A Kinetic Theory of the Gated Self-Evolving LLM Agent

1 INTRODUCTION

• The kinetic description is scoped to plugin-version instances whose DSH architecture has four properties: permutation symmetry, reversibility, acyclicity, and typed contracts.The authors state that a monolithic self-modifying agent lacks these properties and falls outside the theory's scope.Source: §1 · p. 1
• The validation gate acts as an external Maxwell daemon, turning diffusion into optimization while yielding a convergence certificate and statistical signatures.The contribution list identifies the gate as an external selection operator in a master equation over the plugin × version × task grid.Source: §1 · p. 2
• The theory provides an any-time hitting-time certificate, a held-out validation-budget resolution law, and a separation theorem keeping the evaluator outside the population.Measurements use a faithful minimal instance rather than production-scale training, and the authors do not claim dynamical equivalence.Source: §1 · p. 2
• The large plugin population exhibits density-gated fluctuation scaling and non-equilibrium steady-state relaxation as directional evidence.The signatures are invisible at sparse edit density and emerge at the predicted rate under tripled density.Source: §1 · p. 2

Kinetic and physics-flavored theories

• Jung and Asnacios (2026) share the population-level kinetic perspective, while this paper distinguishes its external gate, shared retrieval-pool re-collision, and production-runtime measurements.Mean-field MARL uses mean field computationally, whereas this theory treats it as a statistical description of gated, non-optimizing skill instances.Source: §2 · p. 2

Self-evolving agents

• SkillOpt (Yang et al., 2026) supplies the skill-state, bounded-edit, strict-gate loop, while Reflexion (Shinn et al., 2023) and the G¨odel Agent lack external statistical control.RE-TRAC (Zhu et al., 2026) and Resource2Skill (Fan et al., 2026) are described as orthogonal approaches.Source: §2 · p. 2

Statistical gating

• Anytime-valid statistics address when to stop safely, while this paper focuses on finite-set resolution and multi-candidate selection within a population-level kinetic theory.Planners proposing 5–8 pipelines per round break the single-candidate assumption, requiring the Vovk–Wang merge.Source: §2 · pp. 2–3

3 OVERVIEW

• The diagram traces edits to plugin cards, admission by an external validation gate, and a coarse-grained continuum with density, flow, and temperature.The plugin instances are presented as hard spheres, and the gate as a Maxwell daemon outside the gas.Source: §3 · p. 3
• The right-hand instruments measure predicted −1/2 fluctuation scaling and non-equilibrium steady-state relaxation as directional evidence.The overview frames the fluid interpretation as a bounded analogy.Source: §3 · p. 3
• The overview treats each fluid element as axiom-licensed rather than as a literal claim about fluid dynamics.The figure's continuum picture uses density ρ(x, v, e, t), flow u, and temperature T.Source: §3 · p. 3

4.1 STATE SPACE AND THE KINETIC EQUATION

• Given A1–A4, the population is a multiset of plugin-version instances on a plugin × version × task grid, with density governed by a four-operator kinetic equation.The four architectural properties are permutation symmetry, mount/unmount reversibility, acyclic dependencies, and typed contracts.Source: §4.1 · pp. 3–4

4.2 THE FOUR OPERATORS AND THE EFFECTIVE POPULATION

• Co-activation defines collision, mount/unmount/materialization defines reaction, the task distribution supplies the external field, and the validation gate selectively dissipates rejected edits.The gate retains candidates only when their held-out score strictly improves; all optimization direction comes from this projection.Source: §4.1 · p. 4, §4.2 · p. 5

4.3 DYNAMICS AND STATISTICS OF THE GATED GAS

• Under the stated proposal, validation, and deployment assumptions, the hitting-time certificate bounds expected rounds to a prescribed improvement, with realized drift checkable from run logs.The log counterpart is acceptance rate times mean accepted gain; late saturation is attributed to the resolution floor rather than skill.Source: §4.3 · p. 5
• Under i.i.d. initial conditions and a bounded collision kernel with no dominating instance, the population converges to a deterministic nonlinear Boltzmann-type kinetic equation.The derivation uses propagation of chaos in the McKean–Vlasov sense.Source: §4.3 · p. 5, §4.2 · p. 5
• A plateau is either proposal exhaustion or information exhaustion, distinguishable from logged data; under summable autocorrelation, the predicted fluctuation exponent is β = −1/2.The shared database can violate the autocorrelation-summability condition.Source: §4.3 · pp. 5–6
• The frozen validation set must remain separate from the evolving population for the hitting-time certificate; merging them voids its drift premise, though non-separated designs may still converge.The gate measures strictly improving post-states on the set it evaluates, so an evolving evaluation set makes acceptance self-referential.Source: §4.3 · p. 6

