Simjecture 框架：语言模型智能体以反例驱动完成 Z-pinch 辐射能超额研究

\preprintsty@sw

Bowen Zhu Affiliation: State Key Laboratory of Electrical Insulation and Power Equipment, Xi’an Jiaotong University, Xi’an 710049, China    Jian Wu Email: jxjawj@mail.xjtu.edu.cn Affiliation: State Key Laboratory of Electrical Insulation and Power Equipment, Xi’an Jiaotong University, Xi’an 710049, China

November 7, 2026

Abstract

Recent breakthroughs in artificial-intelligence-assisted mathematics show that open-ended machine search can outperform human mathematicians, yet transferring this paradigm to physics requires refactoring the research workflow. We introduce Simjecture, an open-source framework in which language-model agents pursue human-defined physical conjectures. The framework tackles two central challenges in this transfer: first, because physics is empirical and no model is final, it structures inquiry around counterexample-driven falsification and prospective hypothesis repair; second, because physical theories must interact with the real world, it uses computational physics as an executable testbed while separating the qualification of numerical instruments from scientific claim review so solvers and diagnostics can be adapted within a recorded investigation. We apply the framework in a 136.3 h investigation to tackle the long-standing problem of radiated energy exceeding the imploding-shell kinetic energy in a Z-pinch. The investigation reveals competing effects of energy conversion, retention, and release timing. Stronger coherent helical deformation increases resistive conversion while reducing radiation and increasing retained energy. In a matched fine-grid test, broadening the trailing-plasma profile delays the later radiation maximum by 1.80 ns and recovers an early 153 J bolometric deficit to within 0.08% by 44 ns, with a spectral response that shifts during the pulse. These results provide new insight into energy conversion in high-energy-density plasma physics and demonstrate a new paradigm for physics research.

I Introduction

I.1 From mathematical discovery to computational physics

Recent breakthroughs in artificial intelligence (AI)-assisted mathematics have demonstrated the ability of AI to fill out and even penetrate beyond the convex hull of human knowledge. Over a span of months in 2026, automated search helped construct planar point configurations with at least n1+δ unit-distance pairs (δ>0) that disproved Erdős’s longstanding n1+o⁡(1) unit-distance conjecture [1], uncovered a three-dimensional polynomial counterexample to the Jacobian conjecture whose constant nonzero Jacobian determinant and non-injectivity can be checked directly [2, 10], and extended to mathematical physics through a reported finite-time velocity blowup for the three-dimensional forced incompressible Navier–Stokes equations accompanied by Lean formalizations and broader proof catalogues [24, 25, 26]. These developments share a common characteristic: AI excels at tasks with a strong asymmetry between discovery and verification, where finding a decisive construction requires expansive search while checking a concrete candidate gives an independent verifier a well-defined task.

Translating this paradigm to physics introduces two additional obligations. First, physics is empirical and no model is final: it is extremely difficult to prove a theory universally right, whereas finding a counterexample is much simpler. Second, physical theories must ultimately answer to nature through observation and experiment; however, laboratory campaigns are costly and rarely repeatable under identical microscopic conditions. Counterexample-driven computational physics investigations therefore provide an executable model world in which the driving motive is simple and controlled interventions can be repeated and dissected.

Yet every computational physics investigation rests on two intertwined foundations: a mathematical and numerical core that integrates the chosen equations numerically, and a physical core that selects the governing equations, closures, and boundary conditions used to represent the real system. When a simulation appears to violate a physical conjecture, that discrepancy can originate either from genuine model physics or from numerical effects. Only after the mathematical and numerical core is shown to be reliable can a contradiction falsify a physical claim, motivate a revised hypothesis with new testable predictions, and guide laboratory experiments. Simjecture reflects this duality directly by coupling two distinct workflows: scientific investigation, which tests and revises physical conjectures, and instrument development, which builds, modifies, and qualifies the numerical solvers and diagnostics required for those tests. The framework delegates solver engineering, numerical qualification, and controlled physical comparisons so agents can revise solvers and diagnostics as an investigation develops.

I.2 Autonomous research with humans in the loop

Much of the recent wave of autonomous research in machine learning instead treats inquiry as end-to-end manuscript generation. Karpathy’s autoresearch exemplifies the metric-driven loop across repeated language-model training runs [15], while The AI Scientist automates idea generation, coding, plotting, and paper writing [19]. These systems arrive as conference and preprint ecosystems are already strained: NeurIPS main-track submissions more than tripled from 9,467 in 2020 to 30,709 in 2026 [23, 22], and arXiv introduced per-author monthly submission caps and stricter moderation rules in 2026 after monthly submissions reached 40,363 amid a surge of low-quality automated preprints [3, 4].

End-to-end manuscript automation carries three fundamental scientific risks. First, an agent can game a convenient proxy without touching the underlying physical mechanism. Second, research ideas selected autonomously by AI agents rarely hold up as good scientific taste once executed [29]. Third, automated pipelines tend to gravitate toward narrow, closed-loop studies that leave little room for future improvement or lasting scientific value.

As demonstrated in mathematics and recent physics-focused agentic systems for optics, scattering, accelerator control, and model exploration [16, 17, 21, 20], automated workflows have proved most illuminating when anchored to well-posed domain questions and conjectures. We therefore argue that scientific taste, namely judging which physical conjecture merits sustained compute and what evidence would genuinely change our physical picture, remains a human responsibility. Accordingly, an autonomous research system requires an open-box design so that its decisions remain understandable to human scientists, and its scientific reports must be re-examined by human researchers to resolve disputed or inconclusive results. Human steering should be an integral part of the scientific research system: the goal of an autonomous research system should not be to remove humans from the loop, but to design for effective human–AI collaboration.

II Research framework

II.1 Conjectures, decision rules, and minimal repair

To investigate a physical conjecture efficiently, the system first translates the natural-language question into an explicit mathematical statement. A conjecture H=(M,𝒟,Q,P) fixes a physical model M specifying the governing equations and closures, a domain 𝒟 of admissible simulation setups x together with the physical regime in which the claim is meant to hold, a target observable Q, and a predicted range P⁡(x) for that observable, whether testing an internal prediction of M or agreement with experimental data. Solved exactly with model M, the ideal conjecture asserts

A simulation, however, never returns the exact value Q⁡(x). It measures Q through a numerical instrument I comprising the solver, grid, and diagnostic scripts, and after mesh-refinement, timestep, and diagnostic controls the instrument returns a measurement range IQ​(x). Along the instrument route, the numerical setup I is commissioned and refined through convergence, conservation, and calibration tests tailored to Q. If a capability is missing or a numerical control fails so that IQ​(x) cannot be trusted, the instrument must be revised and requalified (I→I′) before any scientific claim is evaluated; the physical conjecture H remains open. Once the instrument is qualified, a setup x⋆∈𝒟 is an accepted counterexample only if this entire range lies outside the prediction:

Comparing IQ​(x) against P⁡(x) across the tested setups yields one of three outcomes:

• •

  Falsified: a setup x⋆∈𝒟 satisfies Eq. (2), establishing a counterexample that disproves H.

• •

  Supported within the tested scope: every tested setup x∈𝒟 satisfies IQ​(x)⊆P​(x), supporting H across the sampled regime.

• •

  Unresolved: either IQ​(x⋆) straddles the boundary of P⁡(x⋆), or a numerically verified observable combines multiple physical processes without isolating the mechanism named in H. When the evidence leaves H unresolved, the investigation may seek a diagnostic that isolates the quantity named in H, or use the findings to formulate a related hypothesis with a distinct observable and a prediction specified before its test; testing that hypothesis advances the investigation while leaving the status of H unchanged.

