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The emergence and evolution of a referential code in populations of bee-like agents
Grzegorz ChrupaÅa
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 92%
Last extracted: 8/27/2026, 3:46:42 AM
Summary
This paper presents a computational model of the evolution of referential communication in bee-like agents, specifically focusing on the transition from a direct-pointing code (used in species with horizontal combs) to a gravity-referenced code (used in species with vertical combs). The model simulates selection at the colony level, demonstrating that direct pointing evolves when food is moderately hard to find. The transition to the gravity-referenced code is driven by mutation rates, the magnitude of exogenous benefits for vertical combs, and the coupling between sender and receiver mutations, proceeding reliably without communication breakdown under favorable conditions.
Entities (10)
Relation Signals (10)
Apis florea ā uses ā Direct Pointing
confidence 95% Ā· in species with exposed horizontal combs (Apis florea and Apis andreniformis) the bees orient the axis of the waggle dance directly towards the food patch.
Apis andreniformis ā uses ā Direct Pointing
confidence 95% Ā· in species with exposed horizontal combs (Apis florea and Apis andreniformis) the bees orient the axis of the waggle dance directly towards the food patch.
Vertical Comb ā enables ā Gravity-Referenced Code
confidence 92% Ā· species with vertical combs cannot point directly and instead reference the dance to gravity
Horizontal Comb ā enables ā Direct Pointing
confidence 92% Ā· species with horizontal combs point directly at a food source
Vertical Comb ā prevents ā Direct Pointing
confidence 92% Ā· species with vertical combs cannot point directly and instead reference the dance to gravity
Direct Pointing ā evolveswhen ā Moderate Food Difficulty
confidence 90% Ā· direct pointing evolves readily when food is moderately hard to find by random search alone
Micrapis ā possesses ā Horizontal Comb
confidence 90% Ā· the horizontal comb and direct pointing are the ancestral state, extant in the stem Micrapis subgenus.
Gravity-Referenced Code ā ā
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Abstract
Abstract:Communication requires a shared code, and any change to it must be coordinated between senders and receivers to avoid a breakdown of communication. The honeybee waggle dance illustrates this problem: species with horizontal combs point directly at a food source, while species with vertical combs cannot point directly and instead reference the dance to gravity, decoded against the position of the sun. We model the emergence and evolutionary transition between these two codes in populations of bee-like agents, with selection acting at the level of colonies. In a horizontal-comb model, we find that direct pointing evolves readily when food is moderately hard to find by random search alone, whether because sites are few and large or many and small, and fails when food is too sparse to spark dances or so abundant that it is found without signaling. Adding an exogenous benefit for vertical combs, we then find that the transition to the gravity-referenced code is driven mainly by the mutation rate and the magnitude of this benefit, with the coupling between sender and receiver mutations playing a further role at low mutation rates. Given a favorable confluence of these factors, the transition proceeds reliably and without a breakdown of communication.
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- Source: https://arxiv.org/abs/2608.25779v1
- Canonical: https://arxiv.org/abs/2608.25779v1
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The emergence and evolution of a referential code in populations of bee-like agents Grzegorz ChrupaÅa ā thanks: Code, configurations, and result files to reproduce every figure and table in this paper are available at https://github.com/gchrupala/bees. AI coding agents, primarily Claude Code and Codex, were used in developing the codebase and as a writing aid in the preparation of this manuscript. The author reviewed all content, and takes full responsibility for it. Abstract Communication requires a shared code, and any change to it must be coordinated between senders and receivers to avoid a breakdown of communication. The honeybee waggle dance illustrates this problem: species with horizontal combs point directly at a food source, while species with vertical combs cannot point directly and instead reference the dance to gravity, decoded against the position of the sun. We model the emergence and evolutionary transition between these two codes in populations of bee-like agents, with selection acting at the level of colonies. In a horizontal-comb model, we find that direct pointing evolves readily when food is moderately hard to find by random search alone, whether because sites are few and large or many and small, and fails when food is too sparse to spark dances or so abundant that it is found without signaling. Adding an exogenous benefit for vertical combs, we then find that the transition to the gravity-referenced code is driven mainly by the mutation rate and the magnitude of this benefit, with the coupling between sender and receiver mutations playing a further role at low mutation rates. Given a favorable confluence of these factors, the transition proceeds reliably and without a breakdown of communication. 1 Introduction Communication in animal species takes many forms, from the chemical signals of ants to the symbolic, compositional language of humans. Each of these communication systems has its own idiosyncratic properties. What ecological and evolutionary pressures shaped these systems has been of substantial interest to biologists and cognitive scientists. In general, communication assumes some kind of shared code between senders and receivers, and the emergence of such a code requires a degree of alignment between these two roles. Likewise, any change or modification to the code needs to be coordinated between senders and receivers if the breakdown of communication is to be avoided. In many cases it is possible to think of a plausible pathway for the coordinated emergence and development of a shared code, but a verbally formulated scenario leaves many details underspecified and makes it hard to be precise about the specific conditions that make code change trajectories feasible. For this reason computational models of the evolution of communication systems can provide insights which are hard to obtain by other means. Such models have been used extensively to study the emergence of communication and language (Section 2). In this paper we model a transition of a particularly intriguing communicative system in invertebrates: the waggle dance of honey bees. Communication is referential when signals stand for entities or states in the world rather than merely expressing the signalerās internal state. The communicative code of Apis species von Frisch, (1967) is a unique referential system where food sites discovered by foraging scouts are indicated to other members of the colony by means of a figure-of-eight dance whose axis encodes the direction towards the food source, and whose duration stands for the distance to it. In this study we focus on the signaling of direction, as its evolution illustrates some interesting questions about coordination between participants in a communicative system. Different species of Apis vary in the specific dance variant used to encode the direction of the food patch: in species with exposed horizontal combs (Apis florea and Apis andreniformis) the bees orient the axis of the waggle dance directly towards the food patch. In other members of the Apis genus, combs are vertical and in some species also hidden in cavities. Such combs make direct pointing impossible, and these species use gravity as a reference point, and map the azimuth of the food patch relative to the sun onto the angle of the waggle dance with respect to the vertical. Figure 1 illustrates these two versions of the waggle dance.11 1 This is a somewhat simplified overview: in individual species there may be considerable behavioral flexibility in how the dance is performed: e.g. in Apis mellifera, sometimes bees dance on a horizontal platform at the entrance to the beehive and in this case they use direct pointing, even though this species generally employs the gravity code (Esch,, 2012). The direct-pointing variant is iconic: the form of the signalāthe direction of the waggle axisāmirrors the direction of the referent. The ancestral, direct-pointing dance has been characterized as an enactment of the foraging flight, in which the waggle run imitates the departure toward the food (Barron and Plath,, 2017). The gravity-referenced variant is more abstract, relying on an indirect mapping of the sunās azimuth onto an angle measured from the vertical. The waggle dance thus offers an opportunity to study how a referential code can shift from an iconic to a more abstract form, in a relatively simple system. Figure 1: The two variants of the honeybee waggle dance. On a horizontal, exposed comb (left) the bee aims the straight waggle run directly along the true direction to