# NS-DCRP-48 — Coherent Pancake Fokker–Planck Reduction, Viscous Batchelor Floor, and the Sheet-Form Shadowing Barrier

- date: 2026-08-17
- status: research proof checkpoint / genuinely viscous conditional subbranch theorem
- canonical source: UTF-8 Markdown
- canonical math delimiters: `$...$` and `$$...$$`
- objective:
  1. return from the critical Euler sheet monodromy of DCRP-47 to a genuinely Navier--Stokes-specific normal-profile model;
  2. derive the time-dependent similarity viscosity coefficient in the strict exponent window;
  3. identify a coherent one-sign fixed-plane pancake-sheet subbranch on which a tangential vorticity profile reduces exactly to a one-dimensional Fokker--Planck equation;
  4. derive the exact normal variance/thickness equation;
  5. prove a one-period same-parent thickness recurrence;
  6. prove the positive viscous Batchelor/Burgers thickness fixed point:

     $$
     h_n^2\asymp\varepsilon_n;
     $$

  7. prove that asymptotically subdiffusive coherent sheet shadowing:

     $$
     h_n^2/\varepsilon_n\to0
     $$

     is impossible without a same-order second-moment residual;
  8. derive a Fisher-information lower bound producing a positive normalized sheet-diffusion action;
  9. classify all failures of the coherent reduction as explicit rank/plane/tangential/sign/source residuals;
  10. identify the next frontier as upgrading this conditional one-normal-profile theorem to a general material-sheet tube theorem.
- no full Navier--Stokes regularity claim is made.
- principal external primary calibration:
  - T. Gallay, Y. Maekawa, *Three-dimensional stability of Burgers vortices*, arXiv:1002.2489;
  - Y. Maekawa, H. Miura, C. Prange, *On stability of blow-up solutions of the Burgers vortex type for the Navier--Stokes equations with a linear strain*, arXiv:1807.10341.
- internal dependencies:
  - DCRP-34 effective Type-II viscosity scaling;
  - DCRP-41 moving pancake-jet normal form;
  - DCRP-47 critical Euler sheet monodromy.
- no novelty/priority claim is made without independent audit.

---

# 1. Executive result

DCRP-47 identified a completely critical Euler equality manifold for the strongest rank-two pure pancake branch.

The pure Euler monodromy contains:

$$
\boxed{
\mu
=
e^{(1-2\gamma)S_0}
>1,
}
\tag{1.1}
$$

with:

- shear scalar multiplier:

  $$
  \mu;
  $$

- vorticity two-form multiplier:

  $$
  \mu^{-1};
  $$

- normal cotangent quotient multiplier:

  $$
  \mu^{-2}.
  $$

No Euler-side exponent mismatch remained.

DCRP-48 asks the genuinely viscous question:

> can a smooth Navier--Stokes vorticity sheet shadow the pure Euler normal contraction indefinitely when its effective viscosity is positive?

On a precise coherent one-sign fixed-plane sheet subbranch, the answer is:

$$
\boxed{
\textbf{no at subdiffusive thickness}.
}
\tag{1.2}
$$

The Navier--Stokes normal profile obeys an exact Fokker--Planck equation whose variance contains an unavoidable positive diffusion term.

The resulting same-parent recurrence is:

$$
\boxed{
h_{n+1}^2
=
\mu^{-4}h_n^2
+
\varepsilon_n
\mathfrak D_{\rm nor},
}
\tag{1.3}
$$

with:

$$
\boxed{
\mathfrak D_{\rm nor}>0.
}
\tag{1.4}
$$

The effective viscosities satisfy:

$$
\boxed{
\varepsilon_{n+1}
=
\mu^{-1}
\varepsilon_n.
}
\tag{1.5}
$$

Therefore the dimensionless sheet thickness:

$$
\boxed{
\delta_n
=
\frac{
h_n^2
}{
\varepsilon_n
}
}
\tag{1.6}
$$

satisfies:

$$
\boxed{
\delta_{n+1}
=
\mu^{-3}\delta_n
+
\mu
\mathfrak D_{\rm nor}.
}
\tag{1.7}
$$

Hence:

$$
\boxed{
\delta_n
\longrightarrow
\delta_\ast
=
\frac{
\mu
\mathfrak D_{\rm nor}
}{
1-\mu^{-3}
}
>0.
}
\tag{1.8}
$$

