# NS-DCRP-45 — Logarithmic Sheet Capacity, Super-DSS Interface Escape, and Vanishing Inward Tail Portals

- date: 2026-08-17
- status: research proof checkpoint / plateau-degeneration quantification round
- canonical source: UTF-8 Markdown
- canonical math delimiters: `$...$` and `$$...$$`
- objective:
  1. quantify exactly how far a scalar/shear interface must escape in horizontal scale to hide a fixed contrast behind small planar enstrophy;
  2. prove the logarithmic two-dimensional condenser-capacity inequality directly in the DCRP planar-shear variables;
  3. combine pure-cocycle amplitude amplification with capacity to obtain an enstrophy-growth versus double-exponential scale-escape dichotomy;
  4. define a super-DSS interface-escape exponent;
  5. use the strict critical-tail energy envelope to prove that large-radius inward similarity-material portals have vanishing area and vanishing volume flux;
  6. show that a positive-volume tail-to-core replenishment cannot persist through arbitrarily large radii;
  7. combine the two mechanisms into a sheet-replenishment trichotomy: planar enstrophy concentration, super-DSS interface escape, or vanishing-portal concentration;
  8. record a NO-GO against taxing arbitrary Lagrangian deformation by itself;
  9. identify the next frontier as weighted shear/vorticity transport through vanishing inward portals.
- no full Navier--Stokes regularity claim is made.
- external primary calibration:
  - D. S. Agafontsev, E. A. Kuznetsov, A. A. Mailybaev, *Asymptotic solution for high vorticity regions in incompressible 3D Euler equations*, arXiv:1609.07782;
  - A. Enciso, A. J. Fernández, D. Meyer, *Vortex-sheet desingularization for three-dimensional ideal fluids*, arXiv:2607.19233;
  - H. Huang, *Exact Lagrangian Realization and Robust Strain Sensing in Incompressible Flow*, arXiv:2607.26895.
- internal dependencies:
  - DCRP-30/31 strict critical tail energy envelope;
  - DCRP-43 pure anchored shear Poincare cocycle;
  - DCRP-44 coarea NO-GO, slice-mean gauge, and interface-degeneration classification.
- no novelty/priority claim is made without independent audit.

---

# 1. Executive result

DCRP-44 proved that:

$$
\boxed{
\text{infinite scalar reservoir}
\not\Rightarrow
\text{infinite planar-vorticity cost}.
}
\tag{1.1}
$$

A large scalar plateau can hide its physical gradient by moving the transition interface to a large scale, concentrating it, or making it intermittent.

DCRP-45 quantifies the cheapest such escape.

On one fixed horizontal slice define the circular mean of the gauge-dependent shear potential:

$$
\boxed{
m_q(\rho)
=
\frac1{2\pi}
\int_0^{2\pi}
q(\rho,\theta)\,d\theta.
}
\tag{1.2}
$$

Although:

$$
q\mapsto q-h(z,s)
$$

is a gauge freedom, the difference

$$
\boxed{
m_q(R)-m_q(r)
}
\tag{1.3}
$$

is gauge invariant.

Since:

$$
|\nabla_hq|
=
|\Omega_h|,
$$

one has the exact logarithmic capacity estimate

$$
\boxed{
\int_{
B_R\setminus B_r
}
|\Omega_h|^2dx_h
\ge
\frac{
2\pi
\left|
m_q(R)-m_q(r)
\right|^2
}{
\log(R/r)
}.
}
\tag{1.4}
$$

This is the first central theorem.

Therefore, if the coherent shear contrast obeys

$$
\boxed{
\left|
m_q(R)-m_q(r)
\right|
\ge
\tau>0
}
\tag{1.5}
$$

while the annular planar enstrophy is at most

$$
\boxed{
\varepsilon,
}
\tag{1.6}
$$

then necessarily

$$
\boxed{
\frac Rr
\ge
\exp
\left(
\frac{
2\pi\tau^2
}{
\varepsilon
}
\right).
}
\tag{1.7}
$$

Thus:

$$
\boxed{
\textbf{
a fixed scalar contrast can have small physical interface cost only by logarithmically enormous spatial separation.
}
}
\tag{1.8}
$$

The second central result combines this with the DCRP-43 pure scalar cocycle.

