---
title: "Navier–Stokes Diffuse Carrier Rigidity Program 02：Cross-Scale Interaction Graphs、Partner Multiplicity、Comparable-Annulus Recovery、Commutator Coherence 與 Diffuse-Recurrence Rigidity"
short_title: "NS-DCRP 02"
series: "Navier–Stokes Diffuse Carrier Rigidity Program"
cycle: "VIII"
version: "v0.1"
date: "2026-08-16"
author: "Neo.K / EveMissLab"
language: "zh-TW"
status: "Migrated-supply interaction graph / bounded-locality concentration recovery / far-field and commutator rigidity"
epistemic_status: "Follows the DCRP-01 source-migration theorem by representing migrated nonlinear supply as a directed interaction graph between normalized space-scale carrier cells. Proves a generic Partner-Degree Compensation theorem: for trilinear edge bounds of the form |q_ij| <= C c_i e_j^{1/2}, fixed aggregate output with vanishing parent atom share forces child-weighted partner degree to diverge like p_max^{-1/2}; this exponent is sharp in an abstract graph model. For comparable dyadic shell offsets and bounded normalized spatial interaction radius, band-limited Biot-Savart pseudolocality gives uniformly bounded graph degree, so fixed local cross-shell stretching supply forces a fixed-share parent atom. Hence a genuinely diffuse carrier cannot sustain order-one bounded-band/bounded-span cross-scale stretching; its partner-degree growth must be realized through shell-span growth, spatial/far-field span, amplitude escape, or another nonlocal channel. Relative to the prior DRC source census, shell-label multiplicity is routed to dissipation-span/driver debt. Using the external filtered far-field annular inequality, proves a Comparable-Annulus Recovery theorem: under bounded annular reservoirs, order-one far-field output cannot be supported solely by arbitrarily separated annuli and must recur within a logarithmically bounded comparable-annulus band, unless a reservoir amplitude itself escapes. For the commutator channel, proves a conditional Increment-Coherence theorem: if carrier-resolved increment components have quartic mass bounded superlinearly by carrier shares, persistent critical commutator defect with atom collapse forces the L4 overlap/coherence factor to diverge; bounded-overlap diffuse increments therefore reconcentrate. This converts persistent commutator support into either an atomic increment carrier or coherent Young-microstructure recurrence. The paper does not prove a universal carrier decomposition of the full commutator increment field, does not close comparable-annulus signed packing, and does not exclude combined-invisible pressure/flux/work residuals. Minimal diffuse obstruction exclusion, Forest Coercive Budget, Finite Forest Obstruction, atomic CN3, and Navier-Stokes regularity remain OPEN."
canonical_source: "UTF-8 Markdown"
---

# Navier–Stokes Diffuse Carrier Rigidity Program 02

# Cross-Scale Interaction Graphs、Partner Multiplicity、Comparable-Annulus Recovery、Commutator Coherence 與 Diffuse-Recurrence Rigidity

## 0. 本文定位

DCRP-01 proved:

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

Therefore a genuinely diffuse dangerous carrier must move its order-one nonlinear support into:

- cross-scale interaction;
- far-field strain;
- commutator/subfilter defects;
- pressure/flux/work;
- localization/transition residuals.

The present paper follows that migrated supply.

The central question is:

> can the carrier avoid concentration forever merely by changing partners, shell offsets, or spatial locations?

The answer is:

$$
\boxed{
\text{yes only by paying growing interaction complexity or leaving bounded locality}.
}
$$

---

# 1. Bipartite carrier graph

Let:

$$
I
$$

be a child carrier index set and:

$$
J
$$

a parent carrier index set.

Let:

$$
c_i\ge0,
\qquad
\sum_{i\in I}c_i=1,
$$

and:

$$
e_j\ge0,
\qquad
\sum_{j\in J}e_j=1.
$$

Define parent atomicity:

$$
\boxed{
p_{\max}
=
\sup_{j\in J}
e_j.
}
$$

Let:

$$
\mathcal G=(I,J,\mathcal E)
$$

be a directed bipartite interaction graph.

