# DCRP105 / X72-R88 — Shear-Polarized TR Geometry, Vanishing-Viscosity Residual Matching, and the No-Forced-Spectral-Migration Audit **Series:** Independent Navier–Stokes Research Series / X72 Bridge **Date:** 2026-08-20 **Status:** proof-development checkpoint / vanishing-viscosity shear-polarization round **Immediate predecessor:** `NS_DCRP104_X72R87_RieszSelfConsistency_ShearPolarization_2026-08-20.md` ## Primary internal dependencies - DCRP95–96 — sign-coherent Kelvin phase slip / nematic second-moment lock. - DCRP102 — recurrent fixed-sign transport–Riesz angular pair cell. - DCRP103 — local adjoint five-ray classification. - DCRP104 — frozen Riesz self-consistency: - coaxial \(L^2\) NO-GO; - simple-shear off-resonance \(L^2\) survivors; - axisymmetric polarization kernels; - exact positive-viscosity frozen \(L^2\) eigen-lock NO-GO. ## Fresh primary-source calibration - G. Seregin, *On potential Type II blowups for the Navier-Stokes equations*, arXiv:2606.29468 (2026). Current Type-II analysis continues to use Euler-scaled limits and Liouville-type exclusions. - R. Yu, *Filtered Vortex Stretching and Subgrid Defects for the Three-Dimensional Navier-Stokes Equations*, arXiv:2606.27560 (2026). The differentiated filtered stress is controlled by scale-invariant velocity-increment defects; at criticality bounded defects produce cylindrical generalized Young profiles. - E. Hess-Childs, M. Rosenzweig, S. Serfaty, *Another look at regularity in transport-commutator estimates*, arXiv:2601.02326 (2026). Generic Riesz transport commutator control retains a sharp regularity burden; soft BMO replacement of Lipschitz control fails in general. - B. Álvarez-Samaniego, W. Álvarez-Samaniego, P. 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. No full Navier–Stokes regularity theorem is claimed. --- # 0. Executive result D104 left three frozen inviscid nonlocal mechanisms: \[ \boxed{ \mathsf K_{\rm sh}^{\rm off-res} \vee \mathsf K_{\rm sh}^{\rm near-res} \vee \mathsf K_{\rm axi-pol}. } \] It also proved that for every fixed: \[ \varepsilon>0, \] the exact globally frozen whole-space \(L^2\) viscous eigen-lock is zero. D105 audits whether this positive-viscosity NO-GO becomes a useful **uniform** obstruction as: \[ \varepsilon\to0. \] It does not. Let: \[ \mathscr M_0(n) \] be the frozen inviscid adjoint symbol on one D104 shear/polarization kernel and let: \[ \widehat\Phi_0(\xi) \] satisfy: \[ \boxed{ \mathscr M_0(\widehat\xi)\widehat\Phi_0(\xi)=0. } \tag{0.1} \] The viscous symbol is: \[ \boxed{ \mathscr M_\varepsilon(\xi) = \mathscr M_0(\widehat\xi) + \varepsilon|\xi|^2I. } \tag{0.2} \] Therefore the **same inviscid normalized profile** has residual: \[ \boxed{ \widehat R_\varepsilon = \mathscr M_\varepsilon\widehat\Phi_0 = \varepsilon|\xi|^2\widehat\Phi_0. } \tag{0.3} \] Hence, for every: \[ \Phi_0\in H^2, \] \[ \boxed{ \|R_\varepsilon\|_2 = \varepsilon \|\Delta\Phi_0\|_2. } \tag{0.4} \] Thus a fixed normalized frequency profile survives as an \(O(\varepsilon)\) approximate eigen-lock. There is **no forced frequency migration** at the natural \(O(\varepsilon)\) residual scale. The true critical quantity is: \[ \boxed{ \Theta_{\rm vis} := \frac{ \eta_\varepsilon }{ \varepsilon }, } \tag{0.5} \] where: \[ \eta_\varepsilon = \|\mathscr M_\varepsilon\Phi_\varepsilon\|_2 \] is the adjoint eigen-lock residual. If a nontrivial kernel component remains in a fixed normalized radial band: \[ \rho_- \le |\xi| \le \rho_+, \qquad \rho_->0, \] then: \[ \boxed{ \eta_\varepsilon \ge \varepsilon \rho_-^2 \|\Phi_\varepsilon\|_2. } \tag{0.6} \] Hence order-one fixed-band mass requires: \[ \boxed{ \eta_\varepsilon \gtrsim \varepsilon. } \tag{0.7} \] Conversely, if: \[ \boxed{ \eta_\varepsilon=o(\varepsilon), } \tag{0.8} \] then every normalized kernel component is forced toward: \[ \boxed{ |\xi|\to0. } \] More precisely, if: \[ \|\Phi_\varepsilon\|_2=1, \] and the state lies exactly in the inviscid kernel bundle, then for every fixed: \[ \rho>0, \] \[ \boxed{ \int_{|\xi|\ge\rho} |\widehat\Phi_\varepsilon|^2d\xi \le \frac1{\rho^4} \left( \frac{\eta_\varepsilon}{\varepsilon} \right)^2. } \tag{0.9} \] So: \[ \eta_\varepsilon/\varepsilon\to0 \] forces low-normalized-frequency migration. In the same-parent compact compiler this is a large-spatial-scale / scale-state escape unless the profile vanishes. Therefore: # Main viscosity-rate trichotomy \[ \boxed{ \begin{array}{rcl} \eta_\varepsilon=o(\varepsilon) &\Longrightarrow& \text{low-frequency / large-scale escape or vanishing}, \\[1mm] \eta_\varepsilon\asymp\varepsilon &\Longrightarrow& \text{fixed-scale inviscid shear/polarization may survive}, \\[1mm] \eta_\varepsilon\gg\varepsilon &\Longrightarrow& \text{eigen-lock residual itself is active}. \end{array} } \tag{0.10} \] This is the main D105 result. The positive-viscosity exact NO-GO is a **singular perturbation**: \[ \boxed{ \ker\mathscr M_\varepsilon=\{0\} \quad(\varepsilon>0), \qquad \ker\mathscr M_0\neq\{0\}. } \tag{0.11} \] The smallest singular value on a fixed inviscid-kernel frequency band collapses linearly like: \[ \boxed{ \sigma_{\min} \sim \varepsilon|\xi|^2. } \tag{0.12} \] D105 also checks the D102 TR angular cone and the D95 Kelvin nematic lock. They do **not** eliminate the simple-shear survivor locally. For the trace-free Riesz kernel: \[ K_0(z) = c|z|^{-3} (I-3n\otimes n), \qquad n=z/|z|, \] and shear tensor: \[ H_{ij} = e_i\otimes e_j+e_j\otimes e_i, \] the exact pair projection is: \[ \boxed{ \begin{aligned} & [(v\cdot\nabla)K_0(rn)] : H_{ij} \\ &\qquad= -6c\,r^{-4} \left[ v_i n_j + v_j n_i - 5(v\cdot n)n_in_j \right]. \end{aligned} } \tag{0.13} \] For: \[ H_{13}, \qquad n=e_3, \qquad v=e_1, \] this equals: \[ \boxed{ -6c\,r^{-4}\neq0. } \tag{0.14} \] At the same time choose Kelvin detector: \[ A_\Gamma = \operatorname{diag}(1,-1,0) \] and increment covariance: \[ Q=e_1\otimes e_1. \] Then: \[ \boxed{ A_\Gamma:Q=1>0. } \tag{0.15} \] Choosing the sign of: \[ \delta q \] appropriately gives a positive TR factor simultaneously. Thus: \[ \boxed{ \text{Kelvin nematic lock} + \text{simple-shear TR angular lock} } \] is locally algebraically compatible. The late compact survivor therefore becomes: \[ \boxed{ \mathsf C_{\rm vm\mbox{-}shear/pol} } \] = a **viscosity-matched shear/polarization conveyor** carrying: - \(O(\varepsilon)\) adjoint eigen-lock residual; - fixed normalized spectral support or axisymmetric polarization; - fixed-sign TR angular pair cell; - Kelvin nematic second-moment lock. --- # 1. Exact shear projection of the Riesz derivative Use: \[ K_{ab}(z) = c \left[ r^{-3}\delta_{ab} - 3r^{-5}z_az_b \right], \qquad r=|z|. \] Let: \[ z=rn. \] Differentiate in the direction: \[ v. \] A direct calculation gives: \[ \boxed{ (v\cdot\nabla)K_0(rn) = -3cr^{-4} \left[ (v\cdot n)I + v\otimes n + n\otimes v - 5(v\cdot n)n\otimes n \right]. } \tag{1.1} \] For: \[ H_{ij} = e_i\otimes e_j+e_j\otimes e_i, \] one has: \[ I:H_{ij}=0, \] \[ (v\otimes n+n\otimes v):H_{ij} = 2(v_in_j+v_jn_i), \] \[ (n\otimes n):H_{ij} = 2n_in_j. \] Hence: ## Theorem D105.1 — Simple-shear TR angular scalar \[ \boxed{ [(v\cdot\nabla)K_0(rn)] : H_{ij} = -6cr^{-4} [ v_in_j+v_jn_i-5(v\cdot n)n_in_j ]. } \tag{1.2} \] This scalar is generically sign-indefinite. So the recurrent D102 TR angular cell is a real orientation/correlation constraint, not a pointwise positivity identity. --- # 2. Local Kelvin/TR compatibility Take: \[ i=1, \qquad j=3, \] \[ n=e_3, \qquad v=e_1. \] Then: \[ v\cdot n=0, \] and D105.1 gives: \[ \boxed{ [(e_1\cdot\nabla)K_0(re_3)] : H_{13} = -6cr^{-4}. } \tag{2.1} \] Now let the Kelvin covariance be: \[ Q=e_1\otimes e_1. \] Take: \[ A_\Gamma = \operatorname{diag}(1,-1,0). \] Then: \[ \boxed{ A_\Gamma:Q=1. } \tag{2.2} \] If: \[ c>0, \] choose: \[ \delta q<0. \] Then: \[ \boxed{ [(e_1\cdot\nabla)K_0(re_3)] : H_{13} \, \delta q > 0. } \tag{2.3} \] Thus: ## Theorem D105.2 — Local Kelvin/TR compatibility witness The same velocity-increment direction can satisfy: - positive Kelvin nematic covariance projection; - positive simple-shear TR angular projection. So no generic local second-moment/angular-sign contradiction exists. ### Scope The Fourier direction controlling D104 Riesz self-consistency and the pair-separation direction \(n=z/|z|\) in D105.1 are different variables. D105 does not identify them. --- # 3. Viscous residual on an inviscid kernel Let: \[ P(n) \] be a smooth inviscid shear/polarization kernel bundle on one compact angular patch away from D104 spectral degeneracies. Let: \[ \widehat\Phi = P(n)\widehat\Phi. \] Then: \[ \mathscr M_0(n)\widehat\Phi=0. \] The prelimit frozen operator is: \[ \mathscr M_\varepsilon(\xi) = \mathscr M_0(n) + \varepsilon|\xi|^2I. \] Therefore: ## Theorem D105.3 — Exact viscosity residual on the inviscid kernel \[ \boxed{ \mathscr M_\varepsilon(\xi)\widehat\Phi = \varepsilon|\xi|^2\widehat\Phi. } \tag{3.1} \] Consequently: \[ \boxed{ \|\mathscr M_\varepsilon\Phi\|_2 = \varepsilon \||\xi|^2\widehat\Phi\|_2 = \varepsilon \|\Delta\Phi\|_2. } \tag{3.2} \] This equality is exact for a fixed frozen inviscid-kernel field. --- # 4. Fixed radial band lower bound Suppose: \[ \operatorname{supp}\widehat\Phi \subset \{ \rho_-\le|\xi|\le\rho_+ \}, \] where: \[ 0<\rho_-\le\rho_+<\infty. \] Then: \[ \rho_-^2 \|\Phi\|_2 \le \||\xi|^2\widehat\Phi\|_2 \le \rho_+^2 \|\Phi\|_2. \] Therefore: ## Theorem D105.4 — Fixed-band viscosity matching \[ \boxed{ \varepsilon\rho_-^2\|\Phi\|_2 \le \|\mathscr M_\varepsilon\Phi\|_2 \le \varepsilon\rho_+^2\|\Phi\|_2. } \tag{4.1} \] For normalized nontrivial mass: \[ \|\Phi\|_2\ge m_0>0, \] one necessarily has: \[ \boxed{ \eta_\varepsilon \ge m_0\rho_-^2\varepsilon. } \tag{4.2} \] Thus fixed normalized scale requires an \(O(\varepsilon)\) residual. --- # 5. Subviscous residual forces low-frequency migration Assume: \[ \|\Phi_\varepsilon\|_2=1, \] and: \[ \mathscr M_0(\widehat\xi)\widehat\Phi_\varepsilon=0. \] Set: \[ \eta_\varepsilon = \|\mathscr M_\varepsilon\Phi_\varepsilon\|_2. \] Then: \[ \frac{\eta_\varepsilon^2}{\varepsilon^2} = \int |\xi|^4 |\widehat\Phi_\varepsilon|^2d\xi. \] For every: \[ \rho>0, \] \[ \begin{aligned} \int_{|\xi|\ge\rho} |\widehat\Phi_\varepsilon|^2d\xi &\le \rho^{-4} \int |\xi|^4 |\widehat\Phi_\varepsilon|^2d\xi \\ &= \rho^{-4} \left( \frac{\eta_\varepsilon}{\varepsilon} \right)^2. \end{aligned} \] Thus: ## Theorem D105.5 — Subviscous residual migration If: \[ \boxed{ \eta_\varepsilon/\varepsilon\to0, } \] then: \[ \boxed{ \forall\rho>0: \quad \int_{|\xi|\ge\rho} |\widehat\Phi_\varepsilon|^2 \to0. } \tag{5.1} \] The normalized Fourier mass collapses toward zero frequency. For a same-parent compact spatial package, persistent order-one low-frequency concentration is a large-normalized-scale / tail/state issue, not a fixed-core recurrent mode. --- # 6. The \(O(\varepsilon)\) rate is achievable Let: \[ \Phi_0 \] be any nonzero smooth frozen inviscid shear/polarization survivor from D104 with: \[ \Phi_0\in H^2. \] Use the same normalized profile for every \(\varepsilon\): \[ \Phi_\varepsilon:=\Phi_0. \] Then: \[ \boxed{ R_\varepsilon = \mathscr M_\varepsilon\Phi_0 = -\varepsilon\Delta\Phi_0 } \] up to Fourier sign convention, and therefore: \[ \boxed{ \|R_\varepsilon\|_2 = \varepsilon \|\Delta\Phi_0\|_2. } \tag{6.1} \] So: ## Theorem D105.6 — No forced spectral migration at natural viscosity order A fixed normalized shear/polarization spectrum can approach the inviscid eigen-lock with residual exactly \(O(\varepsilon)\). Therefore: \[ \boxed{ \varepsilon\to0 \not\Rightarrow \text{frequency migration}. } \tag{6.2} \] The frozen positive-viscosity exact NO-GO is nonuniform in the inviscid limit. --- # 7. Singular-value interpretation On a fixed radial band and a smooth inviscid-kernel angular patch, the viscous symbol restricted to the inviscid kernel is simply: \[ \varepsilon|\xi|^2I. \] Thus: \[ \boxed{ \sigma_{\min}^{\rm ker} ( \mathscr M_\varepsilon ) = \varepsilon|\xi|^2. } \tag{7.1} \] On: \[ \rho_-\le|\xi|\le\rho_+, \] \[ \boxed{ \varepsilon\rho_-^2 \le \sigma_{\min}^{\rm ker} \le \varepsilon\rho_+^2. } \tag{7.2} \] The inverse condition number therefore diverges like: \[ \boxed{ \varepsilon^{-1}. } \tag{7.3} \] This is the exact singular-perturbation mechanism behind: \[ \ker\mathscr M_\varepsilon=\{0\} \quad (\varepsilon>0), \] but: \[ \ker\mathscr M_0\neq\{0\}. \] --- # 8. Near-resonance simple-shear debt D104 simple shear