# NS-DCRP-41 — Planar Covariance Shape Disk, Hyperbolic Shear Action, and the Moving Pancake-Jet Normal Form - date: 2026-08-17 - status: research proof checkpoint / rank-two shape-rigidity round - canonical source: UTF-8 Markdown - canonical math delimiters: `$...$` and `$$...$$` - objective: 1. refine the DCRP-40 rank-two planar Floquet equality by separating covariance magnitude, shape anisotropy, and in-plane orientation; 2. derive an exact two-dimensional disk equation for the normalized in-plane covariance; 3. invert that equation to reconstruct the in-plane deviatoric strain from covariance-shape motion; 4. derive a positive hyperbolic shape/phase action; 5. prove that zero shape-action forces pointwise isotropic affine extension on the vorticity plane; 6. reconstruct the full three-dimensional affine tensor on that equality branch as a moving pancake jet; 7. obtain a universal periodic reproduction-action lower bound from the strict DSS normal-compression condition; 8. calibrate, but not identify, the final branch against exact Euler pancake/vortex-sheet mechanisms; 9. identify the next frontier as the scalar/potential--shear dynamics inside the moving pancake jet. - no full Navier--Stokes regularity claim is made. - principal external primary calibration: - D. S. Agafontsev, E. A. Kuznetsov, A. A. Mailybaev, *Asymptotic solution for high vorticity regions in incompressible 3D Euler equations*, arXiv:1609.07782; - A. Enciso, A. J. Fernández, D. Meyer, *Vortex-sheet desingularization for three-dimensional ideal fluids*, arXiv:2607.19233. - internal dependencies: - DCRP-36 affine-jet reproduction action; - DCRP-38 covariance matrix ledger; - DCRP-40 rank-two planar covariance / Floquet compression. - no novelty/priority claim is made without independent audit. --- # 1. Executive result DCRP-40 reduced the exact rank-two zero-covariance-residual branch to $$ \boxed{ B' = AB + BA - c_\gamma B, } \tag{1.1} $$ where $$ \boxed{ c_\gamma = 2-3\gamma > 0, } \tag{1.2} $$ and $$ \boxed{ \operatorname{rank}B=2. } \tag{1.3} $$ Let $$ n(s) $$ be the unit normal spanning $$ \ker B(s). $$ Then $$ \boxed{ n' = -An + (n\cdot An)n. } \tag{1.4} $$ The period-averaged normal compression is $$ \boxed{ \frac1{S_0} \int_0^{S_0} n\cdot An\,ds = -c_\gamma. } \tag{1.5} $$ DCRP-41 now separates the **in-plane covariance shape** from its magnitude. Choose a Fermi--Walker orthonormal frame $$ E(s) = (e_1(s),e_2(s)) $$ for the plane $$ n(s)^\perp, $$ satisfying $$ \boxed{ E^TE=I_2, \qquad E^Tn=0, \qquad E^TE'=0. } \tag{1.6} $$ Define the positive in-plane covariance $$ \boxed{ C = E^TBE. } \tag{1.7} $$ Then the Fermi-frame connection drops out of the covariance equation and $$ \boxed{ C' = A_EC + CA_E - c_\gamma C, } \tag{1.8} $$ where $$ \boxed{ A_E = E^TAE. } \tag{1.9} $$ Let $$ \boxed{ m = \operatorname{tr}C } \tag{1.10} $$ and define the normalized planar covariance $$ \boxed{ P = \frac{C}{m}. } \tag{1.11} $$ Then $$ P>0, \qquad \operatorname{tr}P=1. $$ Decompose the in-plane affine strain as $$ \boxed{ A_E = a I_2 + S, } \tag{1.12} $$ where $$ \boxed{ a = \frac12 \operatorname{tr}A_E = -\frac12 n\cdot An, } \tag{1.13} $$ and $$ \boxed{ \operatorname{tr}S=0. } \tag{1.14} $$ The normalized covariance satisfies the exact equation $$ \boxed{ P' = SP + PS - 2(S:P)P. } \tag{1.15} $$ Thus: $$ \boxed{ \textbf{ normal compression and isotropic planar extension do not change covariance shape. } } \tag{1.16} $$ Only the in-plane deviatoric strain $$ S $$ changes the normalized planar covariance. This is the first main result. The second main result identifies the shape space with the open unit disk. Every positive symmetric $$ 2\times2 $$ matrix of trace one can be written uniquely as $$ \boxed{ P = \frac12 \left[ I_2 + Z \right], } \tag{1.17} $$ where $$ \boxed{ Z = \begin{pmatrix} z_1&z_2\\ z_2&-z_1 \end{pmatrix}, } \tag{1.18} $$ and $$ \boxed{ |z|^2 = z_1^2+z_2^2 < 1. } \tag{1.19} $$ Likewise write $$ \boxed{ S = \begin{pmatrix} s_1&s_2\\ s_2&-s_1 \end{pmatrix}. } \tag{1.20} $$ Then (1.15) becomes the exact two-dimensional equation $$ \boxed{ z' = 2 \left[ s - (s\cdot z)z \right]. } \tag{1.21} $$ Equivalently, $$ \boxed{ z' = 2 \left( I_2-zz^T \right)s. } \tag{1.22} $$ Because $$ |z|<1, $$ the matrix $$ I_2-zz^T $$ is positive definite. Therefore $$ \boxed{ s = \frac12 \left( I_2-zz^T \right)^{-1} z'. } \tag{1.23} $$ This is the central inversion formula of DCRP-41. It means: $$ \boxed{ \textbf{ inside the rank-two interior, the in-plane deviatoric affine strain is completely determined by covariance-shape motion. } } \tag{1.24} $$ The planar shear is not an independent hidden degree of freedom. The third main result is the hyperbolic shape action. Define the rank-two anisotropy parameter $$ \boxed{ \vartheta_2 = \frac{ 4\det C }{ (\operatorname{tr}C)^2 }. } \tag{1.25} $$ In disk coordinates, $$ \boxed{ \vartheta_2 = 1-|z|^2. } \tag{1.26} $$ Thus: -: $$ |z|=0 $$ is isotropic covariance in the plane; -: $$ |z|\uparrow1 $$ is rank-one collapse. Parameterize $$ \boxed{ z = r \left( \cos2\theta, \sin2\theta \right), \qquad 0\le r<1. } \tag{1.27} $$ The angle $$ \theta $$ is the principal-axis angle of the covariance tensor, modulo the usual $$ \pi $$ axis symmetry. Using (1.23), $$ \boxed{ |S|_F^2 = \frac{ (r')^2 }{ 2(1-r^2)^2 } + 2r^2 (\theta')^2. } \tag{1.28} $$ Thus the in-plane deviatoric strain pays exactly for: 1. anisotropy-amplitude motion: $$ r'; $$ 2. in-plane covariance-axis rotation: $$ \theta'. $$ This is the precise finite-dimensional phase-action formula. The singular factor $$ (1-r^2)^{-2} $$ shows that changing anisotropy near rank-one collapse is increasingly expensive in the shape metric. The fourth result is the zero-shape-action rigidity. Define $$ \boxed{ \mathcal A_{\rm shape} = \int_0^{S_0} |S(s)|_F^2ds. } \tag{1.29} $$ Then $$ \boxed{ \mathcal A_{\rm shape}=0 } \tag{1.30} $$ if and only if $$ \boxed{ S(s)\equiv0. } \tag{1.31} $$ Equivalently, $$ \boxed{ A|_{n^\perp} = a(s)I_{n^\perp} } \tag{1.32} $$ pointwise in similarity time. Thus every exact zero-shape-action rank-two branch has **isotropic planar affine extension at every instant**, not merely on average. The fifth main result reconstructs the full affine tensor. From the plane-normal equation: $$ n' = -An + (n\cdot An)n, $$ one gets $$ \boxed{ P_{n^\perp}An = -n'. } \tag{1.33} $$ Because $$ A $$ is symmetric and trace free, on the zero-shape-action branch it is uniquely determined by: - one scalar: $$ a(s); $$ - the moving plane normal: $$ n(s). $$ The exact formula is $$ \boxed{ A_{\rm