A general mechanism for collective foraging: percolating recruitment cascades coupled to a maximum-entropy

A general mechanism for collective foraging: percolating recruitment cascades coupled to a maximum-entropy field

arXiv:2610.24865v1 [physics.soc-ph] 17 Oct 2026

Galen J. Wilkerson1 1

Komplexity AI LLC

(Dated: October 20, 2026)

1

Abstract Foraging recruitment is a contagion: an agent that finds a resource triggers a linear-threshold cascade—the model built for rumour and adoption spread on social networks—embedded in space. Because activation is degree-normalized, the same rule that spreads the cascade also stifles it as neighbourhoods crowd (density-dependent blocking), so the swarm aggregates on a resource and then releases to explore again. The cascade runs on the communication network, but acting on it moves agents in physical space, and that motion rewires the network: network and physical space are one coupled system, and this feedback—not any tuning—is what makes the coordination self-limiting. Because recruitment chains along the graph rather than reaching near neighbours alone, whether a discovery stays local or spans the swarm is a matter of percolation, and past a critical connectivity the swarm holds itself in a cascade window —locally near-critical though globally subcritical, a self-organized, sandpile-like drive–avalanche cycle. Nothing is tuned, yet behaviours foraging biology documents separately emerge together —recruitment to discoveries, marginal-value patch-leaving, and Lévy search with a switch to local exploitation. These are not coincidences but one least-committal principle: the resource field and the search that exploits it both follow from maximum entropy—Lévy search matched to the field’s sparsity, recruitment to its clustering, so active-matter order and optimal foraging are one system seen in two research domains. The gains are real but conditional : up to 7.6× the harvest of solitary search when resources are sparse and clustered, shrinking to nothing when search is already easy. Coordination through percolating cascades on the network yields a large return on minimal communication. And that sparse, clustered field is the generic one—resources almost inevitably aggregate, and maximum entropy makes the clustered field the least-committal one once they do—so one untuned rule should recur widely.

I.

INTRODUCTION

Many living systems solve the same problem: a group must find and gather resources scattered unpredictably across space, using only local sensing and local communication, with no central controller and no global map. Ants, bees, microbes, and foraging birds face it [1–3]; so do engineered swarms searching for disaster survivors, crop-pest outbreaks, or wildfire hotspots [4]. In every case the targets are typically sparse and patchy, and the 2

searchers cheap, numerous, and short-sighted. What local rule lets such a swarm coordinate well without central control? We study one, biologically motivated and deliberately minimal: agents explore on their own, and an agent is recruited to a discovery once a threshold fraction of its neighbours are already active. Coordination requires bridging scales—carrying microscopic, local behaviour up to the macroscopic—and the rule does so in two coupled spaces (Fig. 4): its exploration is a Lévy search in physical space (maximum entropy), and its recruitment spreads as a cascade on the communication network (percolation).

The whole loop is our organizing picture (Fig. 1; drawn in space in Fig. 3): agents explore with long hops until one finds a resource and recruits its neighbours; they recruit theirs, so the discovery spreads through the crowd; recruits switch to short local moves and converge on the resource; crowding then blocks further recruitment, and agents revert to long-hop exploration.

A single directed logic sits behind this loop. The resource field comes first, and is exogenous: because resources generically aggregate, a sparse, clustered field is the least-committal one a forager faces (§V E), and the agents do not reshape it. That field has two features, and the mechanism answers each: long Lévy hops to cross the voids of the sparse field, local convergence to exploit a cluster once recruitment finds one (Fig. 3). Movement and recruitment then close a loop (Fig. 1a): a cascade pulls explorers into convergence, whose crowding reshapes the network the next cascade travels on. Because the field is the generic one and neither the search nor the recruitment is tuned, one minimal rule should coordinate the swarm across many environments—and, with nothing in it written for foraging, deliver together the behaviours foraging biology documents separately. 3

sca ca

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se explore relea dissipation RGG 2/ϕ zc

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(block + consume)

network / spatial density (sparse → near-critical → dense)

(a) the cycle (schematic) FIG. 1.

(b) the cycle (measured)

The swarm runs a sandpile-like (Bak–Tang–Wiesenfeld) drive–avalanche

cycle, self-organised to a near-critical window—schematic and data, in the same density×activity plane. [5] (a) The cycle in the network-density×activity plane. A resource discovery triggers a recruitment cascade (explore→recruit→consume); a consuming agent then either blocks (crowded, →blocked) or, once the resource is depleted, disperses directly back to exploration, and blocked agents are released to explore—so consumption exits to either state. States coloured recruit (blue), consume (green), blocked (red), explore (grey). The shaded band is the cascade window, bounded by the percolation onset zcRGG and the blocking onset 2/ϕ (set by the activation threshold ϕ; §V A). Inward arrows below the density axis show the swarm is driven into the window from both sides—aggregation (drive) from too-sparse, blocking+consumption (dissipation) from too-dense. (b) Empirical phase portrait (128 replicas): shaded regions mark where each node state is richest (bin-mean occupancy ≥ 1.5× its plane-average; Methods)—explore (grey), recruit (blue), consume (green), blocked (red)—with grey mean-flow streamlines circulating into a stable fixed point (the near-critical operating point) and the ż=0 (mean degree z; cyan) / Ȧ=0 (active fraction A; magenta) nullclines. The measured circulation realises the schematic’s cycle; the relaxation into the fixed point is overdamped and excitable—SOC family, held just below local criticality by dissipative blocking. zcRGG and 2/ϕ mark the window. The consume-rich region is small and sits inside the recruit-rich band near the peak—consumption is brief, the swarm passing through it fast, like the sandpile’s short avalanche.

The loop is simple; what comes out of it is not. We begin from a property of the linear 4

threshold model (LTM), not from a task. In the LTM each node stays inactive until the fraction of its neighbours that are active reaches its threshold ϕ, whereupon it activates and can trigger others [6, 7]. Because activation is on the fraction a/k ≥ ϕ (a active neighbours out of degree k), cascade size is non-monotonic in connectivity (Fig. 2). The lower boundary is a percolation transition [7]: below it the cluster of vulnerable nodes—those a single active neighbour can activate—does not span the graph, so cascades stay local; once that cluster percolates, a discovery can recruit a system-spanning fraction. But as degree grows further the same fraction rule renders high-degree nodes non-vulnerable, and cascades are blocked. Two boundaries, from one rule—and neither inserted by hand [7]. This non-monotonic response of cascade size to connectivity is a control surface (Fig. 2), whose two boundaries drive the cycle the swarm traces (Fig. 1). This control surface—the cascade response to connectivity—is static: it treats connectivity as a fixed, external input. But the nodes are mobile agents whose recruitment draws them together, so connectivity stops being an external dial and becomes endogenous—a state the population sets for itself: cascades pull agents together, aggregation raises degree, rising degree throttles the cascade, and agents are released. The mechanism therefore lives in the coupling of a network space—the communication network, who can recruit whom—and an embedding space (here physical position) whose geometry induces that network—and it is this coupling, not either space alone, that makes connectivity endogenous. Concretely the agents solve a spatial task —find and aggregate at dispersed resources—with only local binary communication; the same task recurs across the biological and engineered settings above, whose targets share one structure (sparse but locally clustered, §V E). We take collective foraging—the biological archetype of this task—as our worked example. a.

Elegant ‘coincidences’. The starting point of our investigation was the non-monotonic

cascade itself—a behaviour that turns on and off, falling out of basic mathematics. Pursuing it we kept finding more ingredients we never chose, each simply what the underlying structure already supplies, and several elegant in a sharper sense: a structure the mathematics forces turns out to be a behaviour foraging biology has long documented, so the rule does not merely permit natural foraging, it predicts it. We flag the claims here and earn each in the section named. They are one inventory read three ways—an untuned list here, a mechanism in motion in the drive–avalanche cycle (Fig. 1), and one principle seen from three fields (§VI C 0 b)—not three competing schemes. 5

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FIG. 2. Recruitment works only in a connectivity window—too sparse to spread below, self-blocked above—and the swarm self-tunes into it. Mean cascade fraction (±1 s.d.) versus observed mean degree z for the linear-threshold rule on a proximity graph. Recruitment turns on near the random-geometric-graph percolation threshold zcRGG ≈ 4.51 (giant-component fraction, dashed, right axis) and is suppressed again at higher density (blocking), peaking at intermediate connectivity—two boundaries from the single fraction rule a/k ≥ ϕ. The giant component visibly grows below zcRGG through finite-size smearing (finite-size scaling places the transition at zcRGG ; Methods). The red arrows below the axis are the restoring connectivity flow—the sign of ż from the cycle (Fig. 1): too sparse, cascades aggregate agents and raise z; too crowded, blocking disperses agents and lowers z; so the population self-tunes into this cascade window. These two boundaries—recruitment on, blocking off—are the drive and dissipation of the cycle (Fig. 1): the window sets the two opposing tendencies whose balance the swarm traces as the drive–avalanche loop. Quantified in §V A.

1. Recruitment is contagion.

The cascade is the linear-threshold model of ru-

mour/adoption spread on networks [6, 7]; foraging recruitment is that model embedded in space. The identification is not merely formal: in real foraging, recruitment spreads through local positive-feedback signals—pheromone trails, dances, quorum cues—by which an agent commits once enough nearby foragers already have [1, 2]: 6

dense network

blocking

exploration Lévy flights seed

recruitment cascade

resources

local Brownian moves

FIG. 3. Spatial schematic of the mechanism’s phases: explore, recruit to a discovery, then self-limit. Agents explore by long-range (Lévy) search; a discovery seeds a linear-threshold recruitment cascade on the proximity graph, drawing neighbours in, while density-dependent blocking self-limits it and releases agents back to exploration.

