Searching for BSM Experimental Signatures with Large Lagrangian Models

Ibrahim Elsharkawy; Victoria Knapp-Perez; Wahid Bhimji; Aishik Ghosh

hep-phSep 29, 2026

Searching for BSM Experimental Signatures with Large Lagrangian Models

Authors: Ibrahim Elsharkawy, Victoria Knapp-Perez, Wahid Bhimji, Aishik Ghosh

Organizations: Department of Physics, University of Toronto and Vector Institute, Toronto, ON, Canada · NERSC, Lawrence Berkeley National Laboratory, Berkeley, California, USA · Department of Physics and Astronomy, University of California, Irvine, CA 92697 · Halluminate, San Francisco, California, USA, 94107 · Georgia Institute of Technology, Atlanta, GA 30332 · Lawrence Berkeley National Laboratory, Berkeley, CA 94720

Abstract

The search for physics Beyond the Standard Model (BSM) is generally limited not by the supply of theory descriptions but by the lack of discriminating experimental observations. A case in point is dark matter, where the overwhelming gravitational evidence only goes so far in distinguishing between models within a vast theory space. Exploring the space of testable model signatures may help identify overlooked experimental observables and indicate the utility of future experiments. A challenge is designing a search through model signatures outside what is found in the literature. Our primary contribution is hAIthem, a framework that combines the self-guided exploration of reinforcement learning (RL) with the broad literature-derived knowledge of LLMs. We build an RL agent that learns to find which portions of a theory's high-dimensional parameter space are not excluded under some subset of constraints by playing a Battleship-style "game" against a suite of phenomenology tools. The agent is built as a Large Lagrangian Model (LLaM), an autoregressive transformer that reads a tokenized Lagrangian, is pretrained at scale (here on ~1 billion tokens from ~10,000 Lagrangians), and is fine-tuned in a live environment. The framework then constructs a decision tree that separates RL-found regions using observables computed with established tools, and passes the remaining degenerate regions to a set of LLM agents that compete to produce realistic signatures. In this proof of concept, RL-search outperforms an evolutionary-algorithm baseline, finding more viable regions with greater physical diversity. In a restricted space of single dark scalar multiplet models, we find that hAIthem proposes interesting combinations of previously studied observables, such as the application of a halo-independent kinematic ratio to paleo-detectors.

Figures & tables

Figure 1 : Illustration of the hAIthem framework. Symbolic Lagrangians are built and fed to the Large Lagrangian Model along with an encoding of what exclusions we consider. The Large Lagrangian Model searches through parameter space over

B

tunrs, where each round it probes parameter space

N

times. The non-excluded regions it finds (non-excluded under our set up, not in nature) are then used to build a decision tree first with observables we compute with community built tools. Remaining degenerate nodes are fed to a set of LLM-agents who compete in producing a realistic signature.

Figure 2 : The similarities between AlphaGo and AlphaZero (left) and hAIthem (right). Both search optimal moves conditioned on the current play turn and past actions. AlphaGo and hAIthem are similar in being pretrained at scale and fine-tuned with RL and, like AlphaZero , hAIthem is not trained on human-generated data. Major differences include the non-discrete nature of the parameter space and the lack of a competitor.

Probe	Observable	Cut	Package
Relic density	$\Omega h^{2}$	$\tau=1\longrightarrow\Omega h^{2}<0.118$ or $\Omega h^{2}>0.126$ [ 25 , 34 ]	micrOMEGAs
Direct detection	$\sigma_{\rm SI},\sigma^{p}_{\rm SD}$	$r>1$ at 90% CL [ 34 ]	micrOMEGAs
Invisible Higgs	$\mathrm{BR}(h\to\mathrm{inv})$	$>0.11$ [ 5 , 44 ]	micrOMEGAs
Fermi-LAT	$\langle\sigma v\rangle_{c^{\star}}$	annihilation-channel and $m_{\rm DM}$ dependent bound [ 23 , 34 ]	micrOMEGAs
IceCube	$\mu_{\rm solar}$	conservative $b\bar{b}$ bound as a function of $m_{\rm DM}$ [ 19 , 17 , 34 ]	micrOMEGAs
LHC pair production	$r_{\rm max}^{\rm LHC}$	$r_{\rm max}^{\rm LHC}>1$	SModelS

Table 1 : Observables that define viability. The relic-density cut is always active with an episode-dependent allowable error given by

\tau

. Each remaining constraint is independently active with probability 1/2 per episode, with the resulting mask provided to the model.

