PRH | Essay | 7.23 • Poles, Universes, and Blur

Poles, Universes, and Blur An Epistemic View of Singularity Aleksandar Perišić November 2025 Abstract When we talk about modeling in science, we instinctively separate model from reality. We know that no model is perfect, and we live in a culture of “better approximations”: each decade brings new theories that correct the old ones at finer scales of blur. But the act of modeling itself is not outside the universe. Our ability to formalize, guess, prove, misjudge, and revise is just one of the processes happening inside the same grand reality that our models try to describe. This note explores a simple but demanding idea: there is a structural indistinguishability between certain mathematical objects and the physical processes we use them to model, once we fix a blur scale. In that framework, poles and blow-ups are not light formalities. They mark points where the model tries to jump rank: to leave the universe in which it was formulated. We develop an epistemic and mathematical understanding of poles, using three ingredients: a primitive Blank object, an explicit blur structure on the space of states, and the principle of indistinguishability. We show how poles can be seen as moments where the cost of information needed to follow the model diverges, and where any naive attempt to “continue the solution” forces us to change the universe itself. 1 Modeling inside the universe When we build models in science, we implicitly say: “Here is the world, and here is a simplified picture of it.” We separate equations from reality, usually with a clear hierarchy: reality is primary, the model is secondary and corrigible. We know that our models are incomplete and we welcome refinements: Newton is excellent at a certain blur scale, Einstein is better at a finer one, and so on. Yet the activity of modeling is itself part of the universe. Our neurons, experiments, blackboards, simulations, the whole machinery of conjecture and refutation, all live in the same physical reality we try to describe. If one is allowed a poetic sentence: The universe uses us as one of the ways to express its regularities. In this sense: •our capacity to model is a process in the universe; •our models are objects we discover, construct, or recognize within the universe; • our false starts are also part of the universe: for one stable planetary system to form, there are countless unstable trajectories that never become systems. Epistemology, philosophy, and even badly wrong theories are not outside the world they talk about; they are episodes within it. The point of this note is not metaphysical; it is operational. If modeling is an internal process, then there is a natural question: 1 Which mathematical structures are indistinguishable—at a given blur level—from the physical processes that realize or test them? To approach this, we first introduce a minimal language of Blank and blur. 2 Blank, blur, and the principle of indistinguishability 2.1 The Blank object We start with an extreme abstraction. Definition 2.1 (Blank).The Blank object, denoted □ , is a conceptual placeholder with zero extractable information. It is not an element of any given set; it has no specified properties, no coordinates, no internal labels. We cannot even assert that □ “exists” or “does not exist” without already assigning a bit of information to it. Formally, one may think of □ as a symbol outside all the structures we manipulate. It is a reminder that before we choose a universe, a language, or a blur scale, there is nothing we are able to say. Any statement about □already presupposes that we have exited it. In practice, we never work directly with □. We work with blur layered on top of it. 2.2 Blur as the outer layer of Blank To examine anything, we must project some structure onto it. We choose a space of states U and a way to measure distances, norms, energies, or probabilities. This is a small but crucial investment of information. Definition 2.2 (Blur structure).Let U be a space of states (e.g. a function space, a finitedimensional phase space, a configuration space). A blur structure on Uis a family {Bℓ:U→Uℓ}ℓ>0 of maps indexed by a resolution parameter ℓ , where Uℓ is a “blurred” version of U , together with a family of seminorms or metrics on Uℓmeasuring differences at scale ℓ. In applications, Bℓmay be: •spatial convolution with a mollifier of width ℓ; •projection onto low Fourier modes up to frequency 1/ℓ; •coarse-graining into cells of size ℓin phase space; •or any other operation that systematically forgets structure at scales < ℓ. The key point is that: •blur is not an approximation error in the usual sense; • blur is an epistemic boundary: it marks what can be resolved, given finite resources, and what must remain behind a veil of uncertainty. 