded frequency relative to the chosen terminal reference scale.

Both may occur simultaneously.

$\square$

---

# 27. State-visible trace route

If the finite-relative-frequency carrier is supported in a uniformly bounded shell range and the corresponding normalized traces satisfy the global:

$$
L^2
$$

bound and spatial-tightness hypotheses of Theorem 23.1, then a nonzero trace profile can be extracted.

Thus:

$$
\boxed{
\text{finite relative scale}
+
\text{spatial tightness}
+
\text{trace norm bound}
\Longrightarrow
\text{TRACE-CARRIER}.
}
$$

---

# 28. Scale-defect route

If:

$$
\sigma_\ast^{sc}(\{\infty\})>0,
$$

the terminal obstruction cannot be represented by one fixed relative-frequency trace profile.

The measure:

$$
\sigma_\ast^{sc}
$$

itself is retained as a native scale-defect coordinate.

This is not a contradiction.

It is the compactified representation of profile/scale escape.

---

# 29. Spatial escape

A similar one-point compactification may be used for normalized spatial carrier measures.

If the normalized carrier is not tight around the selected center, mass can escape to:

$$
\infty_x.
$$

MORP therefore distinguishes:

$$
\boxed{
\text{compact state profile}
}
$$

from:

$$
\boxed{
\text{spatial defect}
}
$$

and:

$$
\boxed{
\text{relative-scale defect}.
}
$$

The compactified defect coordinates prevent loss of total normalized carrier mass, but do not by themselves prove coercivity.

---

# 30. Profile splitting calibration

Critical Navier--Stokes profile decomposition separates asymptotically orthogonal translations and dilations.

Minimal singular-data arguments then require an additional compactness/rigidity step to select a critical element.

MORP has not yet proved an analogous profile decomposition for its full native package.

Therefore:

$$
\boxed{
\text{profile splitting}
}
$$

remains an M-COM obligation whenever the state/trace carrier is not tight in the chosen normalized chart.

---

# 31. Pressure-tail conclusion

The pressure compactness problem now splits cleanly.

### active pressure

$$
\boxed{
\text{strongly compact}.
}
$$

### harmonic pressure quotient

$$
\boxed{
\text{strongly compact modulo }\mathcal H_x.
}
$$

### physical harmonic tail

$$
\boxed{
\text{retained as a separate weak/tail carrier}.
}
$$

Thus pressure tail is not a single undifferentiated compactness failure.

---

# 32. Defect-completed package

Define the MORP compactness package:

$$
\boxed{
\mathfrak D_n^{comp}
=
\left(
u_n,
[p_n]_{\mathcal H},
[h_n],
\nu_{n}^{diss},
\sigma_n^{sc},
\tau_n^{sel},
\mathcal R_n^{tr}
\right).
}
$$

Here:

- $u_n$ uses strong local:
  $$
  L^3
  $$
  topology;
- $[p_n]_{\mathcal H}$ uses strong local:
  $$
  L^{3/2}
  $$
  topology;
- $[h_n]$ uses the selected weak harmonic-tail topology;
- $\nu_n^{diss}$ uses weak-star measure topology;
- $\sigma_n^{sc}$ uses weak-star probability-measure topology;
- $\tau_n^{sel}$ is a selected-time trace coordinate, strong only under explicit trace compactness guards;
- $\mathcal R_n^{tr}$ is a transition/residual coordinate in an explicitly weak compact negative topology.

---

# 33. Compactness guards

The package class is called **defect-completed tight** when:

1. the local suitable-weak bounds of Section 2 hold;
2. translation and parabolic scaling have been normalized;
3. pressure has a fixed harmonic gauge or quotient;
4. selected-time traces satisfy either:
   - a compact trace guard; or
   - an explicit trace-defect completion;
5. relative-frequency and spatial escape are represented by compactified probability/defect measures;
6. transition residuals are uniformly bounded in a reflexive or weak-star compact native space.

