# NS × X 積分 × 24/72 範式實戰
## Round 32 — Pure Continuous Source-Participation Dynamics / Singular-Source Renormalization Route

- 日期：2026-08-17
- 版本：v0.1
- 狀態：Proof-Route Experiment / Continuous-Only Source-Participation Branch
- canonical source：UTF-8 Markdown
- canonical math delimiters：inline `$...$`；display `$$...$$`
- 前一輪：`NS_X72_Round31_PureContinuous_PersistentLock_OccupancyCapacity_v0.1_2026-08-17.md`
- 本輪目標：Round 31 已把 persistent-lock occupancy 問題壓成 source participation ratio $\mathfrak J_W$。本輪建立任意 smooth positive source 相對 critical mass 的 exact participation dynamics，並分別套到 determinant source、positive $Q$-growth source與 nonlocal pair source。重點是辨識：哪一些 participation 可被 Fisher mixing 壓制，哪一些因 sign interface、zero set 或 singular kernel 必須先 renormalize。
- 非主張：本文沒有證明 determinant / $Q$-growth / pair participation 無條件有界。本文證明 smooth-positive-source branch 具有 universal anti-concentration structure；剩餘 obstruction 是 source-specific relative production、moving sign interface 與 singular-kernel cancellation。

---

# 0. Round 31 handoff

Round 31 定義：

$$
\boxed{
\mathfrak J_W
=
\frac{\mathbb E_\mu[W^2]}{\mathbb E_\mu[W]^2}
}
$$

若 lock region $L$ 承擔至少 $\beta$ 比例 source，則：

$$
\boxed{
\mu(L)\ge \frac{\beta^2}{\mathfrak J_W}.
}
$$

所以 vanishing occupancy 若仍保持 fixed source fraction，必須：

$$
\boxed{
\mathfrak J_W\to\infty
}
$$

或 source 相對 carrier measure 變成 singular。

Round 31 STOP：

$$
\boxed{
\text{STOP-C35}
=
\text{Persistent-Lock Occupancy / Singular-Concentration Gap}.
}
$$

---

# 1. Base critical-mass equation

Round 21 critical-mass probability density $m=m_Q$ 滿足：

$$
\boxed{
\partial_t m+\operatorname{div}(bm)
=
\nu\Delta m+s\,m,
}
\tag{1.1}
$$

其中：

$$
\boxed{
s=3(G_Q-\overline G_Q),
}
\tag{1.2}
$$

且：

$$
\boxed{
\mathbb E_{\mu_0}[s]=0.
}
$$

本輪記：

$$
d\mu_0=m\,dx.
$$

---

# 2. Generic positive source and continuous source tilt

令 $W(x,t)>0$ smooth，且：

$$
0<Z_1(t)=\mathbb E_{\mu_0}[W]<\infty.
$$

定義 source-weighted probability：

$$
\boxed{
d\mu_1=\frac{W}{Z_1}d\mu_0.
}
\tag{2.1}
$$

再定義：

$$
\boxed{
Z_p=\mathbb E_{\mu_0}[W^p],
}
\tag{2.2}
$$

及 continuous tilt：

$$
\boxed{
d\mu_p=\frac{W^p}{Z_p}d\mu_0,
\qquad p\ge0.
}
\tag{2.3}
$$

participation ratio：

$$
\boxed{
\mathfrak J_W=\frac{Z_2}{Z_1^2}.
}
\tag{2.4}
$$

---

# 3. Exact source relative-rate operator

令：

$$
D_b=\partial_t+b\cdot\nabla.
$$

對 $\zeta_W=Wm$ 直接 product rule：

$$
\boxed{
\partial_t\zeta_W+\operatorname{div}(b\zeta_W)
=
\nu\Delta\zeta_W+(s+\mathcal R_W)\zeta_W,
}
\tag{3.1}
$$

其中：

$$
\boxed{
\mathcal R_W
=
\frac{D_bW-\nu\Delta W-2\nu\nabla\log m\cdot\nabla W}{W}.
}
\tag{3.2}
$$

若 $L_W=\log W$，則：

$$
\boxed{
\mathcal R_W
=
D_bL_W-\nu\Delta L_W-\nu|\nabla L_W|^2
-2\nu\nabla\log m\cdot\nabla L_W.
}
\tag{3.3}
$$

