# NS-DCRP-38 — Covariance Determinant Rigidity, Affine Replicator Alignment, and Low-Rank Vorticity Phase Collapse

- date: 2026-08-17
- status: research proof checkpoint / phase-rigidity correction
- canonical source: UTF-8 Markdown
- canonical math delimiters: `$...$` and `$$...$$`
- objective:
  1. correct the interpretation that persistent affine-jet/vorticity phase alignment is necessarily non-generic;
  2. derive the exact fixed-core vorticity covariance matrix equation in DSS similarity variables;
  3. identify the normalized covariance equation as a matrix replicator/alignment flow;
  4. prove a determinant identity that is independent of the affine strain jet itself;
  5. show that periodic full-rank covariance requires a quantitatively nonzero non-affine/turnover residual;
  6. prove that the exact zero-residual periodic branch must have rank at most two;
  7. classify rank-one and rank-two covariance collapse as axial/columnar and planar vorticity normal forms;
  8. replace the vague "phase-locking mystery" by a sharper low-rank vorticity rigidity frontier.
- no full Navier--Stokes regularity claim is made.
- external calibration:
  - B. Galanti, J. D. Gibbon, M. Heritage, *Vorticity alignment results for the three-dimensional Euler and Navier--Stokes equations*, arXiv:chao-dyn/9709003;
  - A. Encinas-Bartos, G. Haller, *Vorticity Alignment with Lyapunov Vectors and Rate-of-Strain Eigenvectors*, arXiv:2310.17267.
- internal dependencies:
  - DCRP-35 finite-annulus affine strain supplier;
  - DCRP-36 affine-jet reproduction;
  - DCRP-37 affine-jet/vorticity-covariance phase formulation.
- no novelty/priority claim is made without independent audit.

---

# 1. Executive correction

DCRP-37 formulated the strict affine branch as a phase-coherence problem between:

$$
A(s)\in\mathrm{Sym}_0(3)
$$

and the core vorticity covariance:

$$
B(s)
=
\int
\phi(y)
\Omega(y,s)\otimes\Omega(y,s)dy.
$$

The dangerous stretching is:

$$
A:B.
$$

The informal question was:

> why can a very non-generic relative tensor phase keep lining up?

DCRP-38 corrects the premise.

Vorticity alignment is not necessarily an accidental phase coincidence.

The vorticity equation itself contains a directional alignment dynamics.

For:

$$
\xi
=
\frac{\Omega}{|\Omega|},
$$

the similarity material derivative satisfies:

$$
\boxed{
D_s\xi
=
S\xi
-
(\xi\cdot S\xi)\xi,
}
\tag{1.1}
$$

where:

$$
D_s
=
\partial_s
+
(\gamma y+V)\cdot\nabla.
$$

If the core strain is dominated by the affine supplier:

$$
S=A(s)+E(y,s),
$$

then:

$$
\boxed{
D_s\xi
=
A\xi
-
(\xi\cdot A\xi)\xi
+
\mathcal E_\xi.
}
\tag{1.2}
$$

For fixed symmetric:

$$
A
$$

and:

$$
E=0,
$$

the Rayleigh quotient:

$$
q
=
\xi\cdot A\xi
$$

satisfies:

$$
\boxed{
\frac{dq}{ds}
=
2
\left[
\xi\cdot A^2\xi
-
(\xi\cdot A\xi)^2
\right]
=
2
|
(A-qI)\xi
|^2
\ge0.
}
\tag{1.3}
$$

Thus, away from eigenvalue degeneracy, vorticity direction is dynamically driven toward an eigendirection of the local strain.

If the top eigendirection is present in the initial direction and the top eigenvalue is simple, the corresponding component ratio dominates exponentially.

Therefore:

$$
\boxed{
\textbf{
phase locking may be dynamically generated by vortex stretching itself.
}
}
\tag{1.4}
$$

The correct obstruction is not alignment alone.

It is the compatibility of alignment with **periodic three-dimensional covariance reproduction**.

