TOPOLOGY-CERTIFIED SWARM CONTROL: HARMONIC CIRCULATION, A DWELL-TIME GUARANTEE, AND A CERTIFIED SPEED LIMIT
TOPOLOGY-CERTIFIED SWARM CONTROL:
HARMONIC CIRCULATION, A DWELL-TIME GUARANTEE,
AND A CERTIFIED SPEED LIMIT
NICHOLAS SHANE KOUNS, DO
Abstract. We give a distributed multi-agent control law that certifies mission-level topology—
a prescribed Betti vector—under noisy geometry, intermittent communication, and agent loss.
Persistent homology selects a scale-stable homology class; the combinatorial Hodge Laplacian
supplies its harmonic representative; a discrete de Rham map carries that representative to
agent velocities; a projection enforces safety; and a gate authorizes coordinated motion only
while the certificate holds.
Three results distinguish the treatment here from a purely descriptive one. First, under
the midpoint de Rham map, rigid translations are exact cochains and rotations about a
protected loop carry nonzero circulation, so the certified harmonic mode admits a concrete
reading: it is coordinated circulation about the protected feature. Second, combining the
stability theorem for persistence diagrams with a bound on agent speed yields a dwell time
∆tsafe = (persq− τp)/(4vmax) during which the certified Betti vector provably persists; the
bound is independent of the control law, which is what allows it to compose with the gate.
Third, requiring distributed consensus to converge within the dwell time yields a certified speed
limit v
∗= (persq− τp)/(4Tmix) relating topological margin to communication quality.
We record five corrections to an earlier formulation of this material, two of which invalidate
repairs that appear sound on paper, and state the validation scope precisely: the results
below are algebraic and numerical. No claim is made about flight, hardware, or comparative
performance against established controllers.
1. Introduction
A swarm executing a mission often has a topological specification rather than a geometric
one. A search pattern must remain connected; a cordon must enclose a hazard; a corridor must
not pinch shut. These are statements about homology, not about position, and they are exactly
the statements that survive the loss of individual agents.
The formalism developed here takes that observation literally. The mission constraint is a
target Betti vector. The controller’s job is to move the swarm while keeping that Betti vector
intact, and—critically—to know when it can no longer guarantee this and stand down.
The technical ingredients are established: stability of persistence diagrams [2], combinatorial
Hodge theory on simplicial complexes [3, 6, 7], and projection onto intersections of subspaces
[9, 1]. The contribution claimed is the composition, together with the two quantitative
guarantees in Sections 7 and 8.
What this paper does not claim. Section 11 states the validation scope in full. In brief:
every result below is algebraic (verified symbolically or to floating-point tolerance) or numerical
Date: July 25, 2026.
2020 Mathematics Subject Classification. 55N31, 93A16, 05E45, 93D09.
Key words and phrases. persistent homology, combinatorial Hodge theory, multi-agent systems, formal
certification, swarm control.
12 NICHOLAS SHANE KOUNS, DO
(verified over randomized inputs, cross-checked against an independent implementation). Noth-
ing here establishes aerodynamic feasibility, actuator realizability, robustness under correlated
or adversarial link failure, or superiority over any published swarm controller—no baseline is
implemented, so no comparative claim is supportable.
2. State, communication complex, and Hodge Laplacian
Let N agents occupy positions X(t) = {x1(t),...,xN(t)}⊂Rd with velocities vi(t), and let
ε be a communication radius. The communication complex is the Vietoris–Rips complex
Kε(X) = VRε(X),
whose q-th homology encodes the mission-level topology, βq(ε) = dim Hq(Kε; R).
Write Cq for the space of real q-chains, oriented by the vertex ordering, and Bq : Cq →Cq−1
for the boundary maps, so that B1 has shape N ×|E|and B2 has shape |E|×|T|. The
combinatorial Hodge Laplacian in degree one is
∆1 = B⊤
1 B1 + B2B⊤
2 , (1)
a symmetric positive semidefinite operator on C1. The discrete Hodge theorem gives
ker ∆q(ε)∼
= Hq(Kε; R), dim ker ∆q(ε) = βq(ε), (2)
so each homology class has a unique harmonic representative once the inner product is fixed,
and C1 decomposes orthogonally as
C1 = im B⊤
1
⊕im B2
⊕ ker ∆1
. (3)
gradient
curl
harmonic
We write Pharm for the orthogonal projector onto ker ∆1. In exact arithmetic Pharm =
I−∆†
1∆1; in floating point it is better computed from an orthonormal kernel basis obtained by
symmetric eigendecomposition, which loses fewer digits when the spectral gap above the kernel
is small.