Along the scientific route, when H is falsified, a revised hypothesis H′=(M′,𝒟′,Q,P′) can be formulated. Simply widening P′​(x) to cover observed extrema or deleting the failed regime from 𝒟′ is trivially true without adding physical insight; minimal repair therefore requires an explicit physical justification for every changed assumption, scope restriction, or bound in (M,𝒟,P) while preserving unaffected predictions and accounting for the parent counterexamples. Before testing H′, a prospective commitment fixes the revised prediction P′​(x), domain 𝒟′, and planned test cases: only fresh runs executed after this commitment can validate H′, while the parent conjecture H and its counterexamples remain permanently recorded in the hypothesis tree even if candidate repairs subsequently fail.

II.2 Research roles and scientific tools

Simjecture implements these two review routes through the roles shown in Fig. 1. The human researcher defines the question and scope. A director plans campaign strategy, monitors scientific coverage and compute budget, adjusts operational limits and helper-worker allocation, and decides when to stop stalled runs, replan, or call on the human scientist for steering, while an autonomous worker configures simulations, inspects fields, writes code, and searches for counterexamples. Every worker submission along the instrument (orange) and scientific (teal) routes is gated by independent reviewers operating in isolated model contexts. The methods reviewer checks diagnostic definitions, conservation accounting, and mesh or timestep controls to verify that the instrument measures the target observable. The claim reviewer then examines whether the conclusion follows within the declared scope and whether proposed repairs survive prospective tests.

Guided by tool-specific skills, the worker can select and combine installed tools as the physical question requires, including using one solver’s output as input to another. Public demonstrations [34] illustrate both a Qin-1 hardware-design study, in which a magnetic field computed with Elmer is transferred to WarpX [32] as a static background for a hybrid particle-in-cell calculation, and a fluid-to-kinetic study, in which a FLASH current-sheet snapshot initializes a local WarpX calculation. Because computational campaigns on a given facility also build cumulatively across studies, related investigations share a persistent workspace that indexes project instructions, solver documentation, input configurations, prior conversations, and completed study reports. Both human researchers and autonomous agents query this library via retrieval-augmented generation [18]; revision-tracked citations and frozen launch contexts keep cross-study reuse subject to the current study’s evidence contract.

II.3 Open-box execution, minimal evidence harness, and research supervision

Beyond combining existing tools, resolving a computational physics question often requires modifying the instrument itself. Simjecture uses an open-box design in which the worker can inspect raw multi-dimensional fields, write new diagnostics, and modify initializers or physics modules as the physical question evolves. The system automatically records every code modification, input configuration, and simulation output so that the reviewer can audit whether a change alters only numerical settings, tests a new setup within 𝒟, redefines a diagnostic, or modifies the governing model M itself.

To keep open-box execution auditable while leaving the worker free to design experiments and use native coding tools, a minimal host harness enforces three service calls: lab.run() executes a calculation, lab.commit() registers a prospective repair plan, and lab.review() requests an independent verdict. At study launch, the host freezes the baseline evidence contract, permitted instruments, resource budget, and completion policy: answer closes once the root conjecture or a qualified repair is supported or falsified, whereas repair requires a validated replacement whenever the root conjecture is falsified. For each local or remote job, lab.run() snapshots the submitted source tree, input declarations, command bindings, and runtime identities before execution and records the output cryptographic hashes after the job finishes. Reference solver installations and human-supplied commissioning files remain read-only while the worker develops and qualifies code changes on an isolated branch. Because the worker has no interface path to approve its own claims, lab.review() verifies artifact hashes, hypothesis lineage, and post-commitment timestamps before passing the evidence packet to an isolated reviewer. During multi-day campaigns, the director can replan or seek human steering at stage boundaries, while a small supervisory policy, Conjecture 1 [33], can route routine status checks without deciding the scientific verdict.

III Validation study: reconnection-rate scaling

We first apply the framework to a controlled question in two-dimensional compressible resistive magnetohydrodynamics (MHD) island coalescence in FLASH [9]. The human researcher supplies the physical hypothesis, a working FLASH application, and initial diagnostics, while the worker selects the test points and qualifies the measurements. The conjecture predicts that the local log-slope of the reconnection rate remains within [−0.55,−0.45] around the Sweet–Parker Sη−1/2 reference for resolved, pre-plasmoid sheets across the nominal Lundquist-number range Sη=L0​vA​0/η=1/η∈[250,4000], where η is the magnetic diffusivity and the reference length and Alfvén speed satisfy L0=vA​0=1 in normalized code units. The rate is measured across a fixed interval of reconnected magnetic flux, with exact definitions and numerical controls detailed in Appendix A.1.

An exploratory 1282 scan indicates that the reconnection rate falls more slowly with Sη than the reference scaling, and the prospectively declared largest-departure rule selects the contrast Sη=1000→2000 for targeted qualification. Because a stable integrated flux rate can coexist with an under-resolved local current sheet as diffusivity decreases, the worker audits current-sheet profiles and magnetic topology at matched reconnected flux alongside mesh, timestep, sampling-cadence, and independent flux-reconstruction controls. When an intermediate refinement step (2562→3842 at Sη=2000) increases the rate shift, methods review blocks claim evaluation and requests a finer 5122 calculation; only after the instrument passes review does the worker execute fresh production trajectories.

The fresh 3842 production runs yield a local scaling exponent peff=−0.3778 with an empirical numerical sensitivity interval [−0.4314,−0.3247] (Fig. 2(g)), which lies entirely above −0.45 and falsifies the proposed scaling band [Eq. (2)]. Throughout the measurement window, the magnetic-null inventory retains a single X point and two O points in every saved field snapshot, confirming the pre-plasmoid regime at the resolved scales, and independent claim review reproduces the rates and error propagation from the recorded outputs to accept the counterexample.

After falsifying the [−0.55,−0.45] band, the worker proposes two candidate repairs: a transient sheet-width model driven by the measured inflow strain and upstream magnetic field, and an amplitude-budget extension. Without an adjustable rate prefactor, both models underpredict the anchor rates by 25%–31% and fail their prospective calibration checks. The workflow therefore records both failed repair attempts and closes the study with a verified counterexample and an open theoretical question; a companion FLASH-to-WarpX fluid-to-kinetic current-sheet study in the archived repository [34] illustrates the complementary outcome in which a bounded hypothesis is supported across numerical and seed controls.

\preprintsty@sw

IV Investigation of radiated-energy excess in a Z-pinch

We now apply Simjecture to a central open problem in pulsed-power plasma physics: determining how much of the radiated X-ray energy excess above the implosion kinetic energy in a stagnating wire-array Z-pinch is supplied by magnetic reconnection. Because the reconnection share is unknown a priori, we pose an initial working conjecture that non-axisymmetric magnetic reconnection directly accounts for at most 10% of the positive radiation excess above a reference implosion kinetic energy, using this quantitative threshold to launch a four-stage, 136.3 h human–AI collaborative investigation in three-dimensional radiation-MHD. On the fine mesh, total Ohmic heating reaches 12.74% of the radiation excess. However, the volume-integrated ηe​J2 diagnostic combines reconnecting and non-reconnecting dissipation, leaving the 10% conjecture unresolved. Testing this conjecture nonetheless reveals how magnetic conversion, energy retention, and release timing compete. Stronger coherent helical deformation boosts three-dimensional resistive conversion but weakens radial compression and traps more magnetic and thermal energy. Conversely, broadening the trailing-plasma profile shunts current outward and delays the stagnation radiation peak before late emission recovers the early bolometric deficit.

