PiEvo: Nuclear Scattering & Scientific Discovery

• PiEvo is a dual-use concept describing both a nuclear scattering simulator and a framework for principle‐evolvable scientific discovery.
• In nuclear physics, it integrates electroweak pion production, realistic spectral functions, and intranuclear transport to reproduce experimental observables.
• In autonomous discovery, it employs Bayesian optimization over a learnable principle space to trigger theory evolution and enhance discovery efficiency.

Searching arXiv for papers using the term “PiEvo” and closely related context. PiEvo is a name used in two distinct technical senses in recent arXiv literature. In nuclear scattering and transport, it denotes “pion evolution” in the Achilles event simulator: the coupled treatment of the electroweak pion-production amplitude on a bound nucleon, the smearing and attenuation induced by bound-state correlations, and the semi-classical propagation, scattering, charge exchange, and absorption of the produced pion and intermediate resonances in the nuclear medium (Isaacson et al., 26 Aug 2025). In LLM-based scientific agents, PiEvo denotes a principle-evolvable framework for scientific discovery that treats principles as learnable probabilistic objects and casts discovery as Bayesian optimization over an expanding principle space, combining Information-Directed Hypothesis Selection via Gaussian Process with anomaly-driven augmentation (Pu et al., 6 Feb 2026). A common source of confusion is therefore terminological rather than methodological: the same label refers to unrelated programs in nuclear many-body event generation and autonomous scientific discovery.

1. Scope and nomenclature

Within Achilles, PiEvo is not a standalone Monte Carlo component in isolation but the nucleus-level description required for accurate “pion evolution”: tying together the electroweak pion-production amplitude, realistic hole spectral functions, and intranuclear transport with scattering and absorption. The emphasis is on microscopic consistency across the electroweak vertex, the meson-baryon amplitudes used during propagation, and the treatment of the nuclear target configuration (Isaacson et al., 26 Aug 2025).

In the scientific-agent literature, PiEvo is explicitly introduced as “Principle-Evolvable Scientific Discovery via Uncertainty Minimization.” Its motivating claim is that many LLM-based scientific-agent frameworks fix both their prior “principles” and their hypothesis space, which induces restricted novelty, inefficient search, and anomaly blindness. PiEvo departs from this by making scientific principles themselves learnable and probabilistic, so that anomalous outcomes become triggers for principle refinement rather than discarded noise (Pu et al., 6 Feb 2026).

The two usages share no common mathematical substrate. One concerns single-pion production and propagation in nuclei; the other concerns sequential decision-making under epistemic uncertainty in scientific discovery. This suggests that any encyclopedic treatment of PiEvo must begin with explicit disambiguation.

2. PiEvo as pion evolution in Achilles

The Achilles implementation extends the simulator by incorporating the single-pion production mechanism in a fully exclusive fashion. Its electroweak interaction vertex is modeled by combining the ANL–Osaka Dynamical Coupled-Channels (DCC) approach with realistic hole spectral functions, so that both correlations in the initial target state and correlations in the residual spectator system enter the one-nucleon–one-pion response (Isaacson et al., 26 Aug 2025).

The charged-current double-differential cross section is written as

$$\frac{d^2\sigma}{d\Omega\,dE'} \;=\;\frac{G_F^2\cos^2\theta_C}{4\pi^2}\,\frac{|\mathbf{k}'|}{|\mathbf{k}|}\;L_{\mu\nu}\;W^{\mu\nu}_{\rm DCC}\,,$$

with $L_{\mu\nu}$ the leptonic tensor and $W^{\mu\nu}_{\rm DCC}$ built from the full DCC pion-production current. The current is decomposed into non-resonant and resonant contributions,

$$\langle\pi N|\,j^\mu\,|N\rangle =\langle\chi_{\pi N}|j^\mu_{\rm bg}|N\rangle +\sum_{m,n}\langle\chi_{\pi N}|\Gamma|N_m^*\rangle\,[D]_{mn}\,\langle N_n^*|j^\mu|N\rangle\,,$$

where $\chi_{\pi N}$ is a distorted meson-baryon state and $[D]_{mn}$ is the resonance propagator matrix. The background and resonant terms are derived from an energy-independent Hamiltonian $H=H_0+v+\Gamma$ via Feshbach projection and coupled integral equations of the form

$$T = t_{\rm bg} + t_{\rm res}\,,\qquad t_{\rm bg}=V+V\,G_{MB}\,t_{\rm bg}\,,\quad t_{\rm res}=\Gamma\,D\,\Gamma\,.$$

