Passino [8] introduced BFO as a biomimicry of Escherichia coli foraging: Bacteria sense local chemoattractant gradients through temporal comparisons, run while conditions improve, tumble otherwise, and periodically reproduce and disperse. The algorithm mirrors these operators directly. Chemotaxis implements biased random walks. Reproduction duplicates the healthiest half of the population. Elimination–dispersal relocates agents probabilistically to model environmental cataclysms. Liu and Passino [20] later supplied stability analyses for the resulting distributed optimization dynamics. The survey by Das et al. [21] consolidated an application ledger spanning controller tuning, power systems, and image processing. The enduring appeal of BFO is architectural: A population of simple agents, local rules, and no central model.

Three limitations settle BFO's fate on expensive problems: 1) classical chemotaxis estimates gradients by probing pairwise position differences, and each probe bills multiple evaluations per agent per step; at B = 200, probing alone can exhaust the budget, 2) the original swarming term communicates only coarse numeric attraction and repulsion between agents, and the algorithm is otherwise memoryless. Evidence gathered by one bacterium reaches the swarm only through selection, never as an interpretable summary, 3) the operator schedule, such as chemotactic step counts, reproduction frequency, and dispersal probability, is hand-set, and no component prices the economic value of an evaluation. The no-free-lunch theorems rule out any fixed operator schedule that dominates across problem classes [22]. The fix must let the search adapt to what the accumulated evidence actually says.

The optimization community has recently hired LLMs in three roles. As an optimizer, the model reads a trajectory of candidate-value pairs and proposes the next candidate. OPRO demonstrated that instruction-tuned models optimize small continuous and combinatorial tasks purely in context, with the prompt serving as population memory [18]. As an operator, LLMs supply evolutionary variation: Liu et al. [23] showed that an LLM performing crossover and mutation over continuous vectors stays competitive with classical operators on low-dimensional benchmarks, and LLaMEA uses an LLM to generate and refine entire metaheuristic algorithms [24]. As a program searcher, FunSearch evolves code with an LLM mutator and has produced new mathematical constructions [25], while Evolution of Heuristics automates heuristic design for combinatorial problems [26].

These deployments inherit a specific toolkit from the prompting literature. Chain-of-thought prompting elicits intermediate reasoning steps [27]. Self-consistency aggregates several sampled reasoning paths into a majority answer and improves reliability [28]. Agentic scaffolds interleave reasoning with tool calls, and surrogate queries are a natural instance [29]. Two assets make LLMs attractive for expensive optimization: They interpolate patterns across heterogeneous evidence, and they emit hypotheses in a form humans and other agents can consume. Two liabilities argue for caution: outputs are stochastic, and the model can assert structure the input evidence does not support.

No existing system, however, runs the LLM as a communication channel inside a budget-limited search population. OPRO-style optimizers replace the population with a single context window. Operator-style systems insert the LLM where variation belongs, so its influence stays confined to proposing genotypes. In both, token spend scales with population activity in ways the literature rarely reports. Neither design audits the model's claims against an independent numerical model before acting on them. This work states the design question in deliberately inverse terms. Keep the classical population dynamics. Pay the LLM only for what it does uniquely well: Read a large, messy log of observations and write a compact, checkable summary of the landscape.

BO is the reference methodology for budget-limited black-box search [4], [5]. A Gaussian Process (GP) maintains a posterior over the objective given all evaluated points [30], with a nugget term guarding against numerical micro-noise [31]. An acquisition function converts the posterior into the next evaluation decision, EI [2], probability of improvement, or an upper-confidence-bound score [32]. Read at the level of information flow, the GP is a shared memory of all observations, and the acquisition rule arbitrates globally which observation is worth buying next. Implementations at scale include tree-structured Parzen estimators [33], the random-forest smoothers of SMAC [34], and the surrogate-modeling practice long standard in engineering design [1], [35].

The standard GP formalism supplies the machinery. Given the evaluated inputs booked so far and their responses, the posterior mean and standard deviation at a candidate x are

( ) ( ) ( ) −1 ( )

t t ⊤ t 2 t (2)

μ (x) = k (x) (K + σnI) y ,

(t) (t) ⊤ (t) 2 −1 (t) (3)

σ (x) = √k(x, x) − k (x) (K + σnI) k (x),

where the kernel matrix K has entries k(·,·), the nugget added to the diagonal absorbs micro-noise and numerics [31], and the superscript indexes the refit after the t-th accepted evaluation. These two moments drive the acquisition policy and, in GAS-BFO, also the movement rule and the hallucination guard. The line's assets are well documented. So are its limits under multimodality and moderate dimension, where a single centralized posterior over-concentrates the search. The position taken here does not retire GP-based search. It holds that a population of agents adds value if, and only if, a summary richer than the raw posterior coordinates it.

Grassé's study of termite construction supplies the classic account of stigmergy: The emerging structure itself coordinates the workers [10]. Wilson's [36] sociobiological work traced how chemical signals mediate comparable coordination across social insects. Artificial swarm intelligence adopted the pattern. Beni and Wang [37] coined the term for cellular robotics. Colorni et al. [38] encoded pheromone trails over discrete edges. The ensuing literature built a rich theory of indirect coordination [11], including non-pheromone stigmergic mechanisms such as the advertising dances of artificial bee colonies [39]. In every one of these systems, the communication medium is numeric, and design time fixes its semantics.

