January 01, 1970
We prove a finite-window anti-phantom principle for scale-critical Navier–Stokes defect packages and develop a conditional localized transfer framework around it. In a fixed clean quotient, the main compactness theorem shows that if the only defect that is simultaneously invisible, exactly reproducible, and tax-free is a gauge artifact, then observation, reproduction failure, and tax control the clean quotient distance by a positive finite-window gap. We then isolate the additional localized inputs needed to use this clean gap for Navier–Stokes packages: pressure-source observability, enhanced pressure-tail geometry, chart visibility, and residual sub-budgets for localization, reproduction, and gate/tax mismatch. The localized results are conditional finite-window reductions with explicit error constants, including a comparison theorem between the enhanced-tail geometry and the original intrinsic geometry under stated projection and harmonic tail approximation assumptions. The paper should be read as a rigorous finite-window obstruction framework, not as a proof of Navier–Stokes regularity, a construction of a singular solution, or a scale-uniform regularity criterion.
The local regularity theory for the three-dimensional incompressible Navier–Stokes equations turns smallness of scale-critical quantities into regularity. This viewpoint goes back to the Leray–Hopf weak-solution framework and to the partial-regularity theory of Scheffer and Caffarelli–Kohn–Nirenberg, with later refinements and expositions such as Lin’s proof and Seregin’s lecture notes [1]–[7]. A potential singular mechanism must therefore do more than make a single critical norm large. It must persist through a sequence of scales while evading the observations and ledger terms that normally expose pressure, flux, energy, dissipation, and trace defects.
There are two levels of that obstruction. The first is clean: after quotienting out specified gauges, can a non-gauge defect remain invisible to the finite-window detectors? The second is localized and PDE-facing: what pressure, tail, chart, and residual estimates would transfer the clean gap back to a localized Navier–Stokes package?
The terminology of flux, budget, and defect is consistent with the local energy-transfer viewpoint for weak Euler and Navier–Stokes flows [8]–[10]. The pressure terms are kept explicit because local pressure decomposition and pressure regularity are central in Navier–Stokes partial regularity [7], [11], [12]. Critical-space and backward-uniqueness mechanisms provide another perspective on how scale-critical information interacts with possible singular behavior [13].
The word clean means that cutoff leakage, harmonic-pressure artifacts, periodization errors, and localization residuals have been removed from the algebraic model or placed into explicit transfer hypotheses. The word finite-window means that all spaces are finite-dimensional and only a fixed dyadic range is considered. The main clean theorem is therefore an algebraic compactness theorem. The later localized statements are conditional reductions that name the remaining Navier–Stokes inputs.
The framework is motivated by two neighboring lines of work. One line studies one-component and anisotropic regularity criteria for the three-dimensional Navier–Stokes equations [14]–[19]. Another line asks for quantitative regularity information from spatial concentration or weak–strong uniqueness mechanisms [20], [21]. The present manuscript is also a finite-window algebraic companion to the preceding harmonic-pressure, strict-shadow, Schur-visibility, and defect-cascade formulations [22]–[25].
The main contribution of this paper is twofold. First, we prove a clean finite-window anti-phantom theorem. We define a constrained clean defect space \(\mathcal{K}_\Lambda^{\mathrm{cl}}\), a gauge subspace \(\Gamma_\Lambda^{\mathrm{cl}}\), and a quotient distance \[\operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}}) = \inf_{\gamma\in\Gamma_\Lambda^{\mathrm{cl}}}\|d-\gamma\|_{\mathrm{cl}}.\] We then define a computational detector consisting of three parts: \[\text{observation} \quad+\quad \text{reproduction residual} \quad+\quad \text{tax}.\] The theorem is stated for a gauge-compatible clean detector. It says that if the zero set of the induced quotient detector contains no nonzero quotient class, then the detector controls the quotient distance by a positive finite-window constant.
Second, we build a conditional localized transfer layer around the clean gap. The localized part introduces pressure-source quotient observability, enhanced pressure-tail geometry, a chart-visibility decomposition, and four explicit residual sub-budgets: chart mismatch, localization leakage, reproduction drift, and gate/tax mismatch. These pieces are assembled into a finite-window enhanced-tail residual budget. We then prove a conditional comparison between the enhanced-tail distance and the original intrinsic distance, assuming projection and harmonic tail approximation on a common intrinsic representative.
The innovation is not the claim that these localized hypotheses are automatic. It is the organization of the obstruction: the manuscript separates the finite-window algebra that is proved from the PDE-facing compatibility estimates that remain open.
The theorem should not be read as a proof of global regularity for Navier–Stokes. It also does not construct a singular solution, does not claim that singularities are computers, and does not assert any form of undecidability. Its role is narrower: it proves a fixed clean finite-window gap and records conditional mechanisms by which that gap could be transferred to localized defect packages.
The proof narrative has four layers. First, the clean finite-window theorem uses a detector that descends to the quotient by gauge directions and then applies finite-dimensional compactness to obtain a positive detector gap. Second, a conditional local-to-clean transfer theorem records how such a clean gap would imply a localized lower bound in the presence of explicit comparison and residual budget hypotheses. Third, the pressure-source and enhanced-tail sections decompose those hypotheses into concrete finite-window components. Fourth, the final comparison theorem explains when the enhanced-tail geometry is controlled by the original intrinsic geometry, up to explicit projection and harmonic tail errors.
Every localized conclusion in this manuscript is conditional on named structural inputs. No decay of projection tails as \(N\to\infty\), no decay of harmonic tails as \(M\to\infty\), no pressure/tax coercivity theorem, and no scale-uniform moving-window estimate is proved.
2 defines clean finite-window defect spaces, constraints, gauge directions, and quotient distance. 3 records the compactness of the quotient unit sphere. 4 defines observation, reproduction, tax, and the computational anti-phantom constant. 5 proves the kernel-free characterization and the clean finite-window computational anti-phantom inequality. 6 derives a finite-state sector interpretation of the gap. 7 records the conditional localized transfer corollary obtained by using \(\mu_\Lambda^{\rm comp}\) as the clean gap constant. 8 classifies the ways in which the clean mechanism can fail to produce a localized conclusion. 9 develops the pressure-source, enhanced-tail, residual-budget, and intrinsic-comparison components needed for the localized transfer branch. 10 records the next PDE-facing derivation targets and the limitations of the present manuscript.
Fix a finite dyadic window \[\Lambda=\{k_0,k_0+1,\ldots,k_0+L\}, \qquad L<\infty,\] and write \[\Lambda_{\rm adj} := \{k\in\Lambda:\;k+1\in\Lambda\}.\] All constants in this paper may depend on the fixed window \(\Lambda\), the chosen finite-dimensional spaces, and the chosen norms. No scale-uniform statement in \(L\), \(k_0\), or the dyadic radius is asserted.
Definition 1 (Clean finite-window defect space). For each \(k\in\Lambda\), let \(\mathcal{D}_k^{\mathrm{cl}}\) be a finite-dimensional real normed vector space. A clean defect at scale \(k\) is written schematically as \[d_k=(U_k,P_k,R_k,\Pi_k,\Phi_k,\tau_k)\in\mathcal{D}_k^{\mathrm{cl}},\] where the coordinates represent velocity, pressure, covariance or Reynolds stress, interscale flux, energy/trace, and tax or ledger-slack data. The clean finite-window defect space is \[\mathcal{D}_\Lambda^{\mathrm{cl}} := \prod_{k\in\Lambda}\mathcal{D}_k^{\mathrm{cl}}.\] It is equipped with a fixed norm \(\|\cdot\|_{\mathrm{cl}}\).
Definition 2 (Clean constraint space). The clean constraint space \[\mathcal{K}_\Lambda^{\mathrm{cl}}\subset\mathcal{D}_\Lambda^{\mathrm{cl}}\] is a finite-dimensional linear subspace. Its role is to encode the clean finite-dimensional constraints that survive after localization artifacts have been removed, such as projected divergence constraints, projected pressure compatibility, projected momentum compatibility, and projected finite-window energy-flux identities.
Remark 1 (Linear clean model first). The present paper proves the main theorem in the linear clean model, where \(\mathcal{K}_\Lambda^{\mathrm{cl}}\) is a vector subspace. Positivity constraints, semialgebraic admissibility conditions, or Reynolds-stress cones can be added later, but they are not needed for the compactness-based gap theorem below.
Definition 3 (Clean gauge space). Let \(\mathcal{C}_\Lambda^{\mathrm{cl}}\) be a finite-dimensional real normed vector space and let \[G_\Lambda^{\mathrm{cl}}:\mathcal{C}_\Lambda^{\mathrm{cl}}\to\mathcal{D}_\Lambda^{\mathrm{cl}}\] be a linear map. The clean gauge subspace inside the constrained clean space is \[\Gamma_\Lambda^{\mathrm{cl}} := \mathcal{K}_\Lambda^{\mathrm{cl}}\cap \operatorname{Im}G_\Lambda^{\mathrm{cl}}.\] Elements of \(\Gamma_\Lambda^{\mathrm{cl}}\) are the clean removable directions, such as pressure mean gauges, selected harmonic-pressure gauges, finite projection artifacts, and clean periodization artifacts represented at the finite-window level.
Definition 4 (Clean quotient and quotient distance). The clean quotient space is \[\mathcal{Q}_\Lambda^{\mathrm{cl}} := \mathcal{K}_\Lambda^{\mathrm{cl}}/\Gamma_\Lambda^{\mathrm{cl}}.\] For \(d\in\mathcal{K}_\Lambda^{\mathrm{cl}}\), define \[\operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}}) := \inf_{\gamma\in\Gamma_\Lambda^{\mathrm{cl}}}\|d-\gamma\|_{\mathrm{cl}}.\] Equivalently, \[\|[d]\|_{\mathcal{Q}_\Lambda^{\mathrm{cl}}} := \operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}})\] is the quotient norm of the class \([d]\in\mathcal{Q}_\Lambda^{\mathrm{cl}}\).
Remark 2 (Gauge status). The quotient removes only the finite-dimensional clean gauge directions that have been explicitly placed in \(\Gamma_\Lambda^{\mathrm{cl}}\). It does not assert that every localized Navier–Stokes artifact is gauge, and it does not prove a localized gauge-cleaning theorem.
Convention 1 (Nontrivial quotient regime). The main gap statements are formulated in the case \(\mathcal{Q}_\Lambda^{\mathrm{cl}}\ne\{0\}\). If \(\mathcal{Q}_\Lambda^{\mathrm{cl}}=\{0\}\), then every constrained clean defect is already a clean gauge direction, so there is no non-gauge clean finite-window obstruction to detect. In that degenerate case the anti-phantom statement is vacuous rather than a positive-gap theorem.
Lemma 1 (Quotient compactness). The quotient \(\mathcal{Q}_\Lambda^{\mathrm{cl}}\) is finite-dimensional. Its unit sphere \[S_\Lambda^{\mathrm{cl}} := \{[d]\in\mathcal{Q}_\Lambda^{\mathrm{cl}}:\;\|[d]\|_{\mathcal{Q}_\Lambda^{\mathrm{cl}}}=1\}\] is compact.
Proof. Since \(\mathcal{K}_\Lambda^{\mathrm{cl}}\) is finite-dimensional and \(\Gamma_\Lambda^{\mathrm{cl}}\subset\mathcal{K}_\Lambda^{\mathrm{cl}}\) is a linear subspace, the quotient \(\mathcal{Q}_\Lambda^{\mathrm{cl}}\) is finite-dimensional. Every norm on a finite-dimensional vector space induces the usual finite-dimensional topology. The unit sphere of a finite-dimensional normed space is closed and bounded, and hence compact. ◻
Lemma 2 (Distance normalization). If \(d\in\mathcal{K}_\Lambda^{\mathrm{cl}}\) and \(\operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}})>0\), then \[e:=\frac{d}{\operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}})}\] satisfies \[\operatorname{dist}_{\mathrm{cl}}(e,\Gamma_\Lambda^{\mathrm{cl}})=1.\]
Proof. Because \(\Gamma_\Lambda^{\mathrm{cl}}\) is a linear subspace, quotient distance is positively homogeneous: \[\operatorname{dist}_{\mathrm{cl}}(\lambda d,\Gamma_\Lambda^{\mathrm{cl}}) = |\lambda|\operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}})\] for every scalar \(\lambda\). Taking \(\lambda=\operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}})^{-1}\) proves the claim. ◻
Definition 5 (Clean observation detector). Let \(\mathcal{Y}_\Lambda^{P}\), \(\mathcal{Y}_\Lambda^{F}\), \(\mathcal{Y}_\Lambda^{E}\), and \(\mathcal{Y}_\Lambda^{T}\) be finite-dimensional normed observation spaces. The clean observation detector is a linear map \[O_\Lambda^{\mathrm{cl}} = (O_\Lambda^P,O_\Lambda^F,O_\Lambda^E,O_\Lambda^T): \mathcal{K}_\Lambda^{\mathrm{cl}}\to \mathcal{Y}_\Lambda^P\times\mathcal{Y}_\Lambda^F \times\mathcal{Y}_\Lambda^E\times\mathcal{Y}_\Lambda^T.\] The four channels represent active pressure, interscale flux, energy or dissipation, and selected trace or adjoint-trace observations.
Definition 6 (Clean reproduction residual). For each \(k\in\Lambda_{\rm adj}\), fix a linear map \[R_k^{\mathrm{cl}}:\mathcal{D}_k^{\mathrm{cl}}\to\mathcal{D}_{k+1}^{\mathrm{cl}}.\] For \(d=(d_k)_{k\in\Lambda}\in\mathcal{K}_\Lambda^{\mathrm{cl}}\), define \[\operatorname{Rep}_\Lambda^{\mathrm{cl}}(d) := \left( \sum_{k\in\Lambda_{\rm adj}} \|d_{k+1}-R_k^{\mathrm{cl}}d_k\|_{\mathrm{cl},k+1}^2 \right)^{1/2},\] where \(\|\cdot\|_{\mathrm{cl},k+1}\) is a fixed norm on \(\mathcal{D}_{k+1}^{\mathrm{cl}}\).
Remark 3 (Meaning of reproduction). The residual \(\operatorname{Rep}_\Lambda^{\mathrm{cl}}\) measures whether a clean defect is compatible with the chosen adjacent-scale reproduction maps. It does not say that Navier–Stokes actually generates these maps, and it does not prove scale-uniform reproduction.
Definition 7 (Clean tax functional). A clean tax functional is a continuous map \[\operatorname{Tax}_\Lambda^{\mathrm{cl}}:\mathcal{K}_\Lambda^{\mathrm{cl}}\to[0,\infty)\] which is positively homogeneous: \[\operatorname{Tax}_\Lambda^{\mathrm{cl}}(\lambda d)=|\lambda|\operatorname{Tax}_\Lambda^{\mathrm{cl}}(d), \qquad \lambda\in\mathbb{R}.\] It records the finite-window cost assigned to dissipation, flux, pressure, or ledger-slack channels. In this first paper it is an abstract detector, not a proved pressure-tax lower bound.
Definition 8 (Computational detector size). Fix constants \(C_R>0\) and \(C_T>0\). Define \[\mathcal{M}_\Lambda^{\rm comp}(d) := \|O_\Lambda^{\mathrm{cl}}d\| + C_R\operatorname{Rep}_\Lambda^{\mathrm{cl}}(d) + C_T\operatorname{Tax}_\Lambda^{\mathrm{cl}}(d), \qquad d\in\mathcal{K}_\Lambda^{\mathrm{cl}}.\]
Assumption 1 (Clean detector gauge compatibility). The clean detector is constant on clean gauge cosets. More precisely, for every \(d\in\mathcal{K}_\Lambda^{\mathrm{cl}}\) and every \(\gamma\in\Gamma_\Lambda^{\mathrm{cl}}\), \[O_\Lambda^{\mathrm{cl}}(d+\gamma)=O_\Lambda^{\mathrm{cl}}d, \qquad \operatorname{Rep}_\Lambda^{\mathrm{cl}}(d+\gamma)=\operatorname{Rep}_\Lambda^{\mathrm{cl}}(d), \qquad \operatorname{Tax}_\Lambda^{\mathrm{cl}}(d+\gamma)=\operatorname{Tax}_\Lambda^{\mathrm{cl}}(d).\] Equivalently, \(\mathcal{M}_\Lambda^{\rm comp}\) is constant on every affine coset \(d+\Gamma_\Lambda^{\mathrm{cl}}\).
Remark 4 (Why gauge compatibility is included). Continuity and positive homogeneity of \(\mathcal{M}_\Lambda^{\rm comp}\), together with the qualitative kernel-free condition, do not by themselves force a positive gap on the set \(\{d:\operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}})=1\}\). That set is not compact in \(\mathcal{K}_\Lambda^{\mathrm{cl}}\) when gauge directions are nontrivial, and a sequence may escape along \(\Gamma_\Lambda^{\mathrm{cl}}\) while keeping unit quotient distance. The compactness proof below is therefore made on the quotient detector, which is well defined only under 1 or after choosing an equivalent canonical gauge-slice formulation.
Lemma 3 (Continuity and homogeneity of the quotient detector). The map \(\mathcal{M}_\Lambda^{\rm comp}\) is continuous and positively homogeneous: \[\mathcal{M}_\Lambda^{\rm comp}(\lambda d) = |\lambda|\mathcal{M}_\Lambda^{\rm comp}(d).\] Under 1, it descends to a well-defined continuous positively homogeneous map \[\overline{\mathcal{M}}_\Lambda^{\rm comp}: \mathcal{Q}_\Lambda^{\mathrm{cl}}\to[0,\infty), \qquad \overline{\mathcal{M}}_\Lambda^{\rm comp}([d]) := \mathcal{M}_\Lambda^{\rm comp}(d).\]
Proof. The observation map is linear between finite-dimensional normed spaces and is therefore continuous. The reproduction residual is a finite sum of norms of linear expressions in \(d\), hence is continuous and positively homogeneous. The tax functional is continuous and positively homogeneous by definition. The weighted sum of these three terms has the same properties.
If \(d'\) is another representative of \([d]\), then \(d'=d+\gamma\) for some \(\gamma\in\Gamma_\Lambda^{\mathrm{cl}}\). By 1, \(\mathcal{M}_\Lambda^{\rm comp}(d')=\mathcal{M}_\Lambda^{\rm comp}(d)\). Hence the quotient detector is well defined. Choose any linear section \(s:\mathcal{Q}_\Lambda^{\mathrm{cl}}\to\mathcal{K}_\Lambda^{\mathrm{cl}}\) of the quotient map. Since the spaces are finite-dimensional, \(\overline{\mathcal{M}}_\Lambda^{\rm comp}=\mathcal{M}_\Lambda^{\rm comp}\circ s\) is continuous. Homogeneity is inherited from the homogeneity of \(\mathcal{M}_\Lambda^{\rm comp}\). ◻
Definition 9 (Computational anti-phantom constant). Assume the nontrivial quotient regime of 1 and 1. The clean finite-window computational anti-phantom constant is \[\mu_\Lambda^{\rm comp} := \inf_{q\in S_\Lambda^{\mathrm{cl}}} \overline{\mathcal{M}}_\Lambda^{\rm comp}(q).\] Equivalently, because the detector is constant on clean gauge cosets, \[\mu_\Lambda^{\rm comp} = \inf_{\operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}})=1} \mathcal{M}_\Lambda^{\rm comp}(d).\]
Assumption 2 (Kernel-free computational detector modulo gauge). For every \(d\in\mathcal{K}_\Lambda^{\mathrm{cl}}\), \[O_\Lambda^{\mathrm{cl}}d=0, \qquad \operatorname{Rep}_\Lambda^{\mathrm{cl}}(d)=0, \qquad \operatorname{Tax}_\Lambda^{\mathrm{cl}}(d)=0 \quad \Longrightarrow \quad d\in\Gamma_\Lambda^{\mathrm{cl}}.\]
Remark 5 (Status of the kernel-free assumption). 2 is the substantive clean finite-window hypothesis. It says that there is no non-gauge clean defect which is simultaneously invisible, exactly reproducible, and tax-free. The compactness theorem below proves that this qualitative statement is equivalent to a quantitative finite-window gap.
Theorem 1 (Kernel-free characterization of the computational gap). Assume \(\mathcal{Q}_\Lambda^{\mathrm{cl}}\ne\{0\}\) and 1. The following are equivalent.
\(\mu_\Lambda^{\rm comp}>0\).
The kernel-free condition in 2 holds.
Proof. First assume \(\mu_\Lambda^{\rm comp}>0\). Let \(d\in\mathcal{K}_\Lambda^{\mathrm{cl}}\) satisfy \[O_\Lambda^{\mathrm{cl}}d=0, \qquad \operatorname{Rep}_\Lambda^{\mathrm{cl}}(d)=0, \qquad \operatorname{Tax}_\Lambda^{\mathrm{cl}}(d)=0.\] Then \(\overline{\mathcal{M}}_\Lambda^{\rm comp}([d])=0\). If \([d]\ne0\) in \(\mathcal{Q}_\Lambda^{\mathrm{cl}}\), set \[q=\frac{[d]}{\|[d]\|_{\mathcal{Q}_\Lambda^{\mathrm{cl}}}}.\] By 3, the quotient detector is positively homogeneous, so \(\overline{\mathcal{M}}_\Lambda^{\rm comp}(q)=0\). This contradicts the definition of \(\mu_\Lambda^{\rm comp}\) as the infimum of the quotient detector on \(S_\Lambda^{\mathrm{cl}}\). Hence \([d]=0\), equivalently \(d\in\Gamma_\Lambda^{\mathrm{cl}}\).
Conversely, assume 2. Since \(S_\Lambda^{\mathrm{cl}}\) is compact by 1 and \(\overline{\mathcal{M}}_\Lambda^{\rm comp}\) is continuous by 3, the infimum defining \(\mu_\Lambda^{\rm comp}\) is attained at some \(q_\ast\in S_\Lambda^{\mathrm{cl}}\). If \(\mu_\Lambda^{\rm comp}=0\), choose a representative \(d_\ast\in\mathcal{K}_\Lambda^{\mathrm{cl}}\) of \(q_\ast\). Then \[\mathcal{M}_\Lambda^{\rm comp}(d_\ast)= \overline{\mathcal{M}}_\Lambda^{\rm comp}(q_\ast)=0.\] Because all three detector terms are nonnegative, this gives \[O_\Lambda^{\mathrm{cl}}d_\ast=0, \qquad \operatorname{Rep}_\Lambda^{\mathrm{cl}}(d_\ast)=0, \qquad \operatorname{Tax}_\Lambda^{\mathrm{cl}}(d_\ast)=0.\] By 2, \(d_\ast\in\Gamma_\Lambda^{\mathrm{cl}}\), so \(q_\ast=0\), contradicting \(q_\ast\in S_\Lambda^{\mathrm{cl}}\). Therefore \(\mu_\Lambda^{\rm comp}>0\). ◻
Theorem 2 (Clean finite-window computational anti-phantom inequality). Assume \(\mathcal{Q}_\Lambda^{\mathrm{cl}}\ne\{0\}\), 1, and 2. Then \(\mu_\Lambda^{\rm comp}>0\), and every \(d\in\mathcal{K}_\Lambda^{\mathrm{cl}}\) satisfies \[\|O_\Lambda^{\mathrm{cl}}d\| + C_R\operatorname{Rep}_\Lambda^{\mathrm{cl}}(d) + C_T\operatorname{Tax}_\Lambda^{\mathrm{cl}}(d) \ge \mu_\Lambda^{\rm comp} \operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}}).\]
Proof. The positivity of \(\mu_\Lambda^{\rm comp}\) follows from 1. If \(\operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}})=0\), then the right-hand side is zero and the inequality follows from nonnegativity of the detector terms. If \(\operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}})>0\), set \[q=\frac{[d]}{\operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}})} \in S_\Lambda^{\mathrm{cl}}.\] By definition of \(\mu_\Lambda^{\rm comp}\), \[\overline{\mathcal{M}}_\Lambda^{\rm comp}(q) \ge \mu_\Lambda^{\rm comp}.\] Using the positive homogeneity of the quotient detector and the identity \(\overline{\mathcal{M}}_\Lambda^{\rm comp}([d])=\mathcal{M}_\Lambda^{\rm comp}(d)\) gives \[\mathcal{M}_\Lambda^{\rm comp}(d) \ge \mu_\Lambda^{\rm comp} \operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}}),\] which is exactly the displayed inequality. ◻
Remark 6 (Interpretation). 2 says that in a fixed clean quotient, a non-gauge defect must pay in at least one of three ways: it is seen by the combined observation channels, it fails to reproduce across the finite window, or it pays positive tax. This is a finite-dimensional anti-phantom theorem, not a scale-uniform PDE theorem.
The clean gap can be read as a finite-state alternative once the detector is split into its component channels. This section is only a bookkeeping interpretation of the finite-dimensional theorem. It does not assert universal computation, undecidability, or any dynamics of Navier–Stokes itself.
Definition 10 (Finite detector-sector set). Let \[\mathfrak S_\Lambda := \{\mathsf P,\mathsf F,\mathsf E,\mathsf T,\mathsf R,\mathsf{Tax}\}.\] The sectors correspond respectively to pressure observation, flux observation, energy or dissipation observation, trace observation, reproduction failure, and tax.
