sheaf-kernel

name

sheaf-kernel

description

Instantiate a finite capability sheaf for an agent harness: five vertices (Localization, Contract, Ordering, Preservation, Verification), six restriction maps as field projections, exact CSP Φ as the decision rule, relative cohomology as diagnostic only. Generator of the sheaf-* family — the Batruin analogue of meta-prompting. Use when a harness has multiple components that must agree on shared state, or when sheaf-localize / sheaf-preserve / sheaf-glue / sheaf-repair need a typed incidence graph. Trigger on "capability sheaf", "sheaf kernel", "harness incidence graph", "L C O P V", "restriction atlas".

sheaf-kernel

The μ-level of the sheaf-* family. Zhang et al., On Meta-Prompting (arXiv:2312.06562), gave meta-* its generator: an example-agnostic scaffold. Batruin, Capability Sheaves for Compositional Agent-Harness Repair (arXiv:2608.13228, 13 Aug 2026), gives this family its generator: a finite capability sheaf whose global sections are accepted harness runs.

This skill does not produce a patch. It instantiates F.

Depth lives in references/. Read them; do not reinvent them:

• references/THEORY.md — stalks, restrictions, Φ, relative class, honesty.
• references/INCIDENCE.md — the five vertices and six overlaps, copied from the paper.
• references/BIBLIOGRAPHY.md — cite from here.

Sibling skills bind to this kernel:

Skill	Role	Analog
sheaf-localize	stalks on L, C	meta-planning
sheaf-preserve	stalks on O, P	meta-design
sheaf-glue	exact CSP	meta-review
sheaf-repair	quotients + discrete alignment	meta-operad

When to use / when not

Use when two or more harness components (retrieval, routing, state, provenance, verification, skills) share fields and can succeed locally while disagreeing. Do not use for a one-shot edit with no shared state. Do not use to train restriction maps — zero weight updates (see sheaf-repair).

Kernel (compressed)

1. Vertices L, C, O, P, V with typed signature stalks.
2. Edges are the six observed overlaps; restriction = literal field projection.
3. Φ(c) = (∀v, s_v ∈ G_v) ∧ (∀e=vw, ρ_v s_v = ρ_w s_w). Exact finite CSP.
4. Relative class [b − Dx] ∈ coker D is diagnostic / ranking. Necessary for linearized gluing, not sufficient for executability.
5. ACV: critical membership is exact; never average a soft score over a restriction failure.

Workflow

1. Name which harness parts occupy which vertices.
2. Confirm overlaps against the repo (do not invent extra edges).
3. Emit the incidence graph + Φ contract.
4. Hand F to sheaf-localize and sheaf-preserve (parallel). Then sheaf-glue.

Honesty

Present structure as a runtime-checked contract, not a theorem about the sampler. Batruin’s real-repository gate did not pass; do not claim cohomological advantage over exact CSP.