---

# 2. Trilinear edge supply

Let:

$$
q_{ij}
$$

be the normalized contribution carried by the edge:

$$
(i,j)\in\mathcal E.
$$

Assume the trilinear carrier estimate:

$$
\boxed{
|q_{ij}|
\le
C_0
c_i
e_j^{1/2}.
}
$$

This is the carrier scaling generated by one child quadratic mass factor and one parent strain/amplitude square-root factor.

---

# 3. Weighted partner degree

For:

$$
i\in I,
$$

define:

$$
d_i
=
\#\{
j:
(i,j)\in\mathcal E
\}.
$$

Define the child-weighted partner degree:

$$
\boxed{
\mathfrak D_{\rm part}
=
\sum_i
c_i
d_i.
}
$$

The total absolute graph supply is:

$$
\boxed{
\mathfrak Q_{\mathcal G}
=
\sum_{(i,j)\in\mathcal E}
|q_{ij}|.
}
$$

---

# 4. CIV/VIII-2.1 — Partner-Degree Compensation Theorem

## Theorem 4.1

Under Sections 1--3:

$$
\boxed{
\mathfrak Q_{\mathcal G}
\le
C_0
p_{\max}^{1/2}
\mathfrak D_{\rm part}.
}
$$

Consequently, if:

$$
\boxed{
\mathfrak Q_{\mathcal G}
\ge
\sigma>0,
}
$$

then:

$$
\boxed{
\mathfrak D_{\rm part}
\ge
\frac{
\sigma
}{
C_0p_{\max}^{1/2}
}.
}
$$

If in addition:

$$
d_i\le D_0
$$

for every child, then:

$$
\boxed{
p_{\max}
\ge
\left(
\frac{
\sigma
}{
C_0D_0
}
\right)^2.
}
$$

### Proof

For every edge:

$$
e_j^{1/2}
\le
p_{\max}^{1/2}.
$$

Hence:

$$
\begin{aligned}
\mathfrak Q_{\mathcal G}
&\le
C_0
p_{\max}^{1/2}
\sum_i
c_i
\sum_{j:(i,j)\in\mathcal E}
1
\\
&=
C_0
p_{\max}^{1/2}
\mathfrak D_{\rm part}.
\end{aligned}
$$

The remaining statements follow immediately.

$\square$

---

# 5. Interpretation

Carrier atomization may compensate superlinear local suppression by increasing partner multiplicity.

Therefore:

$$
\boxed{
\text{diffuse}
+
\text{fixed output}
\Longrightarrow
\text{interaction degree growth}.
}
$$

Multiplicity is not automatically a contradiction.

The next question is whether Navier--Stokes geometry permits such degree growth inside a bounded local interaction region.

---

# 6. CIV/VIII-2.2 — Sharpness of the Degree Exponent

## Theorem 6.1

The:

$$
p_{\max}^{-1/2}
$$

degree scale in Theorem 4.1 is sharp for the abstract graph inequality.

### Proof

Take:

$$
N
$$

child cells and:

$$
N
$$

parent cells with:

$$
c_i=e_j=1/N.
$$

Let every child have:

$$
d_N
$$

partners.

Set every active edge at the upper model size:

$$
q_{ij}
=
C_0N^{-3/2}.
$$

Then:

$$
\mathfrak Q_{\mathcal G}
=
N
d_N
C_0N^{-3/2}
=
C_0
\frac{
d_N
}{
N^{1/2}
}.
$$

Thus:

$$
d_N\asymp N^{1/2}
=
p_{\max}^{-1/2}
$$

sustains order-one graph supply.

$\square$

### Safety

This is an abstract sharpness model.

It is not a Navier--Stokes construction.

---

# 7. Consequence for rigidity

A diffuse-carrier proof cannot end with:

> partner multiplicity diverges.

The sharpness model shows that such growth can exactly compensate local carrier suppression.

One must use geometric/spectral structure to convert degree growth into another paid channel.

