WHY MATHEMATICS?

Why Mathematics? | BSP for Spatially Variable Genes: Big-Small Patches, Multiscale Local Means, Variance Ratios and 2D/3D Granularity

How BSP compares a gene at two neighbourhood sizes to recognize spatial patterns that persist across granularity, and why radius choice and background-score fitting still require care.

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

Normalize expression and coordinates for comparable patches
Build a small and a large radius neighbourhood around every spot
Compare variance of local means across the two scales
Use the upper tail of a fitted background score distribution

Chapter map

1. 2D and 3D coordinates
2. Expression scaling
3. Density normalization
4. Small radius D1
5. Large radius D2
6. Patch membership
7. Local mean
8. Moving-centre field
9. Variance at one scale
10. Variance ratio
11. Gene variance weight
12. Lognormal score fit
13. Upper-tail probability
14. Low-throughput null genes
15. Dimension-agnostic claim
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. 2D and 3D coordinates

A small example makes the mechanism visible. Spots may have x-y or x-y-z positions.

A concrete situation

Ten serial sections form a volume.

What the mathematics does not license

Anisotropic z spacing must be real. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Record physical scales. 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. Expression scaling

This is where modelling choices begin to affect evidence. Gene expression is normalized across spots.

A concrete situation

Values are mapped to zero through one.

What the mathematics does not license

Min-max scaling is outlier-sensitive. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Inspect extreme 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.

3. Density normalization

The central mathematical idea is concrete. Coordinate ranges and spot density set a common spatial unit.

A concrete situation

Average nearest spacing becomes near one.

What the mathematics does not license

Rectangular or cubic bounds approximate masks. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Test irregular shapes. 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. Small radius D1

Here the geometry becomes operational. The small patch captures nearest local detail.

A concrete situation

D1 equals one normalized unit.

What the mathematics does not license

Sparse edges have few members. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Map patch cardinality. 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. Large radius D2

This step deserves a careful pause. The big patch gives a coarser neighbourhood.

A concrete situation

D2 equals three units.

What the mathematics does not license

Too large a radius erases a thin feature. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Sweep two to five. 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. Patch membership

The useful question is not only how to calculate, but what the calculation assumes. Every centre collects all spots within its radius.

A concrete situation

A central site has eight neighbours.

What the mathematics does not license

Boundary patches are smaller. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Compare edge corrections. 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. Local mean

A small example makes the mechanism visible. Average expression inside each patch smooths noise.

A concrete situation

A cluster remains high after averaging.

What the mathematics does not license

Mixtures blur boundaries. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Plot raw and mean fields. 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. Moving-centre field

This is where modelling choices begin to affect evidence. The local mean is recalculated at every spot.

A concrete situation

Nine centres yield nine local means.

What the mathematics does not license

Overlapping patches are dependent. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Do not treat them as independent 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.

9. Variance at one scale

The central mathematical idea is concrete. Variance across the moving local means quantifies spatial contrast.

A concrete situation

Layer means differ across the tissue.

What the mathematics does not license

Global mean shifts are removed. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

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

10. Variance ratio

Here the geometry becomes operational. Large-scale variance divided by small-scale variance measures persistence.

A concrete situation

A broad band keeps a high ratio.

What the mathematics does not license

Near-zero denominators are unstable. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use documented safeguards. 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. Gene variance weight

This step deserves a careful pause. Intrinsic gene variance is scaled by the maximum across genes.

A concrete situation

A nearly constant gene is downweighted.

What the mathematics does not license

One outlier gene sets the maximum. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Run influence checks. 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. Lognormal score fit

The useful question is not only how to calculate, but what the calculation assumes. Most genes are assumed to form a positive background distribution.

A concrete situation

The observed score histogram is fitted.

What the mathematics does not license

A poor fit miscalibrates tails. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use Q-Q and goodness-of-fit checks. 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. Upper-tail probability

A small example makes the mechanism visible. Large scores are treated as unusual under the fitted background.

A concrete situation

A gene lies beyond the 95th percentile.

What the mathematics does not license

A nominal p-value is not automatic FDR control. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

State multiplicity treatment. 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. Low-throughput null genes

This is where modelling choices begin to affect evidence. Permuted genes supplement a small targeted panel.

A concrete situation

A FISH panel gains shuffled backgrounds.

