--- title: "Navier–Stokes Forest Coercive Budget Program 03: Signed Pressure–Flux Work, Variable-Radius Telescoping, Slow-Scale Critical Lift, Filter-Switch Defects and Model-Cone Recurrence" short_title: "NS-FCBP 03" series: "Navier–Stokes Forest Coercive Budget Program" cycle: "VI" version: "v0.1" date: "2026-08-16" author: "Neo.K / EveMissLab" language: "en" status: "Signed-work critical-lift attempt / slow-scale telescope breakthrough / moving-filter reduction" epistemic_status: "Generalizes the exact pressure-flux endpoint telescope algebra from geometric radii to arbitrary strictly decreasing radii for one common coarse package. This internal Variable-Radius Pressure–Flux Telescope shows that the summable scale weight in the published geometric theorem is not intrinsic to the telescope. Choosing r_k=r_0(k+1)^(-beta), 1/20; scale-relative moving filters preserve critical resolution but break exact endpoint telescoping. An exact Moving-Filter Telescope is proved with an explicit positive filter-switch defect. A chain-dependent finest-scale common-filter strategy removes switch defects but replaces them by uniform-in-chain-length observability/leakage/backscatter requirements. A scale-critical Model-Cone Growth Packet is also proved from the Miller strain balance. The remaining Critical Lift is reduced to filter-switch or uniform common-filter packing, weighted backscatter/leakage closure, and active-work observability on a non-summable set of slow slabs. Forest Coercive Budget, Finite Forest Obstruction, atomic CN3, and Navier-Stokes regularity remain OPEN." canonical_source: "UTF-8 Markdown" --- # Navier–Stokes Forest Coercive Budget Program 03 # Signed Pressure–Flux Work, Variable-Radius Telescoping, Slow-Scale Critical Lift, Filter-Switch Defects and Model-Cone Recurrence ## 0. Document Positioning FCBP-02 proved that the filtered route has a complete conditional compiler but still lacks global critical packing. The dominant barriers were: - comparable-annulus signed packing; - weighted backscatter; - scale-critical commutator recurrence; - observability. The pressure--flux work framework is attractive because it already preserves signs and gives an exact endpoint telescope. The published theorem uses geometric radii: $$ r_k=\theta^k r_0. $$ The first question is: > Is the resulting summable weight $w_k=r_k/r_0$ intrinsic to the PDE telescope, or only to the chosen geometric schedule? The answer is: $$ \boxed{ \textbf{it is not intrinsic to the telescope}. } $$ This creates the first actual non-summable schedule inside the FCBP architecture. --- # 1. External fixed-chain pressure--flux ledger Fix one physical coarse length: $$ \ell>0. $$ Let: $$ U^\ell=S_\ell u, \qquad P^\ell=S_\ell p, $$ $$ R^\ell = S_\ell(u\otimes u) - U^\ell\otimes U^\ell, $$ and: $$ \Pi^\ell = - R^\ell:\nabla U^\ell. $$ Define the signed combined work distribution: $$ \boxed{ G^\ell = \Pi^\ell + \nabla\cdot(P^\ell U^\ell). } $$ The external local coarse energy identity is valid for an arbitrary time interval: $$ I=(t_-,t_+), $$ radius: $$ r>0, $$ and admissible nonnegative weight: $$ \phi: $$ $$ \boxed{ \mathcal W_{I,r}[\phi] + \mathcal D_{I,r}[\phi] = r^{-1} \left( K_\phi^\ell(t_-) - K_\phi^\ell(t_+) \right) + \mathcal