PRH | Essay | 7.9 • Blur at the Finite–Infinite Interface

Blur at the Finite–Infinite Interface The Abel–Blur on Successor Orbits and the Existence of N Aleksandar Perišić November 2025 Abstract We formalize a simple axiom scheme—Abel–blur on a successor orbit—that captures the probabilistic “bridge” between finite and cofinite phenomena along the successor chain. The axiom is purely structural and second-order: it postulates a family of finitely additive probability measures {µq}0<q<1 on the orbit of a basepoint under a successor map, satisfying a resolvent (shift) law and a finite/cofinite detection limit as q↑ 1. We prove that, over the language { 0 , S} , this blur axiom is equivalent to the existence of a Peano orbit, i.e. a copy of ( N, 0 , S ( n ) = n + 1). Thus, in this setting, the passage from finite to infinite is inseparable from a canonical probability profile (the Abel weights); conversely, the existence of the natural numbers is precisely the situation in which such a blur exists. We also extract a general Blur–Uncertainty Schema for complementary ladders (“front” vs. “tail”) and record the additive/multiplicative instance via the Mellin–Fourier dictionary, where Gaussian blur on log x is scale-neutral and yields the usual Heisenberg trade-off. Finally we discuss a constructive/computational reading (working at rational q < 1without taking the q↑ 1limit) and a short remark on physical analogies (e.g. Bose–Einstein condensation) where effective descriptions are inherently distributional. 1 Introduction Between finite and cofinite sets on N there is no finite meeting point: one chain grows by initial segments, the other shrinks by tails. The classical Abel weights wq(n) = (1 −q)qn,0< q < 1, n ∈N, supply an “honest bridge”: the Abel means µq ( A ) = Pn∈Awq ( n )provide a coherent comparison of finite/cofinite features at resolution scale ∼ (1 −q ) −1 , and in the limit q↑ 1they detect finiteness (→0) and cofiniteness (→1). The first purpose of this note is to make the bridge an axiom over a bare successor structure and to show that this axiomatized blur is equivalent to the existence of the natural numbers themselves (as a successor orbit). The second purpose is conceptual: whenever mathematics presents us with complementary ladders (“normal/co-” situations) that cannot be simultaneously sharp, there is a canonical resolvent blur and an associated uncertainty trade–off. This appears additively/multiplicatively via the Mellin–Fourier dictionary and, at a philosophical level, aligns with the idea that some regimes are best thought of distributionally rather than as perfectly sharp objects. 2 Preliminaries: successor orbits and the Peano world Let ( X, S, 0) be a structure with a distinguished basepoint 0and a unary map S : X→X (“successor”). Write the successor orbit O:= {Sn(0) : n= 0,1,2,...} ⊆ X, 1 where S0 := id . We will not assume global injectivity of S on X , nor that 0is not a successor in X; our attention is restricted to O. Definition 2.1 (Peano orbit).We say ( O, 0 , S )is a Peano orbit if the map n7→ Sn (0) is a bijection N→O . Equivalently: Sm (0) = Sn (0) implies m = n (no collisions) and there is no k > 0with Sk(0) = 0 (no cycles). Thus a Peano orbit is precisely a copy of the standard ω -type generated by iterating S from 0. 3 Axiomatizing blur on the orbit We now state blur as an axiom for ( X, S, 0) along O ; it is second-order because it quantifies over all subsets of O. Axiom 3.1 (Abel–blur on a successor orbit).There exists a family {µq}0<q<1 of finitely additive probability measures on P(O)such that for every E⊆O: (Resolvent / shift law) µq(E) = (1 −q)1E(0) + q µq(S−1E),(1) (Normalization of tails) µq({Sk0, Sk+10,...