# NS × X 積分 × 24/72 範式實戰
## Round 44 — Pure Continuous Vorticity-Stress Realizability / Actual Triad Sharpness Route

- 日期：2026-08-17
- 版本：v0.1
- 狀態：Proof-Route Experiment / Continuous-Only Actual-Vorticity-Triad Branch
- canonical source：UTF-8 Markdown
- canonical math delimiters：inline `$...$`；display `$$...$$`
- 前一輪：`NS_X72_Round43_PureContinuous_DivDivFreeStress_FullWaveConePotentialGauge_v0.1_2026-08-17.md`
- 本輪目標：Round 43 已證 generic divdiv-free invisible stress仍可支撐一個完整 transport derivative，但尚未使用 actual NS realizability
  $$
  W
  =
  \omega\otimes\omega-\frac13|\omega|^2I,
  \qquad
  \nabla\cdot\omega=0.
  $$
  本輪直接在 periodic Fourier vorticity space中測試：invisible stress modes、visible/invisible transfer與 one-derivative sharpness能否由真正 divergence-free vorticity modes生成。
- 非主張：本文的 explicit triad是 smooth periodic NS-compatible initial-data witness，不宣稱它自身是一個 stationary whole-space finite-energy NS solution。本文要排除的是「pointwise axisymmetric vorticity-stress cone會自動禁止 Round 43 transfer」這條 algebraic shortcut。

---

# 0. Round 43 handoff

Round 43 對 generic invisible stress：

$$
W_T
$$

得到：

$$
\boxed{
\operatorname{div}\operatorname{div}W_T=0,
}
$$

full wave cone：

$$
\boxed{
\Lambda_{\operatorname{divdiv}}
=
\mathbb S_0,
}
$$

及 symbol-level nonzero transport transfer。

但 actual vorticity stress pointwise滿足：

$$
\boxed{
W(x)
=
\omega(x)\otimes\omega(x)
-
\frac13|\omega(x)|^2I.
}
\tag{0.1}
$$

其 eigenvalues永遠：

$$
\boxed{
\frac23|\omega|^2,
\qquad
-\frac13|\omega|^2,
\qquad
-\frac13|\omega|^2.
}
\tag{0.2}
$$

所以 Round 43 留下：

$$
\boxed{
\text{STOP-C47}
=
\text{Full-Wave-Cone / Vorticity-Realizability Gap}.
}
$$

本輪直接測這個 gap。

---

# 1. Periodic divergence-free vorticity Fourier space

在：

$$
\mathbb T^3,
$$

寫：

$$
\boxed{
\omega(x)
=
\sum_{k\ne0}
\widehat\omega_k
e^{ik\cdot x},
}
\tag{1.1}
$$

with real-field condition：

$$
\widehat\omega_{-k}
=
\overline{
\widehat\omega_k
},
$$

及：

$$
\boxed{
k\cdot\widehat\omega_k=0.
}
\tag{1.2}
$$

對：

$$
k\ne0,
$$

divergence-free inverse curl：

$$
\boxed{
\widehat u_k
=
i
\frac{
k\times\widehat\omega_k
}{
|k|^2
}.
}
\tag{1.3}
$$

所以任意 smooth divergence-free vorticity Fourier datum都對應 smooth divergence-free velocity datum。

---

# 2. Quadratic vorticity-stress convolution

令：

$$
W
=
\omega\otimes\omega
-
\frac13|\omega|^2I.
$$

則：

$$
\boxed{
\widehat W_K
=
\sum_{p+q=K}
\left[
\widehat\omega_p
\otimes
\widehat\omega_q
-
\frac13
(
\widehat\omega_p
\cdot
\widehat\omega_q
)
I
\right].
}
\tag{2.1}
$$

對：

$$
p\ne q,
$$

將 ordered pair：

$$
(p,q),
\qquad
(q,p)
$$

合併，定義 cross-stress amplitude：

$$
\boxed{
\mathcal B(a,b)
=
a\otimes b
+
b\otimes a
-
\frac23
(a\cdot b)I.
}
\tag{2.2}
$$

