# NS × X Integral × 24/72 Paradigm Practice
## Round 27 — Pure Continuous Coherence Dynamics / Angular Phase-Locking Route

- Date: 2026-08-17
- Version: v0.1
- Status: Proof-Route Experiment / Continuous-Only Angular-Dynamics Branch
- canonical source: UTF-8 Markdown
- canonical math delimiters: inline `$...$`; display `$$...$$`
- Previous round: `NS_X72_Round26_PureContinuous_SignedKernel_QuadrupoleCoherence_v0.1_2026-08-17.md`
- Current round objective: Round 26 has formulated the nonlocal virtual coupling as amplitude × anisotropy × coherence, and the static sign has no universal bias. This round investigates the deterministic dynamics of pressure coherence, cross-strain coherence, the strain eigenframe, quotient direction, remote vorticity direction, and line-of-sight, to examine whether rapid angular transport causes time cancellation, or if a dangerous branch can form phase locking.
- Non-claims: This document does not prove that the coherence phase must rotate rapidly, nor does it prove that the dangerous branch must phase-lock. This document establishes exact angular equations, a nonstationary cancellation lemma, and a phase-locking obstruction.

---

# 0. Round 26 handoff

Round 26 obtained:

$$
S:H_p^B
=
\frac{\sqrt6}{4\pi}
|S|A_P\alpha_Pc_P,
$$

and:

$$
\gamma_{B\to x}
=
\frac3{4\pi\sqrt2}
A_S\alpha_Sc_S.
$$

Both signed kernels possess:

$$
\boxed{
\text{zero angular mean + finite angular variance}.
}
$$

But:

$$
\lambda_2>0
$$

does not guarantee a synchronizing sign.

Round 26 STOP:

$$
\boxed{
\text{STOP-C30}
=
\text{Quadrupole-Coherence / Synchronizing-Bias Gap}.
}
$$

---

# 1. Unit directional quadrupole

For:

$$
e\in\mathbb S^2,
$$

define:

$$
\boxed{
\mathbb T(e)
=
\sqrt{\frac32}
\left(
e\otimes e-\frac13I
\right).
}
\tag{1.1}
$$

Then:

$$
|\mathbb T|_F=1,
\qquad
\operatorname{tr}\mathbb T=0.
$$

The pressure pair coherence is:

$$
\boxed{
\psi_P
=
\widehat S:\mathbb T(e)
=
\sqrt{\frac32}\,
e^\top\widehat Se
\in[-1,1].
}
\tag{1.2}
$$

And:

$$
e^\top Se
=
\sqrt{\frac23}|S|\psi_P.
$$

---

# 2. Generic normalized tensor-coherence law

Let the non-zero tensor curves be:

$$
A(t),B(t),
$$

$$
\widehat A=\frac A{|A|},
\qquad
\widehat B=\frac B{|B|},
$$

and:

$$
c=\widehat A:\widehat B.
$$

Then:

$$
\boxed{
\dot{\widehat A}
=
\frac1{|A|}
\Pi_{\widehat A}^\perp\dot A.
}
\tag{2.1}
$$

Therefore:

$$
\boxed{
\dot c
=
\frac{\dot A}{|A|}
:
(\widehat B-c\widehat A)
+
\frac{\dot B}{|B|}
:
(\widehat A-c\widehat B).
}
\tag{2.2}
$$

This is the exact normalized tensor-coherence equation.

If:

$$
|c|<1,
$$

define:

$$
\boxed{
\theta=\arccos c,
}
$$

then:

$$
\boxed{
\dot\theta
=
-\frac{\dot c}{\sqrt{1-c^2}}.
}
\tag{2.3}
$$

---

# 3. Strain-shape dynamics

The Navier–Stokes strain equation is:

$$
\boxed{
D_tS
=
\nu\Delta S
-
S^2
-
\frac14\omega\otimes\omega
+
\frac14|\omega|^2I
-
H_p.
}
\tag{3.1}
$$

Thus:

$$
\boxed{
D_t\widehat S
=
\frac1{|S|}
\Pi_{\widehat S}^\perp
\left[
\nu\Delta S
-
S^2
-
\frac14\omega\otimes\omega
+
\frac14|\omega|^2I
-
H_p
\right].
}
\tag{3.2}
$$