Lemma 4 (Observation norm resolved by channels). There is a finite constant \(A_{\rm obs,\Lambda}<\infty\), depending only on the fixed observation spaces and their chosen product norm, such that for every \(d\in\mathcal{K}_\Lambda^{\mathrm{cl}}\), \[\|O_\Lambda^{\mathrm{cl}}d\| \le A_{\rm obs,\Lambda} \bigl( \|O_\Lambda^P d\| + \|O_\Lambda^F d\| + \|O_\Lambda^E d\| + \|O_\Lambda^T d\| \bigr).\]
Proof. The target of \(O_\Lambda^{\mathrm{cl}}\) is the finite product \[\mathcal{Y}_\Lambda^P\times\mathcal{Y}_\Lambda^F \times\mathcal{Y}_\Lambda^E\times\mathcal{Y}_\Lambda^T.\] On this finite-dimensional product, the chosen product norm is bounded by the sum norm \[\|(y_P,y_F,y_E,y_T)\|_1 := \|y_P\|+\|y_F\|+\|y_E\|+\|y_T\|.\] Thus there exists \(A_{\rm obs,\Lambda}<\infty\) such that \[\|(y_P,y_F,y_E,y_T)\| \le A_{\rm obs,\Lambda}\|(y_P,y_F,y_E,y_T)\|_1\] for every element of the product. Applying this to \[(y_P,y_F,y_E,y_T) = (O_\Lambda^P d,O_\Lambda^F d, O_\Lambda^E d,O_\Lambda^T d)\] gives the displayed estimate. ◻
Definition 11 (Sector detector amplitudes). For \(d\in\mathcal{K}_\Lambda^{\mathrm{cl}}\), define \[\begin{align} D_{\mathsf P}(d) &:= A_{\rm obs,\Lambda}\|O_\Lambda^P d\|,\\ D_{\mathsf F}(d) &:= A_{\rm obs,\Lambda}\|O_\Lambda^F d\|,\\ D_{\mathsf E}(d) &:= A_{\rm obs,\Lambda}\|O_\Lambda^E d\|,\\ D_{\mathsf T}(d) &:= A_{\rm obs,\Lambda}\|O_\Lambda^T d\|,\\ D_{\mathsf R}(d) &:= C_R\operatorname{Rep}_\Lambda^{\mathrm{cl}}(d),\\ D_{\mathsf{Tax}}(d) &:= C_T\operatorname{Tax}_\Lambda^{\mathrm{cl}}(d). \end{align}\] For a non-gauge defect, meaning \(\operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}})>0\), define \(\operatorname{Sector}_\Lambda(d)\) to be the first sector in the fixed order \[\mathsf P\prec \mathsf F\prec \mathsf E \prec \mathsf T\prec \mathsf R\prec \mathsf{Tax}\] at which the maximum of \(\{D_\sigma(d):\sigma\in\mathfrak S_\Lambda\}\) is attained.
Proposition 1 (Finite-state detector alternative). Assume \(\mathcal{Q}_\Lambda^{\mathrm{cl}}\ne\{0\}\), 1, and 2. If \(d\in\mathcal{K}_\Lambda^{\mathrm{cl}}\) and \(\operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}})>0\), then \[D_{\operatorname{Sector}_\Lambda(d)}(d) \ge \frac{\mu_\Lambda^{\rm comp}}{6} \operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}}).\]
Proof. By 4, \[\begin{align} \mathcal{M}_\Lambda^{\rm comp}(d) &= \|O_\Lambda^{\mathrm{cl}}d\| + C_R\operatorname{Rep}_\Lambda^{\mathrm{cl}}(d) + C_T\operatorname{Tax}_\Lambda^{\mathrm{cl}}(d)\\ &\le \sum_{\sigma\in\mathfrak S_\Lambda}D_\sigma(d). \end{align}\] By 2, \[\mu_\Lambda^{\rm comp} \operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}}) \le \mathcal{M}_\Lambda^{\rm comp}(d).\] Combining the two inequalities gives \[\mu_\Lambda^{\rm comp} \operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}}) \le \sum_{\sigma\in\mathfrak S_\Lambda}D_\sigma(d).\] Since \(\mathfrak S_\Lambda\) has six elements, at least one sector satisfies \[D_\sigma(d) \ge \frac{\mu_\Lambda^{\rm comp}}{6} \operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}}).\] The tie-breaking rule in 11 selects one such maximizing sector, and the claim follows. ◻
Corollary 1 (Sector cover of normalized quotient classes). Under the hypotheses of 1, the quotient unit sphere is covered by the six sector sets \[\begin{align} \mathcal{S}_\sigma := \{[d]\in S_\Lambda^{\mathrm{cl}}:& \text{ there is a representative }d \text{ with } \operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}})=1\\ &\text{ and } D_\sigma(d)\ge \mu_\Lambda^{\rm comp}/6\}, \qquad \sigma\in\mathfrak S_\Lambda . \end{align}\]
Proof. Let \([d]\in S_\Lambda^{\mathrm{cl}}\), so \(\operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}})=1\). Applying 1 gives a sector \(\sigma=\operatorname{Sector}_\Lambda(d)\) such that \[D_\sigma(d)\ge\frac{\mu_\Lambda^{\rm comp}}{6}.\] Thus \([d]\in\mathcal{S}_\sigma\), proving the cover. ◻
Remark 7 (Representative dependence). The sector assignment is made for a chosen clean representative \(d\), not for an abstract quotient class alone. This is intentional: the present theorem is a finite-window detector statement before any canonical gauge slice has been selected. A later gauge-slice theorem may turn the sector map into a well-defined map on quotient classes, but that is not claimed here.
Remark 8 (No computational overinterpretation). The word sector means only that, in a fixed finite-dimensional window, one of finitely many detector channels must account for a definite fraction of the clean quotient gap. The result does not encode a Turing machine, does not produce a symbolic dynamics, and does not imply any undecidability statement.
The clean gap can be inserted into a localized transfer theorem only after the localized defect package has been compared with the clean quotient. This section records the algebraic consequence of such a comparison. It does not prove quotient lifting, residual absorption, or any localized Navier–Stokes estimate.
Definition 12 (Abstract localized detector package). Let \(\mathcal{K}_\Lambda^{\mathrm{loc}}\) be a finite-dimensional localized defect space with a localized gauge subspace \(\Gamma_\Lambda^{\mathrm{loc}}\). Define \[\operatorname{dist}_{\mathrm{loc}}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) := \inf_{\gamma\in\Gamma_\Lambda^{\mathrm{loc}}} \|\mathfrak D-\gamma\|_{\mathrm{loc}}.\] Let \[\Theta_\Lambda:\mathcal{K}_\Lambda^{\mathrm{loc}}\to\mathcal{K}_\Lambda^{\mathrm{cl}}\] be a local-to-clean chart, and let \[\mathcal{M}_\Lambda^{\mathrm{loc}}:\mathcal{K}_\Lambda^{\mathrm{loc}}\to[0,\infty)\] be the localized detector size. The quantity \(\mathcal{M}_\Lambda^{\mathrm{loc}}\) is intended to collect the localized pressure, flux, energy, trace, reproduction, and tax observations, but no concrete PDE formula is assumed in this definition.
Assumption 3 (Local-to-clean transfer comparison). There are constants \[0\le\varepsilon_G<1, \qquad \delta_G\ge0,\] and a nonnegative residual functional \[\mathsf{Err}_\Lambda^{\mathrm{loc}}:\mathcal{K}_\Lambda^{\mathrm{loc}}\to[0,\infty)\] such that every \(\mathfrak D\in\mathcal{K}_\Lambda^{\mathrm{loc}}\) satisfies \[\operatorname{dist}_{\mathrm{cl}}(\Theta_\Lambda\mathfrak D,\Gamma_\Lambda^{\mathrm{cl}}) \ge (1-\varepsilon_G) \operatorname{dist}_{\mathrm{loc}}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) - \delta_G\] and \[\mathcal{M}_\Lambda^{\mathrm{loc}}(\mathfrak D) + \mathsf{Err}_\Lambda^{\mathrm{loc}}(\mathfrak D) \ge \mathcal{M}_\Lambda^{\rm comp}(\Theta_\Lambda\mathfrak D).\]
Assumption 4 (Localized residual budget). There are constants \(\eta_\Lambda\ge0\) and \(\Delta_\Lambda\ge0\) such that every \(\mathfrak D\in\mathcal{K}_\Lambda^{\mathrm{loc}}\) satisfies \[\mathsf{Err}_\Lambda^{\mathrm{loc}}(\mathfrak D) \le \eta_\Lambda \operatorname{dist}_{\mathrm{loc}}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) + \Delta_\Lambda.\]
Corollary 2 (Conditional localized transfer from the computational gap). Assume \(\mathcal{Q}_\Lambda^{\mathrm{cl}}\ne\{0\}\), 1, 2, 3, and 4. Then every \(\mathfrak D\in\mathcal{K}_\Lambda^{\mathrm{loc}}\) satisfies \[\begin{align} \mathcal{M}_\Lambda^{\mathrm{loc}}(\mathfrak D) \ge& \bigl( \mu_\Lambda^{\rm comp}(1-\varepsilon_G) - \eta_\Lambda \bigr) \operatorname{dist}_{\mathrm{loc}}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}})\\ &- \mu_\Lambda^{\rm comp}\delta_G - \Delta_\Lambda. \end{align}\] In particular, if \[\eta_\Lambda < \mu_\Lambda^{\rm comp}(1-\varepsilon_G),\] then the localized detector controls the localized quotient distance up to the explicit additive loss \(\mu_\Lambda^{\rm comp}\delta_G+\Delta_\Lambda\).
Proof. By 2 applied to \(\Theta_\Lambda\mathfrak D\), \[\mathcal{M}_\Lambda^{\rm comp}(\Theta_\Lambda\mathfrak D) \ge \mu_\Lambda^{\rm comp} \operatorname{dist}_{\mathrm{cl}}(\Theta_\Lambda\mathfrak D,\Gamma_\Lambda^{\mathrm{cl}}).\] Using the quotient-distance comparison in 3 gives \[\mathcal{M}_\Lambda^{\rm comp}(\Theta_\Lambda\mathfrak D) \ge \mu_\Lambda^{\rm comp} \bigl( (1-\varepsilon_G) \operatorname{dist}_{\mathrm{loc}}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) - \delta_G \bigr).\] The detector comparison in 3 therefore implies \[\mathcal{M}_\Lambda^{\mathrm{loc}}(\mathfrak D) \ge \mu_\Lambda^{\rm comp}(1-\varepsilon_G) \operatorname{dist}_{\mathrm{loc}}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) - \mu_\Lambda^{\rm comp}\delta_G - \mathsf{Err}_\Lambda^{\mathrm{loc}}(\mathfrak D).\] Finally apply 4 to bound \(\mathsf{Err}_\Lambda^{\mathrm{loc}}(\mathfrak D)\) from above. This gives the displayed inequality. The final statement follows by positivity of the coefficient of the localized quotient distance. ◻
Remark 9 (Status of the localized corollary). 2 is only a conditional algebraic transfer statement. It uses \(c_\Lambda^{\mathrm{cl}}=\mu_\Lambda^{\rm comp}\), but it does not prove the local-to-clean chart estimate, the quotient-distance comparison, the residual budget, or the threshold inequality \(\eta_\Lambda<\mu_\Lambda^{\rm comp}(1-\varepsilon_G)\).
The preceding sections separate the clean finite-window obstruction from the localized transfer obstruction. This makes the possible failures explicit.
Definition 13 (Fixed-window computational phantom). A fixed-window computational phantom is an element \[d\in\mathcal{K}_\Lambda^{\mathrm{cl}}\] such that \[\operatorname{dist}_{\mathrm{cl}}(d,\Gamma_\Lambda^{\mathrm{cl}})>0, \qquad O_\Lambda^{\mathrm{cl}}d=0, \qquad \operatorname{Rep}_\Lambda^{\mathrm{cl}}(d)=0, \qquad \operatorname{Tax}_\Lambda^{\mathrm{cl}}(d)=0.\] Equivalently, it is a non-gauge clean defect that is invisible, exactly reproducible, and tax-free in the fixed finite window.
Definition 14 (Moving-window computational gap collapse). A moving-window computational gap collapse is a sequence of finite windows \(\Lambda_n\) and clean detector packages satisfying the fixed-window clean detector gauge compatibility and kernel-free condition for each \(n\), but for which \[\mu_{\Lambda_n}^{\rm comp}\to0.\] This is not a contradiction to the fixed-window theorem. It means only that the positive constants obtained by compactness are not uniform along the chosen sequence of windows.
Definition 15 (Asymptotic invisible reproducible tax-free sequence). An asymptotic invisible reproducible tax-free sequence is a sequence \((\Lambda_n,d_n)\) such that \[\operatorname{dist}_{\mathrm{cl},n}(d_n,\Gamma_{\Lambda_n}^{\mathrm{cl}})=1\] and \[\|O_{\Lambda_n}^{\mathrm{cl}}d_n\| + C_R\operatorname{Rep}_{\Lambda_n}^{\mathrm{cl}}(d_n) + C_T\operatorname{Tax}_{\Lambda_n}^{\mathrm{cl}}(d_n) \to0.\] Such a sequence is an asymptotic version of a clean phantom. It is compatible with every fixed-window theorem unless a window-uniform lower bound is proved.
Definition 16 (Localized transfer-threshold failure). For a localized package satisfying the comparison hypotheses of 3, transfer-threshold failure means that at least one of the following prevents 2 from giving a useful localized lower bound: \[\eta_\Lambda \ge \mu_\Lambda^{\rm comp}(1-\varepsilon_G),\] or the additive loss \[\mu_\Lambda^{\rm comp}\delta_G+\Delta_\Lambda\] is not controlled in the regime under consideration.
Proposition 2 (Exhaustion of finite-window failure modes). Fix a nontrivial clean quotient. Suppose a localized lower bound of the form \[\mathcal{M}_\Lambda^{\mathrm{loc}}(\mathfrak D) \ge c_{\mathrm{loc},\Lambda} \operatorname{dist}_{\mathrm{loc}}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) - \Delta_{\mathrm{loc},\Lambda}\] is not obtained from the clean computational anti-phantom mechanism for the window \(\Lambda\). Then at least one of the following is responsible:
the clean detector gauge compatibility in 1 is unavailable, so the detector has not descended to the clean quotient;
the fixed-window kernel-free condition fails, equivalently a fixed-window computational phantom exists;
the clean gap is positive but too small for the desired window family, indicating moving-window gap collapse as a possible obstruction;
the local-to-clean quotient or detector comparison in 3 is unavailable;
the residual budget in 4 is unavailable;
the transfer threshold or additive-loss control in 16 fails.
Proof. If 1 is unavailable, the clean detector has not been shown to be a quotient detector, so the first alternative is responsible. If gauge compatibility holds but the fixed-window kernel-free condition fails, then 13 gives the second alternative. If both hold, then 1 gives \(\mu_\Lambda^{\rm comp}>0\) for the fixed window. To pass from this clean gap to a localized lower bound, the argument uses exactly the quotient and detector comparison in 3, the residual budget in 4, and the threshold condition in 2. If all these inputs hold with useful constants, the displayed localized lower bound follows from that corollary. Therefore, if the lower bound is not obtained, one of the listed inputs or threshold requirements must fail. Along a family of windows, the additional possibility is that the fixed-window constants are positive but degenerate, which is precisely the moving-window gap-collapse alternative. ◻
Remark 10 (Role of the taxonomy). 2 is a bookkeeping result. It does not say which failure mode actually occurs for Navier–Stokes. Its purpose is to keep future work honest: after the clean theorem is proved, the remaining mathematics is either uniform clean-gap control, local-to-clean comparison, residual absorption, or additive-loss control.
We now isolate the first PDE-facing compatibility input. The purpose of this section is not to prove a pressure estimate. It is to state precisely what must be controlled if the localized quotient distance is to see the active pressure-source mismatch. The model is deliberately aligned with the standard Navier–Stokes pressure identity and with the pressure-splitting techniques used in local regularity theory [7], [11], [12].
Definition 17 (Localized pressure-source datum). A localized pressure-source datum consists, for each \(k\in\Lambda\), of finite-dimensional normed spaces \[\mathcal{X}_{{\rm src},k}, \qquad \mathcal{Z}_{{\rm src},k}, \qquad \mathcal{U}_k, \qquad \mathcal{P}_k^{\rm act}, \qquad \mathcal{R}_k,\] coordinate maps \[U_k:\mathcal{K}_\Lambda^{\mathrm{loc}}\to\mathcal{U}_k, \qquad P_k^{\rm act}:\mathcal{K}_\Lambda^{\mathrm{loc}}\to\mathcal{P}_k^{\rm act}, \qquad R_k:\mathcal{K}_\Lambda^{\mathrm{loc}}\to\mathcal{R}_k,\] an assembly map \[\mathscr A_{{\rm src},k}: \mathcal{P}_k^{\rm act}\times\mathcal{U}_k\times\mathcal{R}_k \to \mathcal{Z}_{{\rm src},k},\] and a fixed finite-dimensional source projection \[\Pi_{{\rm src},k}:\mathcal{Z}_{{\rm src},k} \to \mathcal{X}_{{\rm src},k}.\] The associated pressure-source mismatch is \[S_k^{\rm prs}(\mathfrak D) := \Pi_{{\rm src},k} \mathscr A_{{\rm src},k} \left( P_k^{\rm act}(\mathfrak D), U_k(\mathfrak D), R_k(\mathfrak D) \right),\] where \(\mathscr A_{{\rm src},k}\) represents the finite-dimensional form of \[-\Delta P_k^{\rm act}(\mathfrak D) - \partial_i\partial_j \bigl( U_{k,i}(\mathfrak D)U_{k,j}(\mathfrak D) + R_{k,ij}(\mathfrak D) \bigr) .\]
Remark 11 (Status of the source coordinate). 17 is a finite-window model datum. The notation with \(-\Delta\) and \(\partial_i\partial_j\) records the intended Navier–Stokes pressure identity, but the definition does not prove that the coordinates are produced by a suitable weak solution or that cutoff, projection, harmonic, or truncation errors are small.
Definition 18 (Pressure-source residual). Fix weights \(w_k>0\). Define \[\mathsf{Err}_{\rm src}^{\rm prs}(\mathfrak D) := \left( \sum_{k\in\Lambda} w_k \|S_k^{\rm prs}(\mathfrak D)\|_{\mathcal{X}_{{\rm src},k}}^2 \right)^{1/2}.\] The quotient-compatible pressure-source residual is \[\mathsf{Err}_{\rm src,q}^{\rm prs}(\mathfrak D) := \inf_{\gamma\in\Gamma_\Lambda^{\mathrm{loc}}} \mathsf{Err}_{\rm src}^{\rm prs}(\mathfrak D-\gamma).\]
Remark 12 (Gauge convention). The quotient residual is used because this manuscript has not proved that \(S_k^{\rm prs}\) is invariant under every localized gauge direction. If a later gauge-cleaning theorem proves such invariance, the raw residual \(\mathsf{Err}_{\rm src}^{\rm prs}\) may replace \(\mathsf{Err}_{\rm src,q}^{\rm prs}\).
Assumption 5 (Pressure-source quotient observability). There exist constants \[C_{\rm src}<\infty, \qquad \Delta_{\rm src}\ge0,\] such that every localized package \(\mathfrak D\in\mathcal{K}_\Lambda^{\mathrm{loc}}\) satisfies \[\mathsf{Err}_{\rm src,q}^{\rm prs}(\mathfrak D) \le C_{\rm src} \operatorname{dist}_{\mathrm{loc}}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) + \Delta_{\rm src}.\]
Proposition 3 (Conditional pressure-source quotient observability). Under 5, the active pressure-source residual is controlled by the localized quotient distance up to the additive error \(\Delta_{\rm src}\): \[\mathsf{Err}_{\rm src,q}^{\rm prs}(\mathfrak D) \le C_{\rm src} \operatorname{dist}_{\mathrm{loc}}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) + \Delta_{\rm src}\] for every \(\mathfrak D\in\mathcal{K}_\Lambda^{\mathrm{loc}}\).
Proof. This is exactly 5. No additional pressure estimate is being used. ◻
Remark 13 (Effect on the transfer theorem). This structural-observability model does not change the form of 2; it keeps the original localized quotient distance \(\operatorname{dist}_{\mathrm{loc}}\). Its cost is that 5 becomes an explicit component hypothesis that must be proved or assumed before pressure-source terms can be absorbed into the localized residual budget.
To turn 5 into a theorem, one must control the following components in the source norm \(\mathcal{X}_{{\rm src},k}\):
the active pressure Poisson mismatch generated by the localized pressure splitting;
cutoff commutator leakage from localizing \(-\Delta p=\partial_i\partial_j(u_i u_j)\);
harmonic pressure gauge terms not removed by \(\Gamma_\Lambda^{\mathrm{loc}}\);
finite projection and truncation errors in \(\Pi_{{\rm src},k}\);
stress or covariance mismatch between the projected quadratic velocity term and the \(R_k\)-coordinate.
None of these estimates is proved in this section.
Proposition 4 (Pressure-source observability decision point). The present manuscript establishes only the following conditional implication: if 5 is supplied, then 3 follows. Removing the assumption requires either:
an enhanced quotient distance that includes \(\mathsf{Err}_{\rm src,q}^{\rm prs}\);
a proof of 5 from localized Navier–Stokes pressure geometry; or
a sharpened localized residual norm that includes \(\mathsf{Err}_{\rm src,q}^{\rm prs}\) and a residual-budget estimate for that sharpened norm.
Proof. The three alternatives are precisely the three ways to make the source residual appear in the existing transfer architecture. The first changes the quotient geometry, the second keeps the quotient geometry and proves the structural hypothesis, and the third changes the residual budget. Without at least one of these inputs, the displayed pressure-source estimate is not a consequence of the clean finite-window gap or of the abstract localized transfer corollary. ◻
We now record the enhanced-distance route in a concrete normalized local model. This subsection changes the localized quotient geometry by placing the pressure-source residual directly into the defect distance.
Definition 19 (Normalized pressure-source geometry). Let \[Q_1:=B_1\times(-1,0), \qquad I:=(-1,0),\] with \[B_{1/2}\subset B_{3/4}\subset B_1.\] Fix a cutoff \(\eta\in C_c^\infty(B_1)\) such that \[\eta\equiv1\quad\text{on }B_{3/4}.\] Set \[X_{\rm src} := L^{3/2}\bigl(I;L^{3/2}(B_1)\bigr)^{3\times3}, \qquad Y_{\rm prs} := L^{3/2}\bigl(I;L^{3/2}(B_{1/2})\bigr).\] For a localized velocity \(u\), define \[f_{ij}:=u_i u_j.\] For a clean finite-window package, define the clean source tensor \[F_{ij}^{\mathrm{cl}}:=U_iU_j+R_{ij}.\] The active localized pressure and clean active pressure are \[p^{\rm act}:=R_iR_j(\eta f_{ij}), \qquad p^{\mathrm{cl}}:=P_{\rm prs}^{\mathrm{cl}}R_iR_j(F_{ij}^{\mathrm{cl}}),\] where \(P_{\rm prs}^{\mathrm{cl}}:Y_{\rm prs}\to Y_{\rm prs}\) is a fixed bounded clean pressure projection. All pressures in this subsection are restricted to \(B_{1/2}\) when measured in \(Y_{\rm prs}\). All source tensors are extended by zero outside \(B_1\) before applying the whole-space Riesz transforms.
Definition 20 (Concrete pressure-source mismatch). Let \(h^{\rm harm}\in Y_{\rm prs}\) denote the selected harmonic-gauge residual, with \(h^{\rm harm}=0\) if no harmonic gauge residual is present. Define the concrete pressure-source mismatch by \[\mathfrak P_{\rm src} := p^{\rm act}-p^{\mathrm{cl}}-h^{\rm harm}.\] Define \[\mathsf{Err}_{\rm src}^{\rm prs} := \|\mathfrak P_{\rm src}\|_{Y_{\rm prs}}.\] The quotient-compatible version is \[\mathsf{Err}_{\rm src,q}^{\rm prs}(\mathfrak D) := \inf_{\gamma\in\Gamma_\Lambda^{\mathrm{loc}}} \mathsf{Err}_{\rm src}^{\rm prs}(\mathfrak D-\gamma),\] which agrees with 18 in this concrete one-window pressure model.
Lemma 5 (Pressure-source mismatch decomposition). In the normalized model of 19, the pressure-source mismatch admits the decomposition \[\mathfrak P_{\rm src} = C_\eta(f) + E_{\rm act}^{\rm src} + E_{\rm proj}^{\mathrm{cl}} + E_{\rm harm},\] where \[C_\eta(f) := R_iR_j(\eta f_{ij})-\eta R_iR_j(f_{ij}),\] \[E_{\rm act}^{\rm src} := \eta R_iR_j(f_{ij})-R_iR_j(F_{ij}^{\mathrm{cl}}),\] \[E_{\rm proj}^{\mathrm{cl}} := (I-P_{\rm prs}^{\mathrm{cl}})R_iR_j(F_{ij}^{\mathrm{cl}}), \qquad E_{\rm harm}:=-h^{\rm harm}.\] Consequently, \[\mathsf{Err}_{\rm src}^{\rm prs} \le \|C_\eta(f)\|_{Y_{\rm prs}} + \|E_{\rm act}^{\rm src}\|_{Y_{\rm prs}} + \|E_{\rm proj}^{\mathrm{cl}}\|_{Y_{\rm prs}} + \|E_{\rm harm}\|_{Y_{\rm prs}}.\]
Proof. Using the definitions of \(p^{\rm act}\) and \(p^{\mathrm{cl}}\), \[\begin{align} \mathfrak P_{\rm src} &= R_iR_j(\eta f_{ij}) - P_{\rm prs}^{\mathrm{cl}}R_iR_j(F_{ij}^{\mathrm{cl}}) - h^{\rm harm}\\ &= \bigl(R_iR_j(\eta f_{ij})-\eta R_iR_j(f_{ij})\bigr)\\ &\quad+ \bigl(\eta R_iR_j(f_{ij})-R_iR_j(F_{ij}^{\mathrm{cl}})\bigr)\\ &\quad+ \bigl(R_iR_j(F_{ij}^{\mathrm{cl}}) -P_{\rm prs}^{\mathrm{cl}}R_iR_j(F_{ij}^{\mathrm{cl}})\bigr) - h^{\rm harm}. \end{align}\] This is the stated decomposition. The norm bound follows from the triangle inequality in \(Y_{\rm prs}\). ◻
Lemma 6 (Fixed-geometry commutator bound). There is a constant \(C_\eta<\infty\), depending only on the fixed cutoff and the normalized balls, such that \[\|C_\eta(f)\|_{Y_{\rm prs}} \le C_\eta \|(1-\eta)f\|_{L^{3/2}(I;L^{3/2}(B_1\setminus B_{3/4}))^{3\times3}}.\] This is a fixed-geometry estimate and contains no scale-uniform assertion.