---

# 8. Dyadic child and parent shells

Let:

$$
\omega_k=\Delta_k\omega,
$$

and:

$$
\omega_h=\Delta_h\omega.
$$

Let:

$$
E_k=\|\omega_k\|_2^2,
\qquad
E_h=\|\omega_h\|_2^2.
$$

Partition the child shell into wavelength cubes:

$$
Q
$$

of side:

$$
L2^{-k},
$$

and the parent shell into cubes:

$$
R
$$

of side:

$$
L2^{-h}.
$$

Define normalized carrier shares:

$$
\boxed{
c_{k,Q}
=
\frac{
E_{k,Q}
}{
E_k
},
}
$$

$$
\boxed{
e_{h,R}
=
\frac{
E_{h,R}
}{
E_h
}.
}
$$

---

# 9. Parent-cell strain piece

Let:

$$
\zeta_{h,R}
$$

be a bounded-overlap smooth partition on the parent shell cells.

Define:

$$
\omega_{h,R}
=
\zeta_{h,R}\omega_h,
$$

and:

$$
\boxed{
S_{h,R}
=
T_h\omega_{h,R},
}
$$

where:

$$
T_h
$$

is the band-limited order-zero Biot--Savart/strain multiplier.

For comparable shells:

$$
|h-k|\le L_s,
$$

a parent cell whose normalized enlargement meets:

$$
Q
$$

is called a local partner.

---

# 10. Local cross-shell edge

Define the gross local edge supply:

$$
\boxed{
I_{Q,R}^{k,h}
=
\int_Q
|S_{h,R}(x)|
|\omega_k(x)|^2dx.
}
$$

Define the shellwise normalized edge:

$$
\boxed{
q_{Q,R}^{k,h}
=
\frac{
I_{Q,R}^{k,h}
}{
2^{3h/2}
E_h^{1/2}
E_k
}.
}
$$

---

# 11. CIV/VIII-2.3 — Comparable-Shell Edge Bound

## Theorem 11.1

For:

$$
|h-k|\le L_s
$$

and a local partner pair:

$$
(Q,R),
$$

$$
\boxed{
q_{Q,R}^{k,h}
\le
C_{L,L_s}
c_{k,Q}
e_{h,R}^{1/2}.
}
$$

### Proof

The band-limited kernel has:

$$
\|K_h\|_2
\asymp
2^{3h/2}.
$$

Hence:

$$
\|S_{h,R}\|_{L^\infty}
\le
C
2^{3h/2}
E_{h,R}^{1/2}.
$$

Therefore:

$$
I_{Q,R}^{k,h}
\le
C
2^{3h/2}
E_{h,R}^{1/2}
E_{k,Q}.
$$

Divide by the normalizing denominator.

$\square$

---

# 12. Local partner-degree bound

Fix:

$$
B<\infty
$$

and:

$$
L_s<\infty.
$$

If partners are restricted to:

-:
  $$
  |h-k|\le L_s;
  $$
- parent cells within normalized distance:
  $$
  B
  $$
  wavelengths of the child cell,

then each child cell has at most:

$$
\boxed{
D_0
=
D_0(B,L,L_s)
<
\infty
}
$$

local parent cells across the finite shell band.

This is a geometric packing fact.

---

# 13. Bounded-band local supply

For one parent shell:

$$
h,
$$

define:

$$
\boxed{
\mathfrak Q_{k,h}^{loc}
=
\sum_{
(Q,R)\in\mathcal E_{k,h}^{loc}
}
q_{Q,R}^{k,h}.
}
$$

Let:

$$
\boxed{
p_h^{cell}
=
\sup_R
e_{h,R}.
}
$$

---

# 14. CIV/VIII-2.4 — Bounded-Locality Concentration Recovery

## Theorem 14.1

For every comparable parent shell:

$$
|h-k|\le L_s,
$$

$$
\boxed{
\mathfrak Q_{k,h}^{loc}
\le
C_{B,L,L_s}
\left(
p_h^{cell}
\right)^{1/2}.
}
$$

Consequently, if:

$$
\boxed{
\mathfrak Q_{k,h}^{loc}
\ge
\sigma,
}
$$

then:

$$
\boxed{
p_h^{cell}
\ge
c_{B,L,L_s}
\sigma^2.
}
$$

### Proof

Apply Theorem 4.1 with:

$$
d_i\le D_0(B,L,L_s)
$$

and Theorem 11.1.

$\square$

---

# 15. Finite shell-band version

Suppose:

$$
\boxed{
\sum_{
|h-k|\le L_s
}
\mathfrak Q_{k,h}^{loc}
\ge
\sigma.
}
$$

There are at most:

$$
2L_s+1
$$

shells.