pan}(s) = a(s) \left[ P_{n^\perp} - 2n\otimes n \right] - \left[ n'\otimes n + n\otimes n' \right]. } \tag{1.34} $$ This is the **moving pancake-jet normal form**. It is the rank-two dual of the rank-one Burgers-jet tensor of DCRP-39. When $$ n'=0, $$ $$ \boxed{ A_{\rm pan} = a(s) \left[ P_{n^\perp} - 2n\otimes n \right]. } \tag{1.35} $$ In coordinates $$ n=e_3, $$ $$ \boxed{ A_{\rm pan} = \operatorname{diag} \left( a,a,-2a \right). } \tag{1.36} $$ The periodic pseudo-determinant balance (1.5) becomes $$ \boxed{ \frac1{S_0} \int_0^{S_0} a(s)\,ds = \frac{ c_\gamma }{2}. } \tag{1.37} $$ Thus the moving pancake jet has a strictly positive mean planar-extension rate. The sixth result is a universal normalized reproduction-action lower bound. From (1.34), $$ \boxed{ |A_{\rm pan}|_F^2 = 6a(s)^2 + 2|n'(s)|^2. } \tag{1.38} $$ By Jensen, $$ \boxed{ \int_0^{S_0} a(s)^2ds \ge \frac{ c_\gamma^2 }{4} S_0. } \tag{1.39} $$ Therefore $$ \boxed{ \int_0^{S_0} |A_{\rm pan}|_F^2ds \ge \frac32 c_\gamma^2 S_0 + 2 \int_0^{S_0} |n'|^2ds. } \tag{1.40} $$ DCRP-36 proved for any periodic affine jet: $$ \boxed{ \int_0^{S_0} |A'+A|^2ds = \int_0^{S_0} |A'|^2ds + \int_0^{S_0} |A|^2ds. } \tag{1.41} $$ Hence the zero-shape-action rank-two branch satisfies the universal reproduction lower bound $$ \boxed{ \mathcal A_{\rm rep} \ge \frac32 c_\gamma^2 S_0. } \tag{1.42} $$ If the plane rotates, $$ n'\neq0, $$ the lower bound is strictly larger. Thus the exact pancake equality state is not a source-free tensor. Its annular source dynamics must reproduce a fixed positive amount of normalized pancake strain each period. This is a finite-dimensional visibility theorem. It is not yet a raw physical depletion theorem. The seventh main result classifies every rank-two zero-residual profile into: $$ \boxed{ \textbf{ positive planar shape action} } $$ or $$ \boxed{ \textbf{ moving pancake-jet equality}. } $$ More explicitly: ### shape-dynamic branch $$ \boxed{ \mathcal A_{\rm shape}>0. } \tag{1.43} $$ The in-plane covariance changes anisotropy and/or principal-axis angle. This produces a finite-dimensional shape/phase source requirement. ### shape-static branch $$ \boxed{ \mathcal A_{\rm shape}=0. } \tag{1.44} $$ Then the affine tensor is exactly $$ A_{\rm pan} $$ and has the universal reproduction-action gap (1.42). Therefore no rank-two exact branch remains with both: - zero covariance residual; - zero planar-shape activity; - zero affine reproduction activity. The eighth result is an external calibration NO-GO. The moving pancake geometry is not intrinsically impossible in Euler or Navier--Stokes. Agafontsev--Kuznetsov--Mailybaev exhibit an exact three-dimensional Euler solution combining a shear flow and asymmetric straining flow to model high-vorticity pancake evolution. Their model includes: - one compressed transverse direction; - in-plane stretching directions; - an arbitrary transverse vorticity profile; - infinite global energy on: $$ \mathbb R^3; $$ - a Navier--Stokes extension in which the profile evolves by a heat equation. The model's vorticity is more specialized than the present rank-two covariance branch and should not be identified with it. It shows only that compressed sheet/pancake strain geometries can be exact fluid solutions. Even more strongly, Enciso--Fernández--Meyer (2026) construct exact Euler vorticities in thin tubular neighborhoods of analytic vortex sheets for time intervals uniform in the sheet thickness. Thus: $$ \boxed{ \textbf{ thin sheet-like vorticity geometry itself is not a contradiction. } } \tag{1.45} $$ The DCRP exclusion must use the specific: - same-parent DSS return; - critical tail; - PFET; - covariance-shape; - unforced reproduction; constraints. The corrected strongest rank-two state is therefore: $$ \boxed{ \textbf{ planar potential--shear DSS} + \left[ \textbf{ shape/phase action} \ \vee\ \textbf{ moving pancake jet with reproduction action} \right]. } \tag{1.46} $$ The new exact frontier is $$ \boxed{ \textbf{ Moving Pancake-Jet / Planar Potential--Shear Reproduction Rigidity. } } \tag{1.47} $$ The next PDE question is: > after the covariance-shape dynamics have been removed or quantified, can the local representation > > $$ > V=(\nabla_h\phi,w) > $$ > > with: > > $$ > \Omega_h=J\nabla_hq > $$ > > and the moving pancake affine strain satisfy the strict DSS return without producing: > > - normal-vorticity rank lifting; > - non-affine strain residual; > - material turnover; > - additional pressure/PFET work; > - or a sheet/tail transition defect? This is now the most concrete rank-two PDE frontier. --- # 2. Fermi--Walker frame on the vorticity plane Let $$ n(s) $$ be a smooth unit normal. Choose $$ E(s) = (e_1,e_2) $$ such that $$ E^TE=I_2, \qquad E^Tn=0. $$ There is an arbitrary in-plane $$ SO(2) $$ gauge. Choose the no-in-plane-rotation/Fermi gauge: $$ \boxed{ E^TE'=0. } \tag{2.1} $$ Differentiating $$ E^Tn=0 $$ gives $$ \boxed{ E' = - n n'^T E. } \tag{2.2} $$ Thus every column of $$ E' $$ is parallel to $$ n. $$ --- # 3. In-plane covariance equation Since $$ B=ECE^T $$ and $$ Bn=0, $$ the Fermi-frame derivative terms vanish: $$ E'^TBE = 0, \qquad E^TBE' = 0. $$ Hence $$ \boxed{ C' = E^TB'E. } \tag{3.1} $$ Using $$ B'=AB+BA-c_\gamma B, $$ one obtains $$ \boxed{ C' = A_EC + CA_E - c_\gamma C. } \tag{3.2} $$ Status: $$ \boxed{ \textbf{PROVED}. } $$ --- # 4. Magnitude and shape split Set $$ m=\operatorname{tr}C, \qquad P=C/m. $$ Decompose $$ A_E=aI_2+S, \qquad \operatorname{tr}S=0. $$ Then $$ \boxed{ \frac{m'}m = 2a + 2S:P - c_\gamma. } \tag{4.1} $$ The normalized shape satisfies $$ \boxed{ P' = SP + PS - 2(S:P)P. } \tag{4.2} $$ Status: $$ \boxed{ \textbf{PROVED}. } $$ --- # 5. Shape disk Every positive $$ P $$ with $$ \operatorname{tr}P=1 $$ has $$ P=\frac12(I+Z), $$ where $$ Z $$ is symmetric trace free. Write $$ Z = \begin{pmatrix} z_1&z_2\\ z_2&-z_1 \end{pmatrix}. $$ The eigenvalues of $$ P $$ are $$ \boxed{ \frac{ 1\pm|z| }{2}. } \tag{5.1} $$ Therefore $$ \boxed{ P>0 \Longleftrightarrow |z|<1. } \tag{5.2} $$ The rank-two covariance shape space is the open unit disk. --- # 6. Exact disk dynamics Write $$ S = \begin{pmatrix} s_1&s_2\\ s_2&-s_1 \end{pmatrix}. $$ Using $$ SP+PS = S + (s\cdot z)I, $$ and $$ S:P=s\cdot z, $$ equation (4.2) gives $$ \boxed{ \frac12Z' = S - (s\cdot z)Z. } \tag{6.1} $$ Thus in vector form: $$ \boxed{ z' = 2 \left[ s-(s\cdot z)z \right]. } \tag{6.2} $$ Status: $$ \boxed{ \textbf{PROVED}. } $$ --- # 7. Inversion of the planar shear Equation (6.2) is $$ z' = 2 (I-zz^T)s. $$ For $$ |z|<1, $$ the eigenvalues of $$ I-zz^T $$ are $$ 1 $$ and $$ 1-|z|^2. $$ Hence it is invertible. Therefore $$ \boxed{ s = \frac12 (I-zz^T)^{-1} z'. } \tag{7.1} $$ Status: $$ \boxed{ \textbf{PROVED}. } $$ This is the rank-two analogue of a phase-locking reconstruction theorem. --- # 8. Anisotropy parameter The normalized planar pseudo-determinant is $$ \vartheta_2 = 4\det P. $$ Since $$ \det P = \frac{ 1-|z|^2 }{4}, $$ $$ \boxed{ \vartheta_2 = 1-|z|^2. } \tag{8.1} $$ Thus the disk boundary is exactly the rank-one collapse boundary. --- # 9. Log-anisotropy equation Differentiate $$ \vartheta_2=1-|z|^2. $$ Using (6.2), $$ \boxed{ \vartheta_2' = -4 \vartheta_2 (s\cdot z). } \tag{9.1} $$ Hence $$ \boxed{ \frac d{ds} \log\vartheta_2 = -4 s\cdot z. } \tag{9.2} $$ For periodic rank-two covariance: $$ \boxed{ \int_0^{S_0} s\cdot z\,ds = 0. } \tag{9.3} $$ Thus the signed in-plane deviatoric stretching has zero logarithmic-anisotropy average over one return. --- # 10. Polar shape coordinates Write $$ z = r ( \cos2\theta, \sin2\theta ). $$ The factor $$ 2 $$ reflects that a symmetric-tensor principal axis is unchanged under $$ \theta\mapsto\theta+\pi. $$ Then $$ \vartheta_2=1-r^2. $$ The radial direction of $$ z $$ is the covariance anisotropy axis. The angular variable $$ \theta $$ is the in-plane covariance principal-axis phase. --- # 11. NEW THEOREM — Hyperbolic Shape/Phase Action ## Theorem 11.1 The in-plane deviatoric strain satisfies $$ \boxed{ |S|_F^2 = \frac{ (r')^2 }{ 2(1-r^2)^2 } + 2r^2 (\theta')^2. } \tag{11.1} $$ ### Proof The inverse $$ (I-zz^T)^{-1} $$ acts by: -: $$ (1-r^2)^{-1} $$ on the radial direction; -: $$ 1 $$ on the tangent direction. Also $$ z' = r'e_r + 2r\theta'e_\theta. $$ Thus $$ |s|^2 = \frac14 \left[ \frac{ (r')^2 }{ (1-r^2)^2 } + 4r^2 (\theta')^2 \right]. $$ Since $$ |S|_F^2=2|s|^2, $$ the formula follows. $$ \square $$ Status: $$ \boxed{ \textbf{PROVED}. } $$ --- # 12. Interpretation of the shape metric The action contains two positive pieces: $$ \boxed{ \mathcal A_{\rm amp} = \int \frac{ (r')^2 }{ 2(1-r^2)^2 } ds, } \tag{12.1} $$ and $$ \boxed{ \mathcal A_{\rm ang} = 2 \int r^2 (\theta')^2ds. } \tag{12.2} $$ Thus: - anisotropy changes near: $$ r=1 $$ are strongly amplified; - principal-axis rotation is visible whenever: $$ r>0. $$ At: $$ r=0, $$ orientation is correctly unobservable because the covariance is isotropic. This quotient safety is built into the formula. --- # 13. Shape-action gap Define $$ \boxed{ \mathcal A_{\rm shape} = \int_0^{S_0} |S|_F^2ds. } \tag{13.1} $$ Then: $$ \boxed{ \mathcal A_{\rm shape} \ge \frac12 \int_0^{S_0} |z'|^2ds. } \tag{13.2} $$ If $$ \vartheta_2\ge\vartheta_0>0, $$ then $$ \mathcal A_{\rm shape} $$ is quantitatively equivalent to the natural covariance-shape path action on the compact disk $$ |z|^2\le1-\vartheta_0. $$ Thus any nonconstant shape orbit has positive finite-dimensional action. --- # 14. Zero shape-action rigidity If $$ \mathcal A_{\rm shape}=0, $$ then $$ S=0 $$ almost everywhere. Smoothness gives $$ \boxed{ S(s)\equiv0. } \tag{14.1} $$ Hence $$ \boxed{ A_E=aI_2 } \tag{14.2} $$ at every time. Conversely, if $$ A_E=aI_2, $$ then $$ P'=0 $$ in the Fermi frame. Status: $$ \boxed{ \textbf{PROVED}. } $$ --- # 15. Reconstruction of the moving pancake jet Let $$ P_\perp = I-n\otimes n. $$ On the zero-shape-action