PRH | Essay | 7.16 • Blur for the Three–Body Problem

Blur for the Three–Body Problem Finite Families under Coarse Observation Aleksandar Perišić December 2025 Abstract We treat the Newtonian three–body problem as a testbed for the general principle that under blur there are only finitely many effective families of trajectories. We introduce blur operators on both the phase space and the parameter space (masses, gravitational constant), and define blur–families of orbits: classes of solutions whose blurred trajectories remain uniformly close over a fixed time window. On any compact, non–collision energy region and finite time interval, we show that a finite set of blur–families suffices to approximate all trajectories at a given resolution. When parameters are blurred as well, we obtain finite families in the joint parameter–state space, and we describe blur–stable dynamical regimes where qualitative orbit types (bounded/escape) are constant on blur–classes. Analytically, the results are straightforward applications of continuous–dependence estimates. Conceptually, they fit into a broader “blur vs. finiteness” picture that also appears in number–theoretic problems (e.g. Collatz certificates): one trades sharp, global knowledge for a finite number of coarse representatives. The three–body case provides a concrete, finite–dimensional illustration of how blur turns chaotic dynamics into finitely many observable histories. Contents 1 Introduction 2 2 The Newtonian three–body problem 2 2.1 Phase space and vector field .............................. 2 2.2 Non–collision energy shells ............................... 3 3 Blur on phase space and blur–families 3 3.1 Phase–space blur .................................... 3 3.2 Blur–families on a compact energy shell ....................... 4 4 Blurring parameters as well 5 4.1 Parameter space and continuity ............................ 5 4.2 Blur on parameter space ................................ 6 4.3 Joint blur–families in parameter and state ...................... 6 5 Blur–stable dynamical regimes 7 5.1 Coarse orbit types ................................... 7 6 Blur with an Observer: Finite Observable Histories 9 6.1 Observer, measurement map, and observation metric ................ 9 6.2 Observable blur–distance and finite observable histories .............. 9 6.3 Insolvability and the role of irreducible blur ..................... 10 1 7 Discussion and outlook 11 1 Introduction The Newtonian three–body problem is a classical laboratory for chaotic dynamics and structural complexity. Even in the simplest case of three point masses moving under mutual Newtonian attraction, the long–time behaviour of orbits exhibits extreme sensitivity to initial conditions and parameters. From a blur perspective, this sensitivity is a feature, not a bug. Rather than asking for exact pointwise prediction, we ask: Given a blur scale r in phase space (and possibly in parameter space) and a tolerance ε on a finite time horizon [0 , T ], how many distinct blur–families of trajectories does the system really have? The answer is: finitely many. More precisely: •fix a compact non–collision energy region and a time horizon T; • fix blur scales r (state space) and σ (parameters), and a tolerance ε for the blurred trajectories; • then there exists a finite set of representative trajectories such that every (parameter, initial condition) pair in the region is blur–close to one of these representatives on [0, T ]. This is almost tautological from the viewpoint of continuous dependence: the flow is continuous in initial data and parameters; on a compact set we have uniform continuity; therefore, finitely many sample points suffice at any fixed resolution. The point of this note is to: (i) formulate this in an explicit blur language, including blur on parameters; (ii) state transparent finiteness results (blur–families and blur–stable regimes) that can be transplanted to more complicated systems; (iii) isolate the pattern: finite families under blur as a kind of coarse certificate, echoing the finite certificates used in other contexts (e.g. Collatz Lyapunov proofs). The results here are deliberately modest on the analytic side: nothing about global integrability, no regularity miracle. The contribution is a clean blur formalization in a very concrete, classical model. 