Embedding this vertex in a nucleus uses the extended factorization scheme. The one-nucleon–one-pion response is folded with the realistic hole spectral function $S_{t_k}(\mathbf{k},E)$ for protons or neutrons,

$$W^{\mu\nu}_{1N1\pi}(q)\;=\;\sum_{s_k,t_k}\int\!\frac{d^3k}{(2\pi)^3}\,dE\; S_{t_k}(\mathbf{k},E)\, \frac{\langle k|{j^\mu}^\dagger|\pi p\rangle \langle \pi p|j^\nu|k\rangle} {4\,e(\mathbf{k})\,e_\pi(\mathbf{p}_\pi)}\;\times\;\cdots$$

with energy-momentum conserving $L_{\mu\nu}$0-functions enforcing the correct recoil. The spectral functions are taken from correlated-basis-function theory or from $L_{\mu\nu}$1 data on Ar and Ti. In this sense, PiEvo in Achilles is an overview of exclusive electroweak amplitudes with realistic nuclear many-body removal probabilities.

3. Cascade transport and absorption in the nuclear-medium PiEvo

After the hard vertex produces a pion with $L_{\mu\nu}$2 and a nucleon with $L_{\mu\nu}$3, Achilles propagates the hadrons through an intranuclear cascade (INC). Nuclear configurations are drawn from GFMC correlated Monte-Carlo for $L_{\mu\nu}$4C or from measured density distributions for $L_{\mu\nu}$5Ar. Hadrons move in straight-line segments of duration $L_{\mu\nu}$6, and at each step the probability of scattering from a spectator nucleon is

$L_{\mu\nu}$7

where $L_{\mu\nu}$8 is the total meson-baryon cross section at center-of-mass energy $L_{\mu\nu}$9 and $W^{\mu\nu}_{\rm DCC}$0 is an effective local density (Isaacson et al., 26 Aug 2025).

A distinctive feature is the existence of two complementary treatments of pion propagation and absorption. In the “Virtual-Resonances” INC mode, pion–nucleon scattering amplitudes are taken directly from the DCC partial-wave analysis. The channel cross section is

$W^{\mu\nu}_{\rm DCC}$1

and the center-of-mass scattering angle is sampled from the corresponding partial-wave differential distribution. This preserves consistency between the amplitudes at the electroweak production vertex and those used during propagation.

For pion absorption, Achilles provides two models. The first is the optical-potential model of Oset and Salcedo. In the local-density approximation the pion optical potential is written

$W^{\mu\nu}_{\rm DCC}$2

and the in-medium $W^{\mu\nu}_{\rm DCC}$3 self-energy enters through

$W^{\mu\nu}_{\rm DCC}$4

with $W^{\mu\nu}_{\rm DCC}$5 fitted to data. The second is a propagating-resonance model in which the $W^{\mu\nu}_{\rm DCC}$6 is treated as an explicit degree of freedom, analogously to GiBUU. Then a $W^{\mu\nu}_{\rm DCC}$7 collision is generated with a differential cross section involving the $W^{\mu\nu}_{\rm DCC}$8 spectral function

$W^{\mu\nu}_{\rm DCC}$9

and the resonance propagates with lifetime $$\langle\pi N|\,j^\mu\,|N\rangle =\langle\chi_{\pi N}|j^\mu_{\rm bg}|N\rangle +\sum_{m,n}\langle\chi_{\pi N}|\Gamma|N_m^*\rangle\,[D]_{mn}\,\langle N_n^*|j^\mu|N\rangle\,,$$0, can undergo $$\langle\pi N|\,j^\mu\,|N\rangle =\langle\chi_{\pi N}|j^\mu_{\rm bg}|N\rangle +\sum_{m,n}\langle\chi_{\pi N}|\Gamma|N_m^*\rangle\,[D]_{mn}\,\langle N_n^*|j^\mu|N\rangle\,,$$1 scattering by detailed balance, or decay back to $$\langle\pi N|\,j^\mu\,|N\rangle =\langle\chi_{\pi N}|j^\mu_{\rm bg}|N\rangle +\sum_{m,n}\langle\chi_{\pi N}|\Gamma|N_m^*\rangle\,[D]_{mn}\,\langle N_n^*|j^\mu|N\rangle\,,$$2.