---

# 34. CIV/VII-2.9 — Defect-Completed Package Compactness

## Theorem 34.1

Every sequence of defect-completed tight normalized packages has a subsequence converging in the product topology:

$$
\boxed{
\mathfrak D_{n_j}^{comp}
\to
\mathfrak D_\ast^{comp}.
}
$$

The convergence has the following structure:

$$
u_{n_j}
\to
u_\ast
\quad
\text{strong }L^3_{loc},
$$

$$
[p_{n_j}]_{\mathcal H}
\to
[p_\ast]_{\mathcal H}
\quad
\text{strong }L^{3/2}_{loc},
$$

while the harmonic, measure, scale-defect, trace-defect, and transition coordinates converge in their declared weak/weak-star topologies.

### Proof

Use Theorems 5.1, 10.1, and 13.1.

Use Banach--Alaoglu/reflexive weak compactness for the harmonic and residual coordinates.

Use compactness of probability measures on the compactified scale/space carrier domains.

Take a diagonal subsequence.

$\square$

---

# 35. What Theorem 34.1 does not prove

The theorem is a compactness completion.

It does not prove that:

$$
d_{\rm nat}(\mathfrak D_\ast^{comp})>0.
$$

That requires non-tautological XTR plus lower semicontinuity of the chosen native separation.

It also does not prove actual infinite-branch realization.

---

# 36. Lower-semicontinuous native separation

Assume the normalized native obstruction distance:

$$
d_{\rm nat}
$$

extends to the defect-completed package space and is lower semicontinuous:

$$
\boxed{
d_{\rm nat}(D_\ast)
\le
\liminf_n
d_{\rm nat}(D_n)
}
$$

is not the desired direction.

For preservation of a lower bound one needs either continuity or the closed-superlevel property:

$$
\boxed{
\{
D:
d_{\rm nat}(D)\ge1
\}
\text{ is sequentially closed}.
}
$$

This is therefore made an explicit topology-design requirement.

---

# 37. Native-slice closure condition

Define:

$$
\boxed{
\mathscr O_1^{comp}
=
\{
D:
d_{\rm nat}(D)\ge1,
\ \mathcal N_{\rm pkg}(D)\le C_\ast
\}
}
$$

inside the defect-completed topology.

Require:

$$
\boxed{
\mathscr O_1^{comp}
\text{ sequentially closed}.
}
$$

A copied-gate metric is forbidden even if it trivially has this property.

---

# 38. CIV/VII-2.10 — Conditional Minimal-Profile Existence

## Theorem 38.1

Assume:

1. M-XTR produces a nonempty sequence:
   $$
   D_n\in\mathscr O_1^{comp};
   $$
2. the sequence is defect-completed tight;
3.:
   $$
   \mathscr O_1^{comp}
   $$
   is sequentially closed;
4. the obstruction cost:
   $$
   \mathfrak J
   $$
   is lower semicontinuous.

Then:

$$
\boxed{
m_\ast
=
\inf_{
D\in\mathscr O_1^{comp}
}
\mathfrak J(D)
}
$$

is attained by at least one:

$$
\boxed{
D_\ast\in\mathscr O_1^{comp}.
}
$$

### Proof

Take a minimizing sequence.

Theorem 34.1 gives a convergent subsequence.

Sequential closedness preserves native nontriviality.

Lower semicontinuity gives attainment.

$\square$

---

# 39. Status of minimal obstruction existence

The direct-method obstruction is now explicit.

Minimal-profile existence is reduced to:

$$
\boxed{
\text{M-XTR}
+
\text{defect-completed tightness}
+
\text{native-slice closure}.
}
$$

The compactness machinery itself is no longer a single opaque hypothesis.

---

# 40. M-XTR carrier alternatives

The extraction problem now has three useful routes.

### XTR-T — trace route

A selected-time dangerous carrier survives under Theorem 23.1.

### XTR-D — thickened defect route

A positive spacetime packet survives under Theorem 19.1.

### XTR-E — escape route

A fixed fraction of normalized carrier mass survives in explicit spatial/relative-scale defect coordinates.