branch: $$ A|_{n^\perp} = aP_\perp. $$ Trace free gives $$ \boxed{ n\cdot An = -2a. } \tag{15.1} $$ The normal equation gives $$ \boxed{ P_\perp An=-n'. } \tag{15.2} $$ Symmetry then uniquely gives $$ \boxed{ A = a \left( P_\perp-2n\otimes n \right) - \left( n'\otimes n+n\otimes n' \right). } \tag{15.3} $$ Status: $$ \boxed{ \textbf{PROVED}. } $$ This is the moving pancake-jet tensor. --- # 16. Norm of the pancake jet The tensors $$ P_\perp-2n\otimes n $$ and $$ n'\otimes n+n\otimes n' $$ are Frobenius orthogonal. Also $$ \boxed{ \left| P_\perp-2n\otimes n \right|_F^2 = 6, } \tag{16.1} $$ and $$ \boxed{ \left| n'\otimes n+n\otimes n' \right|_F^2 = 2|n'|^2. } \tag{16.2} $$ Therefore $$ \boxed{ |A|_F^2 = 6a^2 + 2|n'|^2. } \tag{16.3} $$ --- # 17. Mean pancake compression DCRP-40 gives $$ \left\langle n\cdot An \right\rangle = -c_\gamma. $$ Since $$ n\cdot An=-2a, $$ $$ \boxed{ \frac1{S_0} \int_0^{S_0} a(s)ds = \frac{ c_\gamma }{2}. } \tag{17.1} $$ Hence the planar extension coefficient has strictly positive mean. --- # 18. Universal pancake strain-action gap Jensen gives $$ \boxed{ \int_0^{S_0} a^2ds \ge \frac{ c_\gamma^2 }{4} S_0. } \tag{18.1} $$ Using (16.3): $$ \boxed{ \int_0^{S_0} |A|_F^2ds \ge \frac32 c_\gamma^2S_0 + 2 \int_0^{S_0} |n'|^2ds. } \tag{18.2} $$ Thus even the most rigid shape-static planar equality state carries a fixed normalized affine-strain action. --- # 19. Universal reproduction-action gap DCRP-36 established for periodic $$ A $$ the identity $$ \boxed{ \int_0^{S_0} |A'+A|^2ds = \int_0^{S_0} |A'|^2ds + \int_0^{S_0} |A|^2ds. } \tag{19.1} $$ Therefore the moving pancake branch satisfies $$ \boxed{ \mathcal A_{\rm rep} \ge \frac32 c_\gamma^2S_0. } \tag{19.2} $$ If the plane normal rotates, $$ n'\neq0, $$ the lower bound is larger. Status: $$ \boxed{ \textbf{PROVED}. } $$ This is a normalized source/reproduction witness, not a raw-energy contradiction. --- # 20. Rank-two master dichotomy Every exact zero-covariance-residual rank-two profile satisfies: $$ \boxed{ \mathcal A_{\rm shape}>0 } $$ or $$ \boxed{ A=A_{\rm pan}. } $$ Thus: $$ \boxed{ \textbf{ rank-two zero-residual} \Longrightarrow \textbf{ shape/phase dynamics} \ \vee\ \textbf{ moving pancake jet}. } \tag{20.1} $$ On the second branch: $$ \boxed{ \mathcal A_{\rm rep} \ge \frac32 c_\gamma^2S_0. } \tag{20.2} $$ No completely source-free planar equality remains. --- # 21. Relation to the conformal Floquet theorem DCRP-40 described the rank-two monodromy on the covariance plane as conformal in the covariance metric. DCRP-41 gives the continuous-time differential version. The in-plane deviatoric strain: $$ S $$ is exactly the generator of covariance-shape deformation. If $$ S=0, $$ the covariance shape is parallel transported in the Fermi frame and the only monodromy left comes from: - isotropic planar expansion; - geometric plane holonomy. Thus the conformal Floquet equality is the integrated form of the moving pancake-jet branch. --- # 22. Relation to the planar potential--shear field DCRP-40 showed that for fixed plane normal: $$ V = (\nabla_h\phi,w), $$ with $$ q=w-\partial_3\phi, $$ and $$ \Omega_h = (\partial_2q,-\partial_1q). $$ DCRP-41 does not yet solve the scalar dynamics of: $$ q. $$ It shows that the affine strain seen by the rank-two