the food, which it can identify as being at a particular azimuth relative to the sun. On a vertical comb (right) the bee cannot directly point at the food. Instead, it uses gravity as a reference: it maps the direction up to the position of the sun, and it rotates the dance axis away from vertical by an angle Ī± equal to the food siteās azimuth relative to the sun. The figure-of-eight loops and the central waggle run are schematic and not to scale. The distribution of comb orientation and waggle dance code throughout the Apis phylogeny (Figure 2) suggests that the horizontal comb and direct pointing are the ancestral state, extant in the stem Micrapis subgenus. Vertical combs and the gravity-referenced code are derived traits, present in the other two Apis subgenera. Additionally, species within the Apis sensu stricto subgenus are cavity nesters. This pattern indicates that the gravity code was linked to vertical combs and that cavity nesting is not required for its emergence (Barron and Plath,, 2017). Nevertheless, the exact sequence and nature of evolutionary events that led to the emergence of the gravity-based waggle dance is not known in detail. It is also not clear how the transition from one version of the code to the other could happen gradually, while maintaining communicative success at all stages, and how this process interacted with the change in the orientation of honeycombs. Because the code is shared, such a transition poses a coordination problem: a change adopted by senders but not matched by the receiversā interpretation, or the reverse, disrupts communication, so that intermediate stages risk a loss of communicative success. A mechanistic account of this transition has been proposed at the neural level: the central-complex circuitry that represents the beeās directional heading (its own compass orientation) is not tied to any particular spatial reference frame, so a gravity reference could in principle substitute for the celestial one without a dedicated switching mechanism (Barron and Plath,, 2017). Such an account addresses how an individual brain can support either version of the code, but not how a population of senders and receivers stays coordinated while the shared code itself evolves, which is the question we address. Figure 2: A simplified illustration of the Apis phylogeny, with the three subgenera Micrapis, Megapis and Apis sensu stricto. Adapted from Lo et al., (2010) The present study investigates this transition by means of a computational model of evolving populations of bee-like agents. We model selection at the level of colonies, and individual behavior at the level of worker bees as colony members. We address two principal questions: (i) under what ecological conditions a direct-pointing referential code emerges and stabilizes, and (i) which factors lead to a gradual transition from this code to its gravity-referenced variant. Our aim is not to reconstruct the precise historical sequence of these changes, but to delineate the circumstances under which such a transition can proceed without a breakdown of communication. Our starting point is colonies with horizontal combs, which show some heritable variation in traits controlling the inclination to direct signaling and attending to signals. Here we show that: communication via direct pointing evolves readily when food is moderately hard to find by random search alone: i.e. when food sites are few in number but large, as well as when there are many small ones. It fails when food is too sparse to spark dances, or so abundant that it is found without signaling. In the second stage we introduce extrinsic evolutionary pressure for vertical combs, which partially degrades the direct-pointing code while making the gravity reference available. We show that in this scenario the transition towards the gravity-referenced code is driven mainly by the mutation rate and the magnitude of the exogenous benefit of the vertical comb. Additionally the strength of the coupling between the signaling behavior of senders and receivers also plays a role, especially in the low-mutation-rate regime. The pattern of results from these simulations indicates that a basic referential code based on an iconic, direct-pointing signal evolves readily given favorable ecological conditions. On the other hand, the transition from this basic ancestral system to the abstract code based on gravity-referenced pointing requires the confluence of several key favorable ecological and evolutionary factors; once these hold, however, the transition proceeds reliably and without a breakdown of communication. 2 Related work The current study builds on several distinct strands of work. Firstly, it draws on research into the emergence and evolution of communication in artificial multi-agent systems, viewed as an abstract computational problem. Secondly, because our work is inspired by and grounded in real honeybee societies, it also builds extensively on models and simulations of bee communication. These studies are methodologically diverse, employing agent-based models, neural models, evolutionary computation, or combinations thereof. 2.1 Communication in agent societies A long line of computational work has studied how communication systems can emerge and stabilize in populations of interacting agents. The signaling game introduced by Lewis, (1969) provides the formal foundation for much of this work, framing a convention of meaning as a solution to a recurring coordination problem between a sender and a receiver. Skyrms, (2010) shows how such signaling conventions can arise without prior agreement through evolutionary and reinforcement-learning dynamics. A distinct route emphasizes cultural transmission: in the iterated-learning framework, linguistic structure emerges as a code is repeatedly learned and reproduced across successive generations of agents, without being built in (Smith et al.,, 2003). Related formal work models how a protolanguage itself evolves under selection, showing for instance how combining a small set of signals into words overcomes the error limit that caps a simple signal repertoire (Nowak and Krakauer,, 1999). Agent-based models in this tradition have aimed to demonstrate that populations playing repeated language games can self-organize shared, grounded vocabularies and categories, a simulation tradition surveyed by Wagner et al., (2003). More recently, deep multi-agent reinforcement learning has been used to show that discrete communication protocols can emerge when agents are trained on cooperative referential tasks (Lazaridou et al.,, 2017). Subsequent work has examined when such emergent codes become compositional and generalize to novel meanings (Chaabouni et al.,, 2020), and how communicative pressure can drive neural agents towards attested language universals (Lian et al.,, 2023); see Lazaridou and Baroni, (2020) for a review of this line of research. 2.2 Computational models of the waggle dance A venerable research tradition asks an ecological question: given that the dance exists, when does it pay off? The main tool here is the agent-based foraging simulation. A colony of many simple forager agents follows fixed behavioral rules. Dance recruitment is present as a switch, with recruits either guided by a successful foragerās spatial information or left to search on their own, while the ecology (the density, quality, spread, and persistence of food patches, and sometimes colony size) is varied across runs. The outcome of interest is colony-level foraging performance: typically the net energy or nectar collected. Importantly, these models represent neither evolution nor within-lifetime learning: the communication strategy is fixed and the agents do not adapt. What they model is the ecological cost-benefit accounting of a given strategy: how a fixed colony fares, with recruitment versus without, in one environment or another. Using such a model, Dornhaus et al., (2006) find that recruitment benefits a colony mainly when resources are patchy, variable, and hard to find, with the advantage depending on colony size, and SchĆ¼rch and GrĆ¼ter, (2014) show that the danceās long-term benefits can outweigh its short-term costs. Bailis et al., (2010) pose a more abstract version of the same question, analyzing when positional communication beats reliance on private information. This body of work is the closest in spirit to our own foraging payoff, but it holds the communication system constant and measures its payoff in a fixed environment; it does not model the code itself as an evolving trait. More recently, the emergence itself of the bee communicative code has also been investigated computationally. Bernard et al., (2023) treat displaced communication as an evolutionary phenomenon: pairs of foraging agents, each controlled by a continuous-time recurrent neural network, undergo experimental evolution over tens of thousands of generations in a one-dimensional circular arena where a receiver must reach one of several sites a sender has perceived as holding food. Their central finding concerns how a displaced signal arises. Although senders could in principle modulate signal amplitude, communication instead evolved from incidental, movement-derived cues (the onset delay and