PRH | Essay | 7.22 • The Blur–Equivalence Principle

The Blur–Equivalence Principle Invariant Blow-Up Under Blur-Compatible Descriptions Aleksandar Perišić December 2025 Abstract We formulate a general blur–equivalence principle for mathematical models. Once a finite blur budget is fixed, all blur–compatible descriptions of the same universe are equivalent with respect to blur–local properties, including finite–time blow-up and the need to pass to a larger “universe” of objects (poles). The formal setting is that of blurred universes, where each state space is equipped with a family of blur operators and a notion of blur–local observables. A morphism between such universes is admissible if it commutes with blur up to the prescribed budget. We show that blur–local, stable and monotone properties are invariant along any chain of admissible descriptions. From this we deduce that, at fixed blur budget, events such as blow-up, the creation of a pole, or the existence of a Lyapunov–type certificate are intrinsic to the blur universe rather than artefacts of a particular coordinate system or parametrization. Either every blur–compatible description sees the event, or none of them do. The principle is formulated abstractly and proved using a minimal fragment of the general category of blur. Concrete applications — to evolution equations, analytic continuation, randomness, and complexity — are developed elsewhere; here we concentrate on the structural statement itself, with only schematic illustrations via poles and certificates. 1 Introduction The standard formulation of mathematical physics and analysis assumes perfect resolution: we write equations on a continuum, impose sharp initial data, and ask whether solutions exist smoothly for all time. Physically, however, we never observe the continuum itself; we only see blurred versions of it, constrained by finite measurement, finite energy, and fundamental uncertainty. The blur program seeks to make this scale–dependence explicit and to build a calculus of blur that interacts sensibly with dynamics, probability, and logic [1,2,3]. In previous work this machinery was instantiated in a variety of contexts: evolution equations with guard– and certificate–type functionals, analytic continuation and poles, randomness and information distance to a blank state, and the broader “no free information” standpoint [ 4 , 5 , 7 ]. What was missing was an explicit principle tying these strands together: once a blur budget is fixed, can a clever change of variables move blow-up from one description to another, or is the verdict itself invariant? The aim of this note is to pin down a clean answer in an abstract setting. We formulate a blur–equivalence principle: Informal principle. Fix a blur budget (resolution, time window, error tolerance). Within that budget, any two blur–compatible descriptions of the same universe are equivalent with respect to blur–local questions such as: •Does a certain observable stay bounded? •Does a relevant norm blow up before time T? 1 •Does the description force a “universe jump” (pole)? •Does there exist a Lyapunov–type certificate with given drift? In particular, either all blur–compatible descriptions stay inside one universe, or all of them demand a universe change, but not both. Put differently: once we accept that we only know a system up to some blur radius, it no longer makes sense to hope that one coordinate system blows up while another does not, unless the two systems are not blur–compatible. At a fixed blur budget there is exactly one answer: either the blow-up is a real object (a universe–level pole), in which case every blur–compatible description sees it, or it is a fake artefact that can be ironed out by working at slightly coarser blur or by refining the universe itself. The core of the paper is a structural theorem that makes this principle precise inside the category of blurred universes. We show that blur–local, stable and monotone properties of trajectories are invariant under blur–equivalence. The proof uses only a minimal subset of the general “category of blur” framework developed in [2,1]. Structure of the paper In Section 2 we define blur indices, blurred universes, and admissible descriptions. Section 3 introduces blur–local properties and states the blur–equivalence principle as a main theorem, with a