Thus the coherent viscous sheet is driven to a **Batchelor/Burgers thickness floor**:

$$
\boxed{
h_n
\asymp
\sqrt{\varepsilon_n}.
}
\tag{1.9}
$$

The pure Euler normal law by itself would instead give:

$$
\boxed{
h_{n+1}^{Euler}
=
\mu^{-2}
h_n^{Euler},
}
\tag{1.10}
$$

and hence:

$$
\boxed{
\frac{
(h_{n+1}^{Euler})^2
}{
\varepsilon_{n+1}
}
=
\mu^{-3}
\frac{
(h_n^{Euler})^2
}{
\varepsilon_n
}
\to0.
}
\tag{1.11}
$$

Therefore:

$$
\boxed{
\textbf{
asymptotically subdiffusive coherent sheet shadowing is incompatible with the exact Navier--Stokes normal-profile equation.
}
}
\tag{1.12}
$$

If an actual same-parent sequence nevertheless satisfies:

$$
h_n^2/\varepsilon_n\to0,
$$

then at least one assumption of the coherent sheet reduction must fail.

Equivalently, a second-moment residual of order:

$$
\varepsilon_n
$$

must cancel the positive diffusive thickness term.

This is the first DCRP sheet theorem in which the obstruction is genuinely produced by positive viscosity rather than by inviscid scaling.

---

# 2. Similarity Navier--Stokes viscosity coefficient

Consider the generalized backward similarity scaling:

$$
u(x,t)
=
(-t)^{-(1-\gamma)}
V(y,s),
$$

with:

$$
y
=
(-t)^{-\gamma}x,
\qquad
s
=
-\log(-t).
$$

The time derivative and nonlinear terms scale as:

$$
(-t)^{-(2-\gamma)}.
$$

The Laplacian scales as:

$$
(-t)^{-(1+\gamma)}.
$$

Therefore the similarity Navier--Stokes equation contains the viscous coefficient:

$$
\boxed{
\nu
(-t)^{1-2\gamma}
=
\nu
e^{-(1-2\gamma)s}.
}
\tag{2.1}
$$

Set:

$$
\boxed{
\lambda
=
1-2\gamma
>
0.
}
\tag{2.2}
$$

For the:

$$
n
$$

th same-parent Type-II root, denote the effective viscosity at phase:

$$
s=0
$$

by:

$$
\boxed{
\varepsilon_n.
}
\tag{2.3}
$$

Then during one normalized DSS period:

$$
\boxed{
\varepsilon_n(s)
=
\varepsilon_n
e^{-\lambda s}.
}
\tag{2.4}
$$

At:

$$
s=S_0,
$$

$$
\boxed{
\varepsilon_n(S_0)
=
\mu^{-1}
\varepsilon_n
=
\varepsilon_{n+1},
}
\tag{2.5}
$$

where:

$$
\mu
=
e^{\lambda S_0}.
$$

Status:

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

---

# 3. Coherent one-sign fixed-plane sheet subbranch

DCRP-41 gives the shape-static fixed-plane pancake affine strain:

$$
\boxed{
A_{\rm pan}(s)
=
a(s)
\left(
P_h
-
2e_3\otimes e_3
\right).
}
\tag{3.1}
$$

Its mean is constrained by:

$$
\boxed{
\frac1{S_0}
\int_0^{S_0}
a(s)ds
=
\frac{
2-3\gamma
}{2}.
}
\tag{3.2}
$$

Define the normal similarity-material drift coefficient:

$$
\boxed{
\sigma(s)
=
\gamma-2a(s).
}
\tag{3.3}
$$

Then:

$$
W_3
=
\sigma(s)z
$$

on the exact affine normal subbranch.

The period average is:

$$
\begin{aligned}
\frac1{S_0}
\int_0^{S_0}
\sigma(s)ds
&=
\gamma
-
(2-3\gamma)
\\
&=
4\gamma-2
\\
&=
-2(1-2\gamma)
\\
&=
-2\lambda.
\end{aligned}
$$

Thus:

$$
\boxed{
\int_0^{S_0}
\sigma(s)ds
=
-2\lambda S_0.
}
\tag{3.4}
$$

The corresponding inviscid normal material contraction factor is:

$$
\boxed{
\exp
\left[
\int_0^{S_0}
\sigma(s)ds
\right]
=
e^{-2\lambda S_0}
=
\mu^{-2}.
}
\tag{3.5}
$$

This matches the DCRP-47 normal cotangent quotient factor.