其中：

$$
a=\widehat\omega_p,
\qquad
b=\widehat\omega_q.
$$

---

# 3. Frequency-direction visibility law

Round 42 longitudinal Riesz projection：

$$
\boxed{
\mathbb P_L(n)F
=
\frac32
m(n)
[
m(n):F
],
}
\tag{3.1}
$$

where：

$$
\boxed{
m(n)
=
\frac13I
-
n\otimes n,
}
\tag{3.2}
$$

and：

$$
|m(n)|^2
=
\frac23.
$$

對 trace-free：

$$
F,
$$

$$
m(n):F
=
-
n^\top Fn.
$$

所以：

$$
\boxed{
\|
\mathbb P_L(n)F
\|_F^2
=
\frac32
|
n^\top Fn
|^2.
}
\tag{3.3}
$$

因此：

$$
\boxed{
F
\text{ is invisible at frequency direction }n
\iff
n^\top Fn=0.
}
\tag{3.4}
$$

visibility不是 tensor amplitude本身的屬性。

它是：

$$
\boxed{
\text{tensor amplitude}
+
\text{frequency direction}
}
$$

的 relational property。

---

# 4. Cross-mode invisibility condition

令：

$$
K=p+q.
$$

對：

$$
a\perp p,
\qquad
b\perp q,
$$

由 (2.2)：

$$
\boxed{
\begin{aligned}
K^\top
\mathcal B(a,b)
K
={}&
2
(K\cdot a)
(K\cdot b)
\\
&-
\frac23
(a\cdot b)
|K|^2.
\end{aligned}
}
\tag{4.1}
$$

因：

$$
p\cdot a=0,
\qquad
q\cdot b=0,
$$

有：

$$
K\cdot a
=
q\cdot a,
$$

$$
K\cdot b
=
p\cdot b.
$$

所以 cross-stress exact invisibility condition：

$$
\boxed{
3
(q\cdot a)
(p\cdot b)
=
(a\cdot b)
|p+q|^2.
}
\tag{4.2}
$$

這是一個單一 scalar relation，

並有大量 nontrivial divergence-free solutions。

---

# 5. Single-mode polarization dichotomy

對 self interaction：

$$
p=q,
$$

令：

$$
a=\widehat\omega_p.
$$

stress second-harmonic coefficient：

$$
\boxed{
B_{\rm self}
=
a\otimes a
-
\frac13
(a\cdot a)I.
}
\tag{5.1}
$$

因：

$$
p\cdot a=0,
$$

在：

$$
2p
$$

direction：

$$
\boxed{
(2p)^\top
B_{\rm self}
(2p)
=
-\frac43
|p|^2
(a\cdot a).
}
\tag{5.2}
$$

因此：

## real linear polarization

若：

$$
a\in\mathbb R^3\setminus\{0\},
$$

則：

$$
a\cdot a>0,
$$

所以 self stress second harmonic必 visible。

## complex circular/null polarization

若：

$$
\boxed{
a\cdot a=0,
}
\tag{5.3}
$$

則 second harmonic invisible。

---

# 6. Helical single-mode invisible stress

取：

$$
\boxed{
p=e_3,
}
$$

$$
\boxed{
a=e_1+ie_2.
}
\tag{6.1}
$$

則：

$$
p\cdot a=0,
$$

$$
a\cdot a=0.
$$

而：

$$
\boxed{
i
p\times a
=
a.
}
\tag{6.2}
$$

所以這是 circular/helical divergence-free polarization。

其：

$$
2p
$$

stress harmonic：

$$
\boxed{
B_{\rm self}
=
a\otimes a
}
\tag{6.3}
$$

滿足：

$$
\boxed{
\mathbb P_L(e_3)
B_{\rm self}
=
0.
}
\tag{6.4}
$$

因此：

$$
\boxed{
\textbf{
actual divergence-free vorticity already admits exactly invisible nonzero stress harmonics.
}
}
\tag{6.5}
$$

