WHY MATHEMATICS?

Why Mathematics? | sepal for Spatial Transcript Profiles: Diffusion Equations, Discrete Laplacians, Entropy Convergence and Pattern Rankings

How sepal ranks organized spatial transcript profiles by asking how long a numerical diffusion process takes to erase their structure, linking calculus, numerical analysis and information theory.

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

Represent a transcript profile as a concentration field
Approximate the spatial Laplacian on the platform grid
Propagate the heat equation with stable time steps
Use entropy convergence time as a pattern metric

Chapter map

1. Spatial domain
2. Transcript field
3. Neighbour sets
4. Inner and boundary points
5. Continuous heat equation
6. Discrete Laplacian
7. Explicit time step
8. Mass behaviour
9. Normalized proportions
10. Shannon entropy
11. Entropy trajectory
12. Convergence threshold
13. Diffusion-time metric
14. Min-max ranking
15. Parallel computation
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. Spatial domain

The central mathematical idea is concrete. The tissue grid discretizes a continuous region.

A concrete situation

A square array covers one section.

What the mathematics does not license

Missing positions make the domain irregular. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Plot supported sites. 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. Transcript field

Here the geometry becomes operational. One gene supplies a nonnegative value at every supported location.

A concrete situation

A layered marker starts concentrated at one edge.

What the mathematics does not license

Normalization changes relative mass. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Version preprocessing. 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. Neighbour sets

This step deserves a careful pause. Platform geometry defines which sites exchange simulated mass.

A concrete situation

A square site has four orthogonal neighbours.

What the mathematics does not license

Hexagonal layouts require another stencil. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use the documented grid. 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. Inner and boundary points

The useful question is not only how to calculate, but what the calculation assumes. Saturated inner sites have the full expected neighbour count.

A concrete situation

An edge site has fewer neighbours.

What the mathematics does not license

Boundary treatment changes diffusion speed. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Report boundary rules. 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. Continuous heat equation

A small example makes the mechanism visible. The time derivative equals diffusion coefficient times the spatial Laplacian.

A concrete situation

A peak spreads and flattens.

What the mathematics does not license

This is an analytical metaphor. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Separate model from biology. 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. Discrete Laplacian

This is where modelling choices begin to affect evidence. Finite differences approximate curvature from neighbour values.

A concrete situation

A centre of ten surrounded by zeros has negative curvature.

What the mathematics does not license

Grid spacing enters the formula. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Keep units consistent. 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. Explicit time step

The central mathematical idea is concrete. The next field equals the current field plus D times Laplacian times dt.

A concrete situation

One step lowers a sharp peak.

What the mathematics does not license

Large dt can become unstable or negative. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Check the stability condition. 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. Mass behaviour

Here the geometry becomes operational. A well-posed update should preserve total concentration under appropriate boundaries.

A concrete situation

Five units remain five after a step.

What the mathematics does not license

Numerical error can drift. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Track totals. 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. Normalized proportions

This step deserves a careful pause. Divide every field value by the current total.

A concrete situation

Values sum to one.

What the mathematics does not license

Zeros require careful logarithms. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use the zero-log-zero limit. 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. Shannon entropy

The useful question is not only how to calculate, but what the calculation assumes. Minus the sum of p log p measures spatial evenness.

A concrete situation

A uniform field has higher entropy than one peak.

What the mathematics does not license

Entropy ignores where values occur. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Pair it with maps. 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. Entropy trajectory

A small example makes the mechanism visible. Entropy is recorded after every update.

A concrete situation

A smooth gradient approaches uniformity slowly.

What the mathematics does not license

Discretization can create nonmonotone artifacts. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Plot the full trace. 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. Convergence threshold

This is where modelling choices begin to affect evidence. Stop when average entropy change over inner sites is below epsilon.

A concrete situation

Two consecutive steps differ negligibly.

What the mathematics does not license

Epsilon changes the stopping time. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Sweep thresholds. 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. Diffusion-time metric

The central mathematical idea is concrete. The convergence time ranks spatial organization.

