Eureka: Task-Conditioned Meta-Agent Orchestration for Scientific Discovery

Alizer Wong; Heng Cui; Yi Tan; Xiongchao Zhan; Liang Lin; Yuxiang Guo; Zhaorong Dai; Zixin Zeng; Wenyuan Li

Source-linked AI summary

Eureka: Task-Conditioned Meta-Agent Orchestration for Scientific Discovery

Alizer Wong, Heng Cui, Yi Tan, Xiongchao Zhan, Liang Lin, Yuxiang Guo, Zhaorong Dai, Zixin Zeng, Wenyuan Li

arXiv:2608.19047v1Aug 19, 2026cs.AImath.NT

TL;DR

Long-horizon scientific tasks require persistent state, revision, and local autonomy beyond static input-output invocation. Eureka dynamically forms task-specific Macro-Agents and reports completion of 170/170 recursive tasks, alongside substantial context and recomputation reductions.

Problem
Static input-output invocation struggles with long-horizon scientific tasks whose mutually dependent states require persistence, revision, local autonomy, and coordination.
Method
Eureka dynamically compiles goals, constraints, and acceptance semantics into obligation graphs, then generates and evolves specialized Macro-Agents with task-specific state, tools, verifiers, and local planning.
Results
170/170 recursive long-horizon tasks completed; 3,948 acceptance certificates produced; active context reduced from 9,490 to 4,005 tokens, and repeated computation avoided 65.38%.
Takeaways & Limitations
The paper suggests scientific-agent capability may depend on forming computational organization from task cognitive structure rather than fixing architecture before scientific reasoning begins.
Takeaways & Limitations
The Suzuki positivity certificate candidate has not undergone independent formal proof replay and does not constitute a proof of the Riemann Hypothesis.

Abstract

from arXiv · show

We present Eureka, a task-conditioned Meta-Agent architecture that compiles long-horizon tasks into dynamic obligation graphs with explicit acceptance semantics. During execution, Eureka forms Macro-Agents with specialized state, memory, operators, tools, verifiers, and local topology via receding-horizon planning, architecture promotion, and minimal-sufficient compilation. When bottlenecks recur, cost-benefit-gated evolution updates the local architecture under constraints. Theoretically, we establish results on regret, planning invalidation, amortization, subtree interfaces, serializability, and verification. Experimentally, Eureka completes 170/170 recursive tasks and generates 3,948 certificates with no false acceptances. Active context compresses median input from 9,490 to 4,005 tokens; incremental processing avoids 65.38% recomputation across 12,000 tasks; 16,000 concurrent executions serialize consistently. The same Meta-Agent instantiates a Theory-Discovery Agent and a Math/Conjecture Agent. The former yields structural results in quantum-process and spacetime theory. The latter identifies bottlenecks in Riemann Hypothesis research and advances a positivity certificate for Suzuki's localized Weil quadratic form to 0 < a <= 69/200 = 0.345, reaching ~99.55% of (log 2)/2. These results suggest that scientific-agent capability depends not only on the base model but on whether an architecture can be formed to match the task's cognitive structure.

Key Findings at a Glance

Eureka combines task-conditioned orchestration with theoretical-discovery agents to produce certified long-horizon results and new structural findings. Its Riemann-Hypothesis progress remains a certificate candidate rather than an RH proof, while system evaluations show strong completion, efficiency, and verification outcomes.

Riemann-Hypothesis Track: 1.38× expansion beyond a ≤ 1/4 reaches approximately 99.55% of the threshold (log 2)/2 ≈ 0.34657359028.The result is not equivalent to a proof of the Riemann Hypothesis and is not a new record for the proportion of zeros on the critical line.
Theoretical Discovery: The full-rank two-setting/two-outcome QSOST gluing candidate uses t = 1/2, parent minimum eigenvalue 1/8, domination cost 4, and joint dual lower bound 65/16 > 4.Further results distinguish closed behavioural equivalence from black-box interface equivalence and define primitive, query, and resource semantics through an operational intervention signature.
System-Level Findings: 170/170 recursive long-horizon tasks completed, with 3,948 acceptance certificates produced.Governed Evolution records the lowest median total cost, 2525.4, and highest success rate, 60.55%, among four evaluated policies.
System-Level Findings: 65.38% of repeated computation was avoided across 12,000 incremental dependency tasks.Compiled active context reduced median model-input context from 9,490 to 4,005 while preserving the success rate.

1 Introduction

Long-horizon scientific and mathematical tasks exceed static language-model calls because they involve dependent intermediate states, external operations, evolving plans, and heterogeneous epistemic structures. Eureka addresses this architecture mismatch by dynamically compiling task-specific obligation structures and forming specialized agents during execution, enabling separate tests of architecture discovery and scientific discovery.