所以不存在 universal positive modewise visibility lower bound。

---

# 7. Cross-stress amplitudes are not confined to the axisymmetric cone

pointwise actual stress：

$$
W(x)
$$

永遠位於 axisymmetric cone：

$$
\mathcal M_\omega.
$$

但 Fourier coefficient：

$$
\widehat W_K
$$

是 convolution sum。

bilinear cross-stress amplitudes：

$$
\mathcal B(a,b)
$$

的 algebraic span已經是 entire：

$$
\boxed{
\mathbb S_0.
}
\tag{7.1}
$$

理由：

- diagonal trace-free basis由：
  $$
  \mathcal B(e_i,e_i)
  =
  2e_i\otimes e_i
  -
  \frac23I
  $$
  生成；
- off-diagonal symmetric basis由：
  $$
  \mathcal B(e_i,e_j)
  =
  e_i\otimes e_j
  +
  e_j\otimes e_i,
  \qquad
  i\ne j,
  $$
  生成。

wavevectors可分別選在：

$$
a^\perp,
\qquad
b^\perp
$$

以滿足 divergence-free mode constraints。

命名：

$$
\boxed{
\textbf{Quadratic Cross-Stress Span Theorem}.
}
$$

---

# 8. Fourier Cone Deconfinement Principle

因此：

$$
\boxed{
W(x)\in\mathcal M_\omega
\quad\forall x
}
$$

不推出：

$$
\boxed{
\widehat W_K\in\mathcal M_\omega.
}
$$

nonlinear pointwise cone constraint不被 Fourier convolution coefficient-wise保留。

命名：

$$
\boxed{
\textbf{Fourier Cone Deconfinement Principle}.
}
$$

所以 Round 43 若把 pointwise axisymmetric realizability直接套到 individual Fourier stress amplitudes，會過度限制 actual quadratic convolution。

---

# 9. Explicit actual-vorticity invisible input mode

取三個 Fourier wavevectors：

$$
\boxed{
p=e_2,
\qquad
q=-2e_1,
\qquad
r=e_1.
}
\tag{9.1}
$$

取 vorticity amplitudes：

$$
\boxed{
a=
\begin{pmatrix}
1\\
0\\
1
\end{pmatrix},
\qquad
b=
\begin{pmatrix}
0\\
1\\
-\frac65
\end{pmatrix},
\qquad
c=
\begin{pmatrix}
0\\
0\\
1
\end{pmatrix}.
}
\tag{9.2}
$$

則：

$$
p\cdot a=0,
$$

$$
q\cdot b=0,
$$

$$
r\cdot c=0.
$$

所以三者皆為合法 divergence-free vorticity modes。

---

# 10. Input cross stress is exactly invisible

input stress frequency：

$$
\boxed{
\ell
=
p+q
=
\begin{pmatrix}
-2\\
1\\
0
\end{pmatrix}.
}
\tag{10.1}
$$

由：

$$
a\cdot b
=
-\frac65,
$$

得到：

$$
\boxed{
B
=
\mathcal B(a,b)
=
\begin{pmatrix}
\frac45 & 1 & -\frac65\\
1 & \frac45 & 1\\
-\frac65 & 1 & -\frac85
\end{pmatrix}.
}
\tag{10.2}
$$

direct：

$$
\boxed{
\ell^\top B\ell
=
0.
}
\tag{10.3}
$$

所以：

$$
\boxed{
\mathbb P_L(\ell)
B
=
0.
}
\tag{10.4}
$$

這是一個真正由 divergence-free vorticity cross interaction生成的 invisible stress Fourier coefficient。

---

# 11. The input coefficient is off the pointwise axisymmetric cone

對：

$$
B,
$$

有：

$$
\boxed{
|B|^2
=
\frac{
268
}{
25
},
}
\tag{11.1}
$$

$$
\boxed{
\det B
=
-\frac{
472
}{
125
}.
}
\tag{11.2}
$$

axisymmetric vorticity-stress cone要求：

$$
54(\det B)^2
=
|B|^6.
$$

但此處：

$$
\boxed{
54(\det B)^2
-
|B|^6
=
-\frac{
7218496
}{
15625
}
\ne0.
}
\tag{11.3}
$$