---

# 4. Exact strain-eigenframe rotation

Assume:

$$
\lambda_1<\lambda_2<\lambda_3
$$

and:

$$
Se_i=\lambda_ie_i.
$$

Material differentiation gives:

$$
\boxed{
D_te_i
=
\sum_{j\ne i}
\frac{
e_j^\top(D_tS)e_i
}{
\lambda_i-\lambda_j
}
e_j.
}
\tag{4.1}
$$

From:

$$
e_j^\top S^2e_i=0
\qquad
(j\ne i),
$$

and the fact that the identity term has no off-diagonal contribution,

we obtain:

$$
\boxed{
D_te_i
=
\sum_{j\ne i}
\frac{
\nu e_j^\top\Delta S\,e_i
-
\frac14
(\omega\cdot e_j)(\omega\cdot e_i)
-
e_j^\top H_pe_i
}{
\lambda_i-\lambda_j
}
e_j.
}
\tag{4.2}
$$

---

# 5. Self-amplification rotation null

The off-diagonal rotation in Equation (4.2) does not contain:

$$
-S^2.
$$

Therefore:

$$
\boxed{
\textbf{
strain self-amplification changes eigenvalues
but does not directly rotate the instantaneous strain eigenframe.
}
}
\tag{5.1}
$$

Thus:

$$
(-\det S)_+>0
$$

cannot automatically imply rapid angular decoherence.

Frame rotation directly depends on:

- viscous off-diagonal forcing;
- vorticity dyad;
- pressure Hessian.

---

# 6. Eigenvalue-gap angular sensitivity

From (4.2):

$$
\boxed{
|D_te_i|
\le
\sum_{j\ne i}
\frac{
\left|
\nu e_j^\top\Delta S e_i
-
\frac14(\omega\cdot e_j)(\omega\cdot e_i)
-
e_j^\top H_pe_i
\right|
}{
|\lambda_i-\lambda_j|
}.
}
\tag{6.1}
$$

Therefore, near a spectral collision:

$$
|\lambda_i-\lambda_j|\downarrow0
$$

will increase the eigenframe sensitivity.

---

# 7. Vorticity-direction dynamics

Let:

$$
\xi=\frac{\omega}{|\omega|}.
$$

From:

$$
D_t\omega
=
S\omega+\nu\Delta\omega,
$$

we have:

$$
\boxed{
D_t\xi
=
P_\xi^\perp
\left[
S\xi
+
\nu\frac{\Delta\omega}{|\omega|}
\right].
}
\tag{7.1}
$$

Equivalently:

$$
\boxed{
D_t\xi
=
P_\xi^\perp S\xi
+
\nu P_\xi^\perp
\left[
\Delta\xi
+
2\nabla\log|\omega|
\cdot\nabla\xi
\right].
}
\tag{7.2}
$$

If inviscid and:

$$
\xi
$$

is a strain eigenvector,

then:

$$
P_\xi^\perp S\xi=0,
$$

forming an angular locking channel.

---

# 8. Optimal quotient-direction dynamics

Round 14 representative equation:

$$
\partial_tv
+
(u\cdot\nabla)v
+
(\nabla u)^\top v
=
\nu\Delta v+\nabla\chi_g.
$$

Let:

$$
v=rn.
$$

Projecting onto:

$$
n^\perp
$$

yields:

$$
\boxed{
\begin{aligned}
D_tn
={}&
\nu P_n^\perp
\left[
\Delta n
+
2\nabla\log r\cdot\nabla n
\right]
\\
&-
P_n^\perp Sn
+
\frac12\omega\times n
+
\frac1r
P_n^\perp\nabla\chi_g.
\end{aligned}
}
\tag{8.1}
$$

Therefore, the quotient direction rotation is jointly determined by:

- viscous direction diffusion;
- strain turning;
- local rigid rotation;
- gauge-maintenance transverse gradient.