specialisation to blow-up and poles. In Section 4 we give schematic illustrations via poles and Lyapunov–type certificates, without committing to any specific problem. Section 5 connects the principle to the “universes and poles” picture and to the no–free–information viewpoint. We end in Section 6 with an outlook on open directions. The exposition is self-contained on the level of definitions and main statements; proofs use only a minimal fragment of the general category of blur. 2 Blur indices, blurred universes, and admissible descriptions We begin by formalising blur at the level where we will need it: as a scale–indexed family of self–maps on a universe of states that act like controlled loss of information. 2.1 Blur indices Definition 2.1 (Blur index).Ablur index is a partially ordered commutative monoid ( I, ⪯,⊕, 0) with: •0a neutral element: ε⊕0 = ε; •monotonicity: if ε1⪯ε2then ε1⊕δ⪯ε2⊕δ; •Ihas a distinguished top element ∞(possibly formal) with ε⪯ ∞ and ε⊕ ∞ =∞. We think of ε∈I as a blur radius or blur budget, with 0perfectly sharp and ∞ completely blurred. Typical examples include I = [0 ,∞ ]with ⊕ = + or I = { 2 −k : k∈N}∪{ 0 } ordered by ≥ with ⊕ given by dyadic addition truncated at 0. The precise structure will not matter; only the compatibility with the blur operators is used. 2 2.2 Blurred universes Definition 2.2 (Blurred universe).Ablurred universe is a tuple U= (X, O,B) consisting of: •a state space X(set, topological space, Banach space, . . . ); • a class O of observables f : X→Yf into target spaces Yf (usually R or C , or vector– or function–valued); •for each ε∈Ia blur operator Bε:X→X(“blur at radius ε”), such that: (U1) Sharp baseline: B0= idX. (U2) Monotone blur: ε1⪯ε2 implies B ε2 factors through B ε1 , i.e. there is Rε2,ε1 : X→X with Bε2=Rε2,ε1◦Bε1. (U3) Subadditivity: B ε⊕δ factors through B ε◦ B δ and vice versa; in particular there are maps Sε,δ, Tε,δ :X→Xwith Bε⊕δ=Sε,δ ◦Bε◦Bδ,Bε◦Bδ=Tε,δ ◦Bε⊕δ. (U4) Observable compatibility: For each f∈ O and each ε there is a blurred observable fε∈ O such that f◦Bεand fε◦Bεcoincide up to the observational tolerance at blur ε. We denote by [x]ε:= Bε(x)the ε–blur avatar of x∈X. The precise form of the maps Rε2,ε1 and Sε,δ is not important; intuitively they express that we can blur in two steps or in one, without leaving the blur universe. In many examples B ε is convolution with a mollifier, or an operator such as (1 −∆)−s/2at scale ε. Remark 2.3 (Blur classes and information distance).Fix ε . One may define an equivalence relation x∼εy if B ε ( x ) = B ε ( y ), and view Xε := X/∼ε as the set of states indistinguishable at blur scale ε . The maps B ε : X→X then factor through the quotient X→Xε and a canonical section. The information distance to the blank state □ can be modelled as a decreasing function of ε, as in [4]. We will not need an explicit metric here. 2.3 Admissible descriptions We think of different mathematical descriptions of the same physical or abstract system as different blurred universes, connected by blur–compatible maps. We now formalise what it means for such a map to be admissible. Definition 2.4 (Admissible description).Let U = ( X, O, B)and V = ( Y, O′, B ′ )be blurred universes over the same blur index I . A map F : X→Y is an admissible description (or blur–compatible description) if: (D1) Observable preservation: For every g∈ O′ there is an f∈ O such that g◦F and f agree on all blur classes accessible within the budget under consideration. (D2) Blur compatibility: For every ε∈Ithere is a controlled blur radius ε′⪰εsuch that F◦Bε∼ε′B′ ε′◦F, (2.1) where ∼ε′denotes equality of all observables in O′at blur ε′. 3 If there exists also G : Y→X admissible with G◦F∼εid and F◦G∼εid for all ε in the budget, we say Uand Vare blur–equivalent universes and write U ≈blur V. Intuitively, (D1) says that every observable in the target description can be read as a blur– distorted version of some observable in the source description, and (D2) says that blurring before or after F has the same observational content up to the allowed budget. In particular, we do not allow Fto sharpen blur without paying for it elsewhere. Example 2.5 (Sharp vs filtered description).Let X be a space of “sharp” states for some system, and let B ε : X→X be a blur operator at scale ε . Consider two blurred universes over the same index set: •Uwith state space Xand blur operators (Bε); •V with the same underlying set Y = X , but interpreted so that all states are already blurred at some fixed scale ε0>0. Let F : X→Y be F ( x )=B ε0 ( x ). With the obvious choice of observables (defined by pulling back along F ), the map F is an admissible description: it preserves what can be observed at or above scale ε0 , and it commutes with further blurring up to the natural reparametrisation of the blur index. 