# DCRP67 / X72-R50 — Aligned Two-Stress Spectral Geometry and the Axisymmetric Orientation–Amplitude Silent Modes

**Series:** Independent Navier–Stokes Research Series / X72 Bridge  
**Date:** 2026-08-18  
**Status:** Proof-development checkpoint / aligned two-stress spectral reduction  
**Immediate predecessor:** `NS_DCRP66_X72R49_CofactorNullRepair_TwoStressCorrelation_2026-08-18.md`

**Primary internal dependencies**
- DCRP-38 — isotropic covariance state
- DCRP-61–66 — aligned/no-turnover X branch and cofactor/vorticity two-stress reduction
- X72 Round36 — self cofactor angular rate and axisymmetric cofactor-shape lock
- X72 Round38–41 — special cofactor transport commutator / Piola–vorticity reduction
- X72 Round42–43 — actual vorticity-stress visibility / realizability

**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.
- Elias Hess-Childs, Matthew Rosenzweig, Sylvia Serfaty, *A sharp commutator estimate for all Riesz modulated energies*, arXiv:2511.13461.

These references calibrate vorticity/strain alignment and current Riesz transport-commutator theory. All branch-specific tensor identities below are derived directly.

No full Navier–Stokes regularity theorem is claimed.

---

# 0. Executive result

DCRP66 reduced the silent X72 correlation branch to the exact two-stress balance

$$
\boxed{
\mathfrak Q_{C\omega}
=
4\mathfrak Q_{CC},
}
$$

where:

- $C=C_S^0$ is the strain cofactor;
- $W_\Omega$ is the actual vorticity stress;
- $q$ is their exact amplitude difference.

DCRP67 resolves the **pointwise aligned spectral geometry** behind this balance.

Assume the exceptional aligned/no-turnover branch:

$$
\boxed{
S\Omega=\lambda(s)\Omega,
}
$$

with $\lambda$ spatially uniform on the covariance observer.

On the active set let

$$
\boxed{
\xi=\frac{\Omega}{|\Omega|},
\qquad
m=|\Omega|^2.
}
$$

Choose a transverse orthonormal frame

$$
e_+,\ e_-\in\xi^\perp.
$$

Define

$$
\boxed{
U_\xi
=
\xi\otimes\xi-\frac13I,
}
$$

and the transverse shape tensor

$$
\boxed{
H
=
e_+\otimes e_+
-
e_-\otimes e_-.
}
$$

Then:

$$
\boxed{
U_\xi:H=0,
\qquad
|U_\xi|^2=\frac23,
\qquad
|H|^2=2.
}
$$

Every aligned trace-free strain has the unique spectral form

$$
\boxed{
S
=
\frac{3\lambda}{2}U_\xi
+
dH,
}
$$

where the single scalar

$$
\boxed{
d
=
\frac{\mu_+-\mu_-}{2}
}
$$

measures transverse strain anisotropy.

The transverse eigenvalues are

$$
\boxed{
-\frac{\lambda}{2}+d,
\qquad
-\frac{\lambda}{2}-d.
}
$$

Thus, after alignment, **all local strain-shape freedom has collapsed to one scalar $d$ plus the transverse frame angle**.

The two native stresses become

$$
\boxed{
W_\Omega
=
mU_\xi,
}
$$

and

$$
\boxed{
C
=
\left(
\frac{3\lambda^2}{4}-d^2
\right)U_\xi
-
\lambda dH.
}
$$

Hence:

## Pointwise coaxiality theorem

$$
\boxed{
[C,W_\Omega]=0.
}
$$

The strain cofactor and actual vorticity stress always share the vorticity eigenvector and are simultaneously diagonalizable pointwise.

The only extra cofactor degree of freedom is the transverse biaxial piece

$$
\boxed{
-\lambda dH.
}
$$

The exact cofactor/vorticity-stress Frobenius angle is

$$
\boxed{
\chi_{C\omega}
=
\frac{C:W_\Omega}{|C||W_\Omega|}
=
\frac{
3\lambda^2-4d^2
}{
3\lambda^2+4d^2
}.
}
$$

Thus the entire relative two-stress orientation is controlled by the single ratio $d/\lambda$.