4.4 THERMODYNAMICS AND TRANSPORT

• With the gate active, reward drive and credit dissipation balance at a non-equilibrium steady state, and escaping a plateau requires changing system parameters.The proposed changes include temperature, task field, or gate strictness.Source: §4.4 · p. 7
• The fluid analogy is bounded: total impulse is not conserved, so the theory claims no Navier–Stokes closure or literal fluid phenomena.Transport coefficients and regime names denote measurable quantities or analogies, not derived constants.Source: §4.4 · p. 7

5 EXPERIMENTS

• The probes test theory signatures on a minimal WebShop loop, using within-design controls rather than external-method comparisons.The setup uses 1,000 products, 6,910 synthetic-attribute goals, and one public LLM API; the faithful minimal instance maps operators to the loop by architectural role.Source: §5 · p. 7, §5.1 · p. 7

5.1 THE FAITHFUL MINIMAL INSTANCE

• The minimal instance uses four-parameter plugin vectors, tag-based retrieval, bounded schema-validated edits, and a strict held-out gate; the protocol fixes population sizes, seeds, retrieval settings, and sweeps before runs.The gate accepts only when the candidate’s score exceeds the incumbent’s on the same held-out set; ties are rejected.Source: §5.1 · pp. 7–8

5.2 FLUCTUATION SCALING IN POPULATION SIZE

• With the population-size baseline, the fitted fluctuation slope is flat at ˆβ = −0.04, so the signature is not detectable at this power.Per-arm standard deviations are 0.044 → 0.025 → 0.037 for N=5/50/500; the fitted slope has r2 = 0.12.Source: §5.2 · p. 8
• At tripled edit density, the measured slope is ˆβ = −0.58 versus the predicted −1/2; at sparse density, it is flat at ˆβ = −0.04.The authors describe this as directional evidence; Figure 6 plots per-arm standard deviations of two seed finals.Source: §5.2 · p. 8, Figure 6 · p. 9
• Fan-3 retrieval reaches mean final 0.495 versus 0.301 for fan-1, a net gain of +0.19 attributed to co-activation.The fan-3 arm credits the best of three instances.Source: §5.2 · p. 8
• The scale probes observe NESS relaxation, while the 24 skill groups traverse only about 10% of the expected hitting interval and late training is limited by the resolution floor.The NESS discriminator fires at r2 = 0.80 < 0.9; the earlier “entrance >500 rounds” bound is retired.Source: §5.2 · p. 8
• Across pre-registered probes, six of eight L1 signatures are supported or observed, but the probed L2 physics-name candidates fail as observables and the Magnus correlation licenses no circulation claim.The supported/borderline/unsupported triage is post-hoc over pre-registered observables and thresholds.Source: §5.2 · pp. 8–9

6 CONCLUSION AND DISCUSSION

• The theory combines a machine-verified hitting-time certificate, measured signatures with a license inventory, and an analogy boundary set by the conservation ledger.The authors position it as a way to price validation budgets, flag silent collapse, and identify designs outside its coverage.Source: §6 · p. 9
• The measurements use two seeds per arm, one model, and one task family, whereas the certificate is instance-independent.The paper reports density-gated scaling under the corrected instrument, with CLT scaling emerging at tripled edit density as directional evidence.Source: §6 · p. 9
• A hosted reflector may drift between versions; released artifacts remain the ground truth for reported numbers, and drivers can be rerun against compatible endpoints.Experiments use CPU, MiniMax-M3, and WebShop, with device code, drivers, raw artifacts, and Lean sources released.Source: §6 · p. 9

A LEAN VERIFICATION RECORD

• The hitting-time certificate’s expectation bound and hitting-time measurability are machine-checked, with 12 theorems and two auxiliary lemmas.The released Lean verification record reports a full green build.Source: A LEAN VERIFICATION RECORD · p. 12
• The geometric autocorrelation limit (1 + ϕ)/(1 − ϕ) for fluctuation scaling is Lean-verified.Source: A LEAN VERIFICATION RECORD · p. 12
• Under the strict gate, the gate score is monotone.The record attributes this to Theorem 5 of the reservoir.Source: A LEAN VERIFICATION RECORD · p. 12
• The formalized reservoir establishes separation necessity: the certificate is conditional on the evaluator being separable from the evolving population.The necessity direction is cited from reservoir Theorem 7.Source: A LEAN VERIFICATION RECORD · p. 12

A.1 PROOF OF THEOREM 4.1

• Under the stated improving-edit and strict held-out gate assumptions, the proof uses a per-round expected gap reduction of at least pmin∆ to bound the expected rounds to threshold.The deployed skill follows the latest accepted version, while rejected proposals change nothing.Source: §A.1 · p. 12