3 Blur-local properties and the blur–equivalence principle We now describe the class of properties that are invariant under blur–compatible descriptions, and state the main principle. 3.1 Blur-local, stable, and monotone properties We fix a blurred universe U = ( X, O, B)and a time horizon T > 0, and consider trajectories on [0, T]valued in X. Definition 3.1 (Blur-local property).Let Γbe a collection of trajectories u : [0 , T ] →X . A predicate P : Γ → {true,false} is called blur–local (up to budget ε∗∈I ) if there exists ε∗ such that: •For any two trajectories u, v with Bε(u(t)) = Bε(v(t)) for all t∈[0, T], ε ⪯ε∗, we have P(u)=P(v). In words: Pdepends only on the blur classes [u(t)]εfor ε⪯ε∗and t∈[0, T]. Typical examples include: “a blurred energy at scale ℓ remains bounded by M on [0 , T ]”, or “a family of guard inequalities holds for all accessible blur scales”, or “no pole is visible in a Laplace–tilted observable up to time Tbeyond rate s∗”. Definition 3.2 (Stable and monotone properties).A blur–local property Pis: • stable under coarse–graining if whenever B ε′ ( u ) ≡ B ε′ ( v )for some ε′⪰ε∗ and P ( u )holds, then P(v)holds for all vwhose blur–local data dominate that of u; •monotone in blur if there is a direction ⪯such that Pis preserved when we replace Bεby Bε′with ε′⪰ε. 4 In practice, we think of P as encoding either a “tame” condition (e.g. Lyapunov decay, absence of poles) that is preserved under additional blur, or a “wild” condition (e.g. existence of a pole, inverse Lyapunov inequality) that persists at nearby scales. The precise form of stability and monotonicity can be adapted to the context. The only properties we will consider satisfy a natural comparison relation: if U ≈blur V and u and v correspond under this equivalence, then the truth of P should be determined by the common blur data. 3.2 The blur–equivalence principle We now state the main principle in a clean form. It is a direct consequence of the general “Grand Lemma” for blur categories in [ 2 ], specialised to properties defined on trajectories and to blur–compatible descriptions. Main Theorem 3.3 (Blur–equivalence principle).Let U = ( X, O, B)and V = ( Y, O′, B ′ )be blurred universes over the same blur index I , and suppose U ≈blur V via admissible descriptions F : X→Y and G : Y→X as in Definition 2.4. Let Γ X and Γ Y be collections of trajectories on [0 , T ]valued in X and Y , respectively, such that F and G induce correspondences between ΓXand ΓYup to blur. Let P be a blur–local, stable and monotone property on Γ X , and let P′ be the induced property on Γ Y obtained by transporting P along F (i.e. P′ ( v ) := P ( Gv )). Then, for every corresponding pair of trajectories u∈ΓXand v∈ΓY, P(u)⇐⇒ P′(v). In particular, the truth value of Pis an invariant of the blur–equivalence class of the universe. Idea of proof. The blur–locality of P ensures that it depends only on the blur data { B ε ( u ( t )) : ε⪯ε∗, t ∈ [0 , T ] } . Admissibility of F and G implies that the blur data for u and F ( u )are equivalent up to the same budget, and similarly for v and G ( v ). The stability and monotonicity assumptions guarantee that intermediate adjustments of the blur radius do not change the truth of P . The Grand Lemma in [ 2 ] formalises this by viewing ( U,V )as objects in a category of blurred universes and F, G as blur–compatible morphisms; the property P then becomes a natural transformation that must agree along equivalent objects. Specialising that general result to trajectories and our definition of admissible description gives the claim. Conceptually, Theorem 3.3 says that within a fixed blur universe, properties that only care about what is visible at that blur — and that do not change unpredictably under small variations of blur — are immune to changes of description, as long as these descriptions respect the blur structure. 