WHY MATHEMATICS?

Why Mathematics? | CellFM for Single-Cell AI: Continuous Value Projection, ERetNet, Masked Gene Recovery and LoRA Adaptation

How CellFM turns scalar gene expression into high-dimensional value vectors, learns gene context with an efficient retention network, reconstructs masked values and adapts an 800-million-parameter model through low-rank updates.

Why mathematics is important becomes easier to see when a real decision depends on it. This guide follows the mechanism from definitions and dimensions to a worked calculation, validation, practical learning and responsible use. It keeps factual evidence separate from interpretation and treats every illustrative scenario as an example rather than a promise of results.

Choose a reading route

Project continuous expression values without hard bins
Understand ERetNet retention, gates and stable normalization
Calculate masked reconstruction and cell-level losses
Adapt with LoRA while testing transfer, error and corpus limits

Chapter map

1. Curating a 100-million-cell corpus
2. A fixed gene vocabulary
3. The 2,048-gene limit
4. Scalar-to-vector projection
5. Gene identity plus value
6. CLS cell token
7. Twenty-percent masking
8. Linear-complexity retention
9. Gated multi-head processing
10. Simple gated linear unit
11. DeepNorm residual stability
12. Token-level reconstruction loss
13. CLS-based reconstruction loss
14. Worked four-gene recovery
15. LoRA matrix arithmetic
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.

1. Curating a 100-million-cell corpus

Here the geometry becomes operational. Files from several repositories must be quality-controlled, gene-standardized and converted to one sparse representation.

A concrete situation

The paper reports 102,304,686 cells across 19,914 samples.

What the mathematics does not license

Large corpora remain uneven by tissue and technology. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Publish cells, samples and exclusions. 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.

2. A fixed gene vocabulary

This step deserves a careful pause. Gene symbols map to 24,079 learned identity vectors under one versioned ordering.

A concrete situation

A symbol alias resolves to one gene ID.

What the mathematics does not license

A one-column shift corrupts every token. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Checksum vocabulary and aliases. 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.

3. The 2,048-gene limit

The useful question is not only how to calculate, but what the calculation assumes. If more than 2,048 genes are expressed, CellFM randomly selects highly expressed genes; shorter cells receive padding.

A concrete situation

A 3,000-gene cell is shortened for parallel computation.

What the mathematics does not license

Sampling can omit rare markers. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Repeat seeds and measure retention. 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.

4. Scalar-to-vector projection

A small example makes the mechanism visible. LeakyReLU, a learned residual coefficient, softmax and matrices map one value through an intermediate width of 256 to dimension 1,536.

A concrete situation

Expression 2.4 becomes a 1,536-number vector.

What the mathematics does not license

A richer vector is still a learned encoding. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Plot vectors across input values. 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.

Did You Know? In this chapter, the most important number is useful only together with its construction rule. Changing scaling, sample membership or validation design can change the numerical answer without changing the arithmetic.

5. Gene identity plus value

This is where modelling choices begin to affect evidence. A learned gene embedding is added to the projected expression vector.

A concrete situation

Value 3 for gene A differs from value 3 for gene B.

What the mathematics does not license

Vector norms can let one component dominate. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Inspect identity and value ablations. 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.

6. CLS cell token

The central mathematical idea is concrete. A learned CLS vector is prepended so later layers can gather whole-cell information.

A concrete situation

The final CLS state supports cell-level prediction.

What the mathematics does not license

One vector may hide competing programmes. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Compare CLS with pooled gene states. 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.

7. Twenty-percent masking

Here the geometry becomes operational. A mask selects 20% of non-padding gene expressions and replaces their value vectors with a learned mask vector.

A concrete situation

Four of twenty eligible genes are hidden.

What the mathematics does not license

Masking only abundant detected genes can bias learning. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Report targets by abundance. 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.

8. Linear-complexity retention

This step deserves a careful pause. ERetNet rearranges query, key and value multiplication so K-transpose V is formed before queries, avoiding an explicit token-by-token matrix.

A concrete situation

Cost scales linearly with sequence length for fixed width.

What the mathematics does not license

Linear in length can still be expensive in width. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Benchmark runtime and memory. 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.

9. Gated multi-head processing

The useful question is not only how to calculate, but what the calculation assumes. Parallel heads produce retained context, then a swish gate controls the combined output.

A concrete situation

One head emphasizes a gene programme.

What the mathematics does not license

A head is not automatically a pathway. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Repeat seeds and ablate heads. 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.

10. Simple gated linear unit

A small example makes the mechanism visible. SGLU multiplies two learned linear projections before an output projection, creating feature interactions.

A concrete situation

Two channels must be jointly active to pass a signal.

