Adjoint Bridges Under Clock Uncertainty

Conceptual map

I-10.01Inverse problems for queries
I-10.02Range, closure, and stability
I-10.03Common bridges
I-10.04Multiple clocks and cohorts
I-10.05Finite windows and boundary tails

1. A query can be recovered through an adjoint equation

Linear observations may discard the latent response while preserving a specified payoff. The payoff is recoverable exactly when its coefficient lies in the adjoint range.

Let a latent response be $g\in \mathbb{R}^{n}$ , let the observed clock-aggregated response be $$y=Kg$$ with $K\in \mathbb{R}^{m\times n}$ , and let the scalar query be $\theta=c^{\top}g$ . A bridge is a vector $b\in \mathbb{R}^{m}$ such that $\theta=b^{\top}y$ for every $$g$$ .

Proposition 1 · Exact finite-dimensional bridge

The following statements are equivalent: (i) a bridge exists; (ii) $K^{\top}b=c$ has a solution; (iii) $$c$$ belongs to $\operatorname{range}(K^{\top})$ ; and (iv) $c^{\top}v=0$ for every $v\in \operatorname{ker}(K)$ .

Proof. Equality $b^{\top}Kg=c^{\top}g$ for every $$g$$ is equivalent to $K^{\top}b=c$ . The fundamental theorem of linear algebra gives $\operatorname{range}(K^{\top})$ =ker $(K)^{\perp}$ . ∎

2. Total response survives a three-to-two clock

Consider

\[K= \begin{bmatrix}1 & \frac{1}{2} & 0 \\ 0 & \frac{1}{2} & 1\end{bmatrix}, \quad c=(1,1,1)^{\top }.\](1)

The middle latent bin is split equally across two recorded bins. The bridge $b=(1,1)^{\top}$ satisfies $K^{\top}b=c$ , so the latent total equals the observed total. The kernel is spanned by $(-1/2,1,-1/2)^{\top}$ , which is orthogonal to $$c$$ .

The first-bin query $d=(1,0,0)^{\top}$ has no exact bridge: the equations require $b_{1}=1$ from column one, $b_{2}=0$ from column three, and ( $b_{1}+b_{2}$ )/2=0 from column two. The last equality fails. Equivalently, $$d$$ has inner product $$-1/2$$ with the kernel vector.

3. Noise selects among exact bridges

Suppose the observed response is $Y=Kg+\varepsilon$ with mean-zero noise covariance $\Omega\succ0$ . Among exact bridges, the minimum-variance bridge solves

\[\operatorname{min}_{b} b^{\top }\Omega b \text{subject to} K^{\top }b=c.\](2)

If the query is in the adjoint range, one solution is $\Omega ^{-1}K(K^{\top}\Omega ^{-1}K)^{+}c$ . The pseudoinverse acts on the estimable subspace. In computation, solve the equality-constrained system using a rank-revealing factorization and verify both the adjoint residual and the constraint rank.

For $\Omega=I_{2}$ in (1), the exact total bridge is unique and its noise variance is two. A small singular value in the relevant adjoint direction can make the bridge norm large even when the range condition holds.

4. Multiple clocks require a common bridge

Let $K_{1},\ldots,K_{L}$ be admissible clock operators with a shared observed coordinate system. A clock-invariant linear report uses one vector $$b$$ and therefore requires $K_{\ell}^{\top}b=c$ for every $\ell$ . Stack these equations before solving.

For $K_{1}=I_{2}$ and $K_{2}=\begin{bmatrix}1 & 1/2 \\ 0 & 1/2\end{bmatrix}$ , the total query $c=(1,1)^{\top}$ has common bridge $b=(1,1)^{\top}$ . The first-period query $c=(1,0)^{\top}$ forces $b=(1,0)^{\top}$ under $K_{1}$ , whose second adjoint image is $(1,1/2)^{\top}$ ; hence a common bridge is absent.

Failure case · Least-squares residual treated as exact recovery

For an approximate bridge, the worst query bias over $\lVert g\rVert \le R$ equals $R\lVert c-K^{\top}b\rVert$ in the matching Euclidean dual norm. Its numerical size depends on the declared latent-response set and norm. A short residual alone supplies no scale-free recovery claim.

A finite observation window creates unobserved tail coordinates. If their query coefficient and response are $c_{T}$ and $g_{T}$ , the missing contribution is bounded by $\lVert c_{T}\rVert \lVert g_{T}\rVert$ once a tail norm bound is justified.

5. Implementation, exercises, and sources

Construct each clock matrix with its column and row conventions documented. Compute rank, null-space residuals, adjoint residuals, bridge norm, noise variance, and tail bound. Compare exact and least-squares solutions under one fixed norm and tolerance.

Download the volume verification script →

Exercises

Verify both query conclusions for matrix (1) by the range and kernel tests.
Derive the minimum-variance formula in Section 3 from the Lagrangian.
For the two clocks in Section 4, characterize every query admitting a common bridge.

Partial solutions

1. Multiplication gives $K^{\top}(1,1)^{\top}=(1,1,1)^{\top}$ . The kernel witness separates the first-bin query. 2. Stationarity gives $\Omega b=K\lambda$ ; substitution into the constraint yields $K^{\top}\Omega ^{-1}K\lambda=c$ . 3. Since $K_{1}$ forces $$b=c$$ , the second equation requires $c_{1}=c_{2}$ .

Gene H. Golub and Charles F. Van Loan, Matrix Computations, chapters on least squares and constrained least squares.Rank-revealing solution of adjoint systems.
Heinz W. Engl, Martin Hanke, and Andreas Neubauer, Regularization of Inverse Problems, Chapters 2–3.Range, generalized inverses, and stability.
III-12 · Ill-Posed Inverse Problems and Regularization.Closure, singular systems, and regularized recovery.
Chae-Yeon Xon (2026), “Fuzzy Clock: Decision-Specific Timing Information.”Related use of common payoff bridges under clock uncertainty.

6. Audit checkpoint

State latent and observed spaces, operator orientation, query, admissible clocks, noise covariance, exact range test, rank tolerance, adjoint residual, bridge variance, latent-set norm for approximation bias, observation window, and tail bound.

7. Scope boundary

The chapter proves finite-matrix query recovery and common-bridge conditions. Infinite-dimensional closure, regularization rates, and influence functions require the operator and inference volumes.