3.3 Blow-up, poles, and universe jumps We now specialise to the kind of properties that motivated this work: regularity versus blow-up and the need to enlarge the universe of discourse. Definition 3.4 (Blow-up / pole properties).Let U = ( X, O, B)be a blurred universe and let u : [0 , T ) →X be a trajectory that solves a given evolution or iteration rule in the sharp limit. We say that: • finite–time blow-up at T occurs within U if there exists a blur–local norm or guard functional L(e.g. critical Sobolev norm, guard energy) such that for some M > 0and every ε∗there is ε⪯ε∗with sup t<T L(Bεu(t)) > M while L(Bεu(t)) <∞for all t < T; 5 • a pole at T in U occurs if the blow-up is not removable by any blur–compatible extension of U , i.e. any attempt to extend u ( t )beyond T within U at the same blur leads to divergence of some blur–local observable. We say ustays inside the universe up to Tif no such pole occurs. In the language of [ 7 ], a pole forces a universe jump, i.e. one must pass to a larger universe e U that contains new states and observables. We now state the blur–equivalence principle in this context. Corollary 3.5 (No hybrid blow-up under blur–equivalence).Let U ≈blur V be blur–equivalent universes connected by admissible descriptions F and G , and let u and v be corresponding trajectories solving the same underlying evolution in the two descriptions. Fix a blur budget ε∗ and a time horizon T > 0. Then, either both u in U and v in V remain inside their respective universes up to time T , or both develop a pole at T (possibly reflected through F and G ), but there is no hybrid scenario in which one description sees a pole while the other does not, as long as both are blur–compatible. Proof. Take as P the property “no pole before time T at blur budget ε∗ ”; this is blur–local (it only looks at blur–local norms), stable (small additional blur cannot suddenly create a pole that was not there), and monotone in εin the natural direction. Apply Theorem 3.3. Remark 3.6 (Interpretation).Corollary 3.5 formalises the intuition that once we fix what we mean by “knowing a system up to blur ε∗ ”, the blow-up versus regularity verdict is no longer negotiable by changing coordinates or introducing additional variables, unless these break blur–compatibility. In particular, for any evolution system that admits several blur–compatible presentations (sharp vs filtered, different coordinate systems, auxiliary variables, . . . ), the question “does a pole occur at time T within the given blur budget?” has a single answer across all such presentations. 4 Schematic illustrations: poles and certificates The previous section stated the blur–equivalence principle in full generality. We now give two schematic illustrations, keeping the discussion free of any specific problem. The point is simply to show that the kinds of events one typically cares about — poles and Lyapunov–type certificates — fit the hypotheses of the principle. 4.1 Poles as blur-local events Poles are already encoded in Corollary 3.5, but it is helpful to spell out how little structure is actually needed. Fix a blurred universe U = ( X, O, B)over a blur index I and a time horizon T > 0. Suppose we have chosen a family of blur–local norms or guard functionals Lα indexed by some set A (for instance, critical norms, Laplace–tilted energies, or other observables detecting loss of control). For each choice of threshold M > 0and index α∈Awe can define a property Pα,M (u) := hsup t<T Lα(Bεu(t)) ≤Mfor all accessible εi. Each Pα,M is blur–local, and the failure of all such properties for all M and α is precisely what we called a pole in the previous section. The existence or nonexistence of such a pole is therefore a blur–local, stable and monotone property, and Corollary 3.5 applies directly. In particular, whenever two descriptions of the same system are connected by admissible maps, the decision “pole vs no pole” cannot be changed by rewriting the equations, as long as the blur budget and the class of admissible observables are kept fixed. 