What the mathematics does not license

Multiplicative gates can amplify scale. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Monitor activations and norms. 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.

11. DeepNorm residual stability

This is where modelling choices begin to affect evidence. Layer normalization and a scaled residual contribution stabilize deep post-norm blocks.

A concrete situation

A residual factor limits gradient growth.

What the mathematics does not license

Stable optimization does not prove good generalization. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Track gradients and held-out loss. 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.

12. Token-level reconstruction loss

The central mathematical idea is concrete. An MLP predicts each masked expression and MSE averages squared residuals over masked positions.

A concrete situation

Prediction 2.6 versus 3 gives squared error 0.16.

What the mathematics does not license

MSE weights large errors strongly. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Report error by gene range. 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.

13. CLS-based reconstruction loss

Here the geometry becomes operational. The mapped CLS state is dotted with target gene embeddings to produce a second expression estimate and MSE term.

A concrete situation

One cell vector predicts several masked genes.

What the mathematics does not license

A bottleneck can smooth rare signals. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Compare token and CLS errors. 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.

14. Worked four-gene recovery

This step deserves a careful pause. Project four values, mask one, calculate two squared errors and add the MSE terms.

A concrete situation

Token and CLS paths make different estimates.

What the mathematics does not license

Toy vectors hide optimization complexity. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Change the masked gene and recompute. 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.

15. LoRA matrix arithmetic

The useful question is not only how to calculate, but what the calculation assumes. A frozen matrix W0 receives a low-rank update BA with r much smaller than the original dimensions.

A concrete situation

A rank-eight update uses far fewer trainable values.

What the mathematics does not license

Too little rank can underfit; too much can overfit. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

Sweep rank with external data. 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.

16. Governance and learning

A small example makes the mechanism visible. Version corpus, genes, masks, ERetNet, losses and LoRA. This joins linear algebra, approximation and AI.

A concrete situation

Students can start with MSE and matrix rank.

What the mathematics does not license

A reconstructed value is not a measurement. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Write a CellFM factsheet. 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.

A seven-day practice lab

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

A CellFM audit begins with scalar expression values and an explicit gene list. A learned multilayer projection turns each value into a 1,536-dimensional vector, a gene-ID embedding identifies the feature, and a CLS token gathers cell-level context. The ERetNet backbone uses a linear-complexity retention calculation, gated multi-head processing, simple gated linear units and DeepNorm to make an 800-million-parameter model trainable across more than 100 million human cells.

Record the 24,079-gene vocabulary, expression preprocessing, 2,048-token limit, high-expression sampling rule, mask vector, gene and value embeddings, ERetNet checkpoint, normalization constants, CLS vector, MSE terms and LoRA rank. CellFM masks 20% of non-padding genes and learns to reconstruct their expression; padding must never enter the loss.

Validate on whole studies, technologies and tissues kept outside tuning; compare value projection with ranks and bins; test full fine-tuning against low-rank adaptation; and report rare-state as well as average error. Large scale can improve a representation while also concentrating corpus, preprocessing and hardware assumptions.

Frequently asked questions

What is CellFM?

CellFM is a human single-cell foundation model with about 800 million parameters. The 2025 Nature Communications paper reports training from 102,304,686 cells collected across 19,914 samples.

What is continuous value projection?

A learned multilayer map converts each scalar expression value into a high-dimensional vector. Gene-ID embeddings are then added so the same numeric value remains distinguishable across genes.

Why use ERetNet?

ERetNet modifies the RetNet idea with a linear-complexity retention calculation, gating, simple gated linear units and DeepNorm, reducing the quadratic burden of ordinary all-pairs attention.

What does CellFM predict during pretraining?

It masks 20% of non-padding gene expressions and reconstructs them. MSE is computed from gene-token outputs and from a CLS-based prediction path.

What is LoRA?

Low-Rank Adaptation freezes a large pretrained matrix and learns a small update BA, where the chosen rank is far smaller than the original dimensions.

Does 800 million parameters guarantee a better biological answer?

No. Scale increases capacity, while evidence still depends on corpus coverage, preprocessing, leakage controls, baselines and external validation.

Primary source and fact-check note

Yuansong Zeng, Jianciong Xie, Ningyuan Shangguan et al., “CellFM: a large-scale foundation model pre-trained on transcriptomics of 100 million human cells,” Nature Communications 16, 4679 (2025), DOI 10.1038/s41467-025-59926-5. 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

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

Corpus audit

Report sample, tissue, technology and donor composition; keep complete studies outside tuning.

Token-retention test

Repeat the high-expression selection and measure rare-gene and cell-state stability.

Value-projection audit

Plot projected vectors over the expression range and compare with bins and ranks.