Even more importantly, X72 Round36's self-induced cofactor angular rate becomes the exact one-variable formula

$$
\boxed{
\Omega_{C,\rm self}
=
\frac{
2\sqrt3\,
|d|\,
|4d^2-9\lambda^2|
}{
4d^2+3\lambda^2
}.
}
$$

Therefore:

$$
\boxed{
\Omega_{C,\rm self}=0
}
$$

if and only if

$$
\boxed{
d=0
}
$$

or

$$
\boxed{
d=\pm\frac{3\lambda}{2}.
}
$$

So the entire aligned branch has only **two axisymmetric self-lock spectral types**.

---

## Type A — Axial-aligned self-lock

$$
\boxed{
d=0.
}
$$

Then

$$
\boxed{
\operatorname{spec}S
=
\left(
\lambda,-\frac{\lambda}{2},-\frac{\lambda}{2}
\right),
}
$$

with the simple strain axis equal to the vorticity direction $\xi$.

The tensors are

$$
\boxed{
S
=
\frac{3\lambda}{2}U_\xi,
}
$$

$$
\boxed{
C
=
\frac{3\lambda^2}{4}U_\xi,
}
$$

$$
\boxed{
W_\Omega
=
mU_\xi.
}
$$

Therefore:

$$
\boxed{
\chi_{C\omega}=1.
}
$$

The cofactor and vorticity stress are positively collinear in the five-dimensional trace-free tensor space.

The pressure source is

$$
\boxed{
q
=
\frac32\lambda^2-\frac12m.
}
$$

Because $\lambda=\lambda(s)$ is spatially uniform,

$$
\boxed{
|C|
=
\frac{\sqrt6}{4}\lambda^2
}
$$

is spatially constant.

Hence

$$
\boxed{
\delta|C|=0,
}
$$

and

$$
\boxed{
\delta q
=
-\frac12\delta m
=
-\sqrt{\frac38}\,
\delta|W_\Omega|.
}
$$

---

## Type B — Equatorial-aligned self-lock

$$
\boxed{
d=\pm\frac{3\lambda}{2}.
}
$$

Then

$$
\boxed{
\operatorname{spec}S
=
(\lambda,\lambda,-2\lambda)
}
$$

up to permutation.

The vorticity lies in the repeated $\lambda$ eigenspace.

Let $\zeta$ be the simple $-2\lambda$ strain axis, so

$$
\boxed{
\zeta\perp\xi.
}
$$

Then

$$
\boxed{
S
=
-3\lambda U_\zeta,
}
$$

$$
\boxed{
C
=
3\lambda^2U_\zeta,
}
$$

while

$$
\boxed{
W_\Omega
=
mU_\xi.
}
$$

Thus

$$
\boxed{
\chi_{C\omega}
=
-\frac12.
}
$$

The two native stress axes are orthogonal in physical space and have the fixed Frobenius angle $120^\circ$ in trace-free tensor space.

The pressure source is

$$
\boxed{
q
=
6\lambda^2-\frac12m.
}
$$

Again:

$$
\boxed{
|C|=\sqrt6\,\lambda^2
}
$$

is spatially constant, so

$$
\boxed{
\delta|C|=0,
}
$$

and the source variation is once again purely vorticity-amplitude variation:

$$
\boxed{
\delta q
=
-\frac12\delta m
=
-\sqrt{\frac38}\delta|W_\Omega|.
}
$$

---

## Consequence for the D66 four-to-one silent balance

On either self-lock type,

$$
\boxed{
\delta|C|=0.
}
$$

Therefore

$$
\boxed{
\mathfrak Q_{CC}=0.
}
$$

The D66 silence condition

$$
\mathfrak Q_{C\omega}=4\mathfrak Q_{CC}
$$

collapses to

$$
\boxed{
\mathfrak Q_{C\omega}=0.
}
$$

Thus every axisymmetric self-lock equality candidate is reduced to a **pure orientation–amplitude cancellation**:

$$
\boxed{
\operatorname{p.v.}
\int
[
\delta V\cdot\nabla K_0
]
:
\delta C
\,
\delta m
=
0.
}
$$

The cofactor increment is purely angular/orientational, while the scalar source increment is purely vorticity-amplitude.

D65 already proves the source is not spatially constant, so:

$$
\boxed{
\delta m\not\equiv0.
}
$$

D66 proves the cofactor is not spatially constant, so in the self-lock modes:

$$
\boxed{
\delta C\not\equiv0
}
$$

must be produced entirely by axis rotation.