Figure 3 : Viability Computation Pipeline

Figure 4 : The Large Lagrangian Model. The Lagrangian, viability configuration and search history are each encoded into tokens (green) that a shared transformer body processes jointly.

H

policy heads read the autoregressive tokens, each outputting its own per-dimension Beta conditionals. The value head reads a learned token to estimate the per-turn value. Two auxiliary heads are added and used during pretraining.

	small	medium
Hidden width $d$	256	512
Attention Blocks	4	12
Attention Heads	8	16
Total parameters	$\sim 4.8$ M	$\sim 44.0$ M

Table 2 : Two model-size configurations. We describe training and model hyperparameters in Appendix B.4 .

Figure 5 : Tokenization of an example Lagrangian composed of a real scalar, Dirac fermion, a complex scalar, and a Majorana fermion. Each dark-sector field is embedded into its own field token from its gauge-eigenbasis quantum numbers, while the symmetry content and per-field summaries (dashed) are embedded into a single global Lagrangian token.

\mathcal{L}_{\rm SM}

is fixed across all models and never encoded.

Figure 6 : A depiction of autoregressive sampling. Each parameter is drawn from a model forward pass and fed back as context for the next forward pass for the next parameter. The model chooses the beta distribution or differential evolution style policy once per probe and fixes the choice for all parameter dimensions.

Figure 7 : The two-stage training workflow. The policy is first pretrained offline (Section II.5 ) on

\sim

10,000 distinct Lagrangians and

\mathcal{O}(1B)

output tokens from earlier RL runs, using the supervised objectives of Equation ( 4 ). The resulting weights warm-start PPO fine-tuning (Section II.4 ), which trains on the live micrOMEGAs and SModelS environment.

Quantity	Count
Distinct Lagrangians	$10,843$
Boards (Lagrangians + viability criteria, computed with cut augmentation)	$\approx 5.6\times 10^{7}$
Probes evaluated	$111,046,650$
Output tokens (probes $\times$ parameter dimension $d$ )	$1,308,007,956$

Table 3 : Pretraining dataset summary. Board counts include cut augmentation (i.e., we include all combinations of cuts that can be on or off per Lagrangian). The relic-density allowed width

\tau

provides another continuous direction of augmentation. The number of probes evaluated and output tokens are independent of augmentation (i.e., we did in fact evaluate 100M probes).

d

here is the Lagrangian parameter dimension.

Figure 8 : Two (very) small example decision trees, for (upper) a Lagrangian composed of a Complex Scalar Doublet and (lower) a Lagrangian composed of a Real Scalar Singlet. Here, viable means passes all experimental constraints in Section II .

Panel of Figure 9 — Figure 9 : Example LLaM-small-RL search episodes at budget $B=50$ for three benchmark Lagrangians (see numbering of Appendix D.1 ), shown early in the episode (left) and at the final turn (right). Grey points are excluded probes, red dots are current turn’s probes, gold stars are the cumulative viable points found, and the shaded ellipses are the policy’s beta distribution heads. The search acts in the full $d$ -dimensional parameter space. We display the three parameters along which its viable points spread the most. Policy heads need not be in the right location when Differential Evolution is chosen by the LLaM . More examples can be found in Appendix D.1 , and interactive visualizations can be found online.