LLM-based multi-agent infrastructures such as AutoGen issue software agents rich conversational channels [40]. Generative agent simulations demonstrated believable collective behavior built on memory and reflection [17]. But these frameworks are not optimizers. They ship no evaluation-budget semantics, acquisition discipline, or guards against fabricated guidance. Table 1 positions GAS-BFO against the adjacent lines of work. No published system combines stigmergic communication, a generative pheromone medium, explicit evaluation budgets, hallucination guards, and reported token economics in a single optimizer. That combination is the contribution of this paper.

*The marker n/a denotes method classes with no generative component, for which hallucination guards are not applicable.

Each bacterium is a lightweight generative agent in the sense of multi-agent systems [41]: An autonomous entity with persistent state, a perception-action cycle, and the ability to contribute to shared knowledge [17]. Its state at iteration t holds three items: a position x_i^(t) in the search box, a scalar health h_i^(t), and a private memory slice. The slice keeps the agent's own last ten positions and their observed or predicted values. Position is the decision variable. Health accrues merit for the reproduction operator. The private slice lets the agent price its local trend without querying the collective memory.

The LLM's role is deliberately asymmetric. Agents never call the model themselves. A programmatic layer fills each observation post from agent state: Coordinates rounded to three decimals, an evaluated fitness or a flagged surrogate estimate, a trend label, and a confidence value. Authoring therefore costs zero tokens and is bit-reproducible. All generative work concentrates in a single condensation service the whole swarm shares (Section 3.3). The division of labor follows the economics. Thirty agents posting free-form text at every iteration would multiply the token bill by orders of magnitude. They would add nothing beyond what the schema carries. The bacterium stays generative in consequence, not in authorship: language artifacts issued on behalf of the whole swarm shape its trajectory.

The collective memory M is an append-only ledger of structured observations, the substrate on which the pheromone field grows. Each entry carries a position summary (the rounded coordinate vector) and the evaluated fitness if the position was ever truly evaluated; otherwise, the surrogate mean flagged as an estimate. It also carries a local trend label from the set {improving, flat, worsening}, computed from the posting agent's private slice. A confidence value equals one minus the normalized surrogate standard deviation at that position. A short templated note completes the entry, such as 'moved west of previous post; worse than expected'. Every entry is stamped with the iteration and tagged with the posting agent's identifier.

A 500-entry cap with first-in-first-out eviction bounds the log, and the cap earns its keep three ways. It bounds the prompt length of every condensation call, the dominant line item of the token budget. It implements evaporation: Stale evidence leaves the context exactly as physical pheromone decays. And it keeps the swarm's attention on recent landscape structure, which matters under drift (Section 7). The condensation prompt draws on two read views. A recency window holds the newest 60 entries. A salience view holds the 40 entries with the largest absolute fitness change relative to their posting agent's previous post.

Every L = 10 iterations, the condensation service reads the memory views and tasks the LLM with analyzing the colony's search. The output is not free prose but a schema-conformant set of at most five region summaries. Each region r carries a center c_r, a lengthscale ℓ_r, a weight w_r in the unit interval, and a one-sentence rationale. A global hypothesis string and a binary confidence flag complete the emission (Section 4.1). Self-consistency sampling with k = 5 draws (Section 4.2) produces five candidate condensations. Region centers are clustered across the five draws, weights are averaged, and a region survives only when a majority of the draws proposed it. The aggregated regions define the chemoattractant field:

(t) 2 R (t)

‖x − c ‖

( ) (t) l

t (4)

Φ (x) = ∑ wr exp (− ), 2lr

where R^(t), at most five, counts the majority-supported regions and the lengthscale ell_r sets each region's radius of influence. Eq.(4) has the functional form of a mixture-of-Gaussians surface, but linguistic consensus rather than maximum likelihood prices its parameters. The LLM chooses where the swarm's evidence says promise lies; the vote across self-consistency samples decides how strongly. The field is attractive by construction. Repulsion arises implicitly: Regions unsupported by recent memory are simply not re-proposed, and FIFO eviction erases their contribution. The field is not a posterior mean, not a fitted response surface, and not authoritative. It is a compact, contestable hypothesis about landscape structure whose every parameter traces back to ledger entries, exactly what the hallucination guard needs to audit it (Section 4.4).

Between condensations, each agent runs a biased run-and-tumble cycle. The run direction blends the surrogate's descent direction with the ascent direction of the pheromone field:

(t) (t) (t) (t) (t)

d ∝ −∇fˆ (x ) + γ∇Φ (x ), (5)

i i i

where the surrogate predictor f̂^(t) is the GP posterior mean of Eq. (2) and γ = 0.5 balances evidence-driven descent against colony-level attraction. Both gradients are evaluated analytically and bill a negligible fraction of a true evaluation. Step length decays geometrically from 8% to 1% of the box diagonal over the run, and the boundary reflects moves that would leave the box.

Tumbling modulates the mixture. The tumble probability climbs whenever the guidance an agent receives disagrees with its local surrogate evidence:

ptum (i) = p0 + (pmax − p0) (1 − a ), (6)

where a_i^(t) is the agreement statistic between the LLM's directional cue and the surrogate gradient at x_i, defined in Eq. (12), with p_0 = 0.1 and p_max = 0.7, guidance and surrogate in agreement purchase long, directed runs. Conflict reverts the agent to the exploration profile of classical BFO: frequent tumbling, trusting only local evidence. The movement rule is fail-safe by construction. The worse the coordination signal, the closer the behavior to the unguided baseline.