---

# 4. Normalized source-measure equation

由：

$$
Z_1'
=
Z_1\langle s+\mathcal R_W\rangle_1,
$$

得到：

$$
\boxed{
\begin{aligned}
\partial_t m_1+\operatorname{div}(bm_1)
={}&\nu\Delta m_1\\
&+\left[s+\mathcal R_W-\langle s+\mathcal R_W\rangle_1\right]m_1.
\end{aligned}
}
\tag{4.1}
$$

所以 $\mu_0$ 與 $\mu_1$ share same deterministic drift and viscosity，只在 relative selection 上不同。

---

# 5. Universal Source-Participation Dynamics

令：

$$
\boxed{
f=\frac{d\mu_1}{d\mu_0}=\frac{W}{Z_1}.
}
$$

則：

$$
\mathfrak J_W=\int f^2d\mu_0.
$$

direct calculation 給：

$$
\boxed{
\begin{aligned}
\frac d{dt}\log\mathfrak J_W
={}&-2\nu\langle|\nabla\log W|^2\rangle_2\\
&+\left[\langle s\rangle_2-2\langle s\rangle_1+\langle s\rangle_0\right]\\
&+2\left[\langle\mathcal R_W\rangle_2-\langle\mathcal R_W\rangle_1\right].
\end{aligned}
}
\tag{5.1}
$$

命名：

$$
\boxed{
\textbf{Universal Source-Participation Dynamics}.
}
$$

---

# 6. Universal viscous anti-concentration

Equation (5.1) 第一項：

$$
\boxed{
-2\nu\langle|\nabla\log W|^2\rangle_2\le0.
}
\tag{6.1}
$$

所以：

$$
\boxed{
\textbf{common viscosity always opposes source-measure separation
for every smooth positive source }W.
}
$$

source participation 要增長，必須由 base selection curvature 或 source-specific relative production bias 打敗 Fisher smoothing。

---

# 7. Continuous source-tilt calculus

令 $L_W=\log W$。對不顯式依賴 $p$ 的 observable $A$：

$$
\boxed{
\frac d{dp}\langle A\rangle_p
=
\operatorname{Cov}_{\mu_p}(A,L_W).
}
\tag{7.1}
$$

再微分：

$$
\boxed{
\frac{d^2}{dp^2}\langle A\rangle_p
=
\operatorname{Cov}_{\mu_p}
\left(A,(L_W-\langle L_W\rangle_p)^2\right).
}
\tag{7.2}
$$

因此：

$$
\boxed{
\begin{aligned}
\langle s\rangle_2-2\langle s\rangle_1+\langle s\rangle_0
=
\int_0^1\int_\tau^{\tau+1}
\operatorname{Cov}_{\mu_\sigma}
\left(s,(L_W-\langle L_W\rangle_\sigma)^2\right)
d\sigma d\tau.
\end{aligned}
}
\tag{7.3}
$$

以及：

$$
\boxed{
\langle\mathcal R_W\rangle_2-\langle\mathcal R_W\rangle_1
=
\int_1^2
\operatorname{Cov}_{\mu_p}(\mathcal R_W,L_W)dp.
}
\tag{7.4}
$$

所以 participation dynamics 仍完全位於 continuous moment-order axis。

---

# 8. Participation–variance bound

有：

$$
\boxed{
\mathfrak J_W-1
=
\chi^2(\mu_1\|\mu_0)
=
\chi^2(\mu_1\|\mu_2).
}
\tag{8.1}
$$

所以：

$$
\boxed{
|\langle A\rangle_1-\langle A\rangle_0|
\le
\sigma_0(A)\sqrt{\mathfrak J_W-1},
}
\tag{8.2}
$$

$$
\boxed{
|\langle A\rangle_2-\langle A\rangle_1|
\le
\sigma_2(A)\sqrt{\mathfrak J_W-1}.
}
\tag{8.3}
$$

定義：

$$
\boxed{
\mathcal A_W
=
\sigma_2(s)+\sigma_0(s)+2\sigma_2(\mathcal R_W),
}
\tag{8.4}
$$

得到：

$$
\boxed{
(\log\mathfrak J_W)'
\le
-2\nu I_W
+\mathcal A_W\sqrt{\mathfrak J_W-1},
}
\tag{8.5}
$$

