# N0755 — exact reconstruction in a closed, anchored thermal network

**Proved:** the conditional mathematical theorem below. **Not proved:** arbitrary recovery of missing physical measurements, the correctness of a particular thermal model, or novelty.

## Original proposal

A physical network could reconstruct missing boundary measurements by enforcing local energy balance.

This needs an identifiability condition. Conservation cannot create information about arbitrary unknown losses or multiple unconstrained boundary fluxes. We prove recovery of missing node temperatures when conductances, net powers at unobserved nodes, and sufficient temperature anchors are known.

## Model

Let a finite undirected graph have symmetric positive edge conductances k_ij. Partition nodes into measured-temperature anchors B and unobserved nodes U. At every node i in U, assume known net applied thermal power p_i and steady conduction:

    sum over neighbors j of k_ij(T_i−T_j)=p_i.

Any external heat loss at an unobserved node must already be included in p_i or represented by a calibrated connection to a known-temperature node. Conductances are constant and known. No unmodeled advection, radiation, storage, or contact drift is allowed.

Let L be the weighted graph Laplacian: L_ii=sum_j k_ij and L_ij=−k_ij for an edge. Then

    L_UU T_U = p_U−L_UB T_B.

A physically unmeasured boundary node can be placed in U only when its net power is independently known; missing temperature and missing heat exchange cannot both be silently assumed recoverable.

## Theorem 1: unique reconstruction

If every connected component containing an unobserved node also contains at least one anchor, L_UU is positive definite and

    T_U=L_UU⁻¹(p_U−L_UB T_B)

is the unique solution.

### Proof

For any vector x on U, extend it by zero on B, obtaining z. Counting each undirected edge once,

    xᵀL_UU x = zᵀLz = sum_edges k_ij(z_i−z_j)² ≥ 0.

If the sum is zero, each edge connects equal values. Every connected component touching U contains an anchor where z=0. Equality propagates along paths, so z=0 throughout that component and hence x=0. Thus the quadratic form is strictly positive for every nonzero x. A finite positive-definite matrix is invertible, giving existence and uniqueness of the displayed solution. ∎

If a connected component has no anchor, its constant-temperature vector lies in the Laplacian nullspace. Depending on source balance there are either no steady solutions or a family differing by an arbitrary constant. Absolute temperature is not uniquely recoverable there.

## Theorem 2: an anchor-error bound

Assume conductances and powers are exact, while every anchor temperature has error at most ε in absolute value. The reconstruction error e obeys

    L_UU e_U = −L_UB e_B.

At each unobserved node, e_i is a conductance-weighted average of neighboring errors. If an unobserved value exceeded ε, a global maximum exceeding ε would have to be shared by all its neighbors; otherwise its weighted average would be smaller. Following a path reaches an anchor, contradicting the anchor bound. Applying the same argument to −e gives

    max_i∈U |e_i| ≤ ε.

Thus harmonic reconstruction does not amplify bounded anchor-temperature error in the maximum norm when the other model inputs are exact. ∎

With power error δp as well, the general Euclidean bound is

    ||δT_U||₂ ≤ [||δp||₂ + ||L_UB||₂ ||δT_B||₂] / λ_min(L_UU).

This follows by subtracting the two linear systems and using the inverse norm of a symmetric positive-definite matrix. Unknown conductance errors require a separate analysis.

## Exact example including an unmeasured boundary

Take four nodes in a chain, every conductance equal to 1. Anchor T₀=0. Apply known power 4 at the other end, p₃=4, with p₁=p₂=0. Solving gives

    T₁=4, T₂=8, T₃=12.

The end temperature T₃ is recoverable without measuring it, because its net heating is known and a temperature anchor exists. These are constructed model values in consistent arbitrary units, not measured device data.

## Explicit failures

- Without any temperature anchor, adding a constant to all temperatures leaves all balances unchanged.
- A single node connected to an anchor with conductance k has T=T_anchor+p/k. Unknown heat input or loss makes its temperature unknown; a small k strongly amplifies power uncertainty.
- Knowing only q₁+q₂=P at a boundary does not determine q₁ and q₂ individually. Conservation alone is insufficient.
- A small balance residual does not prove the model is right: compensating parameter errors may cancel. Validation needs withheld physical measurements and independently calibrated parameters.

The proof supplies exact closure and failure conditions. It does not establish an original research result; related physically informed missing-node reconstruction is already present in the literature.