The silent branch has therefore become an exact **orientation–amplitude orthogonality problem**.

---

## Isotropic covariance forces an independent vorticity-stress increment budget

The finite compensation state has

$$
\boxed{
B
=
\int\phi\,
\Omega\otimes\Omega
=
\rho I.
}
$$

Let

$$
\boxed{
Z
=
\int\phi m
=
3\rho.
}
$$

Then

$$
\boxed{
\int\phi W_\Omega=0.
}
$$

Let

$$
\boxed{
\Phi=\int\phi.
}
$$

The exact weighted pair-variance identity gives

$$
\boxed{
\iint
\phi(x)\phi(y)
|\delta W_\Omega|^2
\,dxdy
=
2\Phi
\int\phi|W_\Omega|^2dy.
}
$$

Since

$$
|W_\Omega|^2=\frac23m^2,
$$

$$
\boxed{
\iint
\phi(x)\phi(y)
|\delta W_\Omega|^2
\,dxdy
=
\frac{4\Phi}{3}
M_4,
}
$$

where

$$
\boxed{
M_4
=
\int\phi m^2.
}
$$

So the actual vorticity stress has a forced nonzero pair-increment budget independently of D64's pressure-defect increment theorem.

Moreover:

$$
\boxed{
\begin{aligned}
|W_\Omega(x)-W_\Omega(y)|^2
={}&
\frac23
[m(x)-m(y)]^2
\\
&+
2m(x)m(y)
\left[
1-(\xi(x)\cdot\xi(y))^2
\right].
\end{aligned}
}
$$

This gives an exact amplitude/orientation split.

Thus the self-lock branch must simultaneously sustain:

1. nonzero vorticity-amplitude variation, because $q$ is nonflat;
2. nonzero orientation variation, because isotropic covariance cannot be rank one/two under a fixed stress axis;
3. exact zero transport correlation between those two active geometries.

This is much narrower than the D66 generic 4:1 balance.

---

# 1. Aligned strain spectral parameterization

On the active set define

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

Because

$$
S\xi=\lambda\xi,
$$

choose an orthonormal basis

$$
(\xi,e_+,e_-)
$$

diagonalizing $S$.

Trace-free strain gives transverse eigenvalues

$$
\boxed{
\mu_\pm
=
-\frac{\lambda}{2}
\pm d.
}
\tag{1.1}
$$

Define

$$
U_\xi
=
\xi\otimes\xi-\frac13I,
$$

and

$$
H
=
e_+\otimes e_+
-
e_-\otimes e_-.
$$

Then:

## Theorem D67.1 — One-Scalar Aligned Strain Shape

$$
\boxed{
S
=
\frac{3\lambda}{2}U_\xi
+
dH.
}
\tag{1.2}
$$

The scalar $d$ is the entire transverse eigenvalue-shape coordinate.

---

# 2. Basic orthogonal tensor geometry

Directly:

$$
\boxed{
U_\xi:H=0,
}
$$

$$
\boxed{
|U_\xi|^2=\frac23,
}
$$

$$
\boxed{
|H|^2=2.
}
\tag{2.1}
$$

Also:

$$
\boxed{
|S|^2
=
\frac32\lambda^2+2d^2.
}
\tag{2.2}
$$

The determinant is

$$
\boxed{
\det S
=
\lambda
\left(
\frac{\lambda^2}{4}-d^2
\right).
}
\tag{2.3}
$$

---

# 3. Cofactor decomposition

Define

$$
C
=
S^2-\frac13|S|^2I.
$$

A direct diagonal calculation gives:

## Theorem D67.2 — Aligned Cofactor Normal Form

$$
\boxed{
C
=
aU_\xi
-
\lambda dH,
}
\tag{3.1}
$$

where

$$
\boxed{
a
=
\frac{3\lambda^2}{4}-d^2.
}
\tag{3.2}
$$

Thus:

- $aU_\xi$ is the axisymmetric component about vorticity;
- $-\lambda dH$ is the purely transverse biaxial component.

---

# 4. Actual vorticity stress

The actual vorticity stress is

$$
\boxed{
W_\Omega
=
mU_\xi,
}
\tag{4.1}
$$

where

$$
m=|\Omega|^2.
$$

Therefore:

## Theorem D67.3 — Pointwise Cofactor/Vorticity Coaxiality

$$
\boxed{
[C,W_\Omega]=0.
}
\tag{4.2}
$$

The two stresses are simultaneously diagonalizable at every point.

