---
title: "Navier–Stokes Diffuse Carrier Rigidity Program 01：Carrier Entropy、Rényi Concentration、Nonlinear Concentration Recovery、Source Migration 與 Atomic Reprofiling Thresholds"
short_title: "NS-DCRP 01"
series: "Navier–Stokes Diffuse Carrier Rigidity Program"
cycle: "VIII"
version: "v0.1"
date: "2026-08-16"
author: "Neo.K / EveMissLab"
language: "zh-TW"
status: "Diffuse-carrier nonlinear rigidity foundation / first concentration-recovery theorem"
epistemic_status: "Starts from the Cycle-VII surviving normal form: a minimal, zero-tax, mechanism-kernel-saturated diffuse carrier. Proves that Shannon entropy divergence alone is not a dynamical obstruction, and identifies the correct nonlinear concentration functional: if a local output is superlinear in normalized carrier mass, fixed output forces a fixed-share atom. For a dyadic vorticity shell, proves a wavelength-cell pseudolocal estimate for the same-shell strain generated by that shell and derives a concentration-recovery theorem for gross/positive same-shell diagonal vortex stretching: a fixed normalized diagonal stretching output forces a fixed fraction of shell enstrophy into one enlarged wavelength-scale cell, modulo arbitrarily small Schwartz-kernel leakage. Hence a genuinely diffuse carrier cannot sustain an order-one same-shell local stretching supply. If a dangerous source/output remains order one, it must migrate into cross-scale, far-field, commutator/subfilter, pressure-flux, leakage, or another nonlocal channel. This is consistent with recent filtered-vortex-stretching and finite-chain bad-scale counting frameworks, which explicitly isolate far-field/commutator/localization and non-tautological concentration/residual channels. The paper proves an entropy-only no-go, a generic superlinear-output concentration-recovery theorem, a shell-diagonal stretching recovery theorem, and a diffuse source-migration compiler. It does not prove that every dangerous Navier-Stokes branch has a fixed same-shell diagonal stretching fraction, does not exclude diffuse cross-scale carriers, and does not prove a Forest Coercive Budget, Finite Forest Obstruction, atomic CN3, or Navier-Stokes regularity."
canonical_source: "UTF-8 Markdown"
---

# Navier–Stokes Diffuse Carrier Rigidity Program 01

# Carrier Entropy、Rényi Concentration、Nonlinear Concentration Recovery、Source Migration 與 Atomic Reprofiling Thresholds

## 0. Program objective

MORP Cycle VII reduced the surviving hypothetical minimal obstruction to:

$$
\boxed{
\textbf{
minimal + zero-tax + kernel-saturated + diffuse carrier.
}
}
$$

Atomic space--scale escape can be reprofiled.

Pure interior dissipation defect has been removed from the zero-slack kernel.

Known ancient Liouville subclasses have been removed.

Thus the remaining question is:

> can Navier--Stokes sustain a dangerous normalized carrier while every individual space--scale atom carries a vanishing share?

DCRP studies the dynamics of this diffuse carrier.

The first lesson is that entropy alone is not enough.

The second is that **superlinear local nonlinear efficiency** can recover concentration.

---

# 1. Normalized carrier distribution

Let:

$$
\mathscr C
$$

be a finite or countable family of space--scale carrier cells.

Let:

$$
e_\alpha\ge0,
\qquad
\alpha\in\mathscr C,
$$

satisfy:

$$
\boxed{
\sum_{\alpha\in\mathscr C}
e_\alpha
=
1.
}
$$

Define the maximal atomic share:

$$
\boxed{
p_{\max}
=
\sup_{\alpha}
e_\alpha.
}
$$

A diffuse sequence satisfies:

$$
\boxed{
p_{\max}\to0.
}
$$

---

# 2. Shannon carrier entropy

Define:

$$
\boxed{
\mathfrak H(e)
=
-
\sum_\alpha
e_\alpha
\log e_\alpha.
}
$$

The min-entropy bound gives:

$$
\boxed{
\mathfrak H(e)
\ge
-\log p_{\max}.
}
$$

Hence:

$$
p_{\max}\to0
\Longrightarrow
\mathfrak H(e)\to\infty.
$$

This is a fragmentation diagnostic.