2 The Newtonian three–body problem 2.1 Phase space and vector field We work in three spatial dimensions; the planar case is a simplification but does not change the blur logic. Let qi∈R3 be the position and pi∈R3 the momentum of the i –th body, i = 1 , 2 , 3. Denote q= (q1, q2, q3)∈R9, p = (p1, p2, p3)∈R9, z = (q, p)∈X:= R18. Let m = ( m1, m2, m3 ) ∈ (0 ,∞ ) 3 be the masses and G > 0the gravitational constant. The Hamiltonian is HG,m(q, p) = 3 X i=1 |pi|2 2mi −GX 1≤i<j≤3 mimj |qi−qj|.(1) 2 The equations of motion are Hamilton’s equations ˙z=FG,m(z)=J∇HG,m(z), with J the standard symplectic matrix. For now we fix G and m and write F = FG,m , H = HG,m . 2.2 Non–collision energy shells Let Cbe the collision set: C=[ 1≤i<j≤3 {z= (q, p):qi=qj}. We restrict attention to a compact, non–collision region in phase space. Definition 2.1 (Non–collision compact region).Fix energy level E∈R , radius R > 0and a collision–avoidance margin δ > 0. Define ΣE,R,δ =z= (q, p)∈R18 :H(z)=E, max i|qi|≤R, max i|pi|≤R, |qi−qj| ≥ δ∀i < j. Proposition 2.2 (Lipschitz vector field on Σ E,R,δ ).On Σ E,R,δ the vector field F is C∞ and Lipschitz: there exists L=L(E, R, δ, G, m)>0such that ∥F(z)−F(z′)∥≤L∥z−z′∥ ∀z, z′∈ΣE,R,δ. Sketch. On Σ E,R,δ all mutual distances |qi−qj| are bounded below by δ , so the potential −Gmimj/|qi−qj| and its derivatives are bounded. The kinetic part is polynomial. Hence ∇H is smooth with bounded first derivatives on ΣE,R,δ, which implies the Lipschitz bound. By standard ODE theory, for each z0∈ Σ E,R,δ there exists a unique solution z ( t ; z0 )defined on some interval around 0. Since Σ E,R,δ is compact and F is Lipschitz on it, there is a uniform existence time T∗ = T∗ ( E, R, δ, G, m )such that solutions starting in Σ E,R,δ remain in Σ E,R,δ for |t| ≤ T∗. For T≤T∗we denote the flow by Φt: ΣE,R,δ →ΣE,R,δ,Φt(z0)=z(t;z0). Proposition 2.3 (Continuous dependence on initial data).For |t|≤T with T≤T∗ and z0, z′ 0∈ΣE,R,δ, ∥Φt(z0)−Φt(z′ 0)∥≤eLT ∥z0−z′ 0∥, where Lis the Lipschitz constant from Proposition 2.2. Proof. Gronwall’s inequality applied to ˙z = F ( z )and the Lipschitz estimate yields the standard bound ∥Φt(z0)−Φt(z′ 0)∥≤eL|t|∥z0−z′ 0∥. 3 Blur on phase space and blur–families We now introduce blur on the phase space and define blur–families of trajectories. 3.1 Phase–space blur Since X= R18 is finite–dimensional, many blur operators are available. For definiteness we use a Gaussian convolution blur on functions, and the associated action on trajectories. 3 Definition 3.1 (Phase–space blur operator).For r > 0, let κr :X → [0 ,∞ )be a Gaussian kernel κr(z) = 1 (2πr2)9exp−∥z∥2 2r2. For a bounded continuous observable φ:X→Rdefine its blurred version (Brφ)(z) = ZX κr(z−z′)φ(z′)dz′. For a trajectory z(t)in Xwe define the blurred trajectory at scale ras the curve t7→ (Brδz(t)), or more concretely via any fixed observable φ : we track t7→ (B rφ )( z ( t )) instead of t7→ φ ( z ( t )). For our finiteness results it is enough to work at the level of the state itself and view blur as a bounded linear map Br:X→Xsatisfying: •∥Br∥ ≤ 1for all r; •Br→Id strongly as r↓0. One may think of B r as projecting onto a coarse grid of mesh r or onto low Fourier modes; the precise choice does not affect the logic. Definition 3.2 (Blur distance on trajectories).Fix r > 0and a time horizon T > 0. For two trajectories z(t),˜z(t)defined on [0, T]we define their blurred distance dr,T (z, ˜z) := sup 0≤t≤T ∥Brz(t)−Br˜z(t)∥. 