其中：

$$
\boxed{
I_W=\langle|\nabla\log W|^2\rangle_2.
}
$$

---

# 9. Generic Poincaré trapping branch

若 $\mu_0$ 滿足：

$$
\operatorname{Var}_{\mu_0}(g)
\le
C_P\int|\nabla g|^2d\mu_0,
$$

取 $g=f=W/Z_1$：

$$
\boxed{
\mathfrak J_W-1
\le
C_P\mathfrak J_WI_W.
}
\tag{9.1}
$$

所以：

$$
\boxed{
(\log\mathfrak J_W)'
\le
-\frac{2\nu}{C_P}
\frac{\mathfrak J_W-1}{\mathfrak J_W}
+\mathcal A_W\sqrt{\mathfrak J_W-1}.
}
\tag{9.2}
$$

令 $y_W=\sqrt{\mathfrak J_W-1}$：

$$
\boxed{
y_W'
\le
-\frac{\nu}{C_P}y_W
+\frac12(1+y_W^2)\mathcal A_W.
}
\tag{9.3}
$$

所以 Round 23 的 intermittency trap 是任意 smooth positive source 的 generic participation trap。

---

# 10. Conditional closure of the Round 31 smooth-source branch

若：

$$
C_P(t)\le C_\ast,
$$

及：

$$
\mathcal A_W(t)\le a_\ast<\nu/C_\ast,
$$

則相應 Riccati barrier 給 finite participation trapping basin。

因此：

$$
\boxed{
\text{persistent source dominance}
+
\text{Poincaré mixing}
+
\text{bounded relative-source variance}
\Rightarrow
\text{positive occupancy and bulk chargeability}.
}
$$

Round 31 trace/occupancy gap在 smooth positive source branch被條件式封閉。

---

# 11. Trace-free cofactor identity

令 $S$ 為 trace-free symmetric $3\times3$ tensor。

Cayley–Hamilton 給：

$$
\boxed{
\operatorname{cof}S
=
S^2-\frac12|S|^2I.
}
\tag{11.1}
$$

且 trace-free $3\times3$ spectrum滿足：

$$
\boxed{
\operatorname{tr}(S^4)
=
\frac12|S|^4.
}
\tag{11.2}
$$

因此：

$$
\boxed{
\operatorname{cof}S:S^2=0.
}
\tag{11.3}
$$

---

# 12. Exact material derivative of $-\det S$

令：

$$
d=-\det S.
$$

使用：

$$
D_t\det S
=
\operatorname{cof}S:D_tS
$$

與 strain equation：

$$
D_tS
=
\nu\Delta S-S^2-rac14\omega\otimes\omega+rac14|\omega|^2I-H_p,
$$

得到：

$$
\boxed{
D_t(-\det S)
=
-\nu\operatorname{cof}S:\Delta S
+\frac14|S\omega|^2
+\operatorname{cof}S:H_p.
}
\tag{12.1}
$$

這是本輪最重要的 NS-specific identity。

---

# 13. Direct self-amplification cancellation

Equation (12.1) 中 $-S^2$ 沒有 direct contribution。

所以：

$$
\boxed{
\textbf{the strain self-amplification term does not directly change }-\det S
\textbf{ at the instantaneous material-derivative level}.
}
$$

determinant material drivers變成：

1. higher derivative：$-\nu\operatorname{cof}S:\Delta S$；
2. nonnegative vorticity coupling：$\frac14|S\omega|^2$；
3. signed pressure coupling：$\operatorname{cof}S:H_p$。

這再次回到 Round 04/05/18 的 obstruction core。

---

# 14. Determinant participation Fisher geometry

在 active dangerous branch：

$$
d=-\det S>0,
\qquad
r>0,
$$

Round 31 determinant source density：

$$
\boxed{
W_D=\frac d{r^3}=a_DK^3,
}
\tag{14.1}
$$

其中：

$$
K=\frac{|S|}{r},
\qquad
a_D=\frac d{|S|^3}.
$$

所以：

$$
\boxed{
\nabla\log W_D
=
3\nabla\log K+\nabla\log a_D.
}
\tag{14.2}
$$

其 universal Fisher tax：

$$
\boxed{
-2\nu
\left\langle
|3\nabla\log K+\nabla\log a_D|^2
\right\rangle_{D,2}.
}
\tag{14.3}
$$