---

# 2. Similarity vorticity equation

Let:

$$
W
=
\gamma y+V.
$$

The strict DSS vorticity equation is:

$$
\boxed{
\partial_s\Omega
+
W\cdot\nabla\Omega
+
\Omega
=
(\Omega\cdot\nabla)V.
}
\tag{2.1}
$$

Write:

$$
\nabla V
=
S+\mathcal R,
$$

where:

$$
S=S^T
$$

and:

$$
\mathcal R^T=-\mathcal R.
$$

The antisymmetric part has axial vector:

$$
\Omega/2.
$$

Hence:

$$
\boxed{
\mathcal R\Omega=0.
}
\tag{2.2}
$$

Therefore:

$$
\boxed{
(\Omega\cdot\nabla)V
=
S\Omega.
}
\tag{2.3}
$$

The vorticity equation becomes:

$$
\boxed{
\partial_s\Omega
+
W\cdot\nabla\Omega
+
\Omega
=
S\Omega.
}
\tag{2.4}
$$

Status:

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

---

# 3. Pointwise direction equation

Let:

$$
\Omega\neq0
$$

and:

$$
\xi=\Omega/|\Omega|.
$$

Then:

$$
D_s|\Omega|
=
(\xi\cdot S\xi-1)
|\Omega|.
$$

Subtract the magnitude evolution from the vector equation.

One obtains:

$$
\boxed{
D_s\xi
=
S\xi
-
(\xi\cdot S\xi)\xi.
}
\tag{3.1}
$$

Status:

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

This equation is independent of the scalar similarity damping term.

---

# 4. Fixed-affine alignment monotonicity

Assume temporarily:

$$
S=A,
$$

with:

$$
A=A^T
$$

constant along the material trajectory.

Define:

$$
q
=
\xi^TA\xi.
$$

Then:

$$
\begin{aligned}
q'
&=
2
\xi^T
A
\left[
A\xi-q\xi
\right]
\\
&=
2
\left(
\xi^TA^2\xi-q^2
\right).
\end{aligned}
$$

Hence:

$$
\boxed{
q'
=
2
|
(A-qI)\xi
|^2
\ge0.
}
\tag{4.1}
$$

Equality occurs exactly when:

$$
\xi
$$

is an eigenvector of:

$$
A.
$$

Thus strain-eigenvector alignment is an invariant/fixed state of the direction dynamics.

---

# 5. Simple spectral-gap attraction

Let:

$$
a_1>a_2\ge a_3
$$

be the eigenvalues of:

$$
A.
$$

Expand:

$$
\Omega
=
\sum_i
\omega_i e_i.
$$

Under:

$$
D_s\Omega
=
(A-I)\Omega,
$$

$$
\boxed{
\omega_i'
=
(a_i-1)\omega_i.
}
\tag{5.1}
$$

If:

$$
\omega_1\neq0,
$$

then:

$$
\boxed{
\frac{
\omega_i(s)
}{
\omega_1(s)
}
=
\frac{
\omega_i(0)
}{
\omega_1(0)
}
e^{-(a_1-a_i)s},
\qquad
i=2,3.
}
\tag{5.2}
$$

Thus the most extensional eigenvector is exponentially attracting in the frozen-affine model.

Status:

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

This is the local mathematical reason persistent phase alignment need not be statistically miraculous.

---

# 6. Fixed-core covariance tensor

Choose:

$$
\phi
\in
C_c^\infty
$$

with:

$$
0\le\phi\le1.
$$

Define:

$$
\boxed{
B(s)
=
\int
\phi(y)
\Omega(y,s)
\otimes
\Omega(y,s)dy.
}
\tag{6.1}
$$

Then:

$$
B(s)
$$

is symmetric positive semidefinite.

Define:

$$
\boxed{
C_\Omega
=
\Omega\otimes\Omega.
}
\tag{6.2}
$$

From (2.4):

$$
\boxed{
\partial_sC_\Omega
+
W\cdot\nabla C_\Omega
+
2C_\Omega
=
SC_\Omega
+
C_\Omega S.
}
\tag{6.3}
$$

---

# 7. Affine/non-affine strain split

On the support of:

$$
\phi,
$$

write:

$$
\boxed{
S(y,s)
=
A(s)
+
E(y,s),
}
\tag{7.1}
$$

where:

- $$
  A(s)\in\mathrm{Sym}_0(3)
  $$

  is the finite-annulus affine supplier from DCRP-35;

- $$
  E
  $$

  contains the non-affine strain remainder.