a threshold contagion. The signal travels the graph farther than agents walk, reaching bodies nowhere near the discovery, so percolation—not mere proximity—governs whether spread stays local or spans the system (spread scaling with size). Spatial embedding then turns contagion into aggregation: exploration and recruitment become two diffusions, through physical space and through the communication network (§II, V A). 2. Blocking is inherited, and does an ecological job. Because activation is degreenormalized, the same rule that spreads the cascade also stifles it once neighbourhoods saturate—the mobile analogue of rumour-stifling in the Daley–Kendall/Maki– Thompson sense [8, 9] (a spreader that meets saturated neighbours stops spreading), so the upper cascade boundary is inherited, not added (blocking comes “for free”). Ecologically, that same down-slope is marginal-value patch-leaving—crowding drives a/k below ϕ and releases agents, with no rate estimation or memory (§VI B) [10]. 3. The two spaces close the loop—recruitment moves agents in space, motion rewires the graph, the graph changes who can recruit, so the population transits the window under its own dynamics—a self-organized, sandpile-like drive–avalanche cycle held just below local criticality—a feedback loop between physical and network space (§V, VI A; Figs. 1, 4). 4. Sparsity dictates the search mode—heavy-tailed (Lévy) exploration is advanta7

geous for sparse, patchy targets, clustering makes local exploitation correct once one is found, and the cascade state selects between them (§V D) [11]. 5. The environment supplies the sparse-clustered structure for free—resources generically aggregate, so a clustered field is the least-committal arrangement; nonaggregation (regularity, or exact independence) is what would require a mechanism (§V E) [12]. 6. Search mode and field structure match without being made to—independently observed Lévy search in real foragers [3] then reads as the same maximum-entropy fit (§VI C 0 b). The through-line: coordination emerges not because parameters were fit to it but because a minimal rule is matched to the statistics the environment almost inevitably has. The behaviours it reproduces are claimed by different fields—collective order by active matter, patch-leaving and Lévy search by foraging ecology—yet they are coupled, not merely copresent: maximum entropy fixes one side (the field and the search, §VI C 0 b), self-organized criticality the other (the cascade, §VI A), and the cascade is the emergent, locally nearcritical coordination that couples them—not a third thing derived from the same principle. That is what makes the coincidences cohere, not merely co-occur: they are not coincidences at all, but one coupled system met many times. Read one way, this is an inference-first account of foraging: the behaviours are inferred as the least-committal response to limited information (maximum entropy, after Jaynes [13]), so those field-claimed behaviours are not separate adaptations but emerge together when a single local recruitment rule realizes the maximum-entropy principles on a moving spatial graph. Inference sets the optima—where resources sit, how to move, how to allocate—the cascade reaches them, and self-organized criticality follows. Read another way, recruitment cascades and Lévy search are two scale-free bridges from local rules to macroscopic behaviour (Fig. 4)—one on the communication network (percolation), one in physical space (maximum entropy)—joined not by a common derivation but by a shared substrate: the communication network is the swarm’s geometry seen as a graph. Behaviourally they are the two opposing flows on connectivity—Lévy exploration disperses the swarm and lowers it, recruitment aggregates and raises it, blocking caps the rise—so their balance self-tunes the swarm onto the percolation window (§V, VI A). 8

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graph ← positions recruitment → motion blocking → dispersal

long Lévy hop

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local rule a/k ≥ ϕ

FIG. 4. Cascades and Lévy flights: two scale-free bridges, one coupled system. Two scale-free bridges lift microscopic behaviour to macroscopic: recruitment cascades on the communication network (left), Lévy search in physical space (right). Network and space are not separate domains but two views of one system—the communication network is the swarm’s spatial configuration seen as a graph. The scale-freeness of the two bridges has different origins—emergent and collective for the cascade, inferred (maximum entropy) and per-agent for the Lévy walk—so they are unified by their coupling (teal, Q3), not a shared derivation.

Because the sparse, clustered field this needs is the generic one—what foraging environments almost inevitably present (§V E)—the conditions the mechanism requires are the common ones, so a single untuned rule should recur widely rather than being fitted to one environment. Its pieces are field-matched, not field-tuned : the optimal search exponent tracks the field’s geometry (α∗ ≈ Df , §V D), the recruitment cascades are critical on these sparse-clustered fields (mean-field branching on the near-CSR field—complete spatial randomness, CSR: the baseline of independent, uniform placement with no attraction or repulsion, a homogeneous Poisson point process; §V B), and the network–space coupling carries coordination across space far faster than the agents move (∼150×, §V C). 9

b.

Research questions. So far these are claims; the rest of the paper substantiates them.

Four questions test the mechanism in the order a sceptic would demand—does it exist, does it self-organize, what causes it, and how efficient it is—and a fifth asks whether the regime where it pays off is common enough for the mechanism to recur. Q1 (structure) Does spatial LTM recruitment exhibit the cascade window —an inverted-U recruitment response to connectivity (the static control surface)? Q2 (self-organization) Does the system selforganize to the near-critical regime, and is it global or local? Q3 (mechanism) Is that self-organization caused by the topology–geometry coupling? Q4 (efficiency) How efficient is the coordinated regime—in harvest, movement, and communication—and what sets each? Q5 (generality) The mechanism pays off only on sparse, clustered fields (Q4); is that field generic rather than special, and does the least-committal search match it, so the mechanism recurs? c.

Reader’s map. Sections III–V build and test the mechanism: the Results answer

Q1–Q4 in turn, then ask whether the regime where it pays off is generic (Q5, §V E). The Discussion (§VI) then reads what that mechanism is: a near-critical, sandpile-like excitable cycle (§VI A) and marginal-value patch-leaving with Lévy-then-exploit foraging (§VI B). The payoff lives in the second half; the first half earns the right to it. All of it is one claim expressed in the languages of two disciplines (physics and ecology), not separate topics—the through-line we return to at each section.

II.

RELATED WORK

Active matter. Like active matter, our rule produces collective order from purely local interaction—so it connects to the flocking literature—but by a different route: where Vicsektype models order through velocity alignment [14–16], order here is recruitment-mediated (a discrete cascade, not alignment), task-driven, and transient (agents aggregate at a resource, deplete it, disperse)—so we reproduce collective order, then add a mechanism. Threshold cascades and percolation. Watts’ LTM on random graphs undergoes a percolation-type onset once a giant component of vulnerable nodes percolates [7] (continuum percolation of the underlying random geometric graph [17–19]); the rule was introduced for social contagion—adoption and rumour spread [6, 7], which recruitment here spatially embeds. We depart twice: the graph is induced by agent positions, which recruit10

ment then reshapes by drawing those agents together (endogenous connectivity), and we use both boundaries—percolation below and degree-blocking above (the mobile analogue of rumour stifling)—as a productive window rather than studying the onset alone. Spatial cascades. Embedding an LTM on a random geometric graph rather than an unstructured one makes recruitment edge-efficient—a small connection radius, and so few links per node, still spreads large, system-spanning cascades [20]; we turn that static spatial cascade into a mobile one. Swarm robotics / task allocation. Response-threshold models switch task when a stimulus crosses an individual threshold [4, 21–24]; ours thresholds the fraction of active neighbours, and it is this normalisation that yields self-limitation for free, where engineered controllers add explicit inhibition. The explore–exploit balance this strikes is the province of reinforcement learning, formalised by the multi-armed bandit [25–27]; a companion paper develops the connection. Foraging theory. Optimal foraging characterises the single searcher—the marginal value theorem (leave a patch as its intake falls to the habitat average [10]) and Lévy random search [11, 28]. Here the same single-searcher behaviours emerge together from one local rule run by a collective of searchers. Self-organized criticality and excitable media. Our swarm sits in the SOC / excitable-media family: like the Bak–Tang–Wiesenfeld sandpile [5, 29, 30] it is slowly driven by exploration and relaxes through near-critical recruitment avalanches. It departs from that sandpile’s conservation: in the Bak–Tang–Wiesenfeld model a toppling merely passes its grains to neighbours—conserving them in the bulk and losing them only at the open lattice boundary, a vanishing fraction of a large system, so the loss becomes negligible and it sits exactly at criticality in that limit. Our blocking instead dissipates in the bulk (recruits are shed throughout the domain—a fixed fraction that never vanishes, and indeed our torus has no boundary at all), as non-conservative SOC models do (Olami–Feder–Christensen earthquakes [31], the Drossel–Schwabl forest fire [32]): so the branching stays intrinsically just below critical rather than approaching 1, and the forest-fire model already shows SOC and excitability are two sides of the same coin (growth = drive, lightning = trigger, fire = avalanche, ash = refractory). So the regime is a driven, dissipative excitable medium [33] (§VI A); its one genuinely new ingredient is a mobile substrate: where the sandpile topples on a fixed lattice, here the avalanche moves the nodes themselves, so topology and geometry 11

co-evolve—a spatially embedded adaptive network [34].

III.

MODEL

We study a foraging cycle on a 2D resource field (Fig. 3): agents explore by heavy-tailed Lévy flight until one discovers a resource and recruits its neighbours in a threshold cascade; recruitment aggregates agents to consume the find, aggregation densifies the neighbourhood until the same threshold blocks further recruitment, and depleted or blocked agents disperse back to exploring. The coordination at its heart is a compact primitive, defined independent of environment and of the search mode: (i) an interaction network topology Gt induced by proximity (a random geometric graph, RGG [17]), (ii) a binary fraction-threshold recruitment rule on Gt —which cascades when the active-neighbour fraction is high and intrinsically self-limits (blocking) as neighbourhoods densify—and (iii) a coupling from activation to motion. The exploration law (Lévy here, Brownian equally; §VI A) and the resource field are the layers this primitive couples to, not part of it—which is why the coordination we find is transport-independent. a.

State and dynamics. A square torus [0, L)2 —periodic boundaries, to avoid edge

effects and keep every location statistically equivalent—holds N agents and M localized resources (minimum-image distances). Inactive agents explore by heavy-tailed Lévy flights (stable index α, 0 < α < 2; step-length tail exponent µ = 1 + α); an agent overlapping a resource becomes a seed and consumes it; depleted resources respawn elsewhere, keeping the field statistically stationary. Agents i, j share an edge when d(xi , xj ) ≤ r (a random geometric graph), rewired as agents move. Because information propagates faster than agents, the LTM rule is iterated to its fixed point each tick: with ai active neighbours and degree ki , each agent’s binary state si ∈ {0, 1} (1=active) updates asynchronously as   1, ki > 0 and ai /ki ≥ ϕ, si ←−  s , otherwise, i

(1)

activating monotonically to the Watts global-cascade fixed point. Recruited agents step a fixed fraction toward the centroid of their recruiters (aggregation solves the task); seeds hold position. Convergence raises local density, so ki grows, ai /ki falls below ϕ, and recruitment is blocked —agents are released. This topology–geometry feedback (cascades reshape geometry, 12

which rewires the graph, which regulates percolation and blocking) is the central mechanism. Per-tick pipeline: move → seed → cascade to fixed point (Eq. 1) → converge → commit.

IV.