Motivation: Complex tasks require more than single conditional generation because correctness depends on mutually dependent intermediate states and changing execution conditions.These tasks may involve hierarchical objectives, partially unknown future steps, cross-stage state dependencies, and varying parallelism.
Motivation: Agent capability jointly depends on model reasoning, tools, environmental observations, persistent state, memory, feedback, and control flow.External operations such as retrieval, exact computation, code execution, database access, and environment modification cannot be reliably performed through language generation alone.
Architecture mismatch: Long-horizon subtasks often share internal state, tools, verification, and local decision policies, so recursive decomposition alone cannot provide sufficient local autonomy.Adjacent subtasks may repeatedly access the same facts, invoke the same operators, and update shared state through a common verifier.
Scientific discovery: Scientific discovery is especially demanding because research combines retrieval, formalization, hypothesis generation, derivation, counterexample search, experimentation, evidence integration, and exclusion of prior explanations.The resulting systems move beyond summarization toward continually forming, modifying, and testing candidate scientific structures.
Eureka: Eureka dynamically compiles goals, constraints, and acceptance semantics into a recursive obligation structure, then generates, runs, and evolves specialized agents during execution.Its task-conditioned design avoids relying solely on a predefined expert-agent library or an immediately selected fixed template.
Research questions: Eureka’s central evaluation separates architecture discovery from scientific discovery: forming locally matched specialized agents, then testing whether they produce valuable theories, mathematics, or other results.The two stages are sequential and separately testable, with architecture decisions informed by structural information revealed during execution.

2 Related Work

Prior work advances long-horizon planning, dynamic and optimizable agent architectures, architecture evolution, and scientific-discovery workflows. Eureka unifies task decomposition, architecture-boundary discovery, task-conditioned agent synthesis, and architecture discovery within online execution, while jointly evaluating architecture and scientific discovery.

Long-Horizon Planning: Long-horizon agent planning now jointly addresses decomposition granularity, execution dependencies, and computational-resource allocation.Compiled execution frameworks dispatch ready tasks as soon as dependencies are satisfied.
Dynamic Architectures: Dynamic teaming and input-conditioned systems vary agent roles, collaboration modes, routing, architectures, or inference resources with task state or queries.This line of work addresses limitations of fixed-role systems and context replication from misaligned agent boundaries.
Architecture Optimization: Agent architecture has become an optimizable object, with prior systems searching recursively composable graphs, prompts, tools, control flows, modules, or complete execution harnesses.Self-improving systems use execution experience, source code, scores, traces, validation, consistency, and preference signals to update candidate systems.
Eureka’s Distinction: Eureka unifies task decomposition, architecture-boundary discovery, and task-conditioned agent synthesis within one online long-horizon execution trajectory.It recursively expands the obligation structure, uses state sharing and dependency density, and avoids both pre-searching one complete specialized agent and creating an independent executor for every subtask.
Architecture Evolution: Long-horizon architecture evolution must decide whether to evolve and which layer to modify because diagnosis, candidate generation, evaluation, and state migration incur costs.Eureka classifies bottlenecks across runtime, prompts/operators, memory/skills, tools, state/verifiers, topology, and model capability, applying the lowest sufficient modification or escalating structural changes.
AI for Science: Scientific-agent research spans hypothesis generation and refinement, executable-program search, discipline-specific workflows, formally verified mathematics, and experience-based adaptation within predefined pipelines.Eureka instead jointly evaluates whether architecture can be formed from task requirements and runtime trajectories and what specialized scientific agent can discover.

3 Theoretical Analysis: From Dynamic Task Information to Verifiable Recursive Orchestration … Lemma 2 (Optimal Early-Planning Decision for a Single Planning Unit)

Eureka’s theory formalizes reliable orchestration as cost minimization under explicit information, recording, and acceptance constraints. It shows when task-conditioned architectures remove fixed-architecture regret and when early planning is optimal under conditional validity and benefit trade-offs.

3 Theoretical Analysis: From Dynamic Task Information to Verifiable Recursive Orchestration: The analysis minimizes expected total cost subject to a fixed reliability constraint, rather than combining tokens, success, and verification into one efficiency score.The cost model can include token use, tools, coordination, replanning, and architecture migration.
3.1 Dynamic Task Processes and Admissible Information: Actions must be adapted to legally available filtration, while task completion must be a stopping time that cannot depend on future observations or posteriorly constructed signals.The formal process includes latent task state, observations, control actions, an initial contract, and resource and permission constraints.
3.1 Dynamic Task Processes and Admissible Information: Eureka requires auditable acceptance, finite conditional expected action costs, and lossless recovery of durable information affecting future decisions or Acceptance Contracts.Completed obligations require replayable receipts; tasks are infeasible when no reliability-satisfying policy has finite expected cost.
Proposition 1 (Structural Renaming Invariance): Structural renaming invariance holds when decomposition, promotion, and synthesis depend only on isomorphic task structure and corresponding available information, not entity names.The condition is testable through task anonymization and structural-consistency experiments rather than automatic for arbitrary natural-language agents.
3.2 Structural Regret Lower Bound for Fixed Agent Architectures: Disjoint near-optimal architecture sets imply every fixed architecture incurs regret above at least one task-specific threshold.The result uses separated near-optimal regions because disjoint exact minimizers alone do not guarantee a uniform positive lower bound in continuous architecture spaces.
Theorem 1 (Fixed-Architecture Regret Lower Bound): 𝔼[ℛ_𝛼(𝒜; 𝑇)] ≥ min{𝑝𝜀_1, (1 − 𝑝)𝜀_2} > 0 for every fixed architecture when the two tasks’ near-optimal sets are mutually exclusive.The theorem is conditional: it does not claim that every task requires a distinct agent, only that structurally separated task optima force positive average excess cost.
Corollary 1 (Task-Conditioned Architectures Can Remove the Lower Bound When Structure Is Identifiable): Task-conditioned architecture selection can remove the regret lower bound when an observed structural variable identifies the task and suitable task-specific architectures exist.If both optima are attained with 𝛿_1 = 𝛿_2 = 0, architecture regret is zero; a universal optimal architecture is a boundary case where promotion is unnecessary.
Proposition 3 (Scaling the Base Model Does Not Automatically Eliminate Independent Architecture Overhead): Scaling the base model does not remove architecture regret when semantic-computation cost separates from architecture overhead that remains task-structured and model-invariant.The conclusion can fail if stronger models change state-recovery costs or verifier availability, making architecture overhead depend on the model.