IV.1 Physical background and model

Fast wire-array Z-pinches convert pulsed electromagnetic energy into imploding plasma motion, thermal energy, and intense soft and K-shell X-ray bursts [28, 11]. Across both multi-megampere facilities and 1 MA university-scale drivers, the radiated X-ray energy frequently exceeds the maximum instantaneous bulk kinetic energy of the imploding load [7, 13], reflecting that instantaneous kinetic energy captures only a transient stage of energy transfer in a continuously driven system. As the core stagnates, magneto-Rayleigh–Taylor (m=0) and kink (m=1) instabilities redistribute current between the dense column and surrounding trailing plasma [8, 6], while Lorentz-force work, compression, shocks, rapid cross-field ion heating [30], and resistive dissipation contribute simultaneously to the energy budget [27].

To anchor the model in a well-diagnosed regime, we follow an eight-wire aluminum array on the 1 MA Zebra pulsed-power generator: 15​μ​m wire diameter, 20 mm axial length, and 76.34​μ​g total mass [13]. In this 1 MA regime, laser shadowgraphy, Faraday rotation, X-ray diodes, and three-dimensional simulations directly resolve the coexistence of a dense stagnating core, trailing plasma, and current redistribution, while aluminum (Z=13) provides well-characterized equations of state and opacities alongside a distinct >1 keV hard-band emission channel above sub-keV thermal radiation. To study late implosion and radiative stagnation without modeling early cold-wire ablation, the calculation begins from a prescribed pre-ablated coaxial distribution: an inward-moving Gaussian core shell centered at 0.8 mm with a 100​km/s speed parameter weighted by the local core fraction, surrounded by a nearly stationary trailing annulus centered at 2.5 mm with equal nominal mass fractions. This post-ablation parameterization isolates stagnation-phase conversion and current shunting from early wire-ablation uncertainties while allowing the column geometry and trailing-plasma breadth to be varied independently at fixed mass and drive.

FLASH 4.8 [9] evolves three-temperature (Ti,Te,Trad) resistive MHD coupled to tabulated aluminum equations of state and opacities, electron–ion exchange, thermal conduction, implicit magnetic diffusion, and 30-group flux-limited radiation diffusion. Three-dimensional Cartesian calculations and axisymmetric r–z references use a constant 1 MA drive, periodic axial boundaries, and transverse circuit/open boundaries through which multigroup radiation escapes, with numerical schemes, domain geometry, and baseline profile parameters detailed in Appendix A.2.

IV.2 Campaign progression overview

Because the investigation spans dozens of baseline, screening, nonlinear, and diagnostic-validation runs across two geometries and three initializer generations, we introduce a compact naming convention to track individual calculations across the timeline and tables. Each label encodes the domain geometry, transverse resolution, and seed family: 3D-N gives the cell count N along each transverse Cartesian axis, whereas RZ-N gives the radial cell count in axisymmetric cylindrical geometry. For three-dimensional runs, Lp denotes an initial density-modulation amplitude of p%, while S and M denote the Single-helix and Multimode initializers, where the suffix d/w specifies a centerline displacement D=0.1​d mm, defined by the root-mean-square (RMS) amplitude Drms for M, and a trailing-plasma radial width σt=0.1​w mm. Appending -P marks a short pilot or control run used to qualify an initializer, verify a diagnostic, or test execution settings. A parenthesized Roman numeral identifies the campaign stage for repeated physical setups. Full mesh dimensions and nonlinear parameters are tabulated in Appendix A.2.

Prior to autonomous execution, the workspace was seeded with two unperturbed commissioning baselines (RZ-64 and 3D-32-L0), a partial RZ-256 checkpoint, a Message Passing Interface (MPI) domain-decomposition benchmark, and baseline diagnostic scripts. The investigation progressed through four experimental stages over 136.3 h in two execution periods (Fig. 3(a)–(d)). The timeline accumulates their recorded durations, omitting the pause between Stages III and IV.

Stage I: Density-seed screening. In Stage I (Fig. 3(a)), the worker first ran short timing pilots (3D-64-L5-P, 3D-96-L5-P) to gauge computational cost and then executed four principal three-dimensional screening trajectories across mesh size and seed amplitude alongside the axisymmetric continuation RZ-256. Despite refining the mesh and raising the density perturbation to 10%, the pinch column remained virtually straight through stagnation (compare 3D-96-L0 and 3D-96-L10 in Fig. 3(a)), yielding negligible non-axisymmetric Ohmic heating and almost unchanged bolometric radiation. Two 3D-96-L5-P checkpoint-replay pilots served as diagnostic checks that compared Ohmic heating reconstructed from the electron resistivity ηe and current density J=|𝐉| as ηe​J2 against the native magnetic-diffusion ledger and prompted an extended cycle of harmonic and current-sheet diagnostic scripts. At the end of this stage, because the pinch column remained overly stable with minimal three-dimensional deformation, Human Steering #1 directed the agent to consult the Z-pinch instability literature and construct a physically disruptive three-dimensional initialization.

Stage II: Single-helix tests. Opening Stage II (Fig. 3(b)), the worker developed the Single-helix initializer to displace the initial column centerline into a coherent m=1 helix coupled with m=0 axial necking, so that magnetic tension and hoop forces act directly on a bent current channel. In parallel, the first literature synthesis (Lit. #1) motivated tracking both sausage and kink modes and separating initial geometric offset from dynamical amplification. Before launching production trajectories, the agent used two pilot jobs (3D-32-S0/4-P and 3D-64-S4/4-P) for instrument and workflow qualification to verify baseline equivalence and short-window integration stability. Steering #2 then noted that three-dimensional unsplit corner-transport upwind integration runs stably at a Courant–Friedrichs–Lewy number of CFL=0.8, accelerating integration fourfold. The resulting principal trajectories (3D-128-S4/4, the coarse resolution control 3D-64-S4/4, and the doubled-amplitude run 3D-128-S8/4) produced pronounced nonlinear helical kinking and beaded dense cores (Fig. 3(b), right). Physically, comparing 3D-128-S8/4 against 3D-128-S4/4 revealed a striking trend: doubling the initial helical displacement reduced total radiated X-ray energy while leaving more magnetic and thermal energy stored in the plasma. While 3D-128-S8/4 was still executing, Steering #3 and a second literature review (Lit. #2) noted that experimental wire-array implosions develop multi-wavelength axial structure embedded in ablated trailing plasma, prompting the worker to prepare the next solver generation in parallel.

Stage III: Multimode comparisons. Built and qualified during the second half of Panel (b) and executed in Stage III (Fig. 3(c)), the Multimode initializer superposed three axial wavelengths with mixed helicities and exposed the trailing-plasma radial width σt as an explicit control parameter. Before the main Stage III runs, 3D-64-M0/4-P verified baseline recovery, while 3D-32-M4/10-P1 and P2 tested two-step initialization and checkpoint handoff. The worker then launched the concurrent coarse pair 3D-64-M4/4 and 3D-64-M4/10 to isolate the effect of broadening the trailing plasma; when a wrapper-level retention issue interrupted the end of their initial 32 ns segment, the agent ran 3D-32-M4/4-P to verify the output-retention fix before launching the fine multimode trajectory 3D-128-M4/4 alongside the two late-time coarse continuations. The resulting density fields (Fig. 3(c), right) and radiation histories showed that both multimode seeding and trailing-plasma broadening reshape the timing and hard-X-ray (>1 keV) spectrum more strongly than the bolometric total. Steering #4 initiated a comprehensive audit of the completed Stage I–III trajectories. The resulting physical synthesis in Secs. IV.3 and IV.4 motivated the next experimental stage.

Stage IV: Timing and numerical controls. The current redistribution and late radiative recovery in Stage III suggested testing an intermediate trailing width, σt=0.7 mm, against a matched 0.4 mm reference on the fine mesh, with an expected ∼2–3 ns delay and subsequent radiative recovery (Fig. 3(d)). This phase extended the physical inquiry to magnetic conversion, retention, and delayed release, using the completed trajectories and their interpretation as starting material. The worker evolved the fine pair through 44 ns and found that absolute late-time thermal inventories remained sensitive to the timestep despite stable integration. It therefore branched checkpoint controls at progressively smaller timesteps and audited the radiation diagnostics. These tests established local contraction of the timestep response, while the retained physical trajectories allowed the intermediate-width timing prediction to be examined in detail.

