Chapter map

1. 1. Spatial nodes
2. 2. KNN edges
3. 3. Adjacency matrix
4. 4. Degree matrix
5. 5. Graph Laplacian
6. 6. Eigenvalues
7. 7. Eigenvectors
8. 8. Graph Fourier coefficients
9. 9. Spectral energy
10. 10. Bandlimit k
11. 11. Low-frequency SVG score
12. 12. Spectral reconstruction
13. 13. Imputation
14. 14. Downstream regularization
15. 15. Subcellular detail
16. 16. Governance and learning

Visit the How Mathematics Works hub for the wider series and its connections to education, reasoning, science, computing, design and careers.

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1. Spatial nodes

Here the geometry becomes operational. Each measured location becomes a graph vertex.

A concrete situation

Four tissue spots form four nodes.

What the mathematics does not license

Quality filtering changes topology. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record included nodes. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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2. KNN edges

This step deserves a careful pause. Spatial nearest neighbours define undirected connections.

A concrete situation

Each spot connects to about square-root n neighbours.

What the mathematics does not license

Large k smooths across barriers. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Plot edge lengths. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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3. Adjacency matrix

The useful question is not only how to calculate, but what the calculation assumes. A records which node pairs are connected.

A concrete situation

A four-node chain has three undirected edges.

What the mathematics does not license

Binary edges discard distance magnitude. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Compare weighted graphs. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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4. Degree matrix

A small example makes the mechanism visible. Diagonal D stores each node’s number or weight of neighbours.

A concrete situation

Middle chain nodes have degree two.

What the mathematics does not license

Boundary degrees differ. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Inspect degree distribution. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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5. Graph Laplacian

This is where modelling choices begin to affect evidence. L equals D minus A or a normalized variant.

A concrete situation

The chain Laplacian penalizes neighbour differences.

What the mathematics does not license

Normalization changes modes. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

State the exact Laplacian. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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6. Eigenvalues

The central mathematical idea is concrete. Nonnegative eigenvalues order modes from smooth to rough.

A concrete situation

Zero belongs to the constant mode in a connected graph.

What the mathematics does not license

Disconnected graphs have multiple zeros. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Check components first. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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7. Eigenvectors

Here the geometry becomes operational. Each eigenvector is a spatial Fourier mode on the graph.

A concrete situation

The second mode separates two broad sides.

What the mathematics does not license

Eigenvector signs are arbitrary. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Compare subspaces, not signs. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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8. Graph Fourier coefficients

This step deserves a careful pause. Inner products project a gene signal onto modes.

A concrete situation

A smooth gradient loads on low modes.

What the mathematics does not license

Coefficient magnitude depends on scaling. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Normalize consistently. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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9. Spectral energy

The useful question is not only how to calculate, but what the calculation assumes. Squared coefficients distribute signal energy by frequency.

A concrete situation

Eighty percent lies in early modes.

What the mathematics does not license

Energy alone ignores mean expression. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Report both. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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10. Bandlimit k

A small example makes the mechanism visible. The first k modes approximate the signal.

A concrete situation

Ten modes reconstruct a tissue layer.

What the mathematics does not license

A chosen k can blur rare structures. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Plot error versus k. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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11. Low-frequency SVG score

This is where modelling choices begin to affect evidence. Strong low-mode representation supports spatial structure.

A concrete situation

A layered marker ranks highly.

What the mathematics does not license

Graph construction affects ranking. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Repeat graph settings. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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12. Spectral reconstruction

The central mathematical idea is concrete. Selected coefficients transform back to node space.

A concrete situation

Low-pass reconstruction smooths measurement noise.

What the mathematics does not license

It also smooths true sharp boundaries. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Inspect residuals at edges. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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13. Imputation

Here the geometry becomes operational. Graph-smooth estimates fill weak or missing signals.

A concrete situation

A low-count spot borrows spectral structure.

What the mathematics does not license

Predictions are not measurements. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Map uncertainty and missingness. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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14. Downstream regularization

This step deserves a careful pause. Spectral features or penalties guide clustering and annotation.

A concrete situation

Low-frequency coordinates support a domain model.

What the mathematics does not license

Better benchmark accuracy may be dataset-specific. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Use held-out tissues. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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15. Subcellular detail

The useful question is not only how to calculate, but what the calculation assumes. Higher modes can represent small organelles or abrupt features.

A concrete situation

A compact landmark uses several frequencies.