METHODS

Notation. Controls: ϕ activation threshold (rule: active-neighbour fraction ai /ki ≥ ϕ),

ρr = N (r/L)2 connectivity density, ρs = M (ℓ/L)2 resource coverage (ℓ patch size; abundance), α Lévy stable index (step-length tail exponent µ=1+α), δϕ threshold heterogeneity. Per agent: ki degree, ai active-neighbour count, si state; A = population active fraction. Connectivity: z = πpρr mean degree (p link probability), zlocal over active agents, zcRGG percolation onset. Criticality: σ branching ratio, χ susceptibility, S avalanche size, Ψ largestcluster fraction. Capitals denote population aggregates; lowercase-with-subscript, per agent. Dispersion is reported as s.d./SEM. Dimensionless controls. Measured against L and one tick, the model’s raw parameters (N, r, M, ℓ, ϕ) reduce to three genuine dimensionless controls—ρr (propagation/percolation), ρs (seeding), and the cascade threshold ϕ (recruitment vs. blocking)—a data collapse: distinct raw settings then behave identically. Threshold heterogeneity δϕ is a further control, held at zero in the baseline and characterised in §V D. Phase diagrams in (ρr , ϕ) at fixed ρs are therefore statements about a class of systems, independent of box size or agent count. Simulation. We simulate the model of §III at concrete scale—N ∼ 102 agents, hori-

zons of order 103 ticks, batches of a few tens of independent replicas (each a fresh random

realisation—new agent positions, graph, and seed nodes). Every observable is a function of the dimensionless controls (above) and is itself dimensionless or intensive (a fraction, ratio, or per-agent rate), so results describe the class of systems, not the particular N ; consistently, the cascade window’s critical point is N -independent across N ∈ {50, . . . , 800} (SI A). Recruiting and no-recruitment conditions (the latter with ϕ raised so no cascade spreads) are compared paired within the same randomised context; each reported quantity is an ensemble average over a batch of independent replicas (batch size per figure). Uncertainty, where shown, is over these replicas: shaded bands are ±1 s.d. or ±1 SEM (as labelled) for curves, error bars likewise for points, 95% bootstrap confidence intervals for scalar estimates (correlations, fitted exponents), and heat-map cells are batch means. Each run tracks four dimensionless per-agent-tick cost ledgers—movement, presence-beacon pings 13

(the once-per-agent-tick local broadcast by which agents within range r discover one another and form the graph Gt ), cascade-activation pings (one message per newly recruited agent as the signal of Eq. 1 propagates), harvest—kept separate and never summed, so no efficiency metric is baked into the dynamics; cost is otherwise reported as work to complete a fixed job (consume a set resource fraction), in steps/target and bits/target. Two Lévy settings appear and should not be conflated: harvest and efficiency results (Q4; Table I, Fig. 9) use α=1, the empirically observed forager optimum (§V D), whereas the criticality diagnostics (Q2; σlocal and the susceptibility, Figs. 5, 18) use α=1.5. Because the nearcriticality is transport-independent (§VI A), the particular exploration law does not affect the Q2 conclusions. Connectivity caveat. The relation z = πpρr is exact only for uniform positions. The swarm aggregates, so realised degree exceeds it (holding z while trading p=1→0.25 changes measured degree by a factor ≈2); and r sets a geometric travel scale z ignores. Topological statistics collapse onto z; harvest and efficiency do not—they vary substantially with the travel scale r—so z alone does not fix efficiency. We hold p=1 and vary connectivity through ρr . High-dimensional efficiency map. The full control space (14 dimensionless parameters) is sampled with a 4096-point Sobol low-discrepancy design [35] (a space-filling quasirandom sequence that covers the space far more evenly than random draws) rather than a grid (414 cells is infeasible); nuisance parameters are randomised not frozen, recovering each marginal at linear cost. Estimators. Each Results quantity is measured as follows. Local branching σlocal is the branching-process (Galton–Watson) offspring-to-parent ratio within a single cascade (σlocal =1 marks criticality). The cluster-size susceptibility is χ = ⟨S 2 ⟩/⟨S⟩ over recruitedcluster sizes S (excluding the largest cluster, as usual in percolation); for finite-size scaling we instead use the order-parameter susceptibility χ = N Var(Ψ), with Ψ the largest-activecluster fraction, swept over N ∈ {50, . . . , 800}. Giving-up density (the marginal-value signature) is the resource remaining when a patch’s occupancy collapses from its peak. Cascade-size finite-size scaling (Fig. 6). We record every cascade’s connectedcomponent size and sweep the swarm size N ∈ {50, . . . , 800} with all dimensionless controls p held fixed (r=L ρr /N auto-rescales; source count ∝ N ). To compare fields at criticality we calibrate the activation threshold per (N, field) so σlocal ≈1 (bisection), then test 14

the collapse sτ −1 P (S≥s) vs s/N . On the near-CSR field—sparse, near-random placement (CSR)—this gives the mean-field branching exponent τ =3/2; strongly clustered fields give a lower, geometry-set exponent, and degenerate point-clumping cannot be brought to σlocal =1 at all. Confidence intervals are bootstrapped over the independent replicas. Signal-vs-body speed (Fig. 8). From an instrumented near-critical run we log, each tick, agent positions and activation state. The recruitment reach is the (periodic) distance from each newly activated agent to the nearest agent active the previous tick—how far coordination propagated across space in one step; the agent displacement is the per-tick move length. We compare the two distributions over all ticks past burn-in. Phase portrait (Fig. 1b). Pooling all replicas over t > tburn , every sample is binned in the (connectivity zlocal , active fraction A) plane and the per-node state fractions (explore/ recruit/consume/blocked, defined in §VI A) averaged within each bin; the binned mean drift (⟨ż⟩, ⟨Ȧ⟩) gives the streamlines and the ż=0, Ȧ=0 nullclines, whose crossing (a fixed point of the mean flow, into which the streamlines flow) marks the stable fixed point. A state’s shaded “rich” region is an enrichment contour of its (smoothed) mean-fraction field—bins where that state’s mean occupancy is at least 1.5× its plane-average, the same factor for all four states (prevalence-aware, so the common explore state and the rare consume state are treated alike)—a visualization of where each state concentrates, not a sharp phase boundary.

V.

RESULTS

The through-line: the mechanism is untuned —its ingredients are forced, not chosen. We take the four mechanism questions in turn (Q1–Q4), then ask whether the regime where it pays off is generic enough for the mechanism to recur (Q5); simulation and estimator details are in §IV. Terminology. We reserve critical for the phase transition itself—the cascade-window boundary, whose criticality we establish by finite-size scaling (§V D, SI A). The swarm’s self-organized operating point we call near-critical : globally it is deeply subcritical (mean branching σ ≈ 0.03), so the swarm as a whole is never at the critical point, while locally the per-avalanche branching sits just below the transition (σlocal ≈ 0.9 < 1)—essentially at it

at small ϕ (σlocal ≈ 0.99; §V B), though never exactly on σ=1. “Marginal” we keep for its foraging meaning alone (the marginal-value theorem, defined in §II). 15

A.

The cascade window: percolation and blocking (Q1)

Two opposing effects act along the connectivity axis. Reach: below a threshold the interaction graph is fragmented and cascades stay local; near the giant-component onset a discovery recruits system-wide. For a 2D random geometric graph this occurs at a critical mean degree zcRGG ≈ 4.51 [18, 19]—a property of RGG connectivity, independent of the task, setting the window’s lower boundary. Blocking: because activation is on the active-neighbour fraction a/k, rising degree lowers it below the threshold ϕ and suppresses propagation— Watts’ upper boundary, high-degree nodes non-vulnerable at k > 1/ϕ [7]. Their competition yields the non-monotonic (inverted-U) response of Fig. 2: cascade fraction rises near zcRGG and peaks at z ≈ 6–7 (peak ≈0.95) before the blocking falloff. Both boundaries are fundamental LTM features, not artifacts, and both follow from degree-normalized activation—the upper brake is not inserted. What is new is that connectivity is set by the agents’ own positions, so the swarm moves within this window as it aggregates. Consistently, the mean degree inside cascaded clusters sits at an approximate blocking boundary 2/ϕ (e.g. ≈10 at ϕ=0.20)—approximate because it is a Watts-vulnerability estimate: the per-node boundary is 1/ϕ, but the mean-degree onset lands near 2/ϕ because a node is typically recruited once ≈2 of its neighbours are active, so a/k first crosses ϕ near k ≈ 2/ϕ.

B.

Self-organization is local, not global (Q2)

Under the resource-search task, the swarm rides a self-limiting explore/exploit cycle, but the global branching ratio—secondary activations per activation, σ=1 at criticality—stays subcritical (mean σ ≈ 0.03, so cascades die out on average): the average is dominated by exploring agents, and—atypically for a spreading process—adding connectivity drives σ down, because higher degree blocks propagation. Resolved locally, the picture inverts. The direct criticality measure is the per-avalanche branching ratio σlocal ≈ 0.9 (from 0.83 at ϕ=0.35 to 0.99 at ϕ=0.20; measured on the swarm’s own Lévy exploration, α=1.5), an order of magnitude nearer criticality (σ=1) than the swarm mean: individual recruitment cascades are near-critical even though the swarm average is deeply subcritical. The low swarm average is that criticality diluted by the non-branching majority, and the shortfall of σlocal below 1 is blocking holding the system on the ordered side—and it tracks the threshold: 16

at ϕ=0.20 (blocking onset 2/ϕ=10) cascades sit essentially at criticality, while the tighter ϕ=0.35 (onset 5.7) pulls them just below, exactly as a stronger blocking constraint should (a corroborating percolation signature—the cluster-size susceptibility peaking at onset—is in SI A). We read criticality off this branching ratio rather than off active-cluster sizes: σlocal is robust to the transport law, whereas the spatial extent of active clusters merely tracks how far agents disperse (a step-scale effect, not a criticality one). The recruitment cascades’ own size distribution nonetheless corroborates σlocal : each cascade’s connected-component size is power-law distributed with an exponent near the mean-field critical-branching value τ =3/2, and—as it must if that power law reflects branching—the exponent tracks σlocal . At ϕ=0.20 (σlocal ≈0.99) the sizes follow the critical s−3/2 form up to the finite-size cutoff at s=N , whereas the more subcritical ϕ=0.35 (σlocal ≈0.83) rolls off steeply below it (Fig. 5). We take this as corroboration only—the fitted exponent is bounded by the s≤N cutoff and shifts with the fit window—so the branching ratio, not this slope, remains our criticality measure. Sitting just below this local critical point is what gives the swarm its dynamic range: most finds trigger small local responses, but near-criticality lets an exceptional find occasionally recruit a large coordinated one—the response scales to the opportunity with no target set anywhere.

a.

Finite-size scaling confirms mean-field branching—on the near-CSR field. The

single-size fit above is only suggestive (the cutoff at s=N bounds it), but finite-size scaling settles it. On the sparse, near-CSR (maximum-entropy) field, holding the swarm at criticality (σlocal ≈1, matched across sizes) and sweeping N ∈ {50, . . . , 800}, the cascade-size

distributions collapse onto the mean-field critical-branching form P (S≥s) ∼ s−1/2 G(s/N )— exponent τ =3/2, cutoff at the swarm size N (Fig. 6)—with the operating point σlocal itself N -independent. The cutoff is thus not a nuisance but the scaling variable, as criticality requires. The exponent is not, however, universal across all fields: as the resource field is made strongly clustered the exponent falls below 3/2—the cascade sizes then inherit the field’s patch geometry rather than the branching dynamics—and for degenerate, point-like clumping the swarm cannot be brought to criticality at all (§IV). So the phenomenon— self-organized, scale-free, near-critical cascades—is robust across resource fields, while the exponent is set by the field: mean-field branching on the near-CSR field, geometry-set on strongly patchy ones. 17

finite-size cutoff s = N

100

P(S s)

10 1

10 2

10 3

= 0.20 ( local 0.99): 1.3 = 0.35 ( local 0.83): 1.9 critical branching s 1/2 ( = 3/2)

101 102 cascade (connected-component) size s FIG. 5. Recruitment cascades are near-critical branching avalanches. How big does a single recruitment chain-reaction get? Each curve is the survival distribution P (S≥s) of cascade size s—the connected-component size of one branching cascade tree in the active subgraph of the radius-r proximity graph (Lévy α=1.5). We vary the activation threshold ϕ because it is the knob that sets how close the cascades sit to criticality: a higher ϕ makes blocking fire sooner, pushing the branching ratio σlocal further below 1. At the lower threshold ϕ=0.20, which leaves the swarm essentially critical (σlocal ≈0.99, blue), cascades come in all sizes—a truncated power law tracking the mean-field critical-branching reference P (S≥s) ∝ s−1/2 (avalanche exponent τ =3/2, dashed) up to the finite-size cutoff at s=N =150 (dashed vertical). The slight excess of blue above the reference just below that cutoff is the pile-up expected at criticality: the intrinsic branching cutoff s∗ ∼ (1 − σlocal )−2 ≫ N , so cascades are capped by the swarm size rather than by branching, and every would-be larger cascade saturates the swarm and lands at s≈N . At the higher threshold ϕ=0.35, which drives it subcritical (σlocal ≈0.83, red), blocking imposes a characteristic size: the tail bends into an exponential and big cascades vanish (τ ≈1.9). The size distribution thus shifts exactly as branching theory predicts from σlocal —corroborating it. But the fitted exponent is bounded by the s≤N cutoff and sensitive to the fit window, so σlocal , not this slope, remains our primary criticality measure.