---

# 9. Pairwise line-of-sight dynamics

Let:

$$
\dot x=u(x,t),
\qquad
\dot y=u(y,t).
$$

Define:

$$
R=|x-y|,
\qquad
e=\frac{x-y}{R},
$$

$$
\delta u=u(x)-u(y).
$$

Then:

$$
\boxed{
\dot R=e\cdot\delta u,
}
\tag{9.1}
$$

and:

$$
\boxed{
\dot e
=
\frac1R
P_e^\perp\delta u.
}
\tag{9.2}
$$

---

# 10. Pairwise pressure-coherence dynamics

$$
\psi_P
=
\widehat S(x):\mathbb T(e).
$$

Along pair trajectories:

$$
\boxed{
\dot\psi_P
=
(D_t\widehat S)(x):\mathbb T(e)
+
2\sqrt{\frac32}\,
\dot e\cdot\widehat S(x)e.
}
\tag{10.1}
$$

That is:

$$
\boxed{
\dot\psi_P
=
(D_t\widehat S):\mathbb T(e)
+
\frac2R
\sqrt{\frac32}
\left(
P_e^\perp\delta u
\right)
\cdot\widehat Se.
}
\tag{10.2}
$$

Thus, the pressure phase rotation splits into:

$$
\boxed{
\text{local strain-frame/shape rotation}
+
\text{line-of-sight rotation}.
}
$$

A large:

$$
R
$$

only suppresses the second term, and does not automatically suppress:

$$
D_t\widehat S.
$$

---

# 11. Sharp normalization of Round 26 cross-strain coherence

Round 26 used:

$$
c_S=n^\top\widehat{\mathbb Q}_Sn.
$$

Since:

$$
\widehat{\mathbb Q}_S
$$

is trace-free symmetric and has a Frobenius norm of 1,

the sharp bound is:

$$
\boxed{
|c_S|
\le
\sqrt{\frac23}.
}
\tag{11.1}
$$

Define:

$$
\boxed{
\mathbb N(n)
=
\sqrt{\frac32}
\left(
n\otimes n-\frac13I
\right)
}
\tag{11.2}
$$

and the normalized coherence:

$$
\boxed{
\widetilde c_S
=
\mathbb N(n):
\widehat{\mathbb Q}_S
=
\sqrt{\frac32}c_S
\in[-1,1].
}
\tag{11.3}
$$

---

# 12. Aggregate cross-strain coherence dynamics

$$
\widetilde c_S
=
\mathbb N(n):
\widehat{\mathbb Q}_S.
$$

Therefore:

$$
\boxed{
\dot{\widetilde c}_S
=
\dot{\mathbb N}:
\widehat{\mathbb Q}_S
+
\mathbb N:
\dot{\widehat{\mathbb Q}}_S.
}
\tag{12.1}
$$

where:

$$
\boxed{
\dot{\mathbb N}
=
\sqrt{\frac32}
(
\dot n\otimes n+n\otimes\dot n
).
}
\tag{12.2}
$$

and:

$$
\boxed{
\dot{\widehat{\mathbb Q}}_S
=
\frac1{|\mathbb Q_S|}
\Pi_{\widehat{\mathbb Q}_S}^\perp
\dot{\mathbb Q}_S.
}
\tag{12.3}
$$

Thus, the aggregate cross-strain phase comes from:

$$
\boxed{
\text{local quotient-direction rotation}
+
\text{remote quadrupole rotation}.
}
$$

---

# 13. Pairwise normalized Biot–Savart phase

Let:

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

$$
\delta
=
|\xi\times n|.
$$

If:

$$
\delta>0,
$$

define:

$$
m
=
\frac{n\times\xi}{\delta}.
$$

Then:

$$
m\perp n,
\qquad
|m|=1.
$$

Round 26 kernel:

$$
X
=
(n\cdot e)
[n\cdot(\xi\times e)].
$$

Since:

$$
n\cdot(\xi\times e)
=
\delta(m\cdot e),
$$

define:

$$
\boxed{
\psi_{BS}
=
2(n\cdot e)(m\cdot e)
\in[-1,1].
}
\tag{13.1}
$$

pair cross-selection:

$$
\boxed{
\gamma_{\rm pair}
=
\frac3{8\pi}
\frac{
|\omega|
\delta
}{
R^3
}
\psi_{BS}.
}
\tag{13.2}
$$