}) = qkfor all k≥0,(2) (Finite/cofinite detection) lim q↑1µq(F) = 0 for every finite F⊂O, lim q↑1µq(C) = 1 for every cofinite C⊂O. (3) Remark 3.2 (Comments on the clauses). • The resolvent identity (1) says µq = (1 −q ) δ0 + q S∗µq , i.e. µq is the law of a geometric mixture of the forward steps starting at 0. • The tail normalization (2) pins the scale consistently with (1) : the mass beyond level k decays as qk. • The detection property (3) is the formal statement that finite/cofinite become the two boundary atoms as resolution is removed (q↑1). 4 Main equivalence: blur ⇐⇒ existence of N Theorem 4.1. Over the language { 0 , S} ,Theorem 3.1 holds on the successor orbit O if and only if (O, 0, S)is a Peano orbit (hence, up to isomorphism, (N,0, n 7→ n+ 1)). We give each direction as a proposition. Proposition 4.2 (Blur ⇒ Peano orbit).Assume Theorem 3.1. Then O is infinite and collision-free under S, hence a Peano orbit. Proof. First, if O were finite then µq ( O ) = 1 for all q , but by (3) finite sets have limit 0as q↑1, a contradiction. Thus Ois infinite. Suppose Sm (0) = Sn (0) with m<n . Then Sn−m (0) = 0, so the forward sequence 0 , S (0) , . . . is eventually periodic, hence the set O is finite (only m + ( n−m )distinct elements), contradicting the previous step. Therefore m = n , so the map n7→ Sn (0) is injective. Surjectivity onto Ois tautological by definition of O. Hence (O, 0, S)is a Peano orbit. 2 Proposition 4.3 (Peano orbit ⇒ Blur).Assume ( O, 0 , S ) ∼ = ( N, 0 , n 7→ n + 1). For E⊆O define µq(E) := X n≥0 (1 −q)qn1E(Sn0),0< q < 1. Then {µq}satisfies Theorem 3.1. Proof. Finitely additive probability is immediate and µq(O) = 1. For (1), µq(E) = (1 −q)1E(0) + X n≥1 (1 −q)qn1E(Sn0) = (1 −q)1E(0) + q µq(S−1E). For (2), with Tk:= {Sk0, Sk+10,...}, µq(Tk) = X n≥k (1 −q)qn=qk. For (3) , if F is finite with max F≤N then µq ( F ) ≤µq ( { 0 , . . . , N} )=1 −qN+1 → 0. If C is cofinite with C⊇Tkthen µq(C)≥µq(Tk) = qk→1. Remark 4.4 (Uniqueness of the Abel profile).On a Peano orbit the resolvent (1) forces the point masses to be geometric. Let pn ( q ) := µq ( {Sn 0 } ). Then p0 ( q ) = (1 −q )and pn+1 ( q ) = q pn ( q ) by testing (1) on singletons. Hence pn ( q ) = (1 −q ) qn and µq is the Abel distribution. Thus the blur is unique under (1) and normalization. 5 Explicit front–tail formulas and the blur scale For n≥ − 1define initial segments Fn := {S0 0 , . . . , Sn 0 } (with F−1 = ∅ ) and tails Gk := {Sk0, Sk+10,...}. By iterating (1) or by a direct sum, µq(Fn) = 1 −qn+1, µq(Gk)=qk, µq(Fn∩Gk) = (qk−qn+1, k ≤n, 0, k > n. (4) Writing q=e−1/L with L>0interprets the blur scale as L≈(1 −q)−1: µq(Fn)=1−e−(n+1)/L, µq(Gk) = e−k/L. At fixed L one cannot make both an initial portion and a far tail simultaneously certain: for any event A⊆O, µq(A∩Fn)≤µq(Fn)=1−qn+1, µq(A∩Gk)≤µq(Gk)=qk.(5) The crossover index is n∼k∼L . This is the finite/cofinite “uncertainty” at the heart of the blur. 6 A general Blur–Uncertainty Schema The previous section can be abstracted to any “forward step” with a basepoint. Definition 6.1 (Resolvent blur of a step operator).Let (Ω ,A )be a measurable space, τ : Ω → Ω a measurable map, and x0∈Ωa basepoint. The resolvent blur is the probability measure µq:= (1 −q)X n≥0 qn(τn)∗δx0,0< q < 1, provided the series converges in the sense of measures. For front/tail families Fn := {τjx0 : 0 ≤ j≤n}and Gk:= {τjx0:j≥k}one has the bounds µq(Fn)=1−qn+1, µq(Gk) = qk, µq(A∩Fn)≤1−qn+1, µq(A∩Gk)≤qk. 3 Thus every complementary ladder generated by a single step comes with a canonical convex geometric blur and the associated front/tail trade–off (5) . The Peano case is obtained by taking τ=Son the successor orbit. 7 Additive vs. multiplicative: the Mellin–Fourier lens Let f : R>0→C be nice (e.g. in L2 ( R>0, d x/x )). With the logarithmic change x = et and g(t) := et/2f(et), the Mellin transform intertwines with the Fourier transform: M[f]1 2+iξ=Z∞ 0 f(x)x(1/2+iξ)−1dx=ZR g(t)e−iξt dt=F[g](ξ). Convolution on dx/x by the log-Gaussian kernel Kσ(x) = 1 √2πσ exp −(log x)2 2σ2 is multiplication in Mellin by e−(σξ)2/2 . This is the scale-neutral blur on the multiplicative line: it preserves dilation structure and realizes the Heisenberg inequality Vart(g)·Varξ(Fg)≥1 4, with equality exactly for Gaussians. In this sense, additive sharpness (small Vart ) demands multiplicative spread (large Varξ ), and vice versa. The discrete Abel blur (1 −q ) qn is the matching neutral choice on the successor chain. 