Search Method	$N_{v}$	$\mathcal{L}_{v}$	$R_{100}$	$N_{\sigma}$	$W$
LLaM-small (pretrained)	$6,105$	$9/21$	1.5	$14$	$0$
LLaM-small (RL)	$20,744$	14/21	$0.9$	29	$3$
LLaM-medium (pretrained)	$3,007$	$13/21$	$0.8$	$17$	$0$
LLaM-medium (RL)	51,802	$13/21$	$0.5$	$23$	12
Differential Evolution	$8,591$	$11/21$	$0.1$	$24$	$0$

Table 4 : Search benchmark composed of 21 representative Lagrangians. Here, summed over all Lagrangians, all budgets, and all three repeats.

N_{v}

is the total number of viable points found,

\mathcal{L}_{v}

the number of Lagrangians where at least one viable point was found,

R_{100}

is the number of regions found via DBSCAN per 100 viable points (a measure of how diverse the viable points are) [ 84 ] ,

N_{\sigma}

the distinct signature classes discovered, and

W

is the win rate, i.e., the number of Lagrangians on which the policy found the most viable points. See Appendix D.1 for information on the Lagrangians used along with per-Lagrangian results. Bold marks the best performance per column.

Figure 10 : A Benchmark of search performance. Total number of viable points found as a function of search budget, summed over the 21 benchmark Lagrangians.

Budget	Search Method	$N_{v}$	$\mathcal{L}_{v}$	$R_{100}$	$N_{\sigma}$	$W$
$B=5$	LLaM-small (pretrained)	$4.3\pm 2.9$	$2.7\pm 1.7$	$0.0\pm 0.0$	$1.7\pm 0.5$	$1.3\pm 1.2$
	LLaM-small (RL)	$7.3\pm 2.1$	$\mathbf{3.0\pm 0.0}$	$0.0\pm 0.0$	$1.7\pm 0.5$	$\mathbf{2.0\pm 0.8}$
	LLaM-medium (pretrained)	$3.0\pm 2.8$	$2.0\pm 1.4$	$0.0\pm 0.0$	$2.0\pm 1.4$	$0.3\pm 0.5$
	LLaM-medium (RL)	$\mathbf{222\pm 183}$	$2.3\pm 0.9$	$\mathbf{1.8\pm 1.7}$	$\mathbf{3.7\pm 1.9}$	$\mathbf{2.0\pm 1.4}$
	Differential Evolution	$3.7\pm 0.5$	$\mathbf{3.0\pm 0.8}$	$0.0\pm 0.0$	$2.3\pm 0.9$	$1.0\pm 0.8$
$B=10$	LLaM-small (pretrained)	$7.3\pm 3.7$	$3.3\pm 1.2$	$0.0\pm 0.0$	$3.0\pm 0.8$	$0.7\pm 0.5$

Table 5 : Search benchmark composed of 21 representative Lagrangians, resolved by search budget:

B=5

10

25

, and

50

turns (that is 640, 1280, 3200, and 6400 probes respectively), with mean and error computed over three repeats. Columns are as in Table 4 . Bold marks the best performance per column within each budget block. See Appendix D.1 for per-Lagrangian results

Figure 11 : Diversity benchmark. Distinct experimental-signature classes discovered (that is, the number of decision-tree leaf nodes; solid, left axis) and DBSCAN -found regions per 100 viable points (striped, right axis) for each search method [ 84 ] .

Figure 12 : Mean number of viable points per benchmark Lagrangian as a function of the Lagrangian parameter-space dimension

d

. The LLaM becomes more effective as the dimension grows, while differential evolution collapses, as expected from the curse of dimensionality. We only plot bins of Lagrangian parameter-space dimension

d

for which we have sufficient statistics.

Lagrangian	$d$	$N_{v}$	$N_{\sigma}$	$N_{R}$
Real Scalar Singlet	$3$	$877$	$3$	$3$
Real Scalar Triplet	$3$	$0$	$0$	$0$
Complex Scalar Doublet	$5$	$20,068$	$4$	$4$
Complex Scalar Doublet + dark $U(1)^{+}$	$7$	$0$	$0$	$0$
Complex Scalar Doublet + dark $U(1)^{-}$	$7$	$24$	$4$	$4$
Complex Scalar Singlet + dark $U(1)^{\pm}$ (secluded)	$8$	$1,752$	$18$	$19$

Table 6 : All 11 distinct Lagrangians in our single-dark-multiplet scan, split by spin and ordered by parameter-space dimension

d

N_{v}

is the total number of viable points found.