What the mathematics does not license

Permutation must preserve valid constraints. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Describe the shuffle. 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. Dimension-agnostic claim

The central mathematical idea is concrete. The radius and means extend to any coordinate dimension.

A concrete situation

A 3D volume uses spheres instead of circles.

What the mathematics does not license

Mathematical extension does not cure registration error. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Validate the volume. 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

Here the geometry becomes operational. Version scaling, D1, D2, fit family, tail and nulls.

A concrete situation

A student computes two patch variances.

What the mathematics does not license

Granularity is a score, not proof. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

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

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

BSP means big-small patch. Around every spatial measurement, it defines a small-radius neighbourhood and a larger-radius neighbourhood. The average expression of one gene is calculated inside each patch, so every original site produces two local means. The method works with two-dimensional coordinates or three-dimensional coordinates because a radius can be applied in either geometry.

For each radius, BSP examines how much the local means vary as the patch centre moves through the tissue. A spatial pattern tends to preserve structured differences after moderate smoothing, whereas unstructured variation is reduced more quickly. The method uses a gene-variance weight times the ratio of the large-patch local-mean variance to the small-patch variance.

Across genes, the score distribution is approximated with a lognormal distribution, with a beta approximation considered in the paper, and an upper-tail probability identifies unusually persistent patterns. This relies on most genes acting as background. For small targeted panels, the published workflow supplements the background with spatially permuted null genes.

A compact calculation can use nine sites on a 3-by-3 grid. For one smooth left-to-right pattern and one shuffled pattern, compute small-patch means, big-patch means, their two variances and the score ratio. Changing the large radius shows why granularity is informative and why a radius larger than a narrow real feature can lose power.

Frequently asked questions

What is BSP?

BSP is the big-small-patch method, a dimension-agnostic approach that scores spatially variable genes by comparing local-mean variability at two neighbourhood radii.

What is a patch?

For a centre spot, a patch is the set of all spots within a chosen Euclidean radius in normalized 2D or 3D coordinates.

What is the BSP score?

It uses a gene-variance weight multiplied by the variance of large-patch local means divided by the variance of small-patch local means.

Why compare two scales?

A coherent spatial pattern can retain structured differences after moderate smoothing, while unstructured local variation tends to disappear faster as the radius grows.

How are p-values obtained?

The distribution of gene scores is approximated with a lognormal distribution, with a beta alternative discussed, and genes in the upper tail receive one-sided probabilities.

What is a major limitation?

The method depends on radius choice and on a suitable background score distribution; very thin patterns, small gene panels or a high share of truly spatial genes need extra sensitivity and null checks.

Primary source and fact-check note

Juexin Wang, Jinpu Li, Skyler T. Kramer et al., “Dimension-agnostic and granularity-based spatially variable gene identification using BSP,” Nature Communications 14, 7367 (2023), DOI 10.1038/s41467-023-43256-5. 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

Coordinate audit

Check density scaling and physical anisotropy.

Patch audit

Map membership and boundary cardinality.

Radius audit

Sweep D1 and D2 against feature size.

Variance audit

Inspect near-zero denominators and outliers.

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.

Fit audit

Use histograms, Q-Q plots and alternatives.

Null audit

Generate valid permuted genes for small panels.

3D audit

Validate slice alignment and z distances.

Final claim

A high ratio supports multiscale persistence.

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: 2D and 3D coordinates

Begin with this mechanism: Spots may have x-y or x-y-z positions. Work through the concrete case: Ten serial sections form a volume. Make the limitation visible: Anisotropic z spacing must be real. Complete the evidence step: Record physical scales. Write the coordinates, adjacency weights, local means, centred values, entropy terms, mixture probabilities, energy contributions, Poisson expectations, variance ratios or cross-correlation values with named units. Calculate one small example by hand, then vary one radius, edge, time step, convergence threshold, component number or smoothness value and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view or genes outside tuning when testing transfer. Keep measurement, numerical representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 2: Expression scaling

Begin with this mechanism: Gene expression is normalized across spots. Work through the concrete case: Values are mapped to zero through one. Make the limitation visible: Min-max scaling is outlier-sensitive. Complete the evidence step: Inspect extreme values. Write the coordinates, adjacency weights, local means, centred values, entropy terms, mixture probabilities, energy contributions, Poisson expectations, variance ratios or cross-correlation values with named units. Calculate one small example by hand, then vary one radius, edge, time step, convergence threshold, component number or smoothness value and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view or genes outside tuning when testing transfer. Keep measurement, numerical representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 3: Density normalization

