Sheaf Cohomology Multiagent Debug

---
name: sheaf-cohomology-multiagent-debug
description: Build a declared finite cellular-sheaf or scalar cochain model and evaluate algebraic compatibility of independently observed local data. Use for scoped residual, rank, and Hodge calculations; not for effect authorization or cause attribution. NOT for inferring agent intent, proving underlying truth, or authorizing an effect.
metadata:
  category: Research & Academic
  tags: [cellular-sheaves, cohomology, cochains, multi-agent, topology]
---

# Sheaf cohomology for modeled compatibility checks

State the finite base complex, orientations, stalk dimensions, restriction maps, coefficient field, metric, observation coverage, and objective before computing a residual. A calculation can report compatibility with that representation. It cannot establish safety, authenticity, freshness, capacity, settlement, intent, a responsible agent, or permission for an external effect.

```mermaid
flowchart LR
  A[Independently observed local reports] --> B[Declare visibility and finite complex]
  B --> C[Declare stalks maps orientations and norm]
  C --> D[Build delta and validate dimensions]
  D --> E[Solve min over x of norm g minus delta x]
  E --> F{Residual within stated tolerance?}
  F -->|yes| G[Compatible with declared linear model]
  F -->|no| H[Report model data or timing mismatch]
  G --> I[Separate domain validation still required]
  H --> I
```

```mermaid
flowchart TD
  A[Scalar triangle: observed g equals 1,1,1] --> B[Signed cycle sum 1 plus 1 minus 1 equals 1]
  B --> C[Not a vertex-potential difference]
  D[Scalar path: any observed edge vector] --> E[Choose root then accumulate potential]
  E --> F[Residual is zero in this graph model]
  C --> G[Neither result attributes a cause]
  F --> G
```

## Reproducible setup

For a linear cochain model `delta:C0 -> C1` and independently observed `g in C1`, report

$$r=\min_x \|g-\delta x\|=\|g-\delta\hat{x}\|.$$

The norm and tolerance are part of the result. If `g` is first defined as `delta x`, then `r=0` by construction; it is not an empirical test. A residual can result from measurement noise, stale versions, missing observations, different units, orientation errors, restriction-map errors, or an inadequate model.

For a finite cochain complex with maps `delta0` from `C0` to `C1` and `delta1` from `C1` to `C2`, the selected sheaf has

$$H^1=\ker(\delta_1)/\operatorname{im}(\delta_0),\qquad \dim H^1=\dim C^1-\operatorname{rank}(\delta_1)-\operatorname{rank}(\delta_0),$$

provided `delta1 delta0=0`. A constant scalar connected graph without faces has `dim H1=beta1`; this does not extend to arbitrary stalk dimensions or restriction maps.

## Predeclare fault signatures

For one versioned feature and one observation watermark, specify which independently sourced edge packets the auditor receives. In a constant scalar graph with incidence matrix `B`, let `Q = I - B B^+`. Project each predeclared fault signature `s_j` to `z_j = Q s_j`, group labels with equal `z_j`, and compute their minimum pairwise separation. Equal signatures are observationally indistinguishable; a positive separation supports classification only within that library and a stated noise bound. The full residual vector carries this information, while its norm alone may not.

For known signed unit single-edge errors in a connected loopless graph, distinct signatures for every edge and sign require three-edge connectivity. For arbitrary errors on at most `k` edge packets, the minimum cut must exceed `2k` for unique recovery modulo compatible reports. These are conditional graph results, not causal diagnoses or general sheaf theorems. Use `scripts/sheaf_observability_study.py` for exact fixture calculations, compare an equal-information direct contract checker, and obtain independent artifact truth before naming a workflow failure.

## Bounded follow-up, not automatic repair

A residual support can suggest which observed coordinates to inspect. Ranking coordinates by residual energy/cost is a **local heuristic**; it is neither an optimal min-cut theorem nor a guarantee that one action reduces `beta1` or drives a residual to zero. Any data correction, schedule change, fence, payment, or other effect requires its own authority and independently verified domain rules.

## Navigation

- [Cellular sheaf model and dimensions](references/cellular-sheaves-for-engineers.md)
- [Observed residual and settlement boundary](references/h1-as-settlement-obstruction.md)
- [Energy and numerical monitoring limits](references/dirichlet-energy-implementation.md)
- [Hansen–Ghrist source scope](references/hansen-ghrist-2021.md)
- [Finite triangle and path walkthrough](references/practical-walkthrough.md)
- [Repair-ranking hypotheses](references/active-repair-and-triadic-cohomology.md)
- [Source access and fixture limits](references/source-access-and-fixtures.md)

The included [finite helper](examples/finite_sheaf_checks.py) and receipt fixture validate matrix identities only. They do not connect to a runtime or authorize a response.

## Bundle navigation

[memory index](memory/INDEX.md).