# NS × X 積分 × 24/72 範式實戰
## Round 27 — Pure Continuous Coherence Dynamics / Angular Phase-Locking Route

- 日期：2026-08-17
- 版本：v0.1
- 狀態：Proof-Route Experiment / Continuous-Only Angular-Dynamics Branch
- canonical source：UTF-8 Markdown
- canonical math delimiters：inline `$...$`；display `$$...$$`
- 前一輪：`NS_X72_Round26_PureContinuous_SignedKernel_QuadrupoleCoherence_v0.1_2026-08-17.md`
- 本輪目標：Round 26 已將 nonlocal virtual coupling寫成 amplitude × anisotropy × coherence，且 static sign無 universal bias。本輪研究 pressure coherence、cross-strain coherence、strain eigenframe、quotient direction、remote vorticity direction與 line-of-sight 的 deterministic dynamics，檢驗 rapid angular transport是否造成 time cancellation，或 dangerous branch可形成 phase locking。
- 非主張：本文沒有證明 coherence phase必快速旋轉，也沒有證明 dangerous branch必 phase-lock。本文建立 exact angular equations、nonstationary cancellation lemma與 phase-locking obstruction。

---

# 0. Round 26 handoff

Round 26 得到：

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

以及：

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

兩種 signed kernel都具有：

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

但：

$$
\lambda_2>0
$$

不保證 synchronizing sign。

Round 26 STOP：

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

---

# 1. Unit directional quadrupole

對：

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

定義：

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

則：

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

pressure pair coherence：

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

並且：

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

---

# 2. Generic normalized tensor-coherence law

令非零 tensor curves：

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

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

及：

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

則：

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

所以：

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

這是 exact normalized tensor-coherence equation。

若：

$$
|c|<1,
$$

定義：

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

則：

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

---

# 3. Strain-shape dynamics

Navier–Stokes strain equation：

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

因此：

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

---

# 4. Exact strain-eigenframe rotation

假設：

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

且：

$$
Se_i=\lambda_ie_i.
$$

material differentiation給：

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

由：

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

以及 identity term無 off-diagonal contribution，

得到：

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

---

# 5. Self-amplification rotation null

Equation (4.2) 的 off-diagonal rotation沒有：

$$
-S^2.
$$

因此：

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

所以：

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

不能自動推出 rapid angular decoherence。

frame rotation直接依賴：

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

---

# 6. Eigenvalue-gap angular sensitivity

由 (4.2)：

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

所以 near spectral collision：

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

會提高 eigenframe sensitivity。

---

# 7. Vorticity-direction dynamics

令：

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

由：

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

有：

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

等價：

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

若 inviscid 且：

$$
\xi
$$

為 strain eigenvector，

則：

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

形成 angular locking channel。

---

# 8. Optimal quotient-direction dynamics

Round 14 representative equation：

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

令：

$$
v=rn.
$$

投影到：

$$
n^\perp
$$

得到：

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

因此 quotient direction rotation由：

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

共同決定。

---

# 9. Pairwise line-of-sight dynamics

令：

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

定義：

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

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

則：

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

以及：

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

---

# 10. Pairwise pressure-coherence dynamics

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

沿 pair trajectories：

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

即：

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

所以 pressure phase rotation拆成：

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

large：

$$
R
$$

只壓低第二項，不會自動壓低：

$$
D_t\widehat S.
$$

---

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

Round 26 使用：

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

因：

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

trace-free symmetric且 Frobenius norm為 1，

sharp bound：

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

定義：

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

與 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

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

所以：

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

其中：

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

以及：

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

所以 aggregate cross-strain phase來自：

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

---

# 13. Pairwise normalized Biot–Savart phase

令：

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

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

若：

$$
\delta>0,
$$

定義：

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

則：

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

Round 26 kernel：

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

因：

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

定義：

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

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

令：

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

則：

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

並且：

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

令：

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

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

故：

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

其中：

- $\dot n$ 由 (8.1)；
- $\dot\xi$ 由 (7.1)；
- $\dot e$ 由 (9.2)。

因此 pairwise BS phase dynamics已完全寫成 continuous multi-frame geometry。

---

# 15. Phase singularity at transverse depletion is removable

當：

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

normalized：

$$
m,\psi_{BS}
$$

可能失去穩定定義。

但 physical amplitude：

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

同時趨近 0。

所以：

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

---

# 16. Angular phases

若：

$$
|\psi_P|<1,
$$

定義：

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

若：

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

定義：

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

對 source sign固定區間，pair couplings可寫：

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

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

其中：

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

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

---

# 17. Nonstationary angular-cancellation lemma

令：

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

於：

$$
[t_0,t_1].
$$

假設：

$$
A,\theta'
$$

absolutely continuous且：

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

integration by parts：

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

因此：

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

命名：

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

---

# 18. Sustained signed coupling requires locking or modulation

large cumulative：

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

至少需要：

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

這與 Round 10 Fourier phase route形成 exact structural parallel。

---

# 19. Self-amplification does not force angular mixing

因：

$$
-S^2
$$

不直接旋轉 strain eigenframe，

可能存在：

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

如果 line-of-sight與 remote quadrupole也慢，

則：

$$
\theta_P
$$

可以 near-lock。

所以：

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

---

# 20. Exact lock conditions

strain eigenframe lock：

若：

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

對所有：

$$
j\ne i,
$$

則：

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

vorticity-direction lock：

若：

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

則：

$$
D_t\xi=0.
$$

quotient-direction lock：

若右側 (8.1) 等於 0，則：

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

所以 persistent nonlocal coupling是一個：

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

---

# 21. Coherence-time carrier

定義：

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

對 aggregate coupling可定義 amplitude-weighted inverse coherence time：

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

並定義：

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

若：

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

phase在 selection累積前快速翻轉。

若：

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

signed virtual interaction具有足夠 persistence。

---

# 22. Round 10 / Round 27 obstruction confluence

Round 10 Fourier triad：

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

Round 27 physical-space nonlocal coupling：

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

兩者的 sustained signed effect都需要：

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

所以 Fourier route與 physical-space nonlocal route再次匯流到：

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

---

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

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

命名：

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

---

# 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

本輪使用：

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

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

以及 continuous material trajectories。

eigenvector labels：

$$
i=1,2,3
$$

只是有限-dimensional spectral notation；

所有 frame dynamics也可用 tensor projector calculus重寫。

因此：

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

---

# 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

下一輪直接研究：

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

問題：

1. strain eigenframe lock受到 perturbation後是 restoring還是 destabilizing；
2. vorticity–strain eigenvector alignment lock是否 stable；
3. quotient-direction lock中的 gauge term是否提供 damping；
4. large separation的 slow line-of-sight rotation是否延長 coherence time；
5. 線性化：
   $$
   \delta\theta'
   =
   a(t)\delta\theta+\cdots;
   $$
6. amplification-sign lock若 unstable，nonstationary cancellation重新獲得力量；
7. stable amplification lock若存在，則成為 persistent nonlocal danger carrier；
8. 仍使用 continuous angular stability，不做 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.
   - strain-rate eigenframe dynamics、eigenframe rotation及 anisotropic pressure Hessian作用的 primary-source背景。

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.
   - Biot–Savart local/nonlocal strain decomposition與 vorticity alignment背景。

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

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

本輪 normalized tensor-coherence law、strain-eigenframe formula、quotient-direction dynamics、pair angular-phase equations、nonstationary cancellation lemma與 phase-lock obstruction均為本文直接推導。

---

# 29. Commit state

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