Chapter map

1. 1. Annotated single-cell counts
2. 2. Shared-gene vocabulary
3. 3. Topic number
4. 4. Dirichlet prior on cell topics
5. 5. Dirichlet prior on gene topics
6. 6. Topic assignment
7. 7. Topic-gene matrix
8. 8. Cell-topic matrix
9. 9. Cell-type topic profile
10. 10. Spatial topic inference
11. 11. Profile-to-type mapping
12. 12. Rare-cell sensitivity
13. 13. Topic-based spatial domains
14. 14. Cross-section similarity
15. 15. Three-dimensional reconstruction
16. 16. End-to-end validation

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1. Annotated single-cell counts

Here the geometry becomes operational. Cells supply gene counts and type labels.

A concrete situation

Thirty thousand cells cover twelve types.

What the mathematics does not license

Reference imbalance affects topic learning. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Balance and document classes. 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. Shared-gene vocabulary

This step deserves a careful pause. Common genes define the topic vocabulary.

A concrete situation

Eight thousand genes remain.

What the mathematics does not license

Missing genes remove distinguishing evidence. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Publish the gene set. 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. Topic number

The useful question is not only how to calculate, but what the calculation assumes. K topics define latent resolution.

A concrete situation

The model uses 30 topics.

What the mathematics does not license

K can merge or split biology. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Compare stability across 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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4. Dirichlet prior on cell topics

A small example makes the mechanism visible. Alpha shapes how many topics each cell uses.

A concrete situation

Small alpha favours sparse mixtures.

What the mathematics does not license

Prior strength can dominate low counts. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Sensitivity-test alpha. 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. Dirichlet prior on gene topics

This is where modelling choices begin to affect evidence. Eta regularizes topic-gene probabilities.

A concrete situation

Every gene starts with symmetric mass.

What the mathematics does not license

Symmetry may not reflect gene prevalence. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Inspect posterior concentration. 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. Topic assignment

The central mathematical idea is concrete. Each molecule selects a latent topic.

A concrete situation

Count token m in cell d uses topic z.

What the mathematics does not license

Tokens are exchangeable within a cell. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

State the bag-of-genes assumption. 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. Topic-gene matrix

Here the geometry becomes operational. Beta gives gene probabilities for every topic.

A concrete situation

Topic 7 loads on immune genes.

What the mathematics does not license

A topic can represent state rather than type. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Use enrichment cautiously. 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. Cell-topic matrix

This step deserves a careful pause. Theta gives topic proportions for each reference cell.

A concrete situation

Cell i is 80% topic 7.

What the mathematics does not license

Posterior uncertainty varies by depth. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Retain uncertainty or repeated fits. 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. Cell-type topic profile

The useful question is not only how to calculate, but what the calculation assumes. Annotated cells are aggregated by type.

A concrete situation

Macrophage profile averages 500 cells.

What the mathematics does not license

Large classes can appear more stable. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Use stratified estimates. 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. Spatial topic inference

A small example makes the mechanism visible. The trained topic model estimates each spot’s topic mixture.

A concrete situation

Spot j is 40% topics 7 and 9.

What the mathematics does not license

Platform shift changes word frequencies. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Test held-out protocols. 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. Profile-to-type mapping

This is where modelling choices begin to affect evidence. Spot topics are expressed through cell-type topic profiles.

A concrete situation

Nonnegative weights sum to one.

What the mathematics does not license

Related types may be collinear. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Inspect identifiability and residuals. 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. Rare-cell sensitivity

The central mathematical idea is concrete. Small weights are compared with known simulated truth.

A concrete situation

A 3% type is recovered in 70% of mixtures.

What the mathematics does not license

Thresholds trade recall and false positives. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Plot both curves. 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. Topic-based spatial domains

Here the geometry becomes operational. Spot mixtures support clustering or signature maps.

A concrete situation

Neighbouring topic profiles form a domain.

What the mathematics does not license

Smooth anatomy can be imposed by preprocessing. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Validate boundaries independently. 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. Cross-section similarity

This step deserves a careful pause. Sections are compared in topic space.

A concrete situation

Adjacent sections have high topic correlation.

What the mathematics does not license

Orientation and sampling gaps confound similarity. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Keep physical registration metadata. 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. Three-dimensional reconstruction

The useful question is not only how to calculate, but what the calculation assumes. Aligned sections form a tissue stack.

A concrete situation

Ten ordered sections define a volume.

What the mathematics does not license

Between-section structure is inferred. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Label interpolation explicitly. 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. End-to-end validation

A small example makes the mechanism visible. Simulations, held-out genes and real anatomy test each stage.

A concrete situation

A complete section is withheld.

What the mathematics does not license

Good deconvolution does not guarantee correct 3D alignment. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Report stage-specific evidence. 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

STRIDE transfers topic structure from annotated single-cell RNA sequencing to spatial mixtures. In latent Dirichlet allocation, each cell is a mixture of topics and each topic is a probability distribution over genes. Dirichlet priors regularize topic weights, while multinomial count models connect latent topics to observed molecules. The topics can capture programs shared across related cell types instead of relying on a short marker list.

