# N0751 — a differential swelling sensor: exact response and limits

**Proved:** the linear two-path model results below. **Not proved:** that a real swelling material obeys this model, chemical specificity, universal detection of arbitrarily fast changes, or novelty.

## Original proposal

A mechanical differentiator could detect rapid chemical changes through the mismatch of two swelling pathways.

A concrete interpretation is a mechanically read difference of two displacement paths responding to the same concentration c(t), with equal calibrated equilibrium gain g>0 and different positive time constants τ_f<τ_s:

    τ_f dx_f/dt+x_f=g c(t),
    τ_s dx_s/dt+x_s=g c(t),
    y=x_f−x_s.

The material, input delivery, mechanical readout, and operating range must be experimentally shown compatible with this approximation. The two paths need equal equilibrium gain, not merely equal dimensions.

## Theorem 1: rejection of a constant concentration

For a constant concentration c₀ at equilibrium, both paths equal gc₀, so y=0. For a step c(t)=A from an initially equilibrated zero baseline,

    y(t)=gA[exp(−t/τ_s)−exp(−t/τ_f)], t≥0.

This follows by solving each first-order equation with an integrating factor and subtracting. For A>0, y is positive at every finite t>0, and tends to zero both initially and at long times. Thus a concentration change produces a transient while a constant equilibrated input is rejected. ∎

The peak occurs at

    t*=τ_f τ_s ln(τ_s/τ_f)/(τ_s−τ_f).

Differentiating y and solving dy/dt=0 gives this expression; the derivative changes from positive to negative there, so it is the unique peak.

For g=A=1, τ_f=1, τ_s=4, t*=(4/3)ln4 and peak y=(3/4)4^(−1/3)≈0.47247. These are arbitrary-unit model values, not material measurements.

## Theorem 2: the exact frequency band

For a steady sinusoid of angular frequency ω, substitution of a complex exponential yields

    H(iω)=g iω(τ_s−τ_f)/[(1+iωτ_f)(1+iωτ_s)].

Therefore

    |H|=g |ω|(τ_s−τ_f)/sqrt[(1+ω²τ_f²)(1+ω²τ_s²)].

The response vanishes at ω=0 and tends to zero as |ω|→∞. This device is a band-pass change sensor, not an unlimited-bandwidth differentiator.

To find the maximum, set z=ω². Apart from a positive constant, squared gain is

    F(z)=z/[(1+τ_f²z)(1+τ_s²z)].

Its derivative has the sign of 1−τ_f²τ_s²z², so the unique positive maximum occurs at

    ω*=1/sqrt(τ_f τ_s),
    |H(iω*)|=g(τ_s−τ_f)/(τ_s+τ_f). ∎

## Theorem 3: a controlled derivative approximation

Assume the stable causal responses are driven by a bounded history with sufficiently smooth c. Integrating the convolution by parts gives

    x_τ(t)=g c(t)−g∫₀∞ exp(−s/τ)c′(t−s)ds.

Thus

    y(t)=g∫₀∞ [exp(−s/τ_s)−exp(−s/τ_f)]c′(t−s)ds.

If |c″|≤K over the relevant history, subtract g(τ_s−τ_f)c′(t). The mean-value bound |c′(t−s)−c′(t)|≤Ks gives

    |y−g(τ_s−τ_f)c′(t)|
       ≤gK∫₀∞ s[exp(−s/τ_s)−exp(−s/τ_f)]ds
       =gK(τ_s²−τ_f²).

The kernel is nonnegative because τ_s>τ_f. This proves a bounded approximation to a scaled derivative for inputs whose derivative changes sufficiently slowly. It is not an approximation for arbitrary sharp inputs; use the exact step or frequency response there. ∎

## Counterexamples and physical limits

- If the time constants are equal, the matched paths produce y=0 for every input: no change signal exists.
- If gains differ, a constant concentration leaves y=(g_f−g_s)c₀. Calibration mismatch mimics a change or drift.
- Arbitrarily high-frequency changes are attenuated, not amplified.
- Temperature sensitivity, hysteresis, diffusion modes, input-delivery lag, changing geometry, and mechanical inertia can invalidate the two-exponential model.
- A synthetic transient does not demonstrate chemical specificity or detection at a stated noise level. Both require independent physical calibration and blinded tests.

The exact output is therefore predictable only within the specified model. Existing swelling kinetics and linear filtering principles are the foundation; no new physical law or verified original device is claimed.


# 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.