Proof. On \(B_{1/2}\), \(\eta=1\). Hence \[C_\eta(f) = R_iR_j(\eta f_{ij})-R_iR_j(f_{ij}) = -R_iR_j((1-\eta)f_{ij}).\] The source \((1-\eta)f\) is supported in \[B_1\setminus B_{3/4},\] which is separated from \(B_{1/2}\). Therefore the kernel of \(R_iR_j\) is smooth and uniformly bounded as a map from the source annulus to \(B_{1/2}\). For each fixed time, \[\|C_\eta(f)(t)\|_{L^{3/2}(B_{1/2})} \le C_\eta \|(1-\eta)f(t)\|_{L^{3/2}(B_1\setminus B_{3/4})^{3\times3}}.\] Taking the \(L^{3/2}\)-norm in time gives the claim. ◻
Remark 14 (Fixed estimates versus structural terms). 6 is the only estimate in this subsection that comes from the fixed separated geometry. The active source residual \(E_{\rm act}^{\rm src}\), the clean projection residual \(E_{\rm proj}^{\mathrm{cl}}\), and the harmonic gauge residual \(E_{\rm harm}\) are not shown to be small or controlled by the original localized quotient distance.
Definition 21 (Enhanced localized quotient distance). Fix \(\alpha_{\rm src}>0\). Define \[\operatorname{dist}_{\mathrm{loc}}^{\sharp} (\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) := \operatorname{dist}_{\mathrm{loc}}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) + \alpha_{\rm src} \mathsf{Err}_{\rm src,q}^{\rm prs}(\mathfrak D).\]
Lemma 7 (Enhanced-distance pressure-source observability). For every localized package \(\mathfrak D\), \[\mathsf{Err}_{\rm src,q}^{\rm prs}(\mathfrak D) \le \alpha_{\rm src}^{-1} \operatorname{dist}_{\mathrm{loc}}^{\sharp} (\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}).\]
Proof. By 21, \[\operatorname{dist}_{\mathrm{loc}}^{\sharp} (\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) \ge \alpha_{\rm src} \mathsf{Err}_{\rm src,q}^{\rm prs}(\mathfrak D).\] Dividing by \(\alpha_{\rm src}>0\) gives the result. ◻
Assumption 6 (Enhanced local-to-clean transfer comparison). There are constants \(0\le\varepsilon_G<1\) and \(\delta_G\ge0\) such that every \(\mathfrak D\in\mathcal{K}_\Lambda^{\mathrm{loc}}\) satisfies \[\operatorname{dist}_{\mathrm{cl}}(\Theta_\Lambda\mathfrak D,\Gamma_\Lambda^{\mathrm{cl}}) \ge (1-\varepsilon_G) \operatorname{dist}_{\mathrm{loc}}^{\sharp}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) - \delta_G\] and \[\mathcal{M}_\Lambda^{\mathrm{loc}}(\mathfrak D) + \mathsf{Err}_\Lambda^{\mathrm{loc}}(\mathfrak D) \ge \mathcal{M}_\Lambda^{\rm comp}(\Theta_\Lambda\mathfrak D).\]
Assumption 7 (Enhanced localized residual budget). There are constants \(\eta_\Lambda\ge0\) and \(\Delta_\Lambda\ge0\) such that every \(\mathfrak D\in\mathcal{K}_\Lambda^{\mathrm{loc}}\) satisfies \[\mathsf{Err}_\Lambda^{\mathrm{loc}}(\mathfrak D) \le \eta_\Lambda \operatorname{dist}_{\mathrm{loc}}^{\sharp}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) + \Delta_\Lambda.\]
Theorem 3 (Conditional enhanced-distance localized transfer). Assume \(\mathcal{Q}_\Lambda^{\mathrm{cl}}\ne\{0\}\), 1, 2, 6, and 7. Then every \(\mathfrak D\in\mathcal{K}_\Lambda^{\mathrm{loc}}\) satisfies \[\begin{align} \mathcal{M}_\Lambda^{\mathrm{loc}}(\mathfrak D) \ge& \bigl( \mu_\Lambda^{\rm comp}(1-\varepsilon_G) - \eta_\Lambda \bigr) \operatorname{dist}_{\mathrm{loc}}^{\sharp} (\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}})\\ &- \mu_\Lambda^{\rm comp}\delta_G - \Delta_\Lambda. \end{align}\]
Proof. The proof is the same algebraic transfer argument as in 2, with \(\operatorname{dist}_{\mathrm{loc}}\) replaced by \(\operatorname{dist}_{\mathrm{loc}}^{\sharp}\). Applying 2 to \(\Theta_\Lambda\mathfrak D\) gives \[\mathcal{M}_\Lambda^{\rm comp}(\Theta_\Lambda\mathfrak D) \ge \mu_\Lambda^{\rm comp} \operatorname{dist}_{\mathrm{cl}}(\Theta_\Lambda\mathfrak D,\Gamma_\Lambda^{\mathrm{cl}}).\] The enhanced quotient comparison in 6 implies \[\mathcal{M}_\Lambda^{\rm comp}(\Theta_\Lambda\mathfrak D) \ge \mu_\Lambda^{\rm comp} \bigl( (1-\varepsilon_G) \operatorname{dist}_{\mathrm{loc}}^{\sharp}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) - \delta_G \bigr).\] Using the detector comparison from the same assumption and then subtracting the residual term gives \[\mathcal{M}_\Lambda^{\mathrm{loc}}(\mathfrak D) \ge \mu_\Lambda^{\rm comp}(1-\varepsilon_G) \operatorname{dist}_{\mathrm{loc}}^{\sharp}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) - \mu_\Lambda^{\rm comp}\delta_G - \mathsf{Err}_\Lambda^{\mathrm{loc}}(\mathfrak D).\] Finally apply 7. This proves the displayed estimate. ◻
Remark 15 (Meaning of enhanced pressure-source observability). 7 proves pressure-source observability only relative to the enhanced quotient geometry. It does not show that the original localized quotient distance \(\operatorname{dist}_{\mathrm{loc}}\) controls \(\mathsf{Err}_{\rm src,q}^{\rm prs}\). It also makes no scale-uniform claim and gives no Navier–Stokes regularity conclusion.
Remark 16 (Tradeoff). The advantage of \(\operatorname{dist}_{\mathrm{loc}}^{\sharp}\) is that pressure-source observability is built into the quotient geometry. The cost is that the localized quotient geometry has changed. A later theorem must either compare \(\operatorname{dist}_{\mathrm{loc}}^{\sharp}\) with the original localized quotient distance or justify the enhanced distance as the natural defect distance for localized Navier–Stokes packages.
The next question is whether the enhanced distance is merely a convenient renorming of the original localized quotient distance, or whether it has added a genuinely new pressure-source coordinate. The present subsection proves only a sufficient condition for one-sided comparison. It does not prove that condition from Navier–Stokes pressure geometry.
Definition 22 (Quotient pressure-source component budget). For \(\gamma\in\Gamma_\Lambda^{\mathrm{loc}}\), let \[C_\eta^\gamma,\qquad E_{{\rm act},\gamma}^{\rm src},\qquad E_{{\rm proj},\gamma}^{\mathrm{cl}},\qquad E_{{\rm harm},\gamma}\] denote the four pressure-source components in 5 evaluated on \(\mathfrak D-\gamma\). Define the quotient component budget by \[\begin{align} B_{\rm src,q}^{\rm prs}(\mathfrak D) := \inf_{\gamma\in\Gamma_\Lambda^{\mathrm{loc}}} \bigl( &\|C_\eta^\gamma\|_{Y_{\rm prs}} + \|E_{{\rm act},\gamma}^{\rm src}\|_{Y_{\rm prs}}\\ &+ \|E_{{\rm proj},\gamma}^{\mathrm{cl}}\|_{Y_{\rm prs}} + \|E_{{\rm harm},\gamma}\|_{Y_{\rm prs}} \bigr). \end{align}\]
Lemma 8 (Component budget controls the quotient pressure residual). For every localized package \(\mathfrak D\), \[\mathsf{Err}_{\rm src,q}^{\rm prs}(\mathfrak D) \le B_{\rm src,q}^{\rm prs}(\mathfrak D).\]
Proof. Fix \(\gamma\in\Gamma_\Lambda^{\mathrm{loc}}\). Applying 5 to \(\mathfrak D-\gamma\) gives \[\begin{align} \mathsf{Err}_{\rm src}^{\rm prs}(\mathfrak D-\gamma) \le& \|C_\eta^\gamma\|_{Y_{\rm prs}} + \|E_{{\rm act},\gamma}^{\rm src}\|_{Y_{\rm prs}}\\ &+ \|E_{{\rm proj},\gamma}^{\mathrm{cl}}\|_{Y_{\rm prs}} + \|E_{{\rm harm},\gamma}\|_{Y_{\rm prs}}. \end{align}\] Taking the infimum over \(\gamma\in\Gamma_\Lambda^{\mathrm{loc}}\) gives the displayed inequality. ◻
Assumption 8 (Original-distance pressure-source component budget). There are constants \[C_{\rm cmp}<\infty, \qquad \Delta_{\rm cmp}\ge0,\] such that every localized package \(\mathfrak D\in\mathcal{K}_\Lambda^{\mathrm{loc}}\) satisfies \[B_{\rm src,q}^{\rm prs}(\mathfrak D) \le C_{\rm cmp} \operatorname{dist}_{\mathrm{loc}}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) + \Delta_{\rm cmp}.\]
Proposition 5 (Conditional one-sided comparison of localized distances). Assume 8. Then every localized package \(\mathfrak D\in\mathcal{K}_\Lambda^{\mathrm{loc}}\) satisfies \[\operatorname{dist}_{\mathrm{loc}}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) \le \operatorname{dist}_{\mathrm{loc}}^{\sharp}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}})\] and \[\operatorname{dist}_{\mathrm{loc}}^{\sharp}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) \le \bigl(1+\alpha_{\rm src}C_{\rm cmp}\bigr) \operatorname{dist}_{\mathrm{loc}}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) + \alpha_{\rm src}\Delta_{\rm cmp}.\] In particular, if \(\Delta_{\rm cmp}=0\), then the enhanced distance is bounded above and below by fixed multiples of the original distance in this finite-window model.
Proof. The lower bound follows directly from 21, since \(\mathsf{Err}_{\rm src,q}^{\rm prs}\ge0\). For the upper bound, combine 21, 8, and 8: \[\begin{align} \operatorname{dist}_{\mathrm{loc}}^{\sharp}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) &= \operatorname{dist}_{\mathrm{loc}}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) + \alpha_{\rm src} \mathsf{Err}_{\rm src,q}^{\rm prs}(\mathfrak D)\\ &\le \operatorname{dist}_{\mathrm{loc}}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) + \alpha_{\rm src} B_{\rm src,q}^{\rm prs}(\mathfrak D)\\ &\le \bigl(1+\alpha_{\rm src}C_{\rm cmp}\bigr) \operatorname{dist}_{\mathrm{loc}}(\mathfrak D,\Gamma_\Lambda^{\mathrm{loc}}) + \alpha_{\rm src}\Delta_{\rm cmp}. \end{align}\] If \(\Delta_{\rm cmp}=0\), this gives the stated two-sided multiplicative comparison. ◻
Remark 17 (Status of the comparison condition). 5 does not prove that the original localized quotient distance controls the pressure-source residual. It identifies the exact componentwise pressure estimate needed for that conclusion. If 8 fails, then the pressure-source mismatch is not merely a bookkeeping term for the original distance; it is evidence that the enhanced pressure-source geometry may be a genuinely different localized defect geometry.
We now make one explicit finite-window norm choice. This is not presented as the unique or canonical localized norm. It is a model geometry whose purpose is to make clear what changes when pressure-source observability is built directly into the localized quotient distance.
Definition 23 (Model A: weighted finite-dimensional localized coordinate norm). For each \(k\in\Lambda\), fix finite-dimensional normed spaces \[V_{U,k},\quad V_{P,k},\quad V_{R,k},\quad V_{\Pi,k},\quad V_{\Phi,k},\quad V_{T,k},\quad V_{s,k},\] with norms \[\|\cdot\|_{U,k},\quad \|\cdot\|_{P,k},\quad \|\cdot\|_{R,k},\quad \|\cdot\|_{\Pi,k},\quad \|\cdot\|_{\Phi,k},\quad \|\cdot\|_{T,k},\quad \|\cdot\|_{s,k}.\] A Model A localized package is a finite-window coordinate vector \[D = (U_k,P_k^{\rm act},R_k,\Pi_k,\Phi_k,T_k,s_k)_{k\in\Lambda},\] where \[\begin{gather} U_k\in V_{U,k},\qquad P_k^{\rm act}\in V_{P,k},\qquad R_k\in V_{R,k},\\ \Pi_k\in V_{\Pi,k},\qquad \Phi_k\in V_{\Phi,k},\qquad T_k\in V_{T,k},\qquad s_k\in V_{s,k}. \end{gather}\] Here \(U_k\) is the localized velocity coordinate, \(P_k^{\rm act}\) is the active pressure coordinate, \(R_k\) is the Reynolds or covariance coordinate, \(\Pi_k\) is the flux coordinate, \(\Phi_k\) is the energy or trace coordinate, \(T_k\) is the selected trace coordinate, and \(s_k\) is the ledger slack coordinate. Fix positive weights \[w_U,w_P,w_R,w_\Pi,w_\Phi,w_T,w_s>0.\] Define \[\begin{align} \|D\|_{\mathrm{loc},0}^2 := \sum_{k\in\Lambda} \bigl( &w_U\|U_k\|_{U,k}^2 + w_P\|P_k^{\rm act}\|_{P,k}^2 + w_R\|R_k\|_{R,k}^2\\ &+ w_\Pi\|\Pi_k\|_{\Pi,k}^2 + w_\Phi\|\Phi_k\|_{\Phi,k}^2 + w_T\|T_k\|_{T,k}^2 + w_s\|s_k\|_{s,k}^2 \bigr). \end{align}\] Let \(\Gamma_\Lambda^{\mathrm{loc}}\) be a linear localized gauge subspace of this finite-dimensional coordinate space. The Model A baseline localized quotient distance is \[\operatorname{dist}_{\mathrm{loc},0}(D,\Gamma_\Lambda^{\mathrm{loc}}) := \inf_{\gamma\in\Gamma_\Lambda^{\mathrm{loc}}} \|D-\gamma\|_{\mathrm{loc},0}.\]
Definition 24 (Model A pressure-source quotient residual). Fix a nonnegative positively homogeneous pressure-source residual \[\mathsf{Err}_{\rm src}^{\rm prs}:D\mapsto[0,\infty).\] In applications this is intended to be the pressure-source residual associated with the concrete pressure mismatch of 20. In Model A, its positive homogeneity is part of the finite-dimensional datum; it is not derived here from the nonlinear Navier–Stokes source \(u_i u_j\). Define \[\mathsf{Err}_{\rm src,q}^{\rm prs}(D) := \inf_{\gamma\in\Gamma_\Lambda^{\mathrm{loc}}} \mathsf{Err}_{\rm src}^{\rm prs}(D-\gamma).\]
Definition 25 (Model A enhanced localized quotient distance). Fix \(\alpha_{\rm src}>0\). Define \[\operatorname{dist}_{\mathrm{loc},\alpha}^{\sharp} (D,\Gamma_\Lambda^{\mathrm{loc}}) := \operatorname{dist}_{\mathrm{loc},0}(D,\Gamma_\Lambda^{\mathrm{loc}}) + \alpha_{\rm src} \mathsf{Err}_{\rm src,q}^{\rm prs}(D).\]
Lemma 9 (Model A pressure-source observability). For every Model A localized package \(D\), \[\mathsf{Err}_{\rm src,q}^{\rm prs}(D) \le \alpha_{\rm src}^{-1} \operatorname{dist}_{\mathrm{loc},\alpha}^{\sharp} (D,\Gamma_\Lambda^{\mathrm{loc}}).\]
Proof. By 25, \[\operatorname{dist}_{\mathrm{loc},\alpha}^{\sharp} (D,\Gamma_\Lambda^{\mathrm{loc}}) \ge \alpha_{\rm src} \mathsf{Err}_{\rm src,q}^{\rm prs}(D).\] Divide by \(\alpha_{\rm src}>0\). ◻
Lemma 10 (Homogeneity and gauge vanishing in Model A). For every \(\lambda\ge0\) and every Model A localized package \(D\), \[\operatorname{dist}_{\mathrm{loc},\alpha}^{\sharp} (\lambda D,\Gamma_\Lambda^{\mathrm{loc}}) = \lambda \operatorname{dist}_{\mathrm{loc},\alpha}^{\sharp} (D,\Gamma_\Lambda^{\mathrm{loc}}).\] Moreover, if \(D\in\Gamma_\Lambda^{\mathrm{loc}}\), then \[\operatorname{dist}_{\mathrm{loc},\alpha}^{\sharp}(D,\Gamma_\Lambda^{\mathrm{loc}})=0.\]
Proof. Because \(\Gamma_\Lambda^{\mathrm{loc}}\) is a linear subspace and \(\|\cdot\|_{\mathrm{loc},0}\) is a norm, the baseline quotient distance is positively homogeneous: \[\operatorname{dist}_{\mathrm{loc},0}(\lambda D,\Gamma_\Lambda^{\mathrm{loc}}) = \lambda \operatorname{dist}_{\mathrm{loc},0}(D,\Gamma_\Lambda^{\mathrm{loc}}) \qquad(\lambda\ge0).\] The same argument applies to the quotient pressure-source residual. For \(\lambda>0\), use the change of variables \(\gamma=\lambda\gamma'\) in the infimum and the positive homogeneity of \(\mathsf{Err}_{\rm src}^{\rm prs}\): \[\begin{align} \mathsf{Err}_{\rm src,q}^{\rm prs}(\lambda D) &= \inf_{\gamma\in\Gamma_\Lambda^{\mathrm{loc}}} \mathsf{Err}_{\rm src}^{\rm prs}(\lambda D-\gamma)\\ &= \lambda \inf_{\gamma'\in\Gamma_\Lambda^{\mathrm{loc}}} \mathsf{Err}_{\rm src}^{\rm prs}(D-\gamma') = \lambda\mathsf{Err}_{\rm src,q}^{\rm prs}(D). \end{align}\] The case \(\lambda=0\) follows from positive homogeneity, which gives \(\mathsf{Err}_{\rm src}^{\rm prs}(0)=0\), and from \(\|0\|_{\mathrm{loc},0}=0\). Adding the two homogeneous terms proves the first claim.
If \(D\in\Gamma_\Lambda^{\mathrm{loc}}\), then choosing \(\gamma=D\) gives \(\operatorname{dist}_{\mathrm{loc},0}(D,\Gamma_\Lambda^{\mathrm{loc}})=0\) and \(\mathsf{Err}_{\rm src,q}^{\rm prs}(D)=0\). Hence \(\operatorname{dist}_{\mathrm{loc},\alpha}^{\sharp}(D,\Gamma_\Lambda^{\mathrm{loc}})=0\). ◻
Remark 18 (Seminorm and gauge status). 10 proves the formal homogeneity and gauge-vanishing properties of Model A. The object \(\operatorname{dist}_{\mathrm{loc},\alpha}^{\sharp}\) should nevertheless be read as a finite-window quotient gauge distance. It is a genuine norm on the intended physical quotient only after the pressure-source residual and the localized gauge subspace are verified to be compatible with the same physical gauge and to separate non-gauge classes. That compatibility is not proved here.
Assumption 9 (Model A enhanced transfer comparison). There are constants \(0\le\varepsilon_G<1\) and \(\delta_G\ge0\) such that every Model A localized package \(D\) satisfies \[\operatorname{dist}_{\mathrm{cl}}(\Theta_\Lambda D,\Gamma_\Lambda^{\mathrm{cl}}) \ge (1-\varepsilon_G) \operatorname{dist}_{\mathrm{loc},\alpha}^{\sharp}(D,\Gamma_\Lambda^{\mathrm{loc}}) - \delta_G\] and \[\mathcal{M}_\Lambda^{\mathrm{loc}}(D) + \mathsf{Err}_\Lambda^{\mathrm{loc}}(D) \ge \mathcal{M}_\Lambda^{\rm comp}(\Theta_\Lambda D).\]
Assumption 10 (Model A enhanced residual budget). There are constants \(\eta_\Lambda\ge0\) and \(\Delta_\Lambda\ge0\) such that every Model A localized package \(D\) satisfies \[\mathsf{Err}_\Lambda^{\mathrm{loc}}(D) \le \eta_\Lambda \operatorname{dist}_{\mathrm{loc},\alpha}^{\sharp}(D,\Gamma_\Lambda^{\mathrm{loc}}) + \Delta_\Lambda.\]
Theorem 4 (Conditional Model A enhanced-distance localized transfer). Assume \(\mathcal{Q}_\Lambda^{\mathrm{cl}}\ne\{0\}\), 1, 2, 9, and 10. Then every Model A localized package \(D\) satisfies \[\begin{align} \mathcal{M}_\Lambda^{\mathrm{loc}}(D) \ge& \bigl( \mu_\Lambda^{\rm comp}(1-\varepsilon_G) - \eta_\Lambda \bigr) \operatorname{dist}_{\mathrm{loc},\alpha}^{\sharp} (D,\Gamma_\Lambda^{\mathrm{loc}})\\ &- \mu_\Lambda^{\rm comp}\delta_G - \Delta_\Lambda. \end{align}\]
Proof. This is the localized transfer argument with the Model A distance \(\operatorname{dist}_{\mathrm{loc},\alpha}^{\sharp}\) in place of the abstract localized distance. By 2, \[\mathcal{M}_\Lambda^{\rm comp}(\Theta_\Lambda D) \ge \mu_\Lambda^{\rm comp} \operatorname{dist}_{\mathrm{cl}}(\Theta_\Lambda D,\Gamma_\Lambda^{\mathrm{cl}}).\] Using 9 gives \[\mathcal{M}_\Lambda^{\rm comp}(\Theta_\Lambda D) \ge \mu_\Lambda^{\rm comp} \bigl( (1-\varepsilon_G) \operatorname{dist}_{\mathrm{loc},\alpha}^{\sharp}(D,\Gamma_\Lambda^{\mathrm{loc}}) - \delta_G \bigr).\] The detector comparison in 9 then implies \[\mathcal{M}_\Lambda^{\mathrm{loc}}(D) \ge \mu_\Lambda^{\rm comp}(1-\varepsilon_G) \operatorname{dist}_{\mathrm{loc},\alpha}^{\sharp}(D,\Gamma_\Lambda^{\mathrm{loc}}) - \mu_\Lambda^{\rm comp}\delta_G - \mathsf{Err}_\Lambda^{\mathrm{loc}}(D).\] Applying 10 gives the stated estimate. ◻
Remark 19 (Model A tradeoff). Model A is a finite-window norm choice. It makes pressure-source observability part of the localized quotient geometry. The remaining PDE problem is to justify that this norm is natural for localized Navier–Stokes packages, or to compare it with a more intrinsic quotient norm derived from pressure splitting and local energy estimates. No uniqueness or canonicality of Model A, no scale-uniformity, and no Navier–Stokes regularity conclusion is claimed.
We next record the minimal comparison statement that would make Model A a controlled finite-window coordinate norm. The intrinsic norm below is still a model datum. Its role is to separate a purely finite-dimensional coordinate comparison from the PDE problem of deriving such a norm from suitable weak solutions, pressure splitting, and the local energy inequality.
Definition 26 (Intrinsic localized package norm candidate). For each \(k\in\Lambda\), let \[\mathcal{U}_k,\quad \mathcal{P}_k,\quad \mathcal{R}_k,\quad \mathcal{F}_k,\quad \mathcal{E}_k,\quad \mathcal{T}_k,\quad \mathcal{S}_k\] be normed spaces for intrinsic localized velocity, active pressure, Reynolds/covariance, flux, energy or trace, selected trace, and slack data. The intended PDE examples are \[\mathcal{U}_k=L^3(Q_k)^3,\qquad \mathcal{P}_k=L^{3/2}(Q_k),\qquad \mathcal{R}_k=L^{3/2}(Q_k)^{3\times3},\] with the remaining factors given by the corresponding finite-window flux, energy, trace, and slack observables. No scale-uniform assertion is attached to these choices.