A.2 THEOREM 4.5 IN THE FORMALIZED RESERVOIR

• If the measured validation set overlaps the evolving population, registered improvements need not track the out-of-sample objective, voiding the certificate’s premises.The formalized reservoir proves this necessity direction.Source: §A.2 · p. 12

A.3 DERIVATION ROUTE FOR PROPOSITION 4.4

• With Neff effectively independent contribution streams and absolutely summable autocorrelation, the summed score has Green–Kubo variance and predicts β = −1/2 on log-log.The geometric-autocorrelation special case is verified in the released Lean sources.Source: §A.3 · p. 13

A.4 REMARK ON PROPOSITION 4.3 (EXHAUSTIVENESS)

• Under the strict gate, persistent acceptances coincide with δgate > 0, making the proposed middle case empty and the dichotomy exhaustive.Positive mean gain remains within the δgate > 0 clause even when gains approach zero.Source: §A.4 · p. 13

• The evaluation uses a reproducible WebShop text-mode simulator with 1,000 products and synthetically generated task goals.Human-instruction goals are not used.Source: B DEVICE SPECIFICATION · p. 13
• Each plugin is a four-parameter vector, and each version instance also stores its tag profile, version number, and usage history.The parameters are search rank, read policy, price floor, and price ceiling.Source: B DEVICE SPECIFICATION · p. 13
• Retrieval samples from the top-k instances, weighting each by exp(−α d_i); k and α control the collision radius.Distance is based on task-query words and instance tags.Source: B DEVICE SPECIFICATION · p. 13
• The reflector proposes up to L single-field edits from per-instance reward and usage evidence; invalid proposals are discarded, and a deterministic heuristic handles LLM failures.Source: B DEVICE SPECIFICATION · p. 13
• The gate replaces the candidate library only when its mean reward on 20 held-out episodes strictly exceeds the incumbent; ties are rejected.Rejected candidates leave the population unchanged, while their evidence informs the next reflection.Source: B DEVICE SPECIFICATION · p. 13

C.3 CLOSURE REPLAY (PREDICTION 3)

• Under the corrected gate, waterline deviations were at most 0.024, and the earlier large first-burst concentration was mostly attributed to the train–validation baseline gap.The replay reports near-additivity, rather than the pre-correction deviations of up to 0.32.Source: §C.3 · p. 14

C.4 GOODHART PROBE (PREDICTION 4)

• Gate deltas averaged +0.0064 while fresh 60-task independent-set deltas averaged −0.0486, the reported Goodhart-precursor pattern.The authors attribute the opposite signs to selection-set inflation rather than evaluation noise or task-distribution shift.Source: §C.4 · p. 14

C.5 CORRECTED-GATE RE-MEASUREMENT (FORMAL8)

• With the corrected instrument at J=2, the b=3 fit was flat (β̂ = −0.043, r2 = 0.12), while b=9 yielded β̂ = −0.576 (r2 = 0.94), reported without presuming convergence to −1/2.Source: §C.5 · pp. 14–15

C.6 COLLISION-CHANNEL PROBE (FORMAL9)

• Fan-3 co-activation reached mean final 0.495 versus 0.301 for fan-1, a pre-registered descriptive net gain of +0.194.Source: §C.6 · p. 15, Figure 8 · p. 15

C.7 GATE-STRICTNESS SWEEP (FORMAL10, CORRECTED GATE)

• The gate-strictness sweep did not meet its pre-registered jump bar, supporting a smooth rather than abrupt control-parameter response.The largest adjacent drop remained below the required 2.0.Source: §C.7 · p. 15

C.8 SCALE PROBES UNDER THE CORRECTED GATE (FORMAL11/12)

• At N=2000, the corrected-gate run gained 0.0898 across eight accepts, while at N=50 the 500-round run reached a plateau with a 0.0029 last-third slope.The N=2000 entrance/plateau pattern persisted within 150 rounds, but its pre-registered laminar-window criterion was not met.Source: §C.8 · pp. 15–16

C.9 SIGNATURE RE-RUNS UNDER THE CORRECTED GATE (FORMAL13A/13B)

• The corrected-gate signature re-runs supported six of eight L1 signatures, while several name-only candidates failed and boundary cases remained mixed.Magnus correlation appeared gate-dependently, and the authors explicitly make no circulation claim.Source: §C.9 · p. 16

C.10 GIANT SCALE PROBE (FORMAL6, PRE-CORRECTION INSTRUMENT)

• The pre-correction giant-scale and dense time-axis probes are superseded by corrected-gate re-runs, which changed the accept counts and plateau readings.The pre-correction dense run did not reach its plateau within 500 rounds; formal12 reached one under the corrected gate.Source: §C.10 · p. 16, §C.11 · p. 16