The divergence of the similarity material velocity is:

$$
\boxed{
\nabla\cdot W
=
3\gamma.
}
\tag{7.2}
$$

---

# 8. NEW THEOREM — Exact Covariance Matrix Ledger

## Theorem 8.1

The fixed-core covariance satisfies:

$$
\boxed{
B'
=
AB
+
BA
-
(2-3\gamma)B
+
R_B,
}
\tag{8.1}
$$

where:

$$
\boxed{
R_B
=
\int
\phi
\left[
EC_\Omega
+
C_\Omega E
\right]dy
+
\int
(W\cdot\nabla\phi)
C_\Omega dy.
}
\tag{8.2}
$$

### Proof

Integrate (6.3) against:

$$
\phi.
$$

For the transport term:

$$
-\int
\phi
W\cdot\nabla C_\Omega
=
\int
\nabla\cdot(\phi W)
C_\Omega.
$$

Use:

$$
\nabla\cdot(\phi W)
=
W\cdot\nabla\phi
+
3\gamma\phi.
$$

The affine part:

$$
A
$$

is independent of:

$$
y
$$

inside the core, so:

$$
\int
\phi
AC_\Omega
=
AB,
$$

and similarly:

$$
\int
\phi
C_\Omega A
=
BA.
$$

Collect terms.

$$
\square
$$

Status:

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

---

# 9. Meaning of the covariance residual

The residual:

$$
R_B
$$

contains exactly two broad mechanisms.

### non-affine strain

$$
\boxed{
R_B^{na}
=
\int
\phi
\left[
EC_\Omega
+
C_\Omega E
\right].
}
\tag{9.1}
$$

### covariance turnover through the core window

$$
\boxed{
R_B^{tr}
=
\int
(W\cdot\nabla\phi)
C_\Omega.
}
\tag{9.2}
$$

Therefore:

$$
\boxed{
R_B=0
}
$$

is the exact affine/no-turnover covariance equality branch.

It is much more precise than "perfect phase locking."

---

# 10. Trace/enstrophy equation

Let:

$$
\boxed{
m(s)
=
\operatorname{tr}B(s)
=
\int
\phi|\Omega|^2.
}
\tag{10.1}
$$

Taking the trace of (8.1):

$$
\boxed{
m'
=
2A:B
-
(2-3\gamma)m
+
\operatorname{tr}R_B.
}
\tag{10.2}
$$

Thus the affine core stretching competes with:

- positive similarity enstrophy demand;
- covariance turnover/non-affine residual.

---

# 11. Normalized covariance shape

Whenever:

$$
m>0,
$$

define:

$$
\boxed{
P
=
\frac{B}{m}.
}
\tag{11.1}
$$

Then:

$$
P\ge0,
\qquad
\operatorname{tr}P=1.
$$

Define:

$$
\boxed{
\widehat R_B
=
\frac{
R_B
}{
m
}
-
\frac{
\operatorname{tr}R_B
}{
m
}
P.
}
\tag{11.2}
$$

Then:

$$
\boxed{
P'
=
AP
+
PA
-
2(A:P)P
+
\widehat R_B.
}
\tag{11.3}
$$

Status:

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

The scalar similarity damping:

$$
2-3\gamma
$$

has disappeared.

This is the core **matrix replicator equation** for vorticity orientation.

---

# 12. Interpretation as a matrix alignment flow

On the zero-residual branch:

$$
\boxed{
P'
=
AP
+
PA
-
2(A:P)P.
}
\tag{12.1}
$$

This equation:

- preserves positive semidefiniteness;
- preserves:

  $$
  \operatorname{tr}P=1;
  $$

- moves covariance weight toward more extensional directions of:

  $$
  A.
  $$

Thus the covariance phase is not an independent arbitrary variable.

It is dynamically slaved to the strain jet up to:

$$
\widehat R_B.
$$

This is the matrix version of the pointwise direction-alignment equation.

