Stable Set of Points (SSP) in Mathematics

• Stable Set of Points (SSP) is a property defining directional regularity for set germs and other configurations, ensuring that limiting directions are asymptotically realized.
• The SSP framework underpins bi-Lipschitz invariance through blow-up processes, enabling precise tangent-cone transport and robust geometric rescaling methods.
• SSP concepts extend to SAT, tropical invariant theory, and topological dynamics, offering diverse applications for exploring point stability and structure in theoretical research.

Searching arXiv for the cited SSP-related papers to ground the article in current records. Stable Set of Points (SSP) is not a single uniform notion across contemporary mathematics and theoretical computer science. In Koike–Paunescu’s note “Notes on (SSP) sets,” the acronym “(SSP)” denotes the Sequence Selection Property for set germs at the origin, a directional regularity condition used to extend Sampaio’s bi-Lipschitz tangent-cone invariance theorem from subanalytic sets to a wider class (Koike et al., 2015). In other literatures, the same initials designate a stable set of falsifying assignments for CNF formulas, a stable complete coordinate system for unordered point multisets, or the set of topologically stable points of a group action (Goldberg, 17 Jul 2025, Kubo, 29 Jun 2026, Khan et al., 2020).

1. Terminological scope and principal usages

A common source of confusion is that “SSP” does not denote the same object in all papers. In local real geometry, the term refers to a property of a germ $A \subset \mathbb{R}^n$ near $0$; in SAT, it refers to a subset of $B^n$ closed under clause-directed Hamming-neighborhood moves; in tropical invariant theory, it refers to a stable representation of unordered point clouds; and in topological dynamics it is identified with the set of topologically stable points. The following table organizes the principal usages appearing in the cited arXiv literature (Koike et al., 2015, Goldberg, 17 Jul 2025, Kubo, 29 Jun 2026, Khan et al., 2020).

Usage	Underlying object	Defining feature
Sequence Selection Property	Germ $A \subset \mathbb{R}^n$ at $0$	Directional selection of sequences asymptotic to rays in $D(A)$
Stable set of points for SAT	$P \subset Z(F)$	Closure under $Nbh(p,g(p))$ for a transport function $g$
Stable coordinates for multisets	Orbit $\mathbb{R}^{nr}/S_n$	Injective, permutation-invariant, bi-Lipschitz coordinate map
SSP of a group action	$0$0	Set of topologically stable points

Within this landscape, the Sequence Selection Property is the most technically developed geometric meaning attached to “(SSP)” in the supplied sources. It is also the setting in which the acronym is introduced explicitly as “Sequence Selection Property,” rather than “stable set of points” (Koike et al., 2015).

2. Sequence Selection Property for set germs

For a set germ $0$1 at $0$2 with $0$3, the direction set is

$0$4

If $0$5, the half-cone generated by $0$6 is

$0$7

and the real tangent cone used in the paper is

$0$8

The Sequence Selection Property is a directional approximation condition. A germ $0$9 satisfies (SSP) if, for any sequence $B^n$0 with $B^n$1 and

$B^n$2

there exists a sequence $B^n$3 such that

$B^n$4

The paper also gives the equivalent geometric criterion

$B^n$5

This criterion makes clear that (SSP) is a raywise first-order regularity condition: every limiting direction of $B^n$6 is asymptotically realized by points of $B^n$7 at error $B^n$8 along the corresponding ray (Koike et al., 2015).

The paper works with $B^n$9 rather than other tangent constructions. For $A \subset \mathbb{R}^n$0 manifolds containing $A \subset \mathbb{R}^n$1, $A \subset \mathbb{R}^n$2. In many tame settings $A \subset \mathbb{R}^n$3 also coincides with the Bouligand tangent cone, but the note is formulated entirely in terms of the cone generated by limiting directions. This focus is essential because the subsequent bi-Lipschitz arguments transport directions rather than arbitrary tangent vectors (Koike et al., 2015).

3. Bi-Lipschitz invariance and tangent-cone transport

The central result of Koike–Paunescu is that Sampaio’s rescaling method extends from subanalytic germs to germs satisfying (SSP). For a bi-Lipschitz map $A \subset \mathbb{R}^n$4, with constants $A \subset \mathbb{R}^n$5 such that

$A \subset \mathbb{R}^n$6

for all $A \subset \mathbb{R}^n$7, the rescaled maps are

$A \subset \mathbb{R}^n$8

By Arzelà–Ascoli on compact neighborhoods of $A \subset \mathbb{R}^n$9, there exists a subsequence $0$0 such that

$0$1

where $0$2 is again bi-Lipschitz with the same constants. This $0$3 is the blow-up of $0$4 at $0$5.