The annotated reference converts cell-level topic weights into cell-type topic profiles. The trained topic-gene distributions are then applied to each spatial spot, producing a spot-level topic mixture. STRIDE infers cell-type proportions by comparing the spot topic mixture with the reference cell-type topic profiles under its declared model and constraints.

Choosing the topic number matters. Too few topics merge distinct cell types and states; too many can split noise or platform effects into apparently meaningful components. Sparse counts, sequencing depth, class imbalance and reference mismatch also affect posterior topic weights. Rare-cell recovery needs explicit sensitivity and specificity checks, not only a visually plausible map.

Topic profiles can support downstream spatial-domain analysis and alignment of successive tissue sections, but a reconstructed three-dimensional architecture is still an inference built from topic similarity and section order. Topic stability, held-out genes, simulated mixtures, real-tissue markers and section-level holdouts should all be examined before making biological claims.

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Frequently asked questions

What is STRIDE?

STRIDE is a topic-model-based method that learns cell-type-associated gene topics from annotated single-cell data and transfers them to spatial spots.

What is a topic?

A topic is a probability distribution over genes; each cell or spot has a probability mixture over topics.

Why use Dirichlet priors?

They regularize multinomial topic weights and encode how concentrated or diffuse mixtures are expected to be.

How do topics become cell types?

Reference-cell topic weights are summarized by annotation to create cell-type topic profiles, which support spatial proportion inference.

Why can topic number change the answer?

Too few topics merge signals; too many can split states, noise or technical effects into separate components.

What does section integration add?

Topic similarity can align successive tissue sections for downstream structure reconstruction, but it does not directly observe the missing volume between sections.

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Primary source and fact-check note

Dongqing Sun, Zhaoyang Liu, Taiwen Li, Qiu Wu, Chenfei Wang and colleagues, “STRIDE: accurately decomposing and integrating spatial transcriptomics using single-cell RNA sequencing,” Nucleic Acids Research 50(7), e42 (2022), DOI 10.1093/nar/gkac150. 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

• STdeconvolve reference-free topics
• SPOTlight topic profiles
• SpatialDWLS weighted mixtures
• 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

Reference audit

Version annotations, depths and class balance.

Topic audit

Publish K, priors, initialization and convergence.

Profile audit

Store cell-level and cell-type topic matrices.

Mapping audit

Inspect collinearity, residuals and simplex constraints.

Rare-type audit

Report sensitivity and specificity by abundance.

Domain audit

Withhold donors, tissues and platforms.

Section audit

Separate deconvolution from geometric reconstruction.

Final claim

Topic similarity supports inference, not direct observation.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Annotated single-cell counts

Begin with this mechanism: Cells supply gene counts and type labels. Recreate the concrete case: Thirty thousand cells cover twelve types. Write count tokens, topic assignments, Dirichlet priors, topic-gene probabilities, cell-topic mixtures, cell-type profiles, spot-topic mixtures, composition weights, section similarities and validation errors with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Reference imbalance affects topic learning. Complete the transfer step: Balance and document classes. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 2: Shared-gene vocabulary

Begin with this mechanism: Common genes define the topic vocabulary. Recreate the concrete case: Eight thousand genes remain. Write count tokens, topic assignments, Dirichlet priors, topic-gene probabilities, cell-topic mixtures, cell-type profiles, spot-topic mixtures, composition weights, section similarities and validation errors with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Missing genes remove distinguishing evidence. Complete the transfer step: Publish the gene set. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 3: Topic number

Begin with this mechanism: K topics define latent resolution. Recreate the concrete case: The model uses 30 topics. Write count tokens, topic assignments, Dirichlet priors, topic-gene probabilities, cell-topic mixtures, cell-type profiles, spot-topic mixtures, composition weights, section similarities and validation errors with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: K can merge or split biology. Complete the transfer step: Compare stability across K. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 4: Dirichlet prior on cell topics

Begin with this mechanism: Alpha shapes how many topics each cell uses. Recreate the concrete case: Small alpha favours sparse mixtures. Write count tokens, topic assignments, Dirichlet priors, topic-gene probabilities, cell-topic mixtures, cell-type profiles, spot-topic mixtures, composition weights, section similarities and validation errors with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Prior strength can dominate low counts. Complete the transfer step: Sensitivity-test alpha. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 5: Dirichlet prior on gene topics

Begin with this mechanism: Eta regularizes topic-gene probabilities. Recreate the concrete case: Every gene starts with symmetric mass. Write count tokens, topic assignments, Dirichlet priors, topic-gene probabilities, cell-topic mixtures, cell-type profiles, spot-topic mixtures, composition weights, section similarities and validation errors with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Symmetry may not reflect gene prevalence. Complete the transfer step: Inspect posterior concentration. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 6: Topic assignment

Begin with this mechanism: Each molecule selects a latent topic. Recreate the concrete case: Count token m in cell d uses topic z. Write count tokens, topic assignments, Dirichlet priors, topic-gene probabilities, cell-topic mixtures, cell-type profiles, spot-topic mixtures, composition weights, section similarities and validation errors with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Tokens are exchangeable within a cell. Complete the transfer step: State the bag-of-genes assumption. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 7: Topic-gene matrix