---

# 4. Declared coherent vorticity profile

The DCRP-48 exact reduction assumes a coherent subbranch with:

1. fixed vorticity plane:

   $$
   n=e_3;
   $$

2. zero in-plane covariance-shape action;

3. canonical affine normal drift:

   $$
   W_3=\sigma(s)z;
   $$

4. a one-sign tangential vorticity component:

   $$
   \zeta_n(z,s)\ge0;
   $$

5. no tangential dependence in the declared profile;

6. finite nonzero normal flux mass:

   $$
   0<
   \int_{\mathbb R}
   \zeta_n(z,s)dz
   <
   \infty;
   $$

7. finite second normal moment;

8. no tangential leakage, rank lifting, or non-affine source inside the declared sheet tube.

Any failure of these assumptions is retained as an explicit alternative residual rather than silently ignored.

This is a **conditional coherent-sheet theorem**.

It is not asserted that every rank-two Type-II survivor satisfies this one-dimensional reduction.

---

# 5. Tangential vorticity equation

Take a tangential vorticity direction:

$$
e_1\in e_3^\perp.
$$

On the isotropic planar affine strain:

$$
A_{\rm pan}e_1
=
a(s)e_1.
$$

The similarity Navier--Stokes vorticity equation is:

$$
\boxed{
\partial_s\Omega
+
W\cdot\nabla\Omega
+
\Omega
=
(\Omega\cdot\nabla)V
+
\varepsilon_n(s)
\Delta\Omega.
}
\tag{5.1}
$$

For:

$$
\Omega
=
\zeta_n(z,s)e_1,
$$

this reduces exactly to:

$$
\boxed{
\partial_s\zeta_n
+
\sigma(s)z
\partial_z\zeta_n
=
[a(s)-1]
\zeta_n
+
\varepsilon_n
e^{-\lambda s}
\partial_{zz}\zeta_n.
}
\tag{5.2}
$$

Status:

$$
\boxed{
\textbf{PROVED ON THE DECLARED COHERENT SUBBRANCH}.
}
$$

---

# 6. Vorticity-flux mass

Define:

$$
\boxed{
M_n(s)
=
\int_{\mathbb R}
\zeta_n(z,s)dz.
}
\tag{6.1}
$$

Assuming sufficient decay:

$$
\int
z
\partial_z\zeta_n
=
-M_n.
$$

Integrating (5.2):

$$
\boxed{
M_n'
=
[
\gamma-a(s)-1
]
M_n.
}
\tag{6.2}
$$

Diffusion does not change the total one-dimensional vorticity-flux mass.

---

# 7. Normalized sheet profile

Define the probability density:

$$
\boxed{
f_n(z,s)
=
\frac{
\zeta_n(z,s)
}{
M_n(s)
}.
}
\tag{7.1}
$$

Then:

$$
\boxed{
f_n\ge0,
\qquad
\int_{\mathbb R}
f_n dz
=
1.
}
\tag{7.2}
$$

Using:

$$
2a-\gamma
=
-\sigma,
$$

the normalized profile equation becomes:

$$
\boxed{
\partial_s f_n
+
\partial_z
\left[
\sigma(s)z
f_n
\right]
=
\varepsilon_n
e^{-\lambda s}
\partial_{zz}f_n.
}
\tag{7.3}
$$

Status:

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

This is the exact one-dimensional Fokker--Planck equation of the coherent sheet.

The vorticity stretching reaction has disappeared after flux normalization.

Only:

- affine normal compression;
- molecular diffusion;

remain.

---

# 8. Mean normal position

Define:

$$
\boxed{
\bar z_n(s)
=
\int
z
f_n(z,s)dz.
}
\tag{8.1}
$$

Multiplying (7.3) by:

$$
z
$$

gives:

$$
\boxed{
\bar z_n'
=
\sigma(s)
\bar z_n.
}
\tag{8.2}
$$

Thus the mean follows the deterministic affine material normal flow.

A centered sheet may therefore be arranged by translating the normal origin.

---

# 9. Normal variance

Define the centered second moment:

$$
\boxed{
h_n^2(s)
=
\int
\left(
z-\bar z_n(s)
\right)^2
f_n(z,s)dz.
}
\tag{9.1}
$$

This is the DCRP-48 coherent sheet thickness.

It is a physical width of the normalized one-sign vorticity-flux profile in the declared normal coordinate.

---

# 10. NEW THEOREM — Exact Viscous Thickness ODE

## Theorem 10.1

The coherent sheet variance satisfies:

$$
\boxed{
\frac d{ds}
h_n^2
=
2\sigma(s)h_n^2
+
2\varepsilon_n
e^{-\lambda s}.
}
\tag{10.1}
$$

### Proof

Multiply the Fokker--Planck equation by:

$$
(z-\bar z_n)^2.
$$

The affine drift contributes:

$$
2\sigma h_n^2.
$$

The diffusion term contributes:

$$
2\varepsilon_n e^{-\lambda s}.
$$

The moving-center terms cancel using:

$$
\bar z_n'=\sigma\bar z_n.
$$

$$
\square
$$

Status:

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

This is the principal genuinely viscous identity of DCRP-48.

