Algebraic topology for agents: compute a declared model

Use topology only after declaring a finite complex, orientations, coefficient spaces, restriction maps, inner products, independently observed cochain, and the question being asked. A residual is evidence about compatibility with that specified model. It does not establish a cause, a responsible party, or permission for an external effect.

flowchart LR
  A[State finite complex and orientations] --> B[Declare stalks maps and metrics]
  B --> C[Validate dimensions and delta1 delta0 equals zero]
  C --> D[Supply independently observed cochain]
  D --> E[Compute stated projection or Hodge split]
  E --> F{Numeric checks pass?}
  F -->|yes| G[Report scoped algebraic result]
  F -->|no or unknown| H[Return model or data limitation]

flowchart TD
  A[Same edge data g equals 0,3,0] --> B[Project gradient equals -1,2,1]
  B --> C[Residual equals 1,1,-1]
  C --> D{Is face 012 in the declared complex?}
  D -->|no: unfilled 3-cycle| E[Residual is harmonic]
  D -->|yes: filled triangle| F[Residual is coexact curl]
  E --> G[No causal interpretation follows]
  F --> G

Choose the model before the computation

Need	Minimum declaration	Computation
Scalar graph residual	oriented incidence D, real scalar edge cochains, Euclidean norm	Pi = I - D D^+, then Pi g
Filled-triangle split	delta0, delta1, face orientation, Euclidean bases	gradient, harmonic, coexact components
Cellular-sheaf cochains	every stalk, restriction map, composition rule, degree metrics	mapped coboundaries and metric adjoints
Weighted calculation	symmetric positive-definite W0,W1,W2	use metric adjoints, not bare transposes

The finite-dimensional cellular-sheaf definitions and Hodge decomposition below are grounded in the targeted sections of Hansen and Ghrist (2019). The targeted source read covered Definitions 2.4–2.5, §2.2.2, §§3.1–3.2 and Theorem 3.1 on 2026-09-24, not the entire paper. See source boundary and fixtures.

Conditional formulas that remain useful

Constant scalar connected graph

For an oriented unweighted incidence matrix D: R^V -> R^E, Euclidean edge norm, and an independently observed g in R^E, the least-squares residual is rho = Pi g with

$$\Pi = I - D D^+.$$

This uses the Moore–Penrose pseudoinverse. D^T D is singular on a connected graph because constants are in the kernel; a grounded solve removes one vertex coordinate, while D^+ keeps the minimum-norm gauge convention.

On a connected tree with constant scalar coefficients, rank(D)=|E|, hence Pi=0 for every edge vector in this particular measurement model. It says the scalar residual cannot distinguish any edge vector from a vertex-potential difference. It does not say a tree, supervisor, or any other monitor cannot detect other kinds of error. An arbitrary sheaf on a tree need not have vanishing H1: for one edge with scalar stalks and both vertex-to-edge restriction maps zero, delta0=0 and H1=R.

Unweighted scalar single-edge sensitivity

If g was otherwise exact and is perturbed by s on edge e, then, only for the same unweighted scalar incidence model and Euclidean edge norm,

$$|\Pi(s e_e)|2 = |s|\sqrt{1-R{\mathrm{eff}}(e)}, \qquad R_{\mathrm{eff}}(e)=b_e^T L^+b_e, \quad L=D^T D.$$

Here L^+ is the full Laplacian pseudoinverse. A grounded inverse may be used internally to solve a gauge-fixed system, but it is not substituted into this formula without a derivation. Weighted graphs and nonconstant sheaves require their own model and derivation.

Conditional fault separation

For one exact feature, artifact version, and watermark, predeclare an independently observed edge cochain and a finite library of fault signatures s_j, including no fault. In the constant scalar graph model above, compute z_j = Pi s_j and the minimum pairwise distance gamma = min_{i != j} ||z_i - z_j||_2. Equal projected signatures cannot be distinguished by this evidence. A positive margin separates only the declared signatures under a stated observation-error bound; the scalar norm ||Pi g||_2 alone does not identify an edge or an agent.

For known signed unit errors on one edge of a connected loopless graph, all labels have distinct projected signatures exactly when the graph is three-edge-connected. For arbitrary errors on at most k edges, unique recovery modulo a compatible vertex-potential difference requires minimum edge-cut size greater than 2k. These graph criteria do not transfer automatically to heterogeneous sheaves or semantic failure labels. Verify the exact finite cases with scripts/sheaf_observability_study.py, then compare with direct checks given the same evidence and an independent artifact or ledger oracle.

Three worked calculations

1. Filled triangle: independently observed edge values

Use edges (01,12,02) and face [012]:

$$\delta_0=\begin{bmatrix}-1&1&0\0&-1&1\-1&0&1\end{bmatrix},\qquad \delta_1=\begin{bmatrix}1&1&-1\end{bmatrix},\qquad \delta_1\delta_0=0.$$

For independently observed g=(0,3,0), the Euclidean projection gives grad=(-1,2,1) and rho=(1,1,-1). With the face present, rho=delta1^T(1) is coexact curl and the harmonic component is zero. This says only that the data have a face-circulation component in this declared complex.

2. Same data on the unfilled 3-cycle

Keep the vertices, oriented edges, g, and delta0, but declare no face, so C2=0 and delta1 has no rows. The same rho=(1,1,-1) lies in ker(delta0^T) and is harmonic. The data did not change; the interpretation changed because the model supplied a face. This is a model-dependence check.

3. Path, triangle, and a scalar sensitivity check

For the path 0-1-2, D has rank two and every two-edge scalar cochain is a gradient, so Pi=0. For the unfilled triangle, rank(D)=2 in a three-dimensional edge space and the residual space is one-dimensional. On the unweighted triangle, R_eff(e)=2/3, so a perturbation of magnitude s has residual norm |s|/sqrt(3). These are rank and projection facts under the stated scalar model, not a detection claim about an external system.

Numerical safeguards

• Verify dimensions, finite inputs, and delta1 @ delta0 before solving.
• Use an explicit tolerance for floating-point reconstruction and orthogonality; no result is exact merely because it prints rounded values.
• A descriptive harmonic residual-energy fraction is ||h||² / (||h||² + ||c||²) after Hodge splitting. Return None when the denominator is within tolerance of zero. It has no validated diagnostic threshold and does not locate a cause.
• Use transformed cochain coordinates whenever an edge orientation flips. Comparing a flipped operator to untransformed data changes the model.

Run the executable triangle fixture for its finite algebra checks, then load the Hodge derivation or the sheaf primer for the assumptions.

Quality gates

• Complex, orientations, dimensions, coefficient spaces, and independently observed data are stated.
• For a sheaf, all face restrictions and their compositions are checked.
• Every metric used for an adjoint is symmetric positive definite.
• Chain condition, reconstruction, orthogonality, and tolerance handling have a fixture.
• The report separates algebraic compatibility from causal, authorization, and effect claims.