L_{I,r}[\phi]. } $$ ### Status $$ \boxed{ \mathrm{EXTERNAL/PROVED}. } $$ --- # 2. Published geometric-chain theorem The external theorem fixes: $$ r_k=\theta^kr_0, \qquad 0<\theta<1, $$ and adjacent slabs: $$ \tau_{k+1}-\tau_k=r_k^2. $$ The spatial cutoffs: $$ \chi_k(x) = \chi \left( \frac{x-x_0}{r_k} \right) $$ satisfy: $$ \boxed{ \chi_{k+1}\le\chi_k. } $$ The active weights have common endpoint traces: $$ \boxed{ \widehat\phi_k(\cdot,\tau_k) = \widehat\phi_k(\cdot,\tau_{k+1}) = \chi_k. } $$ With: $$ w_k=\frac{r_k}{r_0}, $$ the external pressure--flux telescope gives: $$ \boxed{ \sum_{k=0}^{N-1} w_k ( \mathcal W_k^+ + \mathcal D_k ) \le \mathcal E_0^- + \sum_{k=0}^{N-1} w_k|\mathcal L_k| + \sum_{k=0}^{N-1} w_k\mathcal W_k^-. } $$ ### Status $$ \boxed{ \mathrm{EXTERNAL/PROVED}. } $$ --- # 3. What the external proof actually uses The endpoint part of the telescope is: $$ \sum_{k=0}^{N-1} w_k ( \mathcal E_k^- - \mathcal E_k^+ ). $$ The proof uses: $$ \boxed{ w_kr_k^{-1} = r_0^{-1}, } $$ and: $$ \boxed{ \chi_k\le\chi_{k-1}. } $$ The constant geometric ratio: $$ r_{k+1}/r_k=\theta $$ does not enter this endpoint cancellation algebra. This observation permits the following internal extension. --- # 4. Arbitrary decreasing radii Let: $$ \boxed{ r_0>r_1>\cdots>r_N>0 } $$ be arbitrary. Choose adjacent times: $$ \boxed{ \tau_{k+1}-\tau_k=r_k^2. } $$ Define: $$ \chi_k(x) = \chi \left( \frac{x-x_0}{r_k} \right), $$ with the same radial nonincreasing reference cutoff. Then: $$ \boxed{ \chi_k\le\chi_{k-1}. } $$ Use one common coarse package: $$ (U^\ell,P^\ell,R^\ell) $$ for every slab. --- # 5. CIV/VI-3.1 — Variable-Radius Pressure–Flux Telescope ## Theorem 5.1 Suppose every slab carries a nonnegative weight: $$ \widehat\phi_k $$ with endpoint traces: $$ \widehat\phi_k(\cdot,\tau_k) = \widehat\phi_k(\cdot,\tau_{k+1}) = \chi_k. $$ Set: $$ \boxed{ w_k=\frac{r_k}{r_0}. } $$ Then: $$ \boxed{ \sum_{k=0}^{N-1} w_k ( \mathcal E_k^- - \mathcal E_k^+ ) \le \mathcal E_0^-, } $$ and: $$ \boxed{ \sum_{k=0}^{N-1} w_k ( \mathcal W_k^+ + \mathcal D_k ) \le \mathcal E_0^- + \sum_{k=0}^{N-1} w_k|\mathcal L_k| + \sum_{k=0}^{N-1} w_k\mathcal W_k^-. } $$ ### Proof Since: $$ w_kr_k^{-1}=r_0^{-1}, $$ $$ \sum_{k=0}^{N-1} w_k ( \mathcal E_k^- - \mathcal E_k^+ ) = \frac1{r_0} \sum_{k=0}^{N-1} \left[ K_{\chi_k}^{\ell}(\tau_k) - K_{\chi_k}^{\ell}(\tau_{k+1}) \right]. $$ Rearrange: $$ = \frac1{r_0} \left[ K_{\chi_0}^{\ell}(\tau_0) - K_{\chi_{N-1}}^{\ell}(\tau_N) + \sum_{k=1}^{N-1} \left( K_{\chi_k}^{\ell}(\tau_k) - K_{\chi_{k-1}}^{\ell}(\tau_k) \right) \right]. $$ The terminal energy is nonnegative. For every interface: $$ \chi_k\le\chi_{k-1}, $$ hence: $$ K_{\chi_k}^{\ell}(\tau_k) - K_{\chi_{k-1}}^{\ell}(\tau_k) \le0. $$ This proves the endpoint inequality. Apply the local coarse-energy identity slab by slab, multiply by: $$ w_k, $$ sum, and separate: $$ \mathcal W_k = \mathcal W_k^+ - \mathcal W_k^-. $$ $\square$ --- # 6. Importance of Theorem 5.1 The pressure--flux telescope itself does **not** require geometric radii. Therefore the summability: $$ \sum_k \frac{r_k}{r_0} < \infty $$ in the published geometric chain is not an unavoidable consequence of the endpoint cancellation. The radius schedule is a genuine FCBP degree of freedom. --- # 