2 2.3 Models as projections from Blank To “examine blur,” we must commit to a handful of modeling choices: •a universe of discourse U(e.g. continuous functions, L2, smooth manifolds); •an evolution law F(a differential equation, a map, a stochastic rule); •an observational blur Bℓand a family of observables. These choices are small compared to the total complexity of the universe itself, but they are not cost-free. As the Riemann zeta story reminds us, some phenomena are about the modeling choices themselves: choosing the “most perfect” multiplication structure on the integers, without explicitly registering that this choice has consequences. We also carry our history. Previous theories, proofs, and failures are engraved into our scientific culture and into our habits of thought. Present time is the thin interface between this accumulated structure and the next modeling move we are about to make. Hypothesizing is, in a sense, our ability to look deeper into blur than the already extracted information strictly allows: we project an anticipated structure onto future data, in the hope that the universe will cooperate. 2.4 The principle of indistinguishability We now state the central idea informally. [Principle of indistinguishability] At a fixed blur scale, the physically accessible behavior of a system cannot be distinguished from that of any proper mathematical object that predicts future observables within the accepted error. Here “proper mathematical object” means: a model whose predictions at scale ℓ are as good as we currently know how to produce, or as good as we require for a given task. Newtonian mechanics is a proper object for everyday speeds and distances; general relativity is a more refined proper object for GPS satellites; quantum field theory is another one at a different blur regime. Formally, fix a blur structure { B ℓ} and a class of observables Oℓ that depend only on B ℓu for some state u . Then two models M1, M2 are indistinguishable at scale ℓ if all observables in Oℓ agree up to the accepted error tolerance. In that sense: physical reality at blur ℓ↔an equivalence class of proper mathematical models. The principle does not say that mathematics and physics are literally the same object. It says that, at each blur scale, there is a zone where they are operationally indistinguishable. We now give two examples in this language. 3 Two examples of indistinguishability 3.1 Quantum randomness and Blank Experiments suggest that, at the deepest currently accessible level, certain quantum events behave as if they were genuinely random: there is no finer hidden-variable model that predicts individual outcomes without violating other constraints. One way to phrase this in our language is: • at the deepest quantum blur layer we can probe, the effective behavior is indistinguishable from draws from a primitive Blank-like source; 3 • on top of this Blank, a layer of modeling was “chosen” at some early epoch of the universe, selecting certain regularities (symmetries, conservation laws, field content); • our mathematics tries to imitate this choice, using our own much smaller cognitive and experimental resources. We can imagine, without contradiction, a pre-material universe exploring an enormous “space of models” and stabilizing on those that are dynamically durable. Our theories are then miniature recreations of these mini-universes: we pick an action functional, a symmetry group, a space of fields, and we see whether the resulting world is coherent. From the viewpoint of indistinguishability: • the quantum Blank and our mathematical uncertainty principles are operationally the same type of object: regions where no further information can be extracted without breaking the framework; • we cannot know whether these blanks are “really the same” or whether one is a shadow of the other; the question is beyond the blur level at which we operate. 