It is not yet a PDE tax.

---

# 3. Rényi concentration functional

For:

$$
\theta>0,
$$

define:

$$
\boxed{
\mathfrak C_\theta(e)
=
\sum_\alpha
e_\alpha^{1+\theta}.
}
$$

Since:

$$
e_\alpha^\theta
\le
p_{\max}^{\theta},
$$

$$
\boxed{
\mathfrak C_\theta(e)
\le
p_{\max}^{\theta}.
}
$$

Therefore:

$$
\boxed{
p_{\max}\to0
\Longrightarrow
\mathfrak C_\theta(e)\to0.
}
$$

Unlike Shannon entropy, this functional directly controls superlinear local outputs.

---

# 4. Nonlinear effective multiplicity

Define:

$$
\boxed{
\mathfrak M_\theta
=
\mathfrak C_\theta(e)^{-1/\theta}.
}
$$

For a uniform distribution on:

$$
N
$$

cells:

$$
e_\alpha=1/N,
$$

one has:

$$
\boxed{
\mathfrak M_\theta=N.
}
$$

For the trilinear exponent:

$$
\theta=\frac12,
$$

$$
\boxed{
\mathfrak M_{1/2}
=
\left(
\sum_\alpha e_\alpha^{3/2}
\right)^{-2}.
}
$$

---

# 5. CIV/VIII-1.1 — Entropy-Only No-Go

## Theorem 5.1

Shannon entropy divergence, by itself, cannot contradict a linear carrier ledger.

### Proof

Let:

$$
e_\alpha^{(N)}
=
1/N,
\qquad
1\le\alpha\le N.
$$

Then:

$$
\boxed{
\mathfrak H(e^{(N)})
=
\log N
\to\infty.
}
$$

But for any constant linear cost density:

$$
c\ge0,
$$

$$
\boxed{
\sum_{\alpha=1}^{N}
c
e_\alpha^{(N)}
=
c
}
$$

for every:

$$
N.
$$

Likewise total normalized carrier mass remains one.

$\square$

---

# 6. Meaning

Entropy can diverge at zero additional linear mass cost.

Therefore a DCRP obstruction requires at least one of:

- superlinear local output;
- strict interaction tax;
- strict splitting tax;
- coherence penalty;
- nonlinear concentration recovery.

Merely defining an entropy coordinate does not solve the diffuse-carrier problem.

---

# 7. Superlinear local-output model

Let:

$$
q_\alpha
$$

be the contribution of carrier cell:

$$
\alpha
$$

to a nonnegative or absolute nonlinear output.

Assume:

$$
\boxed{
|q_\alpha|
\le
C
e_\alpha^{1+\theta},
}
$$

for some:

$$
\theta>0.
$$

Define total output:

$$
\boxed{
Q
=
\sum_\alpha
|q_\alpha|.
}
$$

---

# 8. CIV/VIII-1.2 — Superlinear Concentration Recovery

## Theorem 8.1

Under Section 7:

$$
\boxed{
Q
\le
C
\mathfrak C_\theta(e)
\le
C
p_{\max}^{\theta}.
}
$$

Consequently, if:

$$
\boxed{
Q\ge\sigma>0,
}
$$

then:

$$
\boxed{
p_{\max}
\ge
\left(
\frac{\sigma}{C}
\right)^{1/\theta}.
}
$$

### Meaning

A fixed nonlinear output with superlinear cell efficiency forces a fixed-share atomic carrier.