An intrinsic localized package is \[\mathcal{D} = (u_k,p_k^{\rm act},r_k,\pi_k,\phi_k,\tau_k,\sigma_k)_{k\in\Lambda},\] with components in the seven spaces above. Fix positive intrinsic weights \[a_U,a_P,a_R,a_\Pi,a_\Phi,a_T,a_s>0,\] and define \[\begin{align} \|\mathcal{D}\|_{\mathrm{loc},{\rm int}}^2 := \sum_{k\in\Lambda} \bigl( &a_U\|u_k\|_{\mathcal{U}_k}^2 + a_P\|p_k^{\rm act}\|_{\mathcal{P}_k}^2 + a_R\|r_k\|_{\mathcal{R}_k}^2\\ &+ a_\Pi\|\pi_k\|_{\mathcal{F}_k}^2 + a_\Phi\|\phi_k\|_{\mathcal{E}_k}^2 + a_T\|\tau_k\|_{\mathcal{T}_k}^2 + a_s\|\sigma_k\|_{\mathcal{S}_k}^2 \bigr). \end{align}\] Let \(\Gamma_{\Lambda}^{\rm int}\) be a linear intrinsic gauge subspace and define \[\operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_{\Lambda}^{\rm int}) := \inf_{\zeta\in\Gamma_{\Lambda}^{\rm int}} \|\mathcal{D}-\zeta\|_{\mathrm{loc},{\rm int}}.\]
Assumption 11 (Bounded coordinate extraction from the intrinsic norm). There is a linear coordinate extraction map \[\mathcal{L}_A:\mathfrak I_\Lambda^{\mathrm{loc}}\to\mathcal{K}_{\Lambda,A}^{\mathrm{loc}}\] from intrinsic packages to Model A coordinate packages, written \[\mathcal{L}_A\mathcal{D} = (U_k,P_k^{\rm act},R_k,\Pi_k,\Phi_k,T_k,s_k)_{k\in\Lambda},\] and constants \[\beta_{U,k},\beta_{P,k},\beta_{R,k}, \beta_{\Pi,k},\beta_{\Phi,k},\beta_{T,k},\beta_{s,k}<\infty\] such that, for each \(k\in\Lambda\), \[\begin{gather} \|U_k\|_{U,k}\le \beta_{U,k}\|u_k\|_{\mathcal{U}_k},\qquad \|P_k^{\rm act}\|_{P,k}\le \beta_{P,k}\|p_k^{\rm act}\|_{\mathcal{P}_k},\\ \|R_k\|_{R,k}\le \beta_{R,k}\|r_k\|_{\mathcal{R}_k},\qquad \|\Pi_k\|_{\Pi,k}\le \beta_{\Pi,k}\|\pi_k\|_{\mathcal{F}_k},\\ \|\Phi_k\|_{\Phi,k}\le \beta_{\Phi,k}\|\phi_k\|_{\mathcal{E}_k},\qquad \|T_k\|_{T,k}\le \beta_{T,k}\|\tau_k\|_{\mathcal{T}_k},\\ \|s_k\|_{s,k}\le \beta_{s,k}\|\sigma_k\|_{\mathcal{S}_k}. \end{gather}\] Assume also that intrinsic gauges map into Model A gauges: \[\mathcal{L}_A\Gamma_{\Lambda}^{\rm int} \subset \Gamma_\Lambda^{\mathrm{loc}}.\]
Lemma 11 (Intrinsic norm controls the Model A baseline distance). Under 11, there is a finite constant \[C_{A\leftarrow{\rm int}}<\infty\] depending only on the finite window, the Model A weights, the intrinsic weights, and the coordinate-extraction constants, such that \[\|\mathcal{L}_A\mathcal{D}\|_{\mathrm{loc},0} \le C_{A\leftarrow{\rm int}} \|\mathcal{D}\|_{\mathrm{loc},{\rm int}}\] and \[\operatorname{dist}_{\mathrm{loc},0}(\mathcal{L}_A\mathcal{D},\Gamma_\Lambda^{\mathrm{loc}}) \le C_{A\leftarrow{\rm int}} \operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_{\Lambda}^{\rm int})\] for every intrinsic package \(\mathcal{D}\).
Proof. Set \[C_{A\leftarrow{\rm int}}^2 := \max_{k\in\Lambda} \max\left\{ \frac{w_U\beta_{U,k}^2}{a_U}, \frac{w_P\beta_{P,k}^2}{a_P}, \frac{w_R\beta_{R,k}^2}{a_R}, \frac{w_\Pi\beta_{\Pi,k}^2}{a_\Pi}, \frac{w_\Phi\beta_{\Phi,k}^2}{a_\Phi}, \frac{w_T\beta_{T,k}^2}{a_T}, \frac{w_s\beta_{s,k}^2}{a_s} \right\}.\] This constant is finite because the window is finite and all listed constants are finite. The coordinate bounds in 11 imply, term by term, that \[\|\mathcal{L}_A\mathcal{D}\|_{\mathrm{loc},0}^2 \le C_{A\leftarrow{\rm int}}^2 \|\mathcal{D}\|_{\mathrm{loc},{\rm int}}^2.\] This proves the norm estimate.
For the quotient estimate, fix \(\zeta\in\Gamma_{\Lambda}^{\rm int}\). Since \(\mathcal{L}_A\zeta\in\Gamma_\Lambda^{\mathrm{loc}}\), \[\begin{align} \operatorname{dist}_{\mathrm{loc},0}(\mathcal{L}_A\mathcal{D},\Gamma_\Lambda^{\mathrm{loc}}) &\le \|\mathcal{L}_A\mathcal{D}-\mathcal{L}_A\zeta\|_{\mathrm{loc},0}\\ &= \|\mathcal{L}_A(\mathcal{D}-\zeta)\|_{\mathrm{loc},0}\\ &\le C_{A\leftarrow{\rm int}} \|\mathcal{D}-\zeta\|_{\mathrm{loc},{\rm int}}. \end{align}\] Taking the infimum over \(\zeta\in\Gamma_{\Lambda}^{\rm int}\) gives the claim. ◻
Assumption 12 (Intrinsic pressure-source control). There are constants \[C_{{\rm src},{\rm int}}<\infty, \qquad \Delta_{{\rm src},{\rm int}}\ge0,\] such that every intrinsic localized package \(\mathcal{D}\) satisfies \[\mathsf{Err}_{\rm src,q}^{\rm prs}(\mathcal{L}_A\mathcal{D}) \le C_{{\rm src},{\rm int}} \operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_{\Lambda}^{\rm int}) + \Delta_{{\rm src},{\rm int}}.\]
Proposition 6 (Conditional intrinsic control of the Model A enhanced distance). Assume 11 and 12. Then every intrinsic localized package \(\mathcal{D}\) satisfies \[\begin{align} \operatorname{dist}_{\mathrm{loc},\alpha}^{\sharp} (\mathcal{L}_A\mathcal{D},\Gamma_\Lambda^{\mathrm{loc}}) \le& \bigl( C_{A\leftarrow{\rm int}} + \alpha_{\rm src}C_{{\rm src},{\rm int}} \bigr) \operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_{\Lambda}^{\rm int})\\ &+ \alpha_{\rm src}\Delta_{{\rm src},{\rm int}}. \end{align}\]
Proof. By 25, \[\operatorname{dist}_{\mathrm{loc},\alpha}^{\sharp} (\mathcal{L}_A\mathcal{D},\Gamma_\Lambda^{\mathrm{loc}}) = \operatorname{dist}_{\mathrm{loc},0}(\mathcal{L}_A\mathcal{D},\Gamma_\Lambda^{\mathrm{loc}}) + \alpha_{\rm src} \mathsf{Err}_{\rm src,q}^{\rm prs}(\mathcal{L}_A\mathcal{D}).\] Apply 11 to the first term and 12 to the second. ◻
Remark 20 (What remains PDE-facing). 6 proves a finite-window implication from two explicit inputs: bounded extraction of Model A coordinates from intrinsic localized data, and intrinsic control of the pressure-source residual. It does not derive either input from the Navier–Stokes equations. The PDE-facing work is to construct the intrinsic package from a suitable weak solution, verify the coordinate bounds from the chosen pressure splitting and local energy quantities, and prove or disprove 12.
We now test the first component in the intrinsic pressure-source control assumption. The cutoff–Riesz commutator is the most favorable term because the source is separated from the observation ball. Even here, the natural estimate is quadratic in the intrinsic velocity size, reflecting the nonlinear source \(u_i u_j\).
Definition 27 (Annular intrinsic velocity quotient). In the normalized pressure-source geometry, set \[A_{3/4,1}:=B_1\setminus B_{3/4}.\] For an intrinsic package \(\mathcal{D}\), let \(u\) denote its velocity component in this normalized window. For \(\zeta\in\Gamma_\Lambda^{\rm int}\), let \(\zeta_U\) denote the velocity component of \(\zeta\). Define \[d_{U,{\rm ann}}(\mathcal{D}) := \inf_{\zeta\in\Gamma_\Lambda^{\rm int}} \|u-\zeta_U\|_{L^3(I;L^3(A_{3/4,1}))}.\]
Lemma 12 (Quadratic cutoff–Riesz commutator bound). There is a constant \(C_\eta<\infty\), depending only on the fixed normalized cutoff and balls, such that for every intrinsic package \(\mathcal{D}\), \[\inf_{\zeta\in\Gamma_\Lambda^{\rm int}} \left\| C_\eta\bigl((u-\zeta_U)\otimes(u-\zeta_U)\bigr) \right\|_{Y_{\rm prs}} \le C_\eta d_{U,{\rm ann}}(\mathcal{D})^2.\]
Proof. Fix \(\zeta\in\Gamma_\Lambda^{\rm int}\) and set \[v:=u-\zeta_U, \qquad f_{ij}:=v_i v_j.\] By the fixed separated-support estimate in 6, \[\|C_\eta(f)\|_{Y_{\rm prs}} \le C_\eta \|(1-\eta)f\|_{L^{3/2}(I;L^{3/2}(A_{3/4,1}))^{3\times3}}.\] Since \(\eta\equiv1\) on \(B_{3/4}\), the factor \(1-\eta\) is supported in the annulus. Hölder’s inequality gives \[\|(1-\eta)v_i v_j\|_{L^{3/2}(I;L^{3/2}(A_{3/4,1}))} \le C_\eta \|v\|_{L^3(I;L^3(A_{3/4,1}))}^2.\] Summing over the finitely many tensor components changes only the constant. Thus \[\left\|C_\eta(v\otimes v)\right\|_{Y_{\rm prs}} \le C_\eta \|u-\zeta_U\|_{L^3(I;L^3(A_{3/4,1}))}^2.\] Taking the infimum over \(\zeta\in\Gamma_\Lambda^{\rm int}\) proves the claim. ◻
Corollary 3 (Finite-amplitude linear commutator control). Assume the intrinsic norm dominates the annular velocity quotient in the sense that \[d_{U,{\rm ann}}(\mathcal{D}) \le C_{U,{\rm ann}} \operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_\Lambda^{\rm int})\] for a finite constant \(C_{U,{\rm ann}}\). If, in addition, \[d_{U,{\rm ann}}(\mathcal{D})\le M,\] then \[\inf_{\zeta\in\Gamma_\Lambda^{\rm int}} \left\| C_\eta\bigl((u-\zeta_U)\otimes(u-\zeta_U)\bigr) \right\|_{Y_{\rm prs}} \le C_\eta M C_{U,{\rm ann}} \operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_\Lambda^{\rm int}).\]
Proof. By 12, \[\inf_{\zeta\in\Gamma_\Lambda^{\rm int}} \left\| C_\eta\bigl((u-\zeta_U)\otimes(u-\zeta_U)\bigr) \right\|_{Y_{\rm prs}} \le C_\eta d_{U,{\rm ann}}(\mathcal{D})^2.\] If \(d_{U,{\rm ann}}(\mathcal{D})\le M\), then \[d_{U,{\rm ann}}(\mathcal{D})^2 \le M d_{U,{\rm ann}}(\mathcal{D}) \le M C_{U,{\rm ann}} \operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_\Lambda^{\rm int}).\] This proves the estimate. ◻
Remark 21 (Quadratic obstruction). 12 is a genuine fixed-scale pressure estimate, but it is quadratic in the annular velocity quotient. It does not by itself prove the linear intrinsic pressure-source control assumption in 12. The linear form in 3 requires a finite-amplitude restriction and is not a scale-uniform smallness statement. The active source residual, clean projection residual, and harmonic gauge residual remain separate components.
We next isolate the active source residual. The point is to avoid hiding a nonlinear covariance mismatch inside the pressure operator. Once the cutoff commutator has been separated, the remaining active source term is controlled by a source-level mismatch through the \(L^{3/2}\)-boundedness of the Riesz transforms.
Definition 28 (Active covariance mismatch). In the normalized pressure-source geometry, define the active source mismatch by \[M_{ij}^{\rm act} := \eta f_{ij}-F_{ij}^{\mathrm{cl}} = \eta u_i u_j-(U_iU_j+R_{ij}),\] with all source tensors extended by zero outside \(B_1\). For an intrinsic package \(\mathcal{D}\), define the quotient source mismatch size \[d_{\rm act,src}(\mathcal{D}) := \inf_{\zeta\in\Gamma_\Lambda^{\rm int}} \|M^{\rm act}(\mathcal{D}-\zeta)\|_ {L^{3/2}(I;L^{3/2}(B_1))^{3\times3}}.\]
Lemma 13 (Active residual source decomposition). In the normalized model, \[E_{\rm act}^{\rm src} = R_iR_j(M_{ij}^{\rm act}) - C_\eta(f).\] Consequently, there is a finite constant \(C_{\rm Riesz}\), depending only on the Riesz-transform bound at exponent \(3/2\), such that \[\|E_{\rm act}^{\rm src}\|_{Y_{\rm prs}} \le C_{\rm Riesz} \|M^{\rm act}\|_{L^{3/2}(I;L^{3/2}(B_1))^{3\times3}} + \|C_\eta(f)\|_{Y_{\rm prs}}.\]
Proof. By definition, \[C_\eta(f)=R_iR_j(\eta f_{ij})-\eta R_iR_j(f_{ij}).\] Thus \[\eta R_iR_j(f_{ij}) = R_iR_j(\eta f_{ij})-C_\eta(f).\] Using the definition of \(E_{\rm act}^{\rm src}\), \[\begin{align} E_{\rm act}^{\rm src} &= \eta R_iR_j(f_{ij})-R_iR_j(F_{ij}^{\mathrm{cl}})\\ &= R_iR_j(\eta f_{ij})-R_iR_j(F_{ij}^{\mathrm{cl}})-C_\eta(f)\\ &= R_iR_j(M_{ij}^{\rm act})-C_\eta(f). \end{align}\] Taking the \(Y_{\rm prs}\)-norm, using the triangle inequality, and applying the \(L^{3/2}\)-boundedness of \(R_iR_j\) gives the estimate. ◻
Proposition 7 (Common-gauge active-source residual control). For every intrinsic package \(\mathcal{D}\), \[\begin{align} &\inf_{\zeta\in\Gamma_\Lambda^{\rm int}} \|E_{\rm act}^{\rm src}(\mathcal{D}-\zeta)\|_{Y_{\rm prs}}\\ &\quad\le \inf_{\zeta\in\Gamma_\Lambda^{\rm int}} \bigl( C_{\rm Riesz} \|M^{\rm act}(\mathcal{D}-\zeta)\|_ {L^{3/2}(I;L^{3/2}(B_1))^{3\times3}} + C_\eta \|u_\zeta\|_{L^3(I;L^3(A_{3/4,1}))}^2 \bigr), \end{align}\] where \(u_\zeta\) is the velocity component of \(\mathcal{D}-\zeta\). In particular, if there is a common representative \(\zeta_\diamond\) and constants \(M_U,C_U,\eta_{\rm act},\Delta_{\rm act}\ge0\) such that, with \(\rho_{\rm int}(\mathcal{D})=\operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_\Lambda^{\rm int})\), \[\|u_{\zeta_\diamond}\|_{L^3(I;L^3(A_{3/4,1}))}\le M_U, \qquad \|u_{\zeta_\diamond}\|_{L^3(I;L^3(A_{3/4,1}))}\le C_U\rho_{\rm int}(\mathcal{D}),\] and \[C_{\rm Riesz} \|M^{\rm act}(\mathcal{D}-\zeta_\diamond)\|_ {L^{3/2}(I;L^{3/2}(B_1))^{3\times3}} \le \eta_{\rm act}\rho_{\rm int}(\mathcal{D})+\Delta_{\rm act},\] then \[\begin{align} \inf_{\zeta\in\Gamma_\Lambda^{\rm int}} \|E_{\rm act}^{\rm src}(\mathcal{D}-\zeta)\|_{Y_{\rm prs}} \le (\eta_{\rm act}+C_\eta M_U C_U) \rho_{\rm int}(\mathcal{D})+\Delta_{\rm act}. \end{align}\]
Proof. Apply 13 to the same representative \(\mathcal{D}-\zeta\) and then use the fixed separated-support estimate for the commutator. This gives, for every \(\zeta\in\Gamma_\Lambda^{\rm int}\), \[\begin{align} \|E_{\rm act}^{\rm src}(\mathcal{D}-\zeta)\|_{Y_{\rm prs}} \le& C_{\rm Riesz} \|M^{\rm act}(\mathcal{D}-\zeta)\|_ {L^{3/2}(I;L^{3/2}(B_1))^{3\times3}}\\ &+ C_\eta \|u_\zeta\|_{L^3(I;L^3(A_{3/4,1}))}^2. \end{align}\] Taking the infimum over the same \(\zeta\) proves the first estimate. The second estimate follows by evaluating this common-gauge upper bound at \(\zeta_\diamond\) and using \(\|u_{\zeta_\diamond}\|^2\le M_UC_U\rho_{\rm int}(\mathcal{D})\). ◻
Remark 22 (Active-source status). 7 reduces the active pressure residual to a source-level covariance mismatch and the already identified cutoff commutator only on a common representative. It deliberately avoids taking separate infima for the covariance mismatch and the commutator, because separate optimizing gauges would not control the sum. Control by the intrinsic quotient distance must therefore be supplied by a same-gauge localized covariance coordinate or treated as an obstruction.
The clean projection residual is the part of the clean active pressure source lost by the chosen finite-dimensional pressure projection. This subsection records only boundedness of that tail. No decay in the projection dimension is claimed.
Assumption 13 (Bounded clean pressure projection). For a fixed finite-dimensional pressure space and an index \(N\), let \[P_{{\rm prs},N}^{\mathrm{cl}}:Y_{\rm prs}\to Y_{\rm prs}\] be a bounded linear projection. Its operator norm is denoted by \[\|P_{{\rm prs},N}^{\mathrm{cl}}\|_{Y_{\rm prs}\to Y_{\rm prs}}.\]
Definition 29 (Clean projection residual). Define \[E_{{\rm proj},N}^{\mathrm{cl}} := (I-P_{{\rm prs},N}^{\mathrm{cl}})R_iR_j(F_{ij}^{\mathrm{cl}}).\] For an intrinsic package \(D\), define the quotient clean projection residual by \[d_{{\rm proj},q}(D) := \inf_{\zeta\in\Gamma_\Lambda^{\rm int}} \|E_{{\rm proj},N}^{\mathrm{cl}}(D-\zeta)\|_{Y_{\rm prs}}.\]
Lemma 14 (Bounded projection-tail estimate). Under 13, \[\|E_{{\rm proj},N}^{\mathrm{cl}}\|_{Y_{\rm prs}} \le \bigl( 1+\|P_{{\rm prs},N}^{\mathrm{cl}}\|_{Y_{\rm prs}\to Y_{\rm prs}} \bigr) \|R_iR_j(F_{ij}^{\mathrm{cl}})\|_{Y_{\rm prs}}.\] If, moreover, \[F^{\mathrm{cl}}\in L^{3/2}(I;L^{3/2}(B_1))^{3\times3},\] with zero extension outside \(B_1\), then \[\|E_{{\rm proj},N}^{\mathrm{cl}}\|_{Y_{\rm prs}} \le \bigl( 1+\|P_{{\rm prs},N}^{\mathrm{cl}}\|_{Y_{\rm prs}\to Y_{\rm prs}} \bigr) C_{\rm CZ} \|F^{\mathrm{cl}}\|_{L^{3/2}(I;L^{3/2}(B_1))^{3\times3}},\] where \(C_{\rm CZ}\) is the Calderon–Zygmund constant for \(R_iR_j\) at exponent \(3/2\), followed by restriction to \(B_{1/2}\).
Proof. By definition, \[E_{{\rm proj},N}^{\mathrm{cl}} = R_iR_j(F_{ij}^{\mathrm{cl}}) - P_{{\rm prs},N}^{\mathrm{cl}}R_iR_j(F_{ij}^{\mathrm{cl}}).\] The triangle inequality and boundedness of \(P_{{\rm prs},N}^{\mathrm{cl}}\) give \[\begin{align} \|E_{{\rm proj},N}^{\mathrm{cl}}\|_{Y_{\rm prs}} &\le \|R_iR_j(F_{ij}^{\mathrm{cl}})\|_{Y_{\rm prs}} + \|P_{{\rm prs},N}^{\mathrm{cl}}R_iR_j(F_{ij}^{\mathrm{cl}})\|_{Y_{\rm prs}}\\ &\le \bigl( 1+\|P_{{\rm prs},N}^{\mathrm{cl}}\|_{Y_{\rm prs}\to Y_{\rm prs}} \bigr) \|R_iR_j(F_{ij}^{\mathrm{cl}})\|_{Y_{\rm prs}}. \end{align}\] If \(F^{\mathrm{cl}}\in L^{3/2}(I;L^{3/2}(B_1))^{3\times3}\), then the Calderon–Zygmund estimate for the zero extension of \(F^{\mathrm{cl}}\) gives \[\|R_iR_j(F_{ij}^{\mathrm{cl}})\|_{Y_{\rm prs}} \le C_{\rm CZ} \|F^{\mathrm{cl}}\|_{L^{3/2}(I;L^{3/2}(B_1))^{3\times3}}.\] Substituting this into the first estimate proves the second. ◻
Proposition 8 (Quotient projection-tail bound). Under 13, every intrinsic package \(D\) satisfies \[\begin{align} d_{{\rm proj},q}(D) \le& \bigl( 1+\|P_{{\rm prs},N}^{\mathrm{cl}}\|_{Y_{\rm prs}\to Y_{\rm prs}} \bigr) \inf_{\zeta\in\Gamma_\Lambda^{\rm int}} \|R_iR_j(F_{ij}^{\mathrm{cl}}(D-\zeta))\|_{Y_{\rm prs}}. \end{align}\] If \(F^{\mathrm{cl}}(D-\zeta)\in L^{3/2}(I;L^{3/2}(B_1))^{3\times3}\) for the relevant representatives, then \[\begin{align} d_{{\rm proj},q}(D) \le& \bigl( 1+\|P_{{\rm prs},N}^{\mathrm{cl}}\|_{Y_{\rm prs}\to Y_{\rm prs}} \bigr) C_{\rm CZ}\\ &\times \inf_{\zeta\in\Gamma_\Lambda^{\rm int}} \|F^{\mathrm{cl}}(D-\zeta)\|_{L^{3/2}(I;L^{3/2}(B_1))^{3\times3}}. \end{align}\]
Proof. Apply 14 to \(D-\zeta\) for an arbitrary \(\zeta\in\Gamma_\Lambda^{\rm int}\), and then take the infimum over \(\zeta\). The Calderon–Zygmund form follows in the same way from the second estimate in 14. ◻
Remark 23 (Projection tail). The clean projection residual is a finite-window tail component. Boundedness of the clean projection only proves that the tail is a well-defined controlled residual. Any decay in \(N\), compactness of the clean pressure source, or absorption into the original localized quotient distance is a separate theorem target.
The final term in the pressure-source decomposition is the harmonic gauge tail. We keep the discussion at a fixed finite-window level. The purpose is only to record that the retained harmonic tail is a well-defined pressure source component once a harmonic gauge projection and observation map have been fixed.
Assumption 14 (Fixed harmonic pressure observation datum). Let \(Y_{\rm harm}\) be a fixed finite-window normed space for harmonic pressure components. Let \[\mathcal{H}_M\subset Y_{\rm harm}\] be a chosen finite-dimensional harmonic gauge space, and let \[\Pi_{\rm harm,M}:Y_{\rm harm}\to \mathcal{H}_M\] be a bounded projection. If the harmonic pressure norm is not already realized inside the pressure observation space \(Y_{\rm prs}\), we also fix a bounded observation map \[J_{\rm harm}:Y_{\rm harm}\to Y_{\rm prs}.\] The operator norms are denoted by \[\|\Pi_{\rm harm,M}\|_{Y_{\rm harm}\to Y_{\rm harm}}, \qquad \|J_{\rm harm}\|_{Y_{\rm harm}\to Y_{\rm prs}}.\]
Definition 30 (Harmonic gauge residual). For a harmonic pressure component \(p_{\rm harm}(D)\in Y_{\rm harm}\), define the retained harmonic tail by \[h_{\rm harm}(D) := (I-\Pi_{\rm harm,M})p_{\rm harm}(D).\] In the direct-observation convention, where the tail is already an element of \(Y_{\rm prs}\), define \[E_{\rm harm}(D):=-h_{\rm harm}(D).\] In the observation-map convention of 14, the same notation means \[E_{\rm harm}(D):=-J_{\rm harm}h_{\rm harm}(D).\] This is the harmonic term \(E_{\rm harm}=-h^{\rm harm}\) appearing in 5.
Lemma 15 (Fixed-window harmonic residual bound). In the direct-observation convention, \[\|E_{\rm harm}(D)\|_{Y_{\rm prs}} = \|h_{\rm harm}(D)\|_{Y_{\rm prs}}.\] Under the observation-map convention of 14, \[\|E_{\rm harm}(D)\|_{Y_{\rm prs}} \le \|J_{\rm harm}\|_{Y_{\rm harm}\to Y_{\rm prs}} \|(I-\Pi_{\rm harm,M})p_{\rm harm}(D)\|_{Y_{\rm harm}},\] and hence \[\|E_{\rm harm}(D)\|_{Y_{\rm prs}} \le \|J_{\rm harm}\|_{Y_{\rm harm}\to Y_{\rm prs}} \bigl( 1+\|\Pi_{\rm harm,M}\|_{Y_{\rm harm}\to Y_{\rm harm}} \bigr) \|p_{\rm harm}(D)\|_{Y_{\rm harm}}.\]
Proof. In the direct-observation convention, \[E_{\rm harm}(D)=-h_{\rm harm}(D),\] so the first identity follows from homogeneity of the norm. In the observation-map convention, \[E_{\rm harm}(D) = -J_{\rm harm}(I-\Pi_{\rm harm,M})p_{\rm harm}(D).\] Boundedness of \(J_{\rm harm}\) gives the first inequality. The second follows from \[\|(I-\Pi_{\rm harm,M})p_{\rm harm}(D)\|_{Y_{\rm harm}} \le \bigl( 1+\|\Pi_{\rm harm,M}\|_{Y_{\rm harm}\to Y_{\rm harm}} \bigr) \|p_{\rm harm}(D)\|_{Y_{\rm harm}}.\] ◻
Definition 31 (Quotient harmonic gauge residual). Define \[d_{{\rm harm},q}(D) := \inf_{\zeta\in\Gamma_\Lambda^{\rm int}} \|E_{\rm harm}(D-\zeta)\|_{Y_{\rm prs}}.\]
Proposition 9 (Quotient harmonic-tail bound). Under 14, \[d_{{\rm harm},q}(D) \le \|J_{\rm harm}\|_{Y_{\rm harm}\to Y_{\rm prs}} \inf_{\zeta\in\Gamma_\Lambda^{\rm int}} \|(I-\Pi_{\rm harm,M})p_{\rm harm}(D-\zeta)\|_{Y_{\rm harm}}.\] Consequently, \[d_{{\rm harm},q}(D) \le \|J_{\rm harm}\|_{Y_{\rm harm}\to Y_{\rm prs}} \bigl( 1+\|\Pi_{\rm harm,M}\|_{Y_{\rm harm}\to Y_{\rm harm}} \bigr) \inf_{\zeta\in\Gamma_\Lambda^{\rm int}} \|p_{\rm harm}(D-\zeta)\|_{Y_{\rm harm}}.\]
Proof. Apply 15 to \(D-\zeta\) and then take the infimum over \(\zeta\in\Gamma_\Lambda^{\rm int}\). The second estimate follows from the second bound in 15 before taking the infimum. ◻
Remark 24 (Harmonic tail). The harmonic gauge residual is a fixed finite-window tail component. Boundedness of the harmonic projection and of the harmonic observation map only proves that the residual is a well-defined controlled component. Any decay in the harmonic degree \(M\), compatibility with the physical pressure gauge, or absorption into the original localized quotient distance is a separate theorem target.