7. Slow-scale schedules Let: $$ \boxed{ r_k = r_0 (k+1)^{-\beta}, } $$ with: $$ \boxed{ \frac12<\beta\le1. } $$ Then: $$ \boxed{ \sum_{k=0}^{\infty} r_k^2 = r_0^2 \sum_{k=0}^{\infty} (k+1)^{-2\beta} < \infty. } $$ Therefore one may choose: $$ \tau_{k+1}-\tau_k=r_k^2 $$ so that: $$ \boxed{ \tau_k\uparrow T_\infty<\infty. } $$ By selecting the initial time appropriately, the accumulation time may be placed at: $$ T_\ast. $$ --- # 8. Non-summable telescope weights For the same schedule: $$ w_k = \frac{r_k}{r_0} = (k+1)^{-\beta}. $$ Since: $$ \beta\le1, $$ $$ \boxed{ \sum_{k=0}^{\infty} w_k = \infty. } $$ Thus the same exact endpoint telescope admits: - finite total physical parabolic time; - non-summable scale weights. --- # 9. CIV/VI-3.2 — Slow-Scale Critical-Lift Window ## Theorem 9.1 For every: $$ \frac12<\beta\le1, $$ there exists a decreasing parabolic scale schedule with: $$ \boxed{ \sum_k (\tau_{k+1}-\tau_k) < \infty } $$ but: $$ \boxed{ \sum_k w_k = \infty. } $$ Therefore the FCBP-01 Summable-Weight No-Go does not apply to the variable-radius pressure--flux telescope on this schedule. ### Meaning A non-summable scale weight can be generated by the **geometry of the radius schedule itself**, without changing the local PDE identity. $\square$ --- # 10. Borderline logarithmic schedule The canonical borderline choice is: $$ \boxed{ \beta=1. } $$ Then: $$ r_k = \frac{r_0}{k+1}, $$ $$ \sum_k r_k^2 < \infty, $$ but: $$ \boxed{ \sum_k w_k = \sum_k \frac1{k+1} = \infty. } $$ This produces exactly the logarithmic non-summable weight anticipated in FCBP-01. --- # 11. Active work extraction The external finite-chain work theorem gives, on every selected slab: $$ \boxed{ |\mathcal W_k| \ge c_k \mathfrak A_k(G^\ell). } $$ The sign is split into: $$ \mathcal W_k^+ $$ and: $$ \mathcal W_k^-. $$ The external theorem does not claim: - uniform: $$ c_k; $$ - small backscatter; - summable leakage; - moving-window observability. All remain separate obligations. --- # 12. Weighted active density For a forward active index set: $$ \mathcal I_+, $$ define: $$ \boxed{ \mathfrak D_{\rm act}(N) = \sum_{ \substack{ k0 $$ on: $$ \mathcal I_+, $$ then it suffices that: $$ \boxed{ \sum_{k\in\mathcal I_+} w_k = \infty. } $$ $\square$ --- # 14. Sign is not an error term The external pressure--flux theorem explicitly warns that a forward-only subchain does not telescope. Negative intermediate work is part of the cancellation mechanism. Therefore: $$ \boxed{ \text{backscatter must remain on the paid side of the ledger}. } $$ The Slow-Scale Lift does not require an **unweighted** backscatter bound. It requires finiteness of the same non-summably weighted backscatter appearing in the telescope: $$ \boxed{ \sum_k w_k \mathcal W_k^- < \infty. } $$ This is a more natural signed target than the unweighted backscatter condition used in the FCBP-02 algebraic lemma. --- # 15. Weighted Backscatter Barrier The external theorem does not provide: $$ \sum_k w_k\mathcal W_k^- < \infty $$ as: $$ N\to\infty. $$ It only records backscatter explicitly on the right-hand side. Thus: $$ \boxed{ \textbf{weighted backscatter closure} } $$ remains an open PDE obligation even after the slow-scale schedule produces non-summable weights. --- # 16. Common physical