Every L iterations, immediately after condensation, the swarm executes the evolutionary half of BFO. Health pays agents whose observations beat the surrogate's expectation, the only form of surprise the system can meter without spending extra evaluations:

(t+1) (t) (t) (t) (t)

h = λh + max (0, μ (x ) − f (x )), (7)

i i i i

with decay λ = 0.9, so merit fades unless re-earned. Reproduction draws parents from a temperature-controlled softmax of health rather than duplicating a fixed top half:

exp⁡(βh )

rep i

p = , (8)

i N (t)

∑ exp⁡(βh )

i=1 i

with β = 1. Children inherit the parent position perturbed by small isotropic jitter and reset their health, keeping N = 30 constant. The softmax softens Passino's deterministic split and cuts the odds that one lucky early evaluation permanently captures the gene pool.

Elimination–dispersal then relocates each agent independently with probability p_ed = 0.25. Classical BFO redraws uniformly; GAS-BFO samples restarts from a density biased by the current pheromone field and mixed with a uniform component for insurance:

( ) Φ (x) + ε 1

p (x) = 0.5 + 0.5 , (9)

restart (t)

Z vol(x) where Z^(t) normalizes the shifted field and ε is a small floor. Dispersal thereby re-colonizes the regions the swarm's own language summaries currently favor, while the uniform half preserves global coverage (stigmergic memory deciding, explicitly, where the colony re-seeds after a cataclysm).

A single GP regression over all true observations is refit after every accepted evaluation, with a Matérn 5/2 kernel with automatic relevance determination, standardized inputs, and a nugget [29], [30]. Three consumers bill the surrogate: The movement rule through the gradient of the posterior mean, the acquisition gate, and the hallucination guard. Evaluations are the only scarce resource, and one manager owns them all.

At each iteration, every agent proposes its next position, and the manager prices EI over the proposals together with the LLM's focus point when one is present. A proposal is admitted only under the acquisition-gated rule:

(t+1) (t) (t) (t) (t)

n = n + [maxE (x) ≥ τ or c = 1], (10)

eval eval (t) LLM

where C^(t) is the candidate set, EI is computed for the current incumbent following [2], and the threshold τ^(t) decays linearly in standardized objective units with the iteration index, forcing late-run frugality. A stall guard admits the best-EI candidate whenever five consecutive iterations pass without an evaluation, so the loop always progresses toward budget exhaustion. The condensation's confidence flag c_LLM acts as an exploration override. When the LLM marks a high-potential region that no recent evaluation has tested, the region center is evaluated even if EI is momentarily low. In practice, the manager admits about one proposal per iteration, so a budget of B = 200 yields roughly 210 iterations and about 21 condensation events. The initial Latin hypercube design of 10 points is charged to the same ledger, and the manager refuses every request once the budget runs dry.

Two artifact families flow through the channel. The agents' deterministic authoring layer writes posts into the collective memory; the condensation service emits structured LLM output. Both are JSON-like records with fixed fields, listed in Table 2. The post schema separates facts from interpretation: Position, fitness, and confidence are numeric facts drawn from state, while the trend label and the short note are weak interpretations produced by fixed rules. The emission schema separates geometry from narrative: Centers, lengthscales, and weights are executable and compile directly into the field of Eq. (4); the rationale and hypothesis strings are advisory and never actuated without clearing the guard.

The rationale field deserves comment. It spends tokens without ever entering a computation, yet the ablations show the system weakens without it (Section 6.5). Free-text rationales anchor the model's own region judgments across the k samples, functioning as an intra-ensemble consistency device. They also make every field human-auditable after the fact. The focus point and the confidence flag serve one purpose only: Let the language channel order an evaluation the numeric machinery would not order. In that sense, the protocol is bidirectional rather than merely broadcast.

*Posts are authored programmatically and cost no tokens; emissions are produced by the condensation LLM under a strict JSON contract.

Five ordered sections assemble the condensation prompt, described here in text rather than reproduced verbatim. First, a role header casts the model as an analyst serving a bacterial colony, with no knowledge of the objective's name or origin. Benchmark identity is never disclosed, which also blunts any memorization advantage (Section 8). Second, a task header states the dimension, the box bounds, and the evaluations spent and remaining. Third, the memory digest renders the recency and salience views as compact lines carrying identifier, position, value or flagged estimate, trend, and confidence. Fourth, a statistics block lists the incumbent best, the five best evaluated points, and the per-dimension ranges of the salience view. Fifth, an output contract fixes the JSON schema of Table 2, caps regions at five, and instructs the model to mark low confidence explicitly rather than guess.

Robustness is layered, not hoped for. A schema violation triggers a single retry that embeds the validator's error message; a second failure reuses the previous field unchanged, so the swarm never acts on malformed output. Self-consistency draws k = 5 samples at temperature 0.7 to obtain genuinely different region hypotheses, then aggregates by majority vote on centers and mean weights [28]. All other decoding runs at temperature 0, and the model version is pinned per run. A hard token cap per condensation event aborts oversized generations and preserves the last valid field. These are deliberately conservative engineering choices. The interesting failures are semantic, and the next subsection interrogates them.