duration of a senderās presence in the shared communication zone), which were subsequently ritualized into signals. The more efficient amplitude-based scheme evolved only when the cheaper cue was experimentally suppressed. This model shares our evolutionary framing but selects at the level of individual senderāreceiver pairs rather than colonies, and it abstracts away the geometry of the dance: the object of study is how any displaced signal bootstraps from behavior, not how a specific directional code is structured. Siregar et al., (2026) instead ask what internal representations and channel properties allow a compositional, waggle-dance-like code to be learned. A single senderāreceiver pair, implemented as graph neural networks operating over map-like graphs of routes between the nest and candidate sites, is trained by gradient descent on a referential game to transmit two real-valued tokens identifying the food node. They ask whether the two tokens factor into independent direction and distance components (compositionality), how this depends on the overlap between the senderās and receiverās mental maps, and whether constraining the channel to the kinematic range of a real dance helps. They find that direction is readily encoded whereas direct distance is not, that representational overlap is the main driver of compositionality, and that channel constraints have only a small effect. This is a within-lifetime study of representation and structure, with no population dynamics or selection. In a similar vein, Portegys, (2020) has an individual agent learn to both produce and follow dance-like movements that encode a nectar location, again treating the dance as a code to be acquired within an individualās lifetime rather than shaped by selection. The two differ in what is left open: Siregar et al., (2026) let the code emerge from a referential game and ask what structure results, whereas Portegys, (2020) fixes the dance in advance and has a single agent learn to reproduce and respond to it, studying acquisition of a known code rather than the emergence of its structure. Approach Code adapts by Level Question Communication in agent societies Signaling games and RL payoff dynamics population emergence Iterated learning transmission chain emergence Deep multi-agent RL learning pair emergence Models of the waggle dance Foraging simulations (fixed) colony payoff Evolved controllers evolution pair emergence Learned representations learning individual/pair learnability This work evolution colony transition Table 1: Computational approaches to how a communicative code arises or changes, and where the present model sits among them. The columns give how the code adapts (if at all), the level it applies to, and the question each line of work asks. Rows are grouped into general models of communication and models specific to the waggle dance; each row is discussed and cited in the text. Our study occupies a different point in this space (Table 1). We model the emergence and in particular the evolutionary transition between two forms of a referential code at the level of colonies, using a small set of interpretable, heritable traits rather than opaque neural controllers. This lets us treat costs, benefits, and comb geometry as explicit parameters and ask which ecological and evolutionary conditions drive the shift from iconic direct pointing to an abstract gravity-referenced code: a question about the conditions and dynamics of a communicative transition that none of the prior work addresses. 3 Method Our model is inspired by bee colonies and their waggle dance, but it is simplified in many important ways, as our goal was not to model bee biology in detail but rather investigate scenarios for the emergence and evolution of referential communicative codes, while adopting the case of the waggle dance as a way to ground the modeling in the real world and ensuring that it is evolutionarily plausible. We model evolution at the level of colonies, and behavior at the level of individual bee-like agents. This treatment is motivated by the reproductive and social organization of real bee colonies, where workers are highly related and sterile, and thus selection acts effectively on heritable colony-level traits rather than on individual workersāthis is the standard kin-selection rationale for treating the honeybee colony as a superorganism (Hamilton,, 1964; Seeley,, 1989). We do not model sexual reproduction: agent colonies reproduce without genetic input from other colonies. Colonies have a number of heritable traits which are transmitted to offspring colonies subject to mutation, and individual agents express these traits with some non-heritable individual variation. In the interest of simplicity and focus, we model the waggle dance as only signaling the direction of a food site: our agents do not indicate distance. 3.1 Colonies, workers, and traits We keep two kinds of quantity visually distinct throughout: a colonyās heritable traits, which evolve under mutation and selection and whose names we set in small capitals, and the modelās fixed or swept simulation parameters, which are chosen by the experimenter, referred to by their mathematical symbols, and listed in Tables 3 and 4. A colony is described by a small set of heritable, real-valued traits, listed in full in Table 2. The prose here introduces the traits needed for the basic horizontal-comb model (Section 3.6); the additional traits and geometry required for tilted combs are developed in Section 3.8. Symbol Name Range Role Behavioral (horizontal-comb model, Section 3.6) b directional bias [0,1][0,1] sharpness of a dance about the food direction a receiver attention [0,1][0,1] probability of following a dance rather than searching at random p dance propensity [0,1][0,1] readiness to recruit others after a successful foraging attempt ā foray distance [0,D][0,D] mean outbound distance of a foray Nest and code choice (vertical transition, Section 3.8) Ī³ comb tilt [0,1][0,1] comb inclination, 00 horizontal to 11 vertical ĻĻ comb orientation [0,P)[0,P) compass heading the comb faces tst_s sender transposition [0,1][0,1] weight for the gravity code when producing a dance trt_r receiver transposition [0,1][0,1] weight for the gravity code when interpreting a dance Table 2: The heritable, colony-level traits of the model. The first group governs behavior on a horizontal comb (Sections 3.1 and 3.6); the second adds the nest geometry and code-choice traits used in the vertical transition (Section 3.8). Workers express b, a, p, tst_s, and trt_r with individual variation (Section 3.1); ā varies through the per-foray draw (Section 3.3), and the nest traits Ī³ and ĻĻ are shared by all of a colonyās workers. Here D is the maximum search distance and P the orientation period. Four traits govern the behavior of a colonyās workers. The directional bias b controls how precisely a dance points; the receiver attention a is the propensity to attend to a dance rather than to search at random; the dance propensity p controls how readily a successful scout recruits others to the patch it just found (Section 3.4); and the foray distance ā sets the mean outbound distance a worker travels on a foray. The first three are bounded to [0,1][0,1] and the foray distance to [0,D][0,D], where D is the maximum search distance, a fixed model parameter (set to D=6D=6 km in all our experiments). Workers express the colony traits with non-heritable individual variation. A worker i realizes each trait xāb,a,pxā\b,a,p\ as xi=clipā”(x+Īµi),Īµiā¼ā”(0,Ī·2),x_i=clip (x+ _i ), _i (0,Ī·^2), (1) where Ī· is a fixed worker-variation scale and clipclip returns the value to its admissible range. The foray distance ā carries no separate worker jitter: within-colony variation in how far workers travel arises instead from the per-foray draw described in Section 3.3. A colony thus behaves as a noisy ensemble of workers sharing the same underlying genotype. 