N_{\sigma}

is the number of leaves of each model’s signature tree plotted in Appendix D.2 .

N_{R}

is the total number of separated parameter-space regions (computed with DBSCAN as in Section III ) [ 84 ] . Models with

Y=0

and Dark-U(1) are identical under a field redefinition between

Q=\pm 1

Dark-U(1) field charge. 11 11 11 Due to the self-conjugate condition, real scalars cannot carry hypercharge, and thus their doublets (which require

Y=\pm\frac{1}{2}

for electric neutrality) are excluded. For the same reason, real scalars cannot carry a charge under a Dark-U(1) and thus those models are excluded. 12 12 12 Since our search space allows for different

\mathbb{Z}_{n}

charges for the different components in a complex scalar field, for each complex scalar model, we have two variants. One in which the complex component has a Higgs portal and one in which it is secluded. See Section II .

Figure 13 : Pie chart of exclusions. Each point that is excluded by some set of constraints is tallied, and we compute the ratio of points that were excluded by that combination from the total exclusion count.

Figure 14 : Decision Tree over all single-scalar-multiplet Lagrangians our space allows. The LLM-agent cites [ 104 , 10 , 124 , 9 , 105 , 97 , 37 , 8 , 45 , 3 , 67 , 118 , 95 , 71 , 46 , 57 , 70 , 116 , 23 , 65 , 32 , 24 , 22 , 81 , 114 , 25 , 125 , 13 , 14 , 112 , 90 , 129 , 89 , 115 , 131 , 5 , 73 ] . Interactive visualizations that show reasoning traces can be found online.

	Literature Splits		Literature projections		Novel Observables		Novel Alternatives
Category	$n$	Percentage	$n$	Percentage	$n$	Percentage	$n$	Percentage
Collider signatures	94	$73.4\%$	50	$42.0\%$	5	$13.9\%$	0	$0.0\%$
Astrophysical	24	$18.8\%$	53	$44.5\%$	29	$80.6\%$	4	$80.0\%$
Dedicated DM experiments	2	$1.6\%$	15	$12.6\%$	2	$5.6\%$	1	$20.0\%$
Cosmology	8	$6.2\%$	1	$0.8\%$	0	$0.0\%$	0	$0.0\%$
Total	128	$44.4\%$	119	$41.3\%$	36	$12.5\%$	5	$1.7\%$

Table 7 : Types of measurements proposed by the LLM agent over global and model trees for all

N_{\rm repeat}=5

repeats.

n

is the number in that category. The bottom row is the total per split-type for the final aggregated Trees in Figure 14 and in Appendix D.2 . Percentages for categories are percentages over category type. Percentages for Totals are percentages over LLM split type.

Panel of Figure 15 — Figure 15 : Diagram of Dark Bremsstrahlung possible with the scalar + dark U(1) models we consider. This process results in a dimuon + MET final state near the Higgs mass.

Appendix figures & tables27 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 16 : Direct-detection Observables

\sigma_{\rm SI}

and

\sigma^{p}_{\rm SD}

bounds interpolated from [ 11 , 12 , 15 , 38 , 26 ] . SuperCDMS combines the Si-HV, Si-iZIP, Ge-HV, and Ge-iZIP constraints.

Figure 17 :

\langle\sigma v\rangle

bounds per channel interpolated from [ 22 , 65 , 23 ] . The WW channel is non-constraining below

\sim m_{W}

, similarly for other channels.

