# NS × X Integral × 24/72 Paradigm Practice ## Round 12 — Pure Continuous Critical Dual Geometry / Cancellation-Tradeoff Route - Date: 2026-08-16 - Version: v0.1 - Status: Proof-Route Experiment / Continuous-Only Critical-Dual Branch - canonical source: UTF-8 Markdown - canonical math delimiters: inline `$...$`; display `$$...$$` - Previous round: `NS_X72_Round11_PureContinuous_DeterministicFunctionalResummation_DualAdjoint_v0.1_2026-08-16.md` - This round's objective: Investigate whether the backward matched dual propagator from Round 11 can recover the exact contraction of $L^2$ in scale-critical dual spaces. The main tests are on $L^{3/2}$ and $\dot H^{-1/2}$, to determine if there is a structural tradeoff among the Leray projection, transport skewness, and the critical metric. - Non-claims: This document only provides a restricted no-go for two natural classes of critical dual geometry and two broad metric subfamilies. It does not claim that all possible nonlinear/nonlocal critical dual geometries cannot be contractive. --- # 0. Round 11 handoff Round 11 resummed the deterministic interaction-order hierarchy: $$ 3\to4\to5\to\cdots $$ into the generating functional: $$ \mathcal Z[\varphi,t] = e^{\langle\varphi,u(t)\rangle}, $$ whose exact functional PDE only requires second-order functional derivatives. Therefore, the interaction order is not an essential discreteness witness. The same round yielded the matched backward dual equation: $$ \boxed{ \partial_t\varphi + \nu\Delta\varphi + P[(u\cdot\nabla)\varphi] = 0, } \tag{0.1} $$ and: $$ \boxed{ \frac d{dt} \langle \varphi(t),u(t) \rangle = 0. } \tag{0.2} $$ If: $$ \varphi(T)=\varphi_T, $$ then: $$ \boxed{ \langle \varphi_T,u(T) \rangle = \langle \varphi(0),u_0 \rangle. } \tag{0.3} $$ Round 11 obtained a backward dual contraction in $L^2$. This round asks: $$ \boxed{ \text{Does critical dual geometry also have the same contraction?} } $$ --- # 1. Forward backward-time formulation Let: $$ \sigma=T-t, $$ $$ \psi(\sigma,x) = \varphi(T-\sigma,x), $$ and: $$ U(\sigma,x) = u(T-\sigma,x). $$ Then: $$ \boxed{ \partial_\sigma\psi = \nu\Delta\psi + P[(U\cdot\nabla)\psi]. } \tag{1.1} $$ where: $$ \nabla\cdot U=0, $$ and if: $$ \nabla\cdot\psi(0)=0, $$ then: $$ \nabla\cdot\psi(\sigma)=0. $$ To simplify notation, we write below: $$ T_U = U\cdot\nabla. $$ --- # 2. Dual critical scaling Navier–Stokes scaling: $$ u_\lambda(x,t) = \lambda u(\lambda x,\lambda^2t). $$ To keep the pairing: $$ \langle \psi,u \rangle $$ scale-invariant, the dual field must scale as: $$ \boxed{ \psi_\lambda(x,t) = \lambda^2 \psi(\lambda x,\lambda^2t). } \tag{2.1} $$ For: $$ L^p $$ we have: $$ \|\psi_\lambda\|_{L^p} = \lambda^{2-\frac3p} \|\psi\|_{L^p}. $$ Thus, the critical dual exponent is: $$ \boxed{ p=\frac32. } \tag{2.2} $$ For homogeneous Sobolev spaces: $$ \|\psi_\lambda\|_{\dot H^s} = \lambda^{s+\frac12} \|\psi\|_{\dot H^s}. $$ Thus, the critical Hilbert dual is: $$ \boxed{ s=-\frac12. } \tag{2.3} $$ Therefore, the two natural critical targets for this round are: $$ \boxed{ L^{3/2} } $$ and: $$ \boxed{ \dot H^{-1/2}. } $$ --- # 3. Why $L^2$ contracts exactly First, recall: $$ \frac12 \frac d{d\sigma} \|\psi\|_2^2 = \nu \langle\Delta\psi,\psi\rangle + \langle P T_U\psi,\psi \rangle. $$ Since: $$ P=P^\ast=P^2 $$ and: $$ P\psi=\psi, $$ we have: $$ \langle P T_U\psi,\psi \rangle = \langle T_U\psi,\psi \rangle. $$ Furthermore, from: $$ \nabla\cdot U=0, $$ $$ \langle T_U\psi,\psi \rangle = 0. $$ Therefore: $$ \boxed{ \frac12 \frac d{d\sigma} \|\psi\|_2^2 + \nu \|\nabla\psi\|_2^2 = 0. } \tag{3.1} $$ The $L^2$ exact contraction simultaneously uses two structures: $$ \boxed{ \text{Projection Compatibility} + \text{Transport Skewness}. } \tag{3.2} $$ --- # 4. Critical local route: $L^{3/2}$ More generally, let: $$ 1 For all divergence-free $\psi$, $J_F(\psi)$ remains divergence-free. We have: $$ \operatorname{div} (g(|\psi|)\psi) = g'(|\psi|) \psi\cdot\nabla|\psi|. $$ For this expression to be identically zero for all admissible divergence-free fields, it must be that: $$ \boxed{ g' = 0. } $$ Therefore: $$ g=\text{constant}. $$ That is: $$ \boxed{ F(r) = cr^2/2 + \text{constant}. } \tag{7.3} $$ Thus, among local isotropic integral metrics: $$ \boxed{ \textbf{ quadratic }L^2\textbf{ geometry is the unique universal geometry that preserves both raw transport cancellation and divergence-free norm gradient}. } } \tag{7.4} $$ This is a restricted theorem. It does not rule out nonlocal / nonlinear / relational critical functionals. --- # 8. Critical Hilbert route: $\dot H^{-1/2}$ Now we switch to: $$ A = \Lambda^{-1}, $$ where: $$ \Lambda=(-\Delta)^{1/2}. $$ Define: $$ \boxed{ \|\psi\|_{\dot H^{-1/2}}^2 = \langle \psi,A\psi \rangle. } \tag{8.1} $$ Since: $$ A $$ is a scalar radial Fourier multiplier, it commutes with: $$ P. $$ If: $$ P\psi=\psi, $$ then: $$ P A\psi = A\psi. $$ Therefore: $$ \boxed{ \langle P T_U\psi, A\psi \rangle = \langle T_U\psi, A\psi \rangle. } \tag{8.2} $$ Thus: $$ \dot H^{-1/2} $$ fully preserves projection compatibility. --- # 9. Exact critical Hilbert commutator identity From: $$ T_U^\ast = -T_U $$ in $L^2$, we have: $$ \boxed{ 2 \langle T_U\psi, A\psi \rangle = \langle \psi, [A,T_U]\psi \rangle. } \tag{9.1} $$ Therefore: $$ \boxed{ \frac12 \frac d{d\sigma} \|\psi\|_{\dot H^{-1/2}}^2 + \nu \|\psi\|_{\dot H^{1/2}}^2 = \frac12 \left\langle \psi, [\Lambda^{-1},T_U]\psi \right\rangle. } \tag{9.2} $$ This is the exact critical Hilbert dual balance. Thus: $$ \boxed{ \dot H^{-1/2} \text{ keeps Leray compatibility but loses exact transport skewness}. } \tag{9.3} $$ The reason is not that the transport itself is not skew. Rather, the critical metric: $$ \Lambda^{-1} $$ does not commute with the transport. --- # 10. Fourier representation of the commutator defect Let: $$ k=p+q. $$ Then: $$ \widehat{T_U\psi}(k) = i \int \left( q\cdot\widehat U(p) \right) \widehat\psi(q) dp. $$ Therefore: $$ \boxed{ \widehat{ [\Lambda^{-1},T_U]\psi }(k) = i \int \left( |k|^{-1} - |q|^{-1} \right) \left( q\cdot\widehat U(p) \right) \widehat\psi(q) dp. } \tag{10.1} $$ The critical defect is therefore a continuous triad weight-gap integral. Moreover: $$ \boxed{ \left| |k|^{-1} - |q|^{-1} \right| = \frac{ \left|| k|-|q| \right| }{ |k||q| } \le \frac{ |p| }{ |k||q| }. } \tag{10.2} $$ This is isomorphic to the no-free-radial-jump mechanism from Round 09. --- # 11. General radial Hilbert metric More generally, let: $$ A=a(\Lambda) $$ be a real radial self-adjoint multiplier. Define: $$ \mathcal E_a(\psi) = \frac12 \langle \psi,A\psi \rangle. $$ Projection compatibility still holds. The transport defect is: $$ \boxed{ \frac12 \langle \psi, [A,T_U]\psi \rangle. } \tag{11.1} $$ The Fourier symbol is: $$ \boxed{ a(|k|) - a(|q|). } \tag{11.2} $$ So if: $$ a $$ is constant, the commutator vanishes pointwise. This is precisely $L^2$. --- # 12. Restricted radial-Hilbert uniqueness Suppose