covariance has only two possibilities: - explicit finite-dimensional shape/shear action; - canonical moving pancake strain. Thus the infinite-dimensional problem has been pushed into the potential--shear carrier rather than the covariance geometry. --- # 23. External exact Euler pancake calibration Agafontsev--Kuznetsov--Mailybaev construct an exact Euler solution modeling pancake high-vorticity regions as a superposition of: - a shear flow; - an asymmetric irrotational straining flow. The solution has one compressed direction and growing vorticity and can exhibit arbitrary power-law relations between vorticity amplitude and pancake thickness. It has infinite energy on: $$ \mathbb R^3. $$ The paper also gives a Navier--Stokes extension in which the transverse profile obeys a heat equation. This shows: $$ \boxed{ \textbf{ pancake compression + shear + strain} } $$ is an exact fluid mechanism. The DCRP moving pancake jet is not identified with that model. The external model is used as a NO-GO against excluding the local geometry by appearance alone. --- # 24. Current vortex-sheet calibration Enciso--Fernández--Meyer (2026) prove that analytic three-dimensional vortex sheets can be desingularized into exact Euler vorticities supported in tubular neighborhoods of thickness $$ O(\varepsilon), $$ on a time interval bounded below independently of $$ \varepsilon. $$ The constructed vorticities are organized by foliations of almost parallel surfaces and divergence-free fields tangent to those surfaces. Therefore: $$ \boxed{ \textbf{ thin, approximately planar/tangent vorticity organization is compatible with exact Euler dynamics. } } \tag{24.1} $$ Again, this is calibration rather than identification with the DSS Type-II branch. --- # 25. Why sheet geometry is not enough The strict DCRP branch has extra constraints absent from generic pancake/sheet models: - same-parent DSS recurrence; - exponent window: $$ 2/5<\gamma<1/2; $$ - raw-energy vanishing; - mandatory DCRP-31 inward PFET; - periodic covariance; - finite-annulus strain reproduction; - zero or controlled transition defects. Thus external exact sheet/pancake solutions do not close or realize the DCRP branch automatically. --- # 26. Shape-action branch If $$ \mathcal A_{\rm shape}>0, $$ then at least one of: $$ r' $$ or $$ r\theta' $$ is nonzero. Hence the planar covariance must continually: - change its eigenvalue anisotropy; - or rotate its principal axis relative to the Fermi plane. The exact source of that motion is the in-plane deviatoric annular strain $$ S. $$ Therefore the rank-two phase problem has been reduced to a positive finite-dimensional source action. No additional abstract phase variable is required. --- # 27. Near-rank-one cost As $$ r\uparrow1, $$ $$ \vartheta_2=1-r^2\downarrow0. $$ The radial action contains: $$ \frac{ (r')^2 }{ (1-r^2)^2 }. $$ Thus a trajectory which repeatedly approaches rank one and returns to the rank-two interior pays a large shape-metric action unless the approach/recovery rate becomes correspondingly slow. This creates a quantitative bridge between: - DCRP-39 rank-one collapse; - DCRP-41 rank-two shape dynamics. A full transition lower bound requires a specified distance excursion and is left for later use. --- # 28. Moving-plane cost On