We now collect the four pressure-source components in the intrinsic quotient geometry. The important point is that the quotient infimum must be taken over a single common gauge representative after the component norms have been summed. Taking separate infima for the four components would use different gauges and would not, by itself, control the pressure-source residual.
Definition 32 (Intrinsic quotient pressure-source residual). For an intrinsic package \(\mathcal{D}\), define \[\mathsf{Err}_{{\rm src},{\rm int}}^{\rm prs}(\mathcal{D}) := \inf_{\zeta\in\Gamma_\Lambda^{\rm int}} \|\mathfrak P_{\rm src}(\mathcal{D}-\zeta)\|_{Y_{\rm prs}},\] where \(\mathfrak P_{\rm src}\) is the concrete pressure-source mismatch from 20, evaluated through the normalized intrinsic pressure-source coordinates.
Definition 33 (Common-gauge intrinsic component budget). For \(\zeta\in\Gamma_\Lambda^{\rm int}\), let \[C_\eta^\zeta,\qquad E_{{\rm act},\zeta}^{\rm src},\qquad E_{{\rm proj},\zeta}^{\mathrm{cl}},\qquad E_{{\rm harm},\zeta}\] denote the cutoff commutator, active source residual, clean projection residual, and harmonic gauge residual evaluated on \(\mathcal{D}-\zeta\). Define \[\begin{align} \mathfrak B_{{\rm src},{\rm int}}^{\rm prs}(\mathcal{D}) := \inf_{\zeta\in\Gamma_\Lambda^{\rm int}} \bigl( &\|C_\eta^\zeta\|_{Y_{\rm prs}} + \|E_{{\rm act},\zeta}^{\rm src}\|_{Y_{\rm prs}}\\ &+ \|E_{{\rm proj},\zeta}^{\mathrm{cl}}\|_{Y_{\rm prs}} + \|E_{{\rm harm},\zeta}\|_{Y_{\rm prs}} \bigr). \end{align}\]
Definition 34 (Resolved intrinsic source-size budget). For \(\zeta\in\Gamma_\Lambda^{\rm int}\), write \(u_\zeta\) for the velocity component of \(\mathcal{D}-\zeta\), \(M_\zeta^{\rm act}\) for the active covariance mismatch of \(\mathcal{D}-\zeta\), \(F_\zeta^{\mathrm{cl}}\) for the clean source \(F^{\mathrm{cl}}\) associated with \(\mathcal{D}-\zeta\), and \(p_{{\rm harm},\zeta}\) for its harmonic pressure component. Define \[\begin{align} \mathfrak S_{{\rm src},{\rm int}}^{\rm prs}(\mathcal{D}) := \inf_{\zeta\in\Gamma_\Lambda^{\rm int}} \bigl(& 2C_\eta \|u_\zeta\|_{L^3(I;L^3(A_{3/4,1}))}^2\\ &+ C_{\rm Riesz} \|M_\zeta^{\rm act}\| _{L^{3/2}(I;L^{3/2}(B_1))^{3\times3}}\\ &+ \bigl( 1+\|P_{{\rm prs},N}^{\mathrm{cl}}\|_{Y_{\rm prs}\to Y_{\rm prs}} \bigr) \|R_iR_j(F_{\zeta,ij}^{\mathrm{cl}})\|_{Y_{\rm prs}}\\ &+ \|J_{\rm harm}\|_{Y_{\rm harm}\to Y_{\rm prs}} \|(I-\Pi_{\rm harm,M})p_{{\rm harm},\zeta}\|_{Y_{\rm harm}} \bigr). \end{align}\] If the clean source is measured directly in \(L^{3/2}(I;L^{3/2}(B_1))^{3\times3}\), the third line may be replaced by \[\bigl( 1+\|P_{{\rm prs},N}^{\mathrm{cl}}\|_{Y_{\rm prs}\to Y_{\rm prs}} \bigr) C_{\rm CZ} \|F_\zeta^{\mathrm{cl}}\|_{L^{3/2}(I;L^{3/2}(B_1))^{3\times3}}.\]
Proposition 10 (Intrinsic pressure-source component budget). Under the fixed-window pressure-source assumptions above, \[\mathsf{Err}_{{\rm src},{\rm int}}^{\rm prs}(\mathcal{D}) \le \mathfrak B_{{\rm src},{\rm int}}^{\rm prs}(\mathcal{D}) \le \mathfrak S_{{\rm src},{\rm int}}^{\rm prs}(\mathcal{D}).\] The same conclusion holds with the Calderon–Zygmund version of \(\mathfrak S_{{\rm src},{\rm int}}^{\rm prs}\) when the clean sources have the stated \(L^{3/2}\) integrability.
Proof. Fix \(\zeta\in\Gamma_\Lambda^{\rm int}\). Applying 5 to \(\mathcal{D}-\zeta\) and then using the triangle inequality gives \[\begin{align} \|\mathfrak P_{\rm src}(\mathcal{D}-\zeta)\|_{Y_{\rm prs}} \le& \|C_\eta^\zeta\|_{Y_{\rm prs}} + \|E_{{\rm act},\zeta}^{\rm src}\|_{Y_{\rm prs}}\\ &+ \|E_{{\rm proj},\zeta}^{\mathrm{cl}}\|_{Y_{\rm prs}} + \|E_{{\rm harm},\zeta}\|_{Y_{\rm prs}}. \end{align}\] Taking the infimum over the same gauge representative \(\zeta\) proves \[\mathsf{Err}_{{\rm src},{\rm int}}^{\rm prs}(\mathcal{D}) \le \mathfrak B_{{\rm src},{\rm int}}^{\rm prs}(\mathcal{D}).\]
For the second inequality, use the fixed component estimates on the same representative \(\mathcal{D}-\zeta\). The separated commutator estimate gives \[\|C_\eta^\zeta\|_{Y_{\rm prs}} \le C_\eta \|u_\zeta\|_{L^3(I;L^3(A_{3/4,1}))}^2.\] 13 gives \[\|E_{{\rm act},\zeta}^{\rm src}\|_{Y_{\rm prs}} \le C_{\rm Riesz} \|M_\zeta^{\rm act}\| _{L^{3/2}(I;L^{3/2}(B_1))^{3\times3}} + \|C_\eta^\zeta\|_{Y_{\rm prs}}.\] Together these two estimates contribute the first two lines in \(\mathfrak S_{{\rm src},{\rm int}}^{\rm prs}\). The clean projection term is bounded by 14, and the harmonic term is bounded by 15. Summing these four bounds and then taking the infimum over \(\zeta\in\Gamma_\Lambda^{\rm int}\) proves \[\mathfrak B_{{\rm src},{\rm int}}^{\rm prs}(\mathcal{D}) \le \mathfrak S_{{\rm src},{\rm int}}^{\rm prs}(\mathcal{D}).\] The Calderon–Zygmund version follows by using the second estimate in 14 for the clean projection term. ◻
Assumption 15 (Intrinsic source-size budget compatibility). There are constants \[\eta_{{\rm src},{\rm int}}\ge0, \qquad \Delta_{{\rm src},{\rm int}}\ge0,\] such that every intrinsic package \(\mathcal{D}\) satisfies \[\mathfrak S_{{\rm src},{\rm int}}^{\rm prs}(\mathcal{D}) \le \eta_{{\rm src},{\rm int}} \operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_\Lambda^{\rm int}) + \Delta_{{\rm src},{\rm int}}.\]
Corollary 4 (Conditional intrinsic pressure-source control). Assume 15. Then \[\mathsf{Err}_{{\rm src},{\rm int}}^{\rm prs}(\mathcal{D}) \le \eta_{{\rm src},{\rm int}} \operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_\Lambda^{\rm int}) + \Delta_{{\rm src},{\rm int}}.\]
Remark 25 (Status of the assembled budget). 10 is a fixed-window assembly statement. It proves that the pressure-source residual is bounded by the common-gauge sum of the four already isolated components, and then by the resolved source-size budget. It does not prove that this budget is small, scale-uniform, or controlled by the original intrinsic quotient distance. 15 is the exact remaining compatibility input needed for the conditional control in 4.
The previous subsection reduces intrinsic pressure-source control to the resolved source-size budget. We now record a sufficient condition for that budget to satisfy the compatibility assumption. This criterion is deliberately same-gauge: all component estimates are required on one representative \(\mathcal{D}-\zeta_\ast\). The cutoff commutator part follows from the finite-amplitude annular velocity control; the other components remain explicit inputs.
Assumption 16 (Same-gauge component compatibility datum). There are constants \[M_U,C_U<\infty,\qquad \eta_{\rm act},\eta_{\rm proj},\eta_{\rm harm}\ge0, \qquad \Delta_{\rm act},\Delta_{\rm proj},\Delta_{\rm harm}\ge0\] such that for every intrinsic package \(\mathcal{D}\) there exists a gauge representative \(\zeta_\ast=\zeta_\ast(\mathcal{D})\in \Gamma_\Lambda^{\rm int}\) with the following properties. Writing \[\rho_{\rm int}(\mathcal{D}) := \operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_\Lambda^{\rm int}),\] the annular velocity component satisfies \[\|u_{\zeta_\ast}\|_{L^3(I;L^3(A_{3/4,1}))}\le M_U, \qquad \|u_{\zeta_\ast}\|_{L^3(I;L^3(A_{3/4,1}))} \le C_U\rho_{\rm int}(\mathcal{D}).\] The active covariance, clean projection, and harmonic tail entries satisfy \[C_{\rm Riesz} \|M_{\zeta_\ast}^{\rm act}\| _{L^{3/2}(I;L^{3/2}(B_1))^{3\times3}} \le \eta_{\rm act}\rho_{\rm int}(\mathcal{D})+\Delta_{\rm act},\] \[\bigl( 1+\|P_{{\rm prs},N}^{\mathrm{cl}}\|_{Y_{\rm prs}\to Y_{\rm prs}} \bigr) \|R_iR_j(F_{\zeta_\ast,ij}^{\mathrm{cl}})\|_{Y_{\rm prs}} \le \eta_{\rm proj}\rho_{\rm int}(\mathcal{D})+\Delta_{\rm proj},\] and \[\|J_{\rm harm}\|_{Y_{\rm harm}\to Y_{\rm prs}} \|(I-\Pi_{\rm harm,M})p_{{\rm harm},\zeta_\ast}\|_{Y_{\rm harm}} \le \eta_{\rm harm}\rho_{\rm int}(\mathcal{D})+\Delta_{\rm harm}.\]
Lemma 16 (Finite-amplitude same-gauge commutator compatibility). Under the annular velocity part of 16, \[2C_\eta \|u_{\zeta_\ast}\|_{L^3(I;L^3(A_{3/4,1}))}^2 \le 2C_\eta M_U C_U \operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_\Lambda^{\rm int}).\]
Proof. By the two annular velocity bounds in 16, \[\|u_{\zeta_\ast}\|_{L^3(I;L^3(A_{3/4,1}))}^2 \le M_U \|u_{\zeta_\ast}\|_{L^3(I;L^3(A_{3/4,1}))} \le M_U C_U \operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_\Lambda^{\rm int}).\] Multiplying by \(2C_\eta\) proves the claim. ◻
Proposition 11 (Same-gauge sufficient condition for source-size compatibility). Assume 16. Then 15 holds with \[\eta_{{\rm src},{\rm int}} = 2C_\eta M_U C_U + \eta_{\rm act} + \eta_{\rm proj} + \eta_{\rm harm}\] and \[\Delta_{{\rm src},{\rm int}} = \Delta_{\rm act} + \Delta_{\rm proj} + \Delta_{\rm harm}.\]
Proof. By definition of \(\mathfrak S_{{\rm src},{\rm int}}^{\rm prs}\), its infimum is bounded above by evaluating the source-size expression at the single representative \(\zeta_\ast(\mathcal{D})\). The cutoff commutator contribution is bounded by 16. The active covariance, projection, and harmonic contributions are bounded by the three remaining estimates in 16. Summing these four estimates gives \[\mathfrak S_{{\rm src},{\rm int}}^{\rm prs}(\mathcal{D}) \le \eta_{{\rm src},{\rm int}} \operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_\Lambda^{\rm int}) + \Delta_{{\rm src},{\rm int}},\] with the displayed constants. ◻
Corollary 5 (Same-gauge conditional pressure-source control). Under 16, \[\mathsf{Err}_{{\rm src},{\rm int}}^{\rm prs}(\mathcal{D}) \le \bigl( 2C_\eta M_U C_U + \eta_{\rm act} + \eta_{\rm proj} + \eta_{\rm harm} \bigr) \operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_\Lambda^{\rm int}) + \Delta_{\rm act} + \Delta_{\rm proj} + \Delta_{\rm harm}.\]
Remark 26 (What the criterion proves). 11 proves only a finite-window implication. The cutoff commutator contribution is handled by finite-amplitude annular velocity control. The active covariance, clean projection, and harmonic tail entries are not proved from the intrinsic distance here; they are named compatibility inputs. No smallness, scale-uniform control, decay in \(N\) or \(M\), or Navier–Stokes regularity is claimed.
We next isolate the active covariance component. There are two logically different choices. One may treat the mismatch \(\eta u_i u_j-(U_iU_j+R_{ij})\) as a residual to be estimated, or one may choose the covariance coordinate so that it records this mismatch exactly. For the finite-window model we first take the second, covariance-resolved choice. This does not prove that the convention is canonical for localized Navier–Stokes packages; it only removes the active source term inside the chosen model geometry.
Definition 35 (Active covariance reconstruction residual). For a package \(D\), define \[\mathcal{R}_{\rm cov}^{\rm act}(D) := \eta u_i u_j-(U_iU_j+R_{ij}) \in L^{3/2}(I;L^{3/2}(B_1))^{3\times3}.\] For a common gauge representative \(\zeta_\ast\), set \[d_{\rm cov}^{\rm act}(D;\zeta_\ast) := C_{\rm Riesz} \|\mathcal{R}_{\rm cov}^{\rm act}(D-\zeta_\ast)\| _{L^{3/2}(I;L^{3/2}(B_1))^{3\times3}}.\]
Convention 2 (Covariance-resolved finite-window package). In the covariance-resolved finite-window model, the Reynolds/covariance coordinate is chosen on the same-gauge representative so that \[R_{ij} = \eta u_i u_j-U_iU_j\] in \(L^{3/2}(I;L^{3/2}(B_1))^{3\times3}\). Equivalently, \[\mathcal{R}_{\rm cov}^{\rm act}(D-\zeta_\ast)=0\] for the representative used in the same-gauge source-size budget.
Lemma 17 (Active covariance compatibility in the resolved model). Under 2, the active covariance entry in 16 holds with \[\eta_{\rm act}=0, \qquad \Delta_{\rm act}=0.\]
Proof. By 2, \[M_{\zeta_\ast}^{\rm act} = \eta u_i u_j-(U_iU_j+R_{ij}) = 0\] for the common representative used in the source-size budget. Therefore \[C_{\rm Riesz} \|M_{\zeta_\ast}^{\rm act}\| _{L^{3/2}(I;L^{3/2}(B_1))^{3\times3}} = 0,\] which is the active covariance compatibility bound with \(\eta_{\rm act}=\Delta_{\rm act}=0\). ◻
Remark 27 (Cost of resolving covariance). The covariance-resolved convention makes the active source residual vanish in the finite-window coordinate model. The cost is that \(R\) is no longer an arbitrary Reynolds coordinate; it must store the localized covariance defect generated by \(\eta u_i u_j-U_iU_j\). A later PDE theorem must justify that this is the right localized package coordinate, or compare it with a more intrinsic covariance variable. No compactness, smallness, or scale-uniform control is obtained from the convention alone.
The clean projection tail is the next unresolved component in the same-gauge criterion. There are two possible routes. A genuine approximation route would prove decay of the tail as \(N\to\infty\) from compactness or additional regularity of the clean pressure source. We do not assume such compactness here. Instead we record the finite-window enhanced-distance route, in which the clean projection tail is built into the intrinsic quotient geometry.
Definition 36 (Intrinsic clean projection-tail size). For an intrinsic package \(\mathcal{D}\) and a representative \(\zeta\in\Gamma_\Lambda^{\rm int}\), define \[\mathcal{T}_{\rm proj}(\mathcal{D};\zeta) := \bigl( 1+\|P_{{\rm prs},N}^{\mathrm{cl}}\|_{Y_{\rm prs}\to Y_{\rm prs}} \bigr) \|R_iR_j(F_{\zeta,ij}^{\mathrm{cl}})\|_{Y_{\rm prs}}.\] The quotient projection-tail size is \[\mathcal{T}_{{\rm proj},q}(\mathcal{D}) := \inf_{\zeta\in\Gamma_\Lambda^{\rm int}} \mathcal{T}_{\rm proj}(\mathcal{D};\zeta).\]
Definition 37 (Projection-tail enhanced intrinsic distance). Fix \(\alpha_{\rm proj}>0\). Define \[\begin{align} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm proj}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) := \inf_{\zeta\in\Gamma_\Lambda^{\rm int}} \bigl( &\|\mathcal{D}-\zeta\|_{\mathrm{loc},{\rm int}}\\ &+ \alpha_{\rm proj} \mathcal{T}_{\rm proj}(\mathcal{D};\zeta) \bigr). \end{align}\]
Lemma 18 (Projection-tail observability in the enhanced geometry). For every intrinsic package \(\mathcal{D}\), \[\mathcal{T}_{{\rm proj},q}(\mathcal{D}) \le \alpha_{\rm proj}^{-1} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm proj}} (\mathcal{D},\Gamma_\Lambda^{\rm int}).\]
Proof. For any \(\zeta\in\Gamma_\Lambda^{\rm int}\), \[\mathcal{T}_{{\rm proj},q}(\mathcal{D}) \le \mathcal{T}_{\rm proj}(\mathcal{D};\zeta) \le \alpha_{\rm proj}^{-1} \bigl( \|\mathcal{D}-\zeta\|_{\mathrm{loc},{\rm int}} + \alpha_{\rm proj} \mathcal{T}_{\rm proj}(\mathcal{D};\zeta) \bigr).\] Taking the infimum over \(\zeta\in\Gamma_\Lambda^{\rm int}\) proves the claim. ◻
Remark 28 (Projection-tail fork). 18 proves clean projection-tail observability only after changing the quotient geometry. It does not prove that the original intrinsic distance \(\operatorname{dist}_{\mathrm{loc},{\rm int}}\) controls the projection tail, and it does not prove decay as \(N\to\infty\). Such decay would require a separate compactness or approximation theorem for the clean pressure source.
The harmonic tail has the same structural status as the clean projection tail. Boundedness of the harmonic projection and observation map makes the tail a well-defined finite-window residual, but does not prove that it is controlled by the original intrinsic quotient distance. We therefore record the enhanced-distance route and leave decay in the harmonic degree \(M\) to a separate harmonic approximation theorem.
Definition 38 (Intrinsic harmonic-tail size). For an intrinsic package \(\mathcal{D}\) and a representative \(\zeta\in\Gamma_\Lambda^{\rm int}\), define \[\mathcal{T}_{\rm harm}(\mathcal{D};\zeta) := \|J_{\rm harm}\|_{Y_{\rm harm}\to Y_{\rm prs}} \|(I-\Pi_{\rm harm,M})p_{{\rm harm},\zeta}\|_{Y_{\rm harm}}.\] The quotient harmonic-tail size is \[\mathcal{T}_{{\rm harm},q}(\mathcal{D}) := \inf_{\zeta\in\Gamma_\Lambda^{\rm int}} \mathcal{T}_{\rm harm}(\mathcal{D};\zeta).\]
Definition 39 (Harmonic-tail enhanced intrinsic distance). Fix \(\alpha_{\rm harm}>0\). Define \[\begin{align} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm harm}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) := \inf_{\zeta\in\Gamma_\Lambda^{\rm int}} \bigl( &\|\mathcal{D}-\zeta\|_{\mathrm{loc},{\rm int}}\\ &+ \alpha_{\rm harm} \mathcal{T}_{\rm harm}(\mathcal{D};\zeta) \bigr). \end{align}\]
Lemma 19 (Harmonic-tail observability in the enhanced geometry). For every intrinsic package \(\mathcal{D}\), \[\mathcal{T}_{{\rm harm},q}(\mathcal{D}) \le \alpha_{\rm harm}^{-1} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm harm}} (\mathcal{D},\Gamma_\Lambda^{\rm int}).\]
Proof. For any \(\zeta\in\Gamma_\Lambda^{\rm int}\), \[\mathcal{T}_{{\rm harm},q}(\mathcal{D}) \le \mathcal{T}_{\rm harm}(\mathcal{D};\zeta) \le \alpha_{\rm harm}^{-1} \bigl( \|\mathcal{D}-\zeta\|_{\mathrm{loc},{\rm int}} + \alpha_{\rm harm} \mathcal{T}_{\rm harm}(\mathcal{D};\zeta) \bigr).\] Taking the infimum over \(\zeta\in\Gamma_\Lambda^{\rm int}\) proves the claim. ◻
Remark 29 (Harmonic-tail fork). 19 proves harmonic-tail observability only after changing the quotient geometry. It does not prove that the original intrinsic distance controls the harmonic tail, does not prove decay as \(M\to\infty\), and does not justify compatibility with the physical pressure gauge. Those are separate theorem targets.
The projection and harmonic tails should enter later transfer statements through a single enhanced quotient geometry. The following definition keeps the two weights separate but takes one common quotient infimum.
Definition 40 (Combined pressure-tail enhanced intrinsic distance). Fix \(\alpha_{\rm proj},\alpha_{\rm harm}>0\). Define \[\begin{align} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) := \inf_{\zeta\in\Gamma_\Lambda^{\rm int}} \bigl( &\|\mathcal{D}-\zeta\|_{\mathrm{loc},{\rm int}}\\ &+ \alpha_{\rm proj}\mathcal{T}_{\rm proj}(\mathcal{D};\zeta) + \alpha_{\rm harm}\mathcal{T}_{\rm harm}(\mathcal{D};\zeta) \bigr). \end{align}\]
Lemma 20 (Simultaneous pressure-tail observability). For every intrinsic package \(\mathcal{D}\), \[\mathcal{T}_{{\rm proj},q}(\mathcal{D}) + \mathcal{T}_{{\rm harm},q}(\mathcal{D}) \le \max\{\alpha_{\rm proj}^{-1},\alpha_{\rm harm}^{-1}\} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}).\] In particular, each tail is separately controlled by the same right-hand side.
Proof. Fix \(\zeta\in\Gamma_\Lambda^{\rm int}\). By definition of the two quotient tail sizes, \[\mathcal{T}_{{\rm proj},q}(\mathcal{D}) + \mathcal{T}_{{\rm harm},q}(\mathcal{D}) \le \mathcal{T}_{\rm proj}(\mathcal{D};\zeta) + \mathcal{T}_{\rm harm}(\mathcal{D};\zeta).\] Since \[\mathcal{T}_{\rm proj}+\mathcal{T}_{\rm harm} \le \max\{\alpha_{\rm proj}^{-1},\alpha_{\rm harm}^{-1}\} \bigl( \alpha_{\rm proj}\mathcal{T}_{\rm proj} + \alpha_{\rm harm}\mathcal{T}_{\rm harm} \bigr),\] we obtain \[\begin{align} \mathcal{T}_{{\rm proj},q}(\mathcal{D}) + \mathcal{T}_{{\rm harm},q}(\mathcal{D}) \le& \max\{\alpha_{\rm proj}^{-1},\alpha_{\rm harm}^{-1}\}\\ &\times \bigl( \|\mathcal{D}-\zeta\|_{\mathrm{loc},{\rm int}} + \alpha_{\rm proj}\mathcal{T}_{\rm proj}(\mathcal{D};\zeta) + \alpha_{\rm harm}\mathcal{T}_{\rm harm}(\mathcal{D};\zeta) \bigr). \end{align}\] Taking the infimum over \(\zeta\) proves the displayed estimate. The individual controls follow because both tail sizes are nonnegative. ◻
Remark 30 (Status of the combined tail geometry). 20 combines the two enhanced-distance bookkeeping steps into one finite-window quotient geometry. It does not compare \(\operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}}\) with the original intrinsic distance, and it does not prove decay in \(N\) or \(M\). A later theorem must justify this enhanced pressure-tail geometry or prove the corresponding projection and harmonic approximation estimates.
We now state pressure-source control in the combined enhanced-tail geometry. The localized defect distance is \(\operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}}\). We do not compare it with the original intrinsic distance, and we do not use projection-tail decay in \(N\) or harmonic-tail decay in \(M\).