18

(b) collapse: mean-field branching = 3/2

(a) cascade-size distributions 100 100

s1/2 P(S s)

P(S s)

10 1 10 2 10 3 10 4 10 5

N = 50 N = 100 N = 200 N = 400 N = 800 s 1/2 ( = 3/2)

101

10 1 10 2 10 3

102

N = 50 ( = 1.02) N = 100 ( = 0.99) N = 200 ( = 1.00) N = 400 ( = 1.00) N = 800 ( = 1.00)

10 2

cascade size s

s/N

10 1

100

FIG. 6. Finite-size scaling: on the near-CSR field the cascades are mean-field branching avalanches. At matched criticality (σlocal ≈1, the activation threshold calibrated per N ; legend) on the near-CSR (maximum-entropy) field. (a) Survival distributions P (S≥s) for N =50–800: the cutoff grows with the swarm size, tracking the mean-field reference s−1/2 (dashed) in the scaling region. (b) The finite-size-scaling collapse: s1/2 P (S≥s) against s/N overlays across sizes onto the mean-field critical-branching form (avalanche exponent τ =3/2; dashed reference). The operating point σlocal is itself N -independent by construction here; the collapse places this tuned transition in the mean-field branching universality class, while self-organization to a near-critical operating point is shown separately (§V B).

C.

The recruitment→motion coupling causes self-limitation (Q3)

To separate cause from correlation, we run a controlled intervention: hold every parameter fixed and toggle the recruitment→motion coupling. With it intact the swarm self-limits to ≈0.58 active (riding the cycle); removing the coupling—agents still activate (the cascade still spreads) but no longer converge on the resource—makes activity saturate toward ≈100% (Fig. 7). The mechanism is direct: without convergence recruits never crowd a resource, so the driver for local degree k to climb is removed—the activated fraction a/k never falls below the threshold ϕ, and blocking never fires. The same uncoupling collapses the resource– exploration correlation to ≈0 (§VI A). Because only one link is cut while all else is untouched, this isolates cause, not association—the strongest evidence the model can furnish. 19

(a)

0.8 0.6 0.4 0.2 0.0

coupled uncoupled

0

200

400

tick

600

800

active fraction A (steady state)

active fraction A

1.0

1.0

(b)

0.8 0.6

coupled uncoupled

0.4 0.2 0.0

0

5

10

15

density

20

FIG. 7. The recruitment→motion coupling causes the self-limitation—a controlled comparison. Only the coupling is toggled. (a) Active fraction A over time (mean ±SEM): with the coupling (green) the swarm self-limits around half active; uncoupled (red), activation floods and saturates near 1. (b) Steady-state distribution (t > 200): coupled occupies a broad sub-saturation band, uncoupled piles against 1.0. Removing one link—not any parameter—abolishes the balance.

a.

The coupling carries coordination faster than the agents move. The intervention

shows the coupling matters; a direct measurement shows what it does. Tracking the nearcritical swarm, we compare, per tick, how far the recruitment signal spreads across space— the distance a newly activated agent sits from the nearest already-active one—against how far an agent physically moves. The signal advances a median ≈197 world units per tick (reaching across the dispersed swarm in a few hops), while an agent moves ≈1.3: the recruitment signal outruns the bodies by two orders of magnitude (∼150×; Fig. 8), and active clusters span most of the arena. Coordination is thus carried by the percolating network, not by locomotion—the concrete content of percolation, not proximity, and the quantitative face of the network–space coupling.

D.

Efficiency: what recruitment buys, and what it costs (Q4)

a.

Harvest: where recruitment pays. Whether recruitment pays off depends entirely

on how resources are distributed. Sweeping patchiness (clump radius/L) and coverage ρs against a no-recruitment baseline—the harvest ratio is the number of resources collected with recruitment divided by the number collected without, over the same number of time 20

P( distance)

100

interaction radius r

10 1

152 ×

10 2 10 3

agent displacement / tick (body) recruitment reach / tick (signal)

10 4 10 3 10 2 10 1 100

101

distance per tick (world units)

102

103

FIG. 8. The recruitment signal outruns the bodies. Survival distributions, per tick, of the recruitment reach (blue; distance from a newly activated agent to the nearest previously active one—how far coordination jumped across space in one step) and the agent displacement (red; how far a body moved). The reach distribution sits ∼150× above the displacement distribution (dotted medians; double arrow); dashed line marks the interaction radius r. Information propagates across the field along the network far faster than agents move—coordination is carried by the network, not locomotion.

steps (Fig. 9, Table I): in abundant or uniform fields recruitment over-concentrates—agents pile onto found sources, cutting harvest to as low as ∼0.8× the no-recruitment baseline while adding movement and messages, so it loses even before communication is priced. In sparse, clumped fields it pays off strongly (harvest up to 7.6× the no-recruitment baseline, 95% CI [6.8, 8.4]): a discovery is informative, so recruiting sends agents to a rich neighbourhood for cheap local exploitation, and blocking makes agents leave a patch before it is emptied (marginal-value patch-leaving, §VI B), so their effort is not wasted over-exploiting a depleting patch but returned to search. Over a 48-cell grid, recruitment wins on harvest in 92% (22/24) of clumped regimes vs. 54% (13/24) of diffuse ones, losing only where resources are both abundant and diffuse—a clean boundary over the resource statistics, which is exactly where real foraging does not operate (patchy resources are the default). This is the benefit side; the costs are priced next. b.

Costs: movement and communication. Exploration and exploitation call for differ-

ent motion, and the cascade selects it: exploring agents take long-range Lévy steps (advantageous on sparse, clustered targets, and breaking the shared oversampling of a cohort 21

hatched: 95% CI of ratio includes 1 (n.s.)

0.16

Urecruit/Ubase (>1: recruitment pays)

1.75

s

0.08 coverage

2.00 1.50 1.25

0.04

1.00

0.02

0.75

0.01

0.50 0.25

0.005 0.025 0.05 0.075 0.12 0.19 0.3 0.5 0.7 patchiness (clump radius / L: tight diffuse)

0.00

FIG. 9. Recruitment beats solitary search only for sparse, clustered targets. Harvest ratio—recruitment harvest divided by the no-recruitment baseline—over patchiness (clump radius/L) and coverage ρs : >1 (warm) means recruitment wins, <1 (red) means it loses, with cells hatched where the 95% CI includes 1 (indistinguishable from baseline). Recruitment pays off for sparse, clustered fields and loses when targets are uniform or abundant. Computed at Lévy stable index α=1; a full sweep shows α—how efficient solitary search already is—decides whether recruitment repays its cost (§V D).

released from one patch), recruited agents take short local steps. The heavy tail thus does double duty: it is the least-committal search matched to a scale-free field (the maximumcaliber optimum, §VI C 0 b), and the long-range escape that carries an agent off a patch once blocking releases it or the resource is consumed—the movement half of patch-leaving—so a released cohort disperses instead of re-crowding the spot it just drained. Read as information, being recruited is a belief update (“resource nearby”) and the switch is the matching change of search scale. Two things distinguish this from single-walker Lévy theory, which supplies optimality but not a trigger [11]: the switch is driven by a social signal (a neighbour’s discovery via the cascade), and the case for long jumps is itself social (breaking the group’s shared oversampling). A full sweep reveals what recruitment does and does not buy, separating its movement efficiency from its communication efficiency. Across the 14-parameter space, the Lévy exponent α is the strongest single driver of movement cost (median ∼105 steps/target at α≈0.55

vs. ∼102 at α≈1.8). Recruitment confers no movement economy—paired against an iden22

coverage ρs

patchiness: tight clumps −→ diffuse

sparse (ρs =0.005) 7.6× 3.0× 2.0×

1.9×

medium (ρs =0.02)

5.1× 1.7× 1.2×

1.1×

abundant (ρs =0.08) 2.4× 1.2× 0.8×

0.8×

TABLE I. Recruitment pays off only where targets are sparse and clumped. Each entry is the harvest ratio—harvest with recruitment divided by the no-recruitment baseline—so >1 means recruitment raises harvest and <1 means it lowers it (e.g. 7.6× = 7.6 times the baseline harvest; 0.8× = a 20% loss). Rows vary coverage ρs ; columns sweep patchiness from tight clumps to diffuse. The gain falls monotonically along both axes, dropping below 1 only in the abundant–diffuse corner (bold min)—a clean boundary over the resource statistics. The patchiness axis is the aggregation strength of §V E: tight clumps are the strongly attractive (β≫0) end, diffuse approaches the CSR (β→0) knife-edge. What the mechanism responds to is the clustering strength, not the generative model—the clump-radius fields used here and the Gibbs ensemble of §V E both span this one axis. Each ratio is a paired comparison over B=48 replicas; 95% bootstrap CIs are ≈ ± 3–12% (the maximum 7.6× has CI [6.8, 8.4]; the minimum 0.8×, [0.78, 0.83]).

tical non-recruiting swarm, the recruit-to-baseline movement cost per target scatters about parity across α (1.02, 0.81, 1.17 at α=1, 1.5, 2: no systematic saving). What it buys is time: it completes the fixed job up to ∼15% faster, but only when solitary search is inefficient (completion-rate ratio 1.15 at α=1 → 1.02 by α→2) or resources are sparse (1.21 → 1.00). The communication premium is a separate, ∼5× message cost on the activation channel that is governed by connectivity and threshold (z, ϕ), not by α—so cost and payoff are set by different controls, and the message cost can be tuned through connectivity without disturbing the regime that decides whether recruitment pays off. (Channel-by-channel numbers, the Sobol marginals, and the size-independence of the window under finite-size scaling—the susceptibility χ peaking at a fixed z ∗ ≈ 7.8 for all N , peak ∝ N —are in SI A.) Threshold heterogeneity δϕ raises cascade size (recruited fraction 0.47 → 0.62 at ϕ=0.20) but leaves efficiency and the cascade exponent unchanged—robustness to threshold heterogeneity, a separate axis from the resource field—so the homogeneous baseline is a genuine simplification, not a favourable special case. 23

c.