filter The pressure--flux chain uses one fixed: $$ \boxed{ \ell>0 } $$ for every slab. This is essential to the exact endpoint telescope because: $$ K_{\chi_k}^{\ell} $$ at adjacent slabs is computed from the same resolved field: $$ U^\ell. $$ --- # 17. Scale-relative resolution The external coarse CKN quantities: $$ \Psi^\ell(r), \qquad \Omega^\ell(r) $$ are Navier--Stokes scale invariant when: $$ \boxed{ \ell/r } $$ is kept fixed. The filtered-vorticity coercive module likewise works in the scale-relative regime: $$ \boxed{ \ell\lesssim r. } $$ For a single fixed physical: $$ \ell>0 $$ and: $$ r_k\to0, $$ $$ \boxed{ \ell/r_k\to\infty. } $$ Thus one fixed infinite-chain filter eventually ceases to be a scale-relative fine filter. --- # 18. CIV/VI-3.4 — Common-Filter / Critical-Resolution Barrier ## Theorem 18.1 A single fixed: $$ \ell>0 $$ cannot satisfy: $$ \ell/r_k \le \rho $$ for all sufficiently large: $$ k $$ along any chain with: $$ r_k\to0. $$ Hence: $$ \boxed{ \text{one common physical filter} } $$ and: $$ \boxed{ \text{uniform scale-relative filtering on an infinite fine-scale chain} } $$ cannot hold simultaneously. $\square$ ### Safety The pressure--flux local identity itself remains valid. What fails is the scale-relative interpretation/compatibility with the filtered critical modules. --- # 19. Moving relative filters Set: $$ \boxed{ \ell_k = \sigma r_k. } $$ For each slab, define its own coarse package: $$ U_k = S_{\ell_k}u, $$ $$ P_k = S_{\ell_k}p, $$ $$ R_k = S_{\ell_k}(u\otimes u) - U_k\otimes U_k. $$ The local coarse energy identity remains valid separately on every slab. The endpoint fields, however, now change with: $$ k. $$ --- # 20. Moving-filter endpoint energy Define: $$ \boxed{ K_{\chi}^{(k)}(t) = \frac12 \int |U_k(x,t)|^2 \chi(x)dx. } $$ Then: $$ \mathcal E_k^- = r_k^{-1} K_{\chi_k}^{(k)}(\tau_k), $$ and: $$ \mathcal E_k^+ = r_k^{-1} K_{\chi_k}^{(k)}(\tau_{k+1}). $$ --- # 21. Filter-switch defect At interface: $$ \tau_k, $$ define: $$ \boxed{ \Delta_k^{filt} = K_{\chi_{k-1}}^{(k)}(\tau_k) - K_{\chi_{k-1}}^{(k-1)}(\tau_k). } $$ This measures the resolved localized kinetic-energy change caused only by switching: $$ \ell_{k-1} \to \ell_k $$ while keeping the larger spatial cutoff fixed. --- # 22. CIV/VI-3.5 — Moving-Filter Telescope ## Theorem 22.1 With: $$ w_k=r_k/r_0, $$ the moving-filter endpoint sum satisfies: $$ \boxed{ \sum_{k=0}^{N-1} w_k ( \mathcal E_k^- - \mathcal E_k^+ ) \le \mathcal E_0^- + \frac1{r_0} \sum_{k=1}^{N-1} [ \Delta_k^{filt} ]_+. } $$ Consequently: $$ \boxed{ \begin{aligned} \sum_{k=0}^{N-1} w_k ( \mathcal W_k^+ + \mathcal D_k ) \le\;& \mathcal E_0^- + \sum_{k=0}^{N-1} w_k|\mathcal L_k| + \sum_{k=0}^{N-1} w_k\mathcal W_k^- \\ &+ \frac1{r_0} \sum_{k=1}^{N-1} [ \Delta_k^{filt} ]_+. \end{aligned} } $$ ### Proof Write: $$ \sum_k w_k ( \mathcal E_k^- - \mathcal E_k^+ ) = \frac1{r_0} \sum_k \left[ K_{\chi_k}^{(k)}(\tau_k) - K_{\chi_k}^{(k)}(\tau_{k+1}) \right]. $$ At interface: $$ \tau_k, $$ split: $$ K_{\chi_k}^{(k)} - K_{\chi_{k-1}}^{(k-1)} = \left( K_{\chi_k}^{(k)} - K_{\chi_{k-1}}^{(k)} \right) + \left( K_{\chi_{k-1}}^{(k)} - K_{\chi_{k-1}}^{(k-1)} \right). $$ The first term is nonpositive because: $$ \chi_k\le\chi_{k-1}. $$ The second is: $$ \Delta_k^{filt}. $$ Drop the nonnegative terminal energy and retain the positive part of the switch defect. Then use the slabwise local energy identities. $\square$ --- # 23. Filter-Switch Packing Problem A scale-relative moving-filter Critical Lift therefore requires: $$ \boxed{ \sum_k [ \Delta_k^{filt} ]_+ < \infty } $$ or a sufficiently weak non-summable-weight-compatible substitute. Define: $$ \boxed{ \textbf{FSP — Filter-Switch Packing}. } $$ No such universal theorem is proved here. --- # 24. No automatic sign Cutoff nesting controls: $$ K_{\chi_k}^{(k)} - K_{\chi_{k-1}}^{(k)} \le0. $$ It does **not** control: $$ \Delta_k^{filt}, $$ because two different resolved fields are compared. Therefore FSP is a genuine new endpoint compatibility problem, not a notational artifact. --- # 25. Slow filter ratios On the power-law slow schedule: $$ r_k=r_0(k+1)^{-\beta}, $$ and: $$ \ell_k=\sigma r_k, $$ one has: $$ \boxed{ \frac{\ell_k}{\ell_{k-1}} = \left( \frac{k}{k+1} \right)^{\beta} = 1-\frac{\beta}{k} + O(k^{-2}). } $$ Thus successive filter switches become infinitesimal in logarithmic scale. This suggests a bounded-variation-in-filter route to FSP. No such localized BV theorem is currently established in the FCBP ledger. --- # 26. Finest-scale common-filter strategy There is a second way to avoid the moving-filter defect. For each finite chain of length: $$ N, $$ choose one common filter: $$ \boxed{ \ell^{(N)} = \sigma r_N. } $$ Then for every: $$ k\le N, $$ $$ \boxed{ \frac{\ell^{(N)}}{r_k} \le \sigma. } $$ The chain therefore keeps a fine common filter while preserving the exact fixed-filter telescope. No filter-switch term appears inside that finite chain. --- # 27. Price of the finest-scale common filter The coarse package now depends on: $$ N: $$ $$ G^{\ell^{(N)}}. $$ To pass: $$ N\to\infty, $$ one needs uniform control of: 1. active-work observability constants; 2. selected detector constants: $$ c_k; $$ 3. weighted leakage; 4. weighted backscatter; 5. the subfilter residual/coarse-resolution split. Thus the filter-switch problem is replaced by a **uniform-in-chain-length coarse-observability problem**. --- # 28. Interaction with filtered stretching The filtered near-field estimate contains a lower-order factor involving: $$ (r/\ell)^5. $$ With: $$ \ell^{(N)} = \sigma r_N, $$ on an earlier slab: $$ k Prove that dangerous/residual-small horizon activity produces a set of forward detected work slabs with divergent slow-schedule weight, or else route the complementary slabs into backscatter, subfilter residual, commutator, or another critical recurrence branch. Current status: $$ \boxed{ \mathrm{OPEN}. } $$ --- # 34. Backscatter-compatible obstruction The pressure--flux telescope already records negative work explicitly. Therefore a slow-scale contradiction does not require: $$ \sum_k \mathcal W_k^- < \infty. $$ It requires only the schedule-compatible condition: $$ \boxed{ \sum_k w_k \mathcal W_k^- < \infty. } $$ This is weaker and better aligned with the signed PDE identity. But it remains unproved. --- # 35. Model-cone recurrence Return to the Miller strain balance: $$ \boxed{ \frac12 \frac d{dt} \|S\|_{\dot H^1}^2 + \|-\Delta S\|_2^2 = - \langle \mathcal R_{SV}, -\Delta S \rangle. } $$ Define: $$ \boxed{ \chi_{SV} = \frac{ \|\mathcal R_{SV}\|_2 }{ \|-\Delta S\|_2 }. } $$ Then: $$ \boxed{ \frac12 \frac d{dt} \|S\|_{\dot H^1}^2 \le ( \chi_{SV}-1 ) \|-\Delta S\|_2^2. } $$ --- # 36. CIV/VI-3.6 — Model-Cone Growth Packet ## Theorem 36.1 For: $$ I=[a,b], $$ $$ \boxed{ \|S(b)\|_{\dot H^1}^2 - \|S(a)\|_{\dot H^1}^2 \le 2 \int_a^b ( \chi_{SV}-1 )_+ \|-\Delta S\|_2^2dt. } $$ If: $$ \boxed{ \|S(b)\|_{\dot H^1}^2 \ge (1+\delta) \|S(a)\|_{\dot H^1}^2, } $$ then: $$ \boxed{ \frac{ 2 }{ \|S(a)\|_{\dot H^1}^2 } \int_a^b ( \chi_{SV}-1 )_+ \|-\Delta S\|_2^2dt \ge \delta. } $$ The normalized packet is Navier--Stokes scale invariant. $\square$ --- # 37. External model-cone necessity Miller's external theorem states that a finite-time blow-up requires: $$ \boxed{ \limsup_{t\uparrow T_\ast} \chi_{SV}(t) \ge1. } $$ Thus a hypothetical singular trajectory cannot remain uniformly inside a strictly subcritical model cone. This supports model-cone excess as a critical recurrence coordinate. It does not provide a universal finite sum of the packets in Theorem 36.1. --- # 38. Model-cone role in FCBP If pressure--flux active work repeatedly fails to be observable while dangerous strain growth persists, the model-cone ledger offers a second critical recurrence channel: $$ \boxed{ \text{work-visible} \vee \text{model-cone excess recurrence}. } $$ A theorem connecting the two quantitatively is not yet proved. --- # 39. Affine-jet versus pressure--flux work Filtered far-field affine-jet stretching measures an enstrophy-production mechanism of the form: $$ J\Omega\cdot\Omega. $$ Pressure--flux work measures resolved kinetic-energy transfer: $$ \Pi^\ell + \nabla\cdot(P^\ell U^\ell). $$ There is no identity in the current external results equating these observables. Therefore the FCBP-02 hoped-for direct bridge: $$ \boxed{ \text{affine-jet stretching} \Longrightarrow \text{pressure--flux active work} } $$ remains open. --- # 40. Affine-to-Pressure–Flux Observability Bridge Define: $$ \boxed{ \textbf{APF — Affine-to-Pressure--Flux Bridge} } $$ as any quantitative implication which converts persistent comparable-annulus signed affine stretching work into: - active pressure--flux work; - backscatter; - subfilter residual; - or another explicitly paid critical channel. Current status: $$ \boxed{ \mathrm{OPEN}. } $$ This is the correct cross-observable formulation. --- # 41. Pressure–flux cancellation barrier The external combined work is: $$ \boxed{ \mathcal W = \mathcal F+\mathcal P, } $$ and the cancellation amount: $$ \boxed{ \mathcal C^{PF} = |\mathcal F| + |\mathcal P| - |\mathcal W| \ge0. } $$ Therefore large separate flux and pressure activity can be almost invisible to the signed combined detector. This is one explicit reason AOB/APF cannot be assumed automatically. --- # 42. Harmonic pressure is physical The external pressure decomposition writes: $$ P=P^a+P^h, $$ with: $$ P^h $$ harmonic in the interior. Only the spatially constant harmonic mode is a gauge. Higher harmonic polynomial modes perform genuine localized pressure work and must remain in the signed ledger. Thus harmonic pressure cannot be discarded to manufacture observability. --- # 43. Main positive breakthrough of FCBP-03 The FCBP-01 summable-weight barrier is **not intrinsic to the