$\square$

---

# 9. Critical consequence for diffuse carriers

If:

$$
p_{\max}\to0,
$$

then every nonlinear channel satisfying Section 7 must obey:

$$
\boxed{
Q\to0.
}
$$

Therefore a dangerous diffuse carrier can survive only by routing its order-one output through channels which are:

- not local in the selected cells;
- not superlinear in the local carrier mass;
- or accompanied by a compensating nonlocal/residual contribution.

---

# 10. Dyadic vorticity shell

Let:

$$
\omega_k
=
\Delta_k\omega.
$$

For divergence-free velocity, the corresponding same-shell strain can be written:

$$
\boxed{
S_k
=
T_k\omega_k,
}
$$

where:

$$
T_k
$$

is a band-limited order-zero Fourier multiplier.

Its kernel has the form:

$$
\boxed{
K_k(x)
=
2^{3k}
K(2^kx),
}
$$

with:

$$
K
$$

Schwartz after the dyadic localization.

---

# 11. Wavelength cells

Fix:

$$
L\ge1.
$$

Partition:

$$
\mathbb R^3
$$

into cubes:

$$
Q
$$

of side:

$$
L2^{-k}.
$$

Define:

$$
\boxed{
E_Q
=
\int_Q
|\omega_k|^2dx,
}
$$

and total shell enstrophy:

$$
\boxed{
E
=
\|\omega_k\|_2^2
=
\sum_Q
E_Q.
}
$$

For:

$$
B\ge2,
$$

let:

$$
Q^{(B)}
$$

be the concentric cube enlarged by factor:

$$
B.
$$

Define:

$$
\boxed{
E_Q^{(B)}
=
\int_{Q^{(B)}}
|\omega_k|^2dx,
}
$$

and enlarged-cell atomicity:

$$
\boxed{
p_{k,B}
=
\sup_Q
\frac{
E_Q^{(B)}
}{
E
}.
}
$$

---

# 12. Pseudolocal kernel split

For:

$$
x\in Q,
$$

write:

$$
S_k(x)
=
\int_{Q^{(B)}}
K_k(x-y)
\omega_k(y)dy
+
\int_{\mathbb R^3\setminus Q^{(B)}}
K_k(x-y)
\omega_k(y)dy.
$$

The first term is the near cell contribution.

The second is the Schwartz-kernel leakage.

---

# 13. CIV/VIII-1.3 — Wavelength-Cell Pseudolocal Strain Bound

## Theorem 13.1

For every:

$$
N<\infty,
$$

there exists:

$$
C_{L,N}
$$

such that:

$$
\boxed{
\|S_k\|_{L^\infty(Q)}
\le
C_L
2^{3k/2}
\left(
E_Q^{(B)}
\right)^{1/2}
+
C_{L,N}
B^{-N}
2^{3k/2}
E^{1/2}.
}
$$

### Proof

For the near part, use Cauchy--Schwarz:

$$
\left|
\int_{Q^{(B)}}
K_k(x-y)
\omega_k(y)dy
\right|
\le
\|K_k\|_2
\|\omega_k\|_{L^2(Q^{(B)})}.
$$

Dyadic scaling gives:

$$
\|K_k\|_2
=
C
2^{3k/2}.
$$

For the far part, Schwartz decay gives:

$$
|K_k(z)|
\le
C_{N}
2^{3k}
(
1+2^k|z|
)^{-N-2}.
$$

Since:

$$
|x-y|
\gtrsim
B2^{-k}
$$

outside:

$$
Q^{(B)},
$$

the far kernel has:

$$
L^2
$$

norm:

$$
O(
B^{-N}
2^{3k/2}
).
$$

Apply Cauchy--Schwarz again.

$\square$

---

# 14. Gross same-shell diagonal stretching

Define:

$$
\boxed{
\mathcal V_k^{diag}
=
\int_{\mathbb R^3}
|S_k(x)|
|\omega_k(x)|^2dx.
}
$$

This is a gross absolute same-shell stretching quantity.