所以：

$$
\boxed{
B\notin\mathcal M_\omega.
}
\tag{11.4}
$$

yet：

$$
B
$$

is an actual Fourier coefficient of a pointwise realizable vorticity stress。

這是 Fourier Cone Deconfinement的 explicit witness。

---

# 12. Actual transport velocity

由：

$$
r=e_1,
\qquad
c=e_3,
$$

Biot–Savart inversion：

$$
\boxed{
\widehat u_r
=
i
\frac{
r\times c
}{
|r|^2
}
=
-ie_2.
}
\tag{12.1}
$$

所以：

$$
\boxed{
i
(
\widehat u_r\cdot\ell
)
=
1.
}
\tag{12.2}
$$

transport by the actual velocity mode at：

$$
r
$$

將 input stress frequency：

$$
\ell
$$

shift到：

$$
\boxed{
m
=
\ell+r
=
\begin{pmatrix}
-1\\
1\\
0
\end{pmatrix}.
}
\tag{12.3}
$$

---

# 13. The same stress amplitude becomes visible after the frequency shift

在：

$$
m
$$

direction：

$$
\boxed{
m^\top Bm
=
-\frac25
\ne0.
}
\tag{13.1}
$$

因此：

$$
\boxed{
\mathbb P_L(m)B
=
\begin{pmatrix}
-\frac1{20} & \frac3{20} & 0\\
\frac3{20} & -\frac1{20} & 0\\
0 & 0 & \frac1{10}
\end{pmatrix}.
}
\tag{13.2}
$$

所以：

$$
\boxed{
\textbf{
transport changes visibility by changing the frequency direction,
even when the stress tensor amplitude itself is unchanged.
}
}
\tag{13.3}
$$

---

# 14. An actual visible output stress exists at the shifted frequency

因 real-field condition包含 mode：

$$
-r
$$

with amplitude：

$$
\overline c=c,
$$

the vorticity modes：

$$
p,
\qquad
-r
$$

generate stress at：

$$
p-r
=
m.
$$

其 coefficient：

$$
\boxed{
C
=
\mathcal B(a,c)
=
\begin{pmatrix}
-\frac23 & 0 & 1\\
0 & -\frac23 & 0\\
1 & 0 & \frac43
\end{pmatrix}.
}
\tag{14.1}
$$

其 visible projection：

$$
\boxed{
\mathbb P_L(m)C
=
\begin{pmatrix}
-\frac16 & \frac12 & 0\\
\frac12 & -\frac16 & 0\\
0 & 0 & \frac13
\end{pmatrix}.
}
\tag{14.2}
$$

而：

$$
\boxed{
\left\langle
\mathbb P_L(m)B,
\mathbb P_L(m)C
\right\rangle_F
=
\frac15.
}
\tag{14.3}
$$

---

# 15. Exact actual-vorticity transfer triad

對 transport projection commutator：

$$
[D_u,\mathbb P_L]W,
$$

frequency：

$$
m=r+\ell
$$

來自：

$$
\widehat u_r,
\qquad
B_\ell
$$

的 contribution：

$$
\boxed{
\begin{aligned}
\widehat{
[D_u,\mathbb P_L]W
}(m)
\supset{}&
i
(
\widehat u_r\cdot\ell
)
\\
&\times
[
\mathbb P_L(\ell)
-
\mathbb P_L(m)
]
B.
\end{aligned}
}
\tag{15.1}
$$

由：

$$
\mathbb P_L(\ell)B=0,
$$

及：

$$
i(\widehat u_r\cdot\ell)=1,
$$

得到：

$$
\boxed{
\widehat{
[D_u,\mathbb P_L]W
}(m)
\supset
-
\mathbb P_L(m)B.
}
\tag{15.2}
$$

與同一 vorticity field已存在的 visible output stress：

$$
\mathbb P_L(m)C
$$

pairing：

$$
\boxed{
\left\langle
\mathbb P_L(m)C,
-
\mathbb P_L(m)B
\right\rangle
=
-\frac15
\ne0.
}
\tag{15.3}
$$