Remark 2.1 (spectral gap). Gate conditions below refer to the spectral gap of ∆q. This must
be read as λβq +1, the smallest eigenvalue strictly above the kernel, not λ2. The distinction is
sharp precisely because βq is itself the tracked quantity.
3. Scale selection
Neighbor graphs flicker under measurement noise, so no single ε is trustworthy. Let Dq(X)
denote the persistence diagram of the Rips filtration. One selects an operating scale by
maximizing a persistence score,
ε∗∈arg max
PersScoreq Dq(X),ε ,
ε
and admits a class for execution only when its lifetime exceeds a threshold τp.
The choice of score is not cosmetic. The obvious candidate—the length of the longest bar
alive at ε—is independent of εon the interior of that bar. Every admissible scale therefore ties,
and the arg max resolves to the smallest admissible ε: the sparsest available communication
graph, which is the worst possible choice, since it maximizes the consensus mixing time of
Section 8. On the ring fixture of Section 9 this costs a factor of 4.8 in mixing time and renders
the distributed implementation infeasible.
Definition 3.1 (margin score).
PersScoreq(D,ε) = max (b,d)∈Dq ,b≤ε<d
min ε−b, d−ε .TOPOLOGY-CERTIFIED SWARM CONTROL 3
This places ε∗ at the midpoint of the longest bar—the scale furthest from both the birth
event (the loop has not yet closed) and the death event (the loop has filled in). It densifies the
communication graph as a side effect, so topological robustness and distributed feasibility pull
in the same direction rather than trading off. That they should align is not obvious a priori,
and it is worth stating explicitly.
4. The de Rham bridge
Control acts on vertices: the actuated quantity is V = (v1,...,vN) ∈C0 ⊗Rd. Topology
lives on edges: ∆1 acts on C1. A formalism that adds a harmonic edge cochain directly to an
agent velocity is not well typed, and the choice of bridge between the two spaces determines
whether the harmonic term carries any information at all. Two natural candidates fail.
Proposition 4.1 (gradient cochains are harmonic-free). im B⊤
1 ⊥ker ∆1. Consequently, if the
edge flow is defined as a difference of vertex potentials—in particular as u([i,j]) = ϕj−ϕi for
any vertex field ϕ—then Pharmu= 0 identically.
Proof. Let h∈ker ∆1. Then 0 = ⟨∆1h,h⟩= ∥B1h∥2 + ∥B⊤
2 h∥2, so B1h= 0. Hence for every
ϕ∈C0, ⟨B⊤
1 ϕ,h⟩= ⟨ϕ,B1h⟩= 0. □
Proposition 4.2 (the strain map annihilates rigid motion). Define R : C0 ⊗Rd →C1 by
(RV)([i,j]) = ⟨vj−vi,
ˆ
eij⟩, whereˆ
eij = (xj−xi)/∥xj−xi∥. If vi = c+ Axi with A⊤=−A,
then RV = 0.
Proof. vj−vi = A(xj−xi), and ⟨Aw,w⟩= 0 for antisymmetric A. □
Proposition 4.1 rules out the most common repair: defining the swarm’s edge flow as a
velocity difference makes it exact, and the harmonic term—the entire content of the formalism—
vanishes. Proposition 4.2 rules out the second: R measures strain rate, so it destroys precisely
the coordinated rigid motion the harmonic term should carry. The correct bridge is the discrete
de Rham map with the midpoint rule.
Definition 4.3 (midpoint de Rham map). W : C0 ⊗Rd →C1,
(WV)([i,j]) =vi+vj
2 , xj−xi ,
the edgewise circulation of the velocity field, i.e. the midpoint approximation of [i,j] v·dℓ.
Proposition 4.4 (translations are exact). If vi = cfor all i, then WV= B⊤
1 ϕwith ϕi = ⟨c,xi⟩,
and hence PharmWV = 0.