Figure 18 : IceCube

b\bar{b}

channel bound on

\mu_{\rm solar}

interpolated from [ 17 , 18 ]

$\tau$	$\Omega h^{2}$ range
1	$[0.118,\ 0.126]$
10	$[0.088,\ 0.169]$
50	$[0.024,\ 0.629]$

Appendix

Table 8 : Example Relic-band

\tau

and the acceptance range it maps to. 17 17 17 Given by

\Omega_{\pm}(\tau)=\sqrt{\Omega_{\rm lo}\Omega_{\rm hi}}\cdot\left(\frac{\Omega_{\rm hi}}{\Omega_{\rm lo}}\right)^{\pm\tau/2}

with

\Omega_{\rm lo}=0.118

\Omega_{\rm hi}=0.126

Panel of Figure 19 — Figure 19 : The three production channel classes. (A) $Z^{\prime}$ -mediated Drell–Yan for DM charged under $U(1)^{\prime}$ . (B) Electroweak Drell–Yan via SM gauge bosons for DM in non-singlet $SU(2)_{L}$ reps. (C) Higgs-portal production (here gluon fusion through a top loop) for SM-singlet DM with no $U(1)^{\prime}$ .

#	Lorentz	$(SU(2)_{L},Y)$	$U(1)^{\prime}$ ?	Channel(s)
1	complex scalar	singlet, $Y{=}0$	yes	A
2	Dirac fermion	singlet, $Y{=}0$	yes	A
3	real scalar	singlet, $Y{=}0$	no	C
4	complex scalar	singlet, $Y{=}0$	no	C
5	Majorana / Dirac fermion	singlet, $Y{=}0$	no	—
6	complex scalar	doublet, $Y{=}\tfrac{1}{2}$	no	B

Appendix

Table 9: Dominant production DM channel in the LHC in the search space of Section II .

#	Model	$\in\mathcal{L}_{\rm BSM}$
1	$\mathbb{Z}_{2}$ real scalar singlet	$-\alpha_{1}H^{\dagger}Hs^{2}-\alpha_{2}s^{4}$
2	$\mathbb{Z}_{2}$ complex scalar singlet	$-\alpha_{1}H^{\dagger}H\|S\|^{2}-\alpha_{2}\|S\|^{4}-\left(\alpha_{3}H^{\dagger}HS^{2}+\alpha_{4}S^{4}+\text{h.c.}\right)$
3	$\mathbb{Z}_{2}$ complex scalar singlet (secluded)	$-\alpha_{1}H^{\dagger}Hs_{1}^{2}-\alpha_{2}s_{1}^{4}-\alpha_{3}s_{1}^{2}s_{2}^{2}-\alpha_{4}s_{2}^{4}$
4	$\mathbb{Z}_{2}$ complex scalar singlet $+$ dark $U(1)^{\prime}$	$-\alpha_{1}H^{\dagger}H\|S\|^{2}-\alpha_{2}\|S\|^{4}+\|(\partial_{\mu}-ig_{Z^{\prime}}qZ^{\prime}_{\mu})S\|^{2}-\tfrac{\epsilon}{2}B_{\mu\nu}Z^{\prime\mu\nu}+\tfrac{1}{2}M_{Z^{\prime}}^{2}Z^{\prime}_{\mu}Z^{\prime\mu}$

Appendix

Table 10 : Representative scalar dark-matter Lagrangians from the agent’s search space.

H

is the Higgs doublet,

s

is a real scalar,

S

is a complex scalar, and

Z^{\prime}

is the

U(1)^{\prime}

gauge boson with charge

q

. Kinetic and mass terms are not shown except for

U(1)^{\prime}

\alpha_{i}

are coupling parameters the agent learns to scan through and

g_{Z^{\prime}}

\epsilon

the dark

U(1)^{\prime}

’s coupling and kinetic mixing terms.

$\mathrm{SU}(2)_{L}$	$Y$	Real scalar	Complex scalar	Majorana	Dirac
Singlet	$0$	✓	✓	✓	✓
Doublet	$\tfrac{1}{2}$	$\times$	✓	$\times$	✓
Triplet	$0$	✓	✓	✓	✓

Appendix

Table 11 : Spin and quantum numbers for a single dark multiplet we consider. Electrical neutrality requires

Y=0

for singlets and

Y=\pm\frac{1}{2}

for doublets and we restrict to

Y=0

for triplets. Real scalars or Majorana fermions cannot carry hypercharge, and similarly cannot carry a dark U(1)’ charge. [ 113 ] .