The two modes encode different microscopic pictures of the same physical sector. The optical-potential description emphasizes in-medium self-energy and local-density absorption rates; the propagating-resonance description emphasizes finite-lifetime effects and three-body kinematics. The paper’s own formulation treats them as complementary rather than mutually exclusive.

4. Validation program for Achilles PiEvo

The Achilles paper validates PiEvo against both hadronic and lepton-nucleus data, with the stated aim of demonstrating a coherent, quantitatively reliable description from the electroweak vertex to final absorption or escape (Isaacson et al., 26 Aug 2025).

Inclusive $$\langle\pi N|\,j^\mu\,|N\rangle =\langle\chi_{\pi N}|j^\mu_{\rm bg}|N\rangle +\sum_{m,n}\langle\chi_{\pi N}|\Gamma|N_m^*\rangle\,[D]_{mn}\,\langle N_n^*|j^\mu|N\rangle\,,$$3 on $$\langle\pi N|\,j^\mu\,|N\rangle =\langle\chi_{\pi N}|j^\mu_{\rm bg}|N\rangle +\sum_{m,n}\langle\chi_{\pi N}|\Gamma|N_m^*\rangle\,[D]_{mn}\,\langle N_n^*|j^\mu|N\rangle\,,$$4C and $$\langle\pi N|\,j^\mu\,|N\rangle =\langle\chi_{\pi N}|j^\mu_{\rm bg}|N\rangle +\sum_{m,n}\langle\chi_{\pi N}|\Gamma|N_m^*\rangle\,[D]_{mn}\,\langle N_n^*|j^\mu|N\rangle\,,$$5Ar, including JLab E12-14-012, are reported to reproduce the quasi-elastic and $$\langle\pi N|\,j^\mu\,|N\rangle =\langle\chi_{\pi N}|j^\mu_{\rm bg}|N\rangle +\sum_{m,n}\langle\chi_{\pi N}|\Gamma|N_m^*\rangle\,[D]_{mn}\,\langle N_n^*|j^\mu|N\rangle\,,$$6 peaks. The paper also states that the missing “dip” region is to be filled by meson-exchange currents and the high-$$\langle\pi N|\,j^\mu\,|N\rangle =\langle\chi_{\pi N}|j^\mu_{\rm bg}|N\rangle +\sum_{m,n}\langle\chi_{\pi N}|\Gamma|N_m^*\rangle\,[D]_{mn}\,\langle N_n^*|j^\mu|N\rangle\,,$$7 tail by multi-pion/DIS. Bubble-chamber $$\langle\pi N|\,j^\mu\,|N\rangle =\langle\chi_{\pi N}|j^\mu_{\rm bg}|N\rangle +\sum_{m,n}\langle\chi_{\pi N}|\Gamma|N_m^*\rangle\,[D]_{mn}\,\langle N_n^*|j^\mu|N\rangle\,,$$8 measurements on hydrogen and deuterium, specifically the ANL/BNL reanalysis, are described as showing excellent reproduction of the energy dependence of channels such as $$\langle\pi N|\,j^\mu\,|N\rangle =\langle\chi_{\pi N}|j^\mu_{\rm bg}|N\rangle +\sum_{m,n}\langle\chi_{\pi N}|\Gamma|N_m^*\rangle\,[D]_{mn}\,\langle N_n^*|j^\mu|N\rangle\,,$$9.

