WHY MATHEMATICS?

Why Mathematics? | CytoSPACE Tissue Cartography: Linear Assignment, Correlation Costs, Sub-Spots and Shortest Augmenting Paths

How CytoSPACE converts cell-type abundance and transcriptomic concordance into a constrained global assignment that maps single cells to spatial sub-spots.

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

Match reference-cell counts to estimated tissue demand
Build a cell-by-sub-spot correlation cost matrix
Solve a one-to-one linear assignment globally
Measure stability because solver optimality is not mapping certainty

Chapter map

1. Single-cell atlas
2. Spatial expression spots
3. Cell-type fraction estimate
4. Cells per spot
5. Sub-spot expansion
6. Reference-cell sampling
7. Shared-gene matrix
8. Expression standardization
9. Correlation concordance
10. Cost matrix
11. Binary assignment variable
12. Cell-use constraint
13. Slot-capacity constraint
14. Global objective
15. Shortest augmenting path
16. Stability and validation

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

1. Single-cell atlas

A small example makes the mechanism visible. Annotated reference cells provide full transcriptomes.

A concrete situation

The atlas has 20,000 cells across eight types.

What the mathematics does not license

Missing states cannot be mapped accurately. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Audit reference coverage. 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. Spatial expression spots

This is where modelling choices begin to affect evidence. Each measured location has expression and coordinates.

A concrete situation

Spot j contains 4,000 detected genes.

What the mathematics does not license

A spot may mix several cells. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Retain raw counts and area. 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. Cell-type fraction estimate

The central mathematical idea is concrete. A deconvolution method estimates each type’s share.

A concrete situation

Spot j is 60% type A and 40% type B.

What the mathematics does not license

Fractions carry model error. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Propagate alternative 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.

4. Cells per spot

Here the geometry becomes operational. Imaging or RNA abundance estimates integer capacity.

A concrete situation

Spot j is assigned five cells.

What the mathematics does not license

Segmentation and RNA content differ. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Compare count methods. 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. Sub-spot expansion

This step deserves a careful pause. Every cell slot becomes one assignment destination.

A concrete situation

Five cells create five sub-spots.

What the mathematics does not license

Sub-spots share one measured profile. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Label them as inferred capacity. 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. Reference-cell sampling

The useful question is not only how to calculate, but what the calculation assumes. Atlas cells are sampled to match tissue type totals.

A concrete situation

Thirty type-A cells are selected.

What the mathematics does not license

A seed can change rare-state coverage. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Repeat stratified samples. 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. Shared-gene matrix

A small example makes the mechanism visible. Only genes observed in both technologies enter cost.

A concrete situation

Two thousand genes remain.

What the mathematics does not license

Gene filtering changes concordance. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Publish the shared 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.

8. Expression standardization

This is where modelling choices begin to affect evidence. Cells and spots are put on comparable scales.

A concrete situation

Each gene is centred and scaled.

What the mathematics does not license

Scaling can amplify noise. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Check raw and transformed costs. 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. Correlation concordance

The central mathematical idea is concrete. A similarity compares each cell with each spot.

A concrete situation

Pearson correlation is 0.82.

What the mathematics does not license

Correlation ignores absolute magnitude. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Compare rank or cosine measures. 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. Cost matrix

Here the geometry becomes operational. Similarity is converted into a minimization cost.

A concrete situation

Cost Cij equals one minus 0.82.

What the mathematics does not license

The transform defines priorities. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

State the formula. 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. Binary assignment variable

This step deserves a careful pause. xij is one only when cell i uses slot j.

A concrete situation

x-3,7 equals one.

What the mathematics does not license

Fractional assignments are excluded. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Explain the discrete choice. 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. Cell-use constraint

The useful question is not only how to calculate, but what the calculation assumes. Each selected cell is assigned exactly once.

A concrete situation

Row i sums to one.

What the mathematics does not license

A poor cell still must go somewhere. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use an explicit unassigned option or filtering rule. 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. Slot-capacity constraint

A small example makes the mechanism visible. Each sub-spot receives exactly one cell.

A concrete situation

Column j sums to one.

What the mathematics does not license

Wrong capacity forces wrong matches. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Sweep cell counts. 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. Global objective

This is where modelling choices begin to affect evidence. Total assignment cost is minimized.

A concrete situation

Four chosen edges sum to 0.63.

What the mathematics does not license

Lowest cost need not be unique. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Inspect near-optimal solutions. 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. Shortest augmenting path

The central mathematical idea is concrete. Reassignments along a path improve the matching.

A concrete situation

Cell A moves so cell B can take a better slot.

What the mathematics does not license

Solver optimality is conditional. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Separate algorithm and biological error. 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. Stability and validation

Here the geometry becomes operational. Repeated inputs and orthogonal maps test assignments.

A concrete situation

A cell remains in one region in 18 of 20 runs.