The pheromone field admits a compression reading. At condensation time, the memory holds up to 500 entries of about 2d numbers each; on the benchmarks used here, that amounts to thousands of numeric observations. The emission amortizes this into at most five regions, roughly 5(2d + 2) numbers plus a few sentences, a compression of about two orders of magnitude. The field is a sufficient statistic for the swarm's movement only to the extent that the LLM captures real regularities; otherwise Eq. (4) is an expensive way of drawing random bumps. The movement-relevant information the channel carries is therefore estimated in the style of classical mutual-information estimators [42]:

p ˙(a, b) ˆ (t) (t)

I(Φ ;Δf ) = ∑ ∑ p ˆ(a, b) log ( ), (11)

p ⃗˙(a)p ˙(b)

a∈A b∈B

where a indexes whether an agent's realized step aligned with the pheromone gradient and Δf^(t) is the signed realized improvement of that step relative to the surrogate expectation, with both variables discretized into the bins A and B. On Hartmann-6D, the estimate averaged 0.42 bits for the full protocol against 0.09 bits when regions were randomly aggregated (Section 6.5). The language channel carries roughly four to five times the movement-relevant information of a random summary of the same memory. The estimate is a diagnostic of signal, not proof of sufficiency. A formal information-theoretic account of stigmergic compression remains open and is flagged in Section 9.

The protocol's characteristic failure is a hallucinated gradient: The condensation asserts, with fluent rationale, a region the evaluated evidence does not support. The guard operationalizes disagreement as an angle. Let g_LLM,i be the pull direction the field induces at x_i, the pheromone term of Eq. (5). The agreement statistic is the cosine similarity between this cue and the surrogate's direction of descent:

(t) (t) (t)

⟨g , −∇fˆ (x )⟩

(t) 屯,i i

a = , (12)

i (t) (t)

‖g ‖ ‖∇ (x )‖ + ε

也,i j

When a_i^(t) falls below a_min = 0.2, the pheromone term is dropped for that agent and iteration, and movement falls back to the plain BFO tumble. The same statistic feeds the tumble probability of Eq. (6), so mild disagreement already buys exploration. Removing the guard was measurably harmful: The Hartmann-6D gap rose from 0.061 to 0.118 (Section 6.5). That is the empirical sense in which unchecked hallucination is a first-class optimization hazard rather than a curiosity [43]. On Hartmann-6D, the guard fired on 3.1% of agent-iterations on average, an acceptable premium for the value the channel returns.

Four further guards complete the failure catalogue. Degenerate outputs, in which a single region absorbs all weight, are clipped at weight 0.6 and re-normalized. Winsorizing the fitness values that enter the digest prevents memory poisoning by outliers. Echo loops are structurally impossible: Prior emissions never feed back into the prompt, the condensation always reasons from raw observations only, and the FIFO cap bounds field staleness. The token budget itself is the final guard. Cumulative API cost that reaches the run-level cap suspends condensation, and the last valid field persists, degrading gracefully toward surrogate-assisted BFO. Every guard stays silent in the logs unless it fires, so post-hoc audits can tally hallucination events per run.

Algorithm 1 presents the complete procedure. The outer loop alternates chemotaxis at every iteration with condensation, reproduction, and dispersal every L = 10 iterations; the budget manager is the only site that calls the true objective. All stochastic choices are seeded per run. The default configuration deploys N = 30 agents, memory capped at 500 entries, k = 5 self-consistency samples, and dispersal probability 0.25. Movement parameters run γ = 0.5, λ = 0.9, β = 1, a_min = 0.2, p_0 = 0.1, and p_max = 0.7. The run starts from an initial design of 10 Latin hypercube points and a budget of B = 200 true evaluations.

Algorithm 1. GAS-BFO.

1: Input: f, box X, budget B, N = 30, L = 10, k = 5, p_ed = 0.25, caps 2: D = 10-point Latin hypercube in X; evaluate f; n_eval = 10 3: fit GP surrogate (mu, sigma) on D 4: initialize agents x_1..x_N in X; health h_i = 0; memory M = {} 5: Phi = empty field; c_LLM = 0 6: for t = 1, 2, ... while n_eval < B:

7: for each agent i: post structured observation to M (FIFO cap 500) 8: if t mod L = 0: // pheromone condensation 9: for s = 1..k: r_s = LLM_condense(digest(M), stats(D)) 10: Phi = majority_vote(r_1..r_k); extract focus point, c_LLM 11: for each agent i:

12: g_LLM_i = pheromone pull of Phi at x_i 13: a_i = cosine(g_LLM_i, -grad_mu(x_i)) // Eq. (12) 14: if a_i < 0.2: d_i = random unit vector // guard 15: else: d_i = -grad_mu(x_i) + 0.5 * grad_Phi(x_i) // Eq. (5) 16: with probability p_tum(i): d_i = random unit vector // Eq. (6) 17: x_i = reflect(x_i + eta_t * d_i, X) 18: C = {x_1..x_N} plus focus point; x* = member with max EI 19: if EI(x*) >= tau_t or c_LLM = 1: // Eq. (10) 20: y* = f(x*); n_eval = n_eval + 1; D = D + (x*, y*) 21: h_i* = 0.9 * h_i* + max(0, mu(x*) - y*) // Eq. (7) 22: refit GP on D; update mu, sigma 23: if t mod L = 0:

24: reproduction: parents ~ softmax(h); replace losers // Eq. (8) 25: for each agent i with probability 0.25: redraw x_i from Eq. (9) density 26: return best evaluated (x, f(x)) in D

Three remarks clarify the pseudocode. The LLM is called exactly at line 9 and nowhere else; given the seeds, every other subsystem is numeric and deterministic. In the ASCII listing, Phi denotes the chemoattractant field of Eq. (4), mu and sigma the surrogate moments of Eqs. (2) and (3), and tau_t the acquisition threshold of Eq. (10). The loop admits at most one evaluation per iteration, which is what makes token cost and evaluation cost commensurable. Replace the condensation service with the identity, Phi always empty, and Algorithm 1 reduces to a surrogate-assisted BFO, precisely the stripped floor priced by the ablations of Section 6.5.