3.2 Blur, history, and reconstruction on the fly The second example concerns how blur interacts with the history of a system. In quantum mechanics, the measurement of a distant star is not a passive reading of a frozen history. Photon, star, detector, and observer form an extended process in which the “path” of the photon is, in some interpretations, reconstructed at the moment of detection, consistent with all constraints. Even in classical chaotic systems, blur behaves in a similar way: •looking at a fractal object under a coarse blur, we see a simple shape; •as we reduce blur, new structure appears that was invisible before; • there is no guarantee that the coarse picture contains any simple encoded hint of what the finer picture will be. One can describe this as follows. A system with two blur scales ℓ1> ℓ2 may have two trajectories t7→ u(1)(t), t 7→ u(2)(t), such that Bℓ1u(1)(t)=Bℓ1u(2)(t)for all t∈[0, T], but B ℓ2u(1) ( t )and B ℓ2u(2) ( t )diverge dramatically for t>T . From the coarse perspective, the two histories are indistinguishable up to time T ; only the future reveals which branch we are on. Thus we cannot distinguish between: •a world in which history is fixed once and for all, and we slowly uncover it; • and a world in which history is reconstructed on the fly, each time we refine the blur, as long as the reconstruction matches what has already been observed at coarser scales. Both are compatible with the same blurred data. The principle of indistinguishability does not force us to pick one; it simply tells us that our choice is a modeling preference, not an experimentally mandated fact. We now turn to one of the most classical mathematical objects that sits exactly at this interface between model and universe: the pole. 4 4 Universes, blow-up, and poles 4.1 Universes as modeling environments We formalize a minimal notion of “universe” for a model. Definition 4.1 (Universe).Auniverse is a triple (U, T,E)where •Uis a set of states (typically a Banach or Hilbert space); •T is a family of trajectories u : [0 , T ) →U allowed by some evolution law (ODE, PDE, map); •Eis a family of observables on U(norms, energies, functionals, blur structures). We think of ( U, T,E )as the environment in which a model lives and can be meaningfully interpreted. A universe comes with an implicit commitment: if we speak about solutions, convergence, stability, blow-up, and so on, we do so with respect to this chosen Uand its observables. Example 4.2. Take U = C∞ ( R ), the smooth real-valued functions, with the usual Frechét topology, and T the trajectories of a PDE like the heat equation. Or take U = L2 ( R3 )with T the Leray solutions to Navier–Stokes. Or U = Rn with T the solutions of a finite-dimensional ODE. 4.2 Blow-up and poles in a universe Inside a fixed universe, blow-up is usually defined via divergence of a norm or observable. Definition 4.3 (Blow-up in a universe).Let ( U, T,E )be a universe and let u : [0 , T ) →U be a trajectory. We say that ublows up at time Tin the observable Φ∈ E if lim sup t↑T Φ(u(t)) = +∞. If E contains a distinguished norm ∥·∥U , we may simply say that u blows up at time T in U if lim supt↑T∥u(t)∥U= +∞. At the level of individual formulas, this is the phenomenon of poles. Example 4.4 (Elementary poles). • The function f ( x ) = 1 x has a pole at x = 0 in the universe U = C0 ( R )with the sup norm on compact sets: ∥f∥[−ε,ε]diverges as ε→0. • The solution of the ODE y′ = y2 , y (0) = 1, given by y ( t ) = 1 1−t , has a pole at t = 1 in the universe U=C0([0,1)): the sup norm on [0, t]diverges as t↑1. Classically, we are trained to say “ x = 0 is a pole” and move on. But from the universes point of view, something deeper happens: • in U = C0 ( R ), f ( x ) = 1 /x is not even an element; we must either remove a point or change U; •in U=C0([0,1)), the trajectory t7→ 1/(1 −t)simply does not extend beyond t= 1. In both cases, the model is trying to leave the universe it started in. 5 4.3 Extending the universe and rank jumps A standard response is to enlarge the universe. Example 4.5 (Riemann sphere).The function f ( z ) = 1 /z is not holomorphic at z = 0 as a map C→C . But if we extend the universe to the Riemann sphere b C = C∪ {∞} , with the usual complex-analytic structure, then f becomes a meromorphic function with a simple pole at 0and a well-defined value f(0) = ∞in the extended sense. This suggests the following language. Definition 4.6 (Extension and rank jump).Let ( U, T,E )and ( e U, e T,e E )be two universes with U⊂e U . A trajectory u : [0 , T ) →U has a rank jump at time T if there exists an extension e u: [0,e T)→e Uwith e T > T such that: •e u(t)=u(t)for all t<T; •there is no extension u′: [0,e T)→Uwith this property. A pole is then a symptom of an attempted rank jump: the model is attempting to continue in a larger universe, but our current universe Ucannot accommodate this continuation. Remark 4.7. In complex analysis, passing from C to b C tames the pole by a one-point compactification. In PDE, passing from a classical function space to a space of distributions or measures can play a similar role. But the price is always the same: we change the rules of the universe. 