What the mathematics does not license

Stability can preserve shared bias. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use independent spatial 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.

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

CytoSPACE treats tissue reconstruction as a global assignment problem. A spatial spot can hold several cells, so the method first estimates cell-type fractions and the number of cells per spot, then expands each spot into a matching number of sub-spots. A single-cell RNA-sequencing atlas is sampled so that the number of reference cells of each type matches the estimated tissue demand. These preparatory counts determine which assignments are mathematically possible.

For every reference cell and eligible sub-spot, CytoSPACE calculates a transcriptomic dissimilarity, commonly from correlation across shared genes. The resulting cost matrix is the heart of the optimization. Binary assignment variables state whether cell i is placed at sub-spot j, while row and column constraints enforce one-to-one use. The algorithm minimizes total cost across the entire tissue instead of choosing the best location for each cell independently.

The assignment is solved with a shortest-augmenting-path approach in the Jonker–Volgenant family. An augmenting path rearranges earlier matches when that produces a lower total cost, which is why global optimization can improve on greedy matching. Optimality is conditional on the supplied cost matrix, cell counts, type fractions and candidate cells. It does not mean that the recovered biological locations are uniquely true.

CytoSPACE has no underlying probabilistic model for mapping uncertainty. Stability therefore has to be measured by repeating reference-cell sampling, changing distance metrics, perturbing fraction estimates and validating against known spatial markers or single-cell spatial assays. The mathematics is important because it makes capacity, cost, constraints and optimization explicit—and it also reveals exactly where uncertainty enters before the solver begins.

Frequently asked questions

What is CytoSPACE?

CytoSPACE is an optimization method that assigns single-cell transcriptomes to spatial locations or sub-spots under capacity and cell-type constraints.

Why create sub-spots?

A spatial spot can contain several cells. Sub-spots turn its estimated cell capacity into individual assignment slots.

What is the assignment cost?

It quantifies transcriptomic discordance between a reference cell and a spatial location, commonly using correlation across shared genes.

What does the shortest augmenting path do?

It improves a matching by tracing a path of reassigned cells and slots that lowers total cost while preserving one-to-one constraints.

Is the optimal assignment biologically certain?

No. It is optimal for the chosen cost matrix and constraints, which may be wrong or uncertain.

How can mapping stability be checked?

Repeat cell sampling, fraction estimation and distance choices, then compare assignments with known markers and independent spatial measurements.

Primary source and fact-check note

Milad R. Vahid, Erin L. Brown, Chloé B. Steen, Wubing Zhang, Hyun Soo Jeon, Minji Kang, Andrew J. Gentles and Aaron M. Newman, “High-resolution alignment of single-cell and spatial transcriptomes with CytoSPACE,” Nature Biotechnology 41, 1543–1548 (2023), DOI 10.1038/s41587-023-01697-9. 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

Count audit

Publish cell-type fractions, cells per spot and rounding.

Sampling audit

Record atlas inclusion rules and random seeds.

Gene audit

Version the shared-gene set and transformations.

Cost audit

State correlation or distance and cost conversion.

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.

Constraint audit

List cell-use and slot-capacity equations.

Solver audit

Separate numerical optimality from biological validity.

Stability audit

Compare seeds, metrics and fraction estimates.

Final claim

An optimal assignment is conditional on its inputs and cost.

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: Single-cell atlas

Begin with this mechanism: Annotated reference cells provide full transcriptomes. Recreate the concrete case: The atlas has 20,000 cells across eight types. Write cell counts, type fractions, sub-spot capacities, shared-gene values, correlations, costs, binary variables, constraint sums and total objective 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 states cannot be mapped accurately. Complete the transfer step: Audit reference coverage. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 2: Spatial expression spots

Begin with this mechanism: Each measured location has expression and coordinates. Recreate the concrete case: Spot j contains 4,000 detected genes. Write cell counts, type fractions, sub-spot capacities, shared-gene values, correlations, costs, binary variables, constraint sums and total objective 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 spot may mix several cells. Complete the transfer step: Retain raw counts and area. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 3: Cell-type fraction estimate

Begin with this mechanism: A deconvolution method estimates each type’s share. Recreate the concrete case: Spot j is 60% type A and 40% type B. Write cell counts, type fractions, sub-spot capacities, shared-gene values, correlations, costs, binary variables, constraint sums and total objective 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: Fractions carry model error. Complete the transfer step: Propagate alternative estimates. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 4: Cells per spot

Begin with this mechanism: Imaging or RNA abundance estimates integer capacity. Recreate the concrete case: Spot j is assigned five cells. Write cell counts, type fractions, sub-spot capacities, shared-gene values, correlations, costs, binary variables, constraint sums and total objective 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: Segmentation and RNA content differ. Complete the transfer step: Compare count methods. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