The unresolved task is to prove that the original ANP/CFOP dangerous horizon package must enter at least one route with a native separation lower bound.

---

# 41. Time-slice danger remains the main XTR gap

The current Type-I/non-Type-I entry theorems provide selected-time dangerous certificates.

They do not universally give:

$$
\boxed{
\int_{Q_1}
\chi
|\nabla u_n|^2
\ge
\eta
}
$$

on a fixed normalized time thickness.

Nor do they universally provide the band-limited/tight selected trace assumptions of Theorem 23.1.

Thus:

$$
\boxed{
M\mbox{-}XTR
:
\mathrm{OPEN}.
}
$$

---

# 42. M-COM update

The compactness obligation has been reduced.

### local state

$$
\boxed{
\mathrm{CLOSED}
}
$$

under the standard local suitable-weak bounds.

### active pressure

$$
\boxed{
\mathrm{CLOSED}.
}
$$

### harmonic pressure

$$
\boxed{
\mathrm{QUOTIENT\ CLOSED}
+
\mathrm{WEAK\ TAIL\ RETAINED}.
}
$$

### dissipation loss

$$
\boxed{
\mathrm{DEFECT\ COMPLETED}.
}
$$

### terminal trace

$$
\boxed{
\mathrm{CLOSED\ UNDER\ BAND/TIGHTNESS\ GUARD}
}
$$

otherwise retained as a trace/scale defect.

### profile splitting

$$
\boxed{
\mathrm{OPEN}
}
$$

for the full custom obstruction package.

---

# 43. Profile-to-actual safety

Even if Theorem 38.1 produces:

$$
D_\ast,
$$

it may lie only in the defect-completed closure:

$$
\overline{\mathcal Y^{NS}}.
$$

It is not automatically one actual finite window of the original solution.

A fortiori, its normalized transition orbit is not automatically shadowed by one actual horizon branch.

Thus:

$$
\boxed{
\text{minimal profile existence}
\neq
\text{actual minimal branch existence}.
}
$$

---

# 44. Next paper

The next paper should attack transition invariance and profile splitting together:

$$
\boxed{
\textbf{
NS-MORP 03 —
Normalized Transition Invariance、
Profile Carrier Selection、
Minimal-Level Dynamics、
Ancient/Defect Normal Forms
與 Rigidity Entry
}.
}
$$

Primary tasks:

1. define the normalized transition map on the defect-completed package;
2. prove which compactness coordinates are preserved by one legal NS transition;
3. obtain a finite-profile carrier or classify diffuse profile splitting;
4. test whether minimality eliminates profile dichotomy;
5. connect a state-visible Type-I minimizer to an ancient-solution profile;
6. connect defect-only minimizers to recurrence/measure transition laws;
7. establish the first nontrivial equality-manifold classification.

---

# 45. Formal status ledger

$$
\boxed{
\begin{aligned}
\text{Local State Compactness}
&:\ \mathrm{PROVED/CLASSICAL\ UNDER\ BOUNDS},\\
\text{Active-Pressure Strong Compactness}
&:\ \mathrm{PROVED},\\
\text{Harmonic-Quotient Pressure Compactness}
&:\ \mathrm{PROVED},\\
\text{Dissipation Defect Measure}
&:\ \mathrm{PROVED},\\
\text{Time-Slice Extraction Barrier}
&:\ \mathrm{PROVED},\\
\text{Thickened State/Defect Carrier}
&:\ \mathrm{PROVED},\\
\text{Band-Limited Tight Trace Compactness}
&:\ \mathrm{PROVED},\\
\text{Relative-Frequency Carrier Compactification}
&:\ \mathrm{PROVED},\\
\text{Defect-Completed Package Compactness}
&:\ \mathrm{PROVED\ UNDER\ EXPLICIT\ GUARDS},\\
\text{Conditional Minimal-Profile Existence}
&:\ \mathrm{PROVED\ CONDITIONAL},\\
M\mbox{-}XTR
&:\ \mathrm{OPEN},\\
M\mbox{-}COM\text{ local state/pressure sector}
&:\ \mathrm{SUBSTANTIALLY\ CLOSED},\\
M\mbox{-}COM\text{ full profile-splitting sector}
&:\ \mathrm{OPEN},\\
M\mbox{-}TR
&:\ \mathrm{OPEN},\\
M\mbox{-}RIG
&:\ \mathrm{OPEN},\\
\text{unconditional minimal NS obstruction existence}
&:\ \mathrm{OPEN},\\
\text{Forest Coercive Budget}
&:\ \mathrm{OPEN},\\
\text{Finite Forest Obstruction}
&:\ \mathrm{OPEN},\\
CN3_{\rm Atomic}
&:\ \mathrm{OPEN},\\
\text{Navier--Stokes regularity}
&:\ \mathrm{NOT\ PROVED}.
\end{aligned}
}
$$