---

# 11. One-period thickness recurrence

Solve (10.1) by variation of constants:

$$
\boxed{
h_n^2(S_0)
=
e^{
2\int_0^{S_0}
\sigma
}
h_n^2(0)
+
2\varepsilon_n
\int_0^{S_0}
e^{-\lambda\tau}
e^{
2\int_\tau^{S_0}
\sigma(s)ds
}
d\tau.
}
\tag{11.1}
$$

By (3.4):

$$
\boxed{
e^{
2\int_0^{S_0}\sigma
}
=
e^{-4\lambda S_0}
=
\mu^{-4}.
}
\tag{11.2}
$$

Define:

$$
\boxed{
\mathfrak D_{\rm nor}
=
2
\int_0^{S_0}
e^{-\lambda\tau}
e^{
2\int_\tau^{S_0}
\sigma(s)ds
}
d\tau.
}
\tag{11.3}
$$

Then:

$$
\boxed{
\mathfrak D_{\rm nor}>0.
}
\tag{11.4}
$$

Hence:

$$
\boxed{
h_n^2(S_0)
=
\mu^{-4}h_n^2(0)
+
\varepsilon_n
\mathfrak D_{\rm nor}.
}
\tag{11.5}
$$

---

# 12. Same-parent root identification

On a coherent same-parent sheet lineage, identify:

$$
\boxed{
h_n^2
=
h_n^2(0)
}
\tag{12.1}
$$

and:

$$
\boxed{
h_{n+1}^2
=
h_n^2(S_0).
}
\tag{12.2}
$$

Then:

$$
\boxed{
h_{n+1}^2
=
\mu^{-4}h_n^2
+
\varepsilon_n
\mathfrak D_{\rm nor}.
}
\tag{12.3}
$$

This is the exact same-parent coherent-sheet thickness recurrence.

Status:

$$
\boxed{
\textbf{PROVED CONDITIONAL ON COHERENT SHEET-LINEAGE IDENTIFICATION}.
}
$$

---

# 13. Dimensionless viscous thickness

DCRP-34 gives:

$$
\boxed{
\varepsilon_{n+1}
=
\mu^{-1}\varepsilon_n.
}
\tag{13.1}
$$

Define:

$$
\boxed{
\delta_n
=
\frac{
h_n^2
}{
\varepsilon_n
}.
}
\tag{13.2}
$$

Divide (12.3) by:

$$
\varepsilon_{n+1}
=
\mu^{-1}\varepsilon_n.
$$

Then:

$$
\boxed{
\delta_{n+1}
=
\mu^{-3}\delta_n
+
\mu
\mathfrak D_{\rm nor}.
}
\tag{13.3}
$$

Status:

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

---

# 14. NEW THEOREM — Viscous Batchelor/Burgers Thickness Fixed Point

## Theorem 14.1

The recurrence (13.3) has the unique positive fixed point:

$$
\boxed{
\delta_\ast
=
\frac{
\mu
\mathfrak D_{\rm nor}
}{
1-\mu^{-3}
}
>0.
}
\tag{14.1}
$$

For every:

$$
\delta_0\ge0,
$$

$$
\boxed{
\delta_n
=
\mu^{-3n}\delta_0
+
\delta_\ast
\left(
1-\mu^{-3n}
\right).
}
\tag{14.2}
$$

Hence:

$$
\boxed{
\delta_n\to\delta_\ast.
}
\tag{14.3}
$$

Equivalently:

$$
\boxed{
h_n^2
\sim
\delta_\ast
\varepsilon_n.
}
\tag{14.4}
$$

Thus:

$$
\boxed{
h_n
\sim
\sqrt{
\delta_\ast
\varepsilon_n
}.
}
\tag{14.5}
$$

$$
\square
$$

Status:

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

This is the coherent-sheet viscous floor.

---

# 15. Why the floor is Burgers/Batchelor-like

The equation:

$$
(h^2)'
=
2\sigma h^2
+
2\varepsilon
$$

is the normal moment balance of:

- compressive affine strain;
- molecular diffusion.

A steady or recurrent normalized profile therefore has thickness of order:

$$
\boxed{
\sqrt{
\varepsilon/
|\sigma|
}.
}
\tag{15.1}
$$

This is the same strain--diffusion scaling underlying classical viscous vortex structures.