Proof. (WV)([i,j]) = ⟨c,xj−xi⟩= ϕj−ϕi. Apply Proposition 4.1. □
Proposition 4.5 (rotations circulate). If vi = Axi with A⊤=−A, then
(WV)([i,j]) = x⊤
jAxi.
In R2 with A= ωJ, J= 0−1
1 0 , this equals ω(xi ×xj), twice the signed area of the triangle
(0,xi,xj). Summing over an oriented cycle γ,
(WV)([i,j]) = 2ωArea(γ) ̸= 0
[i,j]∈γ
for any cycle enclosing nonzero area. Hence WV is not exact, and its harmonic component is
nonzero.4 NICHOLAS SHANE KOUNS, DO
Proof. Expanding, 2(WV)([i,j]) = ⟨Axi + Axj,xj−xi⟩. The terms ⟨Axi,xi⟩and ⟨Axj,xj⟩
vanish by antisymmetry, leaving ⟨Axi,xj⟩−⟨Axj,xi⟩. Since ⟨Axj,xi⟩= x⊤
i Axj=−x⊤
jAxi =
−⟨Axi,xj⟩, this is 2⟨Axi,xj⟩= 2x⊤
jAxi. The cycle sum is Green’s theorem for the discrete
circulation; it is nonzero, so WV / ∈im B⊤
1 , and by (3) the harmonic component cannot vanish
unless the flow is entirely curl, which the cycle sum excludes. □
Propositions 4.4 and 4.5 together supply the interpretation the formalism otherwise lacks.
Corollary 4.6 (reading of the harmonic mode). Under W, the certified harmonic mode is
coordinated circulation about the protected topological feature. Translation contributes nothing to
it; the harmonic component measures exactly the swarm motion that winds around the protected
loop.
Numerically, on the 16-agent ring of Section 9, a rigid rotation is 99.91% harmonic and a
rigid translation is harmonic to within 10−16
.
5. Admissible coordination space
Let Ksafe collect linearized safety, actuator, and conservation constraints, and let Psafe =
I−K†
safeKsafe be the orthogonal projector onto ker Ksafe.
It is tempting to compose Psafe with Pharm. This fails twice over. First, for generic linearized
constraints the two do not commute—measured ∥[Psafe,Pharm]∥≈0.6 on the fixture below—so
their product is neither self-adjoint nor idempotent and is not the projector onto the intersection.
Second, and decisively, after Section 4 the two act on different spaces: Pharm on C1 and Psafe
on C0 ⊗Rd. The composition is not defined.
The repair is to impose “coordination is harmonic” as a linear constraint on the space the
control actually inhabits, and take a single kernel.
Definition 5.1 (admissible coordination space).
Eswarm(t) = ker Ksafe(t)
(I−Pharm(t)) W(t) = ker Ksafe(t) ∩W(t)−1 ker ∆1(ε∗(t)). (4)
This is always a genuine orthogonal projector, and it preserves the reading of the earlier
“ker Ksafe ∩ker ∆1” as the canonical coordination space. The hard control law is V= PEswarm Vgoal.
Remark 5.2 (iterative form). Where an iterative scheme is preferred, alternating projections
converge linearly at rate cos2 θ in the Friedrichs angle θ between the two constraint subspaces
[1]; this is confirmed numerically to within 0.05 of the predicted rate. A small θ is itself a useful
gate signal: it indicates that safety and coordination are nearly incompatible in the current
configuration.
5.1. The variational and algebraic laws coincide. The soft law
V(λ) = arg min
∥V−Vgoal∥2 + λH∥(I−Pharm)WV∥2 + λA∥KsafeV∥2 (5)
V
is often presented alongside the algebraic law as though the two were competing specifications.
They are one specification at different stiffness.
Proposition 5.3. V(λ) →PEswarm Vgoal as λH,λA →∞, at rate O(1/λ).
Proof. The normal equations give V(λ) = (I + λHC⊤C + λAK⊤
safeKsafe)−1Vgoal with C=
(I−Pharm)W. In the eigenbasis of the symmetric operator λHC⊤C + λAK⊤
safeKsafe, the
resolvent acts as 1 on the joint kernel—which is exactly Eswarm—and as (1 + λµ)−1 = O(1/λ)
on each eigenvector with eigenvalue µ>0. □TOPOLOGY-CERTIFIED SWARM CONTROL 5
The finite-λform is the one to implement: it is better conditioned, and it degrades continuously
rather than discontinuously when the two constraint sets nearly conflict.