The signed positive same-shell stretching:

$$
\boxed{
\mathcal P_k^{diag}
=
\int
\left(
S_k\omega_k\cdot\omega_k
\right)_+dx
}
$$

satisfies:

$$
\boxed{
\mathcal P_k^{diag}
\le
\mathcal V_k^{diag}.
}
$$

---

# 15. CIV/VIII-1.4 — Diffuse Diagonal-Stretching Bound

## Theorem 15.1

For every:

$$
N,
$$

$$
\boxed{
\frac{
\mathcal V_k^{diag}
}{
2^{3k/2}
E^{3/2}
}
\le
C_L
p_{k,B}^{1/2}
+
C_{L,N}
B^{-N}.
}
$$

### Proof

On each cell:

$$
\int_Q
|S_k|
|\omega_k|^2
\le
\|S_k\|_{L^\infty(Q)}
E_Q.
$$

Apply Theorem 13.1:

$$
\le
C_L
2^{3k/2}
(E_Q^{(B)})^{1/2}
E_Q
+
C_{L,N}
B^{-N}
2^{3k/2}
E^{1/2}
E_Q.
$$

Since:

$$
E_Q^{(B)}
\le
p_{k,B}E,
$$

sum over:

$$
Q
$$

and use:

$$
\sum_QE_Q=E.
$$

$\square$

---

# 16. Dimensionless diagonal stretching activity

Define:

$$
\boxed{
\mathfrak V_k^{diag}
=
\frac{
\mathcal P_k^{diag}
}{
2^{3k/2}
E^{3/2}
}.
}
$$

This is scale invariant under Navier--Stokes scaling at the shell level.

---

# 17. CIV/VIII-1.5 — Same-Shell Concentration Recovery

## Theorem 17.1

Suppose:

$$
\boxed{
\mathfrak V_k^{diag}
\ge
\sigma>0.
}
$$

Choose:

$$
B
$$

large enough that:

$$
C_{L,N}B^{-N}
\le
\sigma/2.
$$

Then:

$$
\boxed{
p_{k,B}
\ge
c_L
\sigma^2.
}
$$

### Proof

Use:

$$
\mathcal P_k^{diag}
\le
\mathcal V_k^{diag}
$$

and Theorem 15.1.

$\square$

---

# 18. Interpretation

A fixed fraction of scale-normalized same-shell diagonal vortex stretching cannot be sustained by a carrier whose wavelength-cell share tends to zero.

Thus:

$$
\boxed{
\text{order-one local same-shell stretching}
\Longrightarrow
\text{fixed-share wavelength atom}.
}
$$

By MORP-05, such an atom is eligible for secondary reprofiling under the inherited compactness guards.

This is the first DCRP concentration-recovery mechanism.

---

# 19. No claim about full vortex stretching

The full vortex stretching:

$$
S\omega\cdot\omega
$$

contains:

- cross-shell interactions;
- low-high interactions;
- high-high interactions;
- spatially nonlocal strain;
- cancellation across shell components.

Theorem 17.1 controls only the same-shell diagonal channel.

Therefore:

$$
\boxed{
\text{diffuse carrier}
}
$$

does not imply total vortex stretching is small.

It implies only that order-one dangerous supply must migrate away from the local same-shell diagonal channel.

---

# 20. Source/output decomposition

Let a dangerous normalized output satisfy:

$$
\boxed{
\mathfrak Q_k^{tot}
\ge
\sigma.
}
$$

Decompose:

$$
\boxed{
\mathfrak Q_k^{tot}
=
\mathfrak Q_k^{diag}
+
\mathfrak Q_k^{rest}.
}
$$

Assume:

$$
|\mathfrak Q_k^{diag}|
\le
C
\mathfrak V_k^{diag}.
$$

---

# 21. CIV/VIII-1.6 — Diffuse Source-Migration Compiler

## Theorem 21.1

Suppose:

$$
p_{k,B}\to0
$$

along a sequence and:

$$
B
$$

is fixed sufficiently large.