On the pure gauge-completed branch:

$$
\boxed{
\widetilde r(\Phi^ma)
=
\mu_r^m
\widetilde r(a),
\qquad
\mu_r
=
e^{(1-2\gamma)S_0}
>
1.
}
\tag{1.9}
$$

Suppose a coherent same-slice contrast can be tracked through generation

$$
m
$$

so that

$$
\boxed{
\Delta_m
\ge
c_0
\mu_r^m
\Delta_0
}
\tag{1.10}
$$

for some fixed

$$
c_0,\Delta_0>0.
$$

Let the corresponding horizontal interface lie between radii

$$
r_m<R_m.
$$

Then

$$
\boxed{
E_{\omega,m}^{ann}
\ge
\frac{
2\pi c_0^2
\mu_r^{2m}
\Delta_0^2
}{
\log(R_m/r_m)
}.
}
\tag{1.11}
$$

Hence there are only two coherent-interface possibilities.

### bounded planar-enstrophy branch

If

$$
\boxed{
E_{\omega,m}^{ann}
\le
E_\ast
}
\tag{1.12}
$$

uniformly, then

$$
\boxed{
\log(R_m/r_m)
\ge
c
\mu_r^{2m}.
}
\tag{1.13}
$$

Equivalently,

$$
\boxed{
\frac{
R_m
}{
r_m
}
\ge
\exp
\left[
c
e^{
2(1-2\gamma)S_0m
}
\right].
}
\tag{1.14}
$$

The scale ratio must therefore grow **double exponentially in the DSS generation**.

### at-most-geometric interface branch

If

$$
\boxed{
\log(R_m/r_m)
\le
C(1+m),
}
\tag{1.15}
$$

then

$$
\boxed{
E_{\omega,m}^{ann}
\ge
c
\frac{
\mu_r^{2m}
}{
1+m
}.
}
\tag{1.16}
$$

Thus the planar enstrophy grows exponentially in the generation index.

This is the second central theorem:

$$
\boxed{
\textbf{
pure pancake amplification}
\Longrightarrow
\textbf{
planar-enstrophy growth}
\ \vee\
\textbf{
double-exponential interface escape}.
}
}
\tag{1.17}
$$

The statement is conditional on coherent same-slice tracking of the amplified contrast.

Failure of that tracking is itself assigned to:

- slice intermittency;
- angular cancellation;
- plane transition;
- rank lifting;
- or normal-shear residual.

The third central result uses the strict Type-II tail-energy exponent.

Assume the period-integrated critical tail envelope

$$
\boxed{
\int_0^{S_0}
\int_{B_R}
|V(y,s)|^2dyds
\le
C_E
R^\kappa,
\qquad
0<\kappa<1.
}
\tag{1.18}
$$

Recall the similarity material velocity

$$
\boxed{
W
=
\gamma y+V.
}
\tag{1.19}
$$

For every sufficiently large dyadic scale

$$
R,
$$

there exists

$$
\boxed{
\rho\in[R,2R]
}
\tag{1.20}
$$

such that

$$
\boxed{
\int_0^{S_0}
\int_{\partial B_\rho}
|V|^2dSds
\le
C
\rho^{\kappa-1}.
}
\tag{1.21}
$$

Define the inward similarity-material portal

$$
\boxed{
\mathcal I_\rho
=
\left\{
(y,s)\in
\partial B_\rho\times[0,S_0]:
W\cdot n<0
\right\}.
}
\tag{1.22}
$$

On this set:

$$
\gamma\rho+V\cdot n<0,
$$

so

$$
\boxed{
|V|
\ge
\gamma\rho.
}
\tag{1.23}
$$

Hence:

$$
\boxed{
|\mathcal I_\rho|
\le
C
\rho^{\kappa-3}.
}
\tag{1.24}
$$

Here the measure is the surface-time measure:

$$
dSds.
$$

Because:

$$
\kappa<1,
$$

the exponent satisfies

$$
\kappa-3<-2.
$$

Thus the inward portals occupy vanishing surface-time measure.

Define the total inward similarity-material volume flux

$$
\boxed{
\mathcal F_{\rm in}(\rho)
=
\int_0^{S_0}
\int_{\partial B_\rho}
(-W\cdot n)_+
dSds.
}
\tag{1.25}
$$

Then

$$
\boxed{
\mathcal F_{\rm in}(\rho)
\le
C
\rho^{\kappa-2}.
}
\tag{1.26}
$$

Since:

$$
\kappa-2<-1,
$$

$$
\boxed{
\mathcal F_{\rm in}(\rho)\to0
}
\tag{1.27}
$$

along the good-radius sequence.

Thus:

$$
\boxed{
\textbf{
the strict critical tail admits no positive-volume inward similarity-material throughput from infinity.
}
}
\tag{1.28}
$$

Any tail-fed replenishment which reaches the core from arbitrarily large normalized radii must become:

- vanishing-area;
- vanishing-volume-flux;
- singularly high-amplitude;
- or concentrated on increasingly exceptional portals.