What the mathematics does not license

High frequency also contains noise. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use independent microscopy. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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16. Governance and learning

A small example makes the mechanism visible. Version graph, Laplacian, scaling, k and downstream split.

A concrete situation

A student diagonalizes a four-node chain.

What the mathematics does not license

A Fourier mode is a basis, not a mechanism. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Write a SpaGFT spectral card. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

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A seven-day practice lab

1. Day 1: define the decision, observation unit, X variables and Y or reference condition.
2. Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
3. Day 3: build a transparent baseline and write what it assumes.
4. Day 4: fit the main method inside a training-only pipeline; record every parameter.
5. Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
6. Day 6: compare validation metrics with the practical workload created by errors.
7. Day 7: present one supported conclusion, one unresolved uncertainty and one safe next experiment.

Questions for a student, parent or mentor

• Which part of the model is a mathematical definition, and which part is a choice?
• What small calculation would catch a sign, scale or dimension error?
• What evidence comes from training, validation and the untouched test?
• What benign condition could look like a fault or prediction failure?
• How would the answer change for another response, asset, school project or time period?
• What should a human review before acting on the score?

A method-specific audit before deployment

SpaGFT interprets each gene or protein as a signal living on a spatial-neighbour graph. The graph Laplacian has eigenvectors that play the role of Fourier modes on an irregular tissue. Projecting expression onto these modes produces graph Fourier coefficients that reveal how much of the signal lies at smooth low frequencies or rapidly changing high frequencies.

Low-frequency modes vary slowly across connected neighbours and are therefore useful for recognizing spatially variable features as approximately bandlimited graph signals. Selecting the first k modes creates a controllable representation. Signal enhancement can retain low-frequency structure, but a sharp real boundary or tiny organelle may also require higher modes.

The same coefficients support imputation, denoising and features for downstream machine learning. Results depend on graph construction, degree normalization, k and spectral gaps. A frequency component is a mathematical basis direction; it is not a literal biological oscillation or a causal tissue process.

A student can build a four-node chain, write its adjacency, degree and Laplacian matrices, verify the constant lowest-frequency eigenvector and project one expression vector onto the eigenbasis. Reconstructing with one, two and four modes shows the bias-detail trade-off directly.

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Frequently asked questions

What is SpaGFT?

SpaGFT is a graph-signal-processing framework that represents spatial omics features with Fourier coefficients defined by a tissue-neighbour graph.

Why is a graph Fourier transform needed?

Tissue locations are irregular and do not form a simple evenly spaced line or image grid, so graph Laplacian eigenvectors provide geometry-adapted basis modes.

What is a low-frequency graph mode?

It changes slowly across strongly connected neighbouring nodes and has a small Laplacian eigenvalue.

How is one gene transformed?

Its expression vector is projected onto the orthonormal graph Fourier modes, producing one coefficient per mode.

What is a bandlimited signal?

It is approximated well by coefficients in a selected set of low-frequency modes, with higher-frequency coefficients set aside.

Does low frequency always mean biology and high frequency mean noise?

No. Smooth anatomy often occupies low modes, but sharp boundaries and small real structures may require higher modes; graph and validation decide.

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Primary source and fact-check note

Yufeng Chang, Jie Liu, Yuchao Jiang et al., “Graph Fourier transform for spatial omics representation and analyses of complex organs,” Nature Communications 15, 7467 (2024), DOI 10.1038/s41467-024-51590-5. Definitions and core claims in this article were checked against this primary publication. Calculations shown here were independently recomputed. Examples are educational illustrations; they are not reported industrial performance results.

Useful next reading

• SPARK-X kernel geometry
• SpatialPCA spatial covariance
• Hotspot neighbour-graph modules
• How Mathematics Works

Keep the options open. Mathematics education supports clearer questions, transferable problem-solving skills and informed choices across science, engineering, computing, AI, statistics, design and many other careers. It does not guarantee intelligence, income, admission or employment; those outcomes depend on far more than one subject or model.

Extended laboratory: topic-specific validation and responsible use

Graph audit

Compare k and weighted adjacency.

Component audit

Check disconnected regions.

Laplacian audit

State normalized or unnormalized form.

Spectrum audit

Plot eigenvalues and spectral gaps.

Band audit

Show reconstruction across k.

Boundary audit

Inspect high-frequency biological edges.

Transfer audit

Hold out tissues and modalities.