we require: $$ \boxed{ \langle \psi, [A,T_U]\psi \rangle = 0 } \tag{12.1} $$ to hold for all smooth divergence-free: $$ U,\psi. $$ If: $$ a $$ is not constant, we can choose a nondegenerate continuous Fourier triad: $$ k=p+q $$ such that: $$ a(|k|) \neq a(|q|). $$ Then choose divergence-free polarizations and relative phases such that: $$ q\cdot\widehat U(p)\neq0 $$ and the corresponding quadratic commutator pairing is non-zero. In $\mathbb R^3$, one can use smooth Fourier wave packets concentrated near this nondegenerate triad to realize the same symbol-level nonvanishing. Therefore, universal exact cancellation forces: $$ \boxed{ a(r)=\text{constant}. } \tag{12.2} $$ Thus, among radial translation-invariant Hilbert metrics: $$ \boxed{ \textbf{ }L^2\textbf{ is again the unique universal metric with exact transport cancellation}. } \tag{12.3} $$ This is likewise a restricted theorem. --- # 13. Criticality–Cancellation Tradeoff Now the two critical routes form a mirror image. ## Local critical geometry $$ \boxed{ L^{3/2} } $$ preserves: $$ \boxed{ \text{transport chain-rule cancellation} } $$ but loses: $$ \boxed{ \text{projection compatibility}. } $$ ## Hilbert critical geometry $$ \boxed{ \dot H^{-1/2} } $$ preserves: $$ \boxed{ \text{projection compatibility} } $$ but loses: $$ \boxed{ \text{transport cancellation} } $$ Only: $$ \boxed{ L^2 } $$ preserves both simultaneously within the two tested natural metric families. But for the dual NS scaling: $$ \|\psi_\lambda\|_2 = \lambda^{1/2} \|\psi\|_2, $$ so: $$ \boxed{ L^2 \text{ is not scale-critical}. } $$ Therefore, we obtain the: $$ \boxed{ \textbf{Criticality–Cancellation Tradeoff}. } \tag{13.1} $$ --- # 14. Cancellation square We can put the three spaces into a table. | Dual geometry | scale critical | raw transport cancellation | Leray compatibility | defect | |---|---:|---:|---:|---| | $L^2$ | no | yes | yes | none | | $L^{3/2}$ | yes | yes | no | $\mathfrak P_{3/2}$ | | $\dot H^{-1/2}$ | yes | no | yes | commutator $\mathfrak C_{-1/2}$ | where: $$ \boxed{ \mathfrak C_{-1/2}[U,\psi] = \frac12 \langle \psi, [\Lambda^{-1},T_U]\psi \rangle. } \tag{14.1} $$ Thus, the critical dual obstruction is not a single issue of "insufficient estimates". It consists of two distinct geometric defects. --- # 15. Defect pair as a relational state Define the critical dual defect pair: $$ \boxed{ \mathfrak D_{\rm crit} = \left( \mathfrak P_{3/2}, \mathfrak C_{-1/2} \right). } \tag{15.1} $$ These two defects respectively measure: $$ \boxed{ \text{norm-gradient leakage out of divergence-free space} } $$ and: $$ \boxed{ \text{critical metric failure to commute with transport}. } $$ Therefore, selecting only one scalar norm hides the other structure. The observation state of this round naturally elevates again to: $$ \boxed{ \mathsf X_{\rm dual}. } $$ --- # 16. A universal contraction would need both defects controlled If one hopes to find a critical dual functional: $$ \mathfrak N(\psi) $$ such that: $$ \boxed{ \frac d{d\sigma} \mathfrak N(\psi) \le0 } $$ holds for an arbitrary smooth NS drift: $$ U, $$ its variational geometry must simultaneously handle at least: 1. diffusion coercivity; 2. divergence-free constraint; 3. transport invariance / skewness; 4. critical scaling. In the two natural classes of this round: $$ \boxed{ \text{critical scaling} } $$ and: $$ \boxed{ \text{double exact cancellation} } $$ do not appear simultaneously. Therefore, the next