Begin with this mechanism: Coordinate ranges and spot density set a common spatial unit. Work through the concrete case: Average nearest spacing becomes near one. Make the limitation visible: Rectangular or cubic bounds approximate masks. Complete the evidence step: Test irregular shapes. Write the coordinates, adjacency weights, local means, centred values, entropy terms, mixture probabilities, energy contributions, Poisson expectations, variance ratios or cross-correlation values with named units. Calculate one small example by hand, then vary one radius, edge, time step, convergence threshold, component number or smoothness value and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view or genes outside tuning when testing transfer. Keep measurement, numerical representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 4: Small radius D1

Begin with this mechanism: The small patch captures nearest local detail. Work through the concrete case: D1 equals one normalized unit. Make the limitation visible: Sparse edges have few members. Complete the evidence step: Map patch cardinality. Write the coordinates, adjacency weights, local means, centred values, entropy terms, mixture probabilities, energy contributions, Poisson expectations, variance ratios or cross-correlation values with named units. Calculate one small example by hand, then vary one radius, edge, time step, convergence threshold, component number or smoothness value and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view or genes outside tuning when testing transfer. Keep measurement, numerical representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

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: Large radius D2

Begin with this mechanism: The big patch gives a coarser neighbourhood. Work through the concrete case: D2 equals three units. Make the limitation visible: Too large a radius erases a thin feature. Complete the evidence step: Sweep two to five. Write the coordinates, adjacency weights, local means, centred values, entropy terms, mixture probabilities, energy contributions, Poisson expectations, variance ratios or cross-correlation values with named units. Calculate one small example by hand, then vary one radius, edge, time step, convergence threshold, component number or smoothness value and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view or genes outside tuning when testing transfer. Keep measurement, numerical representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 6: Patch membership

Begin with this mechanism: Every centre collects all spots within its radius. Work through the concrete case: A central site has eight neighbours. Make the limitation visible: Boundary patches are smaller. Complete the evidence step: Compare edge corrections. Write the coordinates, adjacency weights, local means, centred values, entropy terms, mixture probabilities, energy contributions, Poisson expectations, variance ratios or cross-correlation values with named units. Calculate one small example by hand, then vary one radius, edge, time step, convergence threshold, component number or smoothness value and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view or genes outside tuning when testing transfer. Keep measurement, numerical representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 7: Local mean

Begin with this mechanism: Average expression inside each patch smooths noise. Work through the concrete case: A cluster remains high after averaging. Make the limitation visible: Mixtures blur boundaries. Complete the evidence step: Plot raw and mean fields. Write the coordinates, adjacency weights, local means, centred values, entropy terms, mixture probabilities, energy contributions, Poisson expectations, variance ratios or cross-correlation values with named units. Calculate one small example by hand, then vary one radius, edge, time step, convergence threshold, component number or smoothness value and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view or genes outside tuning when testing transfer. Keep measurement, numerical representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 8: Moving-centre field

Begin with this mechanism: The local mean is recalculated at every spot. Work through the concrete case: Nine centres yield nine local means. Make the limitation visible: Overlapping patches are dependent. Complete the evidence step: Do not treat them as independent samples. Write the coordinates, adjacency weights, local means, centred values, entropy terms, mixture probabilities, energy contributions, Poisson expectations, variance ratios or cross-correlation values with named units. Calculate one small example by hand, then vary one radius, edge, time step, convergence threshold, component number or smoothness value and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view or genes outside tuning when testing transfer. Keep measurement, numerical representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

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: Variance at one scale

Begin with this mechanism: Variance across the moving local means quantifies spatial contrast. Work through the concrete case: Layer means differ across the tissue. Make the limitation visible: Global mean shifts are removed. Complete the evidence step: Inspect the full distribution. Write the coordinates, adjacency weights, local means, centred values, entropy terms, mixture probabilities, energy contributions, Poisson expectations, variance ratios or cross-correlation values with named units. Calculate one small example by hand, then vary one radius, edge, time step, convergence threshold, component number or smoothness value and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view or genes outside tuning when testing transfer. Keep measurement, numerical representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 10: Variance ratio

