The Harmonic Manifold
The Tikhonov state is meaningful only because the algebraic object underneath it is precise. This page connects the sheaf energy, distributed linear solve, and singularly perturbed planner without collapsing them into one black box.
From edge disagreement to one reference
Let $\mathcal F$ be a cellular sheaf over the communication graph. Splitting the zero-cochain into free agent values $q$ and pinned target values $p$ gives the block sheaf Laplacian
\[L_{\mathcal F}= \begin{bmatrix}H&B\\B^\top&L_{PP}\end{bmatrix}.\]
For fixed boundary data, the free part minimizes the Dirichlet energy
\[\mathscr E(q,p)=\tfrac12q^\top Hq+q^\top Bp+ \tfrac12p^\top L_{PP}p.\]
Stationarity in $q$ yields
\[Hq^\star=-Bp, \qquad q^\star(p)=H^{-1}(-Bp).\]
Under the relative-cohomology or pinned-component condition, $H\succ0$ and the reference is unique. The detailed sheaf derivation is given in Coordination as a harmonic extension.
Why the solve stays separate
For a fixed sheaf, target motion changes only $-Bp(t)$. The expensive symbolic work is therefore paid once:
- construct $H$ and $B$;
- factor $H$ into its clique-tree multifrontal representation;
- partition or cache that representation;
- re-solve for each new target sample.
The guide's running example uses the cached tree workspace. Replacing it with distributed_tree_solve changes where the triangular sweeps execute, not the reference seen by the filter. A conic or differentiable IPM can occupy the same interface so long as it supplies $q^\star(t)$ and, for feedforward, a valid $\dot q^\star(t)$.
Multi-agent communication is contained in the map $p\mapsto q^\star$. The normalized filter acts blockwise on the completed reference, and agent $i$ only needs $x_i$ afterward. No global state is introduced into the onboard execution loop.
Dynamic realization
Let the target motion be slow relative to planner delivery and introduce
\[\epsilon\dot x=-x+q^\star(t).\]
Freezing $t$ and rescaling time by $\tau=t/\epsilon$ gives the boundary layer
\[\frac{dx}{d\tau}=-x+q^\star.\]
Its unique equilibrium is $x=q^\star$. Setting $\epsilon=0$ in the full model recovers the algebraic manifold exactly:
\[0=-x+q^\star(t) \quad\Longrightarrow\quad x=q^\star(t)=H^{-1}(-Bp(t)).\]
The overview figure shows this geometry directly. Its black curve is one agent's two-dimensional harmonic reference generated by the cached clique-tree solve. The four colored planner trajectories begin from different initial states and contract onto that moving reference.
For an agent: the manifold is the sequence of globally consistent slots. A fast fiber is the transient by which your delivered slot catches the current global answer.
Sensitivity supplies reference velocity
When $H$ is fixed, differentiating the harmonic equation gives
\[H\dot q^\star=-B\dot p, \qquad \dot q^\star=H^{-1}(-B\dot p).\]
Position and velocity therefore pass through the same cached factorization. For a differentiable conic program, this role is played by implicit differentiation of its KKT conditions. The filter does not care which solver produced the pair $(q^\star,\dot q^\star)$; it only relies on their meaning.
Fixed topology is a real assumption
If an agent joins, an edge disappears, or a restriction map changes, then $H$ changes. Both the harmonic sensitivity identity and factor reuse must be revisited. A slowly switching topology can be analyzed as an additional input or hybrid mode, but it is not covered by the fixed-manifold theorem used here.