IV.3 Radiation excess and energy conversion

Figure 4 tracks the Stage IV three-dimensional multimode implosion 3D-128-M4/4. Magnetic hoop and tension forces gather the imploding shell into a helically bent filament. Its fastest-converging segments produce an initial radiation flash at 5.56 ns as current shifts outward into the lower-density corona. Overall radial compression then continues until the centroid-relative radius reaches its 0–32 ns minimum at 25.27 ns, by which point 87.4% of the 0–32 ns radiation yield has already escaped. Near this compression minimum, the longitudinal y=0 cut intersects the continuously connected three-dimensional column as a chain of radiatively cooled dense patches near Te∼30 eV. Immediately outside this cool core, azimuthal magnetic fields, Ohmic dissipation, and stored radiation concentrate in a lower-density peripheral sheath around a 0.4–4.0 keV hot spot, before the column expands into larger helical excursions by 44.01 ns.

Across this stagnation sequence, the radiation yield exceeds the shell kinetic reference by a large margin. Over the fixed 3.149–25.029 ns evaluation window, 3D-128-M4/4 radiates 3.086 kJ compared with a tagged-load kinetic energy of Kref=0.498 kJ evaluated at 9.776 ns, the last retained field sample before the declared 10 ns reference time. The radiation yield is 6.20 times Kref, giving an excess of Eexcess=2.588 kJ.

The magnetic reservoir supplies most of the net decrease in stored energy over this interval. Separating the tagged aluminum load from the low-density numerical background, we track the load kinetic energy KL and thermal energy Ug,L=Ue,L+Ui,L, together with magnetic and radiation-field energies integrated over the whole domain. Stored magnetic energy EB falls by 2.498 kJ, load kinetic energy falls by 0.389 kJ, and load thermal energy falls by 24.6 J. Together with stored radiation energy Uγ and escaped radiation energy Eγ,escape, these changes enter the balance

where Δ denotes the change across the stated interval. Stored radiation Uγ changes by less than 1 J, and the inferred net exchange with the load and fields is ℬL=0.174 kJ, as defined in Appendix B.

To test the proposed reconnection contribution on this fixed interval W across the domain volume V, we compare the reconstructed resistive conversion QΩ​(W) with the radiation excess Eexcess​(W),

On the fine mesh, the field reconstruction gives QΩ=0.330 kJ, agreeing with the native diffusion increment and reaching 12.74% of Eexcess.

Across the broader 0–32 ns common observation window, the local conversion pathways follow from the magnetic-energy equation,

where μ0 is the vacuum permeability, 𝐒 is the Poynting flux, and 𝐯 is the plasma velocity. Magnetic forces drive motion that heats the plasma through compression and shocks, while resistive dissipation transfers magnetic energy directly into thermal energy. Figure 5 places magnetic, aluminum-load kinetic, aluminum-load thermal, and stored-radiation energies alongside cumulative radiation escape over 0–32 ns in 3D-128-M4/4. Combining the initial load-and-field energy with the ℬL​(T)=0.485 kJ net exchange inferred through T=32 ns defines a fixed budget of 4.777 kJ, where the gray band ℬL​(T)−ℬL​(t) tracks the remaining net input before 32 ns. As the magnetic reservoir shrinks, the imploding load decelerates through stagnation and its kinetic energy drops to 0.124 kJ at 32 ns, leaving 1.084 kJ stored in the load and fields, including 0.192 kJ of load thermal energy, while 3.694 kJ has escaped as radiation.

Comparing the matched single-helix runs 3D-128-S4/4 and 3D-128-S8/4 over 0–32 ns further shows that higher resistive conversion does not increase the radiated yield. Doubling the initial helical displacement from 0.4 to 0.8 mm raises the 32 ns RMS slice-centroid offset from 0.86 to 1.46 mm and increases the centroid-relative RMS radius a⟂ from 0.686 to 0.956 mm. Stronger helical kinking thus weakens local radial compression and leaves an elongated, lower-density core. Consequently, although the native diffusion-stage gas-energy increment rises from 503 to 622 J, snapshot-integrated magnetic-force work decreases from approximately 3.35 to 3.02 kJ, escaping radiation falls by 197 J, and final whole-domain stored energy rises by 182 J, with combined whole-domain magnetic and thermal storage increasing by 209 J.

These energy budgets establish three conclusions. First, the radiation excess draws on a magnetic energy reservoir much larger than the instantaneous shell kinetic energy, as magnetic forces and resistive dissipation continuously convert magnetic energy while photons escape. Second, although volume-integrated Ohmic heating reaches 12.74% of the radiation excess on the fine mesh, this total cannot be attributed to magnetic reconnection alone because ηe​J2 includes non-reconnecting resistive diffusion and reconnection can also energize bulk flows [27]; isolating the reconnection share requires dedicated topology diagnostics and finer mesh convergence. Third, resistive heating, mechanical compression work, and magnetic and thermal storage evolve together, so increasing Ohmic dissipation while weakening transverse compression reduces the net radiated X-ray energy. Quantitative definitions of the geometric metrics, operator-level energy ledgers, resolution and window sensitivities, and work integrals across all cases are consolidated in Appendices A.2 and A.3.

IV.4 Instability and radiative response

Figure 6 compares how initial instabilities and radial mass profiles reshape the radiation pulse across the campaign. In the unperturbed column and the linear density-seed runs (Fig. 6(a)–(c)), the 5% and 10% density modulations leave both the total radiated energy and the photon spectrum above 1 keV nearly unchanged from the unperturbed reference. Across this baseline group, 78% of the >1 keV emission escapes during the initial 0–8 ns transient, followed by weak hard emission through 24 ns.

Centerline deformation. Introducing explicit centerline deformations produces a stronger early pulse and sustains hard X-ray emission across stagnation (Fig. 6(d)–(f)). Increasing the single-helix displacement from 0.4 to 0.8 mm weakens the first power peak, delays the main stagnation peak by about 1 ns, and lowers the 0–32 ns bolometric yield by 5.34% while changing the >1 keV yield by only 2.17%. Replacing the single 0.4 mm helix with the multimode perturbation breaks the uniform helical pitch into uneven bends and variable filament thickness, exciting strong higher azimuthal harmonics such as m=3 at late time. This geometric fragmentation leaves the bolometric yield within 1.07% of the single-helix run yet reduces the >1 keV yield by 7.08%, with the matched 642×32 pair showing the same pattern of bolometric invariance within 0.18% alongside a 4.74% hard-band reduction. Single-mode displacement therefore lowers total radiated energy mainly by weakening bulk compression, whereas multimode fragmentation selectively softens the high-energy spectrum.

Trailing-plasma width. Broadening the trailing annulus redistributes axial current outward and delays radiation release, as shown for the fine-mesh pair σt=0.4 and 0.7 mm (Figs. 6(g),(h) and 7). In the Stage III coarse screen from σt=0.4 to 1.0 mm, broadening the profile reduced the 12–24 ns inner-current fraction from 0.536 to 0.402, delayed the stagnation power peak by 4.58 ns, and recovered 60% of the 32 ns bolometric deficit by 44 ns while lowering the cumulative >1 keV yield by 6.21%. Testing the intermediate width σt=0.7 mm on the fine mesh in Stage IV reproduces the outward current redistribution and delayed radiation release: between 12 and 24 ns, the mean signed axial-current fraction within 2 mm of the column centroid falls from 0.535 to 0.460 (Fig. 7(a)), delaying the rise of the >300 eV tagged load by 1.64 ns (Fig. 7(b)). The measured stagnation-peak delay was 1.80 ns, slightly below the anticipated 2–3 ns range, and the predicted late radiative recovery was observed (Fig. 7(c)). Specifically, the bolometric deficit reaches 153 J near 19.3 ns, narrows to 27 J at 32 ns, and ends 3.4 J (0.08%) above the 4.15 kJ reference at 44 ns after the wider profile emits 30.49 J more over 32–44 ns at CFL=0.4 and 27.87 J more at CFL=0.2. Meanwhile, the >1 keV fraction shifts from 7.38% versus 10.10% over 8–16 ns to 9.98% versus 7.79% over 16–24 ns, leaving cumulative hard emission within 1.35%. This sequence of distributed current, delayed core heating, and late radiative catch-up parallels Zebra experiments in which trailing plasma diverts current around dense pinch necks before imploding to supply late emission [13, 12, 14].