Theorem 2 (Optimal Receding Horizon Under Monotone Marginal Value) … 3.5 Architecture Hotspots and Macro-Agent Promotion

The paper derives threshold and prefix policies for receding-horizon planning, backpressure, delta communication, and local Macro-Agent promotion under explicit cost and validity conditions. It also establishes that recursively decomposed obligations remain semantically sound and terminating when verifier, merge, dependency, and stopping conditions are satisfied, while preserving direct execution when decomposition is not beneficial.

Corollary 2 (Survival-Probability Threshold Under Equal Costs): When early and deferred costs are equal across units, early planning is favored only when survival probability clears a benefit-dependent threshold.As b→0, only nearly certain future units should be expanded early; substantial latency hiding lowers the required survival probability.
Proposition 4 (Modified Threshold with Reusable Residual Value): Reusable residual value extends the region where early planning is useful, so post-observation invalidation is not necessarily 100% wasted work.The residual value is modeled as s_j∈[0,cE_j].
Proposition 5 (Threshold Structure of Ready-Frontier Backpressure): Convex starvation loss makes the benefit B(q)=L(q)−L(q+m) of one planner batch nonincreasing in ready obligations, yielding an empty-or-prefix threshold region q⋆ where B(q)≥cP.This establishes threshold-structured ready-frontier backpressure.
Proposition 6 (State Equivalence and Communication Complexity of PlanDelta): PlanDelta preserves graph state exactly while reducing communication from Θ(n_t) to Θ(m_t) when m_t=o(n_t), because the delta contains every insertion, deletion, and update.The result concerns state encoding and does not alter the scientific action policy.
3.4 Verifiable Recursive Atomization; Assumption 4 (Leaf-Verifier Soundness); Assumption 5 (Soundness of Local Composition Rules); Lemma 3 (One-Level Semantic Composition); Theorem 3 (Recursive Decomposition Soundness): Recursive atomization is sound only when leaf verifiers and cross-child merge rules are sound; completing all children alone is insufficient without consistency, interface, shared-assumption, and merge checks.Under these conditions, one-level composition and finite-DAG induction establish semantic correctness of the root artifact, while unresolved correct leaves remain inconclusive rather than incorrectly accepted.
Corollary 3 (Acceptance Contracts and Execution Recursion Can Share One DAG); Proposition 7 (Sufficient Condition for Termination of Recursive Atomization): Acceptance contracts and execution recursion can share one obligation DAG, and recursive decomposition terminates when every child strictly decreases a nonnegative complexity rank.Without a decreasing structural quantity, an additional budget boundary or semantic stopping rule is required.
Proposition 8 (Boundary Treatment for Cyclic Dependencies); Proposition 9 (Decomposition Is Not Always Beneficial); 3.5 Architecture Hotspots and Macro-Agent Promotion: Cyclic dependencies require SCC contraction with an independent sound rule for each nontrivial component, while decomposition must retain a DIRECT branch when its added costs provide no compensating benefit.Macro-Agent promotion instead compares generic and specialized continuations under the same Acceptance Contract and failure bound, using certifiable per-service cost differences rather than structural statistics alone.

Assumption 6 (Comparable Acceptance Before and After Promotion) … Proposition 12 (General Computational Complexity of Minimal Sufficient Architecture Search)

The section formalizes when Macro-Agent promotion lowers expected cost without weakening acceptance reliability, and when promotion should be deferred or rejected. It then defines minimal sufficient architectures as constrained component-selection problems, establishes their minimality properties, and notes that exact search is generally computationally hard.

Assumption 6 (Comparable Acceptance Before and After Promotion); Assumption 7 (Conditional Lower Bound on Per-Service Savings): Promotion comparisons require the same local Acceptance Contract and reliability threshold, while per-service savings need only satisfy a common conservative conditional lower bound.The savings need not be independent, identically distributed, or equal across services.
Lemma 4 (Conditional Lower Bound on Cumulative Local Savings); Theorem 4 (Conservative Amortization Threshold for Macro-Agent Promotion): Under the savings lower bound and finite expected service count, cumulative expected savings are bounded below by δ_S E[N_S ∣ ℱ_t], enabling promotion when this amortizes fixed cost F_S.The result follows by conditioning each service on the preceding execution trace and applying the tail-sum identity.
Corollary 4 (Break-Even Horizon for a Deterministic Remaining Service Count); Proposition 10 (A Provable Condition Under Which Promotion Has No Advantage): Promotion is not cost-optimal for a horizon too short to recover one-time synthesis and migration costs, and cannot yield positive expected net savings when the conservative upper bound fails to cover F_S.Together, these results define safe-promotion, safe-generic, and uncertainty regions requiring profiling or local trial execution.
Proposition 11 (Lower Bound on Generic-Execution Cost from Shared-State Reload): Generic execution incurs a lower-bounded shared-state reload cost when sessions cannot losslessly share persistent state, although exact backend state sharing can make this excess close to zero.The bound depends on serialized state length L_S, input-unit cost λ_in, and additional activations J_S − 1.
3.6 Minimal Sufficient Agent Architecture Realization; 3.6.1 From Task Requirements to a Finite Component-Selection Problem; 3.6.2 Lazy Architecture Extension: Eureka compiles a promoted subtree into a finite constrained component-selection problem covering requirements, prerequisites, incompatibilities, and optional components that may be installed lazily when first requested.Deferred installation must finish before use, preserving scientific-operator semantics and verifier reliability.
Assumption 8 (Finite Realizability); Theorem 5 (Existence and Inclusion Minimality of a Minimal Sufficient Architecture): With finitely many positive-cost candidates and at least one feasible architecture, an optimal minimal sufficient architecture exists.The theorem establishes inclusion minimality but not uniqueness; deterministic systems therefore need a reproducible tiebreaking rule.
Lemma 5 (Necessity of a Forced Component): A component is mandatory when it uniquely covers an irreplaceable requirement, and every prerequisite reachable from it is mandatory as well.This provides causal provenance for retained components and excludes unsupported expensive components.
Proposition 12 (General Computational Complexity of Minimal Sufficient Architecture Search): Even without prerequisites or incompatibilities and with unit costs, minimal architecture search contains Set Cover, so exact search is not generally polynomial-time and may require heuristics, branch-and-bound, modular search, or approximation.The requirement set is induced by the currently promoted subtree rather than fixed in advance.