3.2 Producing and interpreting signals On a horizontal comb the dance points directly at the food: this is the iconic, direct-pointing code. A worker that has just foraged successfully at azimuth Ī± produces a dance whose intended direction is Ī± itself. The emitted signal is drawn from a von Mises distribution centered on Ī± with concentration Īŗ=biāĪŗmaxĪŗ=b_i\, _ , plus a small wrapped Gaussian dance noise. The von Mises distribution is the circular analogue of a Gaussian: a signal Ļ centered on Ī¼ with concentration Īŗ has density fā”(Ļ,Ī¼,Īŗ)=expā”(Īŗācosā”(ĻāĪ¼))2āĻāI0ā(Īŗ),f(Ļ;Ī¼,Īŗ)= (Īŗ (Ļ-Ī¼) )2Ļ I_0(Īŗ), (2) where I0I_0 is the modified Bessel function of order zero and Īŗ plays the role of an inverse variance; here Ī¼=Ī±Ī¼=Ī± and Īŗ=biāĪŗmaxĪŗ=b_i\, _ . The directional bias b therefore controls the sharpness of the dance: a low bias leads to a near-uniform, uninformative signal, while a high bias gives rise to a tightly concentrated one that reliably indicates Ī±. A worker that follows a dance reads its direction as a search direction, subject to a small added interpretation noise. Communication is successful when this recovered direction is close enough to a food site to lead the follower to it. 3.3 Foraging environment A colony is evaluated over a fixed number of independent foraging episodes, and its fitness is averaged over them (Section 3.4). Each episode draws a fresh environment (Figure 3), with patch radii in meters and distances from the nest in kilometers. The number of food sites is drawn from a Poisson distribution with mean n, so episodes vary in how much food they offer. Each site is placed at an azimuth drawn uniformly around the colony and at a distance drawn uniformly from [dmin,dmax][d_ ,d_ ], and is a physical circular disk whose radius is drawn independently from a lognormal distribution with median Ī¼r _r and log-scale spread Ļr _r. A forager finds the patch when its straight outbound path crosses the disk, so a fixed patch subtends a smaller angle when farther from the nest, and distant food demands more accurate dances. Each patch has value v per visit and a finite capacity limiting how many foragers it can supply before being exhausted; capacity scales with the patch area, growing with the square of its radius, so that larger patches hold proportionally more food while the per-visit value stays fixed. Throughout, v=1v=1, the log-scale spread is Ļr=0.6 _r=0.6, and sites lie no closer than dmin=0.75d_ =0.75 km; the median radius Ī¼r _r, the capacity at that radius, and the maximum distance dmaxd_ vary by experiment (Table 4). The foray distance trait ā (Section 3.1) sets how far workers travel. Each individual foray draws its own outbound length from a Gamma distribution with mean ā and a fixed shape, capped at the maximum search distance D. The per-foray draw is the source of within-colony variation in travel distance and gives a realistic spread of short and long forays around the colony mean. Each episode also fixes the position of the sun, which serves as the external reference for the gravity-based code introduced in Section 3.8. The sunās azimuth is drawn uniformly from an arc of width Īsun _sun centered on a fixed heading Ī¼sun _sun (in our experiments Ī¼sun=Ļ/2 _sun=Ļ/2 and Īsun=Ļ _sun=Ļ), so it is constant within an episode but varies between episodes. We explain the role of this variable sun azimuth in Section 3.8. Figure 3: Six samples of the foraging environment. Each panel shows the colony (bee) at the centre, the food patches (flowers) at their drawn azimuths and distances, and the sun, which marks the external reference for the gravity-based code. Each patch is a physical disk, and the flower size scales with the sampled radius of the patch; radial rings mark distance from the colony in kilometers. The panels are drawn at representative values chosen to illustrate the geometry: the number of patches per episode is Poisson with mean 66, so panels differ in how many patches they contain, the median radius is 150150 m (lognormal, log-sd 0.60.6), and patches lie out to a maximum distance of 66 km. All three sit within the ranges the experiments explore (Sections 4.2 and 4.3). 3.4 Foraging and colony payoff Within each episode (Section 3.3) the colony makes a sequence of foraging attempts. On each attempt a worker is drawn at random; if at least one dance has already been performed in the current episode, the worker follows a randomly chosen dance with probability equal to its attention aia_i, reading it out as a search direction, and otherwise searches in a uniformly random direction. The worker then sets out along that direction on a foray whose length is drawn as in Section 3.3 and enters the first food patch on its path, among patches that still have remaining capacity and lie within the foray length; if no patch is reached the attempt fails. The fraction of foraging attempts that reach a patch is their foraging success. Payoff accrues per episode. A successful foraging attempt yields the patch value v, minus a travel cost ctravelc_travel per unit of patch distance, and may prompt the worker to perform a dance advertising the patch to later foragers. Recruitment is itself an evolved decision: a successful scout dances with probability 1ā(1āpi)C1-(1-p_i)^C, where pip_i is its dance propensity and C is the patchās remaining capacity after the visit, so an exhausted patch never leads to a dance and a scarcely provisioned one rarely does. A dance, when performed, costs cbase+ccueābic_base+c_cue\,b_i, growing with the precision of the signal. A failed attempt costs travel proportional to the foray length it travelled, and each act of attending to a dance incurs an attention cost cattc_att. Collecting these terms, an episode with successful attempts S (each at distance rkr_k), the subset SdāS_d S of successes that produced a dance (each by a worker with bias bkb_k), failed attempts F (each with foray length āj _j), and nan_a dance-following attempts yields raw payoff Ļraw=ākāS(vāctravelārk)āākāSd(cbase+ccueābk)āājāFctravelāājācattāna, _raw= _kā S (v-c_travel\,r_k )- _kā S_d (c_base+c_cue\,b_k )- _jā Fc_travel\, _j-c_att\,n_a, (3) where the dance-cost sum runs over SdS_d because a success produces a dance only when the recruitment decision is positive. The scaled episode payoff is Ļ=(1+BāĪ³)āĻrawĻ=(1+BĪ³)\, _raw, where Ī³ is the comb tilt and Bā„0Bā„ 0 the vertical-comb benefit; in the horizontal model Ī³=0Ī³=0, so the scaling is inert. We motivate its form in Section 3.7. A colonyās fitness is its mean episode payoff (floored at a small positive value). For diagnostic purposes we also record a recruitment advantage: the difference between the foraging success of workers that followed a dance and that of contemporaneous workers that searched at random while dances were available. A positive recruitment advantage indicates that the code is functioning, since following a dance then beats searching blindly. 3.5 Evolutionary dynamics Evolution proceeds in non-overlapping generations over a fixed-size population of colonies; there is no sexual reproduction. In each generation every colony is evaluated, and the next generation is produced by repeated selection of parent colonies with a probability proportional to fitness. A selected colonyās traits are copied subject to mutation. Each trait is perturbed by an independent Gaussian step and then clipped to its range. The stepās standard deviation is a fixed fraction Ļm _m of that range: Ļm _m for the traits on the unit interval, and ĻmāD _mD for the foray distance, whose range is [0,D][0,D]. The workers of each offspring colony are then regenerated from its traits as in Section 3.1. The mutation rate Ļm _m thus sets the pace of evolutionary change. 3.6 Horizontal comb The horizontal-comb model is exactly as described above: the comb is flat and the only code available is direct pointing. We use it to investigate the conditions under which a referential code emerges and stabilizes, starting from an initial population with little communicative ability (low directional bias and low attention), so that early foragers signal weakly and rarely attend to one another. 3.7 Modeling code transition We ask how a colony can move from the direct-pointing code, available on a horizontal comb, to the gravity-referenced code used on a vertical one. We do not model why combs come to hang vertically; instead we impose an exogenous benefit for vertical building through the payoff scaling (1+BāĪ³)(1+BĪ³) (Section 3.4), which rewards greater comb tilt on its own, and ask whether communication can survive as the comb tilts under that pressure. The scaling is multiplicative because the advantages usually attributed to a vertical comb are architectural,22 2 A vertical comb carries cells on both faces of a single sheet, whereas a horizontal one can open its cells only upward; and a comb attached along its top edge is loaded in its own plane rather than in bending, which wax tolerates better (Seeley and Morse,, 1976; Hepburn et al.,, 2014). and improve the return on what a colony forages rather than paying a subsidy that arrives whether it forages or not. This couples the two pressures, since tilt is then worth most to a colony that already forages well, and therefore has a working direct code to lose. Because comb tilt is itself a trait that starts flat and evolves by small mutational steps, a lineage passes through intermediate, partly tilted combs rather than jumping from horizontal to vertical. Those intermediate tilts are what make the transition non-trivial. As the comb tilts, direct pointing degrades: projecting the food direction onto the sloping surface foreshortens it in a bearing-dependent fashion, leaving less of the direction in the surface to carry the signal. How faithfully the heading is recovered depends on whether the receiver corrects for comb orientation. We thus model two decoders (Section 3.8): a more biologically plausible flatten decoder which applies no such correction and is therefore biased on a tilted comb, and an idealized unproject decoder that corrects exactly and serves as a bias-free reference against which to read the results of flatten. Meanwhile, the growing tilt makes the gravity-referenced code increasingly reliable, so a colony can in principle switch codes as direct pointing weakens. We again model this switch as gradual, through a blend of the two codes (Section 3.8). 