Together, these comparisons separate three physical effects that single-scalar yield metrics conflate: coherent helical displacement governs bulk compression and energy retention, multimode axial fragmentation softens the >1 keV spectrum, and trailing-plasma breadth controls current shunting and radiation release timing, as detailed with the Stage III coarse histories in Appendix A.4.

IV.5 System behavior

Across 136.3 h in two execution periods, the investigation recorded 1,196 computational executions: 974 succeeded, 194 failed, and 28 were cancelled, alongside 131 independent method reviews across four experimental stages. The agents proved effective at sustained numerical engineering and single-pass verification: they diagnosed a silent three-dimensional transport failure, cut three-dimensional runtime by 62.9% through MPI decomposition and radiation-solver tuning, implemented and verified the single-helix and multimode Fortran initializers, cross-checked reconstructed diagnostics against native solver ledgers, and executed fine-mesh timestep-contraction controls while preserving all trajectories. Conversely, when single-scalar tests encountered coupled multi-mechanism physics, four brief human steering interventions supplied the physical taste defined in Sec. I to shift the regime from linear seeds to helical kinking, multimode fragmentation, and trailing-plasma current shunting. Within those steered regimes, the agents uncovered the inverse trend between helical Ohmic heating and radiation yield in Stage II, isolated the trailing-plasma delay in Stage III, and executed the prospective fine-grid timing test in Stage IV, after which author analysis reconstructed the full field and load-versus-background energy histories.

Stages I–III ran on a 16-core host using a DeepSeek Flash worker with GPT-6.1 Sol as director and reviewers, whereas Stage IV ran 32-rank fine-grid calculations with GPT-6.1 Terra as worker, GPT-6 Astra as director, GPT-6.1 Sol as reviewer, and the Conjecture 1 coprocessor providing live supervisory routing. Table 1 reports the combined token usage across all four stages after deduplicating cumulative conversation receipts by role and thread; Appendix A details the stage configurations.

V Summary and outlook

This work presents Simjecture, an autonomous research framework that centers simulation campaigns on falsification, counterexample search, and prospective hypothesis repair. Separating gated instrument qualification from scientific claim review allows language-model agents to adapt solvers, physical closures, and diagnostics during a study without confounding numerical artifacts with physical mechanisms. In the two-dimensional resistive-MHD validation study, this workflow autonomously isolated and verified a counterexample to a Sweet–Parker reconnection-rate scaling conjecture. In the 136.3 h three-dimensional aluminum Z-pinch investigation, combining autonomous multi-resolution execution with human steering and analysis showed that stronger coherent helical deformation weakens transverse compression and leaves more magnetic and thermal energy stored despite larger resistive conversion, whereas broadening the trailing-plasma profile shunts axial current early and delays core assembly. In the fine-grid intermediate-width comparison, additional late emission nearly closes the early bolometric deficit by 44 ns, with little change in cumulative hard X-ray yield.

The recorded study trajectories also provide a testbed for evaluating scientific agents on end-to-end research tasks. Existing benchmarks such as SciCode and ScienceAgentBench assess isolated code generation or static data-analysis pipelines [31, 5]. Constructing replayable diagnostic benchmarks from recorded simulation histories showed that raw numerical coding does not guarantee completed scientific delivery. Models frequently wrote scripts that passed held-out numerical accuracy checks yet still failed the research contract by misreading solver unit conventions, dropping provenance bindings, or exhausting wall-clock budgets before finalizing required verification artifacts. Evaluating agents against recorded simulation stages with independent re-execution separates three capabilities that static coding tests conflate: numerical correctness, scientific judgment when deciding whether a discrepancy survives refinement, and disciplined delivery under fixed resource limits.

A second direction we explored is selective supervision during multi-day investigations. In long campaigns, invoking frontier director or reviewer models on a fixed schedule is token-intensive, yet unmonitored runs can waste hours in late status-polling loops or stalled solver stages. To test whether a small model can route attention more economically, we trained and integrated a compact 4​B-parameter decision policy, released separately as Conjecture 1 [33], that reads compressed workflow state and chooses whether to keep observing quietly, trigger a read-only diagnostic check, or wake the director or review role while leaving scientific approval to the host gates. Across initial workflow trials, this separation showed that lightweight policies can handle routine wakeups and bounded checks, while reward design must balance timely fault recovery against unnecessary interruptions during quiet simulation windows.

Developing the system also required comparing a prescribed classical harness against a minimal harness. In our initial classical mode, the harness enforced a rigid schedule of mandatory plan reviews, stage transitions, and explicit transport bookkeeping before the worker could advance. Development trials showed that this strong harness frequently trapped adequate evidence inside administrative stages and prevented the agent from completing multi-case simulation matrices before the deadline, whereas an unharnessed agent finished the runs quickly but exited without a verified report or prospective repair check. We therefore retired the classical workflow in favor of minimal mode, in which the worker freely owns its experimental plan and native coding tools while the host enforces only durable provenance, source-bound methods review, and guarded claim closure.

These lessons define our next steps in both plasma physics and autonomous research design. Anchoring the three-dimensional model to 1 MA Zebra aluminum parameters reproduces key experimental mechanisms, including magnetic-reservoir dominance, helical hot-spot localization, and trailing-plasma current shunting that delays stagnation emission. At the same time, the present calculations have two main physical and modeling limitations. First, the prescribed post-ablation initialization and constant 1 MA drive omit early wire ablation and generator-circuit coupling. Second, single-fluid Eulerian MHD cannot evolve a true vacuum or capture kinetic reconnection physics. The calculation therefore requires a 10−3​kg/m3 background density floor that accumulates significant late-time energy, while single-fluid resistivity conflates reconnecting sheets with background diffusion. Addressing these limits motivates coupled ablation and circuit models, floor-density and vacuum-resistivity controls, and fluid-to-kinetic handoff into particle-in-cell solvers. On the system side, we will extend the minimal evidence harness to additional simulation-gated physics domains while using recorded campaign trajectories to improve agent scientific taste and workflow supervision. We welcome community contributions of scientific conjectures, solver integrations, and benchmark tasks.

Code and data availability

The Simjecture source code, study records, reduced energy and radiation ledgers, and figure-reproduction scripts are available in the archived software release [34]; raw three-dimensional simulation checkpoint arrays are available from the corresponding author upon reasonable request.

Acknowledgements.

We thank Yuanbo Lu for insightful discussions on Z-pinch energy accounting and numerical vacuum treatment.

Appendix A Study configurations and numerical controls

Table 2 lists the model configurations and supplied starting material across the two investigations, including the role changes in Z-pinch Stage IV. Reasoning effort settings apply to each role, and every director or reviewer call runs in an isolated context. The reconnection study uses Simjecture v0.7.0, with full source and runtime identities retained in the study records.