所以 determinant concentration同時需要處理 normalized-rate intermittency與 spectral-shape intermittency。

---

# 15. Determinant source relative-rate core

令：

$$
L_D=\log d-3\log r.
$$

則：

$$
\boxed{
\mathcal R_D
=
D_bL_D-\nu\Delta L_D-\nu|\nabla L_D|^2
-2\nu\nabla\log m_Q\cdot\nabla L_D.
}
\tag{15.1}
$$

且在 $d>0$：

$$
\boxed{
D_t\log d
=
\frac{
-\nu\operatorname{cof}S:\Delta S
+\frac14|S\omega|^2
+\operatorname{cof}S:H_p
}{d}.
}
\tag{15.2}
$$

所以 determinant participation production重新要求：

- cofactor-weighted higher derivative；
- $|S\omega|^2/d$；
- pressure/cofactor ratio；
- quotient amplitude/gauge derivatives。

---

# 16. Determinant sign interface

global source：

$$
D=(-\det S)_+
$$

在 $\det S=0$ 有 moving sign interface。

直接 $\log D$ 不合法。

可取 smooth positive regularization：

$$
\boxed{
D_\varepsilon
=
\frac12
\left[-\det S+\sqrt{(\det S)^2+\varepsilon^2}\right]
+\varepsilon.
}
\tag{16.1}
$$

對每個 $\varepsilon>0$，generic participation law合法。

真正 global determinant closure需要研究：

$$
\boxed{
\varepsilon\downarrow0
}
$$

時 sign-interface Fisher / relative-source terms是否 uniform。

---

# 17. Positive $Q$-growth source has the same interface problem

Round 31 使用：

$$
W_{G+}=G_+.
$$

可 regularize：

$$
\boxed{
W_{G,\varepsilon}
=
\frac12\left(G+\sqrt{G^2+\varepsilon^2}\right)+\varepsilon.
}
\tag{17.1}
$$

則：

$$
\boxed{
\nabla\log W_{G,\varepsilon}
=
\frac{W_{G,\varepsilon}'(G)}{W_{G,\varepsilon}(G)}\nabla G.
}
\tag{17.2}
$$

在 $G\approx0$ 且 $\varepsilon\downarrow0$ 時可變大。

所以 positive-growth participation自然產生：

$$
\boxed{
\text{source-interface Fisher layer}.
}
$$

要利用它，需要控制 $\nabla G$，而 $G_Q=\gamma_Q-\nu K_D$ 的 gradient已包含更高 derivatives，因此又回到 hierarchy obstruction。

---

# 18. Product-space participation theorem

對 product critical mass：

$$
\boxed{
dM_0=d\mu_Q(x)d\mu_Q(y),
}
\tag{18.1}
$$

其 density obey：

$$
\boxed{
\begin{aligned}
\partial_tM_0
&+\operatorname{div}_x(b_xM_0)+\operatorname{div}_y(b_yM_0)\\
&=\nu(\Delta_x+\Delta_y)M_0+[s(x)+s(y)]M_0.
\end{aligned}
}
\tag{18.2}
$$

所以任意 smooth positive pair source $W(x,y,t)$ 具有同一 generic participation theorem。

其 Fisher term：

$$
\boxed{
-2\nu
\left\langle
|\nabla_x\log W|^2+|\nabla_y\log W|^2
\right\rangle_2.
}
\tag{18.3}
$$

所以 separated-pair source concentration也被 common viscosity反向混合。

---

# 19. Refinement of Round 31 pair occupancy

Round 31 假設 positive pair source：

$$
W_{\rm pair}=\mathcal C_+
$$

屬於 $L^2(\mu_Q\otimes\mu_Q)$。

本輪必須補一個重要限制：raw global Biot–Savart strain / pressure-Hessian pair kernel近對角線 $R=|x-y|\downarrow0$ 時是 $R^{-3}$ 級 signed singular kernel。