---

# 14. Pairwise Biot–Savart phase dynamics

Let:

$$
a=n\cdot e,
\qquad
b=m\cdot e.
$$

Then:

$$
\boxed{
\dot\psi_{BS}
=
2
[
(\dot n\cdot e+n\cdot\dot e)b
+
a(\dot m\cdot e+m\cdot\dot e)
].
}
\tag{14.1}
$$

and:

$$
\boxed{
\dot m
=
\frac{
\dot n\times\xi+n\times\dot\xi
}{
\delta
}
-
m
\frac{\dot\delta}{\delta}.
}
\tag{14.2}
$$

Let:

$$
q=n\cdot\xi,
$$

$$
\delta^2=1-q^2,
$$

thus:

$$
\boxed{
\frac{\dot\delta}{\delta}
=
-
\frac q{\delta^2}
(
\dot n\cdot\xi+n\cdot\dot\xi
).
}
\tag{14.3}
$$

where:

- $\dot n$ is given by (8.1);
- $\dot\xi$ is given by (7.1);
- $\dot e$ is given by (9.2).

Therefore, the pairwise BS phase dynamics is completely formulated in terms of continuous multi-frame geometry.

---

# 15. Phase singularity at transverse depletion is removable

When:

$$
\delta=|\xi\times n|\downarrow0,
$$

the normalized:

$$
m,\psi_{BS}
$$

may lose a stable definition.

But the physical amplitude:

$$
\boxed{
\frac{
|\omega|\delta
}{
R^3
}
}
$$

approaches 0 simultaneously.

Therefore:

$$
\boxed{
\text{normalized phase singularity}
\neq
\text{physical coupling singularity}.
}
\tag{15.1}
$$

---

# 16. Angular phases

If:

$$
|\psi_P|<1,
$$

define:

$$
\boxed{
\theta_P=\arccos\psi_P.
}
\tag{16.1}
$$

If:

$$
|\psi_{BS}|<1,
$$

define:

$$
\boxed{
\theta_{BS}
=
\arccos\psi_{BS}.
}
\tag{16.2}
$$

For intervals with a fixed source sign, the pair couplings can be written as:

$$
\boxed{
\mathcal C_P
=
A_P^{\rm pair}\cos\theta_P,
}
\tag{16.3}
$$

$$
\boxed{
\mathcal C_{BS}
=
A_{BS}^{\rm pair}\cos\theta_{BS},
}
\tag{16.4}
$$

where:

$$
A_P^{\rm pair}
=
\frac{\sqrt6}{4\pi}
\frac{|f_p(y)||S(x)|}{R^3},
$$

$$
A_{BS}^{\rm pair}
=
\frac3{8\pi}
\frac{
|\omega(y)|
|\xi\times n|
}{
R^3
}.
$$

---

# 17. Nonstationary angular-cancellation lemma

Let:

$$
\mathcal C(t)
=
A(t)\cos\theta(t)
$$

on:

$$
[t_0,t_1].
$$

Assume:

$$
A,\theta'
$$

are absolutely continuous and:

$$
\boxed{
|\theta'|
\ge
\Omega>0.
}
\tag{17.1}
$$

Integration by parts yields:

$$
\boxed{
\begin{aligned}
\int_{t_0}^{t_1}
A\cos\theta\,dt
={}&
\left[
\frac{
A\sin\theta
}{
\theta'
}
\right]_{t_0}^{t_1}
\\
&-
\int_{t_0}^{t_1}
\left[
\frac{A'}{\theta'}
-
\frac{
A\theta''
}{
(\theta')^2
}
\right]
\sin\theta\,dt.
\end{aligned}
}
\tag{17.2}
$$

Therefore:

$$
\boxed{
\begin{aligned}
\left|
\int
A\cos\theta\,dt
\right|
\le{}&
\frac{
2\|A\|_\infty
}{
\Omega
}
+
\frac{
\|A'\|_{L^1}
}{
\Omega
}
\\
&+
\frac1{\Omega^2}
\int
|A\theta''|dt.
\end{aligned}
}
\tag{17.3}
$$

Named:

$$
\boxed{
\textbf{Nonstationary Angular-Cancellation Lemma}.
}
$$

---

# 18. Sustained signed coupling requires locking or modulation

A large cumulative:

$$
\int\mathcal C(t)dt
$$

requires at least:

$$
\boxed{
\begin{aligned}
\mathrm{L1}:&
\quad
\text{phase locking / near-locking},
\\
\mathrm{L2}:&
\quad
\text{strong amplitude modulation},
\\
\mathrm{L3}:&
\quad
\text{strong phase acceleration},
\\
\mathrm{L4}:&
\quad
\text{repeated amplitude-zero / sign-transition events}.
\end{aligned}
}
\tag{18.1}
$$

This forms an exact structural parallel with the Round 10 Fourier phase route.