5 Blur and the epistemic cost of approaching a pole We now bring back blur and information. 5.1 Blurred universes and information budgets Let ( U, T,E )be a universe, together with a blur structure { B ℓ : U→Uℓ}ℓ>0 , where each Uℓ is equipped with a metric dℓ. For each trajectory u∈ T we can consider its blurred avatars uℓ(t):=Bℓu(t)∈Uℓ. To quantify information, we do not need a detailed Shannon theory; a simple covering notion suffices. Definition 5.1 (Information budget at scale ℓ ).Fix a time interval [0 , T ]and a tolerance ε > 0. The information budget Iℓ ( ε, T )for a family of trajectories F ⊂ T is the minimal integer N such that there exist trajectories v1, . . . , vN∈ F with the property: ∀u∈ F ∃j∈ {1, . . . , N}: sup 0≤t≤T dℓBℓu(t),Bℓvj(t)≤ε. Intuitively, Iℓ ( ε, T )is the minimal number of prototype blurred histories one needs to store in order to represent all trajectories in F up to accuracy ε on [0 , T ]. It is a very coarse measure of complexity, but it is sufficient for our purposes. 6 5.2 Epistemic blow-up We can now distinguish two different notions of blow-up. Definition 5.2 (Mathematical vs. epistemic blow-up).Let u: [0, T)→Ube a trajectory. • We say that u has a mathematical blow-up at time T if there exists an observable Φ ∈ E such that lim supt↑TΦ(u(t)) = +∞. • We say that u has an epistemic blow-up at time T if, for any sequence of blur scales ℓn↓ 0, the information budget Iℓn ( ε, Tn )required to distinguish u from nearby trajectories on intervals [0, Tn]with Tn↑Tdiverges as n→ ∞, even for fixed tolerance ε>0. Mathematical blow-up is about divergence of a norm. Epistemic blow-up is about divergence of the cost of tracking the trajectory at finer and finer blur. In many tame systems, these two notions coincide: if a norm explodes, then an enormous amount of small-scale structure is required to keep following the trajectory in any reasonable space. But there are also situations where the mathematical blow-up is essentially a choice of universe: by choosing a better compactification, we can keep the epistemic cost under control. 5.3 Poles as attempts to cross an uncertainty barrier We can now reinterpret poles. Definition 5.3 (Pole as an uncertainty barrier).Let ( U, T,E,{ B ℓ} )be a blurred universe and u∈ T a trajectory with a mathematical blow-up at time T . We say that u has a pole at T (in the strong epistemic sense) if: 1. uhas a rank jump at Tfrom Uto some extended universe e U; 2. the information budget to track uup to Tat scales ℓ→0diverges: ∀ε > 0,lim ℓ↓0Iℓ(ε, Tℓ)=+∞ for any Tℓ↑T. In words: • a pole is not just a large value of a function; it is a point where the model asks for infinite resolution and infinite information to stay inside the same universe; •to continue beyond it, we must either: –accept that the model stops at T, or –enlarge the universe so that ucan be continued as a different kind of object. This captures the intuition that a pole is a place where we try to jump over the uncertainty principle of our current modeling environment: we push the universe to a point where its own rules can no longer resolve the requested behavior with finite information. 5.4 Toy examples revisited We revisit Example 4.4 with this language. 7 Example 5.4 (The function 1 /x ).Consider the universe U = C0 ([ − 1 , 1] \ { 0 } ), endowed with the sup norm on compact sets away from zero, and trajectories u ( t, x ) = x with trivial time evolution. The function f(x)=1/x has a pole at x= 0 in the usual sense. Now impose a blur structure given by convolution with a mollifier ϕℓ supported in [ −ℓ, ℓ ], and set (Bℓf)(x) = (ϕℓ∗f)(x). For fixed ℓ>0,Bℓfis bounded (indeed smooth), and the blurred trajectory t7→ Bℓf has no blow-up. However, as ℓ↓ 0, the L∞ norm of B ℓf diverges, and the number of bits needed to encode B ℓf on [ − 1 , 1] to within tolerance ε also diverges: more and more of the profile is spent on approximating the singular spike near 0. In this simplest case, the rank jump is the passage from U to an extended universe e U that allows the value f (0) = ∞ , e.g. a space of functions with values in the extended real line R=R∪ {∞}. The epistemic blow-up is the divergence of the information cost as ℓ→0. Example 5.5 (The ODE y′ = y2 ).Consider the universe U = C0 ([0 , 1)) with trajectories given by solutions of y′=y2,y(0) = 1, namely y(t)=1/(1 −t). At t= 1 we have a pole. If we blur in time with a kernel of width ℓ, (Bℓy)(t) = Z1 0 ψℓ(t−s)y(s)ds, then for each fixed ℓ > 0,B ℓy is bounded and smooth on [0 , 1]. However, as ℓ↓ 0, the information budget to approximate B ℓy on [0 , 