其合法 operator meaning依賴：

- angular zero mean；
- principal-value / Calderón–Zygmund cancellation。

直接先取 positive part會破壞 cancellation。

---

# 20. Positive-Pair Cancellation-Destruction No-Go

在 generic nonvanishing angular sector若：

$$
W_{\rm pair}^+\sim R^{-3},
$$

三維 relative coordinate volume：

$$
dz\sim R^2dR\,d\Omega.
$$

所以 first absolute pair mass：

$$
\boxed{
\int_0^\delta R^{-3}R^2dR
=
\int_0^\delta\frac{dR}{R}
=
\infty.
}
\tag{20.1}
$$

second moment更強：

$$
\boxed{
\int_0^\delta R^{-6}R^2dR
=
\int_0^\delta R^{-4}dR
=
\infty.
}
\tag{20.2}
$$

命名：

$$
\boxed{
\textbf{Positive-Pair Cancellation-Destruction No-Go}.
}
$$

所以 raw global $\mathfrak J_{\rm pair}$ 不能天真地由 positive part singular kernel定義。

---

# 21. Legal pair routes

目前 pair participation有三條合法路徑。

## P1 — separated-region route

若：

$$
|x-y|\ge R_0>0,
$$

kernel smooth bounded，Round 31/32 pair participation theorem合法。

## P2 — truncated route

定義：

$$
\boxed{
W_\delta
=
\mathbf1_{\{|x-y|>\delta\}}W.
}
\tag{21.1}
$$

先研究 $\mathfrak J_{W_\delta}$，再檢驗 $\delta\downarrow0$。

## P3 — signed principal-value route

不取 positive part，保留 signed kernel cancellation，改研究 signed covariance / total variation / renormalized source functional。

這會接到下一輪。

---

# 22. Positive measure and singular-integral cancellation are structurally different

Round 31 occupancy lemma需要：

$$
W\ge0
$$

才能把 source fraction解讀成 probability participation。

但 singular integral合法性依賴：

$$
\boxed{
\text{sign cancellation}.
}
$$

所以：

$$
\boxed{
\textbf{positive-source measure language and singular-integral cancellation
are not automatically compatible.}
}
$$

這不是 $\mathsf C\to\mathsf D$。

它是：

$$
\boxed{
\text{positive measure representation}
\leftrightarrow
\text{signed principal-value representation}
}
$$

之間的 losslessness 問題。

---

# 23. Smooth-source closure versus singular-source leakage

## Smooth positive source branch

若：

- $W>0$ smooth；
- $C_P$ 受控；
- $\mathcal R_W$ variance受控；

則：

$$
\boxed{
\mathfrak J_W
\text{ 有 conditional trapping mechanism}.
}
$$

Round 31 occupancy gap可封閉。

## Singular / sign-changing branch

以下仍會 leak：

- determinant sign interface $\det S=0$；
- positive growth interface $G_Q=0$；
- pair diagonal $x=y$；
- carrier zero set $r=0$。

---

# 24. Source singularization taxonomy

source participation blow-up目前可分：

$$
\boxed{
\begin{aligned}
\mathrm{S1}:&\quad
\text{smooth-source tilt bias beats Fisher mixing},\\
\mathrm{S2}:&\quad
\text{moving sign interface creates singular log-gradient},\\
\mathrm{S3}:&\quad
\text{carrier zero set makes source density singular},\\
\mathrm{S4}:&\quad
\text{positive extraction destroys singular-kernel cancellation}.
\end{aligned}
}
\tag{24.1}
$$

Round 31 的 singular concentration現在被細分成這四種 continuous leakage。

---

# 25. STOP-C36 — Source-Participation Trapping / Singular-Source Renormalization Gap

$$
\boxed{
\begin{aligned}
\text{layer}
&=\mathrm{source\ participation\ dynamics},\\
\text{generic smooth source}
&=W>0,\\
\text{participation law}
&=-2\nu\text{ Fisher}+\text{tilt selection}+\text{relative source bias},\\
\text{Poincaré branch}
&=\text{conditional trapping},\\
\text{determinant material derivative}
&=-\nu\operatorname{cof}S:\Delta S+\frac14|S\omega|^2+\operatorname{cof}S:H_p,\\
\text{direct }-S^2\text{ determinant derivative}
&=0,\\
\text{positive-growth source}
&=\text{moving sign-interface Fisher problem},\\
\text{pair source}
&=\text{positive-part extraction can destroy principal-value cancellation},\\
\text{smooth-source Round31 occupancy}
&=\text{conditionally closed},\\
\text{missing}
&=\text{uniform control / renormalization across sign interfaces, zeros and singular diagonals},\\
T_{\mathsf C\to\mathsf D}
&=\text{NOT REACHED}.
\end{aligned}
}
$$