For pion–nucleus reactions and absorption on $\chi_{\pi N}$0C and $\chi_{\pi N}$1Ar, both the Virtual and Propagating modes are reported to bracket the DUET, Ashery, and LADS data for $\chi_{\pi N}$2 and $\chi_{\pi N}$3, differing modestly near the $\chi_{\pi N}$4 peak. In exclusive $\chi_{\pi N}$5 $\chi_{\pi N}$6 and $\chi_{\pi N}$7 on C, the 0$\chi_{\pi N}$8 and 1p0$\chi_{\pi N}$9 spectra versus

$[D]_{mn}$0

with $[D]_{mn}$1MeV, as well as versus $[D]_{mn}$2 and $[D]_{mn}$3, are reported to show good agreement in the QE and $[D]_{mn}$4 regions, with small deficits where MEC and DIS will add strength.

The neutrino validations are especially diagnostic because they test feed-through between topological samples. T2K CC0$[D]_{mn}$5 and CC1$[D]_{mn}$6 on CH at $[D]_{mn}$7MeV are described as being well reproduced in the transverse-kinematic-imbalance variables $[D]_{mn}$8, with the interplay of resonance absorption feeding into the 0$[D]_{mn}$9 sample and being removed from the 1$H=H_0+v+\Gamma$0 sample providing a tight test of PiEvo. MINER$H=H_0+v+\Gamma$1A CC0$H=H_0+v+\Gamma$2 on CH at $H=H_0+v+\Gamma$3GeV is reported to isolate the resonance-FSI difference between the two INC modes, with the Virtual-Resonances mode giving roughly twice the 0$H=H_0+v+\Gamma$4 feed-through in the high-$H=H_0+v+\Gamma$5 tail. On Ar, MicroBooNE CC1p0$H=H_0+v+\Gamma$6 and NC1$H=H_0+v+\Gamma$7 at $H=H_0+v+\Gamma$8MeV are described as showing NC$H=H_0+v+\Gamma$9 double-differential distributions reproduced to within $$T = t_{\rm bg} + t_{\rm res}\,,\qquad t_{\rm bg}=V+V\,G_{MB}\,t_{\rm bg}\,,\quad t_{\rm res}=\Gamma\,D\,\Gamma\,.$$0, while the CC1p0$$T = t_{\rm bg} + t_{\rm res}\,,\qquad t_{\rm bg}=V+V\,G_{MB}\,t_{\rm bg}\,,\quad t_{\rm res}=\Gamma\,D\,\Gamma\,.$$1 channel exhibits a low-$$T = t_{\rm bg} + t_{\rm res}\,,\qquad t_{\rm bg}=V+V\,G_{MB}\,t_{\rm bg}\,,\quad t_{\rm res}=\Gamma\,D\,\Gamma\,.$$2 deficit that likely signals missing MEC and/or updated axial form factors.

The most discriminating observables are identified explicitly: pion–nucleus reaction and absorption cross sections, and the transverse-imbalance distributions in exclusive $$T = t_{\rm bg} + t_{\rm res}\,,\qquad t_{\rm bg}=V+V\,G_{MB}\,t_{\rm bg}\,,\quad t_{\rm res}=\Gamma\,D\,\Gamma\,.$$3– and $$T = t_{\rm bg} + t_{\rm res}\,,\qquad t_{\rm bg}=V+V\,G_{MB}\,t_{\rm bg}\,,\quad t_{\rm res}=\Gamma\,D\,\Gamma\,.$$4–nucleus scattering. This suggests that PiEvo, in the Achilles sense, is best understood as a transport-validation program organized around observables directly sensitive to $$T = t_{\rm bg} + t_{\rm res}\,,\qquad t_{\rm bg}=V+V\,G_{MB}\,t_{\rm bg}\,,\quad t_{\rm res}=\Gamma\,D\,\Gamma\,.$$5 scattering, charge exchange, and absorption in the medium.