The external Burgers-vortex literature confirms that linear strain plus viscosity can support stable coherent vorticity structures.

DCRP-48's precise recurrence is project-specific to the strict DSS exponent/return architecture.

---

# 16. Pure Euler subdiffusive law

If:

$$
\varepsilon_n=0,
$$

then:

$$
\boxed{
h_{n+1}^2
=
\mu^{-4}h_n^2.
}
\tag{16.1}
$$

Since:

$$
\varepsilon_{n+1}
=
\mu^{-1}\varepsilon_n
$$

on the Navier--Stokes roots, the formal Euler thickness ratio relative to the viscous scale would obey:

$$
\boxed{
\delta_{n+1}^{Euler}
=
\mu^{-3}
\delta_n^{Euler}.
}
\tag{16.2}
$$

Thus:

$$
\boxed{
\delta_n^{Euler}
\to0.
}
\tag{16.3}
$$

This is the subdiffusive Euler normal contraction suggested in DCRP-47.

---

# 17. NEW THEOREM — Subdiffusive Shadowing Barrier

## Theorem 17.1

On the exact coherent Navier--Stokes sheet subbranch:

$$
\boxed{
\liminf_{n\to\infty}
\frac{
h_n^2
}{
\varepsilon_n
}
=
\delta_\ast
>
0.
}
\tag{17.1}
$$

Therefore:

$$
\boxed{
\frac{
h_n
}{
\sqrt{\varepsilon_n}
}
\not\to0.
}
\tag{17.2}
$$

In particular, a coherent Navier--Stokes vorticity sheet cannot shadow the pure Euler normal thickness law in the strong sense:

$$
\boxed{
h_n^2/\varepsilon_n\to0.
}
\tag{17.3}
$$

Status:

$$
\boxed{
\textbf{PROVED ON THE COHERENT ONE-NORMAL-PROFILE SUBBRANCH}.
}
$$

---

# 18. Thickness-residual formulation

For a more general same-parent sequence define the second-moment residual:

$$
\boxed{
\mathcal R_{2,n}
=
h_{n+1}^2
-
\mu^{-4}h_n^2
-
\varepsilon_n
\mathfrak D_{\rm nor}.
}
\tag{18.1}
$$

The exact coherent branch has:

$$
\boxed{
\mathcal R_{2,n}=0.
}
\tag{18.2}
$$

Suppose instead:

$$
\boxed{
h_n^2/\varepsilon_n\to0.
}
\tag{18.3}
$$

Then:

$$
h_{n+1}^2/\varepsilon_n
=
\mu^{-1}
\left[
h_{n+1}^2/\varepsilon_{n+1}
\right]
\to0,
$$

and:

$$
\mu^{-4}h_n^2/\varepsilon_n
\to0.
$$

Therefore:

$$
\boxed{
\frac{
\mathcal R_{2,n}
}{
\varepsilon_n
}
\to
-
\mathfrak D_{\rm nor}.
}
\tag{18.4}
$$

Hence:

$$
\boxed{
\liminf_n
\frac{
|\mathcal R_{2,n}|
}{
\varepsilon_n
}
\ge
\mathfrak D_{\rm nor}
>0.
}
\tag{18.5}
$$

Thus any subdiffusive same-parent shadowing must pay a **viscosity-scale second-moment residual**.

Status:

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

This is the quotient-correct viscous sheet defect.

---

# 19. Fisher-information lower bound

For a probability density:

$$
f\ge0,
\qquad
\int f=1,
$$

with mean:

$$
\bar z
$$

and variance:

$$
h^2,
$$

define the Fisher information:

$$
\boxed{
I(f)
=
\int_{\mathbb R}
\frac{
|\partial_zf|^2
}{
f
}
dz.
}
\tag{19.1}
$$

Integration by parts gives:

$$
\boxed{
1
=
-\int
(z-\bar z)
\partial_zf
dz.
}
\tag{19.2}
$$

By Cauchy--Schwarz:

$$
\boxed{
1
\le
h
I(f)^{1/2}.
}
\tag{19.3}
$$

Therefore:

$$
\boxed{
I(f)
\ge
\frac1{h^2}.
}
\tag{19.4}
$$

Status:

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

---

# 20. Normalized viscous Fisher floor

At the same-parent roots:

$$
\boxed{
\varepsilon_n
I(f_n)
\ge
\frac{
\varepsilon_n
}{
h_n^2
}
=
\delta_n^{-1}.
}
\tag{20.1}
$$

Hence on the coherent viscous fixed-point branch:

$$
\boxed{
\liminf_{n\to\infty}
\varepsilon_n
I(f_n)
\ge
\delta_\ast^{-1}
>0.
}
\tag{20.2}
$$

This gives a positive dimensionless normal-profile diffusion/sharpness signal.