---

# 13. Frozen-affine explicit solution

Assume:

$$
A
$$

is constant and:

$$
R_B=0.
$$

Let:

$$
X(s)
=
e^{As}.
$$

Then:

$$
\boxed{
B(s)
=
e^{-(2-3\gamma)s}
X(s)
B(0)
X(s)^T.
}
\tag{13.1}
$$

Therefore:

$$
\boxed{
P(s)
=
\frac{
X(s)P(0)X(s)^T
}{
\operatorname{tr}
\left[
X(s)P(0)X(s)^T
\right]
}.
}
\tag{13.2}
$$

If the top eigenvalue of:

$$
A
$$

is simple and the initial covariance has nonzero projection on the top eigendirection, the normalized covariance converges to the corresponding rank-one projector.

Thus **low-rank alignment is the natural zero-residual asymptotic**, not a pathology added by hand.

---

# 14. Time-periodic affine cocycle

For time-dependent periodic:

$$
A(s),
$$

let:

$$
\boxed{
X'
=
A(s)X,
\qquad
X(0)=I.
}
\tag{14.1}
$$

Because:

$$
\operatorname{tr}A=0,
$$

$$
\boxed{
\det X(s)=1.
}
\tag{14.2}
$$

On the zero-residual branch:

$$
\boxed{
B(s)
=
e^{-(2-3\gamma)s}
X(s)
B(0)
X(s)^T.
}
\tag{14.3}
$$

Let the one-period monodromy be:

$$
\boxed{
M
=
X(S_0).
}
\tag{14.4}
$$

DSS periodicity of:

$$
B
$$

requires:

$$
\boxed{
M
B(0)
M^T
=
e^{(2-3\gamma)S_0}
B(0).
}
\tag{14.5}
$$

This is a finite-dimensional congruence-eigenmatrix equation.

---

# 15. NEW THEOREM — Full-Rank Periodic Covariance No-Go

## Theorem 15.1

Assume:

$$
B(s)>0
$$

for all:

$$
s,
$$

and:

$$
B(S_0)=B(0).
$$

Then:

$$
\boxed{
\frac d{ds}
\log\det B
=
-3(2-3\gamma)
+
\operatorname{tr}
\left(
B^{-1}R_B
\right).
}
\tag{15.1}
$$

Hence:

$$
\boxed{
\int_0^{S_0}
\operatorname{tr}
\left(
B^{-1}R_B
\right)ds
=
3(2-3\gamma)S_0.
}
\tag{15.2}
$$

### Proof

Differentiate:

$$
\log\det B.
$$

Use:

$$
\frac d{ds}
\log\det B
=
\operatorname{tr}
(
B^{-1}B'
).
$$

Insert (8.1).

By cyclicity:

$$
\operatorname{tr}
(
B^{-1}AB
)
=
\operatorname{tr}A
=
0,
$$

and:

$$
\operatorname{tr}
(
B^{-1}BA
)
=
\operatorname{tr}A
=
0.
$$

Also:

$$
\operatorname{tr}
\left[
B^{-1}
(2-3\gamma)B
\right]
=
3(2-3\gamma).
$$

Integrate one period.

$$
\square
$$

Status:

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

---

# 16. Exact zero-residual contradiction

If:

$$
\boxed{
R_B=0
}
\tag{16.1}
$$

and:

$$
B>0,
$$

then:

$$
\boxed{
\frac d{ds}
\log\det B
=
-3(2-3\gamma).
}
\tag{16.2}
$$

In the strict window:

$$
\gamma<1/2,
$$

so:

$$
2-3\gamma>0.
$$

Therefore:

$$
\det B
$$

strictly decays over one period.

This contradicts:

$$
B(S_0)=B(0).
$$

Hence:

$$
\boxed{
\textbf{
periodic full-rank covariance}
\cap
\{R_B=0\}
=
\varnothing.
}
\tag{16.3}
$$

This is the central rigidity theorem of DCRP-38.

---

# 17. Quantitative residual gap

Assume:

$$
\boxed{
\lambda_{\min}(B(s))
\ge
b_0>0
}
\tag{17.1}
$$

for all:

$$
s.
$$

Then:

$$
\|B^{-1}\|_F
\le
\frac{\sqrt3}{b_0}.
$$

Using (15.2):

$$
\begin{aligned}
3(2-3\gamma)S_0
&\le
\int_0^{S_0}
\left|
\operatorname{tr}
(
B^{-1}R_B
)
\right|ds
\\
&\le
\frac{\sqrt3}{b_0}
\int_0^{S_0}
\|R_B\|_Fds.
\end{aligned}
$$

Therefore:

$$
\boxed{
\int_0^{S_0}
\|R_B\|_Fds
\ge
\sqrt3
(2-3\gamma)
b_0
S_0.
}
\tag{17.2}
$$

Status:

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

Thus a uniformly nondegenerate three-dimensional covariance core must pay a fixed non-affine/turnover residual.