Begin with this mechanism: Beta gives gene probabilities for every topic. Recreate the concrete case: Topic 7 loads on immune genes. Write count tokens, topic assignments, Dirichlet priors, topic-gene probabilities, cell-topic mixtures, cell-type profiles, spot-topic mixtures, composition weights, section similarities and validation errors with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: A topic can represent state rather than type. Complete the transfer step: Use enrichment cautiously. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 8: Cell-topic matrix

Begin with this mechanism: Theta gives topic proportions for each reference cell. Recreate the concrete case: Cell i is 80% topic 7. Write count tokens, topic assignments, Dirichlet priors, topic-gene probabilities, cell-topic mixtures, cell-type profiles, spot-topic mixtures, composition weights, section similarities and validation errors with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Posterior uncertainty varies by depth. Complete the transfer step: Retain uncertainty or repeated fits. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 9: Cell-type topic profile

Begin with this mechanism: Annotated cells are aggregated by type. Recreate the concrete case: Macrophage profile averages 500 cells. Write count tokens, topic assignments, Dirichlet priors, topic-gene probabilities, cell-topic mixtures, cell-type profiles, spot-topic mixtures, composition weights, section similarities and validation errors with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Large classes can appear more stable. Complete the transfer step: Use stratified estimates. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 10: Spatial topic inference

Begin with this mechanism: The trained topic model estimates each spot’s topic mixture. Recreate the concrete case: Spot j is 40% topics 7 and 9. Write count tokens, topic assignments, Dirichlet priors, topic-gene probabilities, cell-topic mixtures, cell-type profiles, spot-topic mixtures, composition weights, section similarities and validation errors with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Platform shift changes word frequencies. Complete the transfer step: Test held-out protocols. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 11: Profile-to-type mapping

Begin with this mechanism: Spot topics are expressed through cell-type topic profiles. Recreate the concrete case: Nonnegative weights sum to one. Write count tokens, topic assignments, Dirichlet priors, topic-gene probabilities, cell-topic mixtures, cell-type profiles, spot-topic mixtures, composition weights, section similarities and validation errors with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Related types may be collinear. Complete the transfer step: Inspect identifiability and residuals. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 12: Rare-cell sensitivity

Begin with this mechanism: Small weights are compared with known simulated truth. Recreate the concrete case: A 3% type is recovered in 70% of mixtures. Write count tokens, topic assignments, Dirichlet priors, topic-gene probabilities, cell-topic mixtures, cell-type profiles, spot-topic mixtures, composition weights, section similarities and validation errors with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Thresholds trade recall and false positives. Complete the transfer step: Plot both curves. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 13: Topic-based spatial domains

Begin with this mechanism: Spot mixtures support clustering or signature maps. Recreate the concrete case: Neighbouring topic profiles form a domain. Write count tokens, topic assignments, Dirichlet priors, topic-gene probabilities, cell-topic mixtures, cell-type profiles, spot-topic mixtures, composition weights, section similarities and validation errors with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Smooth anatomy can be imposed by preprocessing. Complete the transfer step: Validate boundaries independently. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 14: Cross-section similarity

Begin with this mechanism: Sections are compared in topic space. Recreate the concrete case: Adjacent sections have high topic correlation. Write count tokens, topic assignments, Dirichlet priors, topic-gene probabilities, cell-topic mixtures, cell-type profiles, spot-topic mixtures, composition weights, section similarities and validation errors with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Orientation and sampling gaps confound similarity. Complete the transfer step: Keep physical registration metadata. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 15: Three-dimensional reconstruction

Begin with this mechanism: Aligned sections form a tissue stack. Recreate the concrete case: Ten ordered sections define a volume. Write count tokens, topic assignments, Dirichlet priors, topic-gene probabilities, cell-topic mixtures, cell-type profiles, spot-topic mixtures, composition weights, section similarities and validation errors with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Between-section structure is inferred. Complete the transfer step: Label interpolation explicitly. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 16: End-to-end validation

Begin with this mechanism: Simulations, held-out genes and real anatomy test each stage. Recreate the concrete case: A complete section is withheld. Write count tokens, topic assignments, Dirichlet priors, topic-gene probabilities, cell-topic mixtures, cell-type profiles, spot-topic mixtures, composition weights, section similarities and validation errors with direction, units, denominators and assumptions. Calculate the smallest non-trivial example by hand, then reproduce it in a spreadsheet or short program. Challenge this limitation: Good deconvolution does not guarantee correct 3D alignment. Complete the transfer step: Report stage-specific evidence. Change exactly one reference donor, feature filter, normalization, mixture size, weight distribution, latent dimension, divergence convention, domain-loss weight, learning rate, gene-frequency cutoff, overdispersion threshold, topic number, Dirichlet prior, PCA dimension, principal-vector threshold, neighbourhood size, distance metric, section holdout or random seed. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, latent assumption, objective, optimization, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.