To reduce a germwise bi-Lipschitz homeomorphism $0$6 to the global setting required by the rescaling argument, the note invokes Whitney–Banach extensions $0$7 and then defines

$0$8

together with

$0$9

which satisfies

$D(A)$0

This doubling process allows the blow-up construction to be applied to a globally defined bi-Lipschitz map.

The core lemma states that if $D(A)$1 satisfies (SSP), then for the blow-up limit $D(A)$2,

$D(A)$3

If both $D(A)$4 and $D(A)$5 satisfy (SSP), then the reverse inclusion follows by applying the same argument to $D(A)$6, yielding

$D(A)$7

A frequent misconception is that equality of tangent cones under bi-Lipschitz homeomorphism would follow from (SSP) on only one side; the note shows that one-sided (SSP) yields only the forward inclusion, whereas equality requires (SSP) for both germs (Koike et al., 2015).

The same blow-up mechanism controls intersections of direction sets. If $D(A)$8 are germs at $D(A)$9, $P \subset Z(F)$0 is bi-Lipschitz, and $P \subset Z(F)$1 satisfy (SSP), then

$P \subset Z(F)$2

If $P \subset Z(F)$3 and $P \subset Z(F)$4 also satisfy (SSP), then equality holds. This yields invariance of transversality for

$P \subset Z(F)$5

and invariance of weak transversality, defined by $P \subset Z(F)$6, under milder hypotheses (Koike et al., 2015).

4. Structural properties, examples, and limitations

The class of sets satisfying (SSP) is strictly wider than the subanalytic and o-minimal definable classes. The note lists several families of examples: cones $P \subset Z(F)$7 for arbitrary germs $P \subset Z(F)$8, subanalytic sets, sets definable in an o-minimal structure, finite unions of (SSP) sets, and $P \subset Z(F)$9 manifolds through the origin. For $Nbh(p,g(p))$0 manifolds, the tangent cone is the classical tangent space $Nbh(p,g(p))$1. This suggests that (SSP) should be understood as a directional regularity hypothesis rather than as a definability condition in the model-theoretic sense (Koike et al., 2015).

The paper also records sharp limitations. Zigzag curves in $Nbh(p,g(p))$2 oscillating between two rays can fail (SSP). If

$Nbh(p,g(p))$3

with $Nbh(p,g(p))$4 a continuous zigzag and $Nbh(p,g(p))$5 bi-Lipschitz, then the image of the positive $Nbh(p,g(p))$6-axis cannot have (SSP). Conversely, if the zigzag image does satisfy (SSP), then $Nbh(p,g(p))$7 is not bi-Lipschitz, so $Nbh(p,g(p))$8 is not Lipschitz. Another limitation is that (SSP) is not invariant under blow-up in general: the note describes constructions where a set $Nbh(p,g(p))$9 in the blow-up manifold satisfies (SSP) but its image under the blow-up map fails (SSP), and also two-dimensional zigzag examples in which the opposite behavior occurs (Koike et al., 2015).

The note contains an additional one-dimensional “numerical SSP” for strictly decreasing positive sequences $g$0: $g$1 If $g$2, then $g$3 and $g$4 also lie in $g$5. At the same time, (SSP) does not imply polynomial boundedness, and polynomial boundedness does not imply (SSP). The note gives $g$6 as an example satisfying (SSP) but not PB, and describes PB sequences with sparse large jumps for which $g$7, hence not (SSP) (Koike et al., 2015).

Two consequences emphasize the rigidity encoded by $g$8. If $g$9 has a neighborhood bi-Lipschitz homeomorphic to an open set in $\mathbb{R}^{nr}/S_n$0, then $\mathbb{R}^{nr}/S_n$1 is bi-Lipschitz to $\mathbb{R}^{nr}/S_n$2. If $\mathbb{R}^{nr}/S_n$3 has a neighborhood $\mathbb{R}^{nr}/S_n$4 bi-Lipschitz homeomorphic to a cone $\mathbb{R}^{nr}/S_n$5, then $\mathbb{R}^{nr}/S_n$6 and $\mathbb{R}^{nr}/S_n$7 are bi-Lipschitz homeomorphic and

$\mathbb{R}^{nr}/S_n$8

These statements situate (SSP) within a broader program of reading local Lipschitz geometry from tangent data (Koike et al., 2015).