1) increases without bound: the peak near t = 1 becomes sharper and higher. Here the rank jump would be to a universe where we extend the time axis beyond t = 1 by reparametrizing or compactifying time, or by treating y as living in an extended state space that allows infinite values. The pole is the place where the original universe can no longer carry the trajectory without infinite information. These examples are trivial, but they illustrate the general pattern: Poles are not merely large values; they are places where a fixed universe cannot encode the continuation of the model at finite information cost. 6 Poles and physical reality 6.1 No physical pole, only decreasing blur In physical systems, we never have access to infinite resolution. The only thing we can do is decrease blur: make ℓ smaller, improve our instruments, refine our theories. In that process, we see sharper and sharper features, but we never reach ℓ= 0. From the indistinguishability viewpoint: • in reality, there is no single moment where we reach a pole; there is only an increasing cost of describing behavior at smaller and smaller blur; • what we call a “physical pole” is really a regime in which the information budget grows so fast that it becomes practically or fundamentally impossible to track. Thus a pole in a model is a particularly compact way of writing: “If you insist on interpreting this system in this universe, and you insist on decreasing blur, then there is a time at which your information requirements diverge.” 8 6.2 Universes and compatibility with poles Different equations and models have different capacities for poles. Some systems, like certain ODEs, generate poles easily. Others, like the heat equation, tend to smooth out irregularities. The question then becomes: Is a given universe structurally compatible with a pole that carries a strong epistemic blow-up, or would such a pole necessarily indicate a mismatch between the model and the underlying physical process? For example: • an electromagnetic field driven to a genuine pole can, in principle, interact with very large regions of spacetime: it already carries long-range structure in its basic formulation; • the Navier–Stokes equations, on the other hand, look much more local and regular: they dissipate energy and smooth out small scales at fixed blur; it is not obvious whether they can support a pole that simultaneously preserves a coherent physical interpretation. We do not pursue the Navier–Stokes question here. The point is conceptual: before asking whether a specific equation has a pole, we should ask whether the universe of that equation is capable of supporting a rank jump of the required kind. 7 Summary and outlook We have proposed the following picture: • Modeling lives inside the universe; it is not an external activity. Our models, good and bad, are events in the same world they describe. • To speak at all, we must leave Blank and commit to a blur structure, a universe of states, and an evolution law. • The principle of indistinguishability says that, at a fixed blur level, physical processes and proper mathematical models that predict the same observables are operationally the same object. • A pole is not a light formal artifact. It is the visible part of a deeper phenomenon: a rank jump between universes, where the information cost of tracking the trajectory, under decreasing blur, diverges. • There is no physical pole in the strict sense; there is only increasingly expensive structure as we push blur towards zero. To insist on an actual pole is to insist on leaving the current universe of discourse. This framework suggests a program: 1. Classify universes by their compatibility with epistemic blow-ups: in which spaces and with which blur structures can strong poles exist? 2. For important equations (Navier–Stokes, Einstein, nonlinear Schrödinger, etc.), study whether genuine poles would force rank jumps that break the intended physical interpretation. 3. Use the information budget perspective to distinguish: •poles that can be tamed by a natural enlargement of the universe, from 9 We take as universe U=L2 σ(T3),T={Leray–Hopf trajectories on [0, T)},E={∥u(t)∥L2,∥∇u(t)∥L2,...