Capability Sheaves for Compositional Agent-Harness Repair: Controlled Quotients and a Real-Repository Stress Test

Technical Summary: Capability Sheaves for Compositional Agent-Harness Repair

Problem Statement
Language model agents rely on "harnesses"—combinations of instructions, retrieval, routing, state management, provenance, and verification. A critical failure mode occurs when individual subsystems appear locally competent (each can perform its specific task) but fail to cohere into a globally consistent execution. For example, a localization module might identify a file, a contract module might identify an API revision, and a verifier might select a test, yet these components reference incompatible files, revisions, or commits. Current optimization methods typically rank complete candidates via scalar scores but lack a mechanism to diagnose or repair this "local-to-global" gluing failure.

Methodology
The paper proposes modeling agent harnesses using finite capability sheaves.

• Sheaf Construction: The agent's requirements (Localization, Contract, Ordering, Preservation, Verification) are modeled as vertices in a finite incidence graph. Each vertex holds a "stalk" of typed behavior signatures (e.g., file paths, API revisions). Overlaps (edges) represent shared fields, and restriction maps project local signatures to these overlaps.
• Exact CSP (Constraint Satisfaction Problem): A candidate harness is accepted only if it forms a "global section"—meaning all local signatures exist within their valid sets and agree on all overlaps. This is formulated as an exact finite CSP, serving as the semantic decision rule.
• Cohomological Diagnostics: The paper introduces a linearized relative cohomology class ($H^1(X, A; F)$) derived from the cellular sheaf. This class acts as a diagnostic tool: a non-zero class indicates a mismatch in the shared state (a "gluing obstruction"), while a zero class indicates compatibility.
• Hidden-State Quotient: To test invariance, the author introduces "hidden interior mediators" between public requirement vertices. They define a quotient space that factors out the influence of these hidden states (nuisance variables), aiming to create a ranking metric that is invariant to stale or misaligned internal representations.

Key Contributions

1. Formal Framework: Construction of a finite capability sheaf with typed stalks and literal field projections, separating local existence from global gluing.
2. Theoretical Guarantees: Proof of exact gluing conditions and spectral stability. The paper establishes that while a non-zero cohomological class certifies the absence of a solution, a zero class does not guarantee existence (it requires the exact CSP check).
3. Controlled Experiment: A study across 20 task clusters with hidden mediators to test the "invariance mechanism."
4. Real-World Stress Test: An evaluation on a discovery split of the SWE-bench Multilingual pool (PatchFuseBench), involving 160 issues from 20 repositories and 875 candidate patches.

Results

• Controlled Hidden-State Experiment:

  • Invariance: When "stale" hidden mediators were introduced, the raw half-edge scores varied, but the quotient ranking remained fixed.
  • Budget Reduction: The quotient policy reduced the candidate evaluation budget from 2.0 to 1.0 (a 50% reduction) compared to a raw-score policy in all 20 clusters ($p \approx 9.5 \times 10^{-7}$).
  • Ablation: When the hidden state was "aligned" (matching the endpoints), the raw score performed equally well to the quotient. This confirms the gain comes specifically from handling nuisance representatives, not from the cohomological label itself.
  • Equivalence to Exact CSP: The quotient policy performed identically to the exact CSP solver, demonstrating that the quotient is a valid diagnostic for this specific class of problems but does not surpass exact reasoning.

• Real-Repository Stress Test (SWE-bench):

  • Identifiability Failure: A "full-pool" construction resulted in a constant cohomology class for all candidate patches, rendering it incapable of ranking different solutions.
  • Corrected Construction: A candidate-indexed repair (restricting the complex to each candidate's specific atoms) produced nontrivial, varying scores.
  • Performance: The corrected quotient selector resolved 118/160 issues, slightly outperforming a matched non-cohomological selector (116/160). However, this difference was not statistically significant across repositories ($p=0.75$).
  • Abstention Gate: A leave-one-repository-out gate using the quotient reached 127/160, tying a strong anchor selector but failing to exceed it significantly ($p=1.0$).
  • Conclusion: The method failed to pass the development gate (requiring specific issue gains and repository-level significance). Consequently, the confirmatory split of 21 repositories remains sealed.

Significance and Claims
The paper modestly claims to define the boundary of the method rather than a general breakthrough:

1. Invariance Mechanism: The study successfully demonstrates that quotienting hidden-state nuisance variables provides a robust, invariant diagnostic for agent harness failures in controlled settings.
2. No Real-World Advantage: The method does not demonstrate a cohomological advantage over matched semantic evidence or exact solvers in real-world patch fusion. The "global section" requirement remains the decisive control.
3. Diagnostic Utility: Linearized cohomology serves as a useful search score and diagnostic (identifying where gluing fails) but cannot replace the exact CSP or execution verification required to confirm a solution.

The author concludes that while the structural observation channel is valid, the current construction of obligation-and-risk complexes is too weak to improve real-world agent repair beyond existing strong anchors. Future work would require a stronger construction that encodes semantic interfaces capable of gluing across edits, rather than attaching a single global class to an issue.