加入 conjugate modes後得到 real smooth periodic field，

corresponding real transfer仍非零。

命名：

$$
\boxed{
\textbf{Actual-Vorticity Visible–Invisible Transfer Triad}.
}
$$

---

# 16. This is genuine NS-compatible initial geometry

define real periodic vorticity：

$$
\boxed{
\omega(x)
=
2\operatorname{Re}
\left[
a e^{ip\cdot x}
+
b e^{iq\cdot x}
+
c e^{ir\cdot x}
\right].
}
\tag{16.1}
$$

它 smooth、periodic、divergence-free。

令：

$$
\widehat u_k
=
i
\frac{
k\times\widehat\omega_k
}{
|k|^2
}
$$

for each nonzero Fourier mode。

則：

$$
\boxed{
\nabla\cdot u=0,
\qquad
\nabla\times u=\omega.
}
\tag{16.2}
$$

所以這是一個合法 smooth periodic incompressible velocity/vorticity datum。

因此 Section 15 的 transfer不是 arbitrary tensor stress才存在的 algebraic artifact。

---

# 17. High-frequency actual-realizability sharpness

對：

$$
N\in\mathbb N,
$$

scale frequencies：

$$
\boxed{
p_N=Np,
\qquad
q_N=Nq,
\qquad
r_N=Nr.
}
\tag{17.1}
$$

並取 vorticity amplitudes：

$$
\boxed{
a_N=Na,
\qquad
b_N=Nb,
\qquad
c_N=Nc.
}
\tag{17.2}
$$

則 corresponding velocity Fourier amplitudes：

$$
\widehat u_{p_N},
\qquad
\widehat u_{q_N},
\qquad
\widehat u_{r_N}
$$

保持：

$$
O(1).
$$

input/output stress amplitudes：

$$
\boxed{
B_N=N^2B,
\qquad
C_N=N^2C.
}
\tag{17.3}
$$

而 transport frequency factor：

$$
\boxed{
i
(
\widehat u_{r_N}
\cdot
\ell_N
)
=
N.
}
\tag{17.4}
$$

所以 triad transfer contribution：

$$
\boxed{
|\mathcal X_N^{\rm triad}|
\asymp
N^5.
}
\tag{17.5}
$$

while：

$$
\boxed{
\|W_{T,\ell_N}\|
\,
\|W_{L,m_N}\|
\asymp
N^4.
}
\tag{17.6}
$$

因此 normalized transfer rate：

$$
\boxed{
\frac{
|\mathcal X_N^{\rm triad}|
}{
\|W_{T,\ell_N}\|
\|W_{L,m_N}\|
}
\asymp
N.
}
\tag{17.7}
$$

所以：

$$
\boxed{
\textbf{
one full transport derivative survives even under actual quadratic vorticity-stress realizability.
}
}
\tag{17.8}
$$

---

# 18. The pointwise cone does not lower the Fourier endpoint

Round 43 的希望是：

$$
W(x)\in\mathcal M_\omega
$$

might shrink the generic divdiv-free transfer class。

Round 44 shows：

1. actual vorticity modes generate exactly invisible stress coefficients；
2. actual cross coefficients can leave $\mathcal M_\omega$ modewise；
3. actual velocity transport shifts invisible stress into visible directions；
4. actual vorticity modes at the shifted frequency can supply matching visible stress；
5. the normalized transfer keeps one derivative at high frequency。

因此：

$$
\boxed{
\textbf{
pointwise axisymmetric realizability does not by itself improve
the Fourier visible–invisible transfer endpoint.
}
}
\tag{18.1}
$$

---

# 19. Why pointwise realizability and Fourier realizability differ

the cone condition：

$$
54(\det W(x))^2
=
|W(x)|^6
$$

is nonlinear in：

$$
W.
$$

Fourier transform converts pointwise multiplication into convolution。

因此 it does not commute with the algebraic cone constraint：

$$
\boxed{
\mathcal F[
\mathcal M_\omega
]
\neq
\mathcal M_\omega
\text{ coefficientwise}.
}
\tag{19.1}
$$

This is the structural reason cross-mode coefficients deconfine。

---

# 20. Modewise visibility can be zero without vorticity vanishing

Section 6 helical self-mode gives：

$$
\boxed{
\widehat W_{2p}\ne0,
\qquad
\mathbb P_L(2p)\widehat W_{2p}=0.
}
\tag{20.1}
$$