Hence some:

$$
h_\ast
$$

satisfies:

$$
\mathfrak Q_{k,h_\ast}^{loc}
\ge
\frac{
\sigma
}{
2L_s+1
}.
$$

By Theorem 14.1:

$$
\boxed{
p_{h_\ast}^{cell}
\ge
c_{B,L,L_s}
\sigma^2.
}
$$

The constant absorbs the finite shell-band factor.

---

# 16. Local cross-scale consequence

A genuinely diffuse parent carrier cannot sustain order-one stretching supply through:

- bounded relative shell offset;
- bounded normalized spatial partner span.

Therefore:

$$
\boxed{
\text{diffuse CROSS supply}
}
$$

must force at least one of:

$$
\boxed{
\text{SHELL-SPAN growth},
}
$$

$$
\boxed{
\text{SPATIAL-SPAN growth},
}
$$

or:

$$
\boxed{
\text{normalization/amplitude escape}.
}
$$

---

# 17. Interaction-degree geometry compiler

Suppose:

- each shell contributes at most:
  $$
  C_B
  $$
  spatial partners inside a fixed normalized span;
- the number of active shell offsets is:
  $$
  N_{\rm shell}.
  $$

Then:

$$
\boxed{
d_i
\le
C_B
N_{\rm shell}.
}
$$

Hence Theorem 4.1 gives:

$$
\boxed{
\mathfrak Q_{\mathcal G}
\le
C
p_{\max}^{1/2}
N_{\rm shell}.
}
$$

Thus fixed output with:

$$
p_{\max}\to0
$$

forces:

$$
\boxed{
N_{\rm shell}
\gtrsim
p_{\max}^{-1/2}
}
$$

unless spatial partner span also grows.

---

# 18. Interface with the prior DRC source census

The DRC source ledger already classified large canonical high-parent shell multiplicity.

Within that established shell-grouping architecture:

$$
\boxed{
R_{\rm MULT}
\subset
R_{\rm DISS},
}
$$

and the dissipation-range branch is routed into dissipation-boundary/driver ancestry.

Therefore when the DCRP interaction-degree growth is realized by **distinct canonical parent-shell labels**, it is not a new primitive mechanism.

It re-enters the prior:

$$
\boxed{
\text{dissipation-span / driver debt}.
}
$$

### Safety

Many spatial partners inside one shell are not silently identified with DRC shell multiplicity.

---

# 19. Spatial-degree branch

If:

$$
N_{\rm shell}
$$

remains bounded while:

$$
\mathfrak D_{\rm part}\to\infty,
$$

then the interaction graph must leave every bounded normalized spatial neighborhood.

This is a true:

$$
\boxed{
\text{FAR / spatial-span}
}
$$

migration.

Thus bounded-band CROSS has been reduced to:

$$
\boxed{
\text{ATOM}
\vee
\text{DRC shell-span}
\vee
\text{FAR}.
}
$$

---

# 20. External annular far-field inequality

The filtered-vorticity far-field analysis provides scale-invariant annular reservoirs:

$$
\mathfrak A_j,
$$

core profiles:

$$
\mathcal Q_k,
$$

and a reassigned annular far-field bound of the form:

$$
\boxed{
\mu_k^{far,ann}
\le
C_0
\sum_{j=0}^{k}
2^{-(k-j)}
\mathfrak A_j
\mathcal Q_k.
}
$$

It also proves unweighted Carleson closure under conjugate:

$$
\ell^p
\mbox{--}
\ell^q
$$

summability of:

$$
\mathfrak A
$$

and:

$$
\mathcal Q.
$$

### Status

$$
\boxed{
\mathrm{EXTERNAL/PROVED}.
}
$$

---

# 21. Uniform far-field reservoir regime

Assume:

$$
\boxed{
\mathfrak A_j
\le
A_\ast,
\qquad
\mathcal Q_k
\le
Q_\ast.
}
$$

For:

$$
M\ge1,
$$

split:

$$
\mu_k^{far,ann}
=
\mu_k^{nearann}(M)
+
\mu_k^{tail}(M),
$$

where the tail contains:

$$
j\le k-M.
$$

---

# 22. Far-annulus tail bound

The geometric kernel gives:

$$
\boxed{
\mu_k^{tail}(M)
\le
C
A_\ast
Q_\ast
2^{-M}.
}
$$

Thus arbitrarily separated annuli cannot carry a fixed fraction of far-field output under bounded scale-invariant reservoirs.