Assumption 17 (Same-gauge enhanced-tail compatibility). For every intrinsic package \(\mathcal{D}\), there exists a representative \[\zeta_\ast=\zeta_\ast(\mathcal{D})\in\Gamma_\Lambda^{\rm int}\] chosen in the combined enhanced-tail quotient, so that \[\begin{align} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) = &\|\mathcal{D}-\zeta_\ast\|_{\mathrm{loc},{\rm int}}\\ &+ \alpha_{\rm proj}\mathcal{T}_{\rm proj}(\mathcal{D};\zeta_\ast) + \alpha_{\rm harm}\mathcal{T}_{\rm harm}(\mathcal{D};\zeta_\ast). \end{align}\] Assume also that the cutoff commutator and active covariance core satisfy \[\begin{align} &2C_\eta \|u_{\zeta_\ast}\|_{L^3(I;L^3(A_{3/4,1}))}^2\\ &\qquad+ C_{\rm Riesz} \|M_{\zeta_\ast}^{\rm act}\| _{L^{3/2}(I;L^{3/2}(B_1))^{3\times3}}\\ &\le \eta_{\rm core} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) + \Delta_{\rm core}. \end{align}\]
Theorem 5 (Conditional pressure-source control in the enhanced-tail geometry). Assume 17. Then every intrinsic package \(\mathcal{D}\) satisfies \[\mathsf{Err}_{{\rm src},{\rm int}}^{\rm prs}(\mathcal{D}) \le \eta_{\rm src}^{\sharp,{\rm tail}} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) + \Delta_{\rm src}^{\sharp,{\rm tail}},\] where \[\eta_{\rm src}^{\sharp,{\rm tail}} := \eta_{\rm core} + \max\{\alpha_{\rm proj}^{-1},\alpha_{\rm harm}^{-1}\}, \qquad \Delta_{\rm src}^{\sharp,{\rm tail}} := \Delta_{\rm core}.\]
Proof. By the component-budget estimate in 10, evaluated at the common representative \(\zeta_\ast\), \[\begin{align} \mathsf{Err}_{{\rm src},{\rm int}}^{\rm prs}(\mathcal{D}) \le& 2C_\eta \|u_{\zeta_\ast}\|_{L^3(I;L^3(A_{3/4,1}))}^2\\ &+ C_{\rm Riesz} \|M_{\zeta_\ast}^{\rm act}\| _{L^{3/2}(I;L^{3/2}(B_1))^{3\times3}}\\ &+ \mathcal{T}_{\rm proj}(\mathcal{D};\zeta_\ast) + \mathcal{T}_{\rm harm}(\mathcal{D};\zeta_\ast). \end{align}\] The first two terms are bounded by 17. For the two tail terms, the defining property of \(\zeta_\ast\) gives \[\begin{align} &\mathcal{T}_{\rm proj}(\mathcal{D};\zeta_\ast) + \mathcal{T}_{\rm harm}(\mathcal{D};\zeta_\ast)\\ &\le \max\{\alpha_{\rm proj}^{-1},\alpha_{\rm harm}^{-1}\} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}). \end{align}\] This is the same enhanced-tail observability mechanism as 20, now applied on the common representative. Combining the two estimates proves the theorem with the displayed constants. ◻
Assumption 18 (Enhanced-tail local-to-clean transfer comparison). There are constants \(0\le\varepsilon_G<1\) and \(\delta_G\ge0\) such that every intrinsic package \(\mathcal{D}\) satisfies \[\operatorname{dist}_{\mathrm{cl}}(\Theta_\Lambda\mathcal{D},\Gamma_\Lambda^{\mathrm{cl}}) \ge (1-\varepsilon_G) \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) - \delta_G\] and \[\mathcal{M}_\Lambda^{\mathrm{loc}}(\mathcal{D}) + \mathsf{Err}_\Lambda^{\mathrm{loc}}(\mathcal{D}) \ge \mathcal{M}_\Lambda^{\rm comp}(\Theta_\Lambda\mathcal{D}).\]
Assumption 19 (Enhanced-tail localized residual budget). There are constants \[\eta_\Lambda^{\sharp,{\rm tail}}\ge0, \qquad \Delta_\Lambda^{\sharp,{\rm tail}}\ge0\] such that every intrinsic package \(\mathcal{D}\) satisfies \[\mathsf{Err}_\Lambda^{\mathrm{loc}}(\mathcal{D}) \le \eta_\Lambda^{\sharp,{\rm tail}} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) + \Delta_\Lambda^{\sharp,{\rm tail}}.\]
Theorem 6 (Conditional enhanced-tail localized transfer). Assume \(\mathcal{Q}_\Lambda^{\mathrm{cl}}\ne\{0\}\), 1, 2, 18, and 19. Then every intrinsic package \(\mathcal{D}\) satisfies \[\begin{align} \mathcal{M}_\Lambda^{\mathrm{loc}}(\mathcal{D}) \ge& \bigl( \mu_\Lambda^{\rm comp}(1-\varepsilon_G) - \eta_\Lambda^{\sharp,{\rm tail}} \bigr) \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int})\\ &- \mu_\Lambda^{\rm comp}\delta_G - \Delta_\Lambda^{\sharp,{\rm tail}}. \end{align}\]
Proof. This is the same algebraic transfer argument as 2, with the localized distance replaced by \(\operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}}\). By 2, \[\mathcal{M}_\Lambda^{\rm comp}(\Theta_\Lambda\mathcal{D}) \ge \mu_\Lambda^{\rm comp} \operatorname{dist}_{\mathrm{cl}}(\Theta_\Lambda\mathcal{D},\Gamma_\Lambda^{\mathrm{cl}}).\] Using 18, we get \[\mathcal{M}_\Lambda^{\rm comp}(\Theta_\Lambda\mathcal{D}) \ge \mu_\Lambda^{\rm comp} \bigl( (1-\varepsilon_G) \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) - \delta_G \bigr).\] The detector comparison in the same assumption gives \[\mathcal{M}_\Lambda^{\mathrm{loc}}(\mathcal{D}) \ge \mu_\Lambda^{\rm comp}(1-\varepsilon_G) \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) - \mu_\Lambda^{\rm comp}\delta_G - \mathsf{Err}_\Lambda^{\mathrm{loc}}(\mathcal{D}).\] Applying 19 proves the displayed estimate. ◻
Remark 31 (Status of enhanced-tail transfer). 5 and 6 prove pressure-source control and localized transfer only in the combined enhanced-tail geometry. They do not claim \[\operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} \lesssim \operatorname{dist}_{\mathrm{loc},{\rm int}},\] do not prove projection-tail decay as \(N\to\infty\), do not prove harmonic-tail decay as \(M\to\infty\), and do not give scale-uniform control or Navier–Stokes regularity. Comparing this enhanced geometry back to the original intrinsic geometry is a separate PDE-facing approximation theorem.
We now split the two enhanced-tail transfer assumptions into smaller finite-window subclaims. This does not prove the PDE estimates behind those subclaims. Its purpose is to identify the exact estimates that would replace the single opaque assumptions 18 and 19.
For an intrinsic package \(\mathcal{D}\) and a common representative \(\zeta\in\Gamma_\Lambda^{\rm int}\), define the three enhanced-tail visibility channels \[X_0(\mathcal{D};\zeta) := \|\mathcal{D}-\zeta\|_{\mathrm{loc},{\rm int}},\] \[X_{\rm proj}(\mathcal{D};\zeta) := \alpha_{\rm proj}\mathcal{T}_{\rm proj}(\mathcal{D};\zeta), \qquad X_{\rm harm}(\mathcal{D};\zeta) := \alpha_{\rm harm}\mathcal{T}_{\rm harm}(\mathcal{D};\zeta).\] Thus \[\operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) \le X_0(\mathcal{D};\zeta) + X_{\rm proj}(\mathcal{D};\zeta) + X_{\rm harm}(\mathcal{D};\zeta)\] for every common representative \(\zeta\).
Assumption 20 (Three-channel enhanced-tail chart visibility). There is a same-gauge representative selection \[\zeta_\ast(\mathcal{D})\in\Gamma_\Lambda^{\rm int}\] and constants \[\kappa_0,\kappa_{\rm proj},\kappa_{\rm harm}>0, \qquad \delta_0,\delta_{\rm proj},\delta_{\rm harm}\ge0,\] such that every intrinsic package \(\mathcal{D}\) satisfies \[\operatorname{dist}_{\mathrm{cl}}(\Theta_\Lambda\mathcal{D},\Gamma_\Lambda^{\mathrm{cl}}) \ge \kappa_0 X_0(\mathcal{D};\zeta_\ast)-\delta_0,\] \[\operatorname{dist}_{\mathrm{cl}}(\Theta_\Lambda\mathcal{D},\Gamma_\Lambda^{\mathrm{cl}}) \ge \kappa_{\rm proj}X_{\rm proj}(\mathcal{D};\zeta_\ast) -\delta_{\rm proj},\] and \[\operatorname{dist}_{\mathrm{cl}}(\Theta_\Lambda\mathcal{D},\Gamma_\Lambda^{\mathrm{cl}}) \ge \kappa_{\rm harm}X_{\rm harm}(\mathcal{D};\zeta_\ast) -\delta_{\rm harm}.\]
Proposition 12 (Enhanced-tail quotient comparison from channel visibility). Assume 20. Set \[\kappa_\ast := \min\{\kappa_0,\kappa_{\rm proj},\kappa_{\rm harm}\}, \qquad \lambda_G := \min\left\{1,\frac{\kappa_\ast}{3}\right\},\] and \[\delta_G^{\rm vis} := \frac{\delta_0+\delta_{\rm proj}+\delta_{\rm harm}}{3}.\] Then \[\operatorname{dist}_{\mathrm{cl}}(\Theta_\Lambda\mathcal{D},\Gamma_\Lambda^{\mathrm{cl}}) \ge \lambda_G \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) - \delta_G^{\rm vis}\] for every intrinsic package \(\mathcal{D}\). Equivalently, the quotient comparison part of 18 holds with \[1-\varepsilon_G=\lambda_G, \qquad \delta_G=\delta_G^{\rm vis}.\]
Proof. Write \[A(\mathcal{D}) := \operatorname{dist}_{\mathrm{cl}}(\Theta_\Lambda\mathcal{D},\Gamma_\Lambda^{\mathrm{cl}}).\] Adding the three visibility inequalities in 20 gives \[\begin{align} 3A(\mathcal{D}) \ge& \kappa_0X_0(\mathcal{D};\zeta_\ast) + \kappa_{\rm proj}X_{\rm proj}(\mathcal{D};\zeta_\ast) + \kappa_{\rm harm}X_{\rm harm}(\mathcal{D};\zeta_\ast)\\ &- \delta_0-\delta_{\rm proj}-\delta_{\rm harm}. \end{align}\] Since each channel is nonnegative, \[\begin{align} A(\mathcal{D}) \ge& \frac{\kappa_\ast}{3} \bigl( X_0(\mathcal{D};\zeta_\ast) + X_{\rm proj}(\mathcal{D};\zeta_\ast) + X_{\rm harm}(\mathcal{D};\zeta_\ast) \bigr)\\ &- \delta_G^{\rm vis}. \end{align}\] The sum inside the parentheses is bounded below by \(\operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int})\). Replacing \(\kappa_\ast/3\) by the smaller coefficient \(\lambda_G\le\kappa_\ast/3\) if necessary gives the displayed comparison. ◻
Definition 41 (Decomposed enhanced-tail residual ledger). For the enhanced-tail transfer branch, introduce nonnegative residual functionals \[\mathcal{R}_{\rm core},\quad \mathcal{R}_{\rm tail},\quad \mathcal{R}_{\rm chart},\quad \mathcal{R}_{\rm loc},\quad \mathcal{R}_{\rm rep},\quad \mathcal{R}_{\rm gate}.\] The first two are the pressure-source pieces already isolated: \[\mathcal{R}_{\rm core}(\mathcal{D}) := 2C_\eta \|u_{\zeta_\ast}\|_{L^3(I;L^3(A_{3/4,1}))}^2 + C_{\rm Riesz} \|M_{\zeta_\ast}^{\rm act}\| _{L^{3/2}(I;L^{3/2}(B_1))^{3\times3}},\] where \(\zeta_\ast\) is the selected same-gauge representative, and \[\mathcal{R}_{\rm tail}(\mathcal{D}) := \mathcal{T}_{{\rm proj},q}(\mathcal{D}) + \mathcal{T}_{{\rm harm},q}(\mathcal{D}).\] The remaining residuals denote, respectively, chart-commutation loss, localization leakage, reproduction drift, and detector or gate-tax mismatch. They are finite-window residual functionals; no smallness or scale-uniformity is attached to their definition.
Assumption 21 (Enhanced-tail detector and residual sub-budgets). The localized detector comparison is mediated by the residual ledger in 41: \[\begin{align} \mathcal{M}_\Lambda^{\mathrm{loc}}(\mathcal{D}) &+ \mathcal{R}_{\rm core}(\mathcal{D}) + \mathcal{R}_{\rm tail}(\mathcal{D}) + \mathcal{R}_{\rm chart}(\mathcal{D})\\ &+ \mathcal{R}_{\rm loc}(\mathcal{D}) + \mathcal{R}_{\rm rep}(\mathcal{D}) + \mathcal{R}_{\rm gate}(\mathcal{D}) \ge \mathcal{M}_\Lambda^{\rm comp}(\Theta_\Lambda\mathcal{D}). \end{align}\] In addition, for \[a\in\{{\rm chart},{\rm loc},{\rm rep},{\rm gate}\},\] there are constants \(\eta_a,\Delta_a\ge0\) such that \[\mathcal{R}_a(\mathcal{D}) \le \eta_a \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) + \Delta_a.\]
Proposition 13 (Enhanced-tail residual budget from sub-budgets). Assume 17 and 21. Define \[\begin{align} \mathsf{Err}_{\Lambda,{\rm dec}}^{\mathrm{loc}}(\mathcal{D}) := &\mathcal{R}_{\rm core}(\mathcal{D}) + \mathcal{R}_{\rm tail}(\mathcal{D}) + \mathcal{R}_{\rm chart}(\mathcal{D})\\ &+ \mathcal{R}_{\rm loc}(\mathcal{D}) + \mathcal{R}_{\rm rep}(\mathcal{D}) + \mathcal{R}_{\rm gate}(\mathcal{D}). \end{align}\] Then \[\mathcal{M}_\Lambda^{\mathrm{loc}}(\mathcal{D}) + \mathsf{Err}_{\Lambda,{\rm dec}}^{\mathrm{loc}}(\mathcal{D}) \ge \mathcal{M}_\Lambda^{\rm comp}(\Theta_\Lambda\mathcal{D})\] and \[\mathsf{Err}_{\Lambda,{\rm dec}}^{\mathrm{loc}}(\mathcal{D}) \le \eta_{\Lambda,{\rm dec}}^{\sharp,{\rm tail}} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) + \Delta_{\Lambda,{\rm dec}}^{\sharp,{\rm tail}},\] where \[\eta_{\Lambda,{\rm dec}}^{\sharp,{\rm tail}} = \eta_{\rm core} + \max\{\alpha_{\rm proj}^{-1},\alpha_{\rm harm}^{-1}\} + \eta_{\rm chart} + \eta_{\rm loc} + \eta_{\rm rep} + \eta_{\rm gate}\] and \[\Delta_{\Lambda,{\rm dec}}^{\sharp,{\rm tail}} = \Delta_{\rm core} + \Delta_{\rm chart} + \Delta_{\rm loc} + \Delta_{\rm rep} + \Delta_{\rm gate}.\]
Proof. The detector comparison is exactly the first assertion in 21 after substituting the definition of \(\mathsf{Err}_{\Lambda,{\rm dec}}^{\mathrm{loc}}\).
The core residual is bounded by 17. The tail residual is bounded by 20: \[\mathcal{R}_{\rm tail}(\mathcal{D}) \le \max\{\alpha_{\rm proj}^{-1},\alpha_{\rm harm}^{-1}\} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}).\] The chart, localization, reproduction, and gate residuals are bounded by the sub-budget estimates in 21. Summing these six estimates gives the displayed constants. ◻
Corollary 6 (Subclaim criterion for enhanced-tail localized transfer). Assume 20, 17, 21, \(\mathcal{Q}_\Lambda^{\mathrm{cl}}\ne\{0\}\), 1, and 2. Then the enhanced-tail localized transfer estimate holds with \[1-\varepsilon_G = \lambda_G, \qquad \delta_G=\delta_G^{\rm vis},\] and with \[\eta_\Lambda^{\sharp,{\rm tail}} = \eta_{\Lambda,{\rm dec}}^{\sharp,{\rm tail}}, \qquad \Delta_\Lambda^{\sharp,{\rm tail}} = \Delta_{\Lambda,{\rm dec}}^{\sharp,{\rm tail}}.\]
Proof. 12 gives the quotient-distance part of the enhanced-tail local-to-clean comparison. 13 supplies both the detector comparison and the residual budget, with the decomposed residual \(\mathsf{Err}_{\Lambda,{\rm dec}}^{\mathrm{loc}}\). Applying 6 with these constants proves the claim. ◻
Remark 32 (Status of the subclaim decomposition). 6 is a finite-window reduction of the enhanced-tail transfer assumptions to smaller subclaims. It does not prove the three chart-visibility estimates, the chart-commutation residual bound, the localization leakage bound, the reproduction drift bound, or the gate-tax mismatch bound. It also does not compare \(\operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}}\) with \(\operatorname{dist}_{\mathrm{loc},{\rm int}}\), prove decay in \(N\) or \(M\), or claim scale-uniformity or Navier–Stokes regularity.
We next isolate the first chart-visibility channel in 20. The result below is purely finite-dimensional. It says that intrinsic-core visibility follows if the clean quotient chart admits a bounded left inverse up to an explicit additive reconstruction defect and if the chosen same-gauge representative is not much larger than the intrinsic quotient distance.
Definition 42 (Intrinsic and clean quotient classes). Write \[[\mathcal{D}]_{\rm int} \in \mathfrak I_\Lambda^{\mathrm{loc}}/\Gamma_\Lambda^{\rm int}, \qquad [\Theta_\Lambda\mathcal{D}]_{\mathrm{cl}} \in \mathcal{K}_\Lambda^{\mathrm{cl}}/\Gamma_\Lambda^{\mathrm{cl}}\] for the intrinsic and clean quotient classes. Their quotient norms are \[\|[\mathcal{D}]_{\rm int}\|_{\mathrm{loc},{\rm int}/\Gamma} = \operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_\Lambda^{\rm int})\] and \[\|[\Theta_\Lambda\mathcal{D}]_{\mathrm{cl}}\|_{\mathrm{cl}/\Gamma} = \operatorname{dist}_{\mathrm{cl}}(\Theta_\Lambda\mathcal{D},\Gamma_\Lambda^{\mathrm{cl}}).\]
Assumption 22 (Bounded intrinsic-core quotient left inverse). There is a linear map \[L_{\rm core}: \mathcal{K}_\Lambda^{\mathrm{cl}}/\Gamma_\Lambda^{\mathrm{cl}} \to \mathfrak I_\Lambda^{\mathrm{loc}}/\Gamma_\Lambda^{\rm int}\] and constants \(C_{\rm core}<\infty\) and \(\delta_{\rm rec}\ge0\) such that \[\|L_{\rm core}q\|_{\mathrm{loc},{\rm int}/\Gamma} \le C_{\rm core}\|q\|_{\mathrm{cl}/\Gamma}\] for every clean quotient class \(q\), and \[\|[\mathcal{D}]_{\rm int} - L_{\rm core}[\Theta_\Lambda\mathcal{D}]_{\mathrm{cl}}\| _{\mathrm{loc},{\rm int}/\Gamma} \le \delta_{\rm rec}\] for every intrinsic package \(\mathcal{D}\).
Assumption 23 (Controlled same-gauge core representative). The same-gauge representative selection \(\zeta_\ast(\mathcal{D})\) used in 20 satisfies \[X_0(\mathcal{D};\zeta_\ast) \le C_{\rm sel} \operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_\Lambda^{\rm int}) + \delta_{\rm sel}\] for constants \(C_{\rm sel}<\infty\) and \(\delta_{\rm sel}\ge0\).
Lemma 21 (Intrinsic-core chart visibility from a quotient left inverse). Assume 22 and 23. Then the intrinsic-core visibility channel \[\operatorname{dist}_{\mathrm{cl}}(\Theta_\Lambda\mathcal{D},\Gamma_\Lambda^{\mathrm{cl}}) \ge \kappa_0X_0(\mathcal{D};\zeta_\ast)-\delta_0\] holds with \[\kappa_0 = \frac{1}{C_{\rm core}C_{\rm sel}}, \qquad \delta_0 = \frac{\delta_{\rm sel}}{C_{\rm core}C_{\rm sel}} + \frac{\delta_{\rm rec}}{C_{\rm core}}.\]
Proof. By the reconstruction property and the boundedness of \(L_{\rm core}\), \[\begin{align} \operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_\Lambda^{\rm int}) &= \|[\mathcal{D}]_{\rm int}\|_{\mathrm{loc},{\rm int}/\Gamma}\\ &\le \|L_{\rm core}[\Theta_\Lambda\mathcal{D}]_{\mathrm{cl}}\| _{\mathrm{loc},{\rm int}/\Gamma} + \delta_{\rm rec}\\ &\le C_{\rm core} \operatorname{dist}_{\mathrm{cl}}(\Theta_\Lambda\mathcal{D},\Gamma_\Lambda^{\mathrm{cl}}) + \delta_{\rm rec}. \end{align}\] Therefore \[\operatorname{dist}_{\mathrm{cl}}(\Theta_\Lambda\mathcal{D},\Gamma_\Lambda^{\mathrm{cl}}) \ge \frac{1}{C_{\rm core}} \operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_\Lambda^{\rm int}) - \frac{\delta_{\rm rec}}{C_{\rm core}}.\] The controlled-representative assumption gives \[\operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_\Lambda^{\rm int}) \ge \frac{1}{C_{\rm sel}}X_0(\mathcal{D};\zeta_\ast) - \frac{\delta_{\rm sel}}{C_{\rm sel}}.\] Substituting this lower bound into the previous inequality yields the stated constants. ◻
Remark 33 (Status of intrinsic-core visibility). The preceding lemma proves only a finite-dimensional sufficient condition for the intrinsic-core visibility channel. The existence of \(L_{\rm core}\), the size of the reconstruction defect \(\delta_{\rm rec}\), and the controlled same-gauge representative property are not derived from Navier–Stokes data here. The lemma does not address projection-tail visibility or harmonic-tail visibility.
We now record the analogous finite-window mechanism for the two pressure-tail visibility channels. The point is not that clean quotient distance must see these tails automatically. Rather, if the clean quotient carries bounded recovery maps for the selected projection and harmonic tail data, then the remaining two visibility inequalities in 20 follow.
Definition 43 (Projection and harmonic tail data). Let \(\mathcal{Y}_{\rm proj}^{\rm tail}\) and \(\mathcal{Y}_{\rm harm}^{\rm tail}\) be finite-window normed spaces. For an intrinsic package \(\mathcal{D}\) and a common representative \(\zeta\in\Gamma_\Lambda^{\rm int}\), let \[\tau_{\rm proj}(\mathcal{D};\zeta) \in \mathcal{Y}_{\rm proj}^{\rm tail}, \qquad \tau_{\rm harm}(\mathcal{D};\zeta) \in \mathcal{Y}_{\rm harm}^{\rm tail}\] be tail data satisfying \[\|\tau_{\rm proj}(\mathcal{D};\zeta)\| _{\mathcal{Y}_{\rm proj}^{\rm tail}} = \mathcal{T}_{\rm proj}(\mathcal{D};\zeta), \qquad \|\tau_{\rm harm}(\mathcal{D};\zeta)\| _{\mathcal{Y}_{\rm harm}^{\rm tail}} = \mathcal{T}_{\rm harm}(\mathcal{D};\zeta).\]
Assumption 24 (Clean quotient recovery of pressure tails). There are linear maps \[L_{\rm proj}^{\rm tail}: \mathcal{K}_\Lambda^{\mathrm{cl}}/\Gamma_\Lambda^{\mathrm{cl}} \to \mathcal{Y}_{\rm proj}^{\rm tail}, \qquad L_{\rm harm}^{\rm tail}: \mathcal{K}_\Lambda^{\mathrm{cl}}/\Gamma_\Lambda^{\mathrm{cl}} \to \mathcal{Y}_{\rm harm}^{\rm tail},\] constants \(C_{\rm proj},C_{\rm harm}<\infty\), and reconstruction defects \(\delta_{\rm proj}^{\rm rec},\delta_{\rm harm}^{\rm rec}\ge0\), such that \[\|L_{\rm proj}^{\rm tail}q\|_{\mathcal{Y}_{\rm proj}^{\rm tail}} \le C_{\rm proj}\|q\|_{\mathrm{cl}/\Gamma}, \qquad \|L_{\rm harm}^{\rm tail}q\|_{\mathcal{Y}_{\rm harm}^{\rm tail}} \le C_{\rm harm}\|q\|_{\mathrm{cl}/\Gamma}\] for every clean quotient class \(q\). For the selected same-gauge representative \(\zeta_\ast(\mathcal{D})\), assume \[\left\| \tau_{\rm proj}(\mathcal{D};\zeta_\ast) - L_{\rm proj}^{\rm tail} [\Theta_\Lambda\mathcal{D}]_{\mathrm{cl}} \right\|_{\mathcal{Y}_{\rm proj}^{\rm tail}} \le \delta_{\rm proj}^{\rm rec}\] and \[\left\| \tau_{\rm harm}(\mathcal{D};\zeta_\ast) - L_{\rm harm}^{\rm tail} [\Theta_\Lambda\mathcal{D}]_{\mathrm{cl}} \right\|_{\mathcal{Y}_{\rm harm}^{\rm tail}} \le \delta_{\rm harm}^{\rm rec}.\]
Lemma 22 (Pressure-tail chart visibility from recovery). Assume 24. Then \[\operatorname{dist}_{\mathrm{cl}}(\Theta_\Lambda\mathcal{D},\Gamma_\Lambda^{\mathrm{cl}}) \ge \kappa_{\rm proj} X_{\rm proj}(\mathcal{D};\zeta_\ast) - \delta_{\rm proj}\] with \[\kappa_{\rm proj} = \frac{1}{\alpha_{\rm proj}C_{\rm proj}}, \qquad \delta_{\rm proj} = \frac{\delta_{\rm proj}^{\rm rec}}{C_{\rm proj}},\] and \[\operatorname{dist}_{\mathrm{cl}}(\Theta_\Lambda\mathcal{D},\Gamma_\Lambda^{\mathrm{cl}}) \ge \kappa_{\rm harm} X_{\rm harm}(\mathcal{D};\zeta_\ast) - \delta_{\rm harm}\] with \[\kappa_{\rm harm} = \frac{1}{\alpha_{\rm harm}C_{\rm harm}}, \qquad \delta_{\rm harm} = \frac{\delta_{\rm harm}^{\rm rec}}{C_{\rm harm}}.\]
Proof. We prove the projection-tail estimate; the harmonic-tail estimate is identical. By the recovery defect bound and boundedness of \(L_{\rm proj}^{\rm tail}\), \[\begin{align} \mathcal{T}_{\rm proj}(\mathcal{D};\zeta_\ast) &= \|\tau_{\rm proj}(\mathcal{D};\zeta_\ast)\| _{\mathcal{Y}_{\rm proj}^{\rm tail}}\\ &\le \|L_{\rm proj}^{\rm tail} [\Theta_\Lambda\mathcal{D}]_{\mathrm{cl}}\| _{\mathcal{Y}_{\rm proj}^{\rm tail}} + \delta_{\rm proj}^{\rm rec}\\ &\le C_{\rm proj} \operatorname{dist}_{\mathrm{cl}}(\Theta_\Lambda\mathcal{D},\Gamma_\Lambda^{\mathrm{cl}}) + \delta_{\rm proj}^{\rm rec}. \end{align}\] Since \(X_{\rm proj}=\alpha_{\rm proj}\mathcal{T}_{\rm proj}\), this gives \[\operatorname{dist}_{\mathrm{cl}}(\Theta_\Lambda\mathcal{D},\Gamma_\Lambda^{\mathrm{cl}}) \ge \frac{1}{\alpha_{\rm proj}C_{\rm proj}} X_{\rm proj}(\mathcal{D};\zeta_\ast) - \frac{\delta_{\rm proj}^{\rm rec}}{C_{\rm proj}}.\] The same argument with \(L_{\rm harm}^{\rm tail}\) proves the harmonic estimate. ◻
Corollary 7 (Finite-window sufficient condition for three-channel visibility). Assume 22, 23, and 24. Then 20 holds with the constants from 21 and 22.