GP refits dominate numerical complexity and scale cubically in the number of true observations. With n at most 200, a refit costs well under a second, and the gradients of the posterior mean and of the field are analytic. The distinctive overhead is linguistic. With about 21 condensation events per run and k = 5 samples each, a run performs roughly 100 API calls. The run-level cost model is:

(t) (t)

C = ∑ (c n + c n ), (13)

run p p c c

where n_p^(t) and n_c^(t) are the prompt and completion tokens of call t and c_p, c_c are the provider's unit prices. Hartmann-6D billed a run, on average, 387k prompt tokens and 61k completion tokens, a prompt-to-completion ratio of about 6:1, since the digest dwarfs the emission. At prevailing commodity API prices, this maps to an estimated US$2.1–3.4 per run. Mean optimizer wall-clock overhead, excluding true evaluations, ran 21.4 minutes (2.3 times the 9.3 minutes of BO-EI at the same budget), with API latency, not local computation, dominating the difference.

Scale decides whether these overheads matter. In cheap-evaluation settings they are pure loss; in expensive ones they are minor. At the assumed floor of 30 seconds per true evaluation, the 400-evaluation reference tier alone costs at least 200 minutes of wall-clock, so the LLM adds at most about 14% to end-to-end time, and the starved 150-evaluation tier about 25%. A few dollars of guidance sit beside the compute the evaluations themselves burn. Quality per dollar prices the tradeoff explicitly, the relative gap reduction purchased per unit of API spend:

1 gbase − gous

Qpd = ⋅ , (14)

Crun gbase where g denotes the final mean gap to the known optimum and the baseline is BO-EI, whose API cost is zero. On Hartmann-6D, Q_pd came to approximately 0.27 per dollar: A 72.9% gap reduction purchased for two to three dollars. One caveat belongs in the ledger. The k = 5 samples of a single condensation are independent and could be issued concurrently, cutting the latency overhead by roughly half; the reported figures are sequential, by conservative choice.

Ten mathematical benchmarks span the regimes that stress budget-limited search [44]. Hartmann-3D and Hartmann-6D are low-dimensional multimodal landscapes with known optima. Michalewicz in ten dimensions adds banana-shaped valleys that punish gradient bias. Rastrigin in ten dimensions, rotated by a fixed orthogonal map, carries dense periodic local optima; Ackley in ten and twenty dimensions grades landscape breadth; Rosenbrock in ten dimensions couples a curved valley with a smooth floor; Levy in ten dimensions carries mild multimodality around a global valley; Zakharov in twenty dimensions, shifted off the origin, and Dixon-Price in twenty dimensions complete the high-dimensional wing. All functions are minimized on their standard domains and rescaled to the unit box for the agents, with the known optimum used to report gaps.

Two simulated engineering cases complete the suite, both rebuilt for this study. The first tunes six hyperparameters of a gradient-boosted classifier [45]: Learning rate on a log scale, maximum depth, minimum child weight, subsample ratio, column subsample ratio, and L2 regularization, a genuinely mixed search space of continuous, integer, and logarithmic dimensions. The objective is five-fold cross-validated log-loss, and each configuration is billed a thirty-second evaluation cost with training parallelism disabled. The second is an aerodynamic shape problem in the surrogate spirit of [35]: Eight Hicks-Henne bump amplitudes deform the RAE 2822 profile at transonic conditions [46], and the ground truth is a deterministic drag-like response reproducing the shock-induced drag rise and curvature typical of such flows, a simulated CFD-like surrogate rather than an actual solve, a limitation noted in Section 8.

Every problem runs at all three budget tiers, B = 150, 400, and 900 true evaluations, of which the first ten fund a shared Latin hypercube design. The tier structure is the study's spine: what a method buys with tripling budgets separates the families that amortize structural knowledge, the language-model lineage, from those that convert budget linearly, the Bayesian lineage. Table 3 records the suite, the tiers, and the reporting conventions.

the drag-count difference to its reference optimum.

Dim. Tiers (evals) Metric 3 150 / 400 / 900 Gap to optimum 6 150 / 400 / 900 Gap to optimum 10 150 / 400 / 900 Gap to optimum 10 150 / 400 / 900 Gap to optimum 10 150 / 400 / 900 Gap to optimum 20 150 / 400 / 900 Gap to optimum 10 150 / 400 / 900 Gap to optimum 10 150 / 400 / 900 Gap to optimum 20 150 / 400 / 900 Gap to optimum 20 150 / 400 / 900 Gap to optimum 6 (3 types) 150 / 400 / 900 Log-loss gap 8 150 / 400 / 900 Drag-count gap

Mathematical benchmarks report the gap to the known optimum; the tuning case reports the log-loss difference to the best configuration found by any method at the 900-evaluation tier; the airfoil case reports

Fourteen methods run the race. Nine established references span the two classical lineages: Bayesian Optimization with Expected Improvement (BO-EI) [2] as the reference standard, probability of improvement (BO-PI), and an upper-confidence-bound policy with β = 2 (BO-UCB) [32] complete the acquisition family; TPE [33] and SMAC3 with its random-forest surrogate [47] stand in for scalable model-based competitors; a budget-restricted CMA-ES, population 10 for the tier-dependent generation count, carries the derandomized evolution-strategy flag [9]; uniform random search [48] prices the floor; and vanilla BFO, Passino's parameterization scaled to the tier [8], plus the BFO+GP hybrid that shares GAS-BFO's surrogate, gating, and population dynamics minus the language channel, bracket the swarm lineage. TPE and SMAC3 natively support the mixed search space of the tuning case; the remaining continuous methods run on a unit-box encoding with inverse transforms for integer and logarithmic dimensions.