This does not imply spatially constant relative orientation, because the common eigenframe may vary.

---

# 5. Exact two-stress angle

The cofactor action along $\xi$ is

$$
\boxed{
\xi^\top C\xi
=
\frac12\lambda^2-\frac23d^2.
}
\tag{5.1}
$$

Since

$$
C:W_\Omega
=
m\,\xi^\top C\xi,
$$

and

$$
|C|
=
\frac{|S|^2}{\sqrt6},
$$

$$
|W_\Omega|
=
m\sqrt{\frac23},
$$

we obtain:

## Theorem D67.4 — One-Parameter Two-Stress Frobenius Angle

$$
\boxed{
\chi_{C\omega}
=
\frac{C:W_\Omega}{|C||W_\Omega|}
=
\frac{
3\lambda^2-4d^2
}{
3\lambda^2+4d^2
}.
}
\tag{5.2}
$$

This is the full pointwise relative-angle law.

---

# 6. Source in spectral variables

The pressure source is

$$
q
=
|S|^2-\frac12m.
$$

Hence:

$$
\boxed{
q
=
\frac32\lambda^2
+
2d^2
-
\frac12m.
}
\tag{6.1}
$$

Using

$$
a=\frac34\lambda^2-d^2,
$$

equivalently:

$$
\boxed{
q
=
3\lambda^2
-
2a
-
\frac12m.
}
\tag{6.2}
$$

Because $\lambda$ is spatially uniform:

$$
\boxed{
\delta q
=
-2\delta a
-
\frac12\delta m.
}
\tag{6.3}
$$

This is the signed spectral form of D66's Two-Stress Source Identity.

---

# 7. Recover the D66 amplitude formula

The cofactor norm is

$$
|C|
=
\frac{|S|^2}{\sqrt6}.
$$

Using (2.2):

$$
\boxed{
|C|
=
\frac{
\frac32\lambda^2+2d^2
}{
\sqrt6
}.
}
\tag{7.1}
$$

Therefore:

$$
\boxed{
\delta|C|
=
\frac2{\sqrt6}\,
\delta(d^2)
}
\tag{7.2}
$$

at fixed time.

Also:

$$
|W_\Omega|
=
\sqrt{\frac23}m.
$$

Hence

$$
q
=
\sqrt6|C|
-
\sqrt{\frac38}|W_\Omega|,
$$

recovering D66 exactly.

---

# 8. Exact self cofactor angular rate in aligned variables

X72 Round36 gives

$$
\Omega_{C,\rm self}
=
\sqrt6|S|
\sqrt{
1-
\frac{
54(\det S)^2
}{
|S|^6
}
}.
$$

Using (2.2) and (2.3), one obtains the factorization

$$
\boxed{
1-
\frac{
54(\det S)^2
}{
|S|^6
}
=
\frac{
4d^2
(2d-3\lambda)^2
(2d+3\lambda)^2
}{
(4d^2+3\lambda^2)^3
}.
}
\tag{8.1}
$$

Therefore:

## Theorem D67.5 — Aligned Cofactor Self-Angular Rate

$$
\boxed{
\Omega_{C,\rm self}
=
\frac{
2\sqrt3
|d|
|4d^2-9\lambda^2|
}{
4d^2+3\lambda^2
}.
}
\tag{8.2}
$$

This is exact.

---

# 9. Axisymmetric self-lock classification

Theorem D67.5 vanishes iff

$$
d=0
$$

or

$$
d=\pm\frac{3\lambda}{2}.
$$

Thus:

## Theorem D67.6 — Complete Aligned Self-Lock Spectral Classification

The only aligned strain spectra with zero self-induced cofactor angular motion are:

### Type A

$$
\boxed{
(\lambda,-\lambda/2,-\lambda/2)
}
$$

with vorticity along the simple eigenaxis.

### Type B

$$
\boxed{
(\lambda,\lambda,-2\lambda)
}
$$

up to permutation, with vorticity in the repeated eigenspace.

No other aligned spectrum is self-locking.