Empirical grounding. Both ingredients—the forager’s heavy-tailed search and the

sparse, scale-free resource fields it exploits—are documented, and—crucially—independently. Diverse marine predators move by long-range Lévy search with heavy tails (step-length exponent µ ≈ 2 in P (∆) ∝ ∆−µ (∆ the step length), i.e. stable index α = µ − 1 ≈ 1—the value used here, so our baseline sits at the empirically observed value) [3] (though strict power-law claims in movement data warrant care over visual fits [36]); measured resource fields are sparse and scale-free (Kalahari canopy clusters, cluster-size exponent βres < 2 [12], rainforest gaps [37], turbulent fields [38]). The forager exponent (µ ≈ 2) and the field exponents (βres < 2) were measured by different investigators studying different quantities, with no fit tying them; that a heavy-tailed search exponent and the fields it searches are mutually compatible is a falsifiable fact, not a definition (§VI C 0 b argues it is no coincidence). Driving the model with genuinely scale-free fields confirms recruitment still pays off (4.2–5.7× the baseline, more on more-clustered fields), and the cascades stay near-critical there—but the exponent reflects the field rather than being universal: on these clustered fields τactive ≈ 1.25–1.30, below the mean-field 3/2 the near-CSR field gives (§V B), because the cascade sizes then carry the field’s patch geometry rather than the branching dynamics. So the swarm self-organizes to near-criticality on any such field (the phenomenon is robust), while its exponent is set by the field, not a constant it imposes. Heavy-tailed search is meanwhile advantageous on clustered targets (SI A), the regime beyond the scattered (Poisson) targets for which inverse-square Lévy is provably not optimal in d ≥ 2 [39].

E.

The regime where recruitment pays off is generic, and the search matches it

(Q5)

The mechanism pays off, but conditionally: on sparse, clustered fields it wins (up to 7.6× the baseline, §V D); on uniform or abundant ones it loses. Whether that makes it a general mechanism or a special-case trick turns on two questions about the regime where it pays off—is a sparse, clustered field generic rather than special, and does the least-committal search match it? The first is settled by the payoff itself together with the sign of real fields; the second by maximum caliber. Both are affirmative. The payoff settles the first question almost before we ask where fields come from. It rises monotonically with clustering—from ∼1.9× at the least-committal (near-CSR, maximum24

entropy) field to ∼7.6× at strong aggregation (Table I)—and falls below one only for abundant, diffuse fields, which a forager, by definition searching a sparse one, never meets. So even the most conservative clustered field we can write down already nearly doubles harvest: that is a floor, set not by fine structure but by the mere fact of clustering. What the floor leaves open is only whether real fields clear it or sit on the knife-edge below—and that is a question about one thing: the sign of the field’s spatial correlation. Sparsity is presupposed—a forager would not search an abundant field—so the real question is the sign: does a resource field deviate from randomness toward aggregation (above the floor) or toward regularity (below it)? One interaction sign sweeps the whole spectrum. Treat a resource field as a point pattern with an effective interaction, written as the least-committal (maximum-entropy) distribution given a mean clumping ⟨C⟩ (with C the number of close pairs): P (field) ∝ eβC , a Gibbs point process [40, 41]. Its single parameter β—the sign of the interaction—fixes the arrangement (Fig. 12): repulsion (β<0) gives a regular, fluctuation-suppressed field (hyperuniform [42])— the exception, because it needs an active dispersive mechanism; independence (β=0) gives complete spatial randomness (CSR), which already departs from regularity but clusters only by fluctuation (a knife-edge, and the weak limit—recruitment barely pays off there, §V D); and attraction (β>0) gives a genuinely clustered field. (The intuition that random placement should look even conflates independent placement with regular spacing: independent placement clumps, and it is the regularity, not the clumping, that would need a mechanism.) Attraction is the generic sign. Independent routes all push a resource field the same way—toward aggregation. Resource is laid down where resource already is: offspring beside parents (reproduction and dispersal, the textbook origin of clustered point patterns [40, 41]); regrowth and accumulation on a shared substrate of water, nutrients, or light (autocatalytic renewal [43, 44]); and purely physical concentration by transport, as turbulent mixing and flow convergence gather passive matter into patches [38]—all on a substrate that is itself heterogeneous. Each drives β>0, and they compound rather than cancel. The one arrangement none of them produces is regularity, which needs an active dispersive mechanism; exact CSR needs exact independence, a measure-zero knife-edge. Many routes lead to clustering, one narrow route to regularity—so non-aggregation, not aggregation, is what takes a mechanism. So the least-committal field consistent with the one fact resources almost always satisfy— that they aggregate—is a clustered one, and that is the field the mechanism needs. Because 25

the argument is combinatorial and physical rather than biological, it carries to engineered and abstract target fields alike. This separates three things that are easy to run together: maximum entropy fixes the family (a Gibbs field is the least-committal distribution at a given mean clumping), the physics of renewal fixes the sign (β>0), and the magnitude of β—how strongly clustered—is what the particular environment sets. Sparsity amplifies the magnitude. That magnitude is not a free parameter in the regime that matters, because the sparsity foraging presupposes drives it up. Write the effective clustering strength as the dilute-limit multiplier that reproduces a field’s close-pair count, βeff ≈ log(⟨C⟩/⟨C⟩CSR )—the log-ratio of its close pairs to those of a CSR field at the same density. Autocatalytic renewal fixes the numerator (offspring appear beside parents, so close pairs scale with reproduction, ∼N µ for clump occupancy µ), while the CSR denominator

falls with density (∼N 2 ); so βeff ≈ log µ − log(density) + const grows logarithmically as the field thins (SI A: confirmed on a renewal cluster field, fitted slope 0.91 against the predicted 1, independent of µ). The one condition that makes search necessary thus also sharpens the clustering recruitment exploits. What stays environment-set is not the contrast but the clump occupancy µ—how rich each clump grows. What the data show. Measured resource fields are clustered across systems—forestgap and canopy-cluster power laws [12, 37, 45], turbulent scalar intermittency [38], sparse patches separated by voids—direct evidence that attractive renewal is the norm. The strongly clustered, scale-free end—where driven, dissipative dynamics self-organize to criticality (SOC [5]), leaving power-law patches with no characteristic scale—is a special, strong limit: attraction fixes the sign, but reaching this maximal-clustering end takes further conditions of the SOC type and is not entailed by attraction alone. It is an environmental criticality of the field, distinct from the swarm’s (which stays globally subcritical, §V), and the end where the search–field match is sharpest (α∗ ≈ Df , §V D). This is the climb the floor sets up: the measured fields above are not merely clustered but strongly, often scale-free clustered (∼1.9× → ∼7.6× across Table I), so the large-β regime, not just β>0, is what the data report—magnitude is carried by evidence, not assumed. The search obeys the same principle. The field is maximum-entropy; the search that matches it is its maximum-caliber twin—the least-committal law over paths rather than positions. Which law depends on what scale you commit to: fix a typical step length and the least-committal path is Gaussian (Brownian); refuse to fix any length scale—the 26

honest choice when no single search scale is privileged—and it is scale-free and heavy-tailed, exactly Lévy exploration. So environment and search share one root—maximum entropy over where resources sit, maximum caliber over how to move (developed fully in §VI C 0 b)— and the field’s two features map onto a two-mode search: Lévy exploration exploits the sparsity, recruitment the clustering, complementary responses to one structure rather than two separate adaptations. The match is quantitative as well as formal, and it is the central signature of the mechanism’s generality: the search is not tuned to a particular field but adapts to whatever field it faces—its optimal exponent tracking the field’s fractal dimension across the sparse-clustered range (α∗ ≈ Df ; §V D, Fig. 16). So one field geometry does double duty: it makes scale-free search optimal (α∗ ≈ Df , maximum caliber) and, seen through the agents’ proximity graph, is what lets recruitment percolate and self-limit (§V C)—binding exploration and cascade into one coupled cycle rather than two separate adaptations.

VI.

DISCUSSION

The Results showed an untuned mechanism—a self-organized, locally near-critical cascade from which the foraging behaviours fall out—that pays off on the sparse, clustered fields attractive renewal makes generic and maximum entropy makes least-committal (§V E). We now read what that mechanism is—one thing seen several ways: a near-critical excitable cycle and a foraging strategy here, and a reinforcement-learning policy in a companion paper. Across these readings the same structure recurs: the coordination lives in the coupling of a percolating recruitment network to the physical space the agents search, and because neither half is tuned the mechanism is general —it adapts to whatever field it meets rather than being built for one.

A.

An excitable medium poised near criticality

Here the mechanism starts to show what it is: the same untuned rule, read dynamically, is a self-organized excitable cycle—the drive–avalanche loop of the Bak–Tang–Wiesenfeld sandpile (the canonical model of self-organized criticality [5]; introduced in §II), but running on the swarm’s moving proximity graph rather than the sandpile’s fixed lattice, so the avalanche reshapes the very substrate it spreads on. In effect the swarm is a moving, local 27

sandpile: rather than a single pile toppling at fixed sites, a transient pile builds at each discovery, avalanches through its neighbourhood, and dissolves as blocking disperses the agents—so the (local, near-critical) dynamics travel with the swarm rather than pinning to lattice sites. As a standing feedback, blocking produces an emergent explore/exploit balance with the exploring fraction self-balancing rather than tuned. As defined in §V, the criticality is local and finite-size: on the near-CSR field the cascade ensemble obeys finite-size scaling with the mean-field branching exponent τ =3/2 (Fig. 6), while the swarm as a whole stays globally subcritical—so the claim is self-organized, locally near-critical branching, not thermodynamic-limit global criticality, and the exponent is set by the field rather than being a universal constant. Read positively, the regime is a task-driven excitable medium [33]: exploration is the rest state—not still, since agents search continuously, but quiescent in the collective recruitment-activity order parameter A; it is itself the slow drive, steadily loading the field toward the next discovery; a discovery is the trigger; the cascade is a bounded excitation; blocking is the refractory step preventing runaway; dispersal resets it—a relaxation oscillation, whose global subcriticality is not a shortfall but the reservoir of un-mobilised search capacity that keeps it near-critical. Concretely it runs the sandpile’s own cycle—slow drive, threshold, fast near-critical avalanche, reset—with each piece named (Fig. 1): exploration and resource respawn are the slow drive; a discovery is the threshold crossing; the recruitment cascade is the avalanche (near-critical recruited cascades, §V B); consumption (which depletes the patch) and blocking (which chokes the front) are its brief, two-channel discharge; and dispersal returns the swarm to rest. This unifies the Q1–Q3 verdicts: the self-limitation is real and coupling-caused (Q3), and the accompanying criticality is genuine but local (per-avalanche σlocal ≈ 0.9, Q2). Both share the drive–avalanche cycle; the one departure is that our dissipation acts in the bulk (blocking throughout, not at a boundary), so the stationary per-avalanche branching stays intrinsically just below critical (σlocal ≈ 0.9, not 1; §V B) instead of approaching 1 as it would under the sandpile’s bulk conservation—the non-conservative near-criticality of Olami–Feder–Christensen or forest-fire SOC [31, 32], not a categorical break from it. The productive middle also appears statically, as the cascade-window peak (§V A), toward which the swarm is driven from both sides—recruitment aggregating it up from too-sparse, blocking dispersing it down from too-dense (Fig. 1). Heuristically this two-sided drive reads 28