Then:

$$
\boxed{
\mathfrak Q_k^{diag}
\to0
}
$$

up to the fixed arbitrarily small pseudolocal leakage.

Consequently, if:

$$
\mathfrak Q_k^{tot}\ge\sigma>0,
$$

then for all sufficiently late indices:

$$
\boxed{
|\mathfrak Q_k^{rest}|
\ge
\sigma/2.
}
$$

after choosing the leakage tolerance below:

$$
\sigma/4.
$$

### Meaning

A diffuse dangerous carrier must migrate its supply into the non-diagonal/nonlocal output channels.

$\square$

---

# 22. The migrated channels

The remainder may contain:

- cross-scale stretching;
- far-field strain;
- differentiated commutator/subfilter stress;
- pressure/flux work;
- localization leakage;
- transition/reproduction source;
- other package-specific channels.

DCRP-01 does not declare these channels impossible.

It identifies them as the only places where a truly diffuse minimal carrier can still obtain order-one nonlinear support.

---

# 23. External filtered-stretching calibration

Recent filtered-vorticity work proves:

$$
\boxed{
\text{positive near-field stretching}
\to
\text{diffusion-absorbed direction defect}
}
$$

up to a lower-order reservoir.

It then assigns the remaining positive surplus to:

- far-field strain;
- commutator forcing;
- localization residuals.

The far-field is reduced to weighted packing/conditional annular Carleson embedding.

The commutator is reduced to a scale-invariant increment defect.

This external architecture is consistent with Theorem 21.1: once local near/same-scale mechanisms are removed or diffuse, dangerous supply must move into explicit nonlocal/defect channels.

### Status

$$
\boxed{
\mathrm{EXTERNAL\ CALIBRATION}.
}
$$

---

# 24. External flux-locality calibration

Dascaliuc--Grujić prove physical-space energy-flux locality under their inertial-range hypotheses.

This shows that, in an established cascade regime, aggregate flux can be supported by interactions local in physical scale.

It does not prove that a hypothetical singular MORP/DCRP carrier lies in that inertial-range regime.

Thus it is calibration, not a universal concentration-recovery theorem.

---

# 25. External profile-decomposition calibration

Gallagher--Koch--Planchon develop critical Navier--Stokes profile decomposition with asymptotically orthogonal translation/dilation parameters and nonlinear profile evolution.

Their work shows that critical noncompactness naturally decomposes into multiple scale/space carriers and that multilinear profile interactions require explicit control.

It does not imply a diffuse MORP carrier must reconcentrate.

This is exactly why DCRP needs a nonlinear concentration-recovery tax.

---

# 26. External finite-chain counting calibration

Recent finite-chain CKN-bad scale counting proves a non-tautological finite-chain bound in which persistent bad scales are paid by explicit standard-PDE channels including:

- vertical one-component concentration;
- annular leakage;
- pressure tails;
- pressure--flux--energy residuals.

The closing mechanism uses one-component compactness rather than inserting the full CKN badness into the detector.

This independently supports the DCRP principle:

$$
\boxed{
\text{persistent badness}
\Longrightarrow
\text{concentration channel}
\vee
\text{explicit residual cost}.
}
$$

### Status

$$
\boxed{
\mathrm{EXTERNAL/FINITE\mbox{-}CHAIN}.
}
$$

---

# 27. Entropy is a routing diagnostic

For a diffuse carrier:

$$
\mathfrak H\to\infty,
\qquad
\mathfrak C_\theta\to0.
$$

The first quantity records fragmentation.

The second predicts decay of superlinear local outputs.

Therefore DCRP uses:

$$
\boxed{
(\mathfrak H,\mathfrak C_\theta)
}
$$

as a two-coordinate fragmentation diagnostic.

Neither is assumed monotone under Navier--Stokes evolution.

---

# 28. CIV/VIII-1.7 — No General Entropy-Monotonicity Principle

## Theorem 28.1

There is no purely measure-theoretic monotonicity law for:

$$
\mathfrak H
$$

under an arbitrary mass-preserving carrier transition.