3.2 Blur–families on a compact energy shell We now define blur–families on ΣE,R,δ. Definition 3.3 (( r, ε, T )–blur–family).Let K⊂ Σ E,R,δ be compact. Fix a blur scale r > 0, tolerance ε > 0and a time horizon T≤T∗ . A family of reference initial data {zj}N j=1 ⊂K is called an (r, ε, T)–blur–family for Kif for every z0∈Kthere exists some jsuch that dr,T Φ·(z0),Φ·(zj)= sup 0≤t≤T ∥BrΦt(z0)−BrΦt(zj)∥≤ε. The corresponding blurred trajectories t7→ BrΦt(zj)are called blur–family representatives. In words: within the blur resolution r and tolerance ε , every trajectory starting in K is indistinguishable from one of the representative trajectories up to time T. The main observation is that such families always exist and are finite. Theorem 3.4 (Finite blur–families on a compact region).Let Σ E,R,δ and T≤T∗ be as above, and let K⊂ Σ E,R,δ be compact. For any blur scale r > 0and tolerance ε > 0there exists a finite (r, ε, T)–blur–family for K. More precisely, one may choose reference points {zj}N j=1 such that K⊂SN j=1 B(zj, δ0)with δ0=ε e−LT , where L is the Lipschitz constant of F on Σ E,R,δ , and N is the covering number of K by balls of radius δ0. 4 Proof. Let Lbe the Lipschitz constant from Proposition 2.2. By Proposition 2.3, ∥Φt(z0)−Φt(z′ 0)∥≤eLT ∥z0−z′ 0∥ for all 0≤t≤Tand z0, z′ 0∈K. Since ∥Br∥ ≤ 1, ∥BrΦt(z0)−BrΦt(z′ 0)∥≤∥Φt(z0)−Φt(z′ 0)∥≤eLT ∥z0−z′ 0∥. Choose a finite δ0 –net {zj}N j=1 in K with δ0 = εe−LT ; such a net exists because K is compact. Then for any z0∈Kthere is a jwith ∥z0−zj∥≤δ0, and therefore dr,T Φ·(z0),Φ·(zj)≤eLT δ0=ε. So {zj} is an ( r, ε, T )–blur–family. The bound on N is immediate from the covering definition. The theorem confirms the intuitive picture: on a compact, non–collision energy region and finite time horizon, there are only finitely many blur–families at any fixed blur and tolerance. As we refine blur (smaller r or ε ), the required number of families grows with the covering number of Kand exponentially with the time window T(through eLT ). Remark 3.5 (Dependence on blur scale).Note that the bound for N in Theorem 3.4 does not depend explicitly on r : we did not use that B r is approximating the identity as r→ 0. To incorporate that, one may let the tolerance ε = ε ( r )depend on r , shrinking as r→ 0, to reflect the fact that coarser blur allows larger tolerance in the sharp norm; the existence result then adapts with a rescaled δ0(r). 4 Blurring parameters as well The three–body flow depends not only on the initial condition but also on parameters ( G, m1, m2, m3 ). We now introduce blur on parameter space and obtain finiteness in the joint parameter–state space. 4.1 Parameter space and continuity Let the parameter space be P= [Gmin, Gmax]×[m1,min, m1,max]×[m2,min, m2,max]×[m3,min, m3,max], with all bounds positive and finite. For p = ( G, m1, m2, m3 ) ∈ Pwe write Fp and Φ p t for the vector field and flow associated to that parameter tuple. Proposition 4.1 (Uniform Lipschitz and continuous dependence on p ).Let K⊂ Σ E,R,δ be compact and let Pbe as above, with all mi bounded away from 0and infinity and G in a compact interval. Then: (a) There exists LK>0such that for all p∈P, ∥Fp(z)−Fp(z′)∥≤LK∥z−z′∥ ∀z, z′∈K. (b) For each T≤T∗ , the flow ( p, z0 ) 7→ Φ p t ( z0 )is uniformly continuous on P ×K for 0 ≤t≤T . Sketch. On K× P, the masses and G are bounded and bounded away from 0, and the positions are collision–separated by δ . The Hamiltonian and its derivatives up to second order are bounded uniformly in ( p, z ), so the Lipschitz constant in z can be chosen uniformly in p . For (b), Fp depends smoothly on p , so the solution depends continuously on both p and z0 ; on a compact domain P×Kand finite time interval, this implies uniform continuity. 