3.8 Dancing on a tilted comb The tilted-comb model extends the horizontal one with the machinery for comb tilt and the gravity-referenced code. It introduces two further colony-level traits describing the nest: the comb tilt Ī³ā[0,1]Ī³ā[0,1], where Ī³=0Ī³=0 is a horizontal comb and Ī³=1Ī³=1 a vertical one, and the comb orientation ĻĻ, the compass heading the comb faces. Like the behavioral traits these are heritable and mutate by an independent Gaussian step each generation, but they are properties of the nest and so are shared by all of a colonyās workers rather than being expressed with individual variation. Because the orientation ĻĻ is an angle defined on a circle of period P, its mutation step has standard deviation ĻmāP _mP rather than Ļm _m. Comb geometry. The comb is a flat surface on which dances are performed, and its tilt determines which reference directions a dance can use. We represent the comb by its unit normal, the direction perpendicular to its face. When the comb is horizontal this normal points straight up; as the comb tilts, the normal swings down toward the horizon in the compass direction ĻĻ, until on a vertical comb it lies flat. Writing the tilt as a physical angle Īø=Ī³āĻ/2Īø=Ī³\,Ļ/2 (so Īø=0Īø=0 for a horizontal comb and Īø=Ļ/2Īø=Ļ/2 for a vertical one), the normal has eastward, northward, and upward components =(sinā”Īøācosā”Ļāeast,sinā”Īøāsinā”Ļānorth,cosā”Īøāup).n= (\, Īø Ļ_east,\; Īø Ļ_north,\; Īø_up\, ). (4) At Īø=0Īø=0 this reduces to (0,0,1)(0,0,1), pointing straight up, and at Īø=Ļ/2Īø=Ļ/2 it lies in the horizontal plane pointing along ĻĻ. The two codes. Both codes express the food direction as an angle in the comb surface; they differ in the reference the angle is measured from and in how much of the direction the surface can carry. In the direct (iconic) code the worker projects the world direction of the food, =(cosā”Ī±,sinā”Ī±,0)f=( Ī±, Ī±,0), orthogonally onto the comb and dances along the resulting in-surface vector p (Figure 4). Figure 4: The direct codeās projection and the flatten decodeās bias. The tilted hexagonal comb and the world-horizontal plane (the grid) share the hiveās eastānorth position. The food direction f (blue) lies in the world plane; its orthogonal projection onto the comb, p (red), found by carrying its start and end points along the combās normal n (black) until they meet the surface, shown by the dashed rays, has in-surface angle Ī“dir _dir, the encoded signal. Decoding that signal back to a world heading, the flatten variant recovers r (orange), which drifts off the true direction by a systematic angle on any tilted comb, whereas the unproject variant would recover f (blue) exactly. The flower marks the food site. In the combās orthonormal surface axes 1,2e_1,e_2, this projection has coordinates (p1,p2)=(1ā ,2ā )(p_1,p_2)=(e_1Ā·f,\;e_2Ā·f), and the encoded signal is its angle Ī“dir=atan2ā”(p2,p1) _dir=atan2(p_2,p_1). The codeās strength is how much of the unit direction survives the projection, sdir=ā„s_dir= . What the projection discards is the part of f along the comb normal, ā =sinā”Īøācosā”(Ī±āĻ)fĀ·n= Īø (Ī±-Ļ), leaving sdir=1āsin2ā”Īøācos2ā”(Ī±āĻ).s_dir= 1- ^2Īø\, ^2(Ī±-Ļ). (5) On a horizontal comb sdir=1s_dir=1; that is the direction lies entirely within the surface and is represented faithfully. As the comb tilts sdirs_dir shrinks, reaching its minimum cosā”Īø Īø for the food site in the direction the comb faces (Ī±=ĻĪ±=Ļ). Only on a fully vertical comb does this minimum fall to zero: the facing direction then projects to nothing and its heading cannot be recovered, while every other direction stays informative. In the gravity-referenced code the worker instead projects the vertical direction =(0,0,1)g=(0,0,1) onto the comb to obtain an in-surface reference, and encodes the food azimuth, measured relative to the sun, as an angle from that reference. Its strength depends only on tilt, not on the bearing to the food site, sgrav=sinā”Īø=sinā”(Ī³āĻ/2),s_grav= Īø= (Ī³\,Ļ/2), (6) behaving oppositely to sdirs_dir: it is zero on a horizontal comb, where gravity is perpendicular to the surface and gives no in-surface reference, and grows to 11 as the comb becomes vertical. The two trade off: a horizontal comb supports only direct pointing, a vertical comb makes the gravity reference available for every food direction, and at intermediate tilts both carry information. When code strength is below 11, the direction it recovers is correspondingly degraded, modeled as a circular interpolation between the true angle and a uniformly random one with weight equal to the strength. Decoding the direct code. The direct encoding is linear in the horizontal food vector: (p1,p2)ā¤=Mā(cosā”Ī±,sinā”Ī±)ā¤(p_1,p_2) =M( Ī±, Ī±) , where M is the 2Ć22Ć 2 matrix whose rows are the eastānorth parts of 1,2e_1,e_2, with detM=cosā”Īø M= Īø. Recovering the world heading exactly requires inverting this map, and we model two receivers that differ in whether they do so (Figure 4). The inversion asks a lot of a receiver: it must know comb tilt and comb orientation, and apply a correction that varies with the direction danced. The uncorrected reading asks nothing, since the dance is a direction in space and the food is on the ground: a receiver that flies along the danceās compass bearing drops the vertical component simply by flying. That reading is also exactly right on a horizontal comb (the ancestral state), which is why we regard it as the more plausible of the two: it leaves the ancestral rule unchanged, whereas the correction would have had no function to select it before combs tilted. Unproject The receiver applies Mā1M^-1, exactly inverting the encoding; since detM=cosā”Īø>0 M= Īø>0 for Īø<Ļ/2Īø<Ļ/2, the true heading is recovered without bias. Flatten The receiver reads the dance as a direction in space and drops its vertical component, taking the compass bearing of what is left. Where the sender projected the food direction onto the comb, the receiver projects the danced direction back onto the horizontal plane: a second projection rather than an undoing of the first, so the recovered heading r is systematically biased on a tilted comb.33 3 Formally, the receiver lifts Ī“ to the in-surface vector cosā”Ī“ā1+sinā”Ī“ā2 Ī“\,e_1+ Ī“\,e_2 and keeps its eastānorth part, which applies Mā¤M rather than Mā1M^-1. On a flat comb M is a rotation, so the two coincide and the decodes are identical; they differ only once the comb tilts. The gravity code, by contrast, is always decoded by exact inversion: the projected gravity reference is available to any receiver, so the sun-relative food azimuth is read back directly. Transposition and blending the codes. Which code a worker uses is governed by two further heritable behavioral traits, the sender transposition tst_s and receiver transposition trt_r (both in [0,1][0,1] and expressed with individual variation as in Section 3.1). These set how much weight a worker places on the gravity-referenced code rather than the direct code when, respectively, producing and interpreting a dance.44 4 The alternative is a binary trait, with a worker either pointing directly or referring to gravity. We use graded weights because they let transposition change by the same small mutational steps as every other trait here. Additionally, there is some biological support for blending: bees put into conflict between a light reference and gravity dance at compromise angles (Edrich,, 1977). Blending is nonetheless the more permissive assumption: a binary trait would make mismatched senders and receivers mutually unintelligible, so a population would pay a coordination cost at intermediate frequencies that a blending one avoids. Both roles blend the codes by the same rule. Given the angles Ī“dir _dir and Ī“grav _grav the two codes assign, and a transposition t, each angle is turned into a unit vector, scaled by its weight, the two vectors are summed, and the blend is the heading of the resultant: blend(Ī“dir,Ī“grav;t)=atan2(ākwksinĪ“k,ākwkcosĪ“k),kādir,grav,blend( _dir, _grav;\,t)=atan2 ( _kw_k _k,\; _kw_k _k ), kā\dir,grav\, (7) where the weights combine the transposition with each channelās geometric strength, wdir=(1āt)āsdir,wgrav=tāsgrav.w_dir=(1-t)\,s_dir, w_grav=t\,s_grav. (8) A sender blends the in-surface angles the two codes assign to the food just visited, giving the intended dance direction Ī“=blendā”(Ī“dir,Ī“grav,ts,i)Ī“=blend( _dir, _grav;\,t_s,i), and emits a signal from a von Mises distribution about Ī“ as in Section 3.2. A receiver decodes the observed signal under both codes and blends the two recovered world directions the same way, with tr,it_r,i in place of ts,it_s,i. A worker with ti=0t_i=0 therefore relies purely on direct pointing and one with ti=1t_i=1 purely on the gravity reference, while the strength factors ensure that a code with no geometric support (such as the gravity code on a horizontal comb) contributes nothing regardless of t. The geometric strength thus enters in two distinct roles: it degrades the world direction each channel recovers, interpolating it toward a uniformly random heading as described above, and it scales that channelās weight in the blend, so a channel with weak geometric support is at once less reliable and less relied upon. Communication is accurate to the extent that senders and receivers weight the two codes alike: the further tr,it_r,i sits from ts,it_s,i, the more the receiver reads the dance