This is the third central theorem.

Combining the capacity and portal theorems gives the DCRP-45 normal form:

$$
\boxed{
\textbf{
pure pancake sheet replenishment}
\Longrightarrow
\textbf{
planar enstrophy concentration}
\ \vee\
\textbf{
super-DSS interface escape}
\ \vee\
\textbf{
vanishing inward-portal concentration}
\ \vee\
\textbf{
existing scalar/plane/rank residual}.
}
}
\tag{1.29}
$$

The final branch has now become a **singular sheet-conveyor problem**.

The escaping interface cannot remain a positive-volume smooth supply channel.

It must funnel through asymptotically negligible inward portals.

The fourth result is a methodological NO-GO.

One should not declare a large Lagrangian deformation gradient or complicated sheet folding impossible by itself.

Recent exact realization results show substantial flexibility of the endpoint Lagrangian deformation in smooth incompressible flows; in particular, a 2026 primary source constructs unforced analytic periodic Euler/Navier--Stokes examples with prescribed one-particle terminal derivative in

$$
SL(3,\mathbb R).
$$

This does not realize the DCRP Type-II singular branch.

It does show that:

$$
\boxed{
\textbf{
Lagrangian deformation magnitude alone is not a safe obstruction.
}
}
\tag{1.30}
$$

The DCRP closure must retain the coupling to:

- strict tail energy;
- scalar amplification;
- planar vorticity;
- PFET;
- rank/plane structure;
- and same-parent recurrence.

The new frontier is therefore

$$
\boxed{
\textbf{
Vanishing Inward Portals /
Weighted Shear--Vorticity Throughput Rigidity.
}
}
\tag{1.31}
$$

The next question is:

> can the fixed scalar/material replenishment required by the recurrent pancake core be transported through portals whose unweighted material volume flux tends to zero, without forcing the shear amplitude, planar vorticity, sheet curvature, or non-affine strain to concentrate?

That is now the principal rank-two tail-replenishment problem.

---

# 2. Circular-mean gauge invariant

Fix one horizontal slice

$$
(z,s).
$$

For a scalar representative

$$
q(x_h,z,s),
$$

define

$$
\boxed{
m_q(\rho;z,s)
=
\frac1{2\pi}
\int_0^{2\pi}
q(
\rho,\theta,z,s
)
d\theta.
}
\tag{2.1}
$$

Under:

$$
q\mapsto q-h(z,s),
$$

$$
m_q(\rho)
\mapsto
m_q(\rho)-h(z,s).
$$

Therefore:

$$
\boxed{
m_q(R)-m_q(r)
}
\tag{2.2}
$$

is gauge invariant.

This contrast is preferable to a point value in horizontal dimension two.

---

# 3. Radial-mean derivative

Differentiate:

$$
m_q(\rho)
=
\frac1{2\pi}
\int_0^{2\pi}
q(
\rho,\theta
)
d\theta.
$$

Then:

$$
\boxed{
m_q'(\rho)
=
\frac1{2\pi}
\int_0^{2\pi}
\partial_\rho q(
\rho,\theta
)
d\theta.
}
\tag{3.1}
$$

By Cauchy--Schwarz:

$$
\boxed{
|m_q'(\rho)|^2
\le
\frac1{2\pi}
\int_0^{2\pi}
|\partial_\rho q|^2d\theta.
}
\tag{3.2}
$$

---

# 4. NEW THEOREM — Logarithmic Annular Capacity

## Theorem 4.1

For every:

$$
0<r<R,
$$

$$
\boxed{
\int_{
B_R\setminus B_r
}
|\nabla_hq|^2dx_h
\ge
\frac{
2\pi
|m_q(R)-m_q(r)|^2
}{
\log(R/r)
}.
}
\tag{4.1}
$$

Equivalently:

$$
\boxed{
\int_{
B_R\setminus B_r
}
|\Omega_h|^2dx_h
\ge
\frac{
2\pi
|m_q(R)-m_q(r)|^2
}{
\log(R/r)
}.
}
\tag{4.2}
$$

### Proof

Integrate:

$$
m_q(R)-m_q(r)
=
\int_r^R
m_q'(\rho)d\rho.
$$

Using (3.2) and weighted Cauchy--Schwarz:

$$
\begin{aligned}
|m_q(R)-m_q(r)|
&\le
\left[
\frac1{2\pi}
\int_r^R
\int_0^{2\pi}
|\partial_\rho q|^2
\rho d\theta d\rho
\right]^{1/2}
\\
&\qquad\times
\left[
\int_r^R
\frac{d\rho}{\rho}
\right]^{1/2}.
\end{aligned}
$$

The radial part of the Dirichlet energy is bounded by the full horizontal gradient energy.