Proof. The intrinsic-core visibility inequality follows from 21. The projection-tail and harmonic-tail visibility inequalities follow from 22. ◻
Remark 34 (Status of pressure-tail recovery). 22 is a finite-window observability statement conditional on the clean quotient carrying tail recovery maps. It does not prove projection-tail decay as \(N\to\infty\), does not prove harmonic-tail decay as \(M\to\infty\), and does not compare the enhanced-tail distance with the original intrinsic distance.
We now replace the placeholder chart residual \(\mathcal{R}_{\rm chart}\) in 41 by explicit finite-window chart mismatch components. This subsection is a chart compatibility module. It does not introduce a new Navier–Stokes pressure estimate.
Assumption 25 (Enhanced-tail same-gauge selector). For every intrinsic package \(\mathcal{D}\), fix a same-gauge representative \[\zeta_\ast(\mathcal{D})\in\Gamma_\Lambda^{\rm int}\] and write \[\mathcal{D}_\ast:=\mathcal{D}-\zeta_\ast(\mathcal{D}).\] There is a selector defect \(\delta_{\rm sel}^{\sharp}\ge0\) such that \[\begin{align} &\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}} + \alpha_{\rm proj} \mathcal{T}_{\rm proj}(\mathcal{D};\zeta_\ast) + \alpha_{\rm harm} \mathcal{T}_{\rm harm}(\mathcal{D};\zeta_\ast)\\ &\le \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) + \delta_{\rm sel}^{\sharp}. \end{align}\] In the exact minimizer case one may take \(\delta_{\rm sel}^{\sharp}=0\).
Definition 44 (Finite-window chart residual operators). Let \[Y_{\rm chart}^{O},\qquad Y_{\rm chart}^{\rm Rep},\qquad Y_{\rm chart}^{\rm Tax},\qquad Y_{\rm chart}^{\rm Tail}\] be finite-dimensional normed spaces. Fix bounded finite-window operators \[K_{\rm chart}^{O}: \mathfrak I_\Lambda^{\mathrm{loc}}\to Y_{\rm chart}^{O}, \qquad K_{\rm chart}^{\rm Rep}: \mathfrak I_\Lambda^{\mathrm{loc}}\to Y_{\rm chart}^{\rm Rep},\] \[K_{\rm chart}^{\rm Tax}: \mathfrak I_\Lambda^{\mathrm{loc}}\to Y_{\rm chart}^{\rm Tax}, \qquad K_{\rm chart}^{\rm Tail}: \mathfrak I_\Lambda^{\mathrm{loc}}\to Y_{\rm chart}^{\rm Tail}.\] The operator \(K_{\rm chart}^{O}\) records observation-coordinate chart mismatch, namely the mismatch between localized observation channels and clean observation channels after applying the local-to-clean chart. The operator \(K_{\rm chart}^{\rm Rep}\) records reproduction-coordinate chart mismatch between localized reproduction residuals and clean adjacent-scale reproduction residuals after charting. The operator \(K_{\rm chart}^{\rm Tax}\) records tax or gate-coordinate chart mismatch. The operator \(K_{\rm chart}^{\rm Tail}\) records pressure-tail chart mismatch between projection/harmonic tail data measured in the enhanced-tail intrinsic geometry and the corresponding clean tail coordinates recovered in the clean quotient.
These are finite-window model-level residual operators. Their boundedness is part of the chart datum, not a PDE estimate.
Definition 45 (Component chart residuals). For an intrinsic package \(\mathcal{D}\), define \[\mathcal{R}_{\rm chart}^{O}(\mathcal{D}) := \|K_{\rm chart}^{O}\mathcal{D}_\ast\|_{Y_{\rm chart}^{O}},\] \[\mathcal{R}_{\rm chart}^{\rm Rep}(\mathcal{D}) := \|K_{\rm chart}^{\rm Rep}\mathcal{D}_\ast\| _{Y_{\rm chart}^{\rm Rep}},\] \[\mathcal{R}_{\rm chart}^{\rm Tax}(\mathcal{D}) := \|K_{\rm chart}^{\rm Tax}\mathcal{D}_\ast\| _{Y_{\rm chart}^{\rm Tax}},\] and \[\mathcal{R}_{\rm chart}^{\rm Tail}(\mathcal{D}) := \|K_{\rm chart}^{\rm Tail}\mathcal{D}_\ast\| _{Y_{\rm chart}^{\rm Tail}}.\] The total chart residual is \[\mathcal{R}_{\rm chart}(\mathcal{D}) := \mathcal{R}_{\rm chart}^{O}(\mathcal{D}) + \mathcal{R}_{\rm chart}^{\rm Rep}(\mathcal{D}) + \mathcal{R}_{\rm chart}^{\rm Tax}(\mathcal{D}) + \mathcal{R}_{\rm chart}^{\rm Tail}(\mathcal{D}).\]
Lemma 23 (Componentwise chart residual bounds). Set \[C_{\rm chart}^{O} := \|K_{\rm chart}^{O}\|_{\mathfrak I_\Lambda^{\mathrm{loc}} \to Y_{\rm chart}^{O}}, \qquad C_{\rm chart}^{\rm Rep} := \|K_{\rm chart}^{\rm Rep}\|_{\mathfrak I_\Lambda^{\mathrm{loc}} \to Y_{\rm chart}^{\rm Rep}},\] \[C_{\rm chart}^{\rm Tax} := \|K_{\rm chart}^{\rm Tax}\|_{\mathfrak I_\Lambda^{\mathrm{loc}} \to Y_{\rm chart}^{\rm Tax}}, \qquad C_{\rm chart}^{\rm Tail} := \|K_{\rm chart}^{\rm Tail}\|_{\mathfrak I_\Lambda^{\mathrm{loc}} \to Y_{\rm chart}^{\rm Tail}}.\] Then \[\mathcal{R}_{\rm chart}^{O}(\mathcal{D}) \le C_{\rm chart}^{O}\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}},\] \[\mathcal{R}_{\rm chart}^{\rm Rep}(\mathcal{D}) \le C_{\rm chart}^{\rm Rep}\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}},\] \[\mathcal{R}_{\rm chart}^{\rm Tax}(\mathcal{D}) \le C_{\rm chart}^{\rm Tax}\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}},\] and \[\mathcal{R}_{\rm chart}^{\rm Tail}(\mathcal{D}) \le C_{\rm chart}^{\rm Tail}\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}}.\]
Proof. Each estimate is the operator-norm bound for the corresponding finite-window chart residual operator, evaluated at \(\mathcal{D}_\ast\). ◻
Lemma 24 (Enhanced-tail selector bound for the core norm). Under 25, \[\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}} \le \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) + \delta_{\rm sel}^{\sharp}.\]
Proof. The defining inequality in 25 is a sum of three nonnegative terms on the left-hand side. Dropping the two tail terms gives the stated bound. ◻
Proposition 14 (Chart residual sub-budget). Assume the enhanced-tail same-gauge selector and the chart operators above. Set \[C_{\rm chart} := C_{\rm chart}^{O} + C_{\rm chart}^{\rm Rep} + C_{\rm chart}^{\rm Tax} + C_{\rm chart}^{\rm Tail}.\] Then \[\mathcal{R}_{\rm chart}(\mathcal{D}) \le C_{\rm chart} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) + C_{\rm chart}\delta_{\rm sel}^{\sharp}.\] Equivalently, the chart residual satisfies \[\mathcal{R}_{\rm chart}(\mathcal{D}) \le \eta_{\rm chart} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) + \Delta_{\rm chart},\] with \[\eta_{\rm chart}=C_{\rm chart}, \qquad \Delta_{\rm chart}=C_{\rm chart}\delta_{\rm sel}^{\sharp}.\] In the exact minimizer case, \(\delta_{\rm sel}^{\sharp}=0\), and hence \(\Delta_{\rm chart}=0\).
Proof. Summing the four estimates in 23 gives \[\mathcal{R}_{\rm chart}(\mathcal{D}) \le C_{\rm chart}\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}}.\] Apply 24. ◻
Proposition 15 (Affine chart residual variant). Suppose that for each \[a\in\{O,{\rm Rep},{\rm Tax},{\rm Tail}\}\] one has an affine finite-window estimate \[\mathcal{R}_{\rm chart}^{a}(\mathcal{D}) \le C_{\rm chart}^{a}\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}} + \Delta_{\rm chart}^{a}\] with \(C_{\rm chart}^{a},\Delta_{\rm chart}^{a}\ge0\). Then \[\mathcal{R}_{\rm chart}(\mathcal{D}) \le \eta_{\rm chart} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) + \Delta_{\rm chart},\] where \[\eta_{\rm chart} = C_{\rm chart}^{O} + C_{\rm chart}^{\rm Rep} + C_{\rm chart}^{\rm Tax} + C_{\rm chart}^{\rm Tail},\] and \[\Delta_{\rm chart} = \eta_{\rm chart}\delta_{\rm sel}^{\sharp} + \Delta_{\rm chart}^{O} + \Delta_{\rm chart}^{\rm Rep} + \Delta_{\rm chart}^{\rm Tax} + \Delta_{\rm chart}^{\rm Tail}.\]
Proof. Sum the four affine component estimates and then use 24. ◻
Corollary 8 (Chart contribution to the enhanced-tail residual ledger). In the decomposed enhanced-tail residual budget from 13, the chart contribution may be taken to be \[\eta_{\rm chart} = C_{\rm chart}^{O} + C_{\rm chart}^{\rm Rep} + C_{\rm chart}^{\rm Tax} + C_{\rm chart}^{\rm Tail}\] and \[\Delta_{\rm chart} = \eta_{\rm chart}\delta_{\rm sel}^{\sharp} + \Delta_{\rm chart}^{O} + \Delta_{\rm chart}^{\rm Rep} + \Delta_{\rm chart}^{\rm Tax} + \Delta_{\rm chart}^{\rm Tail},\] with all \(\Delta_{\rm chart}^{a}=0\) in the purely linear operator-norm case. Consequently \[\begin{align} \eta_{\Lambda,{\rm dec}}^{\sharp,{\rm tail}} = &\eta_{\rm core} + \max\{\alpha_{\rm proj}^{-1},\alpha_{\rm harm}^{-1}\} + \eta_{\rm chart}\\ &+ \eta_{\rm loc} + \eta_{\rm rep} + \eta_{\rm gate}, \end{align}\] and \[\Delta_{\Lambda,{\rm dec}}^{\sharp,{\rm tail}} = \Delta_{\rm core} + \Delta_{\rm chart} + \Delta_{\rm loc} + \Delta_{\rm rep} + \Delta_{\rm gate}.\]
Remark 35 (Status of the chart residual sub-budget). This subsection proves only a finite-window chart residual sub-budget. Once the chart mismatch operators and same-gauge representative selection are fixed, the chart residual is controlled by the enhanced-tail quotient geometry. The result does not prove local-to-clean quotient lifting, the chart-visibility inequalities, scale-uniformity, smallness, or Navier–Stokes regularity. It also does not compare \[\operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}}\] with the original intrinsic distance \[\operatorname{dist}_{\mathrm{loc},{\rm int}}.\] That comparison remains a separate approximation or geometry theorem.
We next replace the placeholder localization leakage residual \(\mathcal{R}_{\rm loc}\) in 41. The convention in this finite-window bookkeeping step is to separate energy, flux, pressure, and momentum leakage. These are localization-channel residuals, not new Navier–Stokes estimates.
Definition 46 (Finite-window localization leakage operators). Let \[Y_{\mathrm{loc}}^{\rm en},\qquad Y_{\mathrm{loc}}^{\rm flux},\qquad Y_{\mathrm{loc}}^{\rm prs},\qquad Y_{\mathrm{loc}}^{\rm mom}\] be finite-dimensional normed spaces. Fix bounded finite-window operators \[K_{\mathrm{loc}}^{\rm en}: \mathfrak I_\Lambda^{\mathrm{loc}}\to Y_{\mathrm{loc}}^{\rm en}, \qquad K_{\mathrm{loc}}^{\rm flux}: \mathfrak I_\Lambda^{\mathrm{loc}}\to Y_{\mathrm{loc}}^{\rm flux},\] \[K_{\mathrm{loc}}^{\rm prs}: \mathfrak I_\Lambda^{\mathrm{loc}}\to Y_{\mathrm{loc}}^{\rm prs}, \qquad K_{\mathrm{loc}}^{\rm mom}: \mathfrak I_\Lambda^{\mathrm{loc}}\to Y_{\mathrm{loc}}^{\rm mom}.\] The operator \(K_{\mathrm{loc}}^{\rm en}\) records localization leakage in the energy or trace ledger. The operator \(K_{\mathrm{loc}}^{\rm flux}\) records leakage from flux terms generated by a cutoff. The operator \(K_{\mathrm{loc}}^{\rm prs}\) records the pressure contribution to the localized energy or flux budget. The operator \(K_{\mathrm{loc}}^{\rm mom}\) records the mismatch in localized momentum residuals. Boundedness of these operators is a finite-window localization datum.
Definition 47 (Localization leakage residuals). For an intrinsic package \(\mathcal{D}\), define \[\mathcal{R}_{\mathrm{loc}}^{\rm en}(\mathcal{D}) := \|K_{\mathrm{loc}}^{\rm en}\mathcal{D}_\ast\|_{Y_{\mathrm{loc}}^{\rm en}},\] \[\mathcal{R}_{\mathrm{loc}}^{\rm flux}(\mathcal{D}) := \|K_{\mathrm{loc}}^{\rm flux}\mathcal{D}_\ast\|_{Y_{\mathrm{loc}}^{\rm flux}},\] \[\mathcal{R}_{\mathrm{loc}}^{\rm prs}(\mathcal{D}) := \|K_{\mathrm{loc}}^{\rm prs}\mathcal{D}_\ast\|_{Y_{\mathrm{loc}}^{\rm prs}},\] and \[\mathcal{R}_{\mathrm{loc}}^{\rm mom}(\mathcal{D}) := \|K_{\mathrm{loc}}^{\rm mom}\mathcal{D}_\ast\|_{Y_{\mathrm{loc}}^{\rm mom}}.\] The total localization leakage residual is \[\mathcal{R}_{\rm loc}(\mathcal{D}) := \mathcal{R}_{\mathrm{loc}}^{\rm en}(\mathcal{D}) + \mathcal{R}_{\mathrm{loc}}^{\rm flux}(\mathcal{D}) + \mathcal{R}_{\mathrm{loc}}^{\rm prs}(\mathcal{D}) + \mathcal{R}_{\mathrm{loc}}^{\rm mom}(\mathcal{D}).\]
Lemma 25 (Componentwise localization leakage bounds). Set \[C_{\mathrm{loc}}^{\rm en} := \|K_{\mathrm{loc}}^{\rm en}\|_{\mathfrak I_\Lambda^{\mathrm{loc}} \to Y_{\mathrm{loc}}^{\rm en}}, \qquad C_{\mathrm{loc}}^{\rm flux} := \|K_{\mathrm{loc}}^{\rm flux}\|_{\mathfrak I_\Lambda^{\mathrm{loc}} \to Y_{\mathrm{loc}}^{\rm flux}},\] \[C_{\mathrm{loc}}^{\rm prs} := \|K_{\mathrm{loc}}^{\rm prs}\|_{\mathfrak I_\Lambda^{\mathrm{loc}} \to Y_{\mathrm{loc}}^{\rm prs}}, \qquad C_{\mathrm{loc}}^{\rm mom} := \|K_{\mathrm{loc}}^{\rm mom}\|_{\mathfrak I_\Lambda^{\mathrm{loc}} \to Y_{\mathrm{loc}}^{\rm mom}}.\] Then \[\mathcal{R}_{\mathrm{loc}}^{\rm en}(\mathcal{D}) \le C_{\mathrm{loc}}^{\rm en}\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}},\] \[\mathcal{R}_{\mathrm{loc}}^{\rm flux}(\mathcal{D}) \le C_{\mathrm{loc}}^{\rm flux}\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}},\] \[\mathcal{R}_{\mathrm{loc}}^{\rm prs}(\mathcal{D}) \le C_{\mathrm{loc}}^{\rm prs}\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}},\] and \[\mathcal{R}_{\mathrm{loc}}^{\rm mom}(\mathcal{D}) \le C_{\mathrm{loc}}^{\rm mom}\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}}.\]
Proof. Each inequality is the boundedness estimate for the corresponding finite-window localization leakage operator, evaluated on the selected representative \(\mathcal{D}_\ast\). ◻
Proposition 16 (Localization leakage sub-budget). Assume the enhanced-tail same-gauge selector from 25 and the finite-window localization leakage operators above. Set \[C_{\mathrm{loc}} := C_{\mathrm{loc}}^{\rm en} + C_{\mathrm{loc}}^{\rm flux} + C_{\mathrm{loc}}^{\rm prs} + C_{\mathrm{loc}}^{\rm mom}.\] Then \[\mathcal{R}_{\rm loc}(\mathcal{D}) \le C_{\mathrm{loc}} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) + C_{\mathrm{loc}}\delta_{\rm sel}^{\sharp}.\] Equivalently, the localization leakage residual satisfies \[\mathcal{R}_{\rm loc}(\mathcal{D}) \le \eta_{\mathrm{loc}} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) + \Delta_{\mathrm{loc}},\] with \[\eta_{\mathrm{loc}}=C_{\mathrm{loc}}, \qquad \Delta_{\mathrm{loc}}=C_{\mathrm{loc}}\delta_{\rm sel}^{\sharp}.\] If the selector is an exact enhanced-tail minimizer, then \(\delta_{\rm sel}^{\sharp}=0\) and \(\Delta_{\mathrm{loc}}=0\).
Proof. Summing the componentwise estimates in 25 gives \[\mathcal{R}_{\rm loc}(\mathcal{D}) \le C_{\mathrm{loc}}\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}}.\] The selector bound in 24 gives the stated estimate. ◻
Proposition 17 (Affine localization leakage variant). Suppose that each localization leakage component satisfies an affine finite-window estimate \[\mathcal{R}_{\mathrm{loc}}^{a}(\mathcal{D}) \le C_{\mathrm{loc}}^{a}\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}} + \Delta_{\mathrm{loc}}^{a}, \qquad a\in\{{\rm en},{\rm flux},{\rm prs},{\rm mom}\}.\] Then \[\mathcal{R}_{\rm loc}(\mathcal{D}) \le \eta_{\mathrm{loc}} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) + \Delta_{\mathrm{loc}},\] where \[\eta_{\mathrm{loc}} = C_{\mathrm{loc}}^{\rm en} + C_{\mathrm{loc}}^{\rm flux} + C_{\mathrm{loc}}^{\rm prs} + C_{\mathrm{loc}}^{\rm mom},\] and \[\Delta_{\mathrm{loc}} = \eta_{\mathrm{loc}}\delta_{\rm sel}^{\sharp} + \Delta_{\mathrm{loc}}^{\rm en} + \Delta_{\mathrm{loc}}^{\rm flux} + \Delta_{\mathrm{loc}}^{\rm prs} + \Delta_{\mathrm{loc}}^{\rm mom}.\]
Proof. Sum the four affine estimates and then use 24. ◻
Corollary 9 (Localization contribution to the enhanced-tail residual ledger). In the decomposed enhanced-tail residual budget from 13, the localization contribution may be taken to be \[\eta_{\mathrm{loc}} = C_{\mathrm{loc}}^{\rm en} + C_{\mathrm{loc}}^{\rm flux} + C_{\mathrm{loc}}^{\rm prs} + C_{\mathrm{loc}}^{\rm mom}\] and \[\Delta_{\mathrm{loc}} = \eta_{\mathrm{loc}}\delta_{\rm sel}^{\sharp} + \Delta_{\mathrm{loc}}^{\rm en} + \Delta_{\mathrm{loc}}^{\rm flux} + \Delta_{\mathrm{loc}}^{\rm prs} + \Delta_{\mathrm{loc}}^{\rm mom},\] with all \(\Delta_{\mathrm{loc}}^{a}=0\) in the purely linear operator-norm case. Thus the decomposed residual-budget constants remain \[\begin{align} \eta_{\Lambda,{\rm dec}}^{\sharp,{\rm tail}} = &\eta_{\rm core} + \max\{\alpha_{\rm proj}^{-1},\alpha_{\rm harm}^{-1}\} + \eta_{\rm chart}\\ &+ \eta_{\mathrm{loc}} + \eta_{\rm rep} + \eta_{\rm gate}, \end{align}\] and \[\Delta_{\Lambda,{\rm dec}}^{\sharp,{\rm tail}} = \Delta_{\rm core} + \Delta_{\rm chart} + \Delta_{\mathrm{loc}} + \Delta_{\rm rep} + \Delta_{\rm gate}.\]
Remark 36 (Status of the localization leakage sub-budget). This subsection proves only a finite-window localization leakage sub-budget. It says that, once the localization leakage operators and the same-gauge representative selection are fixed, the localization residual is controlled by the enhanced-tail quotient geometry. It does not prove a local energy inequality estimate, a pressure localization estimate, a momentum residual estimate, smallness, scale-uniform localization control, or Navier–Stokes regularity. It also does not compare \[\operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}}\] with \[\operatorname{dist}_{\mathrm{loc},{\rm int}}.\]
We next replace the placeholder reproduction residual \(\mathcal{R}_{\rm rep}\) in 41. This is a finite-window bookkeeping module. It records the drift between adjacent scale packages after a reproduction model has been fixed; it does not prove that Navier–Stokes dynamically generates the reproduction maps, and it does not prove scale-uniform reproduction.
Definition 48 (Finite-window reproduction drift operators). Let \[\Lambda_{\rm adj} := \{k\in\Lambda:\;k+1\in\Lambda\}\] be the adjacent-scale index set. For each \(k\in\Lambda_{\rm adj}\), let \(Y_k^{\rm rep}\) be a finite-dimensional normed space and fix a bounded finite-window operator \[K_k^{\rm rep}: \mathfrak I_\Lambda^{\mathrm{loc}}\to Y_k^{\rm rep}.\] The quantity \(K_k^{\rm rep}\mathcal{D}_\ast\) records the model-level drift between the selected intrinsic package at scale \(k+1\) and the package predicted from scale \(k\) by the chosen adjacent-scale reproduction map. Schematically, it represents a coordinate-extracted or projected term of the form \[\mathcal{D}_{\ast,k+1} - \mathcal{R}_k^{\rm int}\mathcal{D}_{\ast,k},\] possibly after clean/local charting. The map \(\mathcal{R}_k^{\rm int}\) is fixed finite-window reproduction data here; no Navier–Stokes generation of this map is asserted.