5 4.2 Blur on parameter space We now define blur on parameter space analogously. Definition 4.2 (Parameter blur).Let B (P) σ :P → Pbe a family of blur operators at scales σ > 0, for instance given by convolution with a smooth compactly supported kernel on the box P(extended by reflection at the boundary), or by projection to a coarse grid of mesh σ . We require: •∥B(P) σ∥≤1; •B(P) σ→Id as σ↓0. We consider the joint blur operator on parameter–state pairs (p, z)∈P×X: Bσ,r(p, z) := B(P) σp, Brz. 4.3 Joint blur–families in parameter and state We now define blur–families in the joint space P×K. Definition 4.3 (( σ, r, ε, T )–joint blur–family).Fix compact K⊂ Σ E,R,δ , parameter box P, blur scales σ > 0,r > 0, tolerance ε > 0and T≤T∗. A finite collection of representatives {(pj, zj)}N j=1 ⊂P×K is a (σ, r, ε, T )–joint blur–family if for every (p, z0)∈P×Kthere exists jsuch that sup 0≤t≤T Bσ,rp, Φp t(z0)−Bσ,rpj,Φpj t(zj) ≤ε. In words: under the observer’s blur on parameters and phase space, every realised three–body configuration over [0 , T ]is indistinguishable from one of the representative parameter–trajectory pairs. Theorem 4.4 (Finite joint blur–families).Under the assumptions of Proposition 4.1, for any blur scales σ > 0, r > 0, tolerance ε > 0and time horizon T≤T∗ , there exists a finite (σ, r, ε, T)–joint blur–family for P×K. More concretely, there exists a finite δ –net { ( pj, zj ) }N j=1 in P ×K such that the above property holds, where δ = δ ( ε )is determined by the uniform continuity modulus of ( p, z0 ) 7→ (Bσ,rp, Bσ,rΦp t(z0)) on P×Kand [0, T]. Proof. By Proposition 4.1, the map (p, z0, t)7→ B(P) σp, BrΦp t(z0) is continuous on the compact set P ×K× [0 , T ], hence uniformly continuous. Thus there exists a modulus ω(·)such that ∥(p, z0)−(p′, z′ 0)∥ ≤ δ=⇒sup 0≤t≤T Bσ,rp, Φp t(z0)−Bσ,rp′,Φp′ t(z′ 0) ≤ω(δ). Pick δ > 0with ω ( δ ) ≤ε , and choose a finite δ –net { ( pj, zj ) }N j=1 in P ×K (possible by compactness). Then for any ( p, z0 )there exists j with ∥ ( p, z0 ) − ( pj, zj ) ∥≤δ , which gives the desired inequality. The theorem expresses precisely the finite–families–under–blur intuition in the joint parameter– state space. 6 5 Blur–stable dynamical regimes The finiteness theorems above concern trajectories as curves. Often one is more interested in qualitative orbit types: bounded motion, escape, collision, and refinements thereof (e.g. binary formation, ejection of one body). We sketch how blur–families interact with coarse classification maps. 5.1 Coarse orbit types Fix a classification time horizon Tmax >0. Define a coarse orbit type map C:P×K→ {1,2,3}, where, say, • C( p, z0 ) = 1 if the orbit remains in a bounded region (no escape, no collision) up to time Tmax; •C(p, z0)=2if a collision occurs before Tmax; • C( p, z0 )=3if some body escapes beyond a large fixed distance Resc before Tmax without collision. The precise definitions are not crucial; the key point is that Cis a map from ( p, z0 )to a finite set of labels. In general Cmay be highly discontinuous on P ×K due to chaotic boundaries between regimes. Nevertheless, there are regions where it is locally constant. Definition 5.1 (Blur–stable regime).Let ( p∗, z∗ ) ∈ P ×K . We say that ( p∗, z∗ )lies in a blur–stable regime at scales (σ, r)if there exists ε > 0such that C(p, z0)=C(p∗, z∗) for all (p, z0)with  Bσ,r(p, z0)−Bσ,r(p∗, z∗) ≤ε. In other words, within the blur–class around ( p∗, z∗ )the coarse orbit type is constant; the regime is stable under the observer’s blur. Proposition 5.2 (Local blur–stability under structural stability).Suppose Cis locally constant in a (sharp) neighbourhood U×V of ( p∗, z∗ )in P ×K . Then there exist σ0, r0> 0and ε > 0 such that ( p∗, z∗ )is blur–stable at all blur scales ( σ, r )with 0 < σ ≤σ0 ,0 < r ≤r0 , with the same label C(p∗, z∗). Proof. If Cis locally constant near ( p∗, z∗ ), there exists η > 0such that C( p, z0 ) = C( p∗, z∗ ) whenever ∥ ( p, z0 ) − ( p∗, z∗ ) ∥ ≤ η . Choose σ0, r0> 0so small that ∥ B σ,r ( p, z0 ) − ( p∗, z∗ ) ∥ ≤ η whenever ∥ ( p, z0 ) − ( p∗, z∗ ) ∥≤η/ 2and 0 < σ ≤σ0 ,0 < r ≤r0 ; this is possible since B σ,r →Id as σ, r → 0. Then take ε = η/ 2; the blur–ball of radius ε around B σ,r ( p∗, z∗ )in blurred norm sits inside the sharp ball of radius ηaround (p∗, z∗), where Cis constant. Combined with Theorem 4.4, this implies that on any compact region where Cis piecewise locally constant, the number of distinct blur–stable orbit types at a given blur resolution is finite. Each blur–family representative in the joint space can be labelled by its orbit type, and the blur–stable regimes correspond to clusters of representatives with the same label. While this is still completely in the realm of classical continuity and structural stability, the blur language turns it into a clean statement: at any finite resolution, the three–body system has finitely many blur–stable fates. 7 Conceptual takeaway: why blur is not optional From a distance, blur may look like a repackaging of things we already know: error bars, coarse grids, numerical tolerances. The point of the present analysis is that, for genuinely history–dependent systems such as the three–body problem, blur is not just convenient, it is mathematically unavoidable. A deterministic three–body system with exact parameters and exact initial data has a unique history. In practice, however, the parameters and initial states live in a small but irreducible uncertainty region: measurement noise, modelling error, unresolved forces, and numerical truncation. Because the dynamics is strongly sensitive to small perturbations, this uncertainty region does not remain small when transported along the flow; it stretches, folds, and threads itself through phase space in a way that makes sharp histories meaningless beyond a short horizon. The blur viewpoint says: do not fight this, build it into the definition of the object you are studying. Fix from the start a precise blur scale in parameter and state space—that is, a region of uncertainty that you accept as constitutive. Instead of one exact trajectory, consider the entire cloud of trajectories compatible with that blur. Our analysis shows that, at any fixed blur scale and any fixed finite time horizon, this cloud breaks into only finitely many blurred families of histories: equivalence classes of solutions that are indistinguishable within the accepted error for all observables we are willing to pay for. Inside each family, the system still has genuine microscopic freedom, but that freedom no longer propagates into new macroscopic scenarios; it only changes which representative of the same blurred history we see. In this sense, blur trades microscopic uniqueness for macroscopic finiteness. The infinite history of the system is not a single fragile curve in phase space but a compatible sequence of blurred histories, one for each time window and each resolution. Once the blur scale is honestly declared, the usual “pictures” of the evolution (configurations, energy exchanges, qualitative regimes) become stable objects: they are read as invariants of these blurred families rather than as shadows of an unattainable exact trajectory. Far from being a weakness, this is precisely the level at which the three–body problem is observable in the real world. Principle 5.3 (Conservation of blur budget).Fix a system with flow Φ t on a state space X , and a blur scale r that encodes an accepted initial uncertainty (a small, honest chunk of missing information in parameter–state space). Suppose this uncertainty is modelled by a blur operator Brthat is