Type	Examples	Maximum Allowed Range
DM or mediator mass	$M_{\rm DM}$ , $M_{Z^{\prime}}$	1 GeV to 10 TeV
Dimensionless Coupling	$a_{2}$ , $\lambda_{i}$ , $g_{Z^{\prime}}$	$10^{-2}$ to $4\pi$
Kinetic Mixing Parameter	$\epsilon$	$10^{-6}$ to $10^{-1}$

Appendix

Table 12 : Continuous parameters sampled by the agent and their full ranges. Each episode further restricts every type to a randomly drawn sub-range of the full range below. 22 22 22 Our EFT scale range is

\Lambda\in[10^{-1},10^{3}]

GeV for the two non-renormizable operators we consider. No points are accepted that do not satisfy

m_{\rm DM}\leq\Lambda

. This is not relevant for our scan of scalar models who contain only renormalizable operators.

Experiment	Observable	Threshold	Software
XLZD	$\sigma_{\rm SI}$	bound on $\sigma_{\rm SI}$ as a function of $m_{\rm DM}$ [ 14 ]	micrOMEGAs
LZ full exposure	$\sigma_{\rm SI}$	bound on $\sigma_{\rm SI}$ as a function of $m_{\rm DM}$ [ 30 ]	micrOMEGAs
DarkSide-20k	$\sigma_{\rm SI}$	bound on $\sigma_{\rm SI}$ as a function of $m_{\rm DM}$ [ 15 ]	micrOMEGAs
SuperCDMS	$\sigma_{\rm SI}$	bound on $\sigma_{\rm SI}$ as a function of $m_{\rm DM}\leq 10$ GeV [ 26 ]	micrOMEGAs
PICO-500	$\sigma^{p}_{\rm SD}$	bound on $\sigma^{p}_{\rm SD}$ as a function of $m_{\rm DM}$ [ 39 ]	micrOMEGAs
CTA	$\langle\sigma v\rangle_{c^{\star}}$	annihilation-channel dependent bound [ 22 ]	micrOMEGAs

Appendix

Table 13 : Projected experiments scored as testability targets. Bounds are interpolated log-log in

m_{\rm DM}

when needed. HL-LHC and FCC-ee Invisible Higgs thresholds are flat in

m_{\rm DM}

. The HL-LHC collider testability flag is computed by rescaling the SModelS

r_{\rm max}

to projected HL-LHC luminosity. How we interpolate bounds from literature values is given in App. A.1 .

Figure 20 : The probe-history encoder.

M

learned queries cross-attend, with a learned recency bias, over the embeddings of every probe taken so far, and the

M

outputs join the input token set as history tokens.

Figure 21 : ( Upper ) Effect of the

\nu

window on a two-parameter toy parameter space.

1\sigma

contours of the broadest (cyan) and sharpest (black) densities each window permits, for a broad and a sharp head at the start, middle and end of an episode. ( Lower ) Per-head concentration windows over an episode with

T=50

turns per episode. Each head’s hard

\nu

window slides from broad (exploration) to sharp (exploitation), with a stagger for a division of labor across heads.

	small	medium
Token hidden dimension $d_{\rm tok}$	256	512
Body depth	4	12
Body attention heads	8	16
MLP width	$4d_{\rm tok}=1024$	$4d_{\rm tok}=2048$
Total parameters	$4.8$ M	$44.0$ M

Appendix

Table 14: LLaM-small and LLaM-medium hyperparameters. Parameters shared between the two sizes are given in Table 15 .

Shared hyperparameters
Architecture		PPO and Training Parameters
Policy heads $H$	4	Learning rate (AdamW, RL and Pre)	$1\times 10^{-4}$
History query tokens $M$	64	LR warmup episodes for RL	1000
Attention head dim	32	Clip $\epsilon$	0.1
Max Lagrangian parameter dim $d_{\max}$	128
Probes per turn $N$	128

Appendix

Table 15 : Hyperparameters shared between the two model sizes.