Definition 49 (Reproduction drift residuals). For an intrinsic package \(\mathcal{D}\), define the component reproduction drift residuals by \[\mathcal{R}_k^{\rm rep}(\mathcal{D}) := \|K_k^{\rm rep}\mathcal{D}_\ast\|_{Y_k^{\rm rep}}, \qquad k\in\Lambda_{\rm adj}.\] The total reproduction drift residual is \[\mathcal{R}_{\rm rep}(\mathcal{D}) := \sum_{k\in\Lambda_{\rm adj}} \mathcal{R}_k^{\rm rep}(\mathcal{D}).\]
Lemma 26 (Componentwise reproduction drift bounds). For each \(k\in\Lambda_{\rm adj}\), set \[C_k^{\rm rep} := \|K_k^{\rm rep}\|_{\mathfrak I_\Lambda^{\mathrm{loc}}\to Y_k^{\rm rep}}, \qquad C_{\rm rep} := \sum_{k\in\Lambda_{\rm adj}}C_k^{\rm rep}.\] Then \[\mathcal{R}_k^{\rm rep}(\mathcal{D}) \le C_k^{\rm rep}\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}}, \qquad k\in\Lambda_{\rm adj},\] and hence \[\mathcal{R}_{\rm rep}(\mathcal{D}) \le C_{\rm rep}\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}}.\]
Proof. The component estimate is the operator-norm bound for \(K_k^{\rm rep}\), evaluated at the selected same-gauge representative \(\mathcal{D}_\ast\). Summing over \(k\in\Lambda_{\rm adj}\) gives the second estimate. ◻
Proposition 18 (Reproduction drift sub-budget). Assume the enhanced-tail same-gauge selector from 25 and the finite-window reproduction drift operators above. Then \[\mathcal{R}_{\rm rep}(\mathcal{D}) \le C_{\rm rep} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) + C_{\rm rep}\delta_{\rm sel}^{\sharp}.\] Equivalently, the reproduction drift residual satisfies \[\mathcal{R}_{\rm rep}(\mathcal{D}) \le \eta_{\rm rep} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) + \Delta_{\rm rep},\] with \[\eta_{\rm rep}=C_{\rm rep}, \qquad \Delta_{\rm rep}=C_{\rm rep}\delta_{\rm sel}^{\sharp}.\] If the selector is an exact enhanced-tail minimizer, then \(\delta_{\rm sel}^{\sharp}=0\) and \(\Delta_{\rm rep}=0\).
Proof. By 26, \[\mathcal{R}_{\rm rep}(\mathcal{D}) \le C_{\rm rep}\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}}.\] The enhanced-tail selector bound in 24 gives the displayed sub-budget. ◻
Proposition 19 (Affine reproduction drift variant). Suppose that each adjacent-scale reproduction component satisfies an affine finite-window estimate \[\mathcal{R}_k^{\rm rep}(\mathcal{D}) \le C_k^{\rm rep}\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}} + \Delta_k^{\rm rep}, \qquad k\in\Lambda_{\rm adj}.\] Then \[\mathcal{R}_{\rm rep}(\mathcal{D}) \le \eta_{\rm rep} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) + \Delta_{\rm rep},\] where \[\eta_{\rm rep} = \sum_{k\in\Lambda_{\rm adj}}C_k^{\rm rep},\] and \[\Delta_{\rm rep} = \eta_{\rm rep}\delta_{\rm sel}^{\sharp} + \sum_{k\in\Lambda_{\rm adj}}\Delta_k^{\rm rep}.\]
Proof. Sum the affine component estimates and use 24. ◻
Corollary 10 (Reproduction contribution to the enhanced-tail residual ledger). In the decomposed enhanced-tail residual budget from 13, the reproduction contribution may be taken to be \[\eta_{\rm rep} = \sum_{k\in\Lambda_{\rm adj}}C_k^{\rm rep},\] and \[\Delta_{\rm rep} = \eta_{\rm rep}\delta_{\rm sel}^{\sharp} + \sum_{k\in\Lambda_{\rm adj}}\Delta_k^{\rm rep},\] with all \(\Delta_k^{\rm rep}=0\) in the purely linear operator-norm case. Therefore the decomposed residual-budget constants remain \[\begin{align} \eta_{\Lambda,{\rm dec}}^{\sharp,{\rm tail}} = &\eta_{\rm core} + \max\{\alpha_{\rm proj}^{-1},\alpha_{\rm harm}^{-1}\} + \eta_{\rm chart}\\ &+ \eta_{\mathrm{loc}} + \eta_{\rm rep} + \eta_{\rm gate}, \end{align}\] and \[\Delta_{\Lambda,{\rm dec}}^{\sharp,{\rm tail}} = \Delta_{\rm core} + \Delta_{\rm chart} + \Delta_{\mathrm{loc}} + \Delta_{\rm rep} + \Delta_{\rm gate}.\]
Proof. Insert either reproduction sub-budget, the linear one from 18 or the affine one from 19, into the decomposed ledger 13. ◻
Remark 37 (Status of the reproduction drift sub-budget). This subsection proves only a finite-window reproduction drift sub-budget. It says that, once the adjacent-scale reproduction drift operators and the same-gauge representative selection are fixed, the reproduction drift residual is controlled by the enhanced-tail quotient geometry. It does not prove that Navier–Stokes generates the reproduction maps, exact reproduction, scale-uniformity, smallness, or Navier–Stokes regularity. It also does not compare \[\operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}}\] with \[\operatorname{dist}_{\mathrm{loc},{\rm int}}.\]
We finally replace the placeholder gate residual \(\mathcal{R}_{\rm gate}\) in 41. This subsection records finite-window mismatch between localized gate data, clean detector gates, and tax or ledger-slack coordinates. It is not a lower bound for a concrete Navier–Stokes pressure, flux, or dissipation tax.
Definition 50 (Finite-window gate/tax mismatch operators). Let \[Y_{\rm gate}^{\rm det},\qquad Y_{\rm gate}^{\rm tax},\qquad Y_{\rm gate}^{\rm slack}\] be finite-dimensional normed spaces. Fix bounded finite-window operators \[K_{\rm gate}^{\rm det}: \mathfrak I_\Lambda^{\mathrm{loc}}\to Y_{\rm gate}^{\rm det}, \qquad K_{\rm gate}^{\rm tax}: \mathfrak I_\Lambda^{\mathrm{loc}}\to Y_{\rm gate}^{\rm tax},\] and \[K_{\rm gate}^{\rm slack}: \mathfrak I_\Lambda^{\mathrm{loc}}\to Y_{\rm gate}^{\rm slack}.\] The operator \(K_{\rm gate}^{\rm det}\) records detector-gate mismatch, namely the mismatch between localized gate coordinates and the clean detector gate after applying the local-to-clean chart. The operator \(K_{\rm gate}^{\rm tax}\) records tax-functional mismatch, namely the finite-window discrepancy between localized tax data and the clean tax functional \(\operatorname{Tax}_\Lambda^{\mathrm{cl}}\). The operator \(K_{\rm gate}^{\rm slack}\) records ledger-slack mismatch. Their boundedness is part of the finite-window gate/tax datum.
Definition 51 (Gate/tax mismatch residuals). For an intrinsic package \(\mathcal{D}\), define \[\mathcal{R}_{\rm gate}^{\rm det}(\mathcal{D}) := \|K_{\rm gate}^{\rm det}\mathcal{D}_\ast\|_{Y_{\rm gate}^{\rm det}},\] \[\mathcal{R}_{\rm gate}^{\rm tax}(\mathcal{D}) := \|K_{\rm gate}^{\rm tax}\mathcal{D}_\ast\|_{Y_{\rm gate}^{\rm tax}},\] and \[\mathcal{R}_{\rm gate}^{\rm slack}(\mathcal{D}) := \|K_{\rm gate}^{\rm slack}\mathcal{D}_\ast\| _{Y_{\rm gate}^{\rm slack}}.\] The total gate/tax mismatch residual is \[\mathcal{R}_{\rm gate}(\mathcal{D}) := \mathcal{R}_{\rm gate}^{\rm det}(\mathcal{D}) + \mathcal{R}_{\rm gate}^{\rm tax}(\mathcal{D}) + \mathcal{R}_{\rm gate}^{\rm slack}(\mathcal{D}).\]
Lemma 27 (Componentwise gate/tax mismatch bounds). Set \[C_{\rm gate}^{\rm det} := \|K_{\rm gate}^{\rm det}\|_{\mathfrak I_\Lambda^{\mathrm{loc}} \to Y_{\rm gate}^{\rm det}}, \qquad C_{\rm gate}^{\rm tax} := \|K_{\rm gate}^{\rm tax}\|_{\mathfrak I_\Lambda^{\mathrm{loc}} \to Y_{\rm gate}^{\rm tax}},\] and \[C_{\rm gate}^{\rm slack} := \|K_{\rm gate}^{\rm slack}\|_{\mathfrak I_\Lambda^{\mathrm{loc}} \to Y_{\rm gate}^{\rm slack}}.\] Then \[\mathcal{R}_{\rm gate}^{\rm det}(\mathcal{D}) \le C_{\rm gate}^{\rm det}\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}},\] \[\mathcal{R}_{\rm gate}^{\rm tax}(\mathcal{D}) \le C_{\rm gate}^{\rm tax}\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}},\] and \[\mathcal{R}_{\rm gate}^{\rm slack}(\mathcal{D}) \le C_{\rm gate}^{\rm slack}\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}}.\]
Proof. Each estimate is the operator-norm bound for the corresponding operator, evaluated at \(\mathcal{D}_\ast\). ◻
Proposition 20 (Gate/tax mismatch sub-budget). Assume the enhanced-tail same-gauge selector from 25 and the finite-window gate/tax mismatch operators above. Set \[C_{\rm gate} := C_{\rm gate}^{\rm det} + C_{\rm gate}^{\rm tax} + C_{\rm gate}^{\rm slack}.\] Then \[\mathcal{R}_{\rm gate}(\mathcal{D}) \le C_{\rm gate} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) + C_{\rm gate}\delta_{\rm sel}^{\sharp}.\] Equivalently, the gate/tax mismatch residual satisfies \[\mathcal{R}_{\rm gate}(\mathcal{D}) \le \eta_{\rm gate} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) + \Delta_{\rm gate},\] with \[\eta_{\rm gate}=C_{\rm gate}, \qquad \Delta_{\rm gate}=C_{\rm gate}\delta_{\rm sel}^{\sharp}.\] If the selector is an exact enhanced-tail minimizer, then \(\delta_{\rm sel}^{\sharp}=0\) and \(\Delta_{\rm gate}=0\).
Proof. Summing the componentwise estimates in 27 gives \[\mathcal{R}_{\rm gate}(\mathcal{D}) \le C_{\rm gate}\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}}.\] The enhanced-tail selector bound in 24 gives the displayed estimate. ◻
Proposition 21 (Affine gate/tax mismatch variant). Suppose that each gate/tax mismatch component satisfies an affine finite-window estimate \[\mathcal{R}_{\rm gate}^{a}(\mathcal{D}) \le C_{\rm gate}^{a}\|\mathcal{D}_\ast\|_{\mathrm{loc},{\rm int}} + \Delta_{\rm gate}^{a}, \qquad a\in\{{\rm det},{\rm tax},{\rm slack}\}.\] Then \[\mathcal{R}_{\rm gate}(\mathcal{D}) \le \eta_{\rm gate} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) + \Delta_{\rm gate},\] where \[\eta_{\rm gate} = C_{\rm gate}^{\rm det} + C_{\rm gate}^{\rm tax} + C_{\rm gate}^{\rm slack},\] and \[\Delta_{\rm gate} = \eta_{\rm gate}\delta_{\rm sel}^{\sharp} + \Delta_{\rm gate}^{\rm det} + \Delta_{\rm gate}^{\rm tax} + \Delta_{\rm gate}^{\rm slack}.\]
Proof. Sum the three affine component estimates and use 24. ◻
Corollary 11 (Gate/tax contribution to the enhanced-tail residual ledger). In the decomposed enhanced-tail residual budget from 13, the gate/tax contribution may be taken to be \[\eta_{\rm gate} = C_{\rm gate}^{\rm det} + C_{\rm gate}^{\rm tax} + C_{\rm gate}^{\rm slack},\] and \[\Delta_{\rm gate} = \eta_{\rm gate}\delta_{\rm sel}^{\sharp} + \Delta_{\rm gate}^{\rm det} + \Delta_{\rm gate}^{\rm tax} + \Delta_{\rm gate}^{\rm slack},\] with all \(\Delta_{\rm gate}^{a}=0\) in the purely linear operator-norm case. Therefore all four finite-window residual sub-budgets in the decomposed enhanced-tail ledger have explicit coefficients, and \[\begin{align} \eta_{\Lambda,{\rm dec}}^{\sharp,{\rm tail}} = &\eta_{\rm core} + \max\{\alpha_{\rm proj}^{-1},\alpha_{\rm harm}^{-1}\} + \eta_{\rm chart}\\ &+ \eta_{\mathrm{loc}} + \eta_{\rm rep} + \eta_{\rm gate}, \end{align}\] while \[\Delta_{\Lambda,{\rm dec}}^{\sharp,{\rm tail}} = \Delta_{\rm core} + \Delta_{\rm chart} + \Delta_{\mathrm{loc}} + \Delta_{\rm rep} + \Delta_{\rm gate}.\]
Proof. Insert either the linear gate/tax sub-budget from 20 or the affine variant from 21 into the decomposed ledger 13. The chart contribution is supplied by 8. The localization and reproduction contributions are supplied by 9 and 10. ◻
Remark 38 (Status of the gate/tax mismatch sub-budget). This subsection proves only a finite-window gate/tax mismatch sub-budget. It says that, once the detector-gate, tax-functional, and ledger-slack mismatch operators and the same-gauge representative selection are fixed, the gate/tax residual is controlled by the enhanced-tail quotient geometry. It does not prove a positive pressure tax, a positive flux tax, coercivity of \(\operatorname{Tax}_\Lambda^{\mathrm{cl}}\), scale-uniformity, smallness, or Navier–Stokes regularity. It also does not compare \[\operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}}\] with \[\operatorname{dist}_{\mathrm{loc},{\rm int}}.\]
We now collect the four finite-window sub-budgets recorded above. This is an assembly step only: it does not derive the enhanced-tail comparison hypothesis, does not compare the enhanced-tail distance with the original intrinsic distance, and does not prove any scale-uniform estimate.
Corollary 12 (Assembled finite-window enhanced-tail residual budget). Assume the core and tail estimates in 17 and 20. Assume also the finite-window chart, localization, reproduction, and gate/tax sub-budgets in 14, 16, 18, and 20. Then the decomposed enhanced-tail residual satisfies \[\mathsf{Err}_{\Lambda,{\rm dec}}^{\mathrm{loc}}(\mathcal{D}) \le \eta_{\Lambda,{\rm dec}}^{\sharp,{\rm tail}} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) + \Delta_{\Lambda,{\rm dec}}^{\sharp,{\rm tail}},\] where \[\begin{align} \eta_{\Lambda,{\rm dec}}^{\sharp,{\rm tail}} = &\eta_{\rm core} + \max\{\alpha_{\rm proj}^{-1},\alpha_{\rm harm}^{-1}\} + \eta_{\rm chart}\\ &+ \eta_{\mathrm{loc}} + \eta_{\rm rep} + \eta_{\rm gate}, \end{align}\] and \[\Delta_{\Lambda,{\rm dec}}^{\sharp,{\rm tail}} = \Delta_{\rm core} + \Delta_{\rm chart} + \Delta_{\mathrm{loc}} + \Delta_{\rm rep} + \Delta_{\rm gate}.\] Consequently, if the detector comparison in 21 holds with the same decomposed residual, then the enhanced-tail residual-budget hypothesis is realized with the displayed constants.
Proof. The definition of \(\mathsf{Err}_{\Lambda,{\rm dec}}^{\mathrm{loc}}\) in 13 is the sum of the core, tail, chart, localization, reproduction, and gate/tax residuals. The core estimate is 17. The pressure-tail estimate is 20. The four finite-window sub-budgets are supplied by the four propositions listed in the statement. Adding the six estimates gives the displayed constants. ◻
Remark 39 (Status of the assembled criterion). 12 is a bookkeeping consequence of the finite-window sub-budgets. It does not prove that the enhanced-tail detector comparison holds for localized Navier–Stokes packages, does not prove the local-to-clean comparison, and does not compare \(\operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}}\) with \(\operatorname{dist}_{\mathrm{loc},{\rm int}}\). Those are the next PDE-facing problems.
We now compare the combined enhanced-tail geometry with the original intrinsic geometry. This is the first PDE-facing step after the residual ledger has been assembled. The comparison is conditional: the projection and harmonic tails must be controlled on a common intrinsic representative.
Lemma 28 (One-sided comparison with the intrinsic distance). For every intrinsic package \(\mathcal{D}\), \[\operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_\Lambda^{\rm int}) \le \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}).\]
Proof. For every \(\zeta\in\Gamma_\Lambda^{\rm int}\), the tail terms in 40 are nonnegative, so \[\|\mathcal{D}-\zeta\|_{\mathrm{loc},{\rm int}} \le \|\mathcal{D}-\zeta\|_{\mathrm{loc},{\rm int}} + \alpha_{\rm proj}\mathcal{T}_{\rm proj}(\mathcal{D};\zeta) + \alpha_{\rm harm}\mathcal{T}_{\rm harm}(\mathcal{D};\zeta).\] Taking the infimum over \(\zeta\) gives the claim. ◻
Assumption 26 (Common intrinsic representative for tail approximation). For every intrinsic package \(\mathcal{D}\), there is a representative \[\zeta_{\rm int}(\mathcal{D})\in\Gamma_\Lambda^{\rm int}\] and a representative-selection error \(\delta_{\rm int}\ge0\) such that \[\|\mathcal{D}-\zeta_{\rm int}\|_{\mathrm{loc},{\rm int}} \le \operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_\Lambda^{\rm int}) + \delta_{\rm int}.\] On this same representative, assume that the projection and harmonic tails satisfy \[\mathcal{T}_{\rm proj}(\mathcal{D};\zeta_{\rm int}) \le C_{\rm proj}^{\rm app} \operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_\Lambda^{\rm int}) + \Delta_{{\rm proj},N},\] and \[\mathcal{T}_{\rm harm}(\mathcal{D};\zeta_{\rm int}) \le C_{\rm harm}^{\rm app} \operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_\Lambda^{\rm int}) + \Delta_{{\rm harm},M}.\] Here \(C_{\rm proj}^{\rm app},C_{\rm harm}^{\rm app}<\infty\), while \(\Delta_{{\rm proj},N},\Delta_{{\rm harm},M}\ge0\) are finite-window approximation errors. No decay of these errors as \(N\to\infty\) or \(M\to\infty\) is assumed here.
Theorem 7 (Conditional enhanced-tail/intrinsic comparison). Assume 26. Then, for every intrinsic package \(\mathcal{D}\), \[\begin{align} &\operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int})\\ &\le \bigl( 1 + \alpha_{\rm proj}C_{\rm proj}^{\rm app} + \alpha_{\rm harm}C_{\rm harm}^{\rm app} \bigr) \operatorname{dist}_{\mathrm{loc},{\rm int}}(\mathcal{D},\Gamma_\Lambda^{\rm int})\\ &\quad+ \delta_{\rm int} + \alpha_{\rm proj}\Delta_{{\rm proj},N} + \alpha_{\rm harm}\Delta_{{\rm harm},M}. \end{align}\]
Proof. Use the representative \(\zeta_{\rm int}(\mathcal{D})\) from 26 as a competitor in the infimum defining the enhanced-tail distance. Then \[\begin{align} &\operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int})\\ &\le \|\mathcal{D}-\zeta_{\rm int}\|_{\mathrm{loc},{\rm int}} + \alpha_{\rm proj}\mathcal{T}_{\rm proj} (\mathcal{D};\zeta_{\rm int}) + \alpha_{\rm harm}\mathcal{T}_{\rm harm} (\mathcal{D};\zeta_{\rm int}). \end{align}\] Substitute the three bounds in 26 and collect terms. ◻
Corollary 13 (Two-sided conditional geometry comparison). Under 26, one has \[\operatorname{dist}_{\mathrm{loc},{\rm int}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) \le \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int})\] and \[\operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) \le C_{{\rm tail}/{\rm int}} \operatorname{dist}_{\mathrm{loc},{\rm int}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) + \Delta_{{\rm tail}/{\rm int}},\] where \[C_{{\rm tail}/{\rm int}} := 1 + \alpha_{\rm proj}C_{\rm proj}^{\rm app} + \alpha_{\rm harm}C_{\rm harm}^{\rm app},\] and \[\Delta_{{\rm tail}/{\rm int}} := \delta_{\rm int} + \alpha_{\rm proj}\Delta_{{\rm proj},N} + \alpha_{\rm harm}\Delta_{{\rm harm},M}.\]
Corollary 14 (Intrinsic lower bound inherited from enhanced-tail transfer). Assume that \(c_\Lambda^{\sharp,{\rm tail}}\ge0\) and that an enhanced-tail localized transfer estimate has the form \[\mathcal{M}_\Lambda^{\mathrm{loc}}(\mathcal{D}) \ge c_\Lambda^{\sharp,{\rm tail}} \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) - \Delta_\Lambda^{\sharp,{\rm tail}}.\] Then \[\mathcal{M}_\Lambda^{\mathrm{loc}}(\mathcal{D}) \ge c_\Lambda^{\sharp,{\rm tail}} \operatorname{dist}_{\mathrm{loc},{\rm int}} (\mathcal{D},\Gamma_\Lambda^{\rm int}) - \Delta_\Lambda^{\sharp,{\rm tail}}.\]
Proof. By 28, the enhanced-tail distance is at least the original intrinsic distance. Multiplication by the nonnegative constant \(c_\Lambda^{\sharp,{\rm tail}}\) preserves the inequality, and substitution into the assumed enhanced-tail transfer estimate gives the result. ◻
Remark 40 (Meaning of the reverse comparison). The lower-bound corollary uses only the immediate comparison \(\operatorname{dist}_{\mathrm{loc},{\rm int}}\le \operatorname{dist}_{\mathrm{loc},{\rm int}}^{\sharp,{\rm tail}}\). The reverse comparison in 7 is needed for a different reason: it justifies that the enhanced-tail geometry is not adding an uncontrolled new pressure-tail coordinate. That justification is conditional on the projection and harmonic tail approximation assumptions in 26.
Remark 41 (Status of the conditional comparison). This subsection does not prove tail approximation, pressure/tax coercivity, scale-uniformity, or Navier–Stokes regularity. It proves only a conditional geometric implication. If the projection and harmonic tails are controlled on a common intrinsic representative, then the enhanced-tail distance is comparable to the original intrinsic distance up to explicit additive approximation errors.
The preceding sections establish a finite-window reduction rather than a complete regularity theorem. In relation to the preceding defect-cascade formulation [25], the present paper isolates the clean finite-window compactness and transfer bookkeeping that such a cascade analysis would require. The purpose of this final section is to make the logical status of the results explicit: which statements have been proved inside the finite-window framework, which inputs remain conditional, and which PDE estimates would turn the framework into a stronger Navier–Stokes result.
Within the definitions fixed in this paper, the main conclusions are the following.
Clean compactness gap. Under the clean detector gauge compatibility assumption in 1 and the kernel-free detector condition in 2, the finite-dimensional clean compactness argument gives a positive computational anti-phantom gap; see 2.
Finite sector reduction. The detector family decomposes the possible non-gauge clean defect into finitely many observable sectors. Equivalently, a normalized clean defect cannot remain simultaneously invisible to all pressure, flux, energy, trace, reproduction, and tax channels; see 1.
Conditional localized transfer. A clean finite-window gap yields a localized lower bound once the local-to-clean comparison and the localized residual budget are assumed; see [ass:local-to-clean-transfer-comparison,ass:localized-residual-budget] and 2.
Enhanced pressure-tail bookkeeping. The later sections refine the transfer mechanism by separating the intrinsic core, the clean projection tail, and the harmonic pressure tail. The resulting enhanced-tail transfer theorem and its subclaim criterion isolate the chart-visibility and residual-budget inputs needed for a pressure-tail version of the localized lower bound; see [thm:enhanced-tail-localized-transfer,cor:subclaim-criterion-enhanced-tail-localized-transfer].
Conditional comparison with the intrinsic geometry. If the clean projection tail and the harmonic tail are controlled on a common intrinsic representative, then the enhanced-tail distance is comparable with the original intrinsic distance up to explicit additive errors; see 7.
Thus the paper proves a precise finite-window implication: persistent localized badness must be visible either through a clean detector sector or through one of the explicitly recorded residual channels, provided the stated comparison and budget hypotheses are available.
The framework deliberately separates algebraic compactness from the PDE estimates needed to realize that compactness inside the Navier–Stokes local geometry. In particular, the present paper does not prove the following inputs unconditionally:
the construction of canonical clean defect spaces, gauge spaces, and gauge-compatible detector families directly from suitable weak solutions;
the local-to-clean quotient lifting and the corresponding transfer comparison in 3;
the localized residual budget in 4, and its enhanced-tail analogue in 19;
the enhanced-tail comparison in 18;
projection-tail and harmonic-tail approximation estimates strong enough to remove the additive errors in 7;
a coercive lower bound for a concrete pressure, flux, or gate/tax functional in the original PDE variables.
These are not cosmetic omissions. They mark the points at which the abstract finite-window architecture must be connected to genuine Navier–Stokes estimates, especially pressure decomposition, cutoff commutators, quotient selection, and same-gauge comparison.
The most immediate continuation is to prove pressure-tail approximation in a fixed localized intrinsic package. Concretely, one should estimate the clean projection error and the harmonic residual on a common representative and show that \[\Delta_{{\rm proj},N}\to0, \qquad \Delta_{{\rm harm},M}\to0,\] under hypotheses that are natural for suitable weak solutions. This would turn the conditional comparison in 7 into a usable bridge between the enhanced-tail geometry and the original intrinsic quotient distance.
A second direction is to justify the enhanced-tail local-to-clean comparison and residual-budget assumptions in a concrete intrinsic model. The sub-budget decomposition developed above reduces this task to three chart-visibility channels and four finite-window residual budgets: chart mismatch, localization leakage, reproduction drift, and gate/tax mismatch. The role of these sub-budgets is to identify exactly where a PDE estimate is needed rather than to hide it inside a single global assumption.
A third, more coercive direction is to develop a normalized pressure/tax model that gives an actual positive lower bound for a detector sector along expanding finite windows. This is the step that would move the framework from a finite-window accounting theorem toward a mechanism capable of excluding specific persistent defect scenarios.
No result in this paper proves Navier–Stokes global regularity, rules out all possible singularity mechanisms, or constructs a singular solution. The paper also does not claim a symbolic-dynamics, universal-computation, or undecidability theorem. Its contribution is narrower and more structural: it turns a persistent finite-window defect into an explicit accounting problem, where the possible escape routes are cleanly separated into detector sectors, pressure-tail errors, and localized residual budgets. The usefulness of the framework therefore depends on whether the conditional inputs isolated above can be proved in intrinsic Navier–Stokes geometries.