---

# 10. Type A tensor geometry

Set

$$
d=0.
$$

Then

$$
S
=
\frac{3\lambda}{2}U_\xi,
$$

$$
C
=
\frac{3\lambda^2}{4}U_\xi,
$$

and

$$
W_\Omega
=
mU_\xi.
$$

Hence:

$$
\boxed{
C
=
\frac{3\lambda^2}{4m}
W_\Omega.
}
\tag{10.1}
$$

Pointwise tensor alignment is exact:

$$
\boxed{
\chi_{C\omega}=1.
}
\tag{10.2}
$$

The source is

$$
\boxed{
q
=
\frac32\lambda^2-\frac12m.
}
\tag{10.3}
$$

---

# 11. Type B tensor geometry

Take, for definiteness,

$$
d=\frac{3\lambda}{2}.
$$

Then the eigenvalues are

$$
(\lambda,\lambda,-2\lambda).
$$

Let $\zeta=e_-$ denote the simple $-2\lambda$ eigenaxis.

Then

$$
\boxed{
\zeta\perp\xi.
}
$$

The strain and cofactor are

$$
\boxed{
S
=
-3\lambda U_\zeta,
}
\tag{11.1}
$$

$$
\boxed{
C
=
3\lambda^2U_\zeta.
}
\tag{11.2}
$$

while

$$
W_\Omega
=
mU_\xi.
$$

Since

$$
U_\zeta:U_\xi
=
(\zeta\cdot\xi)^2-\frac13
=
-\frac13,
$$

and

$$
|U_\zeta|^2=|U_\xi|^2=\frac23,
$$

we get:

$$
\boxed{
\chi_{C\omega}
=
-\frac12.
}
\tag{11.3}
$$

The source is

$$
\boxed{
q
=
6\lambda^2-\frac12m.
}
\tag{11.4}
$$

The $d=-3\lambda/2$ branch is the same after swapping the transverse axes.

---

# 12. Constant cofactor amplitude on both self-lock types

Because $\lambda$ is spatially uniform:

### Type A

$$
\boxed{
|C|
=
\frac{\sqrt6}{4}\lambda^2.
}
\tag{12.1}
$$

### Type B

$$
\boxed{
|C|
=
\sqrt6\,\lambda^2.
}
\tag{12.2}
$$

Thus on either type:

$$
\boxed{
\delta|C|=0.
}
\tag{12.3}
$$

So:

$$
\boxed{
\mathfrak Q_{CC}=0.
}
\tag{12.4}
$$

---

# 13. Silent 4:1 balance collapses

D66's exact silent balance is

$$
\mathfrak Q_{C\omega}
=
4\mathfrak Q_{CC}.
$$

On either self-lock type:

$$
\mathfrak Q_{CC}=0.
$$

Therefore:

## Theorem D67.7 — Self-Lock Silent-Correlation Reduction

$$
\boxed{
\mathfrak Q_{TR}=0
\quad\Longrightarrow\quad
\mathfrak Q_{C\omega}=0
}
\tag{13.1}
$$

inside the self-lock class.

Equivalently:

$$
\boxed{
\operatorname{p.v.}
\int_0^{S_0}
\iint
[
\delta V\cdot\nabla K_0
]
:
\delta C
\,
\delta|W_\Omega|
\,dxdy\,ds
=
0.
}
\tag{13.2}
$$

Since

$$
\delta|W_\Omega|
=
\sqrt{\frac23}\delta m,
$$

this is a pure orientation–amplitude correlation.

---

# 14. Source variation is purely vorticity amplitude on self-lock types

At fixed time, the $\lambda$-dependent term in $q$ is spatially constant.

Therefore:

$$
\boxed{
\delta q
=
-\frac12\delta m.
}
\tag{14.1}
$$

D65 proves:

$$
\delta q\not\equiv0.
$$

Hence:

## Corollary D67.8 — Forced Vorticity-Amplitude Variation

$$
\boxed{
\delta m\not\equiv0
}
\tag{14.2}
$$

on every nonzero self-lock equality branch.

So the amplitude factor in (13.2) is genuinely active.

---

# 15. Cofactor variation is purely orientational

On the self-lock types:

$$
|C|
$$

is spatially constant.

D66 proves:

$$
\delta C\not\equiv0.
$$

Therefore every nonzero cofactor increment is purely a tensor-orientation increment.

### Type A

The varying axis is $\xi$ itself.