---

# 19. Self-amplification does not force angular mixing

Since:

$$
-S^2
$$

does not directly rotate the strain eigenframe,

there may exist:

$$
\boxed{
\text{large strain self-amplification}
+
\text{slow eigenframe rotation}.
}
$$

If the line-of-sight and remote quadrupole are also slow,

then:

$$
\theta_P
$$

can near-lock.

Therefore:

$$
\boxed{
\textbf{
dangerous amplitude growth does not automatically generate
the oscillation required for time cancellation.
}
}
\tag{19.1}
$$

---

# 20. Exact lock conditions

Strain eigenframe lock:

If:

$$
\boxed{
\nu e_j^\top\Delta S e_i
-
\frac14
(\omega\cdot e_j)(\omega\cdot e_i)
-
e_j^\top H_pe_i
=
0
}
\tag{20.1}
$$

for all:

$$
j\ne i,
$$

then:

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

Vorticity-direction lock:

If:

$$
\boxed{
P_\xi^\perp
\left[
S\xi
+
\nu\frac{\Delta\omega}{|\omega|}
\right]
=
0,
}
\tag{20.2}
$$

then:

$$
D_t\xi=0.
$$

Quotient-direction lock:

If the right side of (8.1) equals 0, then:

$$
\boxed{
D_tn=0.
}
\tag{20.3}
$$

Therefore, persistent nonlocal coupling is a:

$$
\boxed{
\textbf{multi-frame angular-locking problem}.
}
$$

---

# 21. Coherence-time carrier

Define:

$$
\boxed{
\Omega_P=|\dot\theta_P|,
\qquad
\Omega_{BS}=|\dot\theta_{BS}|.
}
\tag{21.1}
$$

For aggregate coupling, we can define the amplitude-weighted inverse coherence time:

$$
\boxed{
\tau_{\rm coh}^{-1}
=
\frac{
\iint
A(x,y)
\Omega(x,y)
\,dxdy
}{
\iint
A(x,y)\,dxdy
}.
}
\tag{21.2}
$$

and define:

$$
\boxed{
\mathfrak R_{\rm lock}
=
\Lambda_{\rm sel}
\tau_{\rm coh}.
}
\tag{21.3}
$$

If:

$$
\mathfrak R_{\rm lock}\ll1,
$$

the phase flips rapidly before selection accumulates.

If:

$$
\mathfrak R_{\rm lock}\gg1,
$$

the signed virtual interaction has sufficient persistence.

---

# 22. Round 10 / Round 27 obstruction confluence

Round 10 Fourier triad:

$$
\mathcal T
=
\mathcal A\sin\Phi.
$$

Round 27 physical-space nonlocal coupling:

$$
\mathcal C
=
A\cos\theta.
$$

The sustained signed effect of both requires:

$$
\boxed{
\text{phase/coherence locking}
\vee
\text{strong modulation}.
}
$$

Thus, the Fourier route and the physical-space nonlocal route converge once again at the:

$$
\boxed{
\textbf{phase-locking obstruction core}.
}
$$

---

# 23. STOP-C31 — Angular Phase-Locking / Coherence-Persistence Gap

$$
\boxed{
\begin{aligned}
\text{pressure phase}&=\theta_P,
\\
\text{BS phase}&=\theta_{BS},
\\
\text{self-amplification direct frame rotation}&=0,
\\
\text{rapid phase}&\Rightarrow\text{time cancellation},
\\
\text{sustained coupling}
&\Rightarrow
\text{locking}\vee\text{modulation},
\\
\text{missing}
&=
\text{unconditional lower bound on phase speed
or upper bound on lock duration},
\\
T_{\mathsf C\to\mathsf D}
&=
\text{NOT REACHED}.
\end{aligned}
}
$$