It is a natural candidate second-order sheet observable.

---

# 21. Vorticity form of Fisher information

Since:

$$
f_n
=
\zeta_n/M_n,
$$

$$
\boxed{
I(f_n)
=
\frac1{M_n}
\int
\frac{
|\partial_z\zeta_n|^2
}{
\zeta_n
}
dz
}
\tag{21.1}
$$

on the one-sign branch.

Thus:

$$
\boxed{
\varepsilon_n I(f_n)
}
$$

is a viscosity-weighted normal vorticity-gradient concentration observable.

It is higher order than ordinary vorticity amplitude.

The one-sign assumption is essential for this exact Fisher representation.

---

# 22. Entropy identity

Define:

$$
\boxed{
\mathcal H(f)
=
\int
f\log f\,dz.
}
\tag{22.1}
$$

For the Fokker--Planck equation:

$$
f_s+\partial_z(\sigma zf)
=
\varepsilon_n(s)f_{zz},
$$

one computes:

$$
\boxed{
\frac d{ds}
\mathcal H(f)
=
-\sigma(s)
-
\varepsilon_n(s)
I(f).
}
\tag{22.2}
$$

Thus the affine compression creates profile entropy while diffusion removes it through Fisher dissipation.

This is another exact strain--diffusion ledger.

Status:

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

---

# 23. Interpretation of the entropy ledger

If viscosity were absent:

$$
\mathcal H'
=
-\sigma.
$$

The negative mean:

$$
\langle\sigma\rangle=-2\lambda
$$

would create:

$$
2\lambda S_0
$$

of entropy concentration per period.

Viscosity counters this through:

$$
\int
\varepsilon_n(s)
I(f_n(s))ds.
$$

Thus the viscous floor is not merely a second-moment artifact.

It is the natural balance between:

- sheet-normal compression;
- vorticity-profile diffusion.

---

# 24. Exact Gaussian calibration

For constant:

$$
\sigma<0,
\qquad
\varepsilon>0,
$$

the normalized Fokker--Planck equation admits the stationary Gaussian:

$$
\boxed{
f_\ast(z)
=
\frac1{
\sqrt{2\pi h_\ast^2}
}
\exp
\left(
-\frac{
z^2
}{
2h_\ast^2
}
\right),
}
\tag{24.1}
$$

with:

$$
\boxed{
h_\ast^2
=
-\frac{
\varepsilon
}{
\sigma
}.
}
\tag{24.2}
$$

This saturates the variance balance:

$$
0
=
2\sigma h_\ast^2+2\varepsilon.
$$

This is the simplest exact model of the viscous thickness floor.

---

# 25. Relation to Burgers vortex theory

Classical Burgers vortices are exact stationary Navier--Stokes structures in which linear strain and molecular viscosity balance to create a coherent vorticity core.

Rigorous stability theory shows that such strain--diffusion vortex structures are mathematically legitimate.

Time-dependent linear-strain Burgers-vortex-type blow-up profiles have also been studied rigorously in Navier--Stokes systems with prescribed linear strain.

These sources are used only as calibration.

DCRP-48 does not identify the strict rank-two sheet with a Burgers vortex.

---

# 26. What can break the Fokker--Planck reduction

The exact variance theorem fails if any of the following survives:

### sign change

The selected tangential vorticity component is not one sign, so normalized probability-profile reduction is unavailable.

### tangential leakage

The vorticity profile has significant tangential dependence/transport.

### non-affine normal drift

The normal velocity is not:

$$
\sigma(s)z.
$$

### moving-plane action

The vorticity plane rotates and contributes finite-dimensional frame terms.

### rank lifting

A normal vorticity component appears.

### source/commutator residual

The declared coherent sheet profile exchanges vorticity with neighboring sheets/modes.

These are not failures of the proof.

They are the explicit complementary branches.

---

# 27. Robust residual recurrence

A perturbed profile may satisfy:

$$
\boxed{
h_{n+1}^2
=
\mu^{-4}h_n^2
+
\varepsilon_n
\mathfrak D_{\rm nor}
+
\mathcal R_{2,n}.
}
\tag{27.1}
$$

If:

$$
\boxed{
\frac{
\mathcal R_{2,n}
}{
\varepsilon_n
}
\to0,
}
\tag{27.2}
$$

then the dimensionless thickness still satisfies:

$$
\boxed{
\delta_n
\to
\delta_\ast.
}
\tag{27.3}
$$

Thus the viscous floor is stable under:

$$
o(\varepsilon_n)
$$

second-moment errors.