命名：

$$
\boxed{
\textbf{STOP-C36:
Source-Participation Trapping / Singular-Source Renormalization Gap}.
}
$$

---

# 26. 24/72 Ledger — Round 32

| Step | object | $B$ | $U$ | $O$ | $L$ | status |
|---|---|---|---|---|---|---|
| C453 | generic source tilt $\mu_p$ | $\mathsf C$ | continuous measure | profile | $\mathsf F$ | FORM |
| C454 | generic relative-source operator | $\mathsf C$ | drift/diffusion | relational | $\mathsf F$ | EXACT |
| C455 | normalized source-measure PDE | $\mathsf C$ | selection/diffusion | measure | $\mathsf F$ | EXACT |
| C456 | Universal Source-Participation Dynamics | $\mathsf C$ | measure coupling | scalar | $\mathsf F$ | EXACT |
| C457 | generic Fisher anti-concentration | $\mathsf C$ | viscosity | targeted | $\mathsf F$ | EXACT |
| C458 | source-tilt calculus | $\mathsf C$ | continuous $p$ | profile | $\mathsf F$ | EXACT |
| C459 | participation–variance bound | $\mathsf C$ | $\chi^2$ geometry | scalar | $\mathsf F$ | PROVED |
| C460 | generic Poincaré trap | $\mathsf C$ | spectral gap | targeted | $\mathsf F$ | CONDITIONAL |
| C461 | cofactor strain identity | $\mathsf C$ | algebraic | relational | $\mathsf F$ | EXACT |
| C462 | determinant material derivative | $\mathsf C$ | strain PDE | targeted | $\mathsf F$ | EXACT |
| C463 | self-amplification determinant cancellation | $\mathsf C$ | algebraic/PDE | targeted | $\mathsf F$ | EXACT |
| C464 | determinant Fisher geometry | $\mathsf C$ | source tilt | scalar | $\mathsf F$ | EXACT |
| C465 | determinant sign-interface regularization | $\mathsf C$ | smooth approximation | profile | $\mathsf F$ | FORM |
| C466 | $G_+$ interface regularization | $\mathsf C$ | smooth approximation | profile | $\mathsf F$ | FORM |
| C467 | positive-growth derivative escalation | $\mathsf C$ | hierarchy | targeted | $\mathsf F$ | IDENTIFIED |
| C468 | product-space participation theorem | $\mathsf C$ | pair diffusion | measure | $\mathsf F$ | EXACT |
| C469 | pair relative Fisher | $\mathsf C$ | product geometry | scalar | $\mathsf F$ | EXACT |
| C470 | raw positive pair integrability | $\mathsf C$ | singular kernel | targeted | $\mathsf F$ | REFUTED |
| C471 | separated/truncated pair routes | $\mathsf C$ | renormalization | relational | $\mathsf F$ | LEGAL |
| C472 | singular-source global closure | $\mathsf C$ | coupled NS | targeted | $\mathsf F$ | OPEN / STOP-C36 |

---

# 27. Continuous-versus-discrete status

本輪出現：

- source probability measures；
- continuous source tilt $p$；
- sign-interface regularization $\varepsilon>0$；
- pair truncation radius $\delta>0$；
- product-space diffusion；
- principal-value singular kernels。

全部仍是 continuous parameters與 continuous operators。

pair singular kernel的問題不是需要 graph / atom，而是：

$$
\boxed{
\text{signed cancellation不能被 losslessly 替換成 positive source measure}.
}
$$

因此：

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

---

# 28. Strongest results of Round 32

## R32-A — Universal Source-Participation Dynamics

$$
\boxed{
\begin{aligned}
(\log\mathfrak J_W)'
={}&-2\nu\langle|\nabla\log W|^2\rangle_2\\
&+[\langle s\rangle_2-2\langle s\rangle_1+\langle s\rangle_0]\\
&+2[\langle\mathcal R_W\rangle_2-\langle\mathcal R_W\rangle_1].
\end{aligned}
}
$$