as an entropic force—the dynamical counterpart of the maximum-entropy fields and paths (§VI C 0 b), pushing the swarm up an entropy gradient into the cascade window—with the residual circulation of the measured cycle (Fig. 1b) the non-gradient part a full maximumcaliber treatment would separate. This near-criticality is transport-independent: the swarm self-limits just as critically whether agents explore by Lévy or Brownian walks, because it arises from recruitment and blocking, not from how agents move. Heavy-tailed Lévy exploration’s contribution is separate—it is foraging efficiency, the least-committal search for a scale-free field (§VI C 0 b, Figs. 15, 16)—so a Brownian swarm would be no less critical; it would merely forage worse on clustered fields. A temporal signature sharpens—and qualifies—this reading. Bak, Tang and Wiesenfeld originally advanced self-organized criticality as an explanation of 1/f noise [5], so a nearcritical, sandpile-like cycle invites the question of the power spectrum S(f ) of the global recruitment-activity A(t). That link is famously observable- and geometry-dependent: for the canonical sandpile the flow (dissipation) signal has a Lorentzian ∼ 1/f 2 spectrum, not 1/f [46], and genuine 1/f emerges only under special conditions—e.g. the integrated totalmass observable on narrow, locally driven stripes, where an exponentially long configuration memory yields a very broad band of time scales [47]. Our mechanism sits on the 1/f 2 side of this divide, for a structural reason: the nearcriticality is local, not global (σlocal ≈ 0.9 per cascade against a globally subcritical mean, §V B), and dissipation is non-conservative and acts in the bulk (blocking, plus consumption),

so global activity is a superposition of bounded, short-lived excitations rather than a scalefree stationary process—as the measured phase portrait shows directly (an overdamped relaxation to a stable fixed point, Fig. 1b: a relaxation with a characteristic timescale, not scale-free stationarity), with the recruitment cascades themselves near-critically sizedistributed (τ ≈ 3/2, Fig. 6). We therefore expect SA (f ) to be Lorentzian—flat below a corner frequency set by the activity relaxation time—rather than broadband 1/f ; a clean 1/f in A(t) would instead signal a globally near-critical, long-memory regime we do not claim. Measured directly, this holds (SI Fig. 19): at a near-critical operating point in the stationary (replenished) task, SA (f ) is Lorentzian—a flat, near-white plateau below a corner at fc ≈ 0.08 (relaxation time ≈2–3 ticks, matching the activity autocorrelation) with a roll-off 29

between 1/f and 1/f 2 above it—and is decisively not broadband 1/f : both the white lowfrequency plateau and a spectral exponent βspec ≈ 0.2 (from detrended fluctuation analysis) exclude it. The short memory follows structurally from the spatial, co-moving substrate: whereas the fixed-lattice sandpile’s 1/f requires an exponentially long spatial configuration memory [47], our proximity graph is reshaped by motion each tick, which erases that memory and keeps temporal correlations short—so the corner is set by the cascade/episode lifetime (SI Fig. 19c), and the spectral exponent is a direct, falsifiable discriminant between this local-criticality account and a global-SOC one.

B.

Active matter and foraging

Turn the same untuned rule toward physics and ecology and two more facets appear. Our rule produces collective order too—which connects it to active matter—but of a distinct kind: not alignment-driven, standing flocking, but transient, task-driven, recruitment-mediated aggregation that disperses once a resource is consumed. Read as collective foraging, recruitment is social area-restricted search, and blocking reproduces patch-leaving—but precisely. Charnov prescribes leaving when intake falls to the habitat average [10]; our rule leaves when crowding drops a/k below ϕ. These fire together but are not the same statement: the mechanism reaches the marginal-value outcome while estimating no rate, keeping no memory, and computing no habitat average—a functional equivalence that is stronger than an identity, and a side-effect of the very fraction rule that drives recruitment. It rests on one assumption, that crowding-blocking proxies a patch being worked out, which we test directly: tracking patches from discovery to depletion, they are abandoned before empty (giving-up density ≈40%, only a small fraction drained to empty), and the leaving is caused by blocking and tunable—the blocking-driven fraction of departures rises from 0 to ≈0.6 as ϕ tightens 2/ϕ, while the giving-up density itself is ϕ-independent (Fig. 10). So the equivalence is measured, not asserted. At the collective scale the same rule distributes foragers across patches—a problem that has some characteristics of the multi-armed bandit, a canonical model in reinforcement learning where agents balance exploration and exploitation of resources. Whether it can be formalized as reinforcement learning is taken up in a companion paper. 30

(a)

drained ( 0)

0.0

0.2

1.0

median 0.40

0.4

0.6

fraction of departures caused by blocking

number of patches

70 60 50 40 30 20 10 0

0.8

0.8 0.6 0.4 0.2 0.0

1.0

resource remaining at departure (giving-up density)

(b)

0.05 0.10 0.15 0.20 0.25 0.30 0.35

recruitment threshold

FIG. 10. Patch-leaving is marginal-value-like, and blocking is the measured cause. (a) Left before empty. Giving-up density—resource remaining when a patch’s occupancy collapses from its peak—centred near 0.40, with only a small fraction drained to empty: patches are abandoned before drained (the leave-when-returns-fall signature). (b) Blocking drives leaving. The blocking-driven fraction of departures (an agent squeezed below threshold while it still has active neighbours) rises with the recruitment threshold ϕ: crowding, tunably, releases agents. Each point is the fraction of n=820 departures caused by blocking; bars are the Wilson 95% confidence interval on that proportion. C.

One principle, two vocabularies

a.

Five ingredients, none tuned. What stands out is how little was put in by hand;

taking the rows of Table II in order: the cascade window is a network contagion whose fraction rule both spreads and stifles the cascade, so its upper boundary (blocking) is inherited, not added; that same blocking reproduces Charnov’s patch-leaving behaviour with none of its machinery; connectivity is endogenous—the recruitment→motion coupling carries the population through the window under its own dynamics; search is two-mode—sparsity favours Lévy exploration and clustering local exploitation, the cascade selecting between them; the sparse, clustered targets it needs are what attractive, autocatalytic renewal generically produces—the least-committal field once resources aggregate (§V E)—and real foragers’ independently observed Lévy statistics read as the same fit. None was fitted; each is a consequence of the fraction rule meeting the statistics the environment almost inevitably has. Read as ecology, these are behaviours foraging biology documented separately over decades, emerging here together from one rule, so the model does not merely permit natural 31

Ingredient (ours)

Is, classically

Supplied by—not Earned tuned

Cascade window (on, network contagion: LTM per- the

fraction

rule §V A

colation + degree-blocking [6, a/k ≥ ϕ

then off)

7] Blocking

(self- marginal-value

limitation)

leaving [10]

Endogenous

patch- the same fraction §VI B rule

connec- topology–geometry feedback mobility

tivity Two-mode search

+ §V

recruitment→motion Lévy foraging optimum [11]

target sparsity + §V D clustering

Sparse, clustered tar- generic once resources aggre- attractive renewal, §V E gets

gate [12]

β>0 (max-ent field)

TABLE II. The mechanism is untuned: each ingredient is forced by structure already present. The kind of correspondence varies—the cascade window is an exact identity; blocking-aspatch-leaving is a functional equivalence (the outcome with none of the machinery, stronger than an identity); the Lévy/field row is an empirical compatibility. What coheres them is that none was put in by hand.

foraging but predicts it—the sense in which the coincidences are elegant. And because every ingredient follows from the one rule meeting generic field statistics—nothing fitted to a particular environment—the same coupled network–space mechanism is general : it is what one should expect wherever cheap, short-sighted searchers recruit on sparse, patchy resources. b.

The environment and the search are one principle: maximum entropy. We can now

make the maximum-entropy argument of §V E precise. The clumped field and the Lévy search are the same variational principle over different spaces. On the environment side, maximising entropy over configurations [13] makes the clustered field the least-committal state once resources aggregate (§V E). On the movement side, a truncated Lévy walk is what maximum entropy in its path form—Jaynes’ maximum caliber [13]—yields for movement 32

given only partial information [48, 49]: among trajectory ensembles consistent with limited knowledge, the heavy-tailed one is the least-committal—it commits to no characteristic step scale, mirroring a field that has none. Formally, a configuration partition function P Z = eβC over arrangements—multiplier β favouring the clumping measure C (the number P of close pairs), attractive for β>0—and a path “caliber” e−λΦ over trajectories—multiplier λ, path functional Φ—are the same construction, taken once over where targets are and once over how to move among them. So the searcher’s least-committal strategy (Lévy) is matched to the field’s least-committal structure (sparse-clustered)—directly visible as the optimal search exponent tracking the field’s fractal dimension (α∗ ≈ Df , Fig. 16)—because both are maximum-entropy inferences from partial information; coincidence 6 is not a coincidence. We offer this as a conceptual unification, not a single partition function from which both are derived (they are maximised over different spaces); the load-bearing, falsifiable content stays the independently-measured exponents (the forager’s µ and the fields’ βres , §V D) that were free to disagree and do not. c.

How far does the principle reach? The same reading extends—as a description—

even to the two structures we have called emergent. The swarm’s communication graph is itself a maximum-entropy object—a random graph given only density and range [50], so the homogeneous rule on it is the least-committal model of swarm information spread, with real heterogeneity a structured departure the window survives (§V D)—and the recruitment cascade’s near-critical avalanche-size law (τ ≈ 3/2) is the maximum-entropy distribution under a single “no characteristic scale” constraint; so field, search, substrate, and cascade can all be read as one least-committal form in different spaces. But this unifies outputs, not generators, and the asymmetry is the point: for the field and the search maximum entropy generates the distribution—impose the constraint, derive the law—whereas the cascade’s scale-invariance is self-organized, appearing only at the percolation point the dynamics drive the swarm to; a maximum-entropy substrate off that point gives exponential, not scale-free, cascades (just as the Erdős–Rényi component-size law is scale-free only at pc ). A stronger unification— that the self-organization to criticality is itself a maximum-caliber outcome of the driven, dissipative dynamics [49, 51]—would close the gap and make the cascade maximum-entropygenerated rather than merely so described; whether self-organized criticality admits such a variational account is a long-standing open question [29, 52, 53], and we do not claim it here. 33

d.