5. PiEvo as principle-evolvable scientific discovery

In the scientific-agent literature, PiEvo reformulates discovery as optimization over a growing principle space $$T = t_{\rm bg} + t_{\rm res}\,,\qquad t_{\rm bg}=V+V\,G_{MB}\,t_{\rm bg}\,,\quad t_{\rm res}=\Gamma\,D\,\Gamma\,.$$6 rather than direct search over a fixed hypothesis space. Definition 1 introduces a countable universal principle space of candidate principles $$T = t_{\rm bg} + t_{\rm res}\,,\qquad t_{\rm bg}=V+V\,G_{MB}\,t_{\rm bg}\,,\quad t_{\rm res}=\Gamma\,D\,\Gamma\,.$$7, each assigned prior probability $$T = t_{\rm bg} + t_{\rm res}\,,\qquad t_{\rm bg}=V+V\,G_{MB}\,t_{\rm bg}\,,\quad t_{\rm res}=\Gamma\,D\,\Gamma\,.$$8 of finite entropy $$T = t_{\rm bg} + t_{\rm res}\,,\qquad t_{\rm bg}=V+V\,G_{MB}\,t_{\rm bg}\,,\quad t_{\rm res}=\Gamma\,D\,\Gamma\,.$$9, with an active set $S_{t_k}(\mathbf{k},E)$0 and an unknown true principle $S_{t_k}(\mathbf{k},E)$1 (Pu et al., 6 Feb 2026).

The observation model distinguishes hypotheses $S_{t_k}(\mathbf{k},E)$2 from principles $S_{t_k}(\mathbf{k},E)$3. When testing a hypothesis, the agent observes

$S_{t_k}(\mathbf{k},E)$4

while maintaining posterior beliefs

$S_{t_k}(\mathbf{k},E)$5

For each principle, PiEvo posits a generative likelihood $S_{t_k}(\mathbf{k},E)$6, and discovery is cast as Bayesian optimization where each principle induces an optimal-hypothesis value

$S_{t_k}(\mathbf{k},E)$7

with objective

$S_{t_k}(\mathbf{k},E)$8

To evaluate $S_{t_k}(\mathbf{k},E)$9 efficiently, PiEvo maintains a separate Gaussian Process expert $$W^{\mu\nu}_{1N1\pi}(q)\;=\;\sum_{s_k,t_k}\int\!\frac{d^3k}{(2\pi)^3}\,dE\; S_{t_k}(\mathbf{k},E)\, \frac{\langle k|{j^\mu}^\dagger|\pi p\rangle \langle \pi p|j^\nu|k\rangle} {4\,e(\mathbf{k})\,e_\pi(\mathbf{p}_\pi)}\;\times\;\cdots$$0 for each active principle. The GP uses mean function $$W^{\mu\nu}_{1N1\pi}(q)\;=\;\sum_{s_k,t_k}\int\!\frac{d^3k}{(2\pi)^3}\,dE\; S_{t_k}(\mathbf{k},E)\, \frac{\langle k|{j^\mu}^\dagger|\pi p\rangle \langle \pi p|j^\nu|k\rangle} {4\,e(\mathbf{k})\,e_\pi(\mathbf{p}_\pi)}\;\times\;\cdots$$1 and an RBF kernel

$$W^{\mu\nu}_{1N1\pi}(q)\;=\;\sum_{s_k,t_k}\int\!\frac{d^3k}{(2\pi)^3}\,dE\; S_{t_k}(\mathbf{k},E)\, \frac{\langle k|{j^\mu}^\dagger|\pi p\rangle \langle \pi p|j^\nu|k\rangle} {4\,e(\mathbf{k})\,e_\pi(\mathbf{p}_\pi)}\;\times\;\cdots$$2

Hypotheses and principles are embedded as $$W^{\mu\nu}_{1N1\pi}(q)\;=\;\sum_{s_k,t_k}\int\!\frac{d^3k}{(2\pi)^3}\,dE\; S_{t_k}(\mathbf{k},E)\, \frac{\langle k|{j^\mu}^\dagger|\pi p\rangle \langle \pi p|j^\nu|k\rangle} {4\,e(\mathbf{k})\,e_\pi(\mathbf{p}_\pi)}\;\times\;\cdots$$3, and the semantic feature map is

$$W^{\mu\nu}_{1N1\pi}(q)\;=\;\sum_{s_k,t_k}\int\!\frac{d^3k}{(2\pi)^3}\,dE\; S_{t_k}(\mathbf{k},E)\, \frac{\langle k|{j^\mu}^\dagger|\pi p\rangle \langle \pi p|j^\nu|k\rangle} {4\,e(\mathbf{k})\,e_\pi(\mathbf{p}_\pi)}\;\times\;\cdots$$4