under a mix of references the sender did not use, and the larger the resulting directional error. Alignment only matters where the codes disagree, however, so on a horizontal comb, where sgrav=0s_grav=0 collapses both blends onto the direct code, trt_r is free to drift. The per-episode sun azimuth (Section 3.3) matters through this blending. Within an episode the sun is shared by every worker, so when a dancer and follower both use the gravity code it enters encoding and decoding with opposite sign and cancels, and where it falls does not matter. Because it varies across episodes, however, a colony cannot treat it as a fixed offset: a strategy that only partially commits to the gravity code leaves an uncancelled sun term, so the shifting reference selects for genuine transposition rather than a stable blend of the two codes. Coupling of sender and receiver. A change of code pays off only to the extent that it is matched on the other side, so the two traits may plausibly not mutate independently. We therefore allow the mutations of tst_s and trt_r to be correlated, treating the strength of the coupling as a parameter. Their mutational steps are drawn as a correlated Gaussian pair whose senderāreceiver coupling is the correlation coefficient Ļ, Īts=z1,Ītr=Ļz1+1āĻ2z2,z1,z2ā¼(0,Ļm2), t_s=z_1, t_r=Ļ\,z_1+ 1-Ļ^2\;z_2, z_1,z_2 (0, _m^2), (9) which leaves the marginal mutation scale of each trait unchanged while tuning how tightly the two co-vary. When Ļ is high, a mutation that pushes senders toward the gravity code tends to push receivers by the same amount, so the two sides of the code can shift in a coordinated fashion rather than drifting apart; when Ļ=0Ļ=0 they evolve independently. The comb tilt Ī³, held fixed at 00 in the horizontal model, is free to mutate in this version. 4 Experimental setup We study the model of Section 3 in two stages that mirror the two questions of Section 1. The first stage keeps the comb horizontal and asks under what ecological conditions the direct-pointing code is worth maintaining (Section 4.2). The second stage releases the comb and the transposition traits and asks under what conditions a population can migrate from the direct to the gravity-referenced code as an extrinsic pressure tilts the comb (Section 4.3). 4.1 Fixed model parameters A core set of parameters is held fixed throughout, and only the ecological and evolutionary parameters named below are varied. Table 3 lists the fixed values. The population contains 6060 colonies of 8080 workers each, evolving for 120120 non-overlapping generations; each colony is evaluated over 5050 independent foraging episodes of 1212 foraging attempts, each attempt made by a single worker drawn at random (the worker pool is therefore a sample of the colonyās trait distribution rather than a workforce: its size sets how finely that distribution is discretized, not how much a colony can forage). Workers express colony traits with variation scale Ī· (Section 3.1); dances are performed with maximum concentration Īŗmax _ and are subject to both production and interpretation noise (Section 3.2). These three terms compound: even a colony at maximum directional bias recovers a search direction with a circular standard deviation of about 20ā20 , so the model imposes a floor on communication error and evolved colonies do not approach perfect accuracy. All lengths follow a fixed convention (distances in kilometers, patch radii in meters): the maximum search distance is D=6D=6 km and food patches lie at distances in [0.75,dmax][0.75,d_ ] km. Food patches are physical disks whose radius is drawn from a lognormal and whose capacity scales with patch area, and the number of patches per episode is Poisson (Section 3.3). Each foray draws its outbound distance from a Gamma of fixed shape with the colonyās evolved mean. Foraging carries a per-distance travel cost, a dance cost cbase+ccueābic_base+c_cue\,b_i that is purely bias-dependent (cbase=0c_base=0), and an attention cost cattc_att (Section 3.4). The sun is drawn from an arc of width Ļ centered on Ļ/2Ļ/2, and the comb orientation is treated as axially symmetric (period Ļ). Colonies start with little communicative ability: the initial directional bias is drawn from ā”(0,0.15)U(0,0.15), receiver attention from ā”(0,0.25)U(0,0.25), the foray distance from ā”(0.15āD,0.45āD)U(0.15D,0.45D), the dance propensity from ā”(0.8,1.0)U(0.8,1.0), and both transposition traits start at 00, so early foragers signal weakly, rarely attend to one another, and use only direct pointing. Symbol / name Value Meaning colonies 6060 population size workers per colony 8080 ensemble size per colony episodes per colony 5050 foraging episodes averaged for fitness attempts per episode 1212 foraging attempts per episode Ī· 0.080.08 worker-variation scale (trait sd; traits on [0,1][0,1]) Īŗmax _ 1414 maximum dance concentration (ā16āā 16 scatter at bi=1b_i=1) dance noise 0.180.18 production noise (wrapped Gaussian, rad; ā10āā 10 ) interpretation noise 0.120.12 receiver decoding noise (rad; ā7āā 7 ) D 66 maximum search distance (km) dmind_ 0.750.75 minimum food distance (km) Ļr _r 0.60.6 patch-radius log-scale spread (80%80\% of radii in 7070ā325325 m) foray shape 22 Gamma shape of per-foray distance (coefficient of variation ā0.71ā 0.71) cbase,ccuec_base,\,c_cue 0, 0.020,\ 0.02 dance cost terms (in units of v) cattc_att 0.010.01 attention cost (in units of v) v 11 food value (sets the payoff scale) Ī¼sun,Īsun _sun,\, _sun Ļ/2,Ļ/2,\ Ļ sun arc center and width (rad; a 180ā180 arc) Table 3: Parameters held fixed across all experiments. The parameters that some experiment varies are listed separately in Table 4. The parameters that are not fixed in this way are summarized in Table 4: five ecological parameters (food-site count, patch radius, capacity, maximum food distance, and travel cost) and four evolutionary parameters (the number of generations, B, Ļm _m, and Ļ). That table gives only what each parameter means: the values they take, swept or held fixed, are stated with the experiments themselves in Sections 4.2 and 4.3. Symbol / name Meaning Ecological food-site count Poisson mean number of patches per episode Ī¼r _r median patch radius (m, lognormal) capacity base loads a patch supplies at the median radius dmaxd_ maximum patch distance (km) travel cost cost per km of foray distance (in units of v) Evolutionary generations evolutionary horizon B vertical-comb benefit magnitude Ļm _m mutation step standard deviation Ļ senderāreceiver mutation correlation Table 4: Parameters that are not held fixed across all experiments, and what each means. No single experiment varies all of them; Sections 4.2 and 4.3 give the values each takes in each experiment, whether swept or held fixed. 4.2 Emergence of direct pointing The first stage uses the horizontal-comb model of Section 3.6: comb tilt is fixed at 00 and not allowed to evolve, so the gravity reference has zero strength and the transposition traits are inert, leaving only the direct-pointing code in play. Colonies evolve for 6060 generations under mutation rate Ļm=0.07 _m=0.07, and we vary only the food distribution across 5050 replicate seeds (400400ā449449). The conditions form a full grid crossing the Poisson mean patch count 1,2,3,4,6,8,12,16,24\1,2,3,4,6,8,12,16,24\ against the median patch radius 15,37.5,75,150,300,600\15,37.5,75,150,300,600\ meters, so that the ecology ranges from single small patches (essentially undiscoverable) to many large ones (easily found by random search). The remaining parameters of Table 4 are held fixed: a patch at the median radius has capacity 66, patches lie no farther than dmax=6d_ =6 km, and foraging cost is 0.0270.027 per km travelled. The primary outcome is the recruitment advantage (Section 3.4). Because the follow decision is a randomized coin flip on receiver attention within the same episodes, it is a near-randomized, contemporaneous estimate of what the dance actually buys, and it distinguishes functioning communication from a directional-bias trait that has merely drifted upward. We report it as a mean over the final generations, alongside final-generation mean directional bias, foraging success, and dance propensity. 