To force:

$$
\delta_n\to0,
$$

the residual must be order:

$$
\varepsilon_n.
$$

---

# 28. Strong sheet-form shadowing NO-GO

Define **strong coherent sheet-form shadowing** as a same-parent lineage satisfying:

1. the rank-two pure pancake chart persists;

2. the same coherent one-sign tangential vorticity packet can be identified across roots;

3. tangential leakage and rank/plane residuals are:

   $$
   o(\varepsilon_n)
   $$

   at the second-moment level;

4. the normal material-sheet thickness shadows the Euler factor so strongly that:

   $$
   h_n^2/\varepsilon_n\to0.
   $$

Then Sections 17--18 give a contradiction.

Therefore:

$$
\boxed{
\textbf{
strong coherent sheet-form shadowing cannot persist indefinitely.
}
}
\tag{28.1}
$$

Status:

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

---

# 29. Physical meaning

The Euler equality manifold allows:

$$
\boxed{
\text{material normal contraction}
\sim
\mu^{-2}.
}
$$

Navier--Stokes vorticity does not remain perfectly frozen to those material sheets.

Diffusion spreads the normalized vorticity profile by an additive amount:

$$
\boxed{
O(\varepsilon_n)
}
$$

in variance each return.

Since:

$$
\varepsilon_n
$$

decays only as:

$$
\mu^{-n},
$$

while a pure Euler variance would decay as:

$$
\mu^{-4n},
$$

viscosity eventually dominates the thickness budget.

Thus the Navier--Stokes sheet must:

- diffuse across the Euler material sheet;
- exchange vorticity with neighboring sheet labels;
- lose the pure pancake chart;
- or activate an explicit residual.

This is the first genuinely viscous obstruction to the DCRP-47 Euler equality manifold.

---

# 30. Why this does not yet prove full rank-two closure

The one-dimensional Fokker--Planck theorem requires a very coherent sheet.

A general rank-two sheet may:

- fold;
- rotate;
- change sign;
- carry several vorticity directions inside the plane;
- exchange vorticity tangentially;
- split into multiple layers.

Such geometry can evade a single normal variance.

Therefore DCRP-48 closes a specific equality subbranch.

It does not yet eliminate every rank-two viscous sheet.

---

# 31. Relation to exact thin vortex sheets

Recent Euler desingularization results show that very thin smooth vorticity layers can exist as exact Euler flows over nonzero times.

That does not contradict DCRP-48 because the theorem uses positive Navier--Stokes viscosity and a repeated same-parent thickness recurrence.

The distinction is precisely:

$$
\boxed{
\textbf{
Euler thinness}
\neq
\textbf{
viscous recurrent subdiffusive thinness}.
}
}
\tag{31.1}
$$

---

# 32. Candidate generalization to material tubes

The next theorem should replace the one-dimensional normal profile by a genuine material tube.

Let:

$$
\Sigma_n(s)
$$

be a coherent material sheet and:

$$
d_n(y,s)
$$

its signed normal distance.

Define a normalized one-sign vorticity-flux measure across the tube and its second normal moment:

$$
\boxed{
h_n^2(s)
=
\frac{
\int
d_n^2
\,d\mu_{\omega,n}
}{
\int
d\mu_{\omega,n}
}.
}
\tag{32.1}
$$

A successful generalization would prove:

$$
\boxed{
(h_n^2)'
\ge
2\sigma_{\rm eff}h_n^2
+
c\varepsilon_n
-
\mathcal E_{\rm geom},
}
\tag{32.2}
$$

where:

$$
\mathcal E_{\rm geom}
$$

is explicitly controlled by:

- sheet curvature;
- tangential leakage;
- plane rotation;
- rank lifting;
- non-affine strain.

Then the DCRP-48 floor would extend to general sheets unless one of those geometric defects is active.

This is the correct next goal.

---

# 33. Candidate connection to second-order DCRP defects

The Fisher floor:

$$
\varepsilon_n
I(f_n)
\gtrsim
1
$$

contains:

$$
\partial_z\zeta_n.
$$

Thus it is naturally connected to:

- vorticity-gradient concentration;
- second-order viscous action;
- filtered increment defects;
- the DCRP-28/33 higher-order viscous residues.

A future bridge should convert the one-dimensional Fisher action into one of the already declared native DCRP second-order coordinates.