---

# 23. CIV/VIII-2.5 — Comparable-Annulus Recovery

## Theorem 23.1

Assume:

$$
\mu_k^{far,ann}
\ge
\sigma>0,
$$

and Section 21.

Choose:

$$
M
$$

so that:

$$
C
A_\ast
Q_\ast
2^{-M}
\le
\sigma/2.
$$

Then there exists:

$$
\boxed{
j_\ast\in
\{
k-M+1,\ldots,k
\}
}
$$

such that:

$$
\boxed{
\mathfrak A_{j_\ast}
\mathcal Q_k
\ge
c
\frac{
\sigma
}{
M
}.
}
$$

### Proof

The far-separated tail contributes at most:

$$
\sigma/2.
$$

Hence the last:

$$
M
$$

annular indices carry at least:

$$
\sigma/2.
$$

There are at most:

$$
M
$$

such terms and their geometric coefficients are at most one.

Pigeonhole.

$\square$

---

# 24. Meaning of comparable-annulus recovery

Under bounded annular/core reservoirs:

$$
\boxed{
\text{order-one FAR supply}
}
$$

cannot remain supported only at arbitrarily large relative annular gaps.

It must recurrently enter a logarithmically bounded comparable-annulus band.

Thus FAR is reduced to:

$$
\boxed{
\text{COMPARABLE-ANNULUS}
\vee
\text{RESERVOIR-AMPLITUDE ESCAPE}.
}
$$

---

# 25. Comparable-annulus barrier remains

The external filtered paper explicitly notes that annular reassignment alone does not create unweighted summability.

The coefficient on the:

$$
j=k
$$

channel is order one.

Therefore Theorem 23.1 is a **localization/recovery theorem**, not a packing closure.

The comparable-annulus branch still requires:

- spatial carrier concentration;
- signed cancellation;
- Carleson packing;
- or another interaction tax.

---

# 26. Far-field interaction interpretation

Theorem 23.1 eliminates one possible diffuse strategy:

> continually push every significant far-field source to arbitrarily more distant annuli while all normalized reservoirs remain bounded.

A surviving FAR carrier must instead recur near the comparable-annulus diagonal or make one of the annular/core amplitudes large.

This returns the problem to the same interaction-complexity frontier as CROSS.

---

# 27. Commutator defect

The external filtered theory defines the critical derivative-compatible increment defect:

$$
\boxed{
\widetilde{\mathcal S}^{(3)}
\sim
\int
\|V^\sharp\|^4,
}
$$

schematically, where:

$$
V^\sharp
$$

is the filtered increment field in the derivative-compatible increment space.

Persistent post-near-field surplus can force:

$$
\boxed{
\widetilde{\mathcal S}^{(3)}
\ge
s_0>0.
}
$$

Bounded critical defects admit cylindrical generalized Young profiles.

### Status

$$
\boxed{
\mathrm{EXTERNAL/PROVED}.
}
$$

---

# 28. Carrier-resolved increment model

Suppose the increment field admits a carrier decomposition:

$$
\boxed{
V^\sharp
=
\sum_\alpha
V_\alpha.
}
$$

Let:

$$
e_\alpha\ge0,
\qquad
\sum_\alpha e_\alpha=1.
$$

Assume the component quartic efficiency:

$$
\boxed{
\|V_\alpha\|_{L^4}^4
\le
C_0
e_\alpha^2.
}
$$

Define the quartic overlap/coherence factor:

$$
\boxed{
\Omega_4
=
\frac{
\left\|
\sum_\alpha
V_\alpha
\right\|_{L^4}^4
}{
\sum_\alpha
\|V_\alpha\|_{L^4}^4
}
}
$$

whenever the denominator is nonzero.

