Test whether a summary can answer a question

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Explore the interactive diagram to move inputs with the same summary and change the question you want to recover.

Open the Recovery Laboratory to change the measurement matrix itself, ask several questions, inspect exact recovery rules or collision witnesses, and repair an insufficient summary. Its four-step guided investigation starts with addition, then introduces recoverable combinations, separate questions, and minimal repair. Prediction prompts include explanations. Read the matrix rows as equations, follow a numerical substitution, and compare a fiber picture with the selected answer's graph. Each graph has a visible description and exact value table; the controls work with the keyboard. Open “Notation, everyday meanings, and access help” for the symbols and interaction instructions.

Question: If a summary keeps only the sum of two coordinates, can it recover the first coordinate? This example needs only addition and the idea of a function.

1. Specify the summary and the requested answer

Work over the real numbers. Let

\[R(x_1,x_2)=x_1+x_2,\qquad F(x_1,x_2)=x_1. \]

Exact recovery for every input would require a function \(G\) with \(F=G\circ R\). The factorization glossary entry explains this notation. Example 7 develops the calculation.

2. Find a witness that recovery is impossible

Take \(u=(1,0)\) and \(v=(0,1)\). Their summaries agree, but their requested answers differ:

\[R(u)=R(v)=1,\qquad F(u)=1,\qquad F(v)=0. \]

Any recovery function would need both \(G(1)=1\) and \(G(1)=0\). This contradiction rules out every function of the sum alone, including nonlinear ones. The computation displays the witness. The contradiction explains why the witness settles this recovery question.

For the linear maps above, the difference \(w=u-v=(1,-1)\) gives the same issue in kernel notation:

\[Rw=0,\qquad Fw=1. \]

The summary erases a direction that still changes the desired answer.

3. Check a question the summary can answer

Ask instead for \(H(x_1,x_2)=2x_1+2x_2\). Then \(G_H(s)=2s\) works because \(G_H(R(x))=2(x_1+x_2)=H(x)\) for every input. The program checks the exact coefficient identity \([2][1\ \ 1]=[2\ \ 2]\).

The same summary can therefore be sufficient for one observation and insufficient for another. This example concerns exact recovery for all inputs. It does not measure prediction accuracy for a specified probability distribution.

4. Run the calculation

Follow the download or checkout instructions. From the collection root, run:

python -B examples/komorebi_demonstrations.py recovery

Expected output:

Inputs: (1, 0) and (0, 1)
Same summary: 1
Requested first coordinates: (1, 0)
Kernel witness: (1, -1)
R(w): 0
F(w): 1
Compatible observation coefficients: (2, 2)
PASS: recovery

5. Follow the argument and the wider subject

Continue to	Route and scope
Written argument	The collision argument and the library's factorization criterion.
Papers and books	Realization Boundaries and Paper III develop related questions about compatibility and obstructions.
A further claim	Paper III's nonlinear refinement defect and defect under composition introduce a different, explicitly defined setting.
Written proof of that further claim	Readable composition theorem and proof. Keep its hypotheses when moving beyond this example.
Lean	Paper III coverage and Realization Boundaries coverage. No theorem-to-Lean correspondence is asserted for this standalone demonstration.
Program	Read or download the complete program, then explore the library code and instructions.
More hand calculations	Kernels and fibers, hidden memory and repair directions.

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