PRH | Inter | 5.11 • A Category of Blur and the Grand Lemma

A Category of Blur and the Grand Lemma Aleksandar Perišić August 2025 Abstract We axiomatize blur—a tunable neighbourhood/relaxation—as a categorical construction. A blurred object is a diagram BX : I→ C indexed by blur scales with a reading map ρX : limIBX→X (sharp limit) and exhaustion colimIBX≃ 1. An uncertainty window is a lax retract pair jε X : X→BεX , rε X : BεX→X . We prove a property-transport principle: for properties stable under retracts and filtered limits and monotone for ≾ ,P( X )holds iff P ε ( BεX )holds eventually. Blur-morphisms are natural transformations; the Grand Lemma (Blur–transport lemma) shows sending sharp equals sending blurred then reading, yielding a simple string-diagram calculus and a graded (co)monad structure for scale composition. Examples include analytic blur (Markov/convolution semigroups) and logical blur (quantaleenriched nuclei), linking the construction to domain-theoretic approximation and probabilistic powerdomains. We fix a base category C (e.g. Set , Top , or Meas ). Intuitively, objects are proposition spaces (or answer spaces), and a blur is a tunable neighbourhood/relaxation. Blur index and blurred objects Let ( I, ⪯,⊕, 0 ,∞ )be a directed monoidal poset of blur scales (think: ε∈ (0 ,∞ ], with 0= sharp, ∞ = “no response”), where ⊕ models composing blur (e.g. variance addition), 0is the neutral element, and ∞is absorbing (ε⊕ ∞ =∞). Definition 1 (Blurred object).Ablurred object is a pair X= ( X, BX )where X∈Ob ( C )and BX:I→ C is a functor (a filtered diagram) with: 1. Reading cone (sharp limit): a universal cone ρX : limIBX→X . We say the blur is faithful if ρXis an isomorphism (i.e. X≃limIBX). 2. Exhaustion at infinity: a cocone to a terminal/“no-info” object 1, i.e. colimIBX≃ 1 (“neighbourhood tends to no response”). Write BεXfor BX(ε)and rε X:BεX→Xfor the leg of the limit cone (reading at scale ε). Remark 2 (Relation to domain-theoretic approximation).If we regard I as a directed set of “resolutions”, a faithful blur has the flavour of domain theory: X is recovered as the limit of its approximants ( BεX ) ε∈I . The exhaustion condition plays the role of a bottom element, and the blur diagram is analogous to a directed system of finite approximations whose limit is the full object. The difference is that here the approximation is explicitly parametrized by a physical/epistemic scale ε. Remark 3. Concrete examples: •C = Meas , BεX = X equipped with a Gaussian jitter of variance ε (convolution on functions, or Giry-type randomization on points); 1 •C = Top , BεX the same set with a coarser uniformity/entourage (neighbourhood thickening); •C = Set , BεX the set of “ ε -consistent truth tables” for X with a collapse map to X as ε→0. Weak containment and the uncertainty window The point of blur is that an object is still present while travelling with a neighbourhood, but only in a relaxed sense. We capture this by a lax retract pair. Definition 4 (Uncertainty window / lax retract pair).For each scale ε∈I , we equip X with a pair of natural maps jε X:X−→ BεXand rε X:BεX−→ X, called the injection (thickening) and the reading (deblurring), such that rε X◦jε X= idXand jε X◦rε X≾idBεX. Here ≾ is a chosen ambient preorder on endomorphisms (e.g. pointwise ≤ in Set or Top when available, a.s. ≤ in Meas , or the enrichment order if C is Pos/quantale-enriched). Thus X is contained while blurred (exact on the sharp side, relaxed on the blurred side). We also require naturality in X and that jε X→idX and rε X→idX as ε→ 0(in the sense of your reading cone). Remark 5 (Both–sides neighbourhood and synchronized removal).The pair ( jε X, rε X )says: as long as the Proposition and the Answer carry matching neighbourhoods, the mechanism can pass them with controlled relaxation, and removing the neighbourhood on one side forces the corresponding removal on the other via naturality of ¯ f and the graded (co)monad maps Bε⊕δX→Bε(BδX). Remark 6 (Transmission principle (AC-analogy)).Think of ε as an impedance knob. Sending X through the network at scale ε uses jε X to load X into BεX , transports with ¯ fε , then reads with rε Y . Some invariants (mass, expectations, zero-mode, conserved charges) pass losslessly because ¯ fε and rε preserve them; sharp, fragile invariants are replaced by their relaxed versions while the signal is AC-carried (blurred). As ε↓ 0the impedance vanishes and the relaxation collapses back to sharp equality. Proposition 7 (Property transport under uncertainty).Let Pbe a property of objects that is (i) stable under retracts and filtered limits, and (ii) monotone with respect to the ambient preorder ≾ on endomorphisms. Define its ε -relaxation P ε by “ BεX has Pup to ≾ ”. Then for faithful blurs: P(X)⇐⇒ ∃ε0∀ε⪯ε0:Pε(BεX). In words: an object has Piff it has the relaxed property throughout some uncertainty window, and reading removes the relaxation without loss. Sketch. ( ⇒ ) If P( X )holds and Pis stable under retracts, then BεX inherits P ε via jε X and rε X (lax retract). ( ⇐ ) If P ε ( BεX )holds eventually and ρX : limIBεX∼ = −−→ X , stability under filtered limits transfers Pto X; monotonicity in ≾removes the laxity in the limit. Remark 8 (Blur as a graded comonad).The family ( Bε ) ε∈I comes with canonical comparison maps Bε⊕δX→Bε ( BδX ), natural in X , making it a graded comonad (or dually, a graded monad) on C . In concrete analytic models these maps are realized by Markov semigroups or convolution semigroups; in logical models they correspond to iterating a nucleus or closure operator. Thus the categorical structure behind blur is compatible with standard constructions in probability (Giry/Markov) and in logic (modal/comonadic viewpoints). 2 Morphisms that respect blur Definition 9 (Blur-morphism).Given blurred objects X= ( X, BX )and Y= ( Y, BY ), a blur-morphism ¯ f:X→Yis a natural transformation ¯ f:BX⇒ BY,i.e. ∀ε∈I¯ fε:BεX→BεYand ¯ fε⪯ε′commute with reindexing. Its sharp part is f:= ρY◦limI¯ f◦ρ−1 X:X→Ywhenever the blurs are faithful. Composition is pointwise: ( ¯g◦¯ f ) ε := ¯gε◦¯ fε ; identities are ( id ) ε = id . Thus blurred objects/morphisms form a category Blur(C). Remark 10 (Bidirectional observability).A network is observable in both directions if there exists ¯ f†:Y→Xwith ρX◦lim I(¯ f†◦¯ f)◦ρ−1 X= idX, ρY◦lim I(¯ f◦¯ f†)◦ρ−1 Y= idY, i.e. ¯ f is an isomorphism in Blur ( C ). This formalizes that we may switch Proposition/Answer at will. Unison removal and scale composition We assume reindexing is monoidal: for each ε, δ ∈I there is a canonical comparison µε,δ : Bε⊕δX→Bε ( BδX ), natural in X , making ( Bε ) ε∈I agraded comonad (or graded monad, depending on the concrete model). Naturality of ¯ f means removing neighbourhoods happens in unison: ¯ fεcommutes with the transition maps Bε′X→BεXand Bε′Y→BεY. Standing convention. We fix natural jε X as in Definition 4, so Xjε X −−→ BεXrε X −−→ X is a lax retract pair (object contained with an uncertainty window). Grand Lemma (sequential deblurring = one-shot deblurring) Theorem 11 (Grand Lemma (Blur–transport lemma): sending sharp ⇔ sending blurred and reading stepwise).Let X¯ f −→Y¯g −→Zbe blur-morphisms with faithful blurs. Then for every ε∈I, rε Z◦(¯gε◦¯ fε) = rε Z◦¯gε◦rε Y◦¯ fε= (g◦f)◦rε X, and in particular, after taking the limit ε→0(reading), g◦f=ρZ◦lim I(¯g◦¯ f)◦ρ−1 X=ρZ◦lim I¯g◦ρ−1 Y◦ρY◦lim I ¯ f◦ρ−1 X. Thus sending a Proposition through the network is equivalent to sending it together with its neighbourhood and reading in successive steps. Proof. Pointwise identity rε Z◦¯gε◦¯ fε = ( g◦f ) ◦rε X follows from naturality of the limit cones: rε Y and rε Z are the legs of ρY, ρZ , and ¯ f, ¯g are natural transformations. Taking limits in I yields the equalities with ρX, ρY, ρZby the universal property of limits. Reading at infinity (no response) If colimIBX≃ 1and colimIBY≃ 1, then any blur-morphism ¯ f induces the unique arrow 1 → 1 at ∞ ; i.e. as neighbourhood grows without bound both Proposition and Answer collapse to “no response” in unison. 3 Passing neighbourhoods faithfully Given any neighbourhood ε on the Proposition, the network passes it to the Answer via ¯ fε ; the family {¯ fε}ε∈I ensures the Answer retains the corresponding neighbourhood. Rescaling of neighbourhoods is handled by reindexing I (e.g. ε7→ αε ) and the graded (co)monad coherence. Two canonical models 1. Analytic blur (convolution). C = Meas , BεX acts on measurable functions by convolution with a positive kernel Kε (Gaussian/Poisson/compact mollifier), and on points via randomization with law Kε . Then ¯ f is a Markov kernel family commuting with convolution; ρis the deconvolution limit as ε→0. 