### Reason

A mass-preserving map may:

- split one atom into many atoms, increasing entropy;
- merge many atoms into one atom, decreasing entropy.

Navier--Stokes carrier transitions are not known to be doubly stochastic maps on the selected cell distribution.

Therefore DCRP does not assume:

$$
\mathfrak H_{n+1}\ge\mathfrak H_n.
$$

$\square$

---

# 29. Concentration recovery threshold

For a superlinear output exponent:

$$
1+\theta,
$$

a normalized output lower bound:

$$
\sigma
$$

forces:

$$
\boxed{
p_{\max}
\gtrsim
\sigma^{1/\theta}.
}
$$

For shell-local trilinear stretching:

$$
\theta=\frac12,
$$

the threshold is quadratic:

$$
\boxed{
p_{\max}
\gtrsim
\sigma^2.
}
$$

This gives a quantitative atomic reprofiling threshold.

---

# 30. Minimal diffuse carrier consequence

Let:

$$
D_\ast
$$

be a hypothetical minimal diffuse obstruction.

If its zero-tax recurrence requires a fixed:

$$
\sigma>0
$$

fraction of normalized same-shell diagonal stretching at infinitely many returns, then Theorem 17.1 forces a fixed-share atom at those returns.

Under MORP-05 secondary compactness, that atom can be reprofiled.

Therefore:

$$
\boxed{
\text{minimal diffuse recurrence}
}
$$

cannot be sustained indefinitely by a fixed fraction of local same-shell diagonal stretching.

---

# 31. Surviving diffuse supply

A truly diffuse minimal obstruction must therefore satisfy one of:

$$
\boxed{
\textbf{CROSS}
}
$$

cross-scale nonlinear supply;

$$
\boxed{
\textbf{FAR}
}
$$

far-field strain/tail supply;

$$
\boxed{
\textbf{COM}
}
$$

critical commutator/subfilter supply;

$$
\boxed{
\textbf{WORK}
}
$$

pressure/flux/energy/trace transfer;

or:

$$
\boxed{
\textbf{RES}
}
$$

explicit localization/transition/reproduction residual support.

These labels are routing classes, not new independent physical mechanisms.

---

# 32. Relation to the MORP zero-tax kernel

MORP kernel saturation already constrains several of these channels in selected normal forms.

However the current theory does not prove that all CROSS/FAR/COM/WORK/RES supply vanishes simultaneously on every diffuse minimal carrier.

Therefore DCRP-01 stops at source migration.

It does not yet close the diffuse kernel.

---

# 33. DCRP first rigidity dichotomy

The program now has:

$$
\boxed{
\text{FIXED LOCAL SUPERLINEAR SUPPLY}
\Longrightarrow
\text{ATOMIC REPROFILE},
}
$$

or:

$$
\boxed{
\text{DIFFUSE CARRIER}
\Longrightarrow
\text{OUTPUT MIGRATION}.
}
$$

The next task is to prove that migrated supply itself either reconcentrates or pays a strict diffuse-carrier tax.

---

# 34. Next paper

The next paper should attack the migrated nonlinear supply graph:

$$
\boxed{
\textbf{
NS-DCRP 02 —
Cross-Scale Supply Migration、
Interaction Graph Concentration、
Far-Field/Commutator Carrier Costs
與 Diffuse-Recurrence Rigidity
}.
}
$$

Primary tasks:

1. represent CROSS/FAR/COM source supply as a weighted interaction graph on space--scale carrier cells;
2. seek superlinear recovery for local high-high and high-low interaction clusters;
3. quantify how many partner cells are required when every node share is small;
4. connect large interaction multiplicity to DRC dissipation-span/driver debt;
5. connect far-field supply to annular Carleson packing and comparable-annulus barriers;
6. connect commutator recurrence to critical increment Young profiles;
7. prove either concentration recovery or a strict multiplicity/interaction tax.