### Type B

The varying cofactor symmetry axis is $\zeta\perp\xi$.

Hence:

## Corollary D67.9 — Pure Cofactor-Orientation Variation

The self-lock silent X branch has

$$
\boxed{
\text{cofactor amplitude fixed}
}
$$

but

$$
\boxed{
\text{cofactor axis varying}.
}
$$

The remaining commutator equality is exactly an orientation–amplitude orthogonality.

---

# 16. Isotropic covariance implies zero mean vorticity stress

The finite compensation state satisfies

$$
B
=
\int\phi
\Omega\otimes\Omega
=
\rho I.
$$

Taking traces:

$$
Z
=
\int\phi m
=
3\rho.
$$

Therefore:

$$
\begin{aligned}
\int\phi W_\Omega
&=
B-\frac13ZI
\\
&=
\rho I-\rho I
\\
&=0.
\end{aligned}
$$

Thus:

## Theorem D67.10 — Isotropic-Covariance Vorticity-Stress Centering

$$
\boxed{
\int\phi W_\Omega=0.
}
\tag{16.1}
$$

The actual vorticity stress is automatically centered in the finite compensation observer.

---

# 17. Exact vorticity-stress pair-increment budget

Let

$$
\Phi
=
\int\phi.
$$

For any Hilbert-valued field with weighted mean zero,

$$
\iint
\phi(x)\phi(y)
|F(x)-F(y)|^2
=
2\Phi
\int\phi|F|^2.
$$

Apply this to $W_\Omega$.

Since

$$
|W_\Omega|^2
=
\frac23m^2,
$$

define

$$
M_4
=
\int\phi m^2.
$$

Then:

## Theorem D67.11 — Forced Actual-Stress Increment Budget

$$
\boxed{
\iint
\phi(x)\phi(y)
|\delta W_\Omega|^2
\,dxdy
=
\frac{4\Phi}{3}M_4.
}
\tag{17.1}
$$

This is exact and strictly positive on every nonzero covariance state.

---

# 18. Exact amplitude/orientation split of vorticity-stress increments

Write

$$
W_x=m_xU_{\xi_x},
\qquad
W_y=m_yU_{\xi_y}.
$$

Using

$$
U_{\xi_x}:U_{\xi_y}
=
(\xi_x\cdot\xi_y)^2-\frac13,
$$

one obtains:

## Theorem D67.12 — Actual-Stress Pair Geometry

$$
\boxed{
\begin{aligned}
|W_x-W_y|^2
={}&
\frac23
(m_x-m_y)^2
\\
&+
2m_xm_y
\left[
1-(\xi_x\cdot\xi_y)^2
\right].
\end{aligned}
}
\tag{18.1}
$$

Thus actual-stress variation splits exactly into:

1. amplitude variation;
2. unoriented-axis variation.

---

# 19. Type A: both amplitude and vorticity-axis variation are active

Type A has:

$$
C=c_AU_\xi,
\qquad
c_A=\frac34\lambda^2.
$$

Since $c_A$ is spatially constant,

$$
\boxed{
|C_x-C_y|^2
=
2c_A^2
\left[
1-(\xi_x\cdot\xi_y)^2
\right].
}
\tag{19.1}
$$

D66 says $C$ is nonconstant, so the vorticity axis cannot be spatially fixed.

Together with D67.8:

$$
\boxed{
\delta m\not\equiv0,
\qquad
\delta\xi\not\equiv0.
}
$$

The Type-A silent equality requires exact cancellation between an active orientation field and an active amplitude field.

---

# 20. Type B: cofactor-axis rotation is mandatory

Type B has:

$$
C=c_BU_\zeta,
\qquad
c_B=3\lambda^2,
$$

with

$$
\zeta\perp\xi.
$$

If $\zeta$ were spatially fixed, then every vorticity vector would lie in the fixed plane

$$
\zeta^\perp.
$$

Then the covariance would satisfy

$$
B\zeta=0,
$$

contradicting

$$
B=\rho I,
\qquad
\rho>0.
$$

Therefore:

## Theorem D67.13 — Type-B Cofactor-Axis Rotation Necessity

$$
\boxed{
\zeta
\text{ cannot be spatially constant}.
}
\tag{20.1}
$$

Thus Type B also has a mandatory orientation increment.