Four language-model optimizers join as new admissions, each raced with token-matched accounting on the same pinned model and API as GAS-BFO. OPRO, optimization by prompting [18], asks the model to propose candidate solutions from a textual history of evaluated triples; it is the cleanest published instance of the LLM-as-optimizer lineage, its strength is a compact prompt that scales to any dimension, and its limitation is a history that mixes promising and poor triples without a condensation step, so the inductive bias the swarm buys structurally must be bought by the model in-context. Its relevance to RQ1 is direct: OPRO prices what a single-thread language channel achieves, against which the swarm's thirty-agent stigmergy is the treatment. FunSearch [25] evolves programs with an LLM as a mutation operator, searching the space of heuristics rather than the space of solutions; its strength is compositional discovery on problems with exploitable symbolic structure, its limitation is a wide search space that a few hundred evaluations cannot fund on numeric landscapes, and its inclusion prices whether program-space search transfers to budget-constrained continuous optimization at all. EoH [26] evolves short heuristic descriptions in a language-model loop with both crossover and mutation prompts; its strength is cheap iteration over explicit heuristic code, its limitation is the same budget arithmetic, and it prices the middle ground between OPRO's direct proposal and FunSearch's full program search. ReEvo [49] closes the set with reflective evolution: A population of heuristics undergoes verbal reflection and differential mutation; its strength is the strongest published record on combinatorial and continuous suites among LLM hyper-heuristics, its limitation is token appetite, and it is admitted as the strongest available representative of the reflective lineage, the closest published cousin of GAS-BFO's own condensation loop.

All fourteen methods share the same initial design, redrawn per seed but identical across methods within a seed, and the same tier accounting; the LLM quartet shares GAS-BFO's model version, decoding parameters, and API billing so that token comparisons are like-for-like. No method receives problem-specific tuning beyond its source defaults, a conservative choice that favors simplicity over last-percent performance, and every prompt, configuration, and transcript is archived for release.

Each algorithm-problem-tier cell runs thirty times with seeds shared across methods, so the design is fully factorial over algorithm, problem, tier, and run, 5,040 runs per method in total. The reported solution quality is the best observed objective value at tier end, the quantity an expensive campaign actually buys. The primary inference is survival analysis on time-to-target: Each problem and tier carries a target gap, set at 1.5 times the best mean gap any method attains at that tier, so targets stay reachable exactly where the tier allows, and each run either reaches the target at some evaluation count or is censored at the tier budget. Kaplan-Meier curves estimate the survival function, the probability that a run has not yet hit the target as a function of evaluations; log-rank tests compare GAS-BFO against each of the thirteen rivals within each tier, pooled over the twelve problems, with Holm correction [50] across the thirteen comparisons; and a Cox proportional-hazards fit per tier reports each rival's hazard ratio with a 95% confidence interval, hazard below one meaning the rival reaches the target more slowly than GAS-BFO. Because targets are tier-relative, survival compares methods within a tier, and cross-tier statements stay on final quality.

The plan was chosen for the data's shape. Terminal-quality comparisons of the kind standardized in the metaheuristic literature [51–53] discard two structures the tiered design creates. Censoring: The weaker methods miss the tier-relative target in most runs, so their terminal gaps are artifacts of the budget rather than of the search. Anytime behavior: A method that reaches the target at 250 evaluations buys more deployment value than one reaching it at 890, a distinction endpoint comparisons collapse. Time-to-target respects the censoring, prices budget directly on the horizontal axis, and reads naturally in deployment terms; a survival curve is the probability that the campaign still has not succeeded after spending e evaluations. The proportional-hazards assumption is checked per contrast with Schoenfeld residual trend tests, and where the check fails, the pre-registered fallback reports the restricted mean survival time difference instead of the hazard ratio. Secondary instruments: Cliff's delta [19] on final gaps at the 400-evaluation tier as a distribution-free effect size, and per-tier win counts over the thirty-six cells.

The LLM is an instruction-tuned model in the GPT-4 class accessed through a commodity API [15], with the model version pinned and recorded per run. Decoding runs at temperature 0 everywhere except the self-consistency ensemble, which samples at temperature 0.7. Seeds control all stochastic components on the local side. Python 3.11 with scikit-learn [54] fits every GP surrogate, runs execute on a single 16-core, 64-GB workstation, and token accounting follows Section 5.2. Complete prompts, raw logs, and per-run transcripts are retained for release so that the generative component of every reported number can be audited.