A.1 Reconnection-rate measurement and qualification

Using the initial-state-subtracted reconnected flux ψrec normalized by B0​L0 and time t normalized by τA=L0/vA​0, the worker measures the reconnection rate across ψrec∈[0.01,0.05] and evaluates the pairwise scaling exponent for Sa<Sb:

Following the exploratory 1282 scan across Sη∈{250,500,1000,2000,4000} and the qualification controls described in Sec. III, the fresh 3842 production runs yield R⁡(1000)=0.0496886 and R⁡(2000)=0.0382421 for peff=−0.3778 in Fig. 2(g). Combining the 3842→5122 spatial-refinement shift and 5122 uncertainty at Sη=2000 with 2562 halved-timestep, crossing-bracket, and orthogonal flux-reconstruction sensitivities produces empirical numerical allowances of 1.31% at Sη=1000 and 2.39% at Sη=2000, which propagate to peff∈[−0.4314,−0.3247]. Positivity and net conservation hold across the window.

A.2 Z-pinch initial conditions and diagnostics

The FLASH calculations use a second-order Harten–Lax–van Leer discontinuities Riemann solver with three-dimensional corner-transport upwind integration and 30 photon groups spanning 0.1 eV–100 keV on an 18×18×20​mm3 Cartesian domain. The fine 1282×64 and coarse 642×32 meshes have transverse cell sizes of 0.141 and 0.281 mm, and the baseline radial widths of the core and trailing Gaussian annuli are 0.2 and 0.4 mm.

The single-helix initializer combines a 5 mm-wavelength centerline displacement with a 60% axial density modulation. The multimode initializer combines wavelengths 5, 10, and 2.5 mm with Euclidean-normalized density weights 0.7,0.2,0.1 and displacement weights 0.6,0.3,0.1, using fixed phases and mixed helicities. Table 3 lists the four Stage I screening runs, the six Stage II–III nonlinear displacement cases, and the Stage IV fine pair, which ran at CFL=0.4 with a radiation linear-solver relative tolerance of 10−10 and a 4×4×2 MPI decomposition. Initial tagged masses are 76.3408 and 76.3412​μ​g, respectively. After startup, field snapshots are retained approximately every 0.25 ns.

The centroid-relative RMS radius is

where 𝐫¯⟂​(z) is the weighted centroid of each axial slice. Both fine single-helix runs begin at a⟂=1.935 mm. At 32 ns, the maximum slice-centroid offsets are 2.16 and 1.28 mm for 3D-128-S8/4 and 3D-128-S4/4, with mid-plane density maxima of 10.8 and 18.3​kg/m3, peak Ohmic power densities of 0.87 and 2.19​GW/mm3, and peak stored radiation energy densities of 2.32 and 6.03​mJ/mm3. Across the Stage II–III nonlinear scan, the primary power maximum occurs at 4.65–5.54 ns. In the Stage III 3D-128-M4/4 run, mid-plane azimuthal-mean magnetic partitioning between 3 and 9 ns shows the enclosed-current fraction in dense plasma with ρ>0.10​ρmax dropping from 44% to 8% as the 90% enclosed-current radius r90 reaches 5.6 mm. The Stage IV longitudinal cut at 24.03 ns in Fig. 4 reaches ρmax=33.6​kg/m3, |𝐁|max=192 T, and peak Ohmic power density qΩ=3.98​GW/mm3. In 3D-64-M4/4, late-time mass-weighted azimuthal harmonic amplitudes satisfy A3=1.93​A1, and the coarse trailing-width pair has 32 ns RMS centroid offsets of 0.43 and 0.49 mm for the broader and narrower profiles.

A.3 Energy accounting and numerical sensitivity

The energy inventories separate kinetic, magnetic, electron, ion, and stored radiation components, subtracting stored radiation from FLASH’s combined internal-energy variable to avoid double counting. Table 4 records the common 0–32 ns comparison, with endpoint inventories interpolated between native end-of-step records and escaping photon energy integrated over exact timestep overlaps, where photon groups 21–30 span 1–100 keV. Across the shared 3.149–25.029 ns stagnation interval of the Stage III 3D-128-M4/4 run, the independently recorded native hydro and diffusion increments sum to 3.245650 kJ, matching radiation escape plus the net increase of gas and stored radiation energy to within 2×10−5 J. The whole-domain cumulative balance is ℬ⁡(t)=Estored​(t)−Estored​(0)+Eγ,escape​(0,t), combining net boundary exchange with numerical contributions. Appendix B relates these inventories to the load-and-field partition in Fig. 5.

The Stage IV reference in Figs. 4 and 5 uses this same 3.149–25.029 ns evaluation window; its compression minimum occurs at 25.27 ns. Over the fixed window, reconstructed resistive conversion is 329.78 J versus a native diffusion-stage gas-energy increment of 330.16 J, a 0.12% difference; retaining every second field snapshot changes the reconstruction by 0.0074%. The higher output cadence places the kinetic-reference sample at 9.776 ns, while the Stage III resolution comparisons keep their original reference samples. In the Stage IV per-rank hydrodynamic logs, the recorded hydrodynamic energy adjustment contributes +61.55 J over the stagnation window and +13.01 J over 0–32 ns in 3D-128-M4/4 (IV), compared with +58.88 and +10.64 J in 3D-128-M4/7. Summing the conservative update with this adjustment reproduces the native hydrodynamic energy change with a summed absolute discrepancy below 7×10−5 J over each 0–44 ns history.

In the Stage III resolution audit, reconstructed Ohmic conversion agrees with the native diffusion increment to 2.1×10−4 relative on the fine mesh, with 86.9% inside r<2 mm and 87.0% of the summed absolute fine–coarse mesh difference inside 1 mm. While refining from 642×32 to 1282×64 shifts absolute yields toward harder emission by resolving hotter pinched segments, matched comparisons at fixed resolution preserve the sign and ordering of both the single-to-multimode spectral softening and the trailing-width delay across both grids (Table 3). In a separate window scan on the extended 46.04 ns fine trajectory that uses whole-domain kinetic energy immediately before the hydro update at each window’s leading edge, power thresholds f∈{0.75,0.50,0.25,0.20,0.10} yield excesses of 31.4 J, 318.8 J, 3.033 kJ, 3.283 kJ, and 3.684 kJ. Over 0–32 ns, the native hydro-stage net gas-energy increments for 3D-128-S4/4 and 3D-128-S8/4 are 3.719 and 3.518 kJ, and all snapshot magnetic and kinetic inventories agree with the native histories within 2×10−6 relative tolerance. In the Stage IV fine-pair controls, continuing from 32 to 44 ns at CFL=0.2 gives 27.87 J more radiation in the wider case versus 30.49 J at CFL=0.4, while local checkpoint continuations from 29.52 to 32.5 ns at CFL=0.2, 0.1, and 0.05 contract the 32 ns gas-plus-radiation energy shift from 15.22 to 7.79 J in the narrow case and from 13.52 to 6.98 J in the wider case, measuring the local timestep sensitivity of the evolved gas-plus-radiation inventories. Across both full Stage IV histories, the stepwise decrease of gas-plus-radiation energy during each radiation update matches the recorded escaping radiation with a maximum residual of 2.4​μ​J and a summed absolute residual below 7​μ​J.

A.4 Retained current and radiation histories

The author analysis uses 565 retained three-dimensional field snapshots from seven nonlinear trajectories: 219 in Stages II–III and 346 in Stage IV, together with their complete radiation ledgers and the Stage I screening ledgers in Table 3. Every native radiation step in the original 0–32 ns extracts is reproduced by the longer records without gaps or overlaps, and each escaped group energy equals its power times the native timestep. The common continuation comparison stops at 44 ns, before the shorter trailing-profile trajectory ends. Spectrograms in Fig. 6 retain the original 30 photon groups without smoothing, where color represents group power divided by logarithmic energy width, gray marks spectral power below 0.01 GW per logarithmic interval, and integrated spectra use exact timestep overlaps with group centers connected directly. The >1 keV sum includes both continuum and line emission, and each nonlinear case has a single interior local power maximum between 10 and 32 ns with prominence above 10 GW. Table 5 lists the interval yields and hard fractions for the Stage III and IV trailing-width pairs.