Together with D67.8, its silent equality is again orientation–amplitude cancellation.

---

# 21. Generic non-self-lock branch

If

$$
d\notin
\left\{
0,\pm\frac{3\lambda}{2}
\right\},
$$

then Theorem D67.5 gives

$$
\boxed{
\Omega_{C,\rm self}>0.
}
$$

Therefore the cofactor shape has a nonzero self-induced angular velocity in the five-dimensional trace-free tensor space.

This is exactly the X72 Round36 **cofactor-shape-active branch**.

D67 does not claim that this positive self angular speed alone makes the Round38 transport commutator positive.

Pressure/vorticity forcing can in principle phase-lock or cancel tensor rotation.

But it means the generic aligned spectrum has already left the static cofactor-shape equality manifold.

---

# 22. Refined X-correlation branch split

The D66 two-stress frontier now decomposes as:

## Shape-active branch

$$
\boxed{
\mathsf X_{\rm shape}:
\quad
d\notin
\{0,\pm3\lambda/2\}.
}
$$

Then

$$
\Omega_{C,\rm self}>0.
$$

## Axial self-lock branch

$$
\boxed{
\mathsf X_A:
\quad
d=0.
}
$$

Then:

$$
C\parallel W_\Omega,
$$

and silence reduces to vorticity-axis orientation × amplitude correlation.

## Equatorial self-lock branch

$$
\boxed{
\mathsf X_B:
\quad
d=\pm3\lambda/2.
}
$$

Then the cofactor symmetry axis is perpendicular to vorticity, with

$$
\chi_{C\omega}=-1/2,
$$

and silence again reduces to cofactor-axis orientation × vorticity-amplitude correlation.

Thus:

$$
\boxed{
\mathsf X_{\rm 2stress}
\Longrightarrow
\mathsf X_{\rm shape}
\vee
\mathsf X_A
\vee
\mathsf X_B.
}
\tag{22.1}
$$

---

# 23. Why this is useful

Before D67 the silent equality condition was an indefinite principal-value balance

$$
\mathfrak Q_{C\omega}=4\mathfrak Q_{CC}.
$$

D67 identifies its lowest-defect equality geometries.

If the cofactor wants to avoid its own self-generated angular motion, it has only two spectral options.

And on those options:

$$
\boxed{
\mathfrak Q_{CC}=0.
}
$$

So the 4:1 balance disappears.

The only remaining silence is

$$
\boxed{
\mathfrak Q_{C\omega}=0,
}
$$

with:

- fixed cofactor amplitude;
- rotating cofactor axis;
- varying vorticity amplitude.

That is a much more rigid and geometrically interpretable target.

---

# 24. Relationship to the material alignment dynamics

D61 proves:

$$
D_s\xi=0
$$

along aligned material trajectories.

Therefore the axis rotation required by D67.9 / D67.13 is **spatial**, not material rotation of an individual vorticity direction.

The equality state must continually organize different materially frozen vorticity axes across space while maintaining:

$$
B=\rho I.
$$

This is a strong same-parent spatial texture requirement.

It suggests the next attack should use the integrability of

$$
\nabla V=S+R
$$

rather than another generic commutator estimate.

---

# 25. Natural gradient-compatibility equations

For every incompressible velocity field,

$$
\boxed{
\nabla\cdot S
=
\frac12\Delta V
=
-\frac12\nabla\times\Omega.
}
\tag{25.1}
$$

Also:

$$
\boxed{
\nabla\cdot\Omega=0.
}
\tag{25.2}
$$

On Type A:

$$
S
=
\frac{3\lambda}{2}
U_\xi,
\qquad
\Omega=r\xi.
$$

Therefore:

$$
\boxed{
3\lambda
\nabla\cdot U_\xi
=
-\nabla\times(r\xi).
}
\tag{25.3}
$$

On Type B:

$$
S=-3\lambda U_\zeta,
\qquad
\Omega=r\xi,
\qquad
\xi\perp\zeta,
$$

so:

$$
\boxed{
-6\lambda
\nabla\cdot U_\zeta
=
-\nabla\times(r\xi).
}
\tag{25.4}
$$

These are the natural next PDE constraints on the two axisymmetric silent modes.