With this construction, the GP posterior yields predictive mean $$W^{\mu\nu}_{1N1\pi}(q)\;=\;\sum_{s_k,t_k}\int\!\frac{d^3k}{(2\pi)^3}\,dE\; S_{t_k}(\mathbf{k},E)\, \frac{\langle k|{j^\mu}^\dagger|\pi p\rangle \langle \pi p|j^\nu|k\rangle} {4\,e(\mathbf{k})\,e_\pi(\mathbf{p}_\pi)}\;\times\;\cdots$$5 and variance $$W^{\mu\nu}_{1N1\pi}(q)\;=\;\sum_{s_k,t_k}\int\!\frac{d^3k}{(2\pi)^3}\,dE\; S_{t_k}(\mathbf{k},E)\, \frac{\langle k|{j^\mu}^\dagger|\pi p\rangle \langle \pi p|j^\nu|k\rangle} {4\,e(\mathbf{k})\,e_\pi(\mathbf{p}_\pi)}\;\times\;\cdots$$6, which are then used for posterior updates and experimental design.

Hypothesis selection follows Information-Directed Sampling. At round $$W^{\mu\nu}_{1N1\pi}(q)\;=\;\sum_{s_k,t_k}\int\!\frac{d^3k}{(2\pi)^3}\,dE\; S_{t_k}(\mathbf{k},E)\, \frac{\langle k|{j^\mu}^\dagger|\pi p\rangle \langle \pi p|j^\nu|k\rangle} {4\,e(\mathbf{k})\,e_\pi(\mathbf{p}_\pi)}\;\times\;\cdots$$7, PiEvo defines expected regret

$$W^{\mu\nu}_{1N1\pi}(q)\;=\;\sum_{s_k,t_k}\int\!\frac{d^3k}{(2\pi)^3}\,dE\; S_{t_k}(\mathbf{k},E)\, \frac{\langle k|{j^\mu}^\dagger|\pi p\rangle \langle \pi p|j^\nu|k\rangle} {4\,e(\mathbf{k})\,e_\pi(\mathbf{p}_\pi)}\;\times\;\cdots$$8

and information gain

$$W^{\mu\nu}_{1N1\pi}(q)\;=\;\sum_{s_k,t_k}\int\!\frac{d^3k}{(2\pi)^3}\,dE\; S_{t_k}(\mathbf{k},E)\, \frac{\langle k|{j^\mu}^\dagger|\pi p\rangle \langle \pi p|j^\nu|k\rangle} {4\,e(\mathbf{k})\,e_\pi(\mathbf{p}_\pi)}\;\times\;\cdots$$9

The chosen hypothesis satisfies

$L_{\mu\nu}$00

In practice, $L_{\mu\nu}$01 is approximated with the BALD estimator using GP predictive variances, while a warm-up phase instead maximizes total predictive variance $L_{\mu\nu}$02. The stated conceptual shift is from exploring hypotheses inside a fixed worldview to evolving the worldview itself.

6. Anomaly-driven augmentation, guarantees, and empirical performance

PiEvo’s defining mechanism is anomaly-driven principle augmentation. When the current Maximum-A-Posteriori principle fails to explain new data, the framework computes the anomaly score

$L_{\mu\nu}$03

flags observations with $L_{\mu\nu}$04, and, if anomaly count exceeds a threshold, prompts the Principle Agent to propose a new principle $L_{\mu\nu}$05, after which

$L_{\mu\nu}$06

The new principle inherits its prior from the universal prior, and its GP expert is trained on all past observations by back-filling (Pu et al., 6 Feb 2026).

The algorithmic loop consists of four stages repeated over the budget $L_{\mu\nu}$07: anomaly detection and augmentation, posterior update, hypothesis generation by the Hypothesis Agent with IDS, and execution by the Experiment Agent. This decomposition formalizes the paper’s central claim that anomalies should be treated as drivers of epistemic growth.