## R32-B — Generic source participation trap

$$
\boxed{
(\log\mathfrak J_W)'
\le
-\frac{2\nu}{C_P}
\frac{\mathfrak J_W-1}{\mathfrak J_W}
+\mathcal A_W\sqrt{\mathfrak J_W-1}.
}
$$

## R32-C — Exact determinant evolution

$$
\boxed{
D_t(-\det S)
=
-\nu\operatorname{cof}S:\Delta S
+\frac14|S\omega|^2
+\operatorname{cof}S:H_p.
}
$$

## R32-D — Direct self-amplification determinant cancellation

$$
\boxed{
-S^2
\text{ contributes zero directly to }D_t(-\det S).
}
$$

## R32-E — Positive-pair cancellation-destruction no-go

$$
\boxed{
R^{-3}\text{ signed kernel may be principal-value legal, while its positive part has divergent near-diagonal pair mass}.
}
$$

---

# 29. Next round — Renormalized Signed Source Measures

Round 32 顯示 generic positive-source participation已經有完整 dynamic skeleton。

下一輪直接攻：

$$
\boxed{
\text{signed / singular sources}.
}
$$

核心問題：

1. determinant $d=-\det S$ 的 signed measure，而不是先取 positive part；
2. $G_Q$ 的 signed growth measure；
3. nonlocal pair kernel保留 principal value sign；
4. 定義 positive / negative source balance與 cancellation efficiency；
5. 研究 magnitude concentration 與 sign cancellation能否分離；
6. 對 pair singular kernel研究 truncated principal value $\delta\downarrow0$ 的 cancellation budget；
7. 若 signed cancellation可由 continuous total variation / Jordan-type measure處理，仍不需離散 source atoms。

---

# 30. External primary-source anchors

1. Evan Miller, *On the interaction of strain and vorticity for solutions of the Navier--Stokes equation*, arXiv:2407.02691.
   - strain–vorticity interaction、projected strain structure與 nonlinear depletion背景。

2. Borys Álvarez-Samaniego, Wilson P. Álvarez-Samaniego, Pedro G. Fernández-Dalgo, *On the use of the Riesz transforms to determine the pressure term in the incompressible Navier-Stokes equations on the whole space*, arXiv:2004.02588.
   - whole-space pressure的 Riesz-transform / singular-integral representation背景。

3. Patrick Cattiaux, Arnaud Guillin, Cyril Roberto, *Poincaré inequality and the $L^p$ convergence of semi-groups*, arXiv:1003.0784.
   - diffusion-type Markov semigroup中 Poincaré inequality與 $L^p$ convergence的 primary-source背景。

4. Guillaume Wang, Lénaïc Chizat, *Local convergence of mean-field Langevin dynamics: from gradient flows to linearly monotone games*, arXiv:2602.11999.
   - $\chi^2$ divergence、diffusive dynamics與 Poincaré control的近期 primary-source背景。

本輪 generic participation law、determinant material-derivative identity、direct self-amplification cancellation與 positive-pair cancellation-destruction no-go均為本文直接推導。

---

# 31. Commit state

$$
\boxed{
\begin{aligned}
\text{Route}
&=\mathrm{Pure\ Continuous\ Source\text{-}Participation\ Dynamics},\\
\text{Essential }\mathsf C\to\mathsf D
&=\mathrm{Not\ reached},\\
\text{Generic smooth positive source}
&=\mathrm{viscously\ anti\text{-}concentrated},\\
\text{Poincaré branch}
&=\mathrm{conditionally\ trapped},\\
\text{Round31 occupancy gap}
&=\mathrm{conditionally\ closed\ for\ smooth\ positive\ sources},\\
\text{Determinant dynamics}
&=\mathrm{vorticity}+\mathrm{pressure}+\mathrm{higher\ derivative},\\
\text{Direct }-S^2\text{ determinant growth}
&=0,\\
\text{Positive growth / determinant}
&=\mathrm{sign\text{-}interface\ renormalization},\\
\text{Pair source}
&=\mathrm{principal\text{-}value\ cancellation\ required},\\
\text{STOP-C36}
&=\mathrm{Source\text{-}Participation\ Trapping/Singular\text{-}Source\ Renormalization\ Gap},\\
\text{Next}
&=\mathrm{Renormalized\ Signed\ Source\ Measures}.
\end{aligned}
}
$$