---

# 29. CIV/VIII-2.6 — Increment-Coherence Concentration Compiler

## Theorem 29.1

Under Section 28:

$$
\boxed{
\|V^\sharp\|_{L^4}^4
\le
C_0
\Omega_4
\sum_\alpha
e_\alpha^2
\le
C_0
\Omega_4
p_{\max}.
}
$$

Therefore, if:

$$
\boxed{
\|V^\sharp\|_{L^4}^4
\ge
\sigma>0,
}
$$

then:

$$
\boxed{
p_{\max}
\ge
\frac{
\sigma
}{
C_0\Omega_4
}.
}
$$

Equivalently, if:

$$
p_{\max}\to0
$$

while the critical increment defect remains bounded below, then:

$$
\boxed{
\Omega_4\to\infty.
}
$$

### Meaning

A diffuse commutator carrier can persist only through increasing cross-carrier increment coherence/overlap if the component superlinear bound holds.

$\square$

---

# 30. Bounded-overlap commutator branch

If:

$$
\boxed{
\Omega_4
\le
\Omega_\ast<\infty,
}
$$

then persistent critical increment defect forces:

$$
\boxed{
p_{\max}
\ge
c
\sigma.
}
$$

Thus bounded-overlap increment microstructure reconcentrates.

A genuinely diffuse commutator recurrence must violate bounded-overlap/quasiorthogonality.

---

# 31. Relation to generalized Young profiles

The external Young-profile theorem supplies a compactness object for the persistent critical increment defect.

It also explicitly warns that vanishing commutator stress/covariance does not force a Dirac Young profile because the covariance map is not injective.

Therefore:

$$
\boxed{
\Omega_4\to\infty
}
$$

should be interpreted as a candidate **coherent microstructure recurrence**, not as an already excluded pathology.

A full carrier decomposition satisfying Section 28 is not presently supplied by the external theorem.

---

# 32. Commutator recurrence alternative

The COM branch now has the conditional form:

$$
\boxed{
\text{ATOMIC INCREMENT CARRIER}
}
$$

or:

$$
\boxed{
\text{UNBOUNDED INCREMENT COHERENCE}
}
$$

or:

$$
\boxed{
\text{CARRIER-DECOMPOSITION FAILURE}.
}
$$

This is a rigidity reduction, not a closure theorem.

---

# 33. Combined migrated-supply graph

Collect the migrated channels after DCRP-01:

$$
\boxed{
\mathfrak Q^{rest}
=
\mathfrak Q^{CROSS}
+
\mathfrak Q^{FAR}
+
\mathfrak Q^{COM}
+
\mathfrak Q^{WORK}
+
\mathfrak Q^{RES}.
}
$$

Suppose:

$$
\boxed{
|\mathfrak Q^{rest}|
\ge
\sigma.
}
$$

Then at least one channel carries a fixed fraction:

$$
\boxed{
\ge
\sigma/5
}
$$

in absolute value.

DCRP-02 provides a separate rigidity compiler for the first three channels.

---

# 34. CIV/VIII-2.7 — Diffuse Interaction-Graph Routing Theorem

## Theorem 34.1

Assume a minimal diffuse carrier has:

$$
p_{\max}\to0
$$

while:

$$
|\mathfrak Q^{rest}|
\ge
\sigma>0.
$$

Then, after subsequence extraction, at least one of the following occurs.

### CROSS route

Interaction degree becomes unbounded.

If normalized spatial span is bounded, shell span becomes unbounded and re-enters the DRC dissipation-span/driver census.

If shell span is bounded, normalized spatial span becomes unbounded and enters FAR.

### FAR route

Either annular/core reservoir amplitude escapes or a fixed fraction of far-field supply recurs in a comparable-annulus band.

### COM route

Either an atomic increment carrier is recovered, or the carrier-resolved quartic overlap/coherence diverges, or the required carrier decomposition fails.

### WORK/RES route

The dangerous supply remains in the combined pressure/flux/work or explicit residual channels already isolated by FCBP/MORP.

### Safety

The theorem is a routing theorem.

It does not exclude the WORK/RES or coherent-COM branches.

$\square$

---

# 35. What happened to unrestricted cross-scale migration?

It is no longer a primitive escape description.

For local trilinear supply, fixed output plus atom collapse forces interaction-degree growth.

Bounded geometric locality turns degree growth into shell or spatial span.