Rearrange.

$$
\square
$$

Status:

$$
\boxed{
\textbf{PROVED}.
}
$$

---

# 5. Sharp radial calibration

The logarithmic dependence is sharp.

For the radial harmonic condenser

$$
q(\rho)
=
\Delta
\frac{
\log(\rho/r)
}{
\log(R/r)
},
$$

$$
q(r)=0,
\qquad
q(R)=\Delta,
$$

one has:

$$
\boxed{
\int_{
B_R\setminus B_r
}
|\nabla_hq|^2dx_h
=
\frac{
2\pi\Delta^2
}{
\log(R/r)
}.
}
\tag{5.1}
$$

Thus logarithmic interface cheapness is a genuine two-dimensional capacity effect.

It cannot be removed by improving the elementary estimate.

---

# 6. Quantitative interface escape

If:

$$
|m_q(R)-m_q(r)|
\ge
\tau
$$

and:

$$
\int_{B_R\setminus B_r}
|\Omega_h|^2
\le
\varepsilon,
$$

then:

$$
\boxed{
\log(R/r)
\ge
\frac{
2\pi\tau^2
}{
\varepsilon
}.
}
\tag{6.1}
$$

Hence:

$$
\boxed{
R/r
\ge
\exp
\left(
2\pi\tau^2/\varepsilon
\right).
}
\tag{6.2}
$$

This converts interface escape into an exact scale requirement.

---

# 7. Capacity escape number

Define the dimensionless quantity

$$
\boxed{
\mathfrak C_{r,R}[q]
=
\frac{
\log(R/r)
}{
2\pi
}
\,
\frac{
\int_{B_R\setminus B_r}
|\Omega_h|^2
}{
|m_q(R)-m_q(r)|^2
}.
}
\tag{7.1}
$$

Whenever the denominator is nonzero:

$$
\boxed{
\mathfrak C_{r,R}[q]
\ge1.
}
\tag{7.2}
$$

Equality is achieved by the radial logarithmic condenser.

This is a native gauge-invariant sheet-capacity observable.

---

# 8. Pure cocycle amplification

On the DCRP-43 pure anchored branch:

$$
\boxed{
\widetilde r(\Phi^ma)
=
\mu_r^m
\widetilde r(a),
}
\tag{8.1}
$$

with:

$$
\mu_r>1.
$$

For two material labels in the same pure-cocycle tube:

$$
\boxed{
\widetilde r(\Phi^ma)
-
\widetilde r(\Phi^mb)
=
\mu_r^m
\left[
\widetilde r(a)-\widetilde r(b)
\right].
}
\tag{8.2}
$$

Thus material scalar contrast amplifies by the same multiplier.

---

# 9. Coherent annular-generation hypothesis

To apply the horizontal capacity theorem, the amplified material contrast must remain visible as a same-slice coherent contrast.

Declare the coherent branch as follows.

There exist:

-:

  $$
  c_0>0;
  $$

-:

  $$
  \Delta_0>0;
  $$

- generations:

  $$
  m\to\infty;
  $$

- horizontal slices:

  $$
  (z_m,s_m);
  $$

- radii:

  $$
  0<r_m<R_m;
  $$

such that:

$$
\boxed{
\left|
m_{\widetilde r}(R_m)
-
m_{\widetilde r}(r_m)
\right|
\ge
c_0
\mu_r^m
\Delta_0.
}
\tag{9.1}
$$

Failure of this condition means that the material contrast has been lost through:

- angular cancellation;
- slice separation;
- normal-shear residual;
- plane transition;
- or rank lifting.

Those are retained as explicit alternative channels.

---

# 10. NEW THEOREM — Amplification--Capacity Inequality

## Theorem 10.1

On the coherent annular-generation branch:

$$
\boxed{
E_{\omega,m}^{ann}
\ge
\frac{
2\pi
c_0^2
\mu_r^{2m}
\Delta_0^2
}{
\log(R_m/r_m)
},
}
\tag{10.1}
$$

where:

$$
\boxed{
E_{\omega,m}^{ann}
=
\int_{
B_{R_m}\setminus B_{r_m}
}
|\Omega_h|^2dx_h.
}
\tag{10.2}
$$

Status:

$$
\boxed{
\textbf{PROVED CONDITIONAL ON COHERENT SAME-SLICE TRACKING}.
}
$$

---

# 11. Double-exponential escape theorem

## Theorem 11.1

Assume the coherent branch and:

$$
\boxed{
E_{\omega,m}^{ann}
\le
E_\ast
}
\tag{11.1}
$$

uniformly.