Assumption 9 (Safe Monotone Architecture Extension) … Theorem 8 (Optimistic Lease Serializability)

Eureka’s formal results establish safe architecture growth, lossless subtree interfaces, reopen-triggered validity preservation, and serializable parallel execution. Together, these conditions allow deferred capability installation, compressed parent context, conflict-free concurrency, and commit-order-equivalent durable state.

Assumption 9 (Safe Monotone Architecture Extension): Lazy installation preserves completed obligations and only enlarges future capabilities, making deferred extension semantically safe.Existing certified state migrates without loss, and previously legal operations remain legal after extension.
Theorem 6 (Exact Selection Condition for Lazy Architecture Extension): When future capability use is uncertain and incurs no migration penalty, lazy installation has lower expected cost than upfront installation.If d_j + ℓ_j = 0, lazy installation weakly dominates upfront installation; it is strictly cheaper whenever p_j < 1.
Theorem 7 (Lossless Subtree Compression): A decision-sufficient Subtree ABI can replace the complete internal trajectory without reducing the parent’s optimal continuation capability.The interface must preserve effects on future action sets, costs, transitions, and terminal Acceptance Contracts through Z_S.
Corollary 6 (Hierarchical Bound on Parent Context): Parent context remains independent of total internal transcript length when active Macro-Agent interfaces and ABI sizes are bounded.Complete histories need only be paged in for audit or reopen events, while ordinary decisions use the fixed parent state and serialized ABIs.
Proposition 13 (Decision Sufficiency Cannot Be Omitted): Interfaces that collapse histories with different decision-relevant costs, actions, transitions, or acceptance conditions cannot support lossless parent decisions.Eureka therefore requires typed exports, validity assumptions, verification receipts, and reopen triggers rather than free-form summaries.
Proposition 14 (Export Preservation in the Absence of a Reopen Trigger): A previously valid export remains valid across parent-state transitions whenever the ABI-detectable reopen trigger stays inactive.Reopen completeness requires every valid-to-invalid transition to trigger reopening.
Corollary 7 (Safe Parallelism for Pairwise Conflict-Free Fanout): Conflict-free leases commute, so pairwise conflict-free fanout can execute in parallel and merge typed deltas in any order.Parallel execution requires explicit, auditable read/write dependencies, complete read sets, and controlled pre-commit side effects.
Theorem 8 (Optimistic Lease Serializability): Concurrent leases that satisfy complete read sets, isolated side effects, and validation at commit produce exactly the durable state of serial execution in commit order.The validation rule rejects both read-write and write-write interference, so the theorem does not rely on Snapshot Isolation alone.

Corollary 8 (Default Serial Boundary for Shared Proof State) … Theorem 9 (Governed Evolution Admission Threshold)

The theory sets explicit boundaries for serializable proof-state execution and event coalescing, then governs self-evolution through contract-preserving mutations whose amortized benefit exceeds intervention cost. Evolution is admitted only when recurring bottlenecks support conservative future savings without weakening correctness guarantees.