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: Sub-spot expansion

Begin with this mechanism: Every cell slot becomes one assignment destination. Recreate the concrete case: Five cells create five sub-spots. Write cell counts, type fractions, sub-spot capacities, shared-gene values, correlations, costs, binary variables, constraint sums and total objective 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: Sub-spots share one measured profile. Complete the transfer step: Label them as inferred capacity. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 6: Reference-cell sampling

Begin with this mechanism: Atlas cells are sampled to match tissue type totals. Recreate the concrete case: Thirty type-A cells are selected. Write cell counts, type fractions, sub-spot capacities, shared-gene values, correlations, costs, binary variables, constraint sums and total objective 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 seed can change rare-state coverage. Complete the transfer step: Repeat stratified samples. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 7: Shared-gene matrix

Begin with this mechanism: Only genes observed in both technologies enter cost. Recreate the concrete case: Two thousand genes remain. Write cell counts, type fractions, sub-spot capacities, shared-gene values, correlations, costs, binary variables, constraint sums and total objective 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: Gene filtering changes concordance. Complete the transfer step: Publish the shared set. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 8: Expression standardization

Begin with this mechanism: Cells and spots are put on comparable scales. Recreate the concrete case: Each gene is centred and scaled. Write cell counts, type fractions, sub-spot capacities, shared-gene values, correlations, costs, binary variables, constraint sums and total objective 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: Scaling can amplify noise. Complete the transfer step: Check raw and transformed costs. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

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: Correlation concordance

Begin with this mechanism: A similarity compares each cell with each spot. Recreate the concrete case: Pearson correlation is 0.82. Write cell counts, type fractions, sub-spot capacities, shared-gene values, correlations, costs, binary variables, constraint sums and total objective 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: Correlation ignores absolute magnitude. Complete the transfer step: Compare rank or cosine measures. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 10: Cost matrix

Begin with this mechanism: Similarity is converted into a minimization cost. Recreate the concrete case: Cost Cij equals one minus 0.82. Write cell counts, type fractions, sub-spot capacities, shared-gene values, correlations, costs, binary variables, constraint sums and total objective 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: The transform defines priorities. Complete the transfer step: State the formula. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 11: Binary assignment variable

Begin with this mechanism: xij is one only when cell i uses slot j. Recreate the concrete case: x-3,7 equals one. Write cell counts, type fractions, sub-spot capacities, shared-gene values, correlations, costs, binary variables, constraint sums and total objective 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: Fractional assignments are excluded. Complete the transfer step: Explain the discrete choice. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 12: Cell-use constraint

Begin with this mechanism: Each selected cell is assigned exactly once. Recreate the concrete case: Row i sums to one. Write cell counts, type fractions, sub-spot capacities, shared-gene values, correlations, costs, binary variables, constraint sums and total objective 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 poor cell still must go somewhere. Complete the transfer step: Use an explicit unassigned option or filtering rule. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

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: Slot-capacity constraint

Begin with this mechanism: Each sub-spot receives exactly one cell. Recreate the concrete case: Column j sums to one. Write cell counts, type fractions, sub-spot capacities, shared-gene values, correlations, costs, binary variables, constraint sums and total objective 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: Wrong capacity forces wrong matches. Complete the transfer step: Sweep cell counts. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 14: Global objective

Begin with this mechanism: Total assignment cost is minimized. Recreate the concrete case: Four chosen edges sum to 0.63. Write cell counts, type fractions, sub-spot capacities, shared-gene values, correlations, costs, binary variables, constraint sums and total objective 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: Lowest cost need not be unique. Complete the transfer step: Inspect near-optimal solutions. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 15: Shortest augmenting path

Begin with this mechanism: Reassignments along a path improve the matching. Recreate the concrete case: Cell A moves so cell B can take a better slot. Write cell counts, type fractions, sub-spot capacities, shared-gene values, correlations, costs, binary variables, constraint sums and total objective 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: Solver optimality is conditional. Complete the transfer step: Separate algorithm and biological error. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.

Workshop 16: Stability and validation

Begin with this mechanism: Repeated inputs and orthogonal maps test assignments. Recreate the concrete case: A cell remains in one region in 18 of 20 runs. Write cell counts, type fractions, sub-spot capacities, shared-gene values, correlations, costs, binary variables, constraint sums and total objective 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: Stability can preserve shared bias. Complete the transfer step: Use independent spatial evidence. Change exactly one reference atlas, cell count, type fraction, gene filter, normalization, distance metric, cost transform, solver seed, co-embedding dimension, forest size, graph threshold, marker list, PAGE cutoff, damping value, pseudo-spot size, neighbour count, graph normalization or validation split. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, transformation, constraint, optimization, graph operation, prediction, interpretation and validation on separate notebook lines. Reserve whole donors, sections, tissues or platforms outside tuning when testing transfer.