所以：

$$
\boxed{
\text{nonzero vorticity stress}
\not\Rightarrow
\text{positive Riesz visibility}.
}
$$

Section 15 simultaneously gives an actual mixed state with nonzero visible/invisible transfer。

所以 simple lower bounds：

$$
0<\eta_\ast
\le
\eta_\omega
$$

cannot come from polarization algebra alone。

---

# 21. Quadratic realizability is flexible in Fourier space

the admissible bilinear amplitude map：

$$
(a,b)
\mapsto
\mathcal B(a,b)
$$

already spans：

$$
\mathbb S_0.
$$

Combined with:

$$
p\cdot a=0,
\qquad
q\cdot b=0,
$$

this means divergence-free quadratic convolution retains substantial tensor flexibility。

This does not imply arbitrary stress Fourier data can be prescribed independently at all frequencies，

because different coefficients share the same underlying vorticity modes。

But it rules out a simple coefficientwise axisymmetric-cone rigidity argument。

---

# 22. Static realizability is not the remaining depletion mechanism

after Round 44：

$$
\boxed{
\text{generic divdiv flexibility}
}
$$

and：

$$
\boxed{
\text{actual quadratic vorticity realizability}
}
$$

both permit one-derivative visible/invisible transfer。

Therefore remaining depletion must be dynamical / cumulative：

- quartic vorticity-stress diffusion；
- alignment selection；
- phase persistence；
- visibility fraction dynamics；
- energy transfer between $W_L/W_T$ constrained by total stress budget。

So the proof frontier returns to dynamics rather than static realizability。

---

# 23. Visibility ratio is now a dynamic variable, not an algebraic barrier

Round 42：

$$
\boxed{
\eta_\omega
=
\frac{
\|W_L\|_2^2
}{
\|W\|_2^2
}
=
\frac{
36
\|\mathfrak V_\omega\|_2^2
}{
\|\omega\|_4^4
}.
}
\tag{23.1}
$$

Round 44 shows neither：

$$
\eta_\omega=0
$$

nor mixed：

$$
0<\eta_\omega<1
$$

is algebraically excluded by vorticity stress structure。

So next question is：

$$
\boxed{
\textbf{
how does }\eta_\omega(t)\textbf{ evolve under stretching, diffusion and transfer?}
}
$$

---

# 24. STOP-C48 — Actual-Vorticity Triad / Dynamic-Only Depletion Gap

$$
\boxed{
\begin{aligned}
\text{layer}
&=
\mathrm{actual\ vorticity\text{-}stress\ realizability},
\\
\text{single linear polarization}
&=
\mathrm{self\ harmonic\ visible},
\\
\text{helical/null polarization}
&=
\mathrm{self\ harmonic\ invisible},
\\
\text{cross-mode invisibility}
&=
3(q\cdot a)(p\cdot b)
=
(a\cdot b)|p+q|^2,
\\
\text{cross-stress span}
&=
\mathbb S_0,
\\
\text{pointwise axisymmetric cone}
&\not\Rightarrow
\text{coefficientwise Fourier cone},
\\
\text{actual periodic transfer triad}
&\ne0,
\\
\text{high-frequency normalized transfer}
&\asymp
N,
\\
\text{one transport derivative}
&=
\mathrm{sharp\ under\ actual\ realizability},
\\
\text{static realizability closure}
&=
\mathrm{refuted},
\\
\text{missing}
&=
\mathrm{dynamic\ control\ of\ visibility\ ratio,
quartic\ stress\ alignment,\ diffusion,\ and\ transfer\ persistence},
\\
T_{\mathsf C\to\mathsf D}
&=
\mathrm{NOT\ REACHED}.
\end{aligned}
}
$$

命名：

$$
\boxed{
\textbf{STOP-C48:
Actual-Vorticity Triad / Dynamic-Only Depletion Gap}.
}
$$