Corollary 8 (Default Serial Boundary for Shared Proof State): Proof states with read-after-write, write-after-read, or write-after-write dependencies default to serial execution unless a commutative update or merge algebra is proved.Concurrent commits from different snapshots can violate the serializability condition.
Proposition 15 (Snapshot Isolation Alone Is Insufficient for Theorem 8): Disjoint write sets alone do not ensure serializability: cross-reading leases can both commit and produce a state equivalent to no serial order.The Boolean x,y construction yields x = y = 0 concurrently, whereas either serial order leaves one key unchanged.
Proposition 16 (Parallelism Boundary for Irreversible External Side Effects): Irreversible external operations before validation prevent end-to-end serializability unless delayed commit, idempotent deduplication, or compensation can restore external state.Durable-state validation alone cannot reverse an external change made by a lease that later aborts.
Proposition 17 (Equivalence of Coalescing Commutative Control Events): Commuting non-hard-interrupt events can be batched without changing control state, allowing one Meta-Agent wake-up with a single ControlCapsule.Supported examples include worker-completion events, ready-count updates, and conflict-free receipt insertions.
Proposition 17 (Equivalence of Coalescing Commutative Control Events): Contradictions, verifier mismatches, budget exhaustion, and acceptance-boundary changes are hard interrupts that cannot be coalesced with ordinary events.Equation (43) defines the safety boundary for event coalescing rather than guaranteeing that fewer wake-ups are always correct.
3.9 Amortized Theory of Governed Self-Evolution: Evolution mutations may alter governed runtime, prompt, memory, tool, verifier, or topology components, but cannot modify task contracts, frozen acceptance contracts, source cutoff, or sealed evaluator boundaries.Incumbent and candidate architectures are compared under the same local Acceptance Contract and failure-probability bound.
Assumption 12 (Evolution Contract Invariance): A mutation preserves admissibility only when it satisfies the same local Acceptance Contract and reliability bound; verifier-semantic changes require Meta-Agent contract-level review.The reliability condition is ℙ(S is accepted correctly ∣ ℱ_t) ≥ 1 − α_S.
Assumption 13 (Conservative Gain Lower Bound for a Recurring Bottleneck): A recurring bottleneck supports admission when conservative per-service savings γ_m persist over future services, while a one-off anomaly does not establish γ_m > 0.The assumption does not require strict improvement at every service or independent, identically distributed costs.

Corollary 9 (Break-Even Horizon for Evolution) … Proposition 20 (Forcing INCONCLUSIVE to FAIL Is Unsound in General)

The paper formalizes evolution admission around amortized cost, mutation scope, and optional-stopping guarantees, while enforcing monotone certified state through provenance-aware invalidation and three-valued verification. It proves that uncertified text cannot silently alter active scientific state and that INCONCLUSIVE cannot generally be treated as FAIL.

Corollary 9 (Break-Even Horizon for Evolution): Short remaining horizons can make immediate evolution inferior despite lower candidate steady-state cost because diagnosis, evaluation, and migration add fixed overhead.The break-even condition is expressed through the remaining service count N_S = H_rem and the evolution-cost ratio m/γ_m.
Proposition 18 (Weak Dominance of the Lowest Sufficient Mutation): A lower-level mutation strictly dominates a higher-level one only when both induce identical future behavior distributions, making its one-time evolution cost lower.If a broader mutation enables behavior unavailable to lower-level changes, smaller scope alone does not justify choosing it.
Proposition 19 (Reliability of Evolution Admission Under Optional Stopping): At most β_m is the conditional probability of falsely certifying positive amortized value from sequential micro-evaluation sampling error.Time-uniform confidence sequences preserve coverage under data-dependent stopping, enabling evaluation without fixing sample size in advance.
3.10 Verification Semantics and Monotone Scientific Progress: Certified receipts retain exact inputs, artifacts, certificates, dependencies, and version references in an append-only ledger, while invalidations propagate through dependent descendants.This separates historical monotonicity from revocability of the active certified view.
Lemma 8 (Semantic Soundness of Receipt Addition): Under verifier soundness, every added active receipt satisfies the true semantics on its exact input.Equation (50) requires acceptance by the verifier, which implies Φ_o(i_r, y_r) = 1.
Theorem 10 (Soundness of the Active Certified Ledger): The active certified ledger contains only semantically sound receipts with current dependencies and no provenance path from known invalidation seeds.The result depends on a sound initial ledger, provenance completeness, valid receipt additions, and complete invalidation propagation.
Corollary 10 (Certified State Cannot Be Silently Overwritten by Unverified Text): Unverified natural-language claims cannot enter the certified ledger or remove an active receipt without a valid refutation or invalidation certificate.Thus, certified scientific state cannot be silently rewritten by conflicting language-model output.
Proposition 20 (Forcing INCONCLUSIVE to FAIL Is Unsound in General): Mapping every INCONCLUSIVE result to FAIL is unsound because an incomplete acceptance verifier may miss a semantically true result lacking any sound refutation certificate.The three-valued rule distinguishes certified PASS, certified FAIL, and INCONCLUSIVE; inconsistent verifier outputs trigger a hard interrupt.

Corollary 11 (Error Control for Anytime Statistical Verification) … Theorem 12 (Structure-Driven Specialized-Agent Emergence)

The paper establishes anytime verification error control and formal evaluation-isolation guarantees, then shows that specialized-agent architectures emerge when task instances have disjoint optimal architecture sets. It also separates architecture formation from later scientific discovery and identifies runtime leakage or design-stage contamination as threats to causal validity.