Thus CROSS has been converted into:

$$
\boxed{
\text{DRC shell-span}
\vee
\text{FAR}
\vee
\text{ATOM}.
}
$$

This is the main DCRP-02 reduction.

---

# 36. Interaction multiplicity is not yet a tax

The sharpness model from Theorem 6.1 shows that:

$$
\mathfrak D_{\rm part}
\sim
p_{\max}^{-1/2}
$$

can sustain order-one graph supply.

Therefore DCRP cannot claim:

$$
\boxed{
\mathfrak D_{\rm part}\to\infty
\Longrightarrow
\text{contradiction}.
}
$$

A true obstruction requires:

- bounded geometric degree;
- a shell-span/dissipation debt;
- far-field packing;
- coherence cost;
- or another strict interaction tax.

---

# 37. Interaction entropy

Normalize the nonzero edge supply:

$$
\boxed{
r_{ij}
=
\frac{
|q_{ij}|
}{
\mathfrak Q_{\mathcal G}
}.
}
$$

Define edge effective multiplicity:

$$
\boxed{
\mathfrak M_{\rm edge}
=
\left(
\sum_{(i,j)}
r_{ij}^2
\right)^{-1}.
}
$$

If the largest edge share tends to zero, then:

$$
\mathfrak M_{\rm edge}\to\infty.
$$

This is another diagnostic of diffuse interaction support.

It is not a dynamical tax by itself.

---

# 38. External scale-locality calibration

Sharp-spectral and physical-space locality results show, under their respective inertial-range hypotheses, that aggregate cascade flux is supported mainly by local scale interactions.

These results support the use of bounded-locality interaction graphs.

They are not imported as a theorem for arbitrary hypothetical singular branches.

---

# 39. External finite-scale supply-tax calibration

Recent critical-ledger work proves that persistent badness along a finite admissible chain must be paid by untaxed critical supply or leakage.

The supply is explicitly separated into nonlinear flux, pressure transport, interpolation amplification, and pressure regeneration.

This provides a finite-scale external accounting container for the migrated WORK/RES branch.

It does not provide the diffuse-carrier concentration recovery proved here for the CROSS channel.

---

# 40. DCRP residual after Paper 02

Avoid introducing a new physical taxonomy.

The remaining theorem obligations are compressed to:

$$
\boxed{
\textbf{IG-PACK}
}
$$

— convert large interaction degree/comparable-annulus recurrence into a strict packing or dissipation tax;

$$
\boxed{
\textbf{COM-COH}
}
$$

— classify or exclude coherent critical increment Young microstructure;

$$
\boxed{
\textbf{WORK-RIG}
}
$$

— combine diffuse-carrier recurrence with the PFET/paid-side/invisible-work rigidity developed in FCBP/MORP.

---

# 41. Next paper

The next paper should attack the two surviving nonlinear structures rather than repeat source routing:

$$
\boxed{
\textbf{
NS-DCRP 03 —
Interaction Packing Tax、
Comparable-Annulus Recurrence、
Increment Young-Microstructure Rigidity
與 Diffuse Carrier Closure Attempt
}.
}
$$

Primary tasks:

1. derive a quantitative tax from repeated interaction-degree growth;
2. combine shell-degree growth with dissipation-wavenumber residence;
3. seek an unweighted/comparable-annulus packing theorem on the minimal zero-tax class;
4. formulate carrier entropy directly on increment Young profiles;
5. determine whether recurrent quartic coherence can remain PFET/model-cone invisible;
6. connect coherent COM recurrence to positive defect work;
7. test whether all migrated nonlinear support can finally be forced into an atomic reprofile or a paid critical tax.