6 4.2 Certificates and monotone drift Many arguments in dynamics and probability rely on certificates: functionals that provide a monotone drift in some direction. We now show that the existence of such a certificate is itself a blur–local, stable and monotone property, and hence invariant under blur–equivalence. Definition 4.1 (Certificate of drift).Let U = ( X, O, B)be a blurred universe and Γa class of trajectories on [0 , T ]. A certificate of drift at blur budget ε∗ is a pair (Φ , ρ )with Φ : X→R an observable and ρ>0such that: •Φis blur–local up to ε∗(it only depends on [x]εfor ε⪯ε∗); •for every trajectory u∈Γand all 0≤s<t≤T, Φ(Bεu(t)) ≤Φ(Bεu(s)) −ρ(t−s)for all ε⪯ε∗(4.1) whenever the left-hand side is defined. We think of Φas a Lyapunov–type functional providing a uniform negative drift along the trajectory, visible at blur scales up to ε∗. Proposition 4.2 (Invariance of certificates).Let U ≈blur V be blur–equivalent universes connected by admissible descriptions F and G as in Definition 2.4, and let Γ X, Γ Y be corresponding classes of trajectories. Then the following are equivalent: (i) there exists a certificate of drift (Φ, ρ)for (U,ΓX)at some blur budget ε∗; (ii) there exists a certificate of drift (Ψ , ρ′ )for ( V, Γ Y )at a (possibly different but controlled) blur budget. In particular, the existence of any certificate with nontrivial drift is a blur–equivalence invariant. Proof sketch. If (Φ , ρ )is a certificate on U , define Ψon Y by transporting Φalong G , using admissibility to ensure that Ψis blur–local up to a controlled budget. The drift inequality (4.1) is blur–local and monotone in ε , so it is preserved when we rewrite it in the V –description. Conversely, a certificate on V can be transported back to U . The details are the same as in Theorem 3.3, applied to the property “admits a certificate of drift”. Conceptually, Proposition 4.2 says that once we know a system admits a certificate of drift at a given blur budget, no blur–compatible reparametrisation can take that certificate away without paying elsewhere in the blur structure. Conversely, if no certificate exists in one description, none can exist in any blur–equivalent description. 5 Universes, poles, and no-free-information We briefly connect the blur–equivalence principle to the “universes and poles” picture developed in [7] and to the no–free–information viewpoint of [5]. 5.1 Universes and poles Auniverse U is, informally, a category of mathematical objects (functions, fields, trajectories) together with admissible operations and observables. A pole is an obstruction to extending a trajectory or observable inside U without leaving that category; it signals the need to pass to a larger universe e U. Blur refines this picture by inserting an explicit blur layer between the sharp universe and the observer. The blurred universe U associated to U keeps track only of what is visible at finite resolution. In this language, a pole is not just a blow-up of a norm, but a failure of all blur–compatible extensions inside U. 7 Proposition 5.1 (Universe-level invariance of poles).Let U and V be two universes modelling the same underlying system, and let U and V be their associated blurred universes over a fixed blur index. Suppose U ≈blur V . Then the existence of a pole at T (in the sense of Corollary 3.5) is a universe–level invariant: it does not depend on whether we work in U or V , as long as both are blur–compatible. Proof. A pole is detected by a blur–local blow-up property P in the blurred universe. By Theorem 3.3 and Corollary 3.5, P cannot have different truth values in U and V . Since U and V both project to these blurred universes, the existence of a pole is a property of the underlying system, not of the choice of universe. 5.2 Wild vs tame regimes In many applications one can partition blur–local configurations into tame and wild regimes, usually defined by the sign or size of some guard functional (for example, Lyapunov vs inverse– Lyapunov behaviour). A typical meta–statement then has the following form: • if a solution stays in the wild regime across a descending scale ladder, then some critical quantity blows up in finite time; • conversely, any globally regular solution must escape the wild regime infinitely often by returning to tame values of the guard. Such a statement is a form of no–perpetual–wildness inside the given blur universe. The blur–equivalence principle upgrades this from a coordinate–level statement to a universe– level one: once the blur universe and the tame/wild partition are fixed, perpetual wildness is either impossible in all blur–compatible descriptions, or it forces a blow-up in all of them. There is no