---

# 18. Eigenframe-free nondegeneracy parameter

Define:

$$
\boxed{
\Theta_B
=
\frac{
27\det B
}{
(\operatorname{tr}B)^3
}
\in[0,1].
}
\tag{18.1}
$$

The upper bound follows from arithmetic--geometric mean for the three nonnegative eigenvalues.

Interpretation:

### isotropic/full-rank orientation occupancy

$$
\Theta_B
$$

is bounded away from zero.

### orientation collapse

$$
\Theta_B\to0.
$$

This scalar avoids the eigenframe singularity at repeated eigenvalues.

---

# 19. Quantitative normalized dichotomy

Suppose:

$$
\operatorname{tr}B
\ge
m_0>0
$$

and:

$$
\Theta_B
\ge
\theta_0>0.
$$

If the eigenvalues are:

$$
\lambda_1\ge\lambda_2\ge\lambda_3>0,
$$

then:

$$
\lambda_1\lambda_2
\le
\frac{
(\lambda_1+\lambda_2)^2
}{4}
\le
\frac{
m^2
}{4}.
$$

Since:

$$
\det B
\ge
\frac{
\theta_0
m^3
}{27},
$$

$$
\boxed{
\lambda_{\min}(B)
\ge
\frac{
4\theta_0
}{
27
}
m
\ge
\frac{
4\theta_0
}{
27
}
m_0.
}
\tag{19.1}
$$

Thus Theorem 17.1 gives a uniform covariance-residual gap.

Therefore on a compact normalized class:

$$
\boxed{
\textbf{
full-rank orientation occupancy}
\Longrightarrow
\textbf{
positive covariance residual}.
}
\tag{19.2}
$$

If the residual vanishes, the sequence must enter:

$$
\boxed{
\Theta_B\to0.
}
\tag{19.3}
$$

---

# 20. NEW THEOREM — Exact Zero-Residual Low-Rank Collapse

## Theorem 20.1

Assume:

$$
R_B=0
$$

and:

$$
B(S_0)=B(0).
$$

Then:

$$
\boxed{
\operatorname{rank}B(s)
\le2
}
\tag{20.1}
$$

for every:

$$
s.
$$

### Proof

If:

$$
\operatorname{rank}B=3,
$$

Theorem 16.1 gives a contradiction.

Under:

$$
R_B=0,
$$

the representation:

$$
B(s)
=
e^{-(2-3\gamma)s}
X(s)B(0)X(s)^T
$$

holds with invertible:

$$
X.
$$

Therefore rank is constant in:

$$
s.
$$

$$
\square
$$

Status:

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

---

# 21. Geometric meaning of rank collapse

Let:

$$
\phi>0
$$

on the active core.

If:

$$
n\in\ker B,
$$

then:

$$
\boxed{
0
=
n^TBn
=
\int
\phi
|
n\cdot\Omega
|^2dy.
}
\tag{21.1}
$$

Hence:

$$
\boxed{
n\cdot\Omega(y,s)=0
}
\tag{21.2}
$$

throughout the active core by smoothness.

Therefore:

### rank two

There exists a single spatial direction:

$$
n(s)
$$

such that all core vorticity lies in the common plane:

$$
\boxed{
n(s)^\perp.
}
\tag{21.3}
$$

### rank one

There exists a single axis:

$$
e(s)
$$

such that:

$$
\boxed{
\Omega(y,s)
=
\omega(y,s)e(s)
}
\tag{21.4}
$$

throughout the active core.

Thus exact zero-residual phase locking forces an actual directional dimensional collapse, not merely a statistical alignment bias.

---

# 22. Rank-one columnar consequence

Suppose:

$$
\Omega
=
\omega e(s),
$$

where:

$$
e(s)
$$

is spatially constant on the core.