---

# 42. Formal status ledger

$$
\boxed{
\begin{aligned}
\text{Partner-Degree Compensation}
&:\ \mathrm{PROVED},\\
\text{degree exponent sharpness}
&:\ \mathrm{PROVED\ ABSTRACTLY},\\
\text{Comparable-Shell Edge Bound}
&:\ \mathrm{PROVED},\\
\text{Bounded-Locality Concentration Recovery}
&:\ \mathrm{PROVED},\\
\text{bounded-band CROSS primitive status}
&:\ \mathrm{REMOVED/ROUTED},\\
\text{Comparable-Annulus Recovery}
&:\ \mathrm{PROVED\ FROM\ EXTERNAL\ BOUND},\\
\text{Increment-Coherence Compiler}
&:\ \mathrm{PROVED\ CONDITIONAL},\\
\text{full commutator carrier decomposition}
&:\ \mathrm{OPEN},\\
\text{interaction-degree strict tax}
&:\ \mathrm{OPEN},\\
\text{comparable-annulus unweighted packing}
&:\ \mathrm{OPEN},\\
\text{coherent Young-microstructure rigidity}
&:\ \mathrm{OPEN},\\
\text{WORK/RES diffuse rigidity}
&:\ \mathrm{OPEN},\\
\text{minimal diffuse obstruction exclusion}
&:\ \mathrm{OPEN},\\
\text{Forest Coercive Budget}
&:\ \mathrm{OPEN},\\
\text{Finite Forest Obstruction}
&:\ \mathrm{OPEN},\\
CN3_{\rm Atomic}
&:\ \mathrm{OPEN},\\
\text{Navier--Stokes regularity}
&:\ \mathrm{NOT\ PROVED}.
\end{aligned}
}
$$

---

# 43. Conclusion

DCRP-02 follows the nonlinear supply after local same-shell concentration recovery has failed.

The central graph theorem shows that fixed trilinear supply and vanishing parent atoms can coexist only if interaction degree grows at least like:

$$
p_{\max}^{-1/2}.
$$

This rate is sharp at the abstract graph level.

Navier--Stokes geometry then becomes decisive.

Inside a bounded relative shell band and bounded normalized spatial span, wavelength-cell partner degree is uniformly finite.

Hence order-one local cross-shell stretching again forces a fixed-share atom.

A diffuse carrier must leave bounded locality.

If it does so through shell labels, the prior DRC source census routes the multiplicity into dissipation-span/driver debt.

If it does so spatially, it enters the far-field branch.

The external annular far-field inequality then shows that, under bounded annular reservoirs, order-one far-field supply cannot hide only in arbitrarily separated annuli; it must recur near a comparable annulus or pay amplitude escape.

The commutator route has a parallel concentration/coherence alternative.

Under a carrier-resolved quartic decomposition, persistent critical increment defect with atom collapse forces the quartic overlap factor to diverge.

Thus a diffuse commutator obstruction must become increasingly coherent at the microstructure level rather than merely increasingly numerous.

The diffuse carrier has therefore been compressed again.

It can no longer survive merely by "changing partners."

It must pay through:

$$
\boxed{
\textbf{
shell-span/dissipation debt,
far-field comparable-annulus recurrence,
coherent critical increment microstructure,
or already-isolated WORK/RES channels.
}
}
$$

The next problem is to turn these recurring structures into a strict packing or coherence tax.

---

# References

1. R. Yu, *Filtered Vortex Stretching and Subgrid Defects for the Three-Dimensional Navier--Stokes Equations*, arXiv:2606.27560.
2. R. Yu, *Critical Ledgers and Scale-Defect Cascades for Navier--Stokes*, arXiv:2606.13887.
3. R. Yu, *Invisible Defect Cascades for Navier--Stokes Regularity*, arXiv:2606.12756.
4. R. Yu, *Finite-Chain CKN-Bad Scale Counting for Navier--Stokes: Standard PDE Closure and Canonical Detector Realization*, arXiv:2606.21783.
5. 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.
6. R. Dascaliuc, Z. Grujić, *Energy cascades and flux locality in physical scales of the 3D Navier--Stokes equations*, arXiv:1101.2193.
7. A. Cheskidov, R. Shvydkoy, *A unified approach to regularity problems for the 3D Navier--Stokes and Euler equations: the use of Kolmogorov's dissipation range*, arXiv:1102.1944.
8. H. Aluie, G. L. Eyink, *Localness of energy cascade in hydrodynamic turbulence, II. Sharp spectral filter*, arXiv:0909.2451.
9. `NS_DCRP_01_CarrierEntropy_ConcentrationRecovery_v0.1.md`.
10. `NS_MORP_CYCLE_VII_HANDOFF_v1.0.md`.