Corollary 11 (Error Control for Anytime Statistical Verification): Incorrect certified decisions have probability at most β_o, even when τ_v is selected adaptively from online observations.On the simultaneous coverage event, FAIL cannot be wrong and incorrect decisions are contained in the confidence-sequence failure event.
3.11 Evaluation-Information Isolation and Causal Validity of Execution; Assumption 16 (Runtime Evaluation Isolation): Runtime evaluation isolation requires evaluator-independent policy kernels, inaccessible evaluation-derived information, identical observation kernels across sealed values, and an almost-surely finite freeze time.The source-firewall condition is probabilistic: physical file isolation is insufficient if public post-evaluation pages alter search observations.
Lemma 9 (Evaluator Invariance of Finite-Horizon Production Traces): Under Assumption 16, finite-horizon production traces have identical distributions for any sealed evaluator values, yielding evaluator-invariant execution before freeze.This follows from induction over shared policy and observation kernels.
Theorem 11 (Evaluation-Isolated Counterfactual Invariance): Theorem 11 extends evaluator invariance to the complete production trace stopped at freeze time, with pathwise identity almost surely under common-random-number coupling.The result applies because freeze time is an almost-surely finite stopping time.
Corollary 12 (Sealed-Evaluator Mutation Trace Invariance): A deterministic or seeded runtime preserves the canonical serialization and hash of the entire pre-freeze trace when the sealed evaluator is replaced.This supplies an implementable audit for orchestration, agent promotion, architecture synthesis, and governed evolution.
Proposition 21 (Evaluation-Time Information Leakage Breaks Observation-Kernel Invariance); Proposition 22 (Runtime Isolation Does Not Replace Design-Stage Evaluation Separation): Search-time leakage breaks observation-kernel invariance, while runtime isolation alone does not remove design-stage dependence on the sealed evaluation target.Complete isolation therefore requires source cutoffs and query audits, plus advance freezing or target-independent optimization of design components and rules.
Proposition 23 (Evidence Hierarchy of Sealed Evaluation and Prospective Holdout): Sealed evaluation establishes reachability for a frozen policy using legally available pre-evaluation information, but prospective holdout is needed to address unrecorded design-time selection bias.A public freeze before the target becomes verifiable provides the stated evidence hierarchy.
3.12 Two-Stage Decomposition of Architecture Discovery and Scientific Discovery; Assumption 17 (Architecture-Stage Measurability); Lemma 10 (Information Separation Between the Two Stages); Theorem 12 (Structure-Driven Specialized-Agent Emergence): Architecture is formed at stopping time τ_A from a promoted subtree’s structured instance, while its Macro-Agent later generates the scientific artifact at stopping time τ_D under legally expanded information.The architecture output is measurable at τ_A and stage-two discovery follows the resulting architecture; exact minimum-cost compilation produces different vectors when optimal architecture sets are disjoint.

Corollary 13 (Architecture Differentiation Induced by an Irreplaceable Requirement) … Corollary 14 (Total Regret Bound from Module Decision Error)

The theory establishes when task structure forces distinct agent architectures, how anonymized tasks preserve architecture decisions, and why architecture discovery must be evaluated separately from scientific discovery. It then decomposes reliability-constrained excess cost into module-level effects and bounds total regret using observable decision discrepancies and continuation-cost sensitivity.

Corollary 13 (Architecture Differentiation Induced by an Irreplaceable Requirement): An irreplaceable requirement forces incompatible optimal architecture choices across task subtrees, producing structurally different specialized agents.Unique coverage requires inclusion of the component in one subtree, while positive-cost minimality excludes it from the other.
Proposition 24 (Architecture Causal Trace Under Task Anonymization): Structurally isomorphic tasks with consistently mapped identifiers and tie-break rules yield corresponding architecture solutions under the compiler.The feasible architecture sets map bijectively while preserving costs and tie-break ordering.
Proposition 25 (Architecture Discovery and Scientific Discovery Require Separate Evaluation): Architecture quality and scientific-discovery quality require separate evaluation when architecture search uses only pre-discovery criteria.Optimizing held-out scientific evaluation during architecture search prevents a causal interpretation based solely on task structure available before discovery.
3.13 End-to-End Regret and Cost Decomposition: Reliability-constrained policy comparison preserves root correctness, and hybrid oracle replacements must remain admissible and interface-compatible.If replacing a module breaks verifier semantics or state compatibility, it must be grouped with the coupled module rather than attributed independently.
Theorem 13 (Exact Hybrid Regret Decomposition): Eureka’s system-level excess cost has an exact telescoping decomposition across successive module replacements by same-information oracle kernels.The identity is attributional: individual module differences need not be nonnegative because module interactions can raise the cost of an isolated replacement.
Assumption 19 (Bounded Local Continuation-Cost Oscillation): A bounded continuation-cost oscillation and finite module-activation count make long-horizon effects of local decision discrepancies controllable.The local sensitivity constant L_j bounds how continuation costs respond to different actions at a module-𝑗 decision history.
Lemma 11 (One-Module Performance-Difference Identity): The one-module performance-difference identity isolates the effect of changing only module 𝑗 by treating complete production history as the Markov state.All transitions and action terms from other modules cancel, leaving the action-kernel difference at module activations.
Corollary 14 (Total Regret Bound from Module Decision Error): Total system regret is nonnegative against the same-information optimal comparator and should be attributed through observable action-kernel discrepancies or deterministic module mismatches.The required attribution spans architecture selection, planning, promotion, evolution, coordination, and verifier gating rather than final success rate alone.

Proposition 26 (Policy Invariance Under a Semantics-Preserving Runtime Transformation) … Proposition 34 (A One-Off Bottleneck Does Not Justify Default Self-Evolution)

The propositions define when Eureka’s transformations and architecture switching preserve scientific behavior, remain stable, and amortize their costs. They also identify failure regimes in which promotion, verification, provenance, parallel execution, or self-evolution lose their guarantees.