Final claim

Frequency is graph-relative, not causal.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Spatial nodes

Begin with this mechanism: Each measured location becomes a graph vertex. Work through the concrete case: Four tissue spots form four nodes. Make the limitation visible: Quality filtering changes topology. Complete the evidence step: Record included nodes. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 2: KNN edges

Begin with this mechanism: Spatial nearest neighbours define undirected connections. Work through the concrete case: Each spot connects to about square-root n neighbours. Make the limitation visible: Large k smooths across barriers. Complete the evidence step: Plot edge lengths. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 3: Adjacency matrix

Begin with this mechanism: A records which node pairs are connected. Work through the concrete case: A four-node chain has three undirected edges. Make the limitation visible: Binary edges discard distance magnitude. Complete the evidence step: Compare weighted graphs. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 4: Degree matrix

Begin with this mechanism: Diagonal D stores each node’s number or weight of neighbours. Work through the concrete case: Middle chain nodes have degree two. Make the limitation visible: Boundary degrees differ. Complete the evidence step: Inspect degree distribution. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 5: Graph Laplacian

Begin with this mechanism: L equals D minus A or a normalized variant. Work through the concrete case: The chain Laplacian penalizes neighbour differences. Make the limitation visible: Normalization changes modes. Complete the evidence step: State the exact Laplacian. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 6: Eigenvalues

Begin with this mechanism: Nonnegative eigenvalues order modes from smooth to rough. Work through the concrete case: Zero belongs to the constant mode in a connected graph. Make the limitation visible: Disconnected graphs have multiple zeros. Complete the evidence step: Check components first. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 7: Eigenvectors

Begin with this mechanism: Each eigenvector is a spatial Fourier mode on the graph. Work through the concrete case: The second mode separates two broad sides. Make the limitation visible: Eigenvector signs are arbitrary. Complete the evidence step: Compare subspaces, not signs. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 8: Graph Fourier coefficients

Begin with this mechanism: Inner products project a gene signal onto modes. Work through the concrete case: A smooth gradient loads on low modes. Make the limitation visible: Coefficient magnitude depends on scaling. Complete the evidence step: Normalize consistently. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 9: Spectral energy

Begin with this mechanism: Squared coefficients distribute signal energy by frequency. Work through the concrete case: Eighty percent lies in early modes. Make the limitation visible: Energy alone ignores mean expression. Complete the evidence step: Report both. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 10: Bandlimit k

Begin with this mechanism: The first k modes approximate the signal. Work through the concrete case: Ten modes reconstruct a tissue layer. Make the limitation visible: A chosen k can blur rare structures. Complete the evidence step: Plot error versus k. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 11: Low-frequency SVG score

Begin with this mechanism: Strong low-mode representation supports spatial structure. Work through the concrete case: A layered marker ranks highly. Make the limitation visible: Graph construction affects ranking. Complete the evidence step: Repeat graph settings. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 12: Spectral reconstruction

Begin with this mechanism: Selected coefficients transform back to node space. Work through the concrete case: Low-pass reconstruction smooths measurement noise. Make the limitation visible: It also smooths true sharp boundaries. Complete the evidence step: Inspect residuals at edges. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 13: Imputation

Begin with this mechanism: Graph-smooth estimates fill weak or missing signals. Work through the concrete case: A low-count spot borrows spectral structure. Make the limitation visible: Predictions are not measurements. Complete the evidence step: Map uncertainty and missingness. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 14: Downstream regularization

Begin with this mechanism: Spectral features or penalties guide clustering and annotation. Work through the concrete case: Low-frequency coordinates support a domain model. Make the limitation visible: Better benchmark accuracy may be dataset-specific. Complete the evidence step: Use held-out tissues. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 15: Subcellular detail

Begin with this mechanism: Higher modes can represent small organelles or abrupt features. Work through the concrete case: A compact landmark uses several frequencies. Make the limitation visible: High frequency also contains noise. Complete the evidence step: Use independent microscopy. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 16: Governance and learning

Begin with this mechanism: Version graph, Laplacian, scaling, k and downstream split. Work through the concrete case: A student diagonalizes a four-node chain. Make the limitation visible: A Fourier mode is a basis, not a mechanism. Complete the evidence step: Write a SpaGFT spectral card. Write the coordinates, pair distances, graph weights, node groups, kernels, covariance matrices, eigenvalues, Fourier coefficients, permutation statistics, module correlations or reconstruction errors with named units. Calculate one small example by hand, then vary one assumption and explain which result moves and why. Reserve complete tissues, donors, genes, fields of view or regions outside tuning when testing transfer. Keep measurement, fitted representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.