Then:

$$
\boxed{
\log
\frac{
R_m
}{
r_m
}
\ge
c
\mu_r^{2m},
}
\tag{11.2}
$$

and therefore:

$$
\boxed{
\frac{
R_m
}{
r_m
}
\ge
\exp
\left[
c
\exp
\left(
2(1-2\gamma)S_0m
\right)
\right].
}
\tag{11.3}
$$

Status:

$$
\boxed{
\textbf{PROVED CONDITIONAL}.
}
$$

This is **double-exponential interface escape in the return index**.

---

# 12. At-most-geometric interface theorem

## Theorem 12.1

Assume:

$$
\boxed{
\log(R_m/r_m)
\le
C(1+m).
}
\tag{12.1}
$$

Then:

$$
\boxed{
E_{\omega,m}^{ann}
\ge
c
\frac{
e^{2(1-2\gamma)S_0m}
}{
1+m
}.
}
\tag{12.2}
$$

Thus a geometrically scale-local sheet lineage must carry exponentially growing planar enstrophy.

Status:

$$
\boxed{
\textbf{PROVED CONDITIONAL}.
}
$$

---

# 13. Super-DSS escape exponent

Define:

$$
\boxed{
\chi_{\rm esc}
=
\liminf_{
m\to\infty
}
\frac1m
\log
\log
\frac{
R_m
}{
r_m
}.
}
\tag{13.1}
$$

If the coherent planar enstrophy remains uniformly bounded, Theorem 11.1 gives:

$$
\boxed{
\chi_{\rm esc}
\ge
2
\log\mu_r
=
2(1-2\gamma)S_0.
}
\tag{13.2}
$$

Thus bounded-enstrophy escape is quantitatively **super-DSS**.

A bounded-lag geometric-shell ancestry cannot realize it.

---

# 14. Why the coherent hypothesis matters

Material labels evolve in the full three-dimensional similarity flow.

Two labels whose scalar contrast amplifies need not remain:

- on one horizontal slice;
- within one common rank-two chart;
- in one coherent annular radial ordering.

If coherent tracking fails, the capacity theorem cannot be applied to that pair.

But the failure itself is already meaningful:

$$
\boxed{
\text{slice separation}
\ \vee\
\text{angular cancellation}
\ \vee\
\text{plane transition}
\ \vee\
\text{rank lifting}
\ \vee\
\text{normal-shear residual}.
}
\tag{14.1}
$$

Thus the theorem does not silently assume that every material contrast is a radial sheet contrast.

---

# 15. Critical tail-energy envelope

Assume:

$$
\boxed{
\int_0^{S_0}
\int_{B_R}
|V|^2dyds
\le
C_E
R^\kappa,
}
\tag{15.1}
$$

where:

$$
\boxed{
0<\kappa<1.
}
\tag{15.2}
$$

This is the strict critical DSS tail exponent inherited from DCRP-30/31.

---

# 16. Good-radius shell bound

For:

$$
R
$$

large:

$$
\int_R^{2R}
\int_0^{S_0}
\int_{\partial B_\rho}
|V|^2dSdsd\rho
\le
C
R^\kappa.
$$

Therefore there exists:

$$
\boxed{
\rho\in[R,2R]
}
\tag{16.1}
$$

such that:

$$
\boxed{
\int_0^{S_0}
\int_{\partial B_\rho}
|V|^2dSds
\le
C
R^{\kappa-1}.
}
\tag{16.2}
$$

Since:

$$
\rho\simeq R,
$$

this is:

$$
\boxed{
\int_0^{S_0}
\int_{\partial B_\rho}
|V|^2
\le
C
\rho^{\kappa-1}.
}
\tag{16.3}
$$