4.3 Evolution of the gravity code The second stage uses the full model of Section 3.8: comb tilt Ī³ and orientation ĻĻ are free to evolve, the transposition traits ts,trt_s,t_r are active, and the raw payoff is scaled by the vertical-comb benefit 1+BāĪ³1+BĪ³. Populations always start horizontal, so a successful run must evolve both a vertical comb and a matched senderāreceiver gravity code. The transition is governed by the interaction of the ecology with three evolutionary parameters: the benefit magnitude B, the mutation rate Ļm _m, and the senderāreceiver coupling Ļ. The question we ask in this stage is whether there is any combination of ecology and evolutionary regime which lets a population cross between the two codes without a breakdown of communication, and if so which. The space in which to look is both large and costly to cover: eight parameters interact, the values we admit for them (Table 5) already combine into some 2.32.3 billion distinct settings, and pricing a single setting costs a full evolutionary run on a panel of seeds. We therefore use optimization, which spends its budget where success looks likely instead of spreading it evenly. The search tells us where viable settings lie, but its trials cluster around what already worked, so their density cannot tell us how large a viable region is. The experiments below therefore re-examine what the search finds on fixed grids. They also re-examine it on fresh random seeds, because a search that maximizes over a panel of seeds will fit that panelās noise along with its signal. Decode variants. On a tilted comb (Section 3.8) the direct code can be decoded via flatten or unproject, and we run the experiments below with each variant. The two pipelines are identical in every other respect, so comparing them isolates the effect of the geometric decode assumption on whether and where the transition occurs. Which way that effect should run is not clear a priori. On the one hand, flatten is biased on a tilted comb, and this imprecision may push a colony toward the gravity referenced code. On the other hand, unproject keeps direct pointing accurate as the comb tilts and so keeps transitional colonies more viable, which may instead ease the transition by keeping them alive long enough to complete it. Parameter search. We search eight parameters: food-site count, patch radius, capacity, maximum distance, and travel cost, together with B, Ļm _m, and Ļ. Table 5 gives the interval and step size searched for each, and the sampler proposes values on those steps, so the settings form a grid rather than a continuum. One trial is one such setting, scored by simulating it as described below; colonies evolve for 120120 generations throughout. Trials are proposed by the Tree-structured Parzen Estimator (Bergstra et al.,, 2011) as implemented in Optuna (Akiba et al.,, 2019), which fits per-parameter distributions to the best-scoring trials so far and proposes settings typical of them, so that later trials concentrate where earlier ones succeeded; the first 6464 trials are drawn at random. We run 10241024 trials per decode variant. Scoring a trial. Each trial evaluates a panel of J=10J=10 seeds. For seed j we write Ī³ĀÆj Ī³_j, tĀÆs,j t_s,j and tĀÆr,j t_r,j for the final-generation population means of the comb tilt and the two transposition traits, and mjm_j for the lowest foraging success (Section 3.4, averaged over the colonies) reached at any generation of the run. A seed is stable if Ī³ĀÆjā„Ī³ā Ī³_jā„Ī³ and minā”(tĀÆs,j,tĀÆr,j)ā„tā ( t_s,j, t_r,j)ā„ t , and non-viable if mjā¤mām_jā¤ m . The first two thresholds mark what we count as a completed transition: Ī³ā=0.80Ī³ =0.80 asks for a mostly vertical comb, and tā=0.50t =0.50 asks that both roles weight the gravity reference at least as heavily as direct pointing, so that the code has genuinely tipped over rather than merely drifted. The threshold mā=0.02m =0.02 is a viability floor: near-random search leaves the founding generation with low success, but the minimum can also occur later, when comb tilt outpaces the transposition traits and degrades the ancestral code before the gravity-referenced one takes over. Penalizing mjm_j wherever it falls steers the search away from settings in which foraging is never adequate. A seedās progress measures how near it came to the first two: each trait is expressed as a fraction of the threshold it must reach, and progress is the smallest of those fractions, capped at one, gj=minā”(1,Ī³ĀÆjĪ³ā,tĀÆs,jtā,tĀÆr,jtā),g_j= (1,\; Ī³_jĪ³ ,\; t_s,jt ,\; t_r,jt ), (10) so that gjg_j tracks whichever of the three lags furthest behind. Each fraction reaches one exactly when its trait reaches its threshold, so gj=1g_j=1 holds exactly when seed j is stable, and gj<1g_j<1 grades the near misses. The trial score is then āj=1J[gj=1]āstable count+1Jāāj=1Jgjāmean progressāāj=1J[mjā¤mā]ānon-viable count. _j=1^J1 [g_j=1 ]_stable count\;+\; 1J _j=1^Jg_j_mean progress\;-\; _j=1^J1 [m_jā¤ m ]_non-viable count. (11) Because the progress term is an average of quantities bounded by one, it can never contribute more than a single point, so one further stable seed outweighs any improvement in progress across the whole panel: the search pursues transitions that complete, and uses progress only to rank panels that fail. Symbol / name Interval Step food-site count 11ā88 integer Ī¼r _r (m) 6060ā360360 1515 capacity 22ā1212 integer dmaxd_ (km) 33ā7.57.5 0.250.25 travel cost (per km) 0.0100.010ā0.0600.060 0.00250.0025 B 0.100.10ā0.600.60 0.020.02 Ļm _m 0.040.04ā0.140.14 0.010.01 Ļ 0.00.0ā1.01.0 0.10.1 Table 5: The space searched by the optimization. Each parameter is proposed on a lattice of the given step within its interval, so the space is discrete rather than continuous. Confirmation and held-out validation. Because the optimization ranks trials by their score on only ten seeds, a top-ranked candidate may owe that score to a genuinely robust transition or to a lucky draw of those seeds, and re-scoring it on unseen seeds tells the two apart. The highest-scoring search trials are re-run on a disjoint confirmation panel of 4040 seeds, and the best candidates are then validated on 100100 held-out seeds that were used in neither search nor confirmation. Validation reports, per candidate, the fraction of stable seeds, the non-viable count, and final-generation means of foraging success, comb tilt, and the lower of the two transposition traits. One-parameter sensitivity. The search locates a viable setting but, its trials clustering around what already worked, does not say how wide the viable region around it is. Around the strongest validated candidate we therefore perturb each parameter in turn, holding the others at their validated values, over the same 100100 held-out seeds; sweeping one axis at a time on a regular grid measures how far each parameter can move before the transition degrades, and so which parameters it tolerates freely and which it depends on finely. Evolutionary-parameter interaction. To ask whether mutation parameters can compensate for a weaker architectural benefit, we hold the validated ecology fixed and run a full-factorial grid over the three evolutionary parameters, sweeping the benefit across its full search range while centring the mutation-parameter grids on the values found stable above, Bā0.10,0.30,0.45,0.60Bā\0.10,0.30,0.45,0.60\, Ļmā0.05,0.07,0.09,0.11 _mā\0.05,0.07,0.09,0.11\, and Ļā0.0,0.3,0.6,0.9Ļā\0.0,0.3,0.6,0.9\, evaluating each of the 6464 cells over the same 100100 held-out seeds. Outcome measures. Across the transition experiments we summarize runs by the same thresholds used in the search objective: the stable fraction (vertical comb with a coordinated gravity code) and the non-viable fraction (foraging success ā¤0.02ā¤ 0.02 at its worst generation), alongside final-generation trait and success means. Seed-bootstrap intervals over the per-seed outcomes quantify sampling uncertainty in the reported fractions. 5 Results We report the two stages in turn: the horizontal-comb ecology which fixes when the direct-pointing code is worth maintaining (Section 4.2), and the vertical transition under both decode variants (Section 4.3). 5.1 Emergence of direct pointing On a comb held flat, the evolved directional bias, which sets how precisely a dance points, is what tells us communication is favored. Absent selection for communication it drifts to a low baseline; it rises clearly above that baseline only along a diagonal band of the food grid (Figure 5), and the patch size at which a precise dance becomes worthwhile falls as patches grow more numerous. Below a patch radius of a few tens of meters the dance never evolves, however abundant the food. The recruitment advantage shows what a dance is worth where it does evolve: it is largest for few, large patches and erodes toward both small patches, where successful foragers are too rare to seed dances, and abundant large ones, where independent search already finds the food. Figure 5: Evolved directional bias (left) and recruitment advantage (right) on a flat comb across the grid of Poisson mean site count against median patch radius, 5050 seeds per cell. Recruitment advantage is the followerāsearcher success difference, and shading is normalized within each panel. 