5. Other established meanings of SSP on point sets

In SAT theory, a stable set of points is a non-empty set $\mathbb{R}^{nr}/S_n$9 of complete assignments falsifying a CNF formula $0$00, together with a transport function $0$01 such that for every $0$02,

$0$03

Here $0$04 is the Hamming-distance-one neighborhood that repairs the clause $0$05 falsified by $0$06. The core theorem is exact: $0$07 is unsatisfiable if and only if there exists such an SSP. Because pointwise SSPs can be very large, the paper introduces stable sets of clusters (SSCs), including cube-based clusters and symmetry-based clusters, and proves that the union of an SSC is an SSP. This yields a sound and complete, cluster-based route to SAT and UNSAT certificates, with an explicit emphasis on structure exploitation and parallel computation (Goldberg, 17 Jul 2025).

In tropical invariant theory, SSP refers to a stable complete representation of unordered point sets. A multiset of $0$08 unordered points in $0$09 is treated as an orbit in $0$10, and the paper constructs the $0$11 basic $0$12-symmetric tropical polynomials $0$13, indexed by $0$14 with $0$15, as explicit permutation-invariant coordinates. The family $0$16 separates $0$17-orbits, admits a constructive inversion via the dual transform

$0$18

and defines a bi-Lipschitz embedding of $0$19. The paper further shows that pairwise-supported invariants suffice exactly for $0$20, and that in general one needs invariants supported on at least three columns and of degree strictly less than $0$21 (Kubo, 29 Jun 2026).

In topological dynamics, the SSP of an action is defined by

$0$22

the set of topologically stable points of a first countable Hausdorff group action $0$23 on a compact metric space. A point $0$24 is one for which sufficiently small perturbations $0$25 admit a continuous orbit semiconjugacy $0$26 with $0$27 and $0$28 on the perturbed orbit. The paper develops the related notion of $0$29-persistent points,

$0$30

proves that $0$31 is $0$32, proves that $0$33 is closed when $0$34 is equicontinuous, and shows that every equicontinuous pointwise topologically stable action is $0$35-persistent (Khan et al., 2020).

6. Adjacent stable-set frameworks and broader context

A related but distinct use of stable sets appears in social choice and game theory. For an irreflexive binary relation $0$36 on a possibly infinite set $0$37, the paper “A Characterization Framework for Stable Sets and Their Variants” interprets the stable set as the von Neumann–Morgenstern stable set of points in $0$38: internal stability excludes $0$39-domination inside the set, and external stability requires every outside point to be $0$40-dominated by some selected point. The paper extends this framework to infinite $0$41, together with extended stable sets, socially stable sets, $0$42-stable sets, and $0$43-stable sets, and characterizes their existence via compact topologies, T1-order separation, and Nachbin closedness (Andrikopoulos et al., 13 Aug 2025).

In cooperative game theory, stable sets of allocation points arise in pillage games. Allocations are points

$0$44

and a stable set has the usual internal and external stability properties relative to a domination relation induced by a power function. The combinatorial input is that a stable set cannot contain a strictly monotonic sequence of length $0$45, which leads to the bound

$0$46

and, in dimension $0$47,

$0$48

This is a stability theory for finite point configurations, but not an instance of the Sequence Selection Property (Saxton, 2010).

The acronym conflict is especially strong in combinatorial optimization, where SSP usually denotes the Stable Set Problem rather than any notion about points. The paper on Lovász–Schrijver lift-and-project operators studies compact SDP relaxations for the maximum stable set problem in graphs, compares $0$49, $0$50, clique-lift and nodal-lift relaxations, and reports that clique-based formulations are preferable on sparse graphs while nodal-based ones are stronger on dense graphs (Battista et al., 2024). This literature is conceptually adjacent only through the phrase “stable set.”

Two further geometric lines develop stability of point configurations without adopting SSP as their primary name. “Stability of optimal spherical codes” proves quantitative rigidity for near-optimal spherical point sets, with general $0$51 stability for Delsarte-tight codes and Lipschitz $0$52 stability for the $0$53 and Leech configurations (Böröczky et al., 2017). “Delaunay stability via perturbations” constructs a perturbed finite point set $0$54 whose restricted Delaunay complex is $0$55-generic, with all $0$56-simplices $0$57-thick and $0$58-protected, thereby guaranteeing stability of the Delaunay triangulation under sufficiently small perturbations (Boissonnat et al., 2013). These works show that, even when the SSP acronym is absent, stability of point sets is a recurring theme across discrete and metric geometry.