Status:

$$
\boxed{
\textbf{PROVED}.
}
$$

---

# 17. Inward similarity-material portal

On:

$$
|y|=\rho,
$$

the similarity material radial velocity is:

$$
\boxed{
W\cdot n
=
\gamma\rho
+
V\cdot n.
}
\tag{17.1}
$$

Define:

$$
\boxed{
\mathcal I_\rho
=
\{
W\cdot n<0
\}.
}
\tag{17.2}
$$

On this set:

$$
V\cdot n<-\gamma\rho.
$$

Therefore:

$$
\boxed{
|V|^2
\ge
\gamma^2\rho^2.
}
\tag{17.3}
$$

---

# 18. NEW THEOREM — Vanishing Inward-Portal Measure

## Theorem 18.1

At every good radius:

$$
\boxed{
|\mathcal I_\rho|
\le
\frac{
1
}{
\gamma^2\rho^2
}
\int_0^{S_0}
\int_{\partial B_\rho}
|V|^2dSds.
}
\tag{18.1}
$$

Hence:

$$
\boxed{
|\mathcal I_\rho|
\le
C
\rho^{\kappa-3}.
}
\tag{18.2}
$$

In particular:

$$
\boxed{
|\mathcal I_\rho|
\to0.
}
\tag{18.3}
$$

Status:

$$
\boxed{
\textbf{PROVED}.
}
$$

---

# 19. NEW THEOREM — Vanishing Inward Material Volume Flux

## Theorem 19.1

Define:

$$
\boxed{
\mathcal F_{\rm in}(\rho)
=
\int_0^{S_0}
\int_{\partial B_\rho}
(-W\cdot n)_+
dSds.
}
\tag{19.1}
$$

Then at every good radius:

$$
\boxed{
\mathcal F_{\rm in}(\rho)
\le
C
\rho^{\kappa-2}.
}
\tag{19.2}
$$

Hence:

$$
\boxed{
\mathcal F_{\rm in}(\rho)\to0.
}
\tag{19.3}
$$

### Proof

On:

$$
\mathcal I_\rho,
$$

$$
(-W\cdot n)_+
=
-\gamma\rho-V\cdot n
\le
|V|.
$$

Therefore:

$$
\mathcal F_{\rm in}
\le
\int_{\mathcal I_\rho}
|V|.
$$

By Cauchy--Schwarz:

$$
\mathcal F_{\rm in}
\le
\left(
\int_{\partial B_\rho\times[0,S_0]}
|V|^2
\right)^{1/2}
|\mathcal I_\rho|^{1/2}.
$$

Use Theorem 18.1.

$$
\square
$$

Status:

$$
\boxed{
\textbf{PROVED}.
}
$$

---

# 20. Positive-volume replenishment NO-GO

Suppose a tail-to-core material replenishment mechanism requires a fixed positive amount of ordinary material volume:

$$
\boxed{
\mathcal V_\ast>0
}
\tag{20.1}
$$

to cross inward through every sufficiently large sphere during one DSS period.

Then Theorem 19.1 gives a contradiction.

Therefore:

$$
\boxed{
\textbf{
no strict critical-tail branch can replenish the core from infinity through a positive-volume inward material conveyor.
}
}
\tag{20.2}
$$

Any such replenishment must become singular in volume.

Status:

$$
\boxed{
\textbf{PROVED FOR POSITIVE-VOLUME REPLENISHMENT}.
}
$$

---

# 21. Portal concentration normal form

A tail-fed scalar/shear replenishment surviving Theorem 20.1 must use one or more of:

$$
\boxed{
\text{vanishing portal area}
}
$$

$$
\boxed{
\text{vanishing material volume}
}
$$

$$
\boxed{
\text{growing scalar amplitude}
}
$$

$$
\boxed{
\text{growing vorticity/gradient}
}
$$

$$
\boxed{
\text{increasing sheet multiplicity}
}
$$

or:

$$
\boxed{
\text{rank/plane/normal-shear transition}.
}
$$

Thus the infinite-reservoir escape has been converted into a singular-portal problem.

---

# 22. Coupling capacity escape to portal concentration

The bounded-enstrophy coherent branch requires:

$$
R_m/r_m
$$

to grow double exponentially.

At such large normalized radii, the strict tail provides inward portals with:

$$
|\mathcal I_\rho|
\lesssim
\rho^{\kappa-3}.
$$

Hence the same branch simultaneously requires:

- enormously remote interfaces;
- enormously small inward portal sets.

Thus:

$$
\boxed{
\textbf{
bounded planar enstrophy}
+
\textbf{
pure scalar amplification}
}
$$

forces an extreme tail architecture rather than an ordinary recurrent sheet.

This is the **super-DSS sheet-conveyor normal form**.

---

# 23. What the portal theorem does not control

The portal theorem is unweighted.

It controls:

- portal area;
- ordinary material volume flux.