• DSH’s Cordis kernel supplies four architectural axioms for the kinetic description: permutation symmetry, reversibility, acyclic dependencies, and typed contracts.Source: D STRUCTURAL CORRESPONDENCE: DSH… · p. 17
• The hot-replacement calculus is confluent, making skill substitution order-insensitive and residue-free, a formal foundation for bounded edits.Source: D STRUCTURAL CORRESPONDENCE: DSH… · p. 17
• A shared skill document has no distinct instance pairs, while a singleton-service MCP architecture lacks interchangeable instances and tracked inverses; for these setups, the kinetic description is undefined.Source: D STRUCTURAL CORRESPONDENCE: DSH… · p. 17
• A versioned skill library with tracked inverses, enforced acyclicity, and typed interfaces could satisfy the obligations; ordinary implementations do not provide them by construction.Source: D STRUCTURAL CORRESPONDENCE: DSH… · p. 17, Table 2 · p. 17
• The hard-sphere reading is short-range, binary, and reversible within a single timeslice, not elastic; the required correspondences are grounded in runtime architecture.Source: D STRUCTURAL CORRESPONDENCE: DSH… · p. 17

E INTERPRETATION BOUNDARIES

• Because momentum is not conserved, no Navier–Stokes limit exists; the paper does not claim literal turbulence, vortex streets, Magnus force, Bernoulli law, or wave dispersion.The reported fluid-like readings are proposed observables, not claims that the corresponding classical mechanisms occur.Source: E INTERPRETATION BOUNDARIES · p. 18

F.1 FLOW AND EXTERNAL FIELD

• Execution transports the frozen plugin set along the pipeline, while task weights act as an external field and the inference chain acts as a thermostat.The paper operationalizes temperature as proposal variance under identical history; provider changes are bath changes.Source: §F.1 · p. 18

F.2 COLLISION STRUCTURE, REACTION CHANNELS, CONSERVATION LEDGER

• Co-activation defines collision; reward and edits inject work, while lifecycle operations include append, delete, update, materialize, and unmaterialize.The collision is inelastic in the driven-gas sense because total credit increases rather than being conserved.Source: §F.2 · pp. 18–19

F.3 EFFECTIVE POPULATION GAUGES

• Effective-particle gauges must be frozen before fitting; their divergence at N=500 diagnoses the scaling formula as unsupported outside the gauge-agreement regime.The paper requires non-dominance by any one instance for rarefaction.Source: §F.3 · p. 21

F.4 THEOREM B: ROUTE CHOICE AND THE ENRICHED-STATE ARGUMENT

• The theory takes a mean-field route because retrieval deliberately creates recollisions; the enriched credit state partly records collision history, but residual dependence through the shared database is unproved.Measured non-dominance is a premise check, not proof of propagation of chaos.Source: §F.4 · p. 21

F.5 TRAINING THERMODYNAMICS

• Without selection, edits can remain frequent while learning signal is zero as the population drifts toward the LLM prior and diversity collapses.The account identifies the gate, not collisions, as the source of training signal; edit frequency alone is not learning rate.Source: §F.5 · p. 22
• With the gate active, drive and dissipation balance in a non-equilibrium steady state centered on successful plugins with an exploration halo.The paper treats convergence as relaxation toward this state and acceptance rate as a proxy for entropy production.Source: §F.5 · p. 22
• The Sagawa–Ueda bound is only an information-theoretic analogy here because free energy over capability space is undefined.The quantitative channel instead uses the daemon’s measurement information, operationalized by the δ̂gate − δ̂ind split.Source: §F.5 · p. 22

F.6 REGIME STRUCTURE

• The fluid analogy stops short of Navier–Stokes or literal turbulence because edits and gate projection prevent momentum conservation.Transport coefficients are names for measurable quantities, and the Euler-type moment system is formal rather than a claimed Chapman–Enskog derivation.Source: §F.6 · pp. 22–23
• Gas, liquid, and solid regimes map respectively to volatile exploration, recurring co-activation motifs, and frozen reproducible workflows.The account associates these stages with changing temperature, density, and gate strictness; motif condensation under strong credit injection and weak selection is described as heuristic.Source: §F.6 · pp. 22–23
• Finite-population run-to-run variability is predicted to scale as std ≈ C p T/Neff, with sparse- and dense-regime probes able to falsify the prediction.A powered sparse-density result showing −1/2 decay or a dense-corner result without decay would falsify the stated regime account.Source: §F.6 · p. 23

G DEPLOYED PRACTICE TOOLS

• The practice tools pair validation resolution and hitting-time horizons with reproducibility pricing, plateau diagnosis, and a rule to keep evaluators outside the evolving population.Additional deployables include reproducibility SLAs, silent-collapse alarms, registry staging, and stopping-rule economics.Source: G DEPLOYED PRACTICE TOOLS · p. 24