---

# 25. 24/72 Ledger — Round 44

| Step | object | $B$ | $U$ | $O$ | $L$ | status |
|---|---|---|---|---|---|---|
| C679 | divergence-free vorticity Fourier space | $\mathsf C$ | Fourier continuum/torus modes | relational | $\mathsf F$ | STANDARD |
| C680 | quadratic stress convolution | $\mathsf C$ | bilinear convolution | tensor | $\mathsf F$ | EXACT |
| C681 | frequency-direction visibility law | $\mathsf C$ | Riesz projection | scalar | $\mathsf F$ | EXACT |
| C682 | cross-mode invisibility condition | $\mathsf C$ | algebraic geometry | targeted | $\mathsf F$ | PROVED |
| C683 | single-mode polarization dichotomy | $\mathsf C$ | complex polarization | targeted | $\mathsf F$ | PROVED |
| C684 | helical invisible self-stress | $\mathsf C$ | divergence-free polarization | targeted | $\mathsf F$ | CONSTRUCTED |
| C685 | cross-stress span theorem | $\mathsf C$ | bilinear tensor span | relational | $\mathsf F$ | PROVED |
| C686 | Fourier cone deconfinement | $\mathsf C$ | nonlinear convolution | $\mathsf X$ | $\mathsf F$ | PROVED |
| C687 | explicit invisible input coefficient | $\mathsf C$ | vorticity triad | tensor | $\mathsf F$ | CONSTRUCTED |
| C688 | off-cone Fourier invariant witness | $\mathsf C$ | tensor invariants | targeted | $\mathsf F$ | PROVED |
| C689 | actual transport velocity | $\mathsf C$ | Biot–Savart inversion | relational | $\mathsf F$ | EXACT |
| C690 | shifted visibility | $\mathsf C$ | frequency geometry | targeted | $\mathsf F$ | PROVED |
| C691 | actual visible output stress | $\mathsf C$ | quadratic convolution | tensor | $\mathsf F$ | CONSTRUCTED |
| C692 | exact nonzero actual transfer | $\mathsf C$ | projection commutator | targeted | $\mathsf F$ | PROVED |
| C693 | high-frequency actual sharpness | $\mathsf C$ | frequency dilation | scalar | $\mathsf F$ | PROVED |
| C694 | static realizability endpoint closure | $\mathsf C$ | algebraic stress cone | targeted | $\mathsf F$ | REFUTED |
| C695 | dynamic visibility route | $\mathsf C$ | stress evolution | targeted | $\mathsf F$ | OPEN / STOP-C48 |

---

# 26. Continuous-versus-discrete status

本輪使用 periodic Fourier modes作 exact algebraic witness。

但 proof carriers依然可以表示為 continuous：

- wavevector：
  $$
  k\in\mathbb R^3;
  $$
- transverse polarization plane：
  $$
  k^\perp;
  $$
- quadratic convolution：
  $$
  p+q=K;
  $$
- frequency-direction projection：
  $$
  n=K/|K|.
  $$

integer torus wavevectors只是 convenient periodic witness representation，

不是 essential proof substrate。

相同 triad symbol可在 continuous Fourier variables / wave packets下表達。

因此：

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

---

# 27. Strongest results of Round 44

## R44-A — cross-stress invisibility condition

$$
\boxed{
3
(q\cdot a)
(p\cdot b)
=
(a\cdot b)
|p+q|^2.
}
$$

## R44-B — actual invisible helical stress

$$
\boxed{
p=e_3,
\quad
a=e_1+ie_2
\Rightarrow
\mathbb P_L(2p)
\left(
a\otimes a
\right)
=0.
}
$$

## R44-C — Fourier cone deconfinement

$$
\boxed{
W(x)\in\mathcal M_\omega
\ \forall x
\not\Rightarrow
\widehat W_K\in\mathcal M_\omega.
}
$$

## R44-D — exact actual transfer witness

for the explicit divergence-free triad：

$$
\boxed{
\left\langle
\mathbb P_L(m)C,
-
\mathbb P_L(m)B
\right\rangle
=
-\frac15.
}
$$

## R44-E — actual high-frequency one-derivative sharpness

$$
\boxed{
\frac{
|\mathcal X_N^{\rm triad}|
}{
\|W_{T,\ell_N}\|
\|W_{L,m_N}\|
}
\asymp
N.
}
$$

So nonlinear vorticity-stress realizability does not remove the critical transport derivative.