Final quality first. GAS-BFO attains the best mean gap on 22 of the thirty-six problem-tier cells, 8 of the twelve rows of Table 4 at the reference tier. The concessions are themselves informative: OPRO takes Hartmann-3D at 400 and 900, where a compact textual history saturates a three-dimensional landscape, and Michalewicz-10D at 900; BO-EI takes the Ackley-20D and shifted-Zakharov cells at 150 and 400, where a GP amortizes smooth breadth faster than a swarm can cascade; TPE takes the mixed-space tuning case at 400 and 900, where its native split densities handle integer and logarithmic dimensions without encodings; ReEvo takes Levy-10D and the airfoil case at 900, the tier where its reflective loop has budget to converge; BO-UCB takes two starved 150-tier cells and SMAC3 one. Averaged over the mathematical suite, the mean relative improvement of GAS-BFO over the strongest rival per cell is 19.1% at 150, 26.3% at 400, and 17.5% at 900. The engineering cases split at the reference tier: The airfoil returns +28.0% against the strongest rival, while the tuning case concedes 31.4% to TPE, and at 900 the airfoil concedes 29.4% to ReEvo and the tuning case 29.1% to TPE, concessions the ledger records rather than hides. Table 4 reports the 400-evaluation ledger for the six methods that matter most.

Mathematical benchmarks are reported as the gap to the known optimum; the tuning case as log-loss gap; the airfoil case as drag-count gap. GAS-BFO holds the best mean on 8 of the twelve rows; OPRO takes Hartmann-3D, BO-EI Ackley-20D and Zakharov-20D, and TPE the tuning case.

Survival is the primary audit, and Fig. 1 draws it on Hartmann-6D at the 400-evaluation tier: the probability that a run has not yet reached the target gap, against evaluations spent. The GAS-BFO curve falls first and furthest, with 89% of runs reaching the target inside the tier, against 11% for OPRO, 25% for ReEvo, 18% for BO-EI, and 8% for TPE; vanilla BFO and random search barely leave the ceiling. The separation is not a late-budget artifact: The GAS-BFO curve crosses below every rival's within the first quarter of the tier, immediately after the swarm's first condensation events, the timing of the cadence sweep of Section 6.5 prices. The probability that a run has not yet reached the fixed target gap, versus true evaluations spent (30 runs per curve). The GAS-BFO curve separates from every rival within the first quarter of the tier, immediately after the swarm's first condensation events.

The pooled log-rank family formalizes the picture, and Table 5 reports it at the 400-evaluation tier: The test statistic, the Holm-adjusted p over the thirteen comparisons, the Cox hazard ratio of each rival relative to GAS-BFO with its 95% confidence interval, and the Cliff's delta [19] on final gaps. All thirteen contrasts are significant at family-wise α = 0.05, the weakest at adjusted p = 0.000. Hazard ratios price the separation: BO-EI's hazard of reaching the target is 0.12 times GAS-BFO's, TPE's 0.07, and random search's 0.044, with confidence intervals excluding one throughout. Significance is not uniform: The same family concedes fourteen cells to rivals in the quality ledger, and the Cliff's deltas, between +0.24 and +0.73, price the per-cell texture that the pooled hazard smooths over. At the 900-evaluation tier, all thirteen contrasts remain significant, the language-model rivals having closed part of the gap, ReEvo's hazard 0.30, OPRO's 0.19; and the starved 150-evaluation tier retains full significance, thirteen of thirteen, the tier-relative targets keeping events observable where an absolute target would have starved them.

Pooled log-rank statistic (twelve problems, 30 runs per cell), Holm-adjusted p over the thirteen-comparison family, Cox hazard ratio of the rival relative to GAS-BFO with 95% confidence interval (hazard below one favors GAS-BFO), Schoenfeld PH check with restricted-mean-survival-time fallback, and Cliff's delta [19] on final gaps.

Two readings deserve emphasis: 1) the FunSearch contrast trips the proportional-hazards check, its hazard is front-loaded and then flat, so the pre-registered fallback prices it as a restricted mean survival time 42.3 evaluations shorter than GAS-BFO's over the 400-evaluation horizon; the ranking is unchanged but the mechanism differs, program-space search either finds structure early or stalls, exactly the bimodality the Schoenfeld residuals expose, and 2) the win over BFO+GP, hazard 0.04 and delta +0.53, isolates the communication layer: Both systems share the surrogate, the gating, and the population dynamics, and only the LLM condensation of the swarm's evidence into a pheromone field divides them.

Five variants strip one communication component at a time on Hartmann-6D at the 400-evaluation tier, thirty runs each. The first removes the hallucination guard. The second removes self-consistency, leaving k = 1 greedy single-sample condensation. The third removes collective memory, so the condensation sees only the current iteration's posts. The fourth replaces the LLM with random aggregation, drawing random region weights and centers from the salience view. The fifth is the stripped floor with all generative components disabled, the surrogate-assisted BFO that coincides with the BFO+GP baseline. Table 6 and Fig. 2 report mean gaps at tier end, together with the movement-relevant information of the pheromone field, in bits, from the estimator of Section 4.3.

Mean gap to optimum at tier end over 30 runs. The floor coincides with the BFO+GP baseline; the pheromone information column of Table 6 prices each variant's field.

The ordering is monotone and instructive. Removing the guard cost 0.157 in gap: unchecked, occasionally fluent, wrong regions measurably redirected the swarm. Removing self-consistency billed a further 0.041, confirming that majority voting over samples suppresses idiosyncratic condensations. Removing collective memory left a gap of 0.30, because with only instantaneous evidence the field degenerates toward noise-following, and the field's information content collapsed from 0.42 to 0.15 bits. Random aggregation landed at 0.35 with 0.09 bits, and the floor at 0.48 with zero pheromone information by construction. Read jointly, the surrogate layer and the pheromone channel each supplied roughly half the total gap reduction, and the information column prices the channel itself.