Observable	Channel Dependence	Bin Width in $\log_{10}$	Overflow Bin Edges in $\log_{10}$
$\sigma_{\rm SI}$ for XLZD, LZ, and DarkSide	N/A	1	$\pm 2$
$\sigma_{\rm SD}$ for PICO	N/A	1	$\pm 2$
$\langle\sigma v\rangle$ for CTA	channel ( $b\bar{b}$ , $WW$ , $\tau\tau$ )	1	$\pm 1.5$
$\langle\sigma v\rangle$ for Fermi-LAT (15 yr)	channel ( $b\bar{b}$ , $WW$ , $\tau\tau$ )	1	$\pm 1.5$
Upward muon flux for IceCube-Gen2	N/A	1	$\pm 2$
BR( $h\to$ inv): HL-LHC	N/A	0.5	$\pm 3$

Appendix

Table 16 : Observables that are used to split nodes in the decision tree. Each

\mu_{e}=\log_{10}(O_{\rm pred}/O_{\rm lim})

is a continuous ratio of the experiment’s projected limit in log space. We discretize this

\mu_{e}

into fixed-width bins with overflow bins. All values below the lower overflow bin are merged into a single “far-below-reach” bin, all values above the upper overflow bin into a single “far-above-reach” bin.

#	Field content	parameter dim $d$	$n_{f}$	U(1) ′
1	$\mathbb{Z}_{3}$ : Real Scalar Singlet	3	1	–
2	$\mathbb{Z}_{3}$ : Majorana Singlet	3	1	–
3	$\mathbb{Z}_{3}$ : Majorana Triplet	3	1	–
4	$\mathbb{Z}_{3}$ : Dirac Doublet	4	1	–
5	$\mathbb{Z}_{3}$ : Dirac Triplet	4	1	–
6	$\mathbb{Z}_{3}$ : Complex Scalar Doublet	5	1	–

Appendix

Table 17 : The 21 Lagrangian classes of the benchmark with

n_{f}

number of fields, ordered by parameter-space dimension

d

	LLaM-small (pretrained)			LLaM-small (RL)			LLaM-medium (pretrained)			LLaM-medium (RL)			Differential Evolution
#	$N_{v}$	$R_{100}$	$N_{\sigma}$	$N_{v}$	$R_{100}$	$N_{\sigma}$	$N_{v}$	$R_{100}$	$N_{\sigma}$	$N_{v}$	$R_{100}$	$N_{\sigma}$	$N_{v}$	$R_{100}$	$N_{\sigma}$
1	$0$	–	$0$	418	$0.2$	$2$	$1$	$0.0$	$1$	$0$	–	$0$	$0$	–	$0$
2	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$
3	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$
4	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$
5	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$

Appendix

Table 18 : Per-Lagrangian benchmark results, summed over all budgets and three repeats, where row numbers refer to Table 17 . Bold marks the policy with the most viable points on each Lagrangian (ties bolded jointly).

#	$N_{v}$	$R_{100}$	$N_{\sigma}$	$N_{v}$	$R_{100}$	$N_{\sigma}$	$N_{v}$	$R_{100}$	$N_{\sigma}$	$N_{v}$	$R_{100}$	$N_{\sigma}$	$N_{v}$	$R_{100}$	$N_{\sigma}$
	LLaM-small (pretrained)			LLaM-small (RL)			LLaM-medium (pretrained)			LLaM-medium (RL)			Differential Evolution
1	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$
2	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$
3	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$
4	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$
5	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$

Appendix

Table 19 : Per-Lagrangian benchmark results at budget

B=5

(mean and error over three repeats), where row numbers refer to Table 17 . Bold marks the policy with the most viable points (ties bolded jointly).