---

# 28. Next round — Visibility Replicator / Quartic Alignment Dynamics

Round 44 closes the static realizability hope in the negative direction。

下一輪直接 use Round 42 projected energy equations to derive：

$$
\boxed{
\eta_\omega'
}
$$

exactly。

Core questions：

1. define：
   $$
   E_L=\|W_L\|_2^2,
   \quad
   E_T=\|W_T\|_2^2,
   \quad
   E=E_L+E_T;
   $$

2. derive：
   $$
   \eta_\omega=E_L/E;
   $$

3. separate visibility selection due：
   - stretching；
   - stress diffusion；
   - commutator transfer：
     $$
     \mathcal X_\omega;
     $$

4. normalize quartic vorticity measure：
   $$
   d\mu_{\omega,4}
   =
   |\omega|^4
   \|\omega\|_4^{-4}dx;
   $$

5. compare visible-sector growth against total quartic alignment：
   $$
   \lambda_\omega=\xi^\top S\xi;
   $$

6. ask whether mixed visibility is dynamically attracted to $0$, $1$, or interior states；

7. if transfer is only redistribution, use total quartic budget to cap cumulative Piola-defect exposure；

8. continue entirely in continuous projected stress energy variables。

---

# 29. External primary-source anchors

1. Holger R. Dullin, James D. Meiss, Joachim Worthington, *Poisson Structure of the Three-Dimensional Euler Equations in Fourier Space*, arXiv:1812.09709.
   - formulates 3D periodic Euler vorticity dynamics on the divergence-free Fourier subspace；
   - explicitly uses
     $$
     k\cdot\widehat\omega_k=0
     $$
     and
     $$
     \widehat u_k
     =
     i
     \frac{
     k\times\widehat\omega_k
     }{
     |k|^2
     }.
     $$

2. Evan Miller, *On the interaction of strain and vorticity for solutions of the Navier--Stokes equation*, arXiv:2407.02691.
   - treats $\omega\otimes\omega$ as a central strain–vorticity interaction object and proves exact depletion identities for divergence-free velocity fields.

3. Tristan Buckmaster, Vlad Vicol, *Nonuniqueness of weak solutions to the Navier-Stokes equation*, arXiv:1709.10033.
   - primary-source background showing that highly oscillatory divergence-free structures can play a decisive role in Navier–Stokes constructions；used only as broad oscillatory-flow context, not as a source for the explicit Round 44 triad.

本輪 cross-stress invisibility formula、helical invisible harmonic、Cross-Stress Span Theorem、Fourier Cone Deconfinement witness、explicit actual-vorticity transfer triad與 high-frequency one-derivative sharpness均為本文直接推導。

---

# 30. Commit state

$$
\boxed{
\begin{aligned}
\text{Route}
&=
\mathrm{Pure\ Continuous\ Actual\ Vorticity\text{-}Stress\ Realizability},
\\
\text{Essential }\mathsf C\to\mathsf D
&=
\mathrm{Not\ reached},
\\
\text{Actual invisible stress}
&=
\mathrm{exists},
\\
\text{Pointwise cone rigidity}
&=
\mathrm{not\ coefficientwise\ Fourier\ rigidity},
\\
\text{Actual visible/invisible transfer}
&=
\mathrm{nonzero},
\\
\text{Actual high-frequency transfer}
&=
\mathrm{one\text{-}derivative\ sharp},
\\
\text{Static realizability depletion}
&=
\mathrm{refuted},
\\
\text{Remaining route}
&=
\mathrm{dynamic\ visibility}
+
\mathrm{quartic\ alignment}
+
\mathrm{diffusion},
\\
\text{STOP-C48}
&=
\mathrm{Actual\text{-}Vorticity\ Triad/Dynamic\text{-}Only\ Depletion\ Gap},
\\
\text{Next}
&=
\mathrm{Visibility\ Replicator/Quartic\ Alignment\ Dynamics}.
\end{aligned}
}
$$