carrier cannot simply be: $$ L^{3/2} $$ or: $$ \dot H^{-1/2} $$ under a different name. It must genuinely change the geometry. --- # 17. Projection defect is a nonlinear tangent-space defect For: $$ L^{3/2}, $$ the norm gradient is: $$ J_{3/2}(\psi) = |\psi|^{-1/2}\psi. $$ Even if: $$ \psi $$ lies on the divergence-free manifold, $$ J_{3/2}(\psi) $$ generally does not remain divergence-free under its tangent-cotangent identification. Therefore: $$ P $$ must be re-projected: $$ J_{3/2} \mapsto P J_{3/2}. $$ And the transport cancellation originally holds for: $$ J_{3/2}, $$ not for: $$ P J_{3/2}. $$ Thus: $$ \boxed{ \text{constraint projection} } $$ and: $$ \boxed{ \text{local entropy gradient} } $$ do not commute. The X-order representation is: $$ \boxed{ P \circ J_{3/2} \neq J_{3/2} \circ P. } \tag{17.1} $$ --- # 18. Hilbert defect is a metric–transport commutator For: $$ \dot H^{-1/2}, $$ the constraint projection has no issues. But the critical metric is generated by: $$ A=\Lambda^{-1}. $$ The transport: $$ T_U $$ alters the frequency content. Therefore: $$ \boxed{ [A,T_U] \neq0. } \tag{18.1} $$ This is not a pressure defect. Nor is it the norm gradient leaving the divergence-free space. It is the noncommutativity between: $$ \boxed{ \text{metric} \leftrightarrow \text{transport} } $$ . --- # 19. Two noncommutativities Thus, Round 12 found two exact X-order obstructions: $$ \boxed{ \begin{aligned} \text{Local critical:}\quad& P J \neq J P, \\ \text{Hilbert critical:}\quad& A T \neq T A. \end{aligned} } \tag{19.1} $$ The uniqueness of $L^2$ lies in the fact that: $$ J_2=I, $$ $$ A_2=I, $$ so both commutators simultaneously degenerate to zero. --- # 20. Why a fixed combination does not automatically repair the problem One might consider a composite functional: $$ \mathfrak N = c_1 \|\psi\|_{3/2}^{3/2} + c_2 \|\psi\|_{\dot H^{-1/2}}^2. $$ Its derivative will only yield: $$ \boxed{ c_1 \mathfrak P_{3/2} + c_2 \mathfrak C_{-1/2} } $$ plus dissipations. But currently, there is no identity forcing: $$ \mathfrak P_{3/2} $$ and: $$ \mathfrak C_{-1/2} $$ to cancel each other out. Thus: $$ \boxed{ \text{multi-norm addition} \neq \text{relational closure}. } \tag{20.1} $$ What is truly needed is a new structural relation between the two defects. --- # 21. A candidate critical geometric functional This round naturally derives the next search target. Do not pre-specify: $$ L^p $$ or: $$ H^s. $$ Instead, look for a functional: $$ \boxed{ \mathfrak N_{\rm crit}[\psi] } $$ satisfying: ## C1. Critical homogeneity $$ \boxed{ \mathfrak N_{\rm crit}[\psi_\lambda] = \mathfrak N_{\rm crit}[\psi]. } $$ ## C2. Constraint compatibility Its variational gradient: $$ J_{\rm crit}(\psi) = \frac{\delta\mathfrak N_{\rm crit}}{\delta\psi} $$ is compatible with the Leray projection in the relevant dual pairing. ## C3. Transport cancellation or controlled commutator $$ \boxed{ \langle P T_U\psi, J_{\rm crit}(\psi) \rangle \le \text{coercive diffusion term}. } $$ ## C4. Lossless norm detection Controlling: $$ \mathfrak N_{\rm crit} $$ must be sufficient to control the primal critical continuation quantity. This is a genuine geometric variational problem. --- # 22. STOP-C16 — Criticality / Double-Cancellation Gap Define: $$ \boxed{ \bot_X^{\mathrm{C16}} = \left\langle \begin{array}{l} \text{layer} = \mathrm{critical\ dual\ geometry}, \\ L^{3/2}\text{ preserves} = \mathrm{transport\ chain\ rule}, \\ L^{3/2}\text{ loses} = \mathrm{Leray\ compatibility}, \\ \dot H^{-1/2}\text{ preserves} = \mathrm{Leray\ compatibility}, \\ \dot H^{-1/2}\text{ loses} = \mathrm{transport\ commutation}, \\ L^2\text{ preserves} = \mathrm{both}, \\ L^2\text{ critical} = \mathrm{false}, \\ \text{missing} = \mathrm{critical\ functional\ with\ joint\ cancellation/coercivity}, \\ \text{discrete\ intrusion} = \mathrm{false}. \end{array} \right\rangle. } $$ Name: $$ \boxed{ \textbf{STOP-C16: Criticality / Double-Cancellation Gap}. } $$ --- # 23. This is not a proof that no critical contractive geometry exists The restricted no-go of this round only covers: 1. local isotropic integral metrics; 2. radial translation-invariant Hilbert multiplier metrics. Thus, it cannot be deduced that: $$ \boxed{ \text{no critical continuous dual Lyapunov functional exists}. } $$ There may still exist: - nonlinear nonlocal functionals; - quotient / constrained Finsler geometries; - transport-adapted metrics; - Lagrangian critical metrics; - relational multi-carrier functionals; - dynamically varying dual metrics. Therefore, the Pure-C path is not yet sealed off. --- # 24. 24/72 Ledger — Round 12 | Step | X object / operation | $B$ | $U$ | $O$ | $L$ | status | |---|---|---|---|---|---|---| | C124 | backward-time dual equation | $\mathsf C$ | $\mathsf S/\mathsf P$ | relational | $\mathsf F$ | EXACT | | C125 | dual scaling | $\mathsf C$ | — | scalar | $\mathsf F$ | EXACT | | C126 | $L^{3/2}$ critical route | $\mathsf C$ | local | scalar | $\mathsf F$ | FORM | | C127 | raw $L^p$ transport cancellation | $\mathsf C$ | transport | scalar | $\mathsf F$ | EXACT | | C128 | projection defect $\mathfrak P_{3/2}$ | $\mathsf C$ | constraint | relational | $\mathsf F$ | EXACT | | C129 | local isotropic double-cancellation | $\mathsf C$ | — | local metric | $\mathsf F$ | ONLY QUADRATIC UNIVERSALLY | | C130 | $\dot H^{-1/2}$ critical route | $\mathsf C$ | Hilbert/nonlocal | scalar | $\mathsf F$ | FORM | | C131 | projection compatibility in $\dot H^{-1/2}$ | $\mathsf C$ | constraint | Hilbert | $\mathsf F$ | EXACT | | C132 | commutator defect $\mathfrak C_{-1/2}$ | $\mathsf C$ | transport | relational | $\mathsf F$ | EXACT | | C133 | radial Hilbert exact transport cancellation | $\mathsf C$ | — | multiplier metric | $\mathsf F$ | ONLY CONSTANT MULTIPLIER UNIVERSALLY | | C134 | criticality–cancellation tradeoff | $\mathsf C$ | multi-route | $\mathsf X$ | $\mathsf F$ | PROVED IN TESTED CLASSES | | C135 | universal critical double-cancellation functional | $\mathsf C$ | variational | $\mathsf X$ | $\mathsf F$ | OPEN / STOP-C16 | --- # 25. Continuous-versus-discrete status Round 12 introduced absolutely no: - atomic decomposition; - dyadic shell; - wavelet index; - discrete packet family; - sequence extraction. All defects are formed by continuous operators: $$ P, \quad \Lambda^{-1}, \quad T_U $$ and variational gradients. Therefore: $$ \boxed{ T_{\mathsf C\to\mathsf D} = \text{NOT YET REACHED}. } \tag{25.1} $$ --- # 26. Pure-C path after Round 12 $$ \boxed{ \begin{aligned} \mathsf C_{\rm energy} &\to \mathsf C_{\rm critical} \\ &\to \mathsf C_{\rm relational} \\ &\to \mathsf C_{\rm nonlocal} \\ &\to \mathsf C_{\rm projected} \\ &\to \mathsf C_{\rm gradient\ geometry} \\ &\to \mathsf C_{\rm hierarchy} \\ &\to \mathsf C_{\rm Gevrey} \\ &\to \mathsf C_{\rm covariance} \\ &\to \mathsf C_{\rm triad\ phase} \\ &\to \mathsf C_{\rm phase\ network} \\ &\to \mathsf C_{\rm functional} \\ &\to \mathsf C_{\rm dual\ adjoint} \\ &\to \mathsf C_{\rm critical\ variational\ geometry}. \end{aligned} } \tag{26.1} $$ --- # 27. Strongest result of Round 12 The most important structural result of this round: $$ \boxed{ \begin{array}{c|cc} &\text{Transport cancellation}&\text{Leray compatibility} \\ \hline L^2&\checkmark&\checkmark \\ L^{3/2}&\checkmark&\times \\ \dot H^{-1/2}&\times&\checkmark \end{array} } \tag{27.1} $$ And: $$ L^2 $$ does not possess dual critical scaling. Therefore, among the two most natural classes of critical dual geometry: $$ \boxed{ \textbf{ criticality splits the two exact cancellations that coincide at }L^2. } } \tag{27.2} $$ --- # 28. Next round — projected critical entropy / constrained metric The next round will no longer test more off-the-shelf norms. It will directly attempt to construct: $$ \boxed{ \mathfrak N_{\rm crit}[\psi] } $$ so that its gradient lives in the divergence-free constraint geometry from the very beginning. Candidate paths: 1. constrained entropy gradient: $$ J_{\rm div} = P J_{3/2}; $$ 2. Check if there exists a functional: $$ \mathfrak N $$ such that: $$ \frac{\delta\mathfrak N}{\delta\psi} = P J_{3/2}(\psi); $$ 3. If it does not exist, find the integrability obstruction: whether the projected entropy vector field is not a functional gradient; 4. If it exists, study the transport pairing: $$ \langle T_U\psi, J_{\rm div} \rangle; $$ 5. More generally, search for a critical constrained Finsler metric; 6. If all continuous constrained metrics require atomic / wave-packet decomposition to be defined or estimated, only then reconsider: $$ T_{\mathsf C\to\mathsf D}. $$ This round will directly ask a very specific variational question: $$ \boxed{ \textbf{ Can Leray projection of the critical entropy gradient itself be integrated back into a scalar critical functional? } } $$ --- # 29. External primary-source anchors 1. Dong Li, *On Kato-Ponce and fractional Leibniz*, arXiv:1609.01780. - fractional Leibniz / commutator estimates; - The standard analytical background for the transport–multiplier commutator encountered in the $\dot H^{-1/2}$ route of this round. 2. D. Q. Khai, N. M. Tri, *On the initial value problem for the Navier-Stokes equations with the initial datum in critical Sobolev and Besov spaces*, arXiv:1601.01726. - Navier–Stokes local theory and small-data global framework in critical homogeneous Sobolev spaces. 3. Jean-Yves Chemin, Ping Zhang, *On the critical one component regularity for 3-D Navier-Stokes system*, arXiv:1310.6442. - $\dot H^{1/2}$ scaling-critical Navier–Stokes regularity framework. The projection-defect identity, critical Hilbert commutator identity, two restricted uniqueness results, and the cancellation-tradeoff in this round are all directly derived in this document. --- # 30. Commit state $$ \boxed{ \begin{aligned} \text{Route} &: \mathrm{Pure\ Continuous\ Critical\ Dual\ Geometry}, \\ \text{Essential }\mathsf C\to\mathsf D &: \mathrm{Not\ reached}, \\ L^{3/2}\text{ defect} &: \mathfrak P_{3/2}, \\ \dot H^{-1/2}\text{ defect} &: \mathfrak C_{-1/2}, \\ L^2\text{ double cancellation} &: \mathrm{exact}, \\ L^2\text{ criticality} &: \mathrm{false}, \\ \text{Restricted natural critical classes} &: \mathrm{no\ double\ exact\ cancellation}, \\ \text{STOP-C16} &: \mathrm{Criticality/Double\text{-}Cancellation\ Gap}, \\ \text{Next} &: \mathrm{Projected\ Critical\ Entropy/Constrained\ Metric}. \end{aligned} } $$