6. Execution gate
Definition 6.1 (gate).
G(t) = 1 persq(t) ≥τp, λβq +1(∆q(t)) ≥τλ, min
dij(t) ≥dsafe, ∥KsafeV∥≤τK, βq = βtarget
i̸=j
q ,
˙
xi(t) = G(t) Vswarm
i (t) + 1−G(t) Vfallback
i (t).
The fallback may be hover, controlled dispersion, safe landing, or a locally certified formation
controller. Each epoch emits a certificate recording ε∗, the target and observed Betti vectors,
persistence, scale margin, dim ker ∆q, λβq +1, the safety residual, minimum separation, gate
state, dwell time, and the reason any condition failed.
7. The dwell-time theorem
A statement of the form “the Betti vector is preserved so long as no persistence interval
crosses its gate threshold” is not a theorem: the gate closes exactly when an interval crosses its
threshold, so the claim asserts that topology holds while topology holds. What is needed is a
bound on how fast the certificate can degrade, and that bound must not depend on the control
law, or it cannot be composed with the gate that selects the control law.
Lemma 7.1 (filtration Lipschitz bound). Suppose ∥˙
xi(t)∥≤vmax for all i. Then for all t,t′
,
dB Dq(t), Dq(t′) ≤ 2 vmax |t−t′|.
Proof. Set δ= vmax|t−t′|, so ∥xi(t)−xi(t′)∥≤δ for every i. By the triangle inequality
|dij(t)−dij(t′)|≤2δ. The Rips filtration value of a simplex σ is diam(σ) = maxi,j∈σdij,
a maximum of functions each perturbed by at most 2δ, so the filtration functions satisfy
∥ft−ft′∥∞≤2δ. The stability theorem for persistence diagrams of filtered complexes [2] gives
dB ≤∥ft−ft′∥∞. □
Corollary 7.2 (persistence Lipschitz bound). Under the hypotheses of Lemma 7.1, the lifetime
of any matched bar changes by at most 4vmax|t−t′|.
Proof. A bottleneck matching of cost η moves each birth and each death by at most η in ℓ∞
,
so a lifetime d−b changes by at most 2η≤4vmax|t−t′|. If a bar is matched to the diagonal it
has lifetime at most 2η, which is consistent with the same bound. □
Theorem 7.3 (persistence-gated dwell time). Let the gate be open at time t0 with certified
persistence persq(t0) = p0 >τp, and suppose ∥˙
xi∥≤vmax on [t0,t1]. Define
∆tsafe =
Then for every t∈[t0, t0 + min(∆tsafe, t1−t0)]:
(a) the executed command is admissible, KsafeV(t) = 0;
(b) the certified class retains a harmonic representative in ker ∆q(ε∗(t));
(c) persq(t) ≥τp, so the persistence condition of the gate cannot fail on this interval;
(d) βq(Kε∗(X(t))) = βtarget
q.
Moreover ∆tsafe depends only on vmax and the certified margin—not on the control law.
p0−τp
(6)
4 vmax6 NICHOLAS SHANE KOUNS, DO
Proof. (a) By construction V= PEswarm Vgoal and Eswarm ⊆ker Ksafe, so KsafeV = 0; this is
immediate from idempotence and self-adjointness of the orthogonal projector.
(c) By Corollary 7.2, |persq(t)−p0|≤4vmax(t−t0) ≤4vmax∆tsafe = p0−τp, hence persq(t) ≥
p0−(p0−τp) = τp.
(d) By (c) the selected class is alive throughout the interval, and the margin score of
Definition 3.1 keeps ε∗(t) inside its bar; therefore the class persists in Hq(Kε∗(t); R) and the
observed Betti number in the selected degree equals the target.
(b) By (2) applied at ε∗(t), the class of (d) has a harmonic representative in ker ∆q(ε∗(t)). □
Remark 7.4 (what the theorem does and does not cover). Theorem 7.3 certifies the topological
gate conditions for a computable interval. The separation and residual conditions are instanta-
neous algebraic checks and are not deferred by it; they must still be evaluated each epoch. The
practical content is that persistence and the Hodge spectrum—the expensive quantities—need
recomputing only every ∆tsafe, while the cheap conditions are checked continuously.