Because:

$$
\nabla\cdot\Omega=0,
$$

$$
\boxed{
e(s)\cdot\nabla\omega
=
0.
}
\tag{22.1}
$$

Thus the vorticity magnitude is locally invariant along the common vorticity axis.

This is a columnar/axial local geometry.

It is reminiscent of Burgers/Oseen vortex geometry but is **not** identified with a Burgers/Oseen solution.

Status:

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

---

# 23. Rank-two planar consequence

If:

$$
\operatorname{rank}B=2,
$$

there is a unit vector:

$$
n(s)
$$

with:

$$
\boxed{
n(s)\cdot\Omega(y,s)=0
}
\tag{23.1}
$$

throughout the active core.

Thus vorticity is confined to a common two-dimensional orientation plane.

This is a vortex-sheet/quasi-two-dimensional orientation normal form.

No full two-dimensional velocity reduction is claimed.

Status:

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

---

# 24. Periodic monodromy on the low-rank support

Let:

$$
M
=
X(S_0).
$$

The zero-residual periodicity relation is:

$$
\boxed{
MB_0M^T
=
e^{(2-3\gamma)S_0}
B_0.
}
\tag{24.1}
$$

Hence:

$$
\operatorname{Ran}B_0
$$

is invariant under:

$$
M.
$$

The low-rank covariance support is therefore a finite-dimensional return subbundle.

For rank one, the common vorticity axis is a Floquet eigendirection of the affine cocycle.

For rank two, the common vorticity plane is an invariant Floquet plane.

Thus the exact zero-residual phase-locked state is a low-dimensional Floquet alignment mode.

---

# 25. Relationship to phase locking

DCRP-37 asked whether the relative eigenframes can remain aligned.

DCRP-38 shows a more precise statement.

If the covariance remains genuinely three-dimensional, periodic reproduction needs:

$$
\boxed{
R_B\neq0.
}
$$

If:

$$
R_B=0,
$$

the covariance does not preserve a generic three-dimensional phase distribution.

It collapses into a common plane or axis.

Therefore the equality branch is:

$$
\boxed{
\textbf{
low-rank phase locking}
}
$$

rather than generic three-dimensional phase locking.

This is substantially narrower.

---

# 26. Alignment is not automatically a defect

Classical alignment dynamics already show that positive vortex-stretching alignment can be an attracting state under suitable conditions.

Modern Lagrangian results likewise relate vorticity alignment to principal material-stretching directions.

Therefore:

$$
\boxed{
\textbf{
alignment itself must not be taxed by fiat.
}
}
\tag{26.1}
$$

The valid native alternatives are:

- non-affine strain residual;
- covariance turnover;
- low-rank directional collapse.

---

# 27. Revised phase-defect package

The phase-aware package should therefore prioritize:

$$
\boxed{
\mathfrak D_{\rm cov}
=
\left(
R_B,
\Theta_B,
\text{low-rank support return}
\right).
}
\tag{27.1}
$$

The raw eigenframe phase:

$$
O=Q^TR
$$

is useful only away from eigenvalue degeneracy.

The determinant/rank formulation is globally well defined across degeneracies.

---

# 28. Compact-class dichotomy

Let:

$$
\mathscr C_{\rm cov}
$$

be a compact normalized strict-DSS class with:

- periodic:

  $$
  B(s);
  $$

- $$
  \operatorname{tr}B
  \ge
  m_0;
  $$

- fixed exponent gap:

  $$
  \gamma
  \le
  1/2-\eta;
  $$

- uniform smoothness of the core.

Then for every profile either:

$$
\boxed{
\inf_s
\Theta_B(s)
\le
\theta_0
}
\tag{28.1}
$$

for an arbitrarily selected low-rank threshold, or:

$$
\boxed{
\int_0^{S_0}
\|R_B\|_Fds
\ge
c_{\rm cov}
(
m_0,\theta_0,\eta,S_0
)
>0.
}
\tag{28.2}
$$

Thus the full-rank equality sector has a finite covariance-residual gap.

---

# 29. What remains on the low-rank branch

The exact low-rank branch is not yet excluded.

It contains two principal normal forms.

## R2 — planar covariance

$$
\operatorname{rank}B=2.
$$

All core vorticity directions lie in one common plane.