---

# 46. Conclusion

MORP-02 turns obstruction compactness into an explicit package rather than a black-box assumption.

Under standard local suitable-weak bounds, the velocity is strongly compact in local:

$$
L^3,
$$

and the active pressure is strongly compact in local:

$$
L^{3/2}.
$$

The local pressure is therefore compact in the quotient by spatially harmonic functions.

The noncompact physical information is isolated into explicit carriers:

- harmonic pressure tail;
- dissipation defect measure;
- selected-time trace;
- spatial/relative-frequency escape;
- transition residual.

A terminal-time dangerous certificate can still disappear from a spacetime topology, so M-XTR must preserve it through either a selected-time trace or a time-thickened packet.

Once thickened, the obstruction cannot vanish silently:

$$
\boxed{
\text{STATE-VISIBLE}
\vee
\text{DEFECT-VISIBLE}.
}
$$

Selected traces also become compact if relative frequency and spatial position are tight; otherwise their escape is retained as a compactified scale/spatial defect rather than discarded.

Thus a defect-completed normalized package is sequentially compact under explicit guards.

If non-tautological extraction and native-slice closure are added, the direct method then produces an actual minimal **profile object** in the completed package space.

The remaining difficulty is no longer ordinary local state compactness.

It is:

$$
\boxed{
\textbf{
terminal dangerous extraction
+
profile splitting
+
transition invariance.
}
}
$$

That is MORP-03.

---

# References

1. W. Rusin, V. Šverák, *Minimal initial data for potential Navier--Stokes singularities*, arXiv:0911.0500.
2. H. Jia, V. Šverák, *Minimal $L^3$-initial data for potential Navier--Stokes singularities*, arXiv:1201.1592.
3. C. E. Kenig, G. S. Koch, *An alternative approach to regularity for the Navier--Stokes equations in critical spaces*, arXiv:0908.3349.
4. I. Gallagher, G. S. Koch, F. Planchon, *A profile decomposition approach to the $L^\infty_t(L^3_x)$ Navier--Stokes regularity criterion*, arXiv:1012.0145.
5. H. Jia, V. Šverák, *Local-in-space estimates near initial time for weak solutions of the Navier--Stokes equations and forward self-similar solutions*, arXiv:1204.0529.
6. D. Albritton, T. Barker, *On local Type I singularities of the Navier--Stokes equations and Liouville theorems*, arXiv:1811.00502.
7. R. Yu, *Finite-Scale One-Component Regularity via Harmonic Pressure for the 3D Navier--Stokes Equations*, arXiv:2606.08352.
8. R. Yu, *Finite-Window Singularity Audits and Local-to-Clean Defect Transfer for Navier--Stokes*, arXiv:2606.15086.
9. R. Yu, *Finite-Window Local-to-Clean Transfer and Anti-Phantom Detection for Sharp Navier--Stokes Packages*, arXiv:2606.18476.
10. R. Yu, *A Structural Audit of Navier--Stokes Obstruction Calculus*, arXiv:2606.25341.
11. `NS_MORP_01_MinimalObstruction_Rigidity_v0.1.md`.