Named:

$$
\boxed{
\textbf{STOP-C31:
Angular Phase-Locking / Coherence-Persistence Gap}.
}
$$

---

# 24. 24/72 Ledger — Round 27

| Step | object | $B$ | $U$ | $O$ | $L$ | status |
|---|---|---|---|---|---|---|
| C359 | unit directional quadrupole | $\mathsf C$ | angular geometry | $\mathsf X$ | $\mathsf F$ | FORM |
| C360 | tensor-coherence equation | $\mathsf C$ | tensor dynamics | relational | $\mathsf F$ | EXACT |
| C361 | angular phase | $\mathsf C$ | continuous angle | scalar | $\mathsf F$ | FORM |
| C362 | strain-shape dynamics | $\mathsf C$ | material PDE | $\mathsf X$ | $\mathsf F$ | EXACT |
| C363 | strain eigenframe rotation | $\mathsf C$ | spectral geometry | relational | $\mathsf F$ | EXACT |
| C364 | self-amplification frame rotation | $\mathsf C$ | algebraic | targeted | $\mathsf F$ | ZERO off-diagonal |
| C365 | eigenvalue-gap sensitivity | $\mathsf C$ | spectral geometry | scalar | $\mathsf F$ | PROVED |
| C366 | vorticity-direction dynamics | $\mathsf C$ | material PDE | relational | $\mathsf F$ | EXACT |
| C367 | quotient-direction dynamics | $\mathsf C$ | gauge/material PDE | relational | $\mathsf F$ | EXACT |
| C368 | line-of-sight dynamics | $\mathsf C$ | pair transport | relational | $\mathsf F$ | EXACT |
| C369 | pressure coherence dynamics | $\mathsf C$ | angular transport | scalar | $\mathsf F$ | EXACT |
| C370 | sharp BS coherence normalization | $\mathsf C$ | tensor geometry | scalar | $\mathsf F$ | PROVED |
| C371 | aggregate BS coherence dynamics | $\mathsf C$ | tensor dynamics | relational | $\mathsf F$ | EXACT |
| C372 | pair BS normalized phase | $\mathsf C$ | angular geometry | scalar | $\mathsf F$ | EXACT |
| C373 | pair BS phase dynamics | $\mathsf C$ | multi-frame transport | scalar | $\mathsf F$ | EXACT |
| C374 | phase singularity at zero amplitude | $\mathsf C$ | polar geometry | targeted | $\mathsf F$ | REMOVABLE physically |
| C375 | nonstationary cancellation | $\mathsf C$ | time integration | scalar | $\mathsf F$ | PROVED |
| C376 | self-amplification $\Rightarrow$ rapid phase | $\mathsf C$ | angular feedback | targeted | $\mathsf F$ | REFUTED as automatic |
| C377 | lock conditions | $\mathsf C$ | constraint | relational | $\mathsf F$ | EXACT |
| C378 | coherence-time ratio | $\mathsf C$ | recognition | scalar | $\mathsf F$ | FORM |
| C379 | unconditional phase-speed lower bound | $\mathsf C$ | coupled NS | targeted | $\mathsf F$ | OPEN / STOP-C31 |

---

# 25. Continuous-versus-discrete status

This round uses:

$$
e,n,\xi\in\mathbb S^2,
$$

$$
\theta\in[0,\pi],
$$

and continuous material trajectories.

Eigenvector labels:

$$
i=1,2,3
$$

are merely finite-dimensional spectral notation;

all frame dynamics can also be rewritten using tensor projector calculus.

Therefore:

$$
\boxed{
T_{\mathsf C\to\mathsf D}
=
\text{NOT YET REACHED}.
}
$$

---

# 26. Strongest results

## R27-A — Exact eigenframe rotation

$$
\boxed{
D_te_i
=
\sum_{j\ne i}
\frac{
\nu e_j^\top\Delta S e_i
-
\frac14(\omega\cdot e_j)(\omega\cdot e_i)
-
e_j^\top H_p e_i
}{
\lambda_i-\lambda_j
}
e_j.
}
$$

## R27-B — Self-amplification rotation null

$$
\boxed{
-S^2
\text{ amplifies/reshapes strain but contributes no direct eigenframe rotation.}
}
$$