pressure--flux telescope**. Theorem 9.1 provides an exact slow-scale schedule with: $$ \boxed{ \sum_k r_k^2 < \infty, } $$ but: $$ \boxed{ \sum_k \frac{r_k}{r_0} = \infty. } $$ This is the first genuine non-summable telescope weight produced inside FCBP. --- # 44. Why Critical Lift is still open To convert the schedule breakthrough into a Forest Coercive Budget, one still needs one of the following complete packages. ## Moving-filter package 1. AOB / active-slab density; 2. weighted leakage closure; 3. weighted backscatter closure; 4. Filter-Switch Packing. ## Finest-common-filter package 1. uniform-in-$N$ AOB; 2. uniform weighted leakage/backscatter; 3. uniform residual/coarse-resolution control; 4. compatibility with the filtered residual branch. Neither package is currently proved. --- # 45. Slow-Scale Critical-Lift Compiler ## Theorem 45.1 Let: $$ r_k=r_0(k+1)^{-\beta}, \qquad \frac12<\beta\le1. $$ Assume a fixed-filter or moving-filter pressure--flux chain satisfies: ### SL-1 — active density $$ \sum_{ k\in\mathcal I_+ } w_k c_k \mathfrak A_k = \infty. $$ ### SL-2 — paid residual finiteness $$ \sum_k w_k |\mathcal L_k| + \sum_k w_k \mathcal W_k^- < \infty. $$ ### SL-3 — filter compatibility For a fixed-filter chain there is no switch term. For a moving-filter chain: $$ \sum_k [ \Delta_k^{filt} ]_+ < \infty. $$ Then the chain is impossible. ### Proof Apply Theorem 5.1 or Theorem 22.1 and pass: $$ N\to\infty. $$ The left side diverges by SL-1 while the right side remains finite by SL-2--SL-3. $\square$ --- # 46. Meaning of the compiler This is the first FCBP closure theorem whose **scale weight itself is non-summable**. The remaining gap is no longer the existence of a non-summable schedule. It is the Navier--Stokes control needed to satisfy SL-1--SL-3. --- # 47. Updated Critical-Lift obligations FCBP-01/02 used: $$ CL\mbox{-}DER, \quad CL\mbox{-}REM, \quad CL\mbox{-}PACK. $$ After FCBP-03: ### CL-DER $$ \boxed{ \mathrm{PARTIALLY\ CLOSED}. } $$ ### CL-REM $$ \boxed{ \mathrm{OPEN/PARTIAL}. } $$ ### CL-PACK A non-summable **schedule** is now available. The remaining packing problem is compressed to: $$ \boxed{ \text{FSP} + \text{AOB/ASD} + \text{weighted leakage/backscatter}. } $$ Thus: $$ \boxed{ CL\mbox{-}PACK : \mathrm{PARTIALLY\ OPEN\ WITH\ A\ NEW\ NONSUMMABLE\ ROUTE}. } $$ --- # 48. Next paper The next paper should exploit the new schedule rather than return to geometric scales: $$ \boxed{ \textbf{ NS-FCBP 04 — Moving-Filter Telescoping, Filter-Switch Energy Defects, Slow-Scale Active Windows and Borderline Critical Lift }. } $$ Primary tasks: 1. estimate: $$ \Delta_k^{filt}; $$ 2. test bounded variation of localized resolved energy in filter scale; 3. compare moving-filter and finest-common-filter strategies; 4. seek uniform coarse observability as: $$ \ell^{(N)}\to0; $$ 5. control slow-schedule weighted leakage/backscatter; 6. prove active-window weighted density or route its failure to subfilter/model-cone/commutator recurrence; 7. decide whether SL-1--SL-3 can be obtained from Navier--Stokes structure. --- # 49. Formal status ledger $$ \boxed{ \begin{aligned} \text{external geometric pressure--flux telescope} &:\ \mathrm{EXTERNAL/PROVED},\\ \text{Variable-Radius Pressure--Flux Telescope} &:\ \mathrm{PROVED},\\ \text{Slow-Scale Critical-Lift Window} &:\ \mathrm{PROVED},\\ \text{Non-Summable Work-Depletion Criterion} &:\ \mathrm{PROVED\ CONDITIONAL},\\ \text{Weighted Backscatter Closure} &:\ \mathrm{OPEN},\\ \text{Common-Filter/Critical-Resolution Barrier} &:\ \mathrm{PROVED},\\ \text{Moving-Filter Telescope} &:\ \mathrm{PROVED},\\ \text{Filter-Switch Packing} &:\ \mathrm{OPEN},\\ \text{Finest-Scale Common-Filter Strategy} &:\ \mathrm{DEFINED/OPEN},\\ \text{Active Observability Bridge} &:\ \mathrm{OPEN},\\ \text{Active-Slab Density} &:\ \mathrm{OPEN},\\ \text{Model-Cone Growth Packet} &:\ \mathrm{PROVED},\\ \text{Affine-to-Pressure--Flux Bridge} &:\ \mathrm{OPEN},\\ \text{Slow-Scale Critical-Lift Compiler} &:\ \mathrm{PROVED\ CONDITIONAL},\\ \text{Critical Lift} &:\ \mathrm{OPEN},\\ \text{Forest Coercive Budget} &:\ \mathrm{OPEN},\\ \text{Finite Forest Obstruction} &:\ \mathrm{OPEN},\\ CN3_{\rm Atomic} &:\ \mathrm{OPEN},\\ \text{Navier--Stokes regularity} &:\ \mathrm{NOT\ PROVED}. \end{aligned} } $$ --- # 50. Conclusion FCBP-03 changes the Critical-Lift problem substantially. The weighted pressure--flux telescope was previously treated as scale-weak because the published theorem uses: $$ r_k=\theta^kr_0, $$ so: $$ \sum_k r_k/r_0 < \infty. $$ But the endpoint cancellation itself uses only: $$ w_kr_k^{-1}=r_0^{-1} $$ and nested spatial cutoffs. Therefore arbitrary decreasing radii are allowed at the telescope level. The slow schedule: $$ r_k = r_0(k+1)^{-\beta}, \qquad \frac12<\beta\le1, $$ simultaneously has: $$ \sum_kr_k^2<\infty $$ and: $$ \sum_k r_k/r_0=\infty. $$ Thus FCBP now has a genuine borderline/non-summable pressure--flux telescope. The new obstruction is compatibility. A common physical filter preserves the exact telescope but cannot remain scale relative forever. Moving scale-relative filters preserve critical resolution but introduce explicit filter-switch endpoint defects. A finite-chain finest-scale common filter avoids switch defects but shifts the problem to uniform-in-chain-length observability and residual control. The signed work itself also remains subject to backscatter and pressure--flux cancellation. Finally, the Miller strain balance gives an independent scale-critical Model-Cone Growth Packet whenever dangerous strain growth occurs. The new frontier is therefore no longer: > can one produce a non-summable weight? That has now been answered. The frontier is: $$ \boxed{ \textbf{ can scale-relative observability and endpoint compatibility survive on that non-summable slow schedule? } } $$ That is FCBP-04. --- # References 1. R. Yu, *Coarse-Grained Resolution and Pressure--Flux Work Depletion for Navier--Stokes CKN Badness*, arXiv:2606.25322. 2. R. Yu, *Filtered Vortex Stretching and Subgrid Defects for the Three-Dimensional Navier--Stokes Equations*, arXiv:2606.27560. 3. R. Yu, *Invisible Defect Cascades for Navier--Stokes Regularity*, arXiv:2606.12756. 4. R. Yu, *Critical Ledgers and Scale-Defect Cascades for Navier--Stokes*, arXiv:2606.13887. 5. E. Miller, *On the interaction of strain and vorticity for solutions of the Navier--Stokes equation*, arXiv:2407.02691. 6. `NS_FCBP_01_CriticalForest_Coercivity_v0.1.md`. 7. `NS_FCBP_02_FilteredStretching_CriticalLift_v0.1.md`.