8 Constructive/computational viewpoint Some mathematical programs—call them computationalist or constructivist in spirit—prefer to avoid completed infinities and closure assumptions. The blur framework accommodates this naturally: • Work at a fixed rational q∈ (0 , 1) (resolution L≈ (1 −q ) −1 ). The measure µq is fully computable: for any decidable set A⊆N , partial sums give ε -accurate approximations with a geometric tail bound qN+1/(1 −q). • The q↑ 1limit can be read operationally: given a known finite set F⊆N with max F≤N , one has µq ( F ) ≤ 1 −qN+1 ; given a known tail Gk , one has µq ( Gk ) = qk . Thus finite/cofinite are effectively distinguished at any prescribed tolerance by choosing q close enough to 1 with a computable modulus depending on the data at hand. • If one declines to quantify over all subsets, one may restrict µq to a class of definable predicates (arithmetical or primitive recursive), keeping the resolvent identity (1) as an operator law rather than a set-theoretic axiom. This loses categoricity but preserves the calculable content. In short, the Abel blur is usable and meaningful without committing to a completed infinite; the limit becomes a scheme of arbitrarily fine approximation. 9 A physical aside In regimes such as Bose–Einstein condensation, coherence phenomena make particle descriptions effectively distributional (macroscopic occupation of a single mode, phase coherence, delocalized profiles). The moral for mathematics is modest but clear: where complementary ladders cannot both be sharp, a blur profile is the correct language. The present successor-orbit blur is a minimal and precise instance of such a language on the finite/infinite boundary. 4 10 Conclusions and outlook We have shown that on a successor orbit the existence of a canonical Abel–blur is equivalent to the existence of the natural numbers themselves. This puts probability/distribution directly at the interface between finite and infinite and explains, in a clean axiom, why a controlled “uncertainty” is intrinsic to that passage. The same mechanism appears on the multiplicative line via Mellin–Fourier, where log-Gaussian blur is the scale-neutral mediator obeying Heisenberg. Two directions seem natural: • Categorical packaging. View ( O, µq )as a coalgebra for the endofunctor E7→ (1 − q)δ0+q S∗Eand study morphisms preserving blur. • Arithmetical channels. Apply resolvent blur to Eulerian objects (Dirichlet series, explicit formulas) where discrete/multiplicative structure meets analytic/spectral structure. A Proof details and small calculations Lemma A.1 (Front masses solve a first-order recursion).Under (1) , the sequence an ( q ) := µq(Fn)satisfies a−1(q) = 0 and an(q) = (1 −q)+q an−1(q), hence an(q) = 1 −qn+1. Proof. Since Fn={0}∪S(Fn−1)and the union is disjoint, µq(Fn) = (1 −q)1Fn(0) + q µq(Fn−1) = (1 −q)+q an−1(q), and the stated solution follows by induction. Lemma A.2 (Point masses are geometric).Let pn ( q ) := µq ( {Sn 0 } ). Then p0 ( q ) = (1 −q )and pn+1(q) = q pn(q),sopn(q) = (1 −q)qn. Proof. Apply (1) to singletons: for n = 0, S−1{ 0 } = ∅ so p0 ( q ) = (1 −q ). For n≥ 1, S−1{Sn0}={Sn−10}, giving pn(q) = q pn−1(q). Lemma A.3 (Tail masses).With Gk={Sk0, Sk+10,...}one has µq(Gk)=qk. Proof. Sum the point masses using Theorem A.2:µq(Gk) = Pn≥k(1 −q)qn=qk. Remark A.4 (Heuristic scale).Writing q = e−1/L gives the asymptotics 1 −qn+1 ≈ 1 −e−(n+1)/L and qk≈e−k/L . The number L is the blur length; front and tail cannot both be resolved much better than L. Acknowledgment. The present note isolates and formalizes the “honest bridge” intuition: that probability/distribution is not an optional add-on but is built in at the finite/infinite boundary. 5