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)
- Vertices L, C, O, P, V with typed signature stalks.
- Edges are the six observed overlaps; restriction = literal field projection.
- Φ(c) = (∀v, s_v ∈ G_v) ∧ (∀e=vw, ρ_v s_v = ρ_w s_w). Exact finite CSP.
- Relative class [b − Dx] ∈ coker D is diagnostic / ranking. Necessary for
linearized gluing, not sufficient for executability.
- ACV: critical membership is exact; never average a soft score over a
restriction failure.
Workflow
- Name which harness parts occupy which vertices.
- Confirm overlaps against the repo (do not invent extra edges).
- Emit the incidence graph + Φ contract.
- 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.
1---2name: sheaf-kernel3description: 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".4---56# sheaf-kernel78The **μ-level** of the sheaf-* family. Zhang et al., *On Meta-Prompting*9(arXiv:2312.06562), gave meta-* its generator: an example-agnostic scaffold.10Batruin, *Capability Sheaves for Compositional Agent-Harness Repair*11(arXiv:2608.13228, 13 Aug 2026), gives this family its generator: a finite12capability sheaf whose global sections are accepted harness runs.1314This skill does **not** produce a patch. It instantiates `F`.1516Depth lives in `references/`. Read them; do not reinvent them:1718- `references/THEORY.md` — stalks, restrictions, Φ, relative class, honesty.19- `references/INCIDENCE.md` — the five vertices and six overlaps, copied from the paper.20- `references/BIBLIOGRAPHY.md` — cite from here.2122Sibling skills bind to this kernel:2324| Skill | Role | Analog |25|---|---|---|26| `sheaf-localize` | stalks on L, C | meta-planning |27| `sheaf-preserve` | stalks on O, P | meta-design |28| `sheaf-glue` | exact CSP | meta-review |29| `sheaf-repair` | quotients + discrete alignment | meta-operad |3031## When to use / when not3233**Use** when two or more harness components (retrieval, routing, state, provenance,34verification, skills) share fields and can succeed locally while disagreeing.35**Do not use** for a one-shot edit with no shared state. Do not use to *train*36restriction maps — zero weight updates (see sheaf-repair).3738## Kernel (compressed)39401. **Vertices** L, C, O, P, V with typed signature stalks.412. **Edges** are the six observed overlaps; restriction = literal field projection.423. **Φ(c)** = (∀v, s_v ∈ G_v) ∧ (∀e=vw, ρ_v s_v = ρ_w s_w). Exact finite CSP.434. **Relative class** [b − Dx] ∈ coker D is diagnostic / ranking. Necessary for44 linearized gluing, **not sufficient** for executability.455. **ACV**: critical membership is exact; never average a soft score over a46 restriction failure.4748## Workflow49501. Name which harness parts occupy which vertices.512. Confirm overlaps against the repo (do not invent extra edges).523. Emit the incidence graph + Φ contract.534. Hand F to `sheaf-localize` and `sheaf-preserve` (parallel). Then `sheaf-glue`.5455## Honesty5657Present structure as a **runtime-checked contract**, not a theorem about the58sampler. Batruin’s real-repository gate did not pass; do not claim cohomological59advantage over exact CSP.
manutej