8. Distributed feasibility and a certified speed limit
Persistence diagrams and harmonic bases are global objects. Indeed, by (3) the harmonic
component is by construction the part of an edge flow that no local computation can determine:
it is orthogonal to every gradient and every curl, so it is invisible to any agent inspecting only
its own neighborhood. Distributed realization therefore requires consensus.
Let L0 = B1B⊤
1 be the graph Laplacian of the communication graph and λ2(L0) its al-
gebraic connectivity. Gossip or consensus power iteration reaches relative accuracy tol in
ln(1/tol)/λ2(L0) iterations; with per-iteration radio latency tround,
ln(1/tol)
Tmix =
·tround. (7)
λ2(L0)
Consensus must converge before the topological margin erodes:
Tmix < ∆tsafe. (8)
Solving (8) for speed gives the operationally useful output of the entire formalism.
Theorem 8.1 (certified speed limit). Under the hypotheses of Theorem 7.3, distributed harmonic
tracking at accuracy tol can keep pace with topological drift only if
p0−τp
vmax < v∗
=
(9)
4 Tmix
with Tmix as in (7). Equivalently, v∗ is the unique speed at which (8) is saturated.
Proof. Substitute (6) into (8) and solve; monotonicity of ∆tsafe in vmax gives uniqueness. □
On the ring fixture, v∗= 2.43 m/s on a 100 Hz mesh and 0.243 m/s at 10 Hz: a tenfold
increase in radio latency costs a tenfold reduction in certified speed. Condition (8) is necessary,
not sufficient—it is stated in terms of a mixing-time bound, not an implemented consensus
algorithm—and it is the natural target for the next stage of validation.
9. Numerical validation
All operators—Rips complexes, boundary maps, the Hodge Laplacian, and persistent H1 by
standard column reduction—were implemented from scratch, with no topology library in the
dependency path, so that an established library could serve as a genuine independent oracle
rather than a self-consistency check. Betti numbers, persistence pairs, and bottleneck distances
agree with gudhi [5] to 10−9 on all fixtures.TOPOLOGY-CERTIFIED SWARM CONTROL 7
Fixtures: a 16-agent ring (β1 = 1), a jittered ring, two disjoint rings (β1 = 2), and random
clouds in R2 and R3
.
Claim Section Result
dim ker ∆1 = β1, five fixtures ×scales (2) exact
B1B2 = 0 §2 <10−16
dim ker(∆1 ⊗Id) = dβ1, d= 2,3 §2 exact
KsafePsafe = 0 §5 <10−15
∥PharmB⊤
1 ∥ Prop. 4.1 2 ×10−14
harmonic fraction, rigid translation Prop. 4.4 <10−16
harmonic fraction, rigid rotation Prop. 4.5 0.9991
rank(PharmW) = β1 Cor. 4.6 exact
V(λ) →PEswarm Vgoal, decade ratios Prop. 5.3 9.6, 10.0, 10.0
alternating projection rate vs. cos2 θ §5 within 0.05
|∆pers|≤4vmax∆t, 400 trials Cor. 7.2 worst ratio 0.714
dB ≤2vmax∆t (oracle) Lem. 7.1 worst ratio 0.959
β1 preserved, 20 certified epochs Thm. 7.3 held
gate closes when loop is torn Def. 6.1 step 37
gate closes on separation violation Def. 6.1 held
Table 1. Validation summary. 68 checks, 0 failures.
9.1. Negative controls. A suite that cannot fail establishes nothing. Four errors were injected
and each was detected: dropping B2B⊤
2 from (1) (dim ker = 17 against β1 = 1); replacing B2
by its entrywise absolute value, caught by B1B2 = 0 at residual 11.3; substituting the strain
map of Proposition 4.2 for W (rotation drops from 99.9% to 54.7% harmonic); and weakening
the constant in (6) from 4 to 1, which makes the bound false at worst ratio 3.19. The last
confirms that the constant is load-bearing rather than slack.
Two remarks on tightness. The bound of Corollary 7.2 is attained to within 29%, so it is
conservative but not vacuous. The unsigned-B2 error was not detected by kernel dimension on
the symmetric ring, only on the random cloud—which is why validation sweeps several fixture
families rather than one.