}. Let { B ℓ}ℓ>0 be a blur structure given by convolution with a standard mollifier or heat kernel at spatial scale ℓ, as in the Navier–Stokes blur paper [1]. We write uℓ(t, x) := (Bℓu(t, ·))(x), Eℓ(t) := 1 2ZT3|uℓ(t, x)|2dx. Thus Φℓ(t):=Eℓ(t)is the blurred kinetic energy observable at scale ℓ. Applying B ℓ to the Navier–Stokes equations and testing with uℓ leads to the filtered energy inequality (see [1]) d dtEℓ(t)+νZT3|∇uℓ(t, x)|2dx ≤ZT3Rℓ(t, x) : ∇uℓ(t, x)dx, (11.2) where Rℓ=Bℓ(u⊗u)−uℓ⊗uℓ is the Reynolds stress at scale ℓ . Inequality (11.2) is the PDE analogue of the abstract blur–knob inequality (9.1) in Section 9, with Φℓ(t)=Eℓ(t), rℓ(t)≃ZT3Rℓ:∇uℓ. 11.2 A model Navier–Stokes blur–knob inequality The detailed analysis in [ 1 ] shows that, for each fixed ℓ > 0, one can package the ratio between advective transfer across scale ℓ and viscous dissipation at that scale into a dimensionless Navier–Stokes blur–knob ΛNS(ℓ, t)of the schematic form ΛNS(ℓ, t)≈Uℓ(t)ℓ ν | {z } local Reynolds number + Θ(ℓ, geometry) | {z } alignment, sign coherence +time-blur defect τ | {z } nonlocal memory ,(11.3) where Uℓ ( t )is a characteristic velocity at scale ℓ and Θ( ℓ, · )collects the geometric knobs (vorticity alignment, persistence loss, etc.) defined in [ 1 ]. The precise definition of Λ NS is not important here; what matters is that the filtered energy inequality can be rewritten in the blur–knob template d dtEℓ(t)≤ΛNS(ℓ, t)Eℓ(t)+rℓ(t),(11.4) where rℓ ( t ) ≥ 0can be estimated in terms of lower-order or explicitly integrable quantities (for instance, the tail of the energy spectrum or a controlled Reynolds-stress budget). Formally, (11.4) is exactly of the same shape as the ODE blur–knob inequality (8.5) , with kℓ replaced by ΛNS and Φℓreplaced by the blurred energy Eℓ. 11.3 A conjectural Navier–Stokes no-pole condition In this language, the absence of epistemic poles for Navier–Stokes in its physical universe can be expressed as a blur–subcriticality conjecture. Conjecture 11.1 (Navier–Stokes blur–subcriticality).Let u be a Leray–Hopf solution of the three-dimensional incompressible Navier–Stokes equations on [0 , T )with finite initial energy. Then there exists a choice of blurred energy observable Eℓ ( t )and a Navier–Stokes blur–knob family (ΛNS(ℓ, t), rℓ(t)) satisfying (11.4)such that for every finite horizon T∗< T : 16 1. there is an integrable function K∈L1([0, T∗]) with ZT∗ 0 ΛNS(ℓ, t)dt ≤ZT∗ 0 K(t)dt for all 0< ℓ ≤ℓ0; 2. there is an integrable function R∈L1([0, T∗]) with ZT∗ 0 rℓ(t)dt ≤ZT∗ 0 R(t)dt for all 0< ℓ ≤ℓ0. In other words: the blur regime is subcritical for the blurred energy at every finite time horizon. If Conjecture 11.1 holds, then the general machinery of Section 9implies: • for each finite T∗< T , the blurred energies Eℓ ( t )remain uniformly bounded on [0 , T∗ ]as ℓ↓0; • the information budget Iℓ ( ε, T∗ )needed to represent all blurred trajectories up to time T∗ with tolerance ε > 0stays bounded as ℓ↓0; • consequently, no trajectory can develop an epistemic pole at any finite time: every mathematical blow-up would have to be an artefact of an inappropriate universe or observable, removable by a better compactification or reparametrization. Conversely, any genuine Navier–Stokes pole in the sense of Definition 5.3—a trajectory for which the information budget diverges as ℓ↓ 0and t↑T —would force a supercritical blur regime in the sense of Section 9: for every reasonable choice of blur–knobs, there would exist a sequence ℓn↓0and times Tn↑Twith lim n→∞ ZTn 0 ΛNS(ℓn, t)dt = +∞. The Clay question about Navier–Stokes regularity is then mirrored by an epistemic question about whether such a supercritical blur regime is compatible with the physical universe of incompressible fluid flows at all. Remark 11.2 (Epistemic Clay question).Conjecture 11.1 is deliberately weaker than a full proof of global smoothness: it only asserts that, at every finite blur and every finite horizon, one can organize the dynamics into a subcritical blur–knob regime with finite information budget. If this conjecture is true, then from the point of view of this paper there is no epistemically meaningful Navier–Stokes pole; any mathematical singularity would be a modelling choice, not a feature of the physical universe. If it is false, then any counterexample would provide a concrete instance of an epistemic pole in the sense of Section 5: a trajectory whose behaviour forces a rank jump beyond the intended fluid universe. In this way the blur–knob language closes the circle between the abstract pole philosophy of this note, the fractal and Navier–Stokes blur pictures in [ 2 , 1 ], and the classical PDE formulations of fluid regularity: the missing ingredient is precisely to locate Navier–Stokes on the subcritical–supercritical boundary of its own blurred universe. References [1] A. Perišić, Navier–Stokes, Blur, and Knobs, Zenodo, 2025. [2] A. Perišić, Fractal Dimension as a Mellin Slope under Positive Blur, Zenodo, 2025. [3] A. Perišić, Navier–Stokes, Blur, and Blurrichevsky Geometry, Zenodo, 2025. 17