transported covariantly along the dynamics, Φt ∗◦Br≈Br◦Φt ∗, so that we always look at the system through the same informational lens. Then the initial error does not blow up in the sense of readability: the missing information stays of order r for all times. What grows is the internal complexity inside each blurred cell, not the blur budget itself. In more concrete terms: an initial error that is treated as a genuine reduction of information inside a proper information field does not, by itself, make the system unreadable. What fails is the Platonic ideal of zero error. If we insist on tracking a single sharp trajectory, then even a tiny unknown fragment explodes under a chaotic flow and the description becomes useless. But if we ride together with the error—fix from the start a legitimate blur scale and propagate it with the dynamics—then the informational gap remains bounded: 0 . 1bits of missing information remain 0.1bits of missing information, only rearranged. The usual mistake is to take the first few time steps, where the system is almost sharp, as the reference standard, and then complain when that level of precision is lost as the unknown fragment starts to affect everything else. From the blur point of view, the correct move is the opposite: accept the uncertainty at t = 0 as structural, design the dynamics and observables so that blur is pushed exactly where blur belongs, and then read the system consistently at that 8 scale. The price for reducing blur is that more and more distinct histories appear (eventually too many for any computational method), but for any fixed, tame blur scale we retain a finite collection of macroscopic outcomes, each no more than the accepted error away from the “true” history. Classical, pointlike determinism is recovered only as the singular limit where the blur scale is driven to zero, and it is precisely in that limit that the description becomes unstable. 6 Blur with an Observer: Finite Observable Histories So far we have blurred the three–body problem at the level of phase space and parameters, without explicitly modelling who is looking. In a Blurrichevsky spirit, we now insert an observer and reinterpret the finite blur–families as finite sets of observable histories. 6.1 Observer, measurement map, and observation metric Let O denote an observer equipped with an instrument model. We encode this as a measurement map MO:P×X−→ Y, where Yis a (finite–dimensional) data space; typical examples are: •Ya space of apparent positions and radial velocities (astrometric + spectroscopic data); •Ya space of light curves or time–dependent fluxes (photometric data); •Ya vector of derived quantities (pairwise distances, angles). The map ( p, z ) 7→ MO ( p, z )includes the observer’s location, line of sight, projection effects, and instrument response. We equip Y with a norm ∥·∥Y (e.g. Euclidean on a finite vector of observables) and define the observation metric on P×Xby dO(p, z),(p′, z′):=  MO(p, z)−MO(p′, z′) Y. Thus two parameter–state pairs are close for O if and only if they produce nearly the same raw data at a single instant. We also allow blur at the level of the data: for a blur scale ρ > 0on Ywe let B(Y) ρ:Y→Y be a linear operator with ∥ B (Y) ρ∥ ≤ 1and B (Y) ρ→IdY as ρ↓ 0(e.g. a convolution with a narrow kernel, or a projection to a coarse grid in data space). Composing with our earlier blur on parameters and state we obtain a total observed blur BO σ,r,ρ(p, z) := B(Y) ρMOB(P) σp, Brz. 6.2 Observable blur–distance and finite observable histories For a fixed observer O , blur scales ( σ, r, ρ ), and time horizon T > 0, we define the observable blur–distance between two parameter–trajectory pairs (p, Φp ·(z0)) and (p′,Φp′ ·(z′ 0)) as dO σ,r,ρ;T(p, z0),(p′, z′ 0):= sup 0≤t≤T  BO σ,r,ρp, Φp t(z0)− BO σ,r,ρp′,Φp′ t(z′ 0)  Y. 9