10. Errata to the earlier formulation
An earlier draft of this material, circulated as Persistence-Gated Hodge Swarm Control,
contained five defects. They are recorded here in full. Two of them invalidate repairs that
appear sound on paper, which is the argument for executing a formalism rather than only
reading it.
Erratum 1 (type mismatch). The earlier draft placed u∈C1 and then wrote ˙
xi = Gui, adding
an edge cochain to a vertex velocity. Both obvious repairs fail: velocity-difference flows are exact
and carry no harmonic component (Prop. 4.1), and the strain map annihilates rigid motion
(Prop. 4.2). The correct bridge is Definition 4.3, which additionally supplies Corollary 4.6.
Erratum 2 (tautological invariance claim). The earlier claim that the Betti vector is preserved
“so long as no persistence interval crosses its gate threshold” assumes its conclusion. It is
replaced by Theorem 7.3, which is quantitative and control-law independent.8 NICHOLAS SHANE KOUNS, DO
Erratum 3 (commuting projectors). The earlier draft took PsafePharm as canonical under a
commuting hypothesis. The hypothesis fails generically, and after Erratum 1 the two projectors
act on different spaces, so the product is undefined. Replaced by Definition 5.1.
Erratum 4 (two specifications for one law). The cost functional and the algebraic law were
presented without relation. Proposition 5.3 shows the latter is the stiff limit of the former.
Erratum 5 (unspecified persistence score). PersScore was left undefined, and the obvious
choice is scale-independent, collapsing the arg max onto the sparsest communication graph.
Replaced by Definition 3.1; see Section 3.
Two minor corrections. The earlier theorem’s claims (a) and (d) are the same statement—(d)
holds by construction of the projected command—and should be merged. The gate’s spectral
condition requires the reading of Remark 2.1. Finally, the harmonic representative hq is defined
only up to an O(βq) rotation when βq >1; the control law above uses Pharm throughout and is
therefore basis-free, but a certificate recording a specific basis is not canonical.
11. Scope of validation
We adopt three evidence classes. E0: algebraic identity or theorem, machine-checked
symbolically or to floating-point tolerance. E1: numerical certificate over randomized inputs.
E2+: hardware-in-the-loop or flight.
Everything in Sections 3–7 is E0 or E1. Nothing in this paper is E2 or above.
Explicitly not established: aerodynamic feasibility; actuator realizability; robustness under
correlated, adversarial, or Byzantine link failure; sensor noise models; regulatory compliance;
airworthiness; and any comparative performance claim—no baseline controller is implemented,
so no statement of superiority over Reynolds flocking [8], ORCA, or MPC-based approaches is
supportable by anything here. Condition (8) is a necessary condition derived from a mixing-time
bound; no consensus algorithm is implemented, and no claim is made that any converges on
physical radios.
All results are for q= 1 and for fixtures of 14–20 agents. The formalism is stated for general
q; higher degrees are untested, and the dense linear algebra used here will not reach 100+
agents without sparse operators.
12. Open problems
(1) Distributed harmonic tracking. Implement consensus power iteration on ∆1 and
measure convergence against ∆tsafe, converting (8) from a necessary condition into a
realized one. This is the natural next E1 contribution.
(2) Baselines. Implement Reynolds flocking and an MPC controller under identical
conditions. Until then no comparative claim is available, and the formalism’s practical
value is unmeasured.
(3) Sharper dwell time. Corollary 7.2 is worst-case over adversarial motion. A bound
exploiting the fact that the executed motion lies in Eswarm—and is therefore harmonic,
hence circulating rather than radial—should be substantially tighter.
(4) Higher degree. Extend validation to β2, where the protected feature is an enclosed
void rather than a loop.
(5) Scaling. Sparse boundary operators and Lanczos iteration for ker ∆1, to reach swarm
sizes of practical interest.TOPOLOGY-CERTIFIED SWARM CONTROL 9
Reproducibility
An implementation accompanies this paper: a package providing the operators and control
law, a unit-test suite, and a validation report that regenerates Table 1 and fails if any check fails.
Persistent H1 is implemented independently of any topology library so that gudhi functions as
an external oracle.
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AIMS Research Institute, Las Vegas, Nevada