Retention arithmetic check

Verify Q(K-transpose V), masks, scaling and complexity on a small tensor against a direct calculation.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Two-loss ledger

Publish token MSE and CLS MSE separately by gene abundance and tissue.

LoRA ladder

Compare frozen probes, several low ranks, partial tuning and full tuning under identical splits.

External validation

Use untouched studies and simple baselines for annotation, perturbation and gene-function tasks.

Final claim

Name corpus, value projection, mask, ERetNet and LoRA. Reconstructed values remain estimates.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Curating a 100-million-cell corpus

Start with this article's mechanism: Files from several repositories must be quality-controlled, gene-standardized and converted to one sparse representation. Work through the concrete case: The paper reports 102,304,686 cells across 19,914 samples. Make the boundary visible: Large corpora remain uneven by tissue and technology. Complete the evidence step: Publish cells, samples and exclusions. Write the input table or token list, transformation, output, dimensions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare masks, ranks, value vectors, retention outputs, logits, losses, embeddings, generated profiles, attention networks or calibration as relevant. Use complete donors, studies, tissues or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 2: A fixed gene vocabulary

Start with this article's mechanism: Gene symbols map to 24,079 learned identity vectors under one versioned ordering. Work through the concrete case: A symbol alias resolves to one gene ID. Make the boundary visible: A one-column shift corrupts every token. Complete the evidence step: Checksum vocabulary and aliases. Write the input table or token list, transformation, output, dimensions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare masks, ranks, value vectors, retention outputs, logits, losses, embeddings, generated profiles, attention networks or calibration as relevant. Use complete donors, studies, tissues or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 3: The 2,048-gene limit

Start with this article's mechanism: If more than 2,048 genes are expressed, CellFM randomly selects highly expressed genes; shorter cells receive padding. Work through the concrete case: A 3,000-gene cell is shortened for parallel computation. Make the boundary visible: Sampling can omit rare markers. Complete the evidence step: Repeat seeds and measure retention. Write the input table or token list, transformation, output, dimensions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare masks, ranks, value vectors, retention outputs, logits, losses, embeddings, generated profiles, attention networks or calibration as relevant. Use complete donors, studies, tissues or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 4: Scalar-to-vector projection

Start with this article's mechanism: LeakyReLU, a learned residual coefficient, softmax and matrices map one value through an intermediate width of 256 to dimension 1,536. Work through the concrete case: Expression 2.4 becomes a 1,536-number vector. Make the boundary visible: A richer vector is still a learned encoding. Complete the evidence step: Plot vectors across input values. Write the input table or token list, transformation, output, dimensions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare masks, ranks, value vectors, retention outputs, logits, losses, embeddings, generated profiles, attention networks or calibration as relevant. Use complete donors, studies, tissues or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 5: Gene identity plus value

Start with this article's mechanism: A learned gene embedding is added to the projected expression vector. Work through the concrete case: Value 3 for gene A differs from value 3 for gene B. Make the boundary visible: Vector norms can let one component dominate. Complete the evidence step: Inspect identity and value ablations. Write the input table or token list, transformation, output, dimensions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare masks, ranks, value vectors, retention outputs, logits, losses, embeddings, generated profiles, attention networks or calibration as relevant. Use complete donors, studies, tissues or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 6: CLS cell token

Start with this article's mechanism: A learned CLS vector is prepended so later layers can gather whole-cell information. Work through the concrete case: The final CLS state supports cell-level prediction. Make the boundary visible: One vector may hide competing programmes. Complete the evidence step: Compare CLS with pooled gene states. Write the input table or token list, transformation, output, dimensions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare masks, ranks, value vectors, retention outputs, logits, losses, embeddings, generated profiles, attention networks or calibration as relevant. Use complete donors, studies, tissues or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 7: Twenty-percent masking

Start with this article's mechanism: A mask selects 20% of non-padding gene expressions and replaces their value vectors with a learned mask vector. Work through the concrete case: Four of twenty eligible genes are hidden. Make the boundary visible: Masking only abundant detected genes can bias learning. Complete the evidence step: Report targets by abundance. Write the input table or token list, transformation, output, dimensions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare masks, ranks, value vectors, retention outputs, logits, losses, embeddings, generated profiles, attention networks or calibration as relevant. Use complete donors, studies, tissues or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 8: Linear-complexity retention

Start with this article's mechanism: ERetNet rearranges query, key and value multiplication so K-transpose V is formed before queries, avoiding an explicit token-by-token matrix. Work through the concrete case: Cost scales linearly with sequence length for fixed width. Make the boundary visible: Linear in length can still be expensive in width. Complete the evidence step: Benchmark runtime and memory. Write the input table or token list, transformation, output, dimensions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare masks, ranks, value vectors, retention outputs, logits, losses, embeddings, generated profiles, attention networks or calibration as relevant. Use complete donors, studies, tissues or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 9: Gated multi-head processing