---

# 35. Formal status ledger

$$
\boxed{
\begin{aligned}
\text{Shannon carrier entropy}
&:\ \mathrm{DEFINED},\\
\text{Rényi concentration}
&:\ \mathrm{DEFINED},\\
\text{Entropy-Only no-go}
&:\ \mathrm{PROVED},\\
\text{Superlinear Concentration Recovery}
&:\ \mathrm{PROVED},\\
\text{Wavelength-Cell Pseudolocal Strain Bound}
&:\ \mathrm{PROVED},\\
\text{Diffuse Diagonal-Stretching Bound}
&:\ \mathrm{PROVED},\\
\text{Same-Shell Concentration Recovery}
&:\ \mathrm{PROVED},\\
\text{Diffuse Source-Migration Compiler}
&:\ \mathrm{PROVED},\\
\text{full Navier--Stokes supply concentration recovery}
&:\ \mathrm{OPEN},\\
\text{diffuse cross-scale carrier rigidity}
&:\ \mathrm{OPEN},\\
\text{strict entropy/interaction tax}
&:\ \mathrm{OPEN},\\
\text{minimal diffuse obstruction exclusion}
&:\ \mathrm{OPEN},\\
\text{Forest Coercive Budget}
&:\ \mathrm{OPEN},\\
\text{Finite Forest Obstruction}
&:\ \mathrm{OPEN},\\
CN3_{\rm Atomic}
&:\ \mathrm{OPEN},\\
\text{Navier--Stokes regularity}
&:\ \mathrm{NOT\ PROVED}.
\end{aligned}
}
$$

---

# 36. Conclusion

DCRP-01 changes the diffuse-carrier problem from entropy bookkeeping to nonlinear efficiency.

Entropy divergence alone is free under a linear mass ledger.

The useful quantity is instead a superlinear concentration functional:

$$
\mathfrak C_\theta
=
\sum e_\alpha^{1+\theta}.
$$

Any local nonlinear channel whose cell output is bounded by:

$$
e_\alpha^{1+\theta}
$$

vanishes in the diffuse limit.

For the same-shell diagonal vortex-stretching channel, dyadic pseudolocality gives precisely such a concentration-recovery mechanism.

A fixed dimensionless diagonal stretching output forces:

$$
\boxed{
p_{k,B}
\gtrsim
\sigma^2.
}
$$

Thus a genuinely diffuse minimal obstruction cannot obtain an order-one fraction of its recurrent dangerous supply from local same-shell diagonal stretching.

It must move that supply into cross-scale, far-field, commutator, work, or residual channels.

This is the first nonlinear rigidity theorem aimed directly at the Cycle-VII surviving diffuse carrier.

The next problem is to follow the migrated supply and determine whether Navier--Stokes can keep moving it forever without either reconcentrating or paying a strict multiplicity/interaction tax.

---

# References

1. 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.
2. R. Dascaliuc, Z. Grujić, *Energy cascades and flux locality in physical scales of the 3D Navier--Stokes equations*, arXiv:1101.2193.
3. R. Yu, *Filtered Vortex Stretching and Subgrid Defects for the Three-Dimensional Navier--Stokes Equations*, arXiv:2606.27560.
4. R. Yu, *Invisible Defect Cascades for Navier--Stokes Regularity*, arXiv:2606.12756.
5. R. Yu, *Critical Ledgers and Scale-Defect Cascades for Navier--Stokes*, arXiv:2606.13887.
6. R. Yu, *Finite-Chain CKN-Bad Scale Counting for Navier--Stokes: Standard PDE Closure and Canonical Detector Realization*, arXiv:2606.21783.
7. R. Yu, *A Structural Audit of Navier--Stokes Obstruction Calculus*, arXiv:2606.25341.
8. `NS_MORP_CYCLE_VII_HANDOFF_v1.0.md`.
9. `NS_MORP_05_Escape_Ancient_FinalAudit_v0.1.md`.