The theoretical analysis states an informal regret guarantee. Under assumptions of coherent augmentation, identifiability, and a well-calibrated LLM generator, PiEvo achieves

$L_{\mu\nu}$08

so the expected cumulative regret is sublinear in $L_{\mu\nu}$09, and the posterior converges on $L_{\mu\nu}$10. The comparison with PiFlow is expressed in terms of entropy of the learning space: PiFlow has regret $L_{\mu\nu}$11, while PiEvo has $L_{\mu\nu}$12, with $L_{\mu\nu}$13. The proposed efficiency gain thus derives from optimizing over a compact principle space rather than a large hypothesis space.

Empirically, the framework is evaluated on four benchmarks: Nanomaterial Optical (NHO, 7-D, $L_{\mu\nu}$14–factor up to 2.0), Molecular Bio-activity (MBO, pChEMBL $L_{\mu\nu}$15), Superconductor $L_{\mu\nu}$16 (SPO, 5 Cu-oxide systems, reference 298.5 K), and Transition Metal Complex (TMC, 4-ligand polarizability up to 500). The reported LLM backbones are Qwen3-32B (No-Thinking) and Gemini-2.5-Flash (No-Thinking), and the reported metrics are Solution Quality, Average Pairwise Distance, and Area Under Optimization Curve. Table 1 gives Qwen3-32B Solution Quality values of 57.6/58.2/28.7/64.0 for Vanilla MAS, 96.1/79.7/34.0/70.3 for PiFlow, and 149.1/96.4/37.3/80.5 for PiEvo across MBO/NHO/SPO/TMC, with averages 52.1, 70.0, and 90.8, respectively. The paper further states that PiEvo achieves average solution quality of up to 90.81%~93.15%, representing a 29.7%~31.1% improvement over the state-of-the-art, and attains an 83.3% speedup in convergence step.

The ablations isolate the contributions of the two central mechanisms. Disabling IDS reduces NHO-SQ to 84.0%; disabling principle evolution caps NHO-SQ at 85.9%; and hyperparameter sensitivity with respect to anomaly threshold, noise $L_{\mu\nu}$17, and warm-up length is reported as stable over wide ranges. The case study on nanohelix chirality reports that over $L_{\mu\nu}$18 experiments PiEvo autonomously uncovers the toroidal–electric quadrupole interference mechanism gated by helix angle and skin-depth constraints, with validation by FDTD simulation. In the vocabulary of the paper, this is offered as evidence that principle evolution can yield de novo theory development rather than only accelerated search.

7. Comparative significance of the two PiEvo usages

The Achilles PiEvo and the principle-evolvable PiEvo address different failure modes. In the nuclear setting, the central problem is physical consistency across scales: the electroweak pion-production vertex, bound-state correlations, and final-state interactions must be modeled coherently if exclusive and semi-inclusive observables are to be reproduced. In the scientific-agent setting, the central problem is epistemic rigidity: fixed priors and a static hypothesis space can waste experiments and suppress theory revision when anomalies occur (Isaacson et al., 26 Aug 2025, Pu et al., 6 Feb 2026).

Their methodological signatures are likewise distinct. Achilles PiEvo is built from DCC amplitudes, realistic hole spectral functions, GFMC or measured nuclear configurations, an intranuclear cascade, and either an optical-potential or explicit-$L_{\mu\nu}$19 absorption treatment. Scientific-discovery PiEvo is built from Bayesian posteriors over principles, per-principle GP experts, IDS, BALD-based information estimation, and anomaly-triggered expansion of the active principle set.

What unites them is not subject matter but architecture: each PiEvo is organized around a latent structure that must itself be updated rather than treated as fixed. In Achilles, that latent structure is the coupled medium-modified history of a pion and any intermediate resonances inside a nucleus. In the scientific-agent framework, it is the active set of principles that define the agent’s current worldview. This suggests a broader interpretive point: “PiEvo” functions as a label for systems in which evolution of the underlying explanatory substrate is the primary design principle, but the specific substrate, equations, and validation criteria are entirely domain-dependent.