description in which wild behaviour can be hidden or neutralised without changing the blur budget or the universe itself. This is conceptually aligned with the no–free–information viewpoint: creating “order from chaos” or suppressing wildness for free would amount to a violation of the blur budget, i.e. to smuggling information across the blur layer without paying for it. 6 Outlook We end by summarising what the blur–equivalence principle does and does not say, and by pointing to directions where it may be useful beyond the abstract setting of this note. What is now fixed Within the blur program, the principle pins down a key meta–question: • Once we fix a blur budget and a class of blur–compatible descriptions, questions about blow-up versus regularity, existence of poles, the need for universe jumps, and the existence of monotone certificates are no longer coordinate–dependent. Either the phenomenon happens in all such descriptions, or in none. This removes a degree of freedom from speculation: one can no longer hope that a singular behaviour appears in one convenient formulation but disappears in another, or that a change of variables will magically produce a certificate, unless that change of variables violates blur– compatibility (for example by sharpening information for free or smuggling in a larger universe). 8 What remains open The principle does not resolve any specific hard problem. It clarifies where the remaining work lies in each concrete setting: •identifying a natural blur universe and blur index for the system; • verifying that the property of interest (blow-up, pole, certificate, etc.) is blur–local, stable and monotone; •proving or disproving that property in one convenient blur–compatible description. The last step is where the classical, system–specific difficulty lives. The blur–equivalence principle says that once a single description settles the question inside the chosen blur universe, all blur–compatible descriptions must agree. Beyond specific systems Three directions where the blur–equivalence principle seems particularly promising are: • Discrete dynamics and certificates: blur allows one to formulate certificate–based criteria for termination or boundedness; the equivalence principle suggests that any certificate visible at finite blur is invariant under blur–compatible encodings of the dynamics. • Complexity and encodings: blur and no–free–information provide a language in which to ask whether the existence of certain kinds of efficient algorithms is invariant under admissible representations of problem instances. • Quantum/thermal universes: blur is naturally tied to decoherence and coarse–graining; the equivalence principle may clarify which phenomena are truly basis–independent at a given blur budget, and which depend on how we package the underlying degrees of freedom. The guiding message is that blur is not a nuisance to be eliminated, but a structural feature that forces different descriptions of the same system to agree on the deepest questions — including whether we must leave a universe, and whether there exists a certificate guiding us through it. Appendix A. Minimal axioms from the blur category For completeness we briefly recall the fragment of the general “blur category” used implicitly in this note; see [2,1] for a full development. Definition A.1 (Category of blur (fragment)).Objects are blurred universes U = ( X, O, B) over a fixed blur index I . A morphism F : U → V is an admissible description in the sense of Definition 2.4. Composition is given by composition of maps, and identities are given by idX with B′=Band O′=O. The key structural fact (Grand Lemma) states that blur–local constructions and properties that are stable and monotone lift to natural transformations on this category: they are invariant under isomorphisms and, more generally, under blur–equivalence. Lemma A.2 (Grand Lemma, schematic form).Let Blur be the category above, and let P be a construction assigning to each object U a set of blur–local, stable, monotone properties P ( U ) on trajectories in U , such that for each morphism F : U → V there is a natural transport map F∗:P(U)→ P(V)compatible with composition. Then for any blur–equivalence F:U → V, the transport maps F∗ and F−1 ∗ are inverses. In particular, the truth of P∈ P ( U )is invariant under blur–equivalence. 9