That bridge is not proved in this round.

---

# 34. Corrected final rank-two state

After DCRP-47 the strongest Euler survivor was:

$$
\boxed{
\textbf{
critical codimension-one pure pancake sheet monodromy}.
}
$$

DCRP-48 splits its Navier--Stokes shadow into:

$$
\boxed{
\textbf{
coherent one-sign sheet}
}
$$

or:

$$
\boxed{
\textbf{
geometric/source residual}.
}
$$

The first branch has the viscous floor:

$$
\boxed{
h_n^2/\varepsilon_n
\to
\delta_\ast>0.
}
$$

Thus it cannot remain asymptotically subdiffusive.

The only remaining way to preserve the pure Euler thin-sheet scaling is to activate an order:

$$
\varepsilon_n
$$

second-moment cancellation/source or leave the coherent subbranch.

---

# 35. What DCRP-48 closes

The following conditional branch is closed:

$$
\boxed{
\textbf{
one-sign}
+
\textbf{
one-normal-profile}
+
\textbf{
fixed-plane}
+
\textbf{
canonical affine pancake drift}
+
\textbf{
same-parent coherent lineage}
+
\textbf{
subdiffusive thickness}.
}
}
$$

It cannot persist with positive Navier--Stokes viscosity and:

$$
o(\varepsilon_n)
$$

second-moment residual.

This is a genuine viscosity-based exclusion.

---

# 36. Correct next frontier

The next target is:

$$
\boxed{
\textbf{
General Material-Sheet Tube /
Viscous Thickness Inequality.
}
}
$$

A useful theorem would remove the one-dimensional coherence assumptions and prove a tube-level inequality of the form:

$$
\boxed{
\text{Euler normal contraction}
+
\text{viscosity}
\Longrightarrow
\text{diffusive thickness floor}
\ \vee\
\text{curvature/leakage/rank residual}.
}
$$

The geometric error terms should be compiled into existing DCRP channels.

A second target is:

$$
\boxed{
\textbf{
Fisher Sheet Action}
\Longrightarrow
\textbf{
existing second-order viscous/supplier defect}.
}
}
$$

If both bridges are proved, the rank-two pure sheet equality branch would be much closer to closure.

---

# 37. Source-status audit

Gallay--Maekawa study classical Burgers vortices as exact stationary Navier--Stokes structures formed by a two-dimensional vortical field embedded in an axisymmetric linear strain, and prove three-dimensional stability.

Maekawa--Miura--Prange study Navier--Stokes equations with a time-dependent axisymmetric linear strain and Burgers-vortex-type backward self-similar blow-up profiles, again confirming that linear-strain/vorticity/diffusion balance is a legitimate viscous mechanism.

DCRP-48 does not borrow a thickness theorem from those papers.

Its Fokker--Planck and variance identities are derived directly from the declared coherent strict-DSS pancake subbranch.

---

# 38. End state

The coherent normalized vorticity profile satisfies:

$$
\boxed{
\partial_s f_n
+
\partial_z
[
\sigma(s)zf_n
]
=
\varepsilon_n
e^{-(1-2\gamma)s}
\partial_{zz}f_n.
}
$$

Its normal variance satisfies:

$$
\boxed{
(h_n^2)'
=
2\sigma(s)h_n^2
+
2\varepsilon_n
e^{-(1-2\gamma)s}.
}
$$

The one-period same-parent recurrence is:

$$
\boxed{
h_{n+1}^2
=
\mu^{-4}h_n^2
+
\varepsilon_n
\mathfrak D_{\rm nor},
\qquad
\mathfrak D_{\rm nor}>0.
}
$$

Since:

$$
\boxed{
\varepsilon_{n+1}
=
\mu^{-1}\varepsilon_n,
}
$$

the normalized thickness obeys:

$$
\boxed{
\delta_{n+1}
=
\mu^{-3}\delta_n
+
\mu
\mathfrak D_{\rm nor}.
}
$$

Therefore:

$$
\boxed{
\delta_n
\to
\delta_\ast>0.
}
$$

So:

$$
\boxed{
\textbf{
coherent Navier--Stokes sheets have a viscous}
\ 
h\sim\sqrt{\varepsilon}
\ 
\textbf{floor}.
}
$$

The pure Euler subdiffusive law can be shadowed indefinitely only if an order:

$$
\varepsilon_n
$$

sheet residual cancels diffusion or the coherent sheet geometry breaks.

The next frontier is:

$$
\boxed{
\textbf{
General Material-Sheet Tube /
Viscous Thickness Inequality.
}
}
$$