Why this rule, and what “untuned” means. Our coordination rule—activate when

the active-neighbour fraction a/k reaches the threshold ϕ, then steer toward the recruiters— is minimal at both ends: one bit (the binary recruitment signal), one threshold (ϕ), and— because recruited agents reuse the exploring step-length and change only direction—no added movement, so recruitment is movement-neutral. The behaviours it produces do not hinge on this particular rule: they follow from reward-proportional recruitment and densitydependent self-limitation (analysed in a companion paper), not the fraction rule specifically, so soft thresholds, weighted neighbourhoods, and other heavy-tailed laws share them—the particular choices are representative. “Untuned” does not mean parameter-free: ϕ and α set where in the regime the swarm operates (how much blocking, how heavy-tailed), not which behaviours appear—each holds across a range, and no behaviour was hand-fitted. e.

Status, and what remains open. Q1—answered: the cascade window appears in ge-

ometric space. Q2—local, not global: recruitment cascades are directly near-critical (local branching σlocal ≈ 0.9 vs. swarm 0.03) while the global ratio stays subcritical. Q3—answered by a controlled comparison: the coupling causes the self-limitation. Q4—conditional: recruitment pays off iff resources are sparse and clumped (up to 7.6× the baseline), loses otherwise, and the exploration regime α decides whether it is worth its cost. Now closed: the patch-leaving equivalence is measured (giving-up density ≈40%, blocking the tunable cause), and heavy-tailed search is confirmed optimal for clustered fields—a recruitment-free single-searcher sweep (App. A, Fig. 16) shows the optimal exponent tracks the field’s fractal dimension (α∗ ≈ Df , r=0.97), a numerical tracking rather than a proven identity. This speaks directly to the result that the inverse-square Lévy walk is not an optimal single-searcher strategy for d ≥ 2 [39]: that analysis concerns encounter rate on randomly (Poisson) placed targets, where the exponent is nearly immaterial, whereas our targets are clustered and our payoff is collective harvest—and on clustered fields the recruitment-free sweep (Fig. 16) shows the optimum moves with the field, the Brownian end always worst (Fig. 15). The exponent matters here precisely because the field is clustered. Q5—affirmative: the sparse, clustered regime where recruitment pays off is generic (attractive renewal fixes the sign β>0; sparsity amplifies the magnitude) and the least-committal search matches it (α∗ ≈ Df ), so the mechanism recurs. Still open: per-cluster lifetime scaling; a search-theoretic Lévyoptimality result for clustered fields (which would put §VI C 0 b on a formal footing); and a fair price of anarchy against a centralized planner, which must control for information (an 34

omniscient allocator conflates knowing-where with coordinating), so is deferred. We should also be plain about the word general : we argue generality at the level of the mechanism’s ingredients—heavy-tailed exploration, threshold recruitment, and density-dependent blocking on a clustered field—and demonstrate it in a single model and task family, in simulation; we do not claim it has been shown empirically across foraging taxa, only that each ingredient and the field statistics it needs are independently documented (§VI B). Finally, a deliberate scope choice: resources carry an economy—they deplete and respawn—but agents carry no energy state or starvation, so nothing penalises failing to collect. This isolates the recruitment mechanism; adding a metabolic budget would couple local density to survival—a different model.

VII.

CONCLUSION

Cascade-mediated recruitment is a minimal, untuned coordination primitive: percolation enables it and blocking self-limits it—the LTM cascade window in geometric space— producing an emergent exploration–exploitation balance.

The criticality is local (per-

avalanche branching near critical, not a global self-tuning), the self-limitation is caused by the topology–geometry coupling, and coordination is efficient exactly in the sparse, clumped fields that dominate real foraging, with blocking acting as marginal-value patch-leaving and the cascade as an information channel that switches search from global to local. One threshold rule thus reproduces, from first principles, a suite of behaviours foraging ecology documented separately—recruitment, patch-leaving, and Lévy/area-restricted search—so the mathematics predicts the natural history. The same heavy-tailed search does double duty—the least-committal match to a scale-free field, and the escape that lets a satiated or released cohort leave a drained patch and reset the cycle—an elegant, multifunctional economy of the kind biology repeatedly exploits [54], here bought without any tuning. Its pieces are field-matched, not field-tuned : the search’s optimal exponent tracks the field’s geometry (α∗ ≈ Df ), the cascades are critical on the sparse-clustered fields it is matched to (mean-field branching on the near-CSR field, Fig. 6), and the network–space coupling carries coordination across space far faster than the agents move (Fig. 8). Network and physical space are not two models but one coupled system—the communication graph is the swarm’s geometry seen as a graph. The order is fixed: the field comes first and sets 35

the movement, and movement and cascade then close the loop that makes the coordination work. The deeper unity is a single principle: the configuration counting that makes the clustered field the least-committal one once resources aggregate, and the maximum-caliber heavy-tailed search that exploits it, are the same maximum-entropy variational choice, made once over space and once over paths—so coordination is adaptive not by fine-tuning but because it is matched to the statistics the environment almost inevitably has, which may be why recruitment-based foraging is so widespread.

COMPETING INTERESTS

The author declares a competing financial interest: a U.S. provisional patent application has been filed (assigned to Komplexity AI LLC) covering aspects of the coordination method described.

DATA AND CODE AVAILABILITY

Complete simulation code, precomputed sweep data, and figure-regenerating scripts are at https://github.com/galenwilkerson/collective-foraging-cascades (publicly available).

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38

Supplementary Information Appendix A: Supplementary figures and analyses

These support the genericity (§V E), Q2 criticality (§V B), cost and field-adaptation (α∗ ≈

Df , §V D), balance (§VI A), and foraging (§VI B) results; placed here to keep the main text focused. a.

The resource-field spectrum (§V E). Fig. 12 samples the maximum-entropy (Gibbs)

point process P ∝ eβC (C the number of close pairs) across the sign of the interaction: repulsion (β<0) gives a regular, fluctuation-suppressed field (pair correlation g(r)<1 at short range); independence (β=0) gives complete spatial randomness (g(r)≈1); attraction (β>0) gives a clustered field (g(r)>1, peaked). Regularity is the exception—it requires an active repulsion—while the generic resource-renewing interaction is attractive, so the leastcommittal field is clustered. The regular limit’s cost is explicit as a partition function: for M targets in K=M cells the perfectly regular state has weight M !/M M ∼ e−M and is reached

only as β → −∞, at an extensive free-energy cost ln(M M /M !) = M nats—an entropic force away from regularity. b.

Sparsity amplifies the effective clustering (§V E). The effective clustering strength

of a field is the dilute-limit multiplier that reproduces its mean close-pair count, βeff =
log(⟨C⟩/⟨C⟩CSR ), where ⟨C⟩CSR = N2 πr02 is the count a CSR field of the same density N gives within range r0 . For a renewal (cluster) field of Np clumps with occupancy µ
(total N =Np µ), within-clump pairs dominate: ⟨C⟩ ≈ Np µ2 ≈ 21 N µ, linear in N , whereas

⟨C⟩CSR ≈ 12 N 2 πr02 is quadratic. Hence βeff ≈ log µ − log N − log(πr02 ): at fixed occupancy µ it rises like log(1/density) with unit slope. Fig. 11 confirms this on a Thomas cluster process (Gaussian offspring at scale σc , r0 ≥σc ): the fitted slope is 0.91–0.92, independent of µ ∈ {10, 20, 40}, with the intercept shifting by log µ as predicted (the small constant offset below the idealized value is the finite offspring spread, which leaves some within-clump pairs just beyond r0 ). Two caveats bound the claim: the growth is logarithmic, and it amplifies contrast (g(r), hence βeff ), not the absolute windfall per discovery, which also depends on the occupancy µ that renewal dynamics set. It is also distinct from the fluctuation clumpiness of a sparse CSR field, where g(r)=1: here the excess pair correlation is genuine (g(r)≫1), and it is its measured magnitude that sparsity amplifies. 39

CSR)

effective clustering strength eff = log( /

Sparsity amplifies the effective clustering strength 4.5 4.0 3.5

slope-1 reference = 10 (slope 0.92) = 20 (slope 0.91) = 40 (slope 0.91)

3.0 2.5 2.0 1.5 1.0 8

7

6

5

log(1/density) (sparser )

4

FIG. 11. Sparsity amplifies the effective clustering strength. βeff = log(⟨C⟩/⟨C⟩CSR ) for a Thomas cluster field versus log(1/density), at fixed clump occupancy µ (points: mean±s.d. over 24 realizations; dashed: a slope-1 reference anchored at each series’ densest (leftmost) point). The slope is ≈0.91 and µ-independent, as the derivation requires, while the intercept rises with µ. Sparser fields (right) carry larger effective clustering: the sparsity foraging presupposes sharpens the structure recruitment exploits.

c.

Cost anatomy and communication channels (§V D). The two communication chan-

nels must be separated. Presence beacons are one per agent per tick regardless of behaviour (constant, one per agent, ≈N /tick; rank correlation −0.001 with z)—a clock, not a signal, paid identically by both conditions, so total bits is 93% rank-determined by timeto-completion. The recruitment-specific activation channel is density-dependent (+0.42 with z), tracing the same inverted-U (rising 21 → 91 pings/tick, falling to 86 as blocking sets in). On that channel recruitment costs ∼4–5× more messages (0.254, 0.249, 0.245 at α = 1, 1.5, 2); since activation is one ping per active agent, the ratio is cascade amplification restated, governed by z and ϕ (∼2× at z≈1 to ∼12× at z≈3–13), flat in α. d.

Universality (finite-size scaling). Sweeping z for N ∈ {50, . . . , 800}, the suscepti-

bility χ = N Var(Ψ) (Ψ the largest-cluster fraction) peaks at a fixed dimensionless z ∗ ≈ 7.8 40

for every N (location std 0.17), peak height ∝ N 1.0 , the window sharpening to this N -stable critical point as N grows (Fig. 13)—the cascade transition is a property of the dimensionless control, not system size. (The single-seed FSS setup gives a lower absolute cascade fraction than Fig. 2’s fixed-fraction seeding; what is N -independent is the window, not the peak height. z ∗ ≈ 7.8 sits just above the cascade-fraction peak z ≈ 6–7 because a susceptibility maximum generically lies above where the order parameter turns over.) e.

Scale-free resource fields and Lévy-on-clustered. Driving the model with power-law

clump sizes (realized βres ∈ {1.6, 2.2, 2.7}, R2 = 0.97), recruitment still pays off (4.2–5.7× baseline, rising as the field clumps), while the emergent active-cluster exponent is roughly constant within the scale-free family: τactive ≈ 1.25–1.30 across the whole βres range (overlapping 95% CIs, no significant trend), though this sits below the mean-field 3/2 the near-CSR field gives (§V B). Blocking keeps the cascades near-critical on clustered fields, but at a lower, geometry-set exponent than the near-CSR case—the criticality is robust across fields while the exponent is set by the field, not a constant the swarm imposes (Fig. 14). Sweeping α on clustered vs. scattered fields, harvest peaks at a heavy tail (α ≲ 1) for both, the clustered field paying several-fold more (Fig. 15)—so heavy-tailed search is advantageous for clustered targets too, as the maximum-entropy reading predicts. An independent, recruitment-free check isolates the pure search-optimality claim and quantifies the field: a single destructive forager on scale-free Lévy-dust fields of measured fractal dimension Df ∈ [0.26, 1.12] is heavytailed-optimal throughout (the Brownian end always worst), and the optimum α∗ tracks the

clustering— rising monotonically from ballistic (α∗ ≈ 0.3) for strongly clustered fields to

inverse-square (α∗ ≈ 1.3) for diffuse ones, with α∗ ≈ Df over the clustered range (r=0.97, 95% CI [0.87, 0.99]; Fig. 16). The two ends of this curve are the classically understood limits—the diffuse end recovers the canonical inverse-square Lévy-foraging optimum (µ=2) for sparse targets [11], and strong clustering drives the optimum ballistic—so the curve is anchored at both ends by known results, and the new content is the monotonic interpolation α∗ ≈ Df between them. This is a numerical tracking of search to field structure, not a proven identity, but it is the sharper, quantitative form of the compatibility that §VI C 0 b argues. f.