## R1 — axial covariance

$$
\operatorname{rank}B=1.
$$

All core vorticity directions are parallel to one common axis.

The R1 mode is especially compatible with vortex-filament/Burgers-like local geometry.

Thus the next closure cannot simply declare rank collapse impossible.

---

# 30. Relationship to the annular affine supplier

On the low-rank branch, the positive affine work:

$$
A:B
$$

simplifies dramatically.

### rank one

If:

$$
B
=
m
e\otimes e,
$$

then:

$$
\boxed{
A:B
=
m
e\cdot Ae.
}
\tag{30.1}
$$

The entire tensor phase problem becomes one axis/eigenvalue alignment problem.

### rank two

If the covariance support is:

$$
E_2,
$$

then only the restriction:

$$
A|_{E_2}
$$

contributes.

Thus the five-dimensional affine-jet phase problem reduces to a lower-dimensional subbundle problem.

---

# 31. Potential geometric depletion route

The low-rank state has strong vorticity-direction coherence.

This is exactly the type of geometry in which vortex-stretching depletion and directional regularity criteria become relevant.

However DCRP-38 does not import a global regularity theorem from local rank collapse.

The profile remains:

- local;
- tail-fed;
- same-parent DSS;
- pressure/PFET active.

A dedicated local-to-global directional rigidity theorem is still required.

---

# 32. Correct next frontier

The next target is:

$$
\boxed{
\textbf{
Low-Rank Vorticity Covariance /
Planar--Axial DSS Rigidity.
}
}
$$

A useful theorem would prove that a strict same-parent DSS profile with:

$$
R_B=0
$$

and:

$$
\operatorname{rank}B\le2
$$

must satisfy at least one of:

1. a locally two-dimensional / columnar normal form that is incompatible with the required DCRP-31 inward PFET;
2. a vorticity-direction coherence condition strong enough to force nonlinear depletion;
3. a nonzero pressure/non-affine strain residual needed to rotate the low-rank support;
4. a scale/spatial transition defect of the low-rank support;
5. an exact Burgers/Oseen-type filament mode whose unforced same-parent reproduction can be separately audited.

The rank-one branch should be attacked first because it has the strongest geometry.

---

# 33. Source-status audit

## Galanti--Gibbon--Heritage

The primary source formulates vorticity--strain alignment using:

$$
\alpha
=
\hat\xi\cdot S\hat\xi
$$

and:

$$
\chi
=
\hat\xi\times S\hat\xi.
$$

It derives dynamical equations for the alignment variables and identifies, under stated assumptions, an attracting positive-stretching alignment state.

Burgers-vortex and shear-layer solutions appear as Lagrangian fixed-point examples.

This calibrates the DCRP correction that alignment itself is not necessarily a rare phase accident.

## Encinas-Bartos--Haller

The primary source derives asymptotic vorticity-alignment estimates relative to material stretching/Lyapunov directions.

For inviscid flows under the stated assumptions, vorticity alignment is determined by principal material-stretching geometry.

This independently calibrates the view that alignment may be dynamically generated.

---

# 34. End state

The exact core covariance ledger is:

$$
\boxed{
B'
=
AB
+
BA
-
(2-3\gamma)B
+
R_B.
}
$$

The normalized orientation covariance obeys:

$$
\boxed{
P'
=
AP
+
PA
-
2(A:P)P
+
\widehat R_B.
}
$$

Thus the tensor phase is a matrix replicator, not a free random variable.

For full-rank periodic covariance:

$$
\boxed{
\int_0^{S_0}
\operatorname{tr}
(
B^{-1}R_B
)ds
=
3(2-3\gamma)S_0.
}
$$

Hence:

$$
\boxed{
\textbf{
full-rank periodic covariance}
\Longrightarrow
\textbf{
nonzero covariance residual}.
}
$$

If:

$$
R_B=0,
$$

periodicity forces:

$$
\boxed{
\operatorname{rank}B\le2.
}
$$

Therefore the strongest exact phase-locked branch is no longer a generic tensor synchronization state.

It is:

$$
\boxed{
\textbf{
planar or axial vorticity covariance collapse}.
}
$$

The next single frontier is:

$$
\boxed{
\textbf{
Low-Rank Vorticity Covariance /
Planar--Axial DSS Rigidity.
}
}
$$