Axial current in Fig. 7(a) is reconstructed from Jz=(∂xBy−∂yBx)/μ0 using native-grid central differences and the verified FLASH magnetic-field normalization. At each axial slice, the numerator of the inner-current fraction integrates signed Jz within 2 mm of that slice’s density-weighted centroid; the denominator integrates within 8 mm of the original axis. Currents are averaged over axial slices before taking their ratio. For the Stage III pair, the narrow and broad profiles enclose mean inner currents of 364 and 304 kA over 12–24 ns, giving signed fractions of 0.536 and 0.402; counting only forward current gives 0.544 and 0.421. Density-defined cores at 5%, 10%, and 20% of each slice’s peak density give the same ordering.

The hot-load diagnostic in Fig. 7(b) integrates tracer-weighted mass where Te>300 eV and ρ≥0.01​kg/m3. For the Stage III pair, following the initial transient, the first retained snapshot after 10 ns with more than 0.1% of the load above that threshold occurs at 13.11 ns in the narrow case and 18.02 ns in the broad case, with preceding samples at 12.01 and 17.01 ns, while the interior stagnation power maximum shifts from 18.31 to 22.89 ns, a delay of 4.58 ns. The Stage III broad profile has a 158.6 J bolometric deficit at 32 ns and recovers 60% of that deficit by 44 ns, leaving a 1.37% total-yield difference and a 6.21% lower cumulative >1 keV yield.

For the Stage IV fine-grid pair with σt=0.4 and 0.7 mm across 173 unique field times per case, the first post-10 ns crossing at which 0.1% of tagged mass exceeds 300 eV lies in [12.53,12.77] and [14.02,14.28] ns, giving a linearly interpolated delay of 1.64 ns. The 500 eV crossing is delayed by 1.73 ns, whereas the 1 keV crossing occurs 1.26 ns earlier in the wider case. Minimum centroid-relative radii of 0.514 and 0.492 mm occur at 25.27 and 27.03 ns. Applying the fixed peak rule to 0.1 ns radiation bins with 0.5 ns centered smoothing places the stagnation power maxima at 17.55 and 19.35 ns, a 1.80 ns delay that is unchanged across 0.2 ns bins and 0.3 or 0.7 ns smoothing, with unsmoothed 0–44 ns yields of 4.1483 and 4.1517 kJ and >1 keV yields of 440.8 and 446.7 J.

Appendix B Numerical vacuum treatment and energy accounting

The Eulerian MHD setup represents the exterior vacuum region with a low-density floor material, vacu, of mass fraction Xv. The initial density combines the core shell and trailing annulus, which share the load tracer and total 76.34​μ​g, with a uniform 10−3​kg/m3 background contributing 6.48​μ​g across the 18×18×20​mm3 domain. In the single-fluid update, all species share the fluid velocity and electron and ion temperatures, while the background follows an ideal-gas equation of state with γe=γi=5/3, molar mass Av=26.9815385​g/mol, fixed charge Zv=13, zero opacity and emissivity, and no vacuum-resistivity override.

The background material energies follow from the recorded mass fraction and its equation of state,

where R is the universal gas constant. Subtracting the background components separately for electrons and ions gives the additive load inventories KL=K−Kv and Ug,L=Ug−Uv across all 173 retained field snapshots per Stage IV trajectory. Reconstructed background thermal energies agree with native specific energies in nearly pure background cells to within 3.2×10−6 relative, and interpolating the background integrals to the native ledger cadence changes the nominal 32 ns kinetic, electron, and ion background inventories by less than 0.04 J when subsampled at every second snapshot.

Figure 8 shows the whole-domain budget and the separate material histories. In the nominal Stage IV run, background kinetic energy reaches 0.454 kJ and background thermal energy reaches 0.461 kJ at 32 ns, while the load retains 0.124 and 0.192 kJ, respectively. By 44 ns, the background material holds 1.476 kJ in total. Subtracting Ev=Kv+Uv defines the retained inventory and its net exchange,

Magnetic and radiation-field energies retain their full spatial integrals, including the nominal vacuum region. The gray band in Fig. 5 is therefore the remaining net exchange with the retained load and fields; energy accumulated in background material enters the excluded part of that exchange. At 32 ns, this gives ℬL=0.485 kJ and the fixed budget EL​(0)+ℬL​(32​ns)=4.777 kJ. The whole-domain inventories in Table 4, the operator audits in Appendix A.3, and the volume-integrated QΩ retain their original definitions. Both Stage IV trailing-width runs share this identical background floor, while quantifying sensitivity to the floor density and vacuum resistivity requires separate numerical controls.