Proposition 26; 3.14 Stability, Hysteresis, and Architecture Oscillation: Semantics-preserving runtime transformations retain the same actions, tool arguments, receipts, final artifact, and acceptance event while strictly reducing expected step cost.This boundary applies to exact caching, PlanDelta, prefix reuse, certified common-subexpression elimination, and deterministic closure; changes to the action law require policy-level evaluation.
3.14.1 Promotion-Demotion Hysteresis; Proposition 27: Hysteresis separates promotion and demotion thresholds to prevent estimation noise from repeatedly oscillating the architecture between generic and Macro-Agent modes.The switching law is presented as an exact control law rather than an empirical heuristic.
Assumption 20; Lemma 12; Theorem 14: Under bounded estimation error and true-signal drift, opposite-direction switches require sufficient true-value variation, yielding dwell-time and total-variation switch-count bounds.When ρ = 0, noise alone cannot trigger a second opposite switch.
Corollary 15: Architecture-switch overhead is bounded by the number of switches times the per-switch cost, so increasing hysteresis reduces churn but can delay genuine regime adaptation.Threshold width must balance stability against adaptivity rather than increase without bound.
Proposition 29; Proposition 30: Promotion is strictly worse when persistent savings are absent, while unsound leaf verification can let an erroneous artifact reach an accepted root and invalidate compositional correctness.Verifier soundness is a logical premise, not a statistical preference recoverable through additional agent votes.
Proposition 31; Proposition 32: Incomplete provenance can leave an invalidated receipt active, and incomplete read sets can invalidate parallel-safety and serializability guarantees.Conservative dependency supersets trade increased recomputation for soundness, whereas omitted dependencies can prevent programmatic invalidation.
Proposition 33; Proposition 34; 3.15 Boundary Conditions and Failure Regimes: High migration or evolution costs can make intervention nonpositive, and a one-off bottleneck does not justify default self-evolution.Only a recurring bottleneck expected to reappear within the remaining horizon can provide a positive evolution advantage.

Proposition 35 (Failure to Adapt After a Regime Shift Produces Linear Local Regret) … Corollary 26 (Linear Growth of Dynamic Regret Under Persistent Non-Adaptation After a Genuine Regime Shift)

The section develops falsifiable predictions about Eureka’s adaptation, verification, isolation, scaling, invariance, efficiency, and stability mechanisms. Across these results, failure to adapt or verify can produce linear regret or invalidate guarantees, while correctly governed architecture changes should yield predictable cost, context, and churn effects.

Proposition 35; Corollary 26: Linear excess cost after a genuine regime shift makes irreversible promotion unsafe, requiring demotion, splitting, or re-synthesis as control interfaces.The old architecture incurs at least ε excess cost per epoch, so delaying adaptation for d epochs accumulates linear local and dynamic regret.
Proposition 36: Without a sound acceptance verifier or calibration condition, Eureka may report evidence-supported empirical or support levels, but not formal semantic certification.Self-evaluation and majority agreement among agents cannot substitute for a reliable verifier.
Proposition 37: Evaluation leakage through routing, priors, mutation, broker queries, or retrieval invalidates information-isolation guarantees, and omitted answer text cannot restore them.The corresponding theorem and corollary cease to apply when sealed evaluation content affects the filtration or observation kernel.
3.16; Corollaries 16–17: The theory yields experimentally testable predictions by varying one formally defined quantity while holding other conditions fixed, including task-architecture separation and uncertainty.Greater separation of near-optimal architecture sets should increase fixed-architecture regret, while increasing task-revelation uncertainty should not increase Eureka’s selected planning horizon.
Corollaries 18–19: A longer remaining horizon increases the conservative promotion value, while greater local-state length or restoration demand increases promotion’s provable cost advantage.These monotonicity predictions assume fixed costs, reliability, and unchanged promotion benefit; violations may indicate altered workload or persistent-state sharing.
Corollaries 20–21: Governed evolution should outperform always-evolve and no-evolve baselines when recurring bottlenecks affect more future obligations, while Subtree-ABI parent context should not scale linearly with child transcripts.Fixed concurrent Macro-Agent and ABI bounds prevent transcript accumulation from forcing linear parent-context growth.
Corollaries 22–23: Structure-preserving task anonymization should preserve architecture selection, and sealed-evaluator mutations should leave the pre-freeze trace hash unchanged.Substantial changes under anonymization suggest task-name priors; any pre-freeze hash change falsifies the corresponding isolation implementation.
Corollaries 24–26: Semantics-preserving runtime optimizations should change computational cost rather than discovery distributions, while wider hysteresis should reduce noise-driven churn but may increase adaptation delay and dynamic regret.Experiments should jointly report token, tool, or latency cost; discovery recall; switch count; churn reduction; and delay after genuine regime shifts.

4 Experiments

Eureka achieves reliable long-horizon execution while compiling distinct architectures for theory-discovery and mathematical-conjecture tasks. Experiments show certified completion, substantial efficiency gains, safe concurrency, and successful structural-transfer evaluations across both scientific task families.

Architecture evolution: Governed Evolution achieved the lowest median total cost and highest success rate among the evaluated evolution policies.EvolutionLease reduced median evolution round trips from 12 to 4, a 66.7% reduction, while lowering evolution-control cost by 11.5%.
Efficiency: 57.8% median context reduction lowered model input from 9,490 to 4,005, while incremental execution avoided 65.38% recomputation across 12,000 tasks.Full recomputation and incremental execution produced identical final results, with only 34.62% of nodes requiring actual recomputation.
Concurrency and consistency: 16,000/16,000 concurrent executions produced serial-equivalent final states with 0 unsafe commits, while dependency-local validation reduced unnecessary rebases by 75.0%.Across 10,000 fanout executions, shared immutable snapshot prefixes enabled 64.24% median cache-eligible prefill reuse.
Execution reliability: 170/170 recursive tasks completed, generating 3,948 acceptance certificates with 0 uncertified accepts, false terminal states, or stagnation aborts.Dependency-ready streaming reduced median completion time from 0.5058 s to 0.2724 s, a 1.8569× median speedup.
Scientific discovery: 5/5 Theory-Discovery structures and 3/3 mathematical structures were accepted, with mathematical checks passing at 2,200/2,200 and 960/960.The Theory-Discovery Agent yields five structural classes, while the Math/Conjecture Agent yields three programmatically verifiable structures.
Generality and transfer: 5,000/5,000 architecture decisions remained structurally consistent under entity renaming, while Theory-Discovery transfer passed 8/8 classes and 11/11 capability dimensions.Math/Conjecture generality tests passed 10/10, alongside 2,200/2,200 deterministic composites and 960/960 stochastic checks.