It does not directly control:

$$
\boxed{
\int
|\widetilde r|^p
(-W\cdot n)_+
}
$$

or:

$$
\boxed{
\int
|\Omega_h|^2
(-W\cdot n)_+.
}
$$

A vanishing-area portal may carry finite weighted throughput if the scalar/vorticity amplitude grows sufficiently fast.

Therefore weighted concentration is the exact remaining loophole.

---

# 24. Weighted portal concentration

Let:

$$
g_\rho
$$

be any nonnegative carrier density on the portal.

If:

$$
\int_{\mathcal I_\rho}
g_\rho
(-W\cdot n)_+
\ge
c_0>0
$$

while:

$$
\mathcal F_{\rm in}(\rho)\to0,
$$

then the flux-weighted average of:

$$
g_\rho
$$

must diverge:

$$
\boxed{
\frac{
\int_{\mathcal I_\rho}
g_\rho
(-W\cdot n)_+
}{
\mathcal F_{\rm in}(\rho)
}
\to\infty.
}
\tag{24.1}
$$

Thus fixed weighted replenishment through vanishing volume flux forces carrier-amplitude concentration.

This elementary observation is the exact next bridge.

---

# 25. Candidate carriers

Natural choices for:

$$
g_\rho
$$

include:

$$
\boxed{
|\widetilde r|^p
}
$$

on a declared gauge-completed sheet branch,

or the fully physical:

$$
\boxed{
|\Omega_h|^2,
}
$$

or a sheet/interface density constructed from:

$$
|\nabla_hq|.
$$

The second and third choices are preferable for final parent-level closure because they are physically gauge free.

---

# 26. Lagrangian-deformation NO-GO

A tempting route is to classify excessive material deformation or sheet folding itself as impossible.

This is unsafe.

Recent exact incompressible-flow realization results demonstrate substantial flexibility of the Lagrangian deformation gradient even in smooth unforced settings.

In particular, one 2026 primary source constructs analytic periodic Beltrami solutions for which a prescribed terminal one-particle deformation matrix in:

$$
SL(3,\mathbb R)
$$

is realized exactly.

This setting is not the DCRP strict Type-II setting.

Nevertheless it gives a clear methodological warning:

$$
\boxed{
\textbf{
large or complicated volume-preserving deformation alone cannot be used as a universal obstruction.
}
}
\tag{26.1}
$$

The DCRP sheet route must couple deformation to the scalar/vorticity and critical-tail budgets.

---

# 27. Exact pancake calibration

Exact Euler pancake solutions already show that shear and anisotropic straining can coexist in highly structured high-vorticity regions.

Their existence prevents us from declaring:

$$
\boxed{
\text{rapid sheet anisotropy}
}
$$

or:

$$
\boxed{
\text{pancake thinning}
}
$$

intrinsically contradictory.

The new capacity/portal theorems are stronger because they use the strict DSS amplitude and tail exponents.

---

# 28. Exact vortex-sheet calibration

Recent three-dimensional Euler desingularization results construct smooth exact vorticities in thin neighborhoods of analytic vortex sheets.

Thus vanishing sheet thickness and concentrated sheet geometry can persist in exact Euler for a nonvanishing time interval.

Therefore:

$$
\boxed{
\textbf{
portal/interface concentration itself is not a local impossibility theorem.
}
}
\tag{28.1}
$$

The remaining task is to test whether such concentration can satisfy the **same-parent DSS replenishment ledger** indefinitely.

---

# 29. Combined DCRP-45 branch tree

The pure rank-two pancake branch now satisfies at least one of:

$$
\boxed{
\text{anchored/mean scalar residual}
}
$$

or:

$$
\boxed{
\text{slice/angular coherence failure}
}
$$

or:

$$
\boxed{
\text{rank/plane lifting}
}
$$

or, on the coherent branch:

$$
\boxed{
\text{planar enstrophy growth}
}
$$

or:

$$
\boxed{
\text{double-exponential interface escape}.
}
$$

If the final branch is fed from infinity, then additionally:

$$
\boxed{
\text{vanishing inward-portal area/volume}
}
$$

is mandatory.

Thus the strongest survivor is:

$$
\boxed{
\textbf{
a super-DSS remote sheet reservoir funneled through asymptotically vanishing inward portals.
}
}
\tag{29.1}
$$

---

# 30. Relation to DCRP-31 inward PFET

DCRP-31 requires a finite-radius inward pressure--kinetic energy matching flux.

DCRP-45 shows that a scalar/material tail supply from normalized infinity cannot enter through a positive-volume conveyor.