2. Logical blur (fuzzy/enriched). C enriched over the quantale ([0 , 1] ,≤,·, 1): a proposition has a neighbourhood of truth values; Bε thickens truth via a nucleus (closure operator). Morphisms are [0 , 1]-nonexpansive maps. Limits recover crisp truth; colimits at ∞ give the trivial truth. Remark 12 (Stochastic universality).Instead of fixed kernels, let Uεbe random perturbations with P ( |Uε|> δ ) → 0as ε→ 0. Then Bεf ( x ) := E [ f ( x + Uε )] defines a blurred diagram; all statements above hold verbatim. “Gaussian” is merely a maximally symmetric instance. String-diagram view of blur We now depict blur morphisms as string diagrams. Wires are objects, small boxes are processes, triangles are reading (removing the blur). Symbols Bε Blur at scale εRead rε Grand Lemma as a diagram Sending a proposition X through a network f and then reading is equal to blurring first, sending through the blurred network ¯ f, then reading. This is the commutativity of the diagram: XBεX BεYY rε X¯ fεrε Y f This string-diagram encapsulates the Grand Lemma (Blur–transport lemma): rε Y◦¯ fε◦rε X=f◦rε X. Bidirectionality If ¯ f is an isomorphism in the blur category, the diagram works both ways: we may slide the reading triangle across the network in either direction, expressing the observability in both roles of Proposition and Answer. 4 Conclusion: The Grand Purpose of Blurring The central role of blur is not merely technical but conceptual. Grand purpose. Blurring allows an object to expand into a neighbourhood, becoming a bulk rather than a sharp point. In this relaxed state, the object has more “energy” to interact with its environment: it collects information that may not strictly belong to the object itself but to the network it traverses. The neighbourhood functions as a probe of the medium, capturing how the object and the network resonate together. In this sense, blurring is catalytic. It does not damage the object nor distort the network; instead, it enhances their interaction, allowing latent structure to surface. The messages that survive this joint evolution are then attached back to the object when the neighbourhood is removed (reading). Thus, the philosophy of blur is: Precision is achieved not by resisting relaxation, but by passing through it. Blurring reveals, interaction refines, and reading restores. In this sense, the applicability of blurring is universal. Blur and Undecidability (toy but sharp) We briefly show that blurring does not make hard decision problems easy: even a positive, normalized blur preserves the halting/non–halting gap. Discrete time as a blurred object Work in C = Meas . Let Prog be the set of programs (or initial states) and, for e∈Prog , let ( Fte ) t∈N be its evolution with a halting predicate h : Prog → { 0 , 1 } (true exactly on halting states). For λ∈(0,1) define the time–blur kernel Kλ(t) = (1 −λ)λton N. The blurred halting mass is the read-out Hλ(e) := X t≥0 Kλ(t)h(Fte) = ET∼Geom(1−λ) h(FTe), i.e. convolution in time with a positive, normalized kernel (a Markov/semigroup blur in the sense of our analytic model). Theorem 13 (Undecidability survives blur).For every fixed λ∈ (0 , 1) and every program e with halting time τ(e)∈N∪ {∞}, Hλ(e) =    λτ(e)if τ(e)<∞, 0if τ(e) = ∞.Hence {e:Hλ(e)>0} ≡ HALT. In particular, deciding the positivity of this blurred read-out is undecidable. Sketch in our language. Kλ is a positive-definite mollifier on time (a blur scale). The evolution e7→ ( Fte ) t is a blur-morphism into the path object; the observable h is a nonnegative map. If e halts at time τ , then h ( Fte ) = 0 for t<τ and 1thereafter, so Hλ ( e ) = (1 −λ ) Pt≥τλt = λτ> 0; if it never halts, the sum is 0. Thus the positivity read-out after blurring is many–one equivalent to HALT. 5 Remark 14 (Fits the framework and links to computable analysis).This is a special case of our analytic blur model: Bλ is convolution on the time axis, ¯ fλ is the (Markov) blur–morphism induced by the dynamics, and rλ is the numerical read-out. The undecidability hinges only on positivity, normalization, and a strict positivity gap on halting—exactly the bloorf invariants. Conceptually this is close in spirit to classical encodings of halting into analytic properties of real-valued functions in computable analysis: blur does not eliminate undecidability; it packages it into a robust positivity gap. Continuous time and spatial blur (one-liners) •Continuous time. With Ky(t) = ye−yt on R≥0and halting time τ(e)∈[0,∞], Hy(e) = Z∞ 0 Ky(t)h(Fte)dt =1{τ(e)<∞} e−yτ(e). Again, Hy(e)>0iff ehalts. • Spatial blur. If halting emits a unit-mass bump at a known location xe while non-halting emits 0, then any nonnegative kernel ϕy with ϕy (0) > 0yields ( ϕy∗fe )( xe ) > 0iff e halts. Moral. Blurring simplifies analysis by preserving positivity and behaving well with limits; those same features preserve the halting/non–halting dichotomy as a strict positivity gap. In the categorical language of blur, undecidability is therefore bloorf–robust: blur clarifies, it doesn’t conjure answers. Two guiding examples of blur We briefly sketch two concrete examples that can be read directly in the language of blurred objects and blur–morphisms: (1) addition and multiplication as a blurred pair of channels; (2) Weyl equidistribution on the circle under Poisson blur. In both cases, the role of blur is not decorative: it is forced by the underlying harmonic analysis, and the categorical picture simply packages that necessity. Addition and multiplication as a blurred pair At a very classical level, addition and multiplication live on different backgrounds: •addition acts naturally on (R, dy)via translations Taf(y) = f(y+a); •multiplication acts naturally on ((0,∞), dx/x)via dilations Dcf(x) = f(cx). A standard device is to move to the log-line: with x = ey the multiplicative group (0 ,∞ )with Haar measure dx/x becomes ( R, dy ), and the Mellin transform on functions of x becomes the Fourier transform on functions of y . This is the precise sense in which multiplicative structure is ablurred copy of additive structure. Fix the base category C = Meas (or, if preferred, the corresponding L2 Hilbert spaces). Consider two blurred objects: X×:= (0,∞),B×,X+:= R,B+, where: 6 •B× ε is log–Gaussian blur on the multiplicative line: if U : L2 ((0 ,∞ ) , dx/x ) →L2 ( R, dy )is the log-change isometry (Uf)(y) = ey/2f(ey), and Gεis the Gaussian of variance ε, then B× εf:= U−1Gε∗(Uf), i.e. blur in log-coordinates, pulled back to the x-line; •B+ τis ordinary Gaussian blur on the additive line: B+ τg:= Gτ∗g, g :R→C. The soft addition / soft multiplication picture can then be read as follows. • On the additive side, +is sharp: we are free to work with g1⊕g2 defined by ( g1⊕g2 )( y ) = g1(y) + g2(y)or with convolution in y, depending on the level of structure we track. • On the multiplicative side, × is sharp: we work with pointwise product in x or with multiplicative convolution (f1∗×f2)(x) = Z∞ 0 f1 x uf2(u)du u. • The link between the two channels is the isometry U , which intertwines multiplication on (0,∞)with translation in the spectral variable on the log-line. The crucial fact is that Fourier on the log-line satisfies a genuine Heisenberg-type uncertainty: if b h ( ξ )is the Fourier transform of h ( y ), then one cannot simultaneously have both h and b h sharply localized. Concretely, there is a lower bound of the form Vary(h) Varξ(b h)≥1 4 (up to the usual normalization choices). On the multiplicative side this reads: you cannot make the blur in logx arbitrarily small while also keeping the multiplicative spectral content arbitrarily sharp. In the language of blurred objects: •the families (B× ε)ε>0and (B+ τ)τ>0are blur functors on X×and X+; •the log-change Uinduces a blur-morphism Switch×→+:X×−→ X+,Switch×→+ε:= B+ τ(ε)◦U◦B× ε, where the blur indices are coupled so that ε·τ(ε)≥c > 0for a universal constant. The Heisenberg inequality is exactly the statement that we cannot make both ε and τ ( ε ) arbitrarily small: there is a hard lower bound on the blur budget required to switch between the additive and multiplicative lenses. In other words, any blur-morphism that genuinely intertwines the two channels must pay with a nonzero uncertainty window. Soft addition and soft multiplication then appear as operations defined at the blurred level: •to add multiplicative data “as if” it were additive, we 1. blur in the multiplicative channel at some ε, 2. switch to the additive channel via Switch×→+, 3. perform the native additive operation there, 7 4. and read back sharply. • to multiply additive data “as if” it were multiplicative, we do the symmetric construction using the inverse blur-morphism Switch+→×. The Grand Lemma then says precisely: sending sharp data through this network is equivalent to sending it together with its neighbourhood and reading in successive steps. The analytic content (Fourier–Mellin duality