Begin with this mechanism: Large-scale variance divided by small-scale variance measures persistence. Work through the concrete case: A broad band keeps a high ratio. Make the limitation visible: Near-zero denominators are unstable. Complete the evidence step: Use documented safeguards. Write the coordinates, adjacency weights, local means, centred values, entropy terms, mixture probabilities, energy contributions, Poisson expectations, variance ratios or cross-correlation values with named units. Calculate one small example by hand, then vary one radius, edge, time step, convergence threshold, component number or smoothness value and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view or genes outside tuning when testing transfer. Keep measurement, numerical representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 11: Gene variance weight

Begin with this mechanism: Intrinsic gene variance is scaled by the maximum across genes. Work through the concrete case: A nearly constant gene is downweighted. Make the limitation visible: One outlier gene sets the maximum. Complete the evidence step: Run influence checks. Write the coordinates, adjacency weights, local means, centred values, entropy terms, mixture probabilities, energy contributions, Poisson expectations, variance ratios or cross-correlation values with named units. Calculate one small example by hand, then vary one radius, edge, time step, convergence threshold, component number or smoothness value and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view or genes outside tuning when testing transfer. Keep measurement, numerical representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 12: Lognormal score fit

Begin with this mechanism: Most genes are assumed to form a positive background distribution. Work through the concrete case: The observed score histogram is fitted. Make the limitation visible: A poor fit miscalibrates tails. Complete the evidence step: Use Q-Q and goodness-of-fit checks. Write the coordinates, adjacency weights, local means, centred values, entropy terms, mixture probabilities, energy contributions, Poisson expectations, variance ratios or cross-correlation values with named units. Calculate one small example by hand, then vary one radius, edge, time step, convergence threshold, component number or smoothness value and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view or genes outside tuning when testing transfer. Keep measurement, numerical representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

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: Upper-tail probability

Begin with this mechanism: Large scores are treated as unusual under the fitted background. Work through the concrete case: A gene lies beyond the 95th percentile. Make the limitation visible: A nominal p-value is not automatic FDR control. Complete the evidence step: State multiplicity treatment. Write the coordinates, adjacency weights, local means, centred values, entropy terms, mixture probabilities, energy contributions, Poisson expectations, variance ratios or cross-correlation values with named units. Calculate one small example by hand, then vary one radius, edge, time step, convergence threshold, component number or smoothness value and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view or genes outside tuning when testing transfer. Keep measurement, numerical representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 14: Low-throughput null genes

Begin with this mechanism: Permuted genes supplement a small targeted panel. Work through the concrete case: A FISH panel gains shuffled backgrounds. Make the limitation visible: Permutation must preserve valid constraints. Complete the evidence step: Describe the shuffle. Write the coordinates, adjacency weights, local means, centred values, entropy terms, mixture probabilities, energy contributions, Poisson expectations, variance ratios or cross-correlation values with named units. Calculate one small example by hand, then vary one radius, edge, time step, convergence threshold, component number or smoothness value and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view or genes outside tuning when testing transfer. Keep measurement, numerical representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 15: Dimension-agnostic claim

Begin with this mechanism: The radius and means extend to any coordinate dimension. Work through the concrete case: A 3D volume uses spheres instead of circles. Make the limitation visible: Mathematical extension does not cure registration error. Complete the evidence step: Validate the volume. Write the coordinates, adjacency weights, local means, centred values, entropy terms, mixture probabilities, energy contributions, Poisson expectations, variance ratios or cross-correlation values with named units. Calculate one small example by hand, then vary one radius, edge, time step, convergence threshold, component number or smoothness value and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view or genes outside tuning when testing transfer. Keep measurement, numerical representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.

Workshop 16: Governance and learning

Begin with this mechanism: Version scaling, D1, D2, fit family, tail and nulls. Work through the concrete case: A student computes two patch variances. Make the limitation visible: Granularity is a score, not proof. Complete the evidence step: Write a BSP audit card. Write the coordinates, adjacency weights, local means, centred values, entropy terms, mixture probabilities, energy contributions, Poisson expectations, variance ratios or cross-correlation values with named units. Calculate one small example by hand, then vary one radius, edge, time step, convergence threshold, component number or smoothness value and explain which result moves and why. Reserve complete tissues, donors, sections, fields of view or genes outside tuning when testing transfer. Keep measurement, numerical representation, statistical evidence, biological interpretation and decision on separate notebook lines so a student, parent or reviewer can audit the mathematics.