## R27-C — Exact quotient-direction dynamics

$$
\boxed{
\begin{aligned}
D_tn
={}&
\nu P_n^\perp[
\Delta n+2\nabla\log r\cdot\nabla n]
-
P_n^\perp Sn
\\
&+
\frac12\omega\times n
+
r^{-1}P_n^\perp\nabla\chi_g.
\end{aligned}
}
$$

## R27-D — Exact pair BS phase factorization

$$
\boxed{
\gamma_{\rm pair}
=
\frac3{8\pi}
\frac{
|\omega|
|\xi\times n|
}{
R^3
}
\psi_{BS}.
}
$$

## R27-E — Nonstationary cancellation

$$
\boxed{
|\theta'|\ge\Omega>0
\Rightarrow
\text{cumulative signed coupling is small
unless amplitude/phase-speed modulation is large}.
}
$$

---

# 27. Next round — Lock-Manifold Stability

The next round will directly investigate:

$$
\boxed{
\text{Is the phase-lock manifold stable?}
}
$$

Questions:

1. Is the strain eigenframe lock restoring or destabilizing after a perturbation;
2. Is the vorticity–strain eigenvector alignment lock stable;
3. Does the gauge term in the quotient-direction lock provide damping;
4. Does the slow line-of-sight rotation at large separation extend the coherence time;
5. Linearization:
   $$
   \delta\theta'
   =
   a(t)\delta\theta+\cdots;
   $$
6. If the amplification-sign lock is unstable, nonstationary cancellation regains power;
7. If a stable amplification lock exists, it becomes a persistent nonlocal danger carrier;
8. Continue to use continuous angular stability, without constructing a discrete state machine.

---

# 28. External primary-source anchors

1. Josin Tom, Maurizio Carbone, Andrew D. Bragg, *Exploring the turbulent velocity gradients at different scales from the perspective of the strain-rate eigenframe*, arXiv:2005.04300.
   - Primary-source background on strain-rate eigenframe dynamics, eigenframe rotation, and the role of the anisotropic pressure Hessian.

2. Peter E. Hamlington, Jörg Schumacher, Werner J. A. Dahm, *Direct Assessment of Vorticity Alignment with Local and Nonlocal Strain Rates in Turbulent Flows*, arXiv:0810.3439.
   - Background on Biot–Savart local/nonlocal strain decomposition and vorticity alignment.

3. Dhawal Buaria, Alain Pumir, *Non-local amplification of intense vorticity in turbulent flows*, arXiv:2106.14370.
   - Background on intense vorticity and nonlocal strain alignment/amplification.

4. Maurizio Carbone, Michele Iovieno, Andrew D. Bragg, *Gauge symmetry and dimensionality reduction of the anisotropic pressure Hessian*, arXiv:1911.08652.
   - Background on anisotropic pressure Hessian and strain/vorticity frame geometry.

The normalized tensor-coherence law, strain-eigenframe formula, quotient-direction dynamics, pair angular-phase equations, nonstationary cancellation lemma, and phase-lock obstruction in this round are all directly derived in this document.

---

# 29. Commit state

$$
\boxed{
\begin{aligned}
\text{Route}
&=
\mathrm{Pure\ Continuous\ Coherence\ Dynamics/Angular\ Phase\ Locking},
\\
\text{Essential }\mathsf C\to\mathsf D
&=
\mathrm{Not\ reached},
\\
\text{Static sign}
&\to
\text{dynamic angular phase},
\\
\text{Self-amplification direct frame rotation}
&=
0,
\\
\text{Frame rotation}
&=
\mathrm{pressure}
+
\mathrm{vorticity}
+
\mathrm{viscosity},
\\
\text{Rapid phase}
&\Rightarrow
\mathrm{time\ cancellation},
\\
\text{Persistent coupling}
&\Rightarrow
\mathrm{phase\ locking/modulation},
\\
\text{STOP-C31}
&=
\mathrm{Angular\ Phase\text{-}Locking/Coherence\text{-}Persistence\ Gap},
\\
\text{Next}
&=
\mathrm{Lock\text{-}Manifold\ Stability}.
\end{aligned}
}
$$