5.2 Evolution of the gravity code Parameter search. Stable vertical-gravity transitions, in which a seed ends with a mostly vertical comb and both worker roles committed to the gravity reference (Section 4.3), are common under both decodes, and unproject is slightly broader (Table 6). Both searches concentrate their strongly stable trials, those stable in at least eight of their ten seeds, in much the same region: abundant, large, reachable food paired with a strong tilt incentive and a high mutation scale (Table 7). The main difference is that unproject tolerates food at longer range: this suggests that the bias of flatten works against the transition more than it drives it. Distant food demands accurate dances, and that is just where flattenās bias corrupts the direct code, so under flatten the transition completes only where food is close enough for the biased dance to stay usable. Outcome Flatten trials Unproject trials Stable in all 1010 seeds 1414 2424 Stable in ā„8ā„ 8 seeds 123123 132132 Stable in ā„5ā„ 5 seeds 312312 338338 Stable in ā„1ā„ 1 seed 611611 641641 No stable seed 413413 383383 Table 6: Stability counts over the 10241024 Optuna trials (ten seeds each) under each decode. Flatten Unproject Parameter Median 10thā90th Median 10thā90th Food-site count (Poisson mean) 88 55ā88 88 66ā88 Patch radius (m) 210210 165165ā345345 225225 150150ā360360 Patch capacity 77 22ā88 77 33ā99 Vertical-comb benefit B 0.560.56 0.500.50ā0.560.56 0.560.56 0.500.50ā0.560.56 Max food distance (km) 3.253.25 3.253.25ā5.255.25 5.255.25 3.53.5ā5.255.25 Travel cost per km 0.0250.025 0.0100.010ā0.0250.025 0.0180.018 0.0100.010ā0.0220.022 Mutation scale Ļm _m 0.1100.110 0.0900.090ā0.1100.110 0.1100.110 0.1100.110ā0.1300.130 Correlation Ļ 0.40.4 0.00.0ā0.90.9 0.40.4 0.00.0ā1.01.0 Table 7: Parameter distribution of the strongly stable trials (eight or more of ten stable seeds) under each decode. Held-out validation. The top five distinct candidates from each search, carried through the 4040-seed confirmation panel and rerun on 100100 held-out seeds (200200ā299299), all produced frequent stable transitions and no non-viable seeds; stable rates ran 8080ā9191 of 100100 for flatten and 8282ā9292 for unproject. The strongest candidate of each decode (Table 8) lands on very similar held-out rates and on overlapping ecologies: large patches, short-to-moderate distances, low travel cost, high benefit. Parameter Flatten Unproject (cand. 729) (cand. 541) Food-site count (Poisson mean) 55 77 Patch radius (m) 315315 315315 Patch capacity 44 1212 Vertical-comb benefit B 0.560.56 0.600.60 Max food distance (km) 3.253.25 4.754.75 Travel cost per km 0.0170.017 0.0100.010 Mutation scale Ļm _m 0.0700.070 0.0900.090 Correlation Ļ 0.90.9 0.40.4 Stable seeds (of 100100) 9191 9292 Final mean comb tilt tft_f 0.8450.845 0.8480.848 Table 8: Strongest validated candidate per decode over 100100 held-out seeds. One-parameter sensitivity. A one-parameter sweep around each validated baseline identifies the mutation scale as the sharpest boundary (Figures 6 and 7): lowering it collapses the transition (flatten falls from 91/10091/100 to 49/10049/100 at Ļm=0.05 _m=0.05; unproject holds until a lone drop to 16/10016/100 at 0.040.04), while the validated Ļm _m of 0.070.07ā0.090.09 sits on the plateau. The sharpest ecological boundary is food-site count under flatten (46/10046/100 at a Poisson mean of two, rising to 87/10087/100 at seven); unproject is more robust to count. Vertical-comb benefit and correlation are monotone and moderate, and patch radius, capacity, distance, and travel cost stay within a narrow band. Figure 6: One-parameter sensitivity of the flatten decode. Each parameterās violin is a kernel density over seed-bootstrap stable-rate draws pooled across that parameterās swept values (100100 held-out seeds per value), so a lobeās position is a swept valueās rate and its width that valueās sampling uncertainty; dots mark the per-value rates (orange below, green above baseline) and the dashed line the baseline. Parameters are sorted by worst-case drop (shared order and x-range with Figure 7). Figure 7: One-parameter sensitivity of the unproject decode, drawn as in Figure 6 and sharing its parameter order and x-range. Evolutionary-parameter interaction. Fixing each decodeās validated ecology and sweeping B, Ļm _m, and Ļ across the search range, stable rate falls steeply with benefit: robust near the validated Bā0.56Bā 0.56ā0.600.60, already marginal by B=0.30B=0.30, and collapsing at the search floor B=0.10B=0.10, with unproject again slightly more robust than flatten at every benefit tested (Figure 8). A complete failure to produce even one stable transition among the 100100 seeds is rare and confined to that same floor: it occurs in 33 of the 6464 cells under each decode, all at B=0.10B=0.10 and, in all but one, also at the lowest mutation scale. Within each benefit panel the rate rises with mutation scale and with correlation, and the best cells pair a high mutation scale (0.090.09ā0.110.11) with moderate-to-high correlation, so higher mutation and correlation only partly compensate for weaker architectural benefit. This compensation from correlation is concentrated in the low-mutation regime: at the smallest mutation scale, raising the correlation from 00 to 0.90.9 roughly doubles the stable rate, whereas at the highest mutation scale its effect is much weaker, and slightly negative under unproject. This regime-dependence explains why the search itself left Ļ largely unconstrained, its strongly stable trials spanning almost the whole [0,1][0,1] axis under both decodes (Table 7): the search concentrated at high mutation scale, exactly where correlation barely moves the outcome. Figure 8: Stable transition rate across the evolutionary-parameter interaction grid, holding each decodeās validated ecology fixed. Benefit is swept across the full search range (0.100.10ā0.600.60) as a generic gradient rather than through the validated operating points (flatten B=0.56B=0.56, unproject B=0.60B=0.60). Tiles cross the senderāreceiver mutation correlation Ļ against the mutation scale Ļm _m, faceted by decode (rows) and vertical-comb benefit B (columns); each cell summarizes 100100 held-out seeds, and its label is the percentage of seeds with a stable transition. 6 Discussion and Conclusion The current study poses two questions: under what ecological conditions a direct-pointing referential code emerges and stabilizes, and which factors let a population move from this code to its gravity-referenced variant. Both are feasibility questions rather than probability estimates: they ask what conditions make each code possible at all, not how likely a given ecology or lineage was to realize it. The two stages of the simulation answer them in turn. On a horizontal comb, direct pointing is feasible only within a moderate band of foraging difficulty (Figure 5), and the edges of that band are gradual rather than sharp: too little food never seeds enough dances to be useful, too much makes independent search sufficient on its own. The transition itself turns out to be feasible far more broadly than it is probable. Sweeping the benefit B, the mutation scale Ļm _m, and the senderāreceiver correlation Ļ over their full search ranges at a fixed, favorable ecology, a transition fails to occur in only 33 of 6464 grid cells per decode, each at the lowest benefit and mutation scale tested (Figure 8); everywhere else, some seeds complete it, though often only a minority do. The regime the search converges on, a high mutation scale paired with a large benefit, is not especially biologically plausible, nor is it necessary to allow the code transition to occur. Our approach is to treat the pattern of results here as a transition feasibility demonstration rather than a claim about its likelihood: across nearly all of the range we tested, nothing about the evolutionary parameters rules the transition out. Taken together, these answers complement the mechanistic proposal of Barron and Plath, (2017), which addresses how a single beeās brain can support either code: if the underlying heading representation is indeed indifferent to the spatial reference frame, our results indicate where the remaining barrier lies, not in individual cognition but in whether an evolving population of senders and receivers can stay coordinated as the code itself changes. The broader picture across both stages is the same kind of claim at two different scales: a referential code, and later its replacement, need not be finely tuned into existence. Direct pointing is favored across a real, if bounded, swath of foraging ecologies; the shift to a gravity-referenced code, once that ecology holds and vertical building pays off, fails only at the joint extreme of weak benefit and low mutation scale, and remains open everywhere else we tested. This is a claim about what is possible, not what was likely in the actual evolutionary history of Apis: that would need independent evidence on ancestral foraging ecology, the real benefit of vertical combs, and the genetic basis of dance production and interpretation. Feasibility is, however, enough to answer the coordination problem posed in the introduction: a shared code can be revised without a breakdown of communication across most of the conditions we examined, not just at one finely tuned point among them. 7 Limitations Several aspects of the model bound what these results can support. Only the danceās directional component is modeled, so the results say nothing about how encoding distance would affect the emergence or the transition of the code. Colonies reproduce asexually, with no gene flow between them, and the vertical-comb benefit B is an exogenous, fixed-form term (Section 3.7) rather than one derived from a mechanical or thermal account of comb architecture, so the model is silent on magnitude or origin of the real advantage. Likewise, the mutation scale Ļm _m sets how fast colony traits explore trait space over a fixed number of generations, not a real per-generation mutation rate. Transposition is also blended continuously rather than switched discretely (Section 3.8): a permissive choice that may widen the feasible region we report. The parameter analyses are similarly bounded. 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