has: \[ \widehat r = \frac{ h(n) }{ d(n) } \widehat q, \] where: \[ h(n)=-2n_in_j, \] and: \[ d(n)=1-a(n). \] Let: \[ E_\delta = \{ |d(n)|\le\delta, \quad |h(n)|\ge h_0 \}. \] Then: \[ |\widehat r| \ge \frac{h_0}{\delta} |\widehat q| \] on \(E_\delta\). Therefore: ## Theorem D105.7 — Near-resonance compactness debt \[ \boxed{ \|\widehat q\|_{L^2(E_\delta)} \le \frac{\delta}{h_0} \|\widehat r\|_{L^2(E_\delta)}. } \tag{8.1} \] If: \[ \|r\|_2\le R_*, \] then: \[ \boxed{ \|\widehat q\|_{L^2(E_\delta)} \le \frac{\delta R_*}{h_0}. } \tag{8.2} \] So bounded \(r\) forbids order-one source mass from accumulating on a nondegenerate part of the resonance cone. Near resonance, one must have at least one of: \[ \boxed{ \text{source suppression} \vee \text{numerator/angular degeneration} \vee \text{Riesz amplitude blow-up}. } \tag{8.3} \] The last is: \[ R_{\rm crit} \vee R_{\rm state}. \] --- # 9. Concrete \(S=\operatorname{diag}(1,0,-1)\), \(H_{13}\) near-resonance geometry D104 gives: \[ d(n)=3n_2^2, \] \[ h(n)=-2n_1n_3. \] The resonance set is: \[ \boxed{ n_2=0. } \tag{9.1} \] The numerator vanishes when: \[ \boxed{ n_1n_3=0. } \tag{9.2} \] On the resonance great circle, simultaneous numerator degeneration occurs only at: \[ \boxed{ n=\pm e_1, \qquad n=\pm e_3. } \tag{9.3} \] Therefore a bounded-\(r\) near-resonance sequence with nonvanishing \(q\)-mass can avoid blow-up only by angular concentration toward those isolated endpoint directions or by suppressing \(q\). This compresses the canonical near-resonance branch to a finite angular endpoint set. ### Scope This \(n\) is the Fourier direction. It is not the D102 physical pair-separation direction. --- # 10. Axisymmetric polarization has the same viscosity rate threshold D104 axisymmetric polarization has: \[ r=0, \] and: \[ \mathscr M_0(n)\widehat\Phi=0 \] on an open set of Fourier directions. Therefore D105.3 applies unchanged: \[ \boxed{ \mathscr M_\varepsilon\widehat\Phi = \varepsilon|\xi|^2\widehat\Phi. } \tag{10.1} \] Thus: - exact positive-viscosity \(L^2\) eigen-lock is zero; - \(O(\varepsilon)\) approximate fixed-band polarization survives; - \(o(\varepsilon)\) residual forces low-frequency migration. So the critical residual rate is not special to simple shear. It is common to every open-set inviscid polarization kernel. --- # 11. Why strict DSS recurrence does not by itself kill the \(O(\varepsilon)\) conveyor In the normalized Euler-limit architecture: \[ \varepsilon_n\to0. \] If the same normalized inviscid shear/polarization profile recurs at each generation, then: \[ \eta_n = O(\varepsilon_n). \] Therefore the normalized residual itself becomes smaller with generation. There is no contradiction from this fact alone. In particular, D105 does not infer: \[ \sum_n\eta_n=\infty. \] Depending on the same-parent scaling, \(\varepsilon_n\) may itself be summable. Thus the correct obstruction is not raw residual summation. It is the **rate-matching and first-order solvability** of the recurrent state. --- # 12. Vanishing-viscosity rate compiler Define: \[ \boxed{ \Theta_n = \frac{ \eta_n }{ \varepsilon_n }. } \tag{12.1} \] On a normalized nontrivial shear/polarization kernel sequence: ## Regime I — subviscous \[ \boxed{ \Theta_n\to0. } \] Then: \[ \boxed{ \text{low-frequency migration} \vee R_{\rm scale} \vee R_{\rm tail} \vee R_{\rm state}. } \tag{12.2} \] ## Regime II — viscosity matched \[ \boxed{ 0 Exact frozen positive-viscosity \(L^2\) kernel is zero, therefore the inviscid ray cannot arise as a vanishing-viscosity limit. False. The kernel is singular in \(\varepsilon\), and \(O(\varepsilon)\) residuals approximate inviscid kernel states at fixed normalized frequency. ## False route B > Vanishing viscosity forces high-frequency or low-frequency migration. False at the natural \(O(\varepsilon)\) residual scale. A fixed normalized spectral patch survives. ## True route If the eigen-lock residual is **smaller than viscosity**: \[ o(\varepsilon), \] then normalized Fourier mass must migrate to zero frequency or vanish. ## True near-resonance route Bounded Riesz scalar forbids order-one source mass on nondegenerate resonance sectors. --- # 15. Status ledger ## PROVED this round ### D105-P1 — exact simple-shear directional Riesz projection formula. ### D105-P2 — local Kelvin nematic / simple-shear TR angular compatibility witness. ### D105-P3 — exact viscous residual on an inviscid kernel: \[ R_\varepsilon=\varepsilon|\xi|^2\Phi. \] ### D105-P4 — fixed normalized radial band requires residual \(\gtrsim\varepsilon\). ### D105-P5 — \(o(\varepsilon)\) residual forces low-frequency Fourier migration. ### D105-P6 — \(O(\varepsilon)\) residual is explicitly achievable with a fixed normalized inviscid profile. ### D105-P7 — inviscid kernel is a singular perturbation of the positive-viscosity operator with singular value \(\sim\varepsilon|\xi|^2\). ### D105-P8 — near-resonance source/Riesz amplitude inequality. ### D105-P9 — canonical shear near-resonance set reduces to a great circle with only four numerator-degenerate endpoints. ### D105-P10 — axisymmetric polarization has the same \(O(\varepsilon)\) critical residual rate. ### D105-P11 — strict DSS recurrence plus \(\varepsilon_n\to0\) does not itself contradict viscosity-matched recurrence. ### D105-P12 — the remaining prelimit ray branch is compressed to the viscosity-matched shear/polarization conveyor. --- # 16. What is NOT proved D105 does not prove: - \(\mathsf C_{\rm vm\mbox{-}shear/pol}\) is impossible; - the actual adjoint residual of a Navier–Stokes prelimit must be \(o(\varepsilon)\); - \(\varepsilon_n\) is nonsummable across generations; - low normalized frequency always implies literal spatial-tail escape without an additional localization/Poincaré hypothesis; - the Fourier angle in Riesz self-consistency equals the physical pair angle in the TR detector; - Kelvin nematic lock determines the shear spectral polarization; - near-resonance endpoint concentration is impossible; - global Navier–Stokes regularity. The remaining problem is now a **first-order vanishing-viscosity solvability / spectral-drift problem**. --- # 17. STOP-D105 \[ \boxed{ \begin{minipage}{0.94\linewidth} The exact