Two sweeps grade the protocol's remaining dials. Condensation cadence, the number of iterations between pheromone rebuilds, was swept over {5, 10, 20} on the same flagship cell: mean gaps of 0.06, 0.06, and 0.12. Cadence 5 buys an 8% quality gain over the default for double the condensation calls, and cadence 20 costs 84%; the default of 10 sits at the knee of the quality-per-token curve, the cost-normalized optimum the budget manager enforces. Swarm size was swept over {10, 20, 30, 40} at all three tiers: At the starved 150-evaluation tier, twenty agents beat thirty, 0.31 against 0.36, because each true evaluation is shared across agents and a smaller swarm spends its share sooner; at 400 and 900 the ordering flips, thirty best at 0.06 and 0.02, forty crowding past the optimum, 0.09 and 0.02. The default of thirty is the budget-aware compromise, and the tier dependence is the reason the swarm-size discussion belongs to the budget ledger rather than to a fixed hyperparameter table.

Table 7 reports the measured cost side, tier by tier, means over the mathematical suite. Token counts scale with tier because condensation events accrue: The prompt share held stable at 82-85% throughout, the self-consistency ensemble dominates the completion side, and the tuning case bills 35% more tokens than the mathematical mean, its mixed-space digests running longer, with the airfoil 18% above. The LLM quartet's own ledgers, at the 400-evaluation tier: OPRO 58k prompt and 6k completion tokens at $0.38 per run, ReEvo 294k and 31k at $1.52, EoH 261k and 28k at $1.38, and FunSearch 312k and 34k at $1.62, so GAS-BFO's $1.29 sits second cheapest among the five language-channel methods while holding the best survival record.

API latency, not local computation, dominates the wall-clock overhead, which ranges from 1.8 to 3.1 times BO-EI's.

The economics favor the method exactly where it is meant to be deployed. Under the thirty-second evaluation floor, the 400-evaluation tier spends at least 3.3 hours evaluating; the LLM adds between 12 and 32 minutes and one to three dollars. Expressed through Eq. (14), the airfoil case purchased a 28.0% gap reduction for US$1.5-2.6 of guidance per run, less than the cost of the handful of evaluations it saves, and GAS-BFO reaches at roughly 250 evaluations the final gap BO-EI attains at 400 on the mathematical suite, a saving of about 150 evaluations, seventy-five minutes at the floor. The converse deserves equal stress: On cheap problems the overhead is pure loss, plain BO costs nothing per run and remains the right tool, and the boundary is a deployment decision the cost model prices openly.

Stigmergic coordination lives or dies on its communication channel, and real channels lose and delay messages. Swarm engineering books dropped and delayed messages as a first-class design constraint rather than an afterthought [55], and networked multi-agent theory ties consensus quality directly to the reliability of the link layer [56]. This experiment turns that lens on the pheromone channel. Two stressors are injected, separately and in combination. Bursty message dropout follows a two-state Gilbert-Elliott model, a good state with mean sojourn of eight ticks and a bad state with mean sojourn of three, whose bad-state occupancy sets the steady-state loss rate p in {0, 0.2, 0.4, 0.6}; a bursty model is harsher than independent coin flips because losses arrive in runs that punch holes in the collective memory. A pheromone-write latency of {0, 2, 5} ticks delays each write before it becomes visible, so agents act on fields assembled from stale evidence.

The protocol holds every other knob fixed on Hartmann-6D at the 400-evaluation tier. Dropout and latency act only on pheromone writes; the budget manager, the GP surrogate, the hallucination guard, and the acquisition gate run unchanged. For each configuration, final best-fitness is measured against the mean of the lossless configuration for the same method, and the ratio is reported as a percentage, 100% meaning the stressor cost nothing. Thirty independent runs price each cell, and the reference methods run through the identical harness, stress levied on their observation updates, so the comparison prices channel redundancy rather than implementation luck.

Table 8 and Fig. 3 report the retention ledger. GAS-BFO retained 97.2% of lossless quality at p = 0.2, 90.8% at p = 0.4, and 79.4% at p = 0.6, against 84.6%, 63.7%, and 41.2% for BO-EI over the same grid; the gap widens monotonically with loss, from 12.6 to 38.2 percentage points. Latency acted more gently on its own: At loss 0.2, a five-tick write latency alone left GAS-BFO at 91.7% against 74.8% for BO-EI. The combined worst cell, p = 0.6 with five-tick latency, left 68.9% retention for GAS-BFO against 33.4% for BO-EI.

*Retention relative to each method's lossless mean) and under write latency at loss 0.2. Reference methods are stressed on their observation updates through the identical harness.

The swarm's amortized knowledge tolerates runs of losses that single-model and single-history rivals cannot. The mechanism mirrors the distributional-shift result of Section 7. BO-EI concentrates its knowledge in one GP posterior, so every dropped or delayed observation distorts the single model that drives acquisition, and OPRO's single textual history is one long message chain with the same fragility. GAS-BFO amortizes knowledge across thirty agents and a capped log, and the condensation service tolerates holes: Region summaries aggregate hundreds of posts, so a burst of losses thins the evidence without flipping the summary. The hallucination guard adds a second line of defense, and when punctured evidence misleads the field, cue-surrogate agreement drops, tumble probabilities rise, and the swarm falls back toward local evidence, the fail-safe of Eq. (6). Latency hurts less than dropout because late evidence is still correct evidence, and the operational reading is that a deployment tolerating 79% retention can absorb a 60% bursty loss rate on the pheromone channel without redesign.