#	$N_{v}$	$R_{100}$	$N_{\sigma}$	$N_{v}$	$R_{100}$	$N_{\sigma}$	$N_{v}$	$R_{100}$	$N_{\sigma}$	$N_{v}$	$R_{100}$	$N_{\sigma}$	$N_{v}$	$R_{100}$	$N_{\sigma}$
	LLaM-small (pretrained)			LLaM-small (RL)			LLaM-medium (pretrained)			LLaM-medium (RL)			Differential Evolution
1	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$
2	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$
3	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$
4	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$
5	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$

Appendix

Table 20 : Per-Lagrangian benchmark results at budget

B=10

(mean and error over three repeats), where row numbers refer to Table 17 . Bold marks the policy with the most viable points (ties bolded jointly).

#	$N_{v}$	$R_{100}$	$N_{\sigma}$	$N_{v}$	$R_{100}$	$N_{\sigma}$	$N_{v}$	$R_{100}$	$N_{\sigma}$	$N_{v}$	$R_{100}$	$N_{\sigma}$	$N_{v}$	$R_{100}$	$N_{\sigma}$
	LLaM-small (pretrained)			LLaM-small (RL)			LLaM-medium (pretrained)			LLaM-medium (RL)			Differential Evolution
1	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$
2	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$
3	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$
4	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$
5	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$

Appendix

Table 21 : Per-Lagrangian benchmark results at budget

B=25

(mean and error over three repeats), where row numbers refer to Table 17 . Bold marks the policy with the most viable points (ties bolded jointly).

#	$N_{v}$	$R_{100}$	$N_{\sigma}$	$N_{v}$	$R_{100}$	$N_{\sigma}$	$N_{v}$	$R_{100}$	$N_{\sigma}$	$N_{v}$	$R_{100}$	$N_{\sigma}$	$N_{v}$	$R_{100}$	$N_{\sigma}$
	LLaM-small (pretrained)			LLaM-small (RL)			LLaM-medium (pretrained)			LLaM-medium (RL)			Differential Evolution
1	$0$	–	$0$	$\mathbf{139\pm 197}$	$0.2\pm 0.0$	$0.7\pm 0.9$	$0.3\pm 0.5$	$0.0\pm 0.0$	$0.3\pm 0.5$	$0$	–	$0$	$0$	–	$0$
2	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$
3	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$
4	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$
5	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$	$0$	–	$0$

Appendix

Table 22 : Per-Lagrangian benchmark results at budget

B=50

(mean and error over three repeats), where row numbers refer to Table 17 . Bold marks the policy with the most viable points (ties bolded jointly).

Panel of Figure 22 — Figure 22 : Example LLaM-small-RL search episodes at budget $B=50$ for benchmark Lagrangians (see numbering of Table 17 ), shown early in the episode (left) and at the final turn (right). Grey points are excluded probes, red dots are current turn’s probes, gold stars are the cumulative viable points found, and the shaded ellipses are the policy’s beta distribution heads. The search acts in the full $d$ -dimensional parameter space. We display the three parameters along which its viable points spread the most. Policy heads need not be in the right location when Differential Evolution is chosen by the LLaM.

Panel of Figure 23 — Figure 23 : Example LLaM-small-RL search episodes at budget $B=50$ for benchmark Lagrangians (see numbering of Table 17 ), shown early in the episode (left) and at the final turn (right). Grey points are excluded probes, red dots are current turn’s probes, gold stars are the cumulative viable points found, and the shaded ellipses are the policy’s beta distribution heads. The search acts in the full $d$ -dimensional parameter space. We display the three parameters along which its viable points spread the most. Policy heads need not be in the right location when Differential Evolution is chosen by the LLaM.

Figure 24 : Three Decision Trees from the One-Multiplet Search Space. None meet the condition for the LLM agent to be applied.

Figure 25 : Complex Scalar Singlet Decision Tree.

Figure 26 : Decision Tree for a Complex Scalar Singlet with Dark

U(1)^{\prime}

and a secluded complex component.

Figure 27 : Decision Tree for a Complex Scalar Singlet with Dark

U(1)^{\prime}

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