\preprintsty@sw

References

• [1] N. Alon, T. F. Bloom, W. T. Gowers, D. Litt, W. Sawin, A. Shankar, J. Tsimerman, V. Wang, and M. M. Wood (2026) Remarks on the disproof of the unit distance conjecture. External Links: Document, https://arxiv.org/abs, 2605.20695, Link Cited by: §I.1.
• [2] (2026) Discovering cryptographic weaknesses with Claude. Note: Research announcement, 28 JulyAccessed 25 October 2026 External Links: Link Cited by: §I.1.
• [3] arXiv (2025) 2025 annual report. Note: Annual reportAccessed 16 October 2026 External Links: Link Cited by: §I.2.
• [4] arXiv (2026) Fair moderation, equitable access, and AI: arXiv’s updated rate limit policy. Note: arXiv Blog, 1 OctoberAccessed 31 October 2026 External Links: Link Cited by: §I.2.
• [5] Z. Chen, S. Chen, Y. Ning, Q. Zhang, B. Wang, B. Yu, Y. Li, Z. Liao, C. Wei, Z. Lu, V. Dey, M. Xue, F. N. Baker, B. Burns, D. Adu-Ampratwum, X. Huang, X. Ning, S. Gao, Y. Su, and H. Sun (2025) ScienceAgentBench: toward rigorous assessment of language agents for data-driven scientific discovery. In Proceedings of the Thirteenth International Conference on Learning Representations (ICLR), pp. 96934–96990. External Links: Link Cited by: §V.
• [6] J. P. Chittenden, S. V. Lebedev, C. A. Jennings, S. N. Bland, and A. Ciardi (2004) X-ray generation mechanisms in three-dimensional simulations of wire array Z-pinches. Plasma Phys. Control. Fusion 46, pp. B457–B476. External Links: Document Cited by: §IV.1.
• [7] M. E. Cuneo, E. M. Waisman, S. V. Lebedev, J. P. Chittenden, W. A. Stygar, G. A. Chandler, R. A. Vesey, E. P. Yu, T. J. Nash, D. E. Bliss, G. S. Sarkisov, T. C. Wagoner, G. R. Bennett, D. B. Sinars, J. L. Porter, W. W. Simpson, L. E. Ruggles, D. F. Wenger, C. J. Garasi, B. V. Oliver, R. A. Aragon, W. E. Fowler, M. C. Hettrick, G. C. Idzorek, D. Johnson, K. Keller, S. E. Lazier, J. S. McGurn, T. A. Mehlhorn, T. Moore, D. S. Nielsen, J. Pyle, S. Speas, K. W. Struve, and J. A. Torres (2005) Characteristics and scaling of tungsten-wire-array z-pinch implosion dynamics at 20 MA. Phys. Rev. E 71, pp. 046406. External Links: Document Cited by: §IV.1.
• [8] F. S. Felber and N. Rostoker (1981) Kink and displacement instabilities in imploding wire arrays. Phys. Fluids 24, pp. 1049–1055. External Links: Document Cited by: §IV.1.
• [9] B. Fryxell, K. Olson, P. Ricker, F. X. Timmes, M. Zingale, D. Q. Lamb, P. MacNeice, R. Rosner, J. W. Truran, and H. Tufo (2000) FLASH: an adaptive mesh hydrodynamics code for modeling astrophysical thermonuclear flashes. Astrophys. J. Suppl. Ser. 131, pp. 273–334. External Links: Document, Link Cited by: §III, §IV.1.
• [10] S. Gao (2026) Counterexamples to the Jacobian conjecture in dimensions greater than two. External Links: Document, https://arxiv.org/abs, 2608.00222, Link Cited by: §I.1.
• [11] M. G. Haines (2011) A review of the dense Z-pinch. Plasma Phys. Control. Fusion 53, pp. 093001. External Links: Document Cited by: §IV.1.
• [12] V. V. Ivanov, A. A. Anderson, D. Papp, A. L. Astanovitskiy, V. Nalajala, and O. Dmitriev (2015) Study of magnetic fields and current in the Z pinch at stagnation. Phys. Plasmas 22, pp. 092710. External Links: Document Cited by: §IV.4.
• [13] V. V. Ivanov, A. A. Anderson, D. Papp, A. L. Astanovitskiy, B. R. Talbot, J. P. Chittenden, and N. Niasse (2013) Current redistribution and generation of kinetic energy in the stagnated Z pinch. Phys. Rev. E 88, pp. 013108. External Links: Document Cited by: §IV.1, §IV.1, §IV.4.
• [14] V. V. Ivanov, A. A. Anderson, and D. Papp (2019) Investigation of wire-array Z-pinches by laser probing diagnostics. Matter Radiat. Extremes 4, pp. 017401. External Links: Document Cited by: §IV.4.
• [15] A. Karpathy (2026) Autoresearch. Note: Software repositorySource revision 228791f External Links: Link Cited by: §I.2.
• [16] M. Krenn, J. S. Kottmann, N. Tischler, and A. Aspuru-Guzik (2021) Conceptual understanding through efficient automated design of quantum optical experiments. Phys. Rev. X 11, pp. 031044. External Links: Document, Link Cited by: §I.2.
• [17] J. Landgraf, V. Peano, and F. Marquardt (2025) Automated discovery of coupled-mode setups. Phys. Rev. X 15, pp. 021038. External Links: Document, Link Cited by: §I.2.
• [18] P. Lewis, E. Perez, A. Piktus, F. Petroni, V. Karpukhin, N. Goyal, H. Küttler, M. Lewis, W. Yih, T. Rocktäschel, S. Riedel, and D. Kiela (2020) Retrieval-augmented generation for knowledge-intensive NLP tasks. In Advances in Neural Information Processing Systems, Vol. 33, pp. 9459–9474. External Links: Link Cited by: §II.2.
• [19] C. Lu, C. Lu, R. T. Lange, Y. Yamada, S. Hu, J. Foerster, D. Ha, and J. Clune (2026) Towards end-to-end automation of AI research. Nature 651, pp. 914–919. External Links: Document, Link Cited by: §I.2.
• [20] M. Nägele and F. Marquardt (2026) Agentic exploration of physics models. Phys. Rev. X 16, pp. 031002. External Links: Document, Link Cited by: §I.2.
• [21] D. Neudecker, T. E. Cutler, M. Devlin, P. Brain, N. Gibson, M. J. Grosskopf, M. W. Herman, J. Hutchinson, T. Kawano, A. Khatiwada, N. Kleedtke, E. Leal-Cidoncha, R. C. Little, A. E. Lovell, A. Stamatopoulos, E. C. Thompson, S. A. Vander Wiel, and E. Williamson (2025) Machine learning to select experiments driven by fundamental science and applications for targeted nuclear data improvement. Phys. Rev. X 15, pp. 021086. External Links: Document, Link Cited by: §I.2.
• [22] NeurIPS 2026 Program Chairs (2026) NeurIPS 2026 main track submission and acceptance statistics. Note: Fortieth Conference on Neural Information Processing Systems (NeurIPS 2026)30,709 valid main-track submissions; accessed 31 October 2026 External Links: Link Cited by: §I.2.
• [23] NeurIPS Program Committee Chairs (2025) Reflections on the 2025 review process from the program committee chairs. Note: NeurIPS Blog, 30 SeptemberAccessed 16 October 2026 External Links: Link Cited by: §I.2.
• [24] OpenAI (2026) Finite time blowup for Navier–Stokes. Note: Research manuscript, 8 SeptemberAccessed 25 October 2026 External Links: Link Cited by: §I.1.
• [25] OpenAI (2026) NavierStokesAndEuler: Lean certificates accompanying Navier–Stokes and Euler results. Note: Software repositorySource revision f9e8bc5 External Links: Link Cited by: §I.1.
• [26] OpenAI (2026) OpenAI research catalog: mathematical manuscripts and proof artifacts. Note: Research repository, 6 OctoberSource revision 1581d4c External Links: Link Cited by: §I.1.
• [27] D. I. Pontin and E. R. Priest (2022) Magnetic reconnection: MHD theory and modelling. Living Rev. Sol. Phys. 19, pp. 1. External Links: Document Cited by: §IV.1, §IV.3.
• [28] D. D. Ryutov, M. S. Derzon, and M. K. Matzen (2000) The physics of fast Z pinches. Rev. Mod. Phys. 72, pp. 167–223. External Links: Document Cited by: §IV.1.
• [29] C. Si, T. Hashimoto, and D. Yang (2025) The ideation-execution gap: execution outcomes of LLM-generated versus human research ideas. External Links: Document, https://arxiv.org/abs, 2506.20803, Link Cited by: §I.2.
• [30] C. Stollberg, E. Kroupp, D. Mikitchuk, P. Sharma, V. Bernshtam, M. Cvejić, R. Doron, E. Stambulchik, Y. Maron, A. Fruchtman, I. E. Ochs, N. J. Fisch, and U. Shumlak (2023) Observation of fast anomalous ion heating across the magnetic field in a Z-pinch plasma. Phys. Rev. Lett. 130, pp. 205101. External Links: Document Cited by: §IV.1.
• [31] M. Tian, L. Gao, S. D. Zhang, X. Chen, C. Fan, X. Guo, R. Haas, P. Ji, K. Krongchon, Y. Li, S. Liu, D. Luo, Y. Ma, H. Tong, K. Trinh, C. Tian, Z. Wang, B. Wu, Y. Xiong, S. Yin, M. Zhu, K. Lieret, Y. Lu, G. Liu, Y. Du, T. Tao, O. Press, J. Callan, E. Huerta, and H. Peng (2024) SciCode: a research coding benchmark curated by scientists. In Advances in Neural Information Processing Systems, Vol. 37, pp. 30624–30650. External Links: Document, Link Cited by: §V.
• [32] J.-L. Vay, A. Almgren, J. Bell, L. Ge, D. P. Grote, M. Hogan, O. Kononenko, R. Lehe, A. Myers, C. Ng, J. Park, R. Ryne, O. Shapoval, M. Thévenet, and W. Zhang (2018) Warp-X: a new exascale computing platform for beam–plasma simulations. Nucl. Instrum. Methods Phys. Res. A 909, pp. 476–479. External Links: Document, Link Cited by: §II.2.
• [33] B. Zhu (2026) Conjecture 1. Note: Hugging Face, experimental model releaseVersion 0.1 External Links: Link Cited by: §II.3, §V.
• [34] B. Zhu (2026) Simjecture. Note: Zenodo, software releaseVersion 0.7.0 External Links: Document, Link Cited by: §II.2, §III, Code and data availability.

注意转载授权 · 阅读原文