and Heisenberg uncertainty) ensures that such a network cannot exist with zero blur; the categorical content ensures that, once we fix a blur budget, properties that are stable under retracts and filtered limits can be transported between the two channels. Thus the addition/multiplication pair is a particularly vivid example of a blurred object and blur–morphism that are not optional: blur is not something we choose to add; it is something the harmonic analysis forces us to account for. Weyl equidistribution under Poisson blur As a second example, we take a classical theorem where blur appears almost unavoidably in standard proofs: Weyl’s equidistribution of {nα}modulo 1. Let T = R/ 2 πZ be the circle with Lebesgue probability measure m , and fix α∈R . Consider the orbit xn:= nα (mod 2π)and the empirical measures µN:= 1 N N X n=1 δxn. Weyl’s theorem says: if α/ 2 π is irrational, then µN converges weak-* to m , i.e. the orbit is equidistributed. In our language, we define a blur on Tvia the Poisson kernel Pr(θ) = 1−r2 1−2rcos θ+r2,0< r < 1, and set Brf := Pr∗f for f∈L1 ( T ). The family ( Br ) 0<r<1 is a standard approximate identity: lim r↑1Brf=fin Lp(T) for 1 ≤p < ∞ , and Br annihilates high Fourier modes exponentially: if f ( θ ) = Pk∈Zˆ f ( k ) eikθ , then Brf(θ) = X k∈Z r|k|ˆ f(k)eikθ. Define blurred empirical measures µN,r := µN∗Pr,0< r < 1, so that for a continuous f, ZT f dµN,r =ZT (Brf)dµN=1 N N X n=1 (Brf)(xn). Now set up the blurred object XT:= (T,B), Br:= B(r) : L1(T)→L1(T), with r∈I := (0 , 1), partially ordered by r⪯r′ if r≤r′ . The sharp limit ρ is f7→ limr↑1Brf = f (when it exists), and the exhaustion at “infinity” is the collapse to constants as r↓0. 8 The property of interest is: P(µ•) : µNw∗ −−→ mas N→ ∞ (equidistribution). Its blurred version at scale ris: Pr(µ•) : µN,r w∗ −−→ mas N→ ∞, i.e. equidistribution holds after Poisson blur at fixed r < 1. Classically, Weyl’s criterion reduces P(µ•)to the decay of exponential sums: 1 N N X n=1 eikxn−→ 0 (N→ ∞)for all k∈Z\ {0}. In the blurred picture, we test against Brf instead. On Fourier coefficients, this inserts the extra factor r|k|, making the estimates strictly easier: 1 N N X n=1 (Brf)(xn) = X k∈Z r|k|ˆ f(k)1 N N X n=1 eikxn. For each fixed r < 1, the tail over large |k| is exponentially suppressed, and the analysis reduces to finitely many frequencies with a built-in damping factor r|k| . In other words, P r ( µ• )is strictly easier to verify than P(µ•). From the viewpoint of the property-transport Proposition (transport under uncertainty window), equidistribution is: • stable under convolution with an approximate identity (Poisson blur): if µN,r →m for a fixed r < 1and Br→id as r↑1, then µN→m; • stable under filtered limits in r and monotone with respect to the natural preorder on kernels (Pr≾Pr′if r≤r′in the sense of pointwise domination). Thus, in this example, we may legitimately work at any fixed blur level r < 1, prove P r ( µ• ) in the blurred world (where the Fourier side is better behaved), and then let r↑ 1to recover the sharp statement P( µ• ). The Poisson blur is not an arbitrary smoothing trick: it is precisely a blur functor Brwith a reading map ρthat satisfies the hypotheses of our transport principle. In summary: • addition vs. multiplication illustrates how blur is forced by Fourier–Mellin uncertainty when we try to switch between algebraic channels; • Weyl equidistribution on the circle illustrates how blur is a natural technical lens: we prove a property in a blurred regime where the analysis is easier, and then read back to the sharp regime using stability under limits. Both examples fit cleanly into the categorical framework developed above and show that blur is not merely a philosophical decoration, but a mathematically inevitable mediator in situations where different structures must be made to communicate. Blur at closure interfaces: from successors to N and from Q to R We briefly record two concrete interfaces where blur is not an optional decoration but the mechanism by which an idealized infinite object is constructed from a finitary one: •the passage from a single successor step “+1” to the full set of natural numbers N; •the passage from the rationals Qto the reals Rvia Cauchy completion. Both can be phrased as instances of our general notion of a blurred object X= ( X, BX )with a faithful reading cone ρX: limIBX ∼ = −−→ X. 9