Start with this article's mechanism: Parallel heads produce retained context, then a swish gate controls the combined output. Work through the concrete case: One head emphasizes a gene programme. Make the boundary visible: A head is not automatically a pathway. Complete the evidence step: Repeat seeds and ablate heads. Write the input table or token list, transformation, output, dimensions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare masks, ranks, value vectors, retention outputs, logits, losses, embeddings, generated profiles, attention networks or calibration as relevant. Use complete donors, studies, tissues or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 10: Simple gated linear unit

Start with this article's mechanism: SGLU multiplies two learned linear projections before an output projection, creating feature interactions. Work through the concrete case: Two channels must be jointly active to pass a signal. Make the boundary visible: Multiplicative gates can amplify scale. Complete the evidence step: Monitor activations and norms. Write the input table or token list, transformation, output, dimensions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare masks, ranks, value vectors, retention outputs, logits, losses, embeddings, generated profiles, attention networks or calibration as relevant. Use complete donors, studies, tissues or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 11: DeepNorm residual stability

Start with this article's mechanism: Layer normalization and a scaled residual contribution stabilize deep post-norm blocks. Work through the concrete case: A residual factor limits gradient growth. Make the boundary visible: Stable optimization does not prove good generalization. Complete the evidence step: Track gradients and held-out loss. Write the input table or token list, transformation, output, dimensions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare masks, ranks, value vectors, retention outputs, logits, losses, embeddings, generated profiles, attention networks or calibration as relevant. Use complete donors, studies, tissues or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 12: Token-level reconstruction loss

Start with this article's mechanism: An MLP predicts each masked expression and MSE averages squared residuals over masked positions. Work through the concrete case: Prediction 2.6 versus 3 gives squared error 0.16. Make the boundary visible: MSE weights large errors strongly. Complete the evidence step: Report error by gene range. Write the input table or token list, transformation, output, dimensions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare masks, ranks, value vectors, retention outputs, logits, losses, embeddings, generated profiles, attention networks or calibration as relevant. Use complete donors, studies, tissues or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 13: CLS-based reconstruction loss

Start with this article's mechanism: The mapped CLS state is dotted with target gene embeddings to produce a second expression estimate and MSE term. Work through the concrete case: One cell vector predicts several masked genes. Make the boundary visible: A bottleneck can smooth rare signals. Complete the evidence step: Compare token and CLS errors. Write the input table or token list, transformation, output, dimensions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare masks, ranks, value vectors, retention outputs, logits, losses, embeddings, generated profiles, attention networks or calibration as relevant. Use complete donors, studies, tissues or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 14: Worked four-gene recovery

Start with this article's mechanism: Project four values, mask one, calculate two squared errors and add the MSE terms. Work through the concrete case: Token and CLS paths make different estimates. Make the boundary visible: Toy vectors hide optimization complexity. Complete the evidence step: Change the masked gene and recompute. Write the input table or token list, transformation, output, dimensions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare masks, ranks, value vectors, retention outputs, logits, losses, embeddings, generated profiles, attention networks or calibration as relevant. Use complete donors, studies, tissues or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 15: LoRA matrix arithmetic

Start with this article's mechanism: A frozen matrix W0 receives a low-rank update BA with r much smaller than the original dimensions. Work through the concrete case: A rank-eight update uses far fewer trainable values. Make the boundary visible: Too little rank can underfit; too much can overfit. Complete the evidence step: Sweep rank with external data. Write the input table or token list, transformation, output, dimensions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare masks, ranks, value vectors, retention outputs, logits, losses, embeddings, generated profiles, attention networks or calibration as relevant. Use complete donors, studies, tissues or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.

Workshop 16: Governance and learning

Start with this article's mechanism: Version corpus, genes, masks, ERetNet, losses and LoRA. This joins linear algebra, approximation and AI. Work through the concrete case: Students can start with MSE and matrix rank. Make the boundary visible: A reconstructed value is not a measurement. Complete the evidence step: Write a CellFM factsheet. Write the input table or token list, transformation, output, dimensions, uncertainty and one observation that would contradict the conclusion. Repeat under a nearby defensible setting and compare masks, ranks, value vectors, retention outputs, logits, losses, embeddings, generated profiles, attention networks or calibration as relevant. Use complete donors, studies, tissues or technologies outside training when testing transfer, because random cells from one biological source are not independent evidence. Keep measurement, model estimate, interpretation and decision on separate notebook lines so a student, parent or reviewer can see what the mathematics establishes.