Therefore the final strict state has a striking two-stage architecture:

1. a remote sheet reservoir is compressed into vanishing inward material portals;

2. the resulting concentrated carrier must couple to the finite PFET matching region.

The missing theorem is a weighted concentration/transport bridge between these two regions.

---

# 31. A candidate weighted-flux contradiction template

Suppose one can prove that a fixed amount of physical planar-vorticity carrier:

$$
c_\omega>0
$$

must be transported inward from every sufficiently large radius.

Then:

$$
\boxed{
\int_{\mathcal I_\rho}
|\Omega_h|^2
(-W\cdot n)_+
\ge
c_\omega.
}
\tag{31.1}
$$

Since:

$$
\mathcal F_{\rm in}(\rho)
\to0,
$$

the flux-weighted average planar enstrophy must diverge.

If a separate supplier/strain theorem bounds or taxes this divergence, the rank-two tail branch would close.

This is not yet proved.

---

# 32. Super-DSS escape versus bounded-lag ancestry

The same-parent DSS tree is naturally organized by fixed geometric scale ratios.

A double-exponential:

$$
R_m/r_m
$$

cannot remain within any bounded number of ordinary geometric shell steps per return.

Therefore the bounded-enstrophy coherent branch necessarily creates an explicit unbounded scale-lag transition.

This is a native scale-escape coordinate.

Thus one may record:

$$
\boxed{
\textbf{
bounded-lag sheet ancestry}
\Longrightarrow
\textbf{
planar enstrophy amplification}.
}
}
\tag{32.1}
$$

and:

$$
\boxed{
\textbf{
bounded planar enstrophy}
\Longrightarrow
\textbf{
unbounded/super-DSS scale lag}.
}
}
\tag{32.2}
$$

---

# 33. What DCRP-45 closes

The following vague escape is removed:

> the scalar plateau can simply move its boundary farther away with little cost.

The corrected statement is:

$$
\boxed{
\textbf{
yes, but a fixed amplified contrast with bounded enstrophy requires double-exponential scale escape.
}
}
$$

The following tail route is also removed:

> the remote reservoir can continually feed the core through an ordinary finite-area material stream.

False under the strict critical tail:

$$
\boxed{
\mathcal F_{\rm in}(\rho)\to0.
}
$$

Thus only singular weighted portals remain.

---

# 34. Correct next frontier

The next target is:

$$
\boxed{
\textbf{
Vanishing Inward Portals /
Weighted Shear--Vorticity Throughput Rigidity.
}
}
$$

A useful theorem would show that the fixed recurrent pancake/shear demand cannot be supplied through:

$$
\mathcal F_{\rm in}(\rho)\to0
$$

without forcing at least one of:

1.:

   $$
   |\Omega_h|^2
   $$

   concentration strong enough to activate an existing supplier/strain defect;

2. scalar-sheet amplitude concentration strong enough to imply a physical gradient interface via a capacity/coarea estimate;

3. rank-three lifting;

4. non-affine normal-shear residual;

5. pressure/PFET concentration;

6. a known exact sheet/filament mode incompatible with the finite-energy same-parent Navier--Stokes ancestry.

This is now the sharpest rank-two tail-replenishment problem.

---

# 35. End state

The horizontal logarithmic capacity theorem is:

$$
\boxed{
\int_{
B_R\setminus B_r
}
|\Omega_h|^2
\ge
\frac{
2\pi
|\Delta m_q|^2
}{
\log(R/r)
}.
}
$$

For a pure-cocycle contrast amplified by:

$$
\mu_r^m,
$$

$$
\boxed{
E_{\omega,m}^{ann}
\ge
\frac{
c
\mu_r^{2m}
}{
\log(R_m/r_m)
}.
}
$$

Thus bounded planar enstrophy forces:

$$
\boxed{
R_m/r_m
\ge
\exp
\left[
c
e^{
2(1-2\gamma)S_0m
}
\right].
}
$$

The strict tail energy simultaneously implies good radii with:

$$
\boxed{
|\mathcal I_\rho|
\lesssim
\rho^{\kappa-3}
}
$$

and:

$$
\boxed{
\mathcal F_{\rm in}(\rho)
\lesssim
\rho^{\kappa-2}
\to0.
}
$$

Therefore the strongest coherent rank-two survivor is:

$$
\boxed{
\textbf{
a double-exponentially remote sheet/interface reservoir feeding the recurrent core through asymptotically vanishing inward similarity-material portals.
}
}
$$

The next frontier is:

$$
\boxed{
\textbf{
Vanishing Inward Portals /
Weighted Shear--Vorticity Throughput Rigidity.
}
}
$$