---

# 26. Status ledger

## PROVED this round

### D67-P1 — one-scalar aligned strain spectral parameterization

$$
S=\frac{3\lambda}{2}U_\xi+dH.
$$

### D67-P2 — exact aligned cofactor normal form

$$
C=
\left(
\frac{3\lambda^2}{4}-d^2
\right)U_\xi
-\lambda dH.
$$

### D67-P3 — pointwise cofactor/vorticity-stress commutation

$$
[C,W_\Omega]=0.
$$

### D67-P4 — exact two-stress angle law

$$
\chi_{C\omega}
=
\frac{3\lambda^2-4d^2}
{3\lambda^2+4d^2}.
$$

### D67-P5 — exact aligned self cofactor angular rate

$$
\Omega_{C,\rm self}
=
\frac{
2\sqrt3|d||4d^2-9\lambda^2|
}{
4d^2+3\lambda^2
}.
$$

### D67-P6 — complete self-lock spectral classification

$$
d=0
\quad\text{or}\quad
d=\pm3\lambda/2.
$$

### D67-P7 — Type A stress angle

$$
\chi_{C\omega}=1.
$$

### D67-P8 — Type B stress angle

$$
\chi_{C\omega}=-1/2.
$$

### D67-P9 — self-lock cofactor amplitude is spatially constant.

### D67-P10 — self-lock 4:1 balance collapses to pure $C$-orientation / $\Omega$-amplitude correlation.

### D67-P11 — exact isotropic-covariance vorticity-stress pair budget

$$
\iint
\phi_x\phi_y
|\delta W_\Omega|^2
=
\frac{4\Phi}{3}M_4.
$$

### D67-P12 — exact vorticity-stress amplitude/orientation increment split.

---

# 27. Open routes

## Not closed

The shape-active branch:

$$
\Omega_{C,\rm self}>0.
$$

Pressure/vorticity forcing may still dynamically compensate cofactor self-rotation.

## Not closed

Type-A orientation–amplitude correlation silence.

## Not closed

Type-B orientation–amplitude correlation silence.

But these are now explicit geometric normal forms rather than generic correlation cancellation.

---

# 28. New STOP

$$
\boxed{
\textbf{
STOP-D67:
Aligned cofactor/vorticity correlation has only one transverse spectral coordinate. Generic spectra self-rotate the cofactor in tensor space; the only zero-self-rotation equality geometries are two axisymmetric modes. In both, cofactor amplitude is spatially frozen, source variation is purely vorticity-amplitude variation, and X72 silence reduces to exact orthogonality between mandatory spatial cofactor-axis rotation and mandatory vorticity-amplitude variation.
}
}
$$

---

# 29. Next autonomous step

## DCRP68 / X72-R51 — Axisymmetric Director Integrability

**Working title**

> **Gradient Compatibility of Axial/Equatorial Self-Lock Modes and the Spatial Texture Obstruction**

Primary tasks:

1. Type A:
   $$
   S=\frac{3\lambda}{2}U_\xi,
   \qquad
   \Omega=r\xi;
   $$
2. Type B:
   $$
   S=-3\lambda U_\zeta,
   \qquad
   \Omega=r\xi,
   \quad
   \xi\perp\zeta;
   $$
3. use
   $$
   \nabla\cdot S=-\frac12\nabla\times\Omega,
   \qquad
   \nabla\cdot\Omega=0;
   $$
4. derive the exact director PDEs for $(r,\xi)$ and $(r,\xi,\zeta)$;
5. classify constant-axis and one-dimensional director solutions;
6. test whether isotropic covariance plus the DCRP30 sublinear energy tail permits an entire recurrent director texture;
7. if a smooth nontrivial texture survives, insert it back into the orientation–amplitude Round38 correlation.

Desired endpoint:

$$
\boxed{
\mathsf X_A,\mathsf X_B
\Longrightarrow
\text{director-integrability defect}
\vee
\text{explicit global texture normal form}.
}
$$

---

# 30. One-line checkpoint

The generic two-stress correlation problem has collapsed to one transverse anisotropy parameter: only two axisymmetric cofactor self-lock spectra avoid intrinsic tensor-shape rotation, and on both of them X72 silence is reduced to a pure spatial-axis-rotation versus vorticity-amplitude cancellation problem.

---

**End checkpoint:** DCRP67 / X72-R50  
**Next:** DCRP68 / X72-R51 — Axisymmetric Director Integrability.