Partial feature flows have exact energy and weight-moment identities

The first derivatives of $\varphi ,W$ are bounded. This follows from compactness for bounded $\mathrm{\Theta }$; on the full Euclidean domain, boundedness and Lipschitz differential imply $\mathrm{\parallel }D\varphi (\theta )e\mathrm{\parallel }\le 2\mathrm{\parallel }\varphi {\mathrm{\parallel }}_{\mathrm{\infty }}+\mathrm{Lip}(D\varphi )\mathrm{/}2$ for unit $e$, by Taylor's inequality on a unit segment. The scalar bound is identical. On a finite horizon, W2 continuity bounds the second moments of ${\mu }_t$. Coupling Cauchy–Schwarz and $\mathrm{\parallel }w\varphi (\theta )-w^{\mathrm{\prime }}\varphi ({\theta }^{\mathrm{\prime }})\mathrm{\parallel }\le \mathrm{\parallel }\varphi {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }w-w^{\mathrm{\prime }}\mathrm{\mid }+\mathrm{\parallel }D\varphi {\mathrm{\parallel }}_{\mathrm{\infty }}\mathrm{\mid }{w}^{\mathrm{\prime }}\mathrm{\mid }\mathrm{\mid }\theta -{\theta }^{\mathrm{\prime }}\mathrm{\mid }$ show norm continuity of $m_t$; hence $f_t$ is bounded and continuous there.

Apply [F1]. Along almost every path, the derivative of $w\varphi (\theta )$ is $w^{\mathrm{\prime }}\varphi (\theta )+wD{\varphi }_{\theta }{\theta }^{\mathrm{\prime }}$, with norm at most $C(1+\mathrm{\mid }w\mathrm{\mid })\mathrm{\mid }v\mathrm{\mid }$. Its path/time integral is finite by the uniform second-moment bound and Cauchy–Schwarz. The same bound controls the derivative of $\mathrm{\mid }w\mathrm{\mid }W(\theta )$. For a real AC path, its derivative is zero almost everywhere on its zero set: at a differentiability point with nonzero derivative that zero is isolated, and isolated zeroes form a countable set. Thus the formula $(\mathrm{\mid }w\mathrm{\mid }{)}^{\mathrm{\prime }}=\mathrm{sign}(w)w^{\mathrm{\prime }}$ with sign zero set to zero holds almost everywhere. Integrating the pathwise FTC and using Fubini gives AC representatives of $m_t,P_t$ with the integrated derivatives. Their time derivatives are bounded by $C(1+\sqrt{S(t)})\sqrt{A(t)}$, hence are in $L^2$ on finite intervals; [F3] identifies the norm derivative of $m_t$.

Compose the AC moment with $R$: its differential is bounded on the compact image of the continuous moment path on the finite interval, so the composition is AC, and the ordinary Fréchet difference quotient gives derivative $⟨f_t,m_t^{\mathrm{\prime }}⟩$ almost everywhere. Pair the inclusion [F2] with the characteristic velocity. At $w\mathrm{\ne }0$ the potential derivatives are ${\mathrm{\partial }}_w{P}_t=⟨f_t,\varphi ⟩+\mathrm{sign}(w)W$ and ${\mathrm{\nabla }}_{\theta }{P}_t=wD{\varphi }^{\ast }{f}_t+\mathrm{\mid }w\mathrm{\mid }\mathrm{\nabla }W$. Any domain normal $n$ satisfies $n\cdot {\theta }^{\mathrm{\prime }}=0$ at two-sided differentiability times: test its normal inequality against $\theta (t+h)$ for both signs of $h$. At $w=0$ the angular velocity is zero, and step 2.1 gives $w^{\mathrm{\prime }}=0$ almost everywhere on that zero set. The cusp and normal contributions therefore have zero pairing. The chain rule from step 2.1 yields $F^{\mathrm{\prime }}(t)=-\int (\mathrm{\mid }{w}^{\mathrm{\prime }}{\mathrm{\mid }}^2+\mathrm{\mid }{\theta }^{\mathrm{\prime }}{\mathrm{\mid }}^2)d{\mu }_t=-A(t)$ almost everywhere. The established AC representative gives the first identity at all endpoints. Nonnegativity of $R,V$ gives $F\le E_0$ and ${\int }_0^{\mathrm{\infty }}A\le E_0$.

Off $w=0$, the unconstrained weight equation is $w^{\mathrm{\prime }}=-⟨f_t,\varphi (\theta )⟩-\mathrm{sign}(w)W(\theta )$; the theta-domain normal has no weight component. Consequently $(w^2{)}^{\mathrm{\prime }}=-2(w⟨f_t,\varphi (\theta )⟩+\mathrm{\mid }w\mathrm{\mid }W(\theta ))$, also at almost every zero-set time. The product is integrable on finite intervals by Cauchy–Schwarz, the second-moment bound and bounded forcing. Integrating the pathwise identity gives the exact formula for $S(t)-S(s)$. If moments are uniformly bounded by $M$, the risk-sublevel bound gives $\mathrm{\parallel }{f}_t\mathrm{\parallel }\le D$, and $0\le P_t\le E_0$. Thus $\mathrm{\mid }{S}^{\mathrm{\prime }}(t)\mathrm{\mid }\le 2(DM+E_0)$ almost everywhere; integrating proves the displayed linear growth bound.