5 Advancing the Riemann Hypothesis

The Math/Conjecture Agent advances a direction-correct localized-Weil positivity program for the Riemann Hypothesis by combining low/high operator decomposition, finite local separation, and structured residual bounds. Its certificate reaches 0 < a <= 69/200 = 0.345, but remains a candidate rather than a proof of the hypothesis.

Operator formulation: The proof obligation requires simultaneous control of the low block, high-complement coercivity, and cross coupling; positive finite Ritz values alone are insufficient.This low/high decomposition prevents finite-dimensional positivity from being mistaken for positivity of the complete operator.
Local separation: A selective spectral projection around a hypothetical off-line zero is not directly observable on the prime side, motivating finite local polynomial separators and explicit control of conditioning, cluster separation, and infinite tails.Chebyshev separation can be masked by another off-line pair, so interpolation is used on finite conjugation-symmetric clusters.
Structured certification: The residual certificate retains degree-10 Taylor/Bernoulli structure on the first 12 normalized Legendre modes, while generic norm bounds apply only to unresolved high/cross remainders.The retained low-modal residual block becomes an explicit polynomial in a, with the analytic tail controlled independently.
Certificate result: 1,010 interval cells on [1/4, 69/200] pass interval Cholesky certification, yielding 0 < a <= 69/200 = 0.345 and a 1.38× support-range extension over a <= 1/4.The final cell [0.344995, 0.345] has a smallest lower pivot of approximately 9.7730412×10^-3; the endpoint reaches approximately 99.55% of (log 2)/2.
Remaining obligation: The result does not establish full λ_a > 0 beyond the first-prime threshold, because boundary absorption must still be combined with killing positivity, the low-modal block, and the high-mode cross Gram.The work is an analytic derivation plus an outward-interval certificate candidate, without independent formal proof replay, and does not constitute a proof of the Riemann Hypothesis.

6 Discovering New Theoretical Structures

The Theory-Discovery Agent identifies new theoretical structures by constructing and auditing operational forms, assumptions, interfaces, and equivalence criteria. It produces a full-rank QSOST gluing solution candidate alongside structural normal forms and meta-theoretical distinctions for evaluating quantum-process theories.

QSOST gluing: 4 deterministic-process domination cost per setting and an exact strict margin of 1/16 yield a QSOST gluing solution candidate.The construction preserves minimal two-setting/two-outcome structure, full-rank parents, and strict common-parent separation, but still requires external or formal review.
QSOST gluing: Frozen discrete invariants, interior homotopy, and strict-separator stability reduce conjunction search to continuous deformation within an equivalence class.This avoids re-searching the entire high-dimensional discrete object space.
Assumption auditing: Complete positivity makes an added sector-preservation condition redundant when the relevant population block vanishes, while cross-output coherence transport remains allowed.The analysis distinguishes same-output coherent merges from cross-output transport, removing an assumption without shrinking the original process class.
Acted-set normal form: A monotone acted-set history variable compresses weak closed-lab case distinctions while preserving one black-box slot for each unknown operation.Physical timing, loss, and route history become internal coherent control and buffering; timing is separated from the externally replaceable operation.
Equivalence and interfaces: Behavioural equivalence, interface-substitution equivalence, and query-depth claims remain distinct until uniform transformations, composition congruence, resource budgets, and primitive signatures are fixed.Independent late-bindability determines whether bundling changes the physical interface or merely renames an equivalent primitive.

7 Conclusion

Eureka dynamically compiles long-horizon scientific tasks into specialized Macro-Agent organizations governed by explicit execution and verification structures. Across systems and research settings, it completes recursive tasks, reduces context and recomputation, and forms distinct agents that advance mathematical and theoretical-science results.

Architecture: Eureka compiles open long-horizon tasks into obligation structures and governs specialized Macro-Agents during execution.The architecture adapts persistent state, operator families, verification semantics, parallel structure, and long-term boundaries to task requirements.
System results: 170/170 recursive tasks completed, with 3,948 acceptance certificates produced.Governed Evolution achieved the lowest median total cost and highest success rate among evaluated strategies.
System results: Compiled active context reduced median model-input context from 9,490 to 4,005 without changing task success rate.Dependency-local rebuilding avoided 65.38% of repeated computation, while all 16,000 concurrent-execution tasks remained consistent with valid serialization.
Scientific instantiations: The Math/Conjecture Agent advanced Suzuki’s localized Weil quadratic-form positivity certificate candidate to 0 < a <= 69/200 = 0.345, approximately 99.55% of the first-prime threshold.Boundary-swap analysis compressed the next theorem obligation to a joint certificate retaining the remaining positive multiplication together with the modal Schur structure.