positive-viscosity frozen \(L^2\) eigen-lock NO-GO from D104 is not uniform as viscosity vanishes. On any D104 inviscid shear or axisymmetric polarization kernel, the viscous symbol is simply \(\varepsilon|\xi|^2\) on the kernel, so the same fixed normalized profile has residual \(R_\varepsilon=\varepsilon|\xi|^2\Phi\) and \(\|R_\varepsilon\|_2=\varepsilon\|\Delta\Phi\|_2\). Hence no spectral migration is required at the natural \(O(\varepsilon)\) residual rate. The critical quantity is \(\Theta_\varepsilon=\eta_\varepsilon/\varepsilon\): fixed normalized-band mass forces \(\eta_\varepsilon\gtrsim\varepsilon\), while \(\eta_\varepsilon=o(\varepsilon)\) quantitatively forces all Fourier mass toward \(|\xi|=0\), which becomes a large-scale/state escape in the compact same-parent compiler. Near a simple-shear Riesz resonance, bounded \(r\) forces source suppression or numerator/angular degeneration; for the canonical \(S=\operatorname{diag}(1,0,-1)\), \(H_{13}\) branch, the resonance circle \(n_2=0\) has numerator degeneration only at \(\pm e_1,\pm e_3\). The D102 TR cone and D95 Kelvin nematic lock do not kill the off-resonance shear locally: the same increment direction \(e_1\) gives a positive Kelvin covariance projection and a nonzero \(H_{13}\) Riesz-derivative pairing at pair direction \(e_3\). Thus the sharp remaining prelimit state is a viscosity-matched shear/polarization conveyor with eigen-lock residual of order exactly \(\varepsilon\), fixed-sign TR angular occupancy, and Kelvin second-moment lock. The next question is not whether viscosity kills the inviscid kernel, but whether a recurrent strict-DSS state can satisfy the first-order solvability conditions needed to absorb that \(O(\varepsilon)\) perturbation without radial spectral narrowing or coefficient/eigenframe drift. \end{minipage} } \] --- # 18. Next autonomous step ## DCRP106 / X72-R89 — First-Order Fredholm / Radial Spectral Narrowing **Working title** > **Can the Viscosity-Matched Shear/Polarization Conveyor Absorb the \(O(\varepsilon|\xi|^2)\) Perturbation by a Single Recurring Eigenvalue/State Correction, or Must Its Spectrum Narrow onto a Radial Shell?** Primary tasks: 1. linearize: \[ \beta_\varepsilon = \beta_0 + \varepsilon\beta_1 +o(\varepsilon); \] 2. allow: \[ S_\varepsilon = S_0+\varepsilon S_1+o(\varepsilon); \] 3. project the \(O(\varepsilon)\) equation onto the inviscid kernel/cokernel; 4. derive the Fredholm solvability condition; 5. on a fixed shear polarization, compare: \[ \beta_1 \quad\text{with}\quad |\xi|^2; \] 6. test whether a single \(\beta_1\) can cancel viscosity on a broadband radial spectrum; 7. seek: \[ \text{broadband survivor} \Longrightarrow \text{radial spectral narrowing} \vee \text{coefficient/eigenframe drift} \vee R_{\rm state}; \] 8. audit whether shell narrowing is compatible with spatial localization / strict DSS recurrence. Desired endpoint: \[ \boxed{ \mathsf C_{\rm vm} \Longrightarrow \text{one radial-shell / coefficient-drift normal form} \vee R_{\rm scale} \vee R_{\rm state} \vee R_{\rm crit}. } \] **End checkpoint:** DCRP105 / X72-R88.