Blocking measured directly. Fig. 17: the blocked fraction (agents with active neigh-

bours but a/k < ϕ) grows monotonically with mean degree—0.04 → 0.57 at ϕ=0.18, 0.05 → 0.78 at ϕ=0.30—and is stronger at larger ϕ (lower boundary 2/ϕ), while the recruited fraction traces the inverted-U, confirming blocking as the operative brake. 41

Resource field as a maximum-entropy (Gibbs) point process: one interaction sign sweeps the spectrum repulsive ( < 0): regular

g(r)

pair correlation g(r)

independent ( = 0): CSR

attractive ( > 0): clustered

pair correlation g(r)

pair correlation g(r)

2.0

2.0

1.5

1.5

1.0

1.0

4

0.5

0.5

2

0.0

0.0

0.1

r

0.2

0.3

0.0

g > 1: clustered g < 1: regular

8 6

0.0

0.1

r

0.2

0.3

0

0.0

0.1

r

0.2

0.3

FIG. 12. The resource field as a maximum-entropy (Gibbs) point process: one interaction sign sweeps the spectrum. Samples of P ∝ eβC (C the number of close pairs). Top: point patterns for repulsion (β<0, regular), independence (β=0, CSR), and attraction (β>0, clustered). Bottom: the corresponding pair correlation g(r)—the neighbour density at separation r relative to a completely random field (g>1 clustering, g<1 regularity, g=1 randomness)—below 1 at short range for the regular field, flat (≈ 1) for CSR, and peaked above 1 for the clustered field. Regularity requires an active repulsion and is the exception; the generic, attractive resource-renewing interaction makes the clustered field the least-committal one (§V E).

42

0.7 0.6 0.5 0.4 0.3

zcRGG

z

4

6 8 10 mean degree z

2/

N=50 N=100 N=200 N=400 N=800

175 150 = N Var( )

cascade fraction

200

N=50 N=100 N=200 N=400 N=800

0.8

125 100 75

0.2

50

0.1

25 0

0.0 2

4

6 8 10 mean degree z

12

14

2

12

14

FIG. 13. Finite-size scaling pins a size-independent critical point. (a) Cascade fraction (largest active-cluster fraction Ψ) vs. z for each N . The curves are not N -independent—they undergo finite-size scaling: small systems round the onset below the RGG percolation threshold zcRGG ≈ 4.51 and depress the peak, sharpening toward a fixed window as N grows, and cross near z ∗ ≈ 7.8. What is N -independent is the critical point, not the finite-N curves. (b) That point is pinned by the susceptibility χ = N Var(Ψ) (Ψ largest-cluster fraction), whose peak stays at fixed z ∗ ≈ 7.8 for all N ∈ [50, 800] (location s.d. 0.17) with height ∝ N .

43

harvest ratio 95% bootstrap CI

6 5 4 3 2 1 0

1.6

1.8 2.0 2.2 2.4 resource cluster-size exponent res

emergent active-cluster exponent active

harvest ratio recruit / baseline

7

1.32 1.30 1.28 1.26 1.24 1.22 1.20 1.18

2.6

active

95% bootstrap CI

1.6

1.8 2.0 2.2 2.4 resource cluster-size exponent res

2.6

FIG. 14. On power-law resource fields, the payoff tracks the field, and the cascades stay critical at a field-set exponent. (a) Harvest ratio > 1 across the whole scale-free regime, rising as the field clumps (smaller βres )—the benefit is field-dependent. (b) The emergent cascade exponent τactive is roughly constant within the scale-free family (≈ 1.25–1.30 across the resourceexponent range βres , no significant trend, overlapping 95% CIs), but sits below the mean-field branching value (τ =3/2) the near-CSR field gives (§V B)—the cascades stay near-critical across these fields while the exponent is set by the clustered regime, not a constant the swarm imposes. Both panels: markers, connecting line, and shaded 95% bootstrap-CI band.

44

steady-state harvest

200

clustered field scattered field optimum (heavy-tailed, 95% bootstrap CI

1)

2.0

3.0

150 100 50 0

0.5

1.0

1.5

2.5

Lévy exponent (small = heavier-tailed search)

FIG. 15. Heavy-tailed search is advantageous for clustered targets too. Steady-state harvest vs. α (smaller = heavier tail) for a clustered (3 clumps; Clark–Evans nearest-neighbour index R=0.32) vs. a scattered (15 clumps; R=0.77) field (R<1 indicates clustering); harvest peaks at α ≲ 1 for both, the clustered field paying several-fold more. Bands: 95% bootstrap CI over replicas. (Fig. 16 quantifies this over a continuous range of scale-free fields.)

45

(a) efficiency vs search exponent

(b) the search exponent tracks the field (r=0.97) 1.2

identity * = Df measured *

*

0.9

optimal search exponent

foraging efficiency (normalised)

1.0

0.8 0.7 0.6 0.5 strongly clustered (Df=0.26, * 0.30) intermediate (Df=0.97, * 0.95) diffuse (Df=1.12, * 1.30) optimum * (shifts left as clustering grows)

0.4 0.3

1.0 0.8 0.6 0.4

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 search Lévy stable index (ballistic Brownian)

0.4 0.6 0.8 1.0 1.2 resource-field fractal dimension Df (clustered diffuse)

FIG. 16. A recruitment-free single-searcher sweep: the optimal search exponent tracks the field’s clustering. Destructive foraging (targets per unit path length) on Lévy-dust fields spanning fractal dimension Df ∈ [0.26, 1.12], swept over the Lévy stable index α (tail exponent µ=1+α; small α = heavier tail). (a) Efficiency η(α) for three representative fields (±SEM bands): strongly clustered fields peak ballistic (α → 0), diffuse fields peak near inverse-square (α ≈ 1.2); the Brownian end (α → 2) is worst throughout, so heavy-tailed search is optimal in every case. (b) The optimum α∗ (centre-of-mass of the near-optimal region) rises monotonically with Df , α∗ ≈ Df over the clustered range and saturating near inverse-square for diffuse fields (r=0.97, 95% CI [0.87, 0.99], 192 samples/point; raw-argmax estimate r≈0.86). This is a normative check with no recruitment, complementing the swarm-harvest sweep of Fig. 15 and quantifying its fields (which are only labelled clustered/scattered). It is an empirical, numerical trend—a tracking of search to field structure, not a proven identity—consistent with §VI C 0 b. 12 fields × 16 searchers per point.

46

(a) recruited (cascade) fraction

blocked fraction

0.8

(b) =0.18 =0.30

0.6 0.4 0.2 0.0

101

mean degree z (dotted = blocking boundary 2/ )

=0.18 =0.30

0.6 0.5 0.4 0.3 0.2 0.1 0.0

101

mean degree z (dotted = blocking boundary 2/ )

FIG. 17. Blocking is the brake at high density. (a) The blocked fraction rises with mean degree z, faster for larger ϕ. (b) The recruited fraction peaks in the window then collapses; dotted lines mark z = 2/ϕ.

= S 2 / S (L\'evy =1.5)

6

0.25

5

0.20

4

cascade fraction

3

cascade fraction

local susceptibility

0.30

0.15 0.10

2

0.05

1 0

5

10

15 20 mean degree z

25

30

0.00

FIG. 18. A corroborating percolation signature: local susceptibility peaks inside the cascade window. χ = ⟨S 2 ⟩/⟨S⟩ (largest cluster excluded) vs. mean degree z, for the swarm’s own Lévy exploration (α=1.5), with cascade fraction (dashed): a broad peak at z ≈ 6.6—within the window and near the finite-size-scaling z ∗ ≈ 7.8—the percolation-susceptibility hallmark. Shaded band: 95% bootstrap CI over replicas.

47

(a) power spectrum of activity A(t) 10 1 10 2

CA ( )

SA(f)

100

SA(f) (Welch, 192 replicas) Lorentzian+floor (fc=0.078)

(b) activity autocorrelation (semi-log) CA ( ) e / ac, ac=2.7

10 1

10 3 grey 1/f 2, red 1/f

100 10 1 10 2 10 3 10 4 10 5 10 6

10 2

10 2 0

10 1

frequency f (1/tick)

(c) activity-episode lifetimes (CCDF)

10

20 lag (ticks)

30

40

(d) detrended fluctuation analysis

P( ) tail slope 2.34 =1.5

DFA=0.62 ( =0.23)

100 F(n)

survival P(

)

10 3

10 1 100

101 102 episode lifetime (ticks)

101

102 window n (ticks)

103

FIG. 19. The temporal signature is Lorentzian, not 1/f : activity has short memory— the mark of local, not global, criticality. Power spectrum and companion diagnostics of the global recruitment-activity A(t) (active fraction, sampled per tick, instantaneous—not integrated, which would add a spurious f −2 ), at a near-critical operating point in the stationary replenished task (ρr =1.6, so zlocal ≈ 7.5; ϕ=0.20, α=1.5; 192 replicas, 16,384 ticks after a 2000-tick transient). (a) SA (f ) (replica mean; band 95%) with a Lorentzian+white-floor fit: a flat plateau below a corner fc ≈ 0.08, then a roll-off between the 1/f and 1/f 2 guides—decisively not the broadband 1/f of a globally near-critical system. (b) The activity autocorrelation (semi-log) is dominated by a fast relaxation (τac ≈ 2.7 ticks, so 1/(2πτac ) is of the same order as fc —the fast relaxation sets the Lorentzian corner), with a weak slow tail (the heavy lifetime tail below, and the mild long-memory of panel d). (c) Episode lifetimes τlife (consecutive active ticks per agent) follow a power-law survival function (CCDF, slope ≈ 2.3) with a cutoff beyond ∼100 ticks and a short mean (⟨τlife ⟩ ≈ 1.5): broad but bounded. Through the lifetime→spectrum relation [46], this boundedbut-broad P (τlife ) is what yields the intermediate 1/f -to-1/f 2 roll-off rather than a pristine 1/f 2 . (d) Detrended fluctuation analysis gives αDFA ≈ 0.62, i.e. βspec =2αDFA −1 ≈ 0.2: near-white, sub-1/f long-range scaling. Two timescales are in play (§VI A): the recruitment cascade resolves within a tick and carries the size power law (Fig. 14), whereas the per-tick spectrum here reflects the multi-tick activity episodes.

48

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