the shape-static branch, all in-plane shear vanishes, but plane rotation remains through $$ n'(s). $$ The affine tensor contains $$ - (n'\otimes n+n\otimes n'). $$ Thus moving-plane recurrence is not free at the affine-jet reproduction level. The strain-action contains the explicit term: $$ \boxed{ 2 \int |n'|^2ds. } \tag{28.1} $$ This is the rank-two counterpart of the axis-rotation term in the rank-one Burgers jet. --- # 29. What is still not proved DCRP-41 does not prove that positive normalized: $$ \mathcal A_{\rm shape} $$ or $$ \mathcal A_{\rm rep} $$ is non-summable in raw physical variables. The critical-scaling NO-GOs from earlier rounds still apply. The new quantities are equality-manifold/source classifiers. A global contradiction requires a same-parent return-depletion or a PDE Liouville theorem for the final moving pancake state. --- # 30. Corrected final rank-two state The DCRP-40 survivor $$ \text{planar conformal Floquet vorticity} $$ is now refined to $$ \boxed{ \textbf{ planar potential--shear DSS} + \left[ \textbf{ positive covariance shape/phase action} \ \vee\ \textbf{ moving pancake jet} \right]. } \tag{30.1} $$ The moving pancake jet is completely described by $$ \boxed{ a(s) } $$ and $$ \boxed{ n(s)\in S^2. } $$ This is a three-dimensional finite parameter fiber coupled to the planar potential--shear PDE. --- # 31. Correct next frontier The next target is $$ \boxed{ \textbf{ Moving Pancake-Jet / Planar Potential--Shear Reproduction Rigidity. } } $$ A useful theorem should take the zero-shape-action branch $$ A=A_{\rm pan} $$ and the local representation $$ V=(\nabla_h\phi,w) $$ and derive a scalar/differential-form evolution for $$ q=w-\partial_n\phi. $$ The desired classification is: $$ \boxed{ \text{normal-vorticity rank lifting} \ \vee\ \text{non-affine strain residual} \ \vee\ \text{material/sheet turnover} \ \vee\ \text{pressure/PFET source} \ \vee\ \text{exact pancake eigenmode}. } $$ The last branch should then be compared with: - exact Euler pancake solutions; - vortex-sheet-type Euler solutions; - the strict DSS tail and same-parent finite-energy ancestry. This is now the principal low-rank PDE frontier. --- # 32. End state The normalized planar covariance shape obeys the exact disk equation $$ \boxed{ z' = 2 \left( I-zz^T \right)s. } $$ Therefore $$ \boxed{ s = \frac12 \left( I-zz^T \right)^{-1}z'. } $$ The in-plane deviatoric strain is exactly the covariance-shape velocity. In polar tensor coordinates: $$ \boxed{ |S|_F^2 = \frac{ (r')^2 }{ 2(1-r^2)^2 } + 2r^2 (\theta')^2. } $$ Thus rank-two covariance deformation has a positive hyperbolic amplitude/phase action. If that action vanishes, $$ \boxed{ A = a \left( P_{n^\perp} - 2n\otimes n \right) - \left( n'\otimes n+n\otimes n' \right). } $$ DSS periodicity forces $$ \boxed{ \langle a\rangle = \frac{ 2-3\gamma }{2}. } $$ Therefore the zero-shape-action branch is a moving pancake jet with universal normalized strain/reproduction activity. The rank-two problem has now been reduced from a generic planar Floquet covariance to: $$ \boxed{ \textbf{ a planar potential--shear PDE driven by a three-parameter moving pancake strain. } } $$ The next frontier is: $$ \boxed{ \textbf{ Moving Pancake-Jet / Planar Potential--Shear Reproduction Rigidity. } } $$