A concrete situation

A structured stripe outlasts random speckles.

What the mathematics does not license

Scale and mask affect time. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Compare only defensible cohorts. 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. Min-max ranking

Here the geometry becomes operational. Times are rescaled within a sample between zero and one.

A concrete situation

The slowest gene receives one.

What the mathematics does not license

The scale is relative, not universal. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Preserve raw times. 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. Parallel computation

This step deserves a careful pause. Genes diffuse independently and can use several cores.

A concrete situation

Four profiles run simultaneously.

What the mathematics does not license

Hardware changes speed, not the metric. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Record software and tolerances. 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

The useful question is not only how to calculate, but what the calculation assumes. Version grid, stencil, D, dt, epsilon and normalization.

A concrete situation

A student diffuses a five-site line.

What the mathematics does not license

Long diffusion time is a ranking signal. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Write a sepal numerical 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

sepal approaches spatial patterning as a diffusion problem rather than a conventional gene-by-gene hypothesis test. Each transcript profile is treated as an initial concentration field over a supported spatial grid. A numerical approximation to the heat equation repeatedly moves concentration toward neighbouring locations until the field becomes nearly uniform.

The discrete Laplacian measures how a location differs from its neighbours, and an explicit time step updates the field by a diffusion coefficient times that Laplacian. The implementation accounts for supported square and hexagonal layouts and distinguishes inner from unsaturated boundary points. Stability depends on the grid, neighbourhood and time-step choices.

At every step, the field is normalized into proportions and its spatial entropy is computed. Convergence occurs when the change in entropy over inner points falls below a threshold. The required diffusion time becomes the ranking metric: an organized profile usually takes longer to erase than a scattered one. Min-max normalization makes ranks comparable within a sample, not automatically across unrelated experiments.

Students can start with a five-location line, a concentrated profile and an alternating profile. Compute one Laplacian update and entropy change for each, then continue until the threshold is met. The example connects differential equations, numerical methods, probability distributions and information theory without pretending that diffusion is the biological mechanism that created the gene pattern.

Frequently asked questions

What is sepal?

sepal is a diffusion-based method that ranks spatial transcript profiles by how long their observed spatial structure takes to become uniform under a numerical heat-equation model.

What is diffusing?

The algorithm treats measured transcript abundance as a mathematical concentration field. The simulated diffusion is an analysis device, not a claim that transcripts physically produced the pattern by diffusion.

What does the Laplacian measure?

The discrete Laplacian compares a location with its neighbours and is positive where surrounding values exceed the centre and negative where the centre exceeds its surroundings.

Why use entropy?

After normalizing the evolving field into proportions, entropy summarizes how evenly the mass is spread. It rises toward a maximum as the profile becomes uniform.

What is diffusion time?

It is the time step when the average entropy change over inner locations falls below a chosen threshold; longer times rank profiles as more spatially organized.

Is a high diffusion-time gene automatically significant?

No. sepal supplies a within-sample ranking metric rather than the same hypothesis-testing framework used by many SVG methods. Biological validation and sensitivity checks still matter.

Primary source and fact-check note

Alma Andersson and Joakim Lundeberg, “sepal: identifying transcript profiles with spatial patterns by diffusion-based modeling,” Bioinformatics 37, 2644–2650 (2021), DOI 10.1093/bioinformatics/btab164. 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

Grid audit

Confirm square, hexagonal and missing-site geometry.

Stencil audit

Write the exact discrete Laplacian.

Stability audit

Check D and dt against numerical stability.

Boundary audit

Compare saturated and unsaturated points.

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.

Entropy audit

Track totals, proportions and entropy traces.

Threshold audit

Sweep epsilon and preserve raw diffusion times.

Transfer audit

Compare ranks across matched samples only.

Final claim

Diffusion time is a pattern ranking, not mechanism.

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: Spatial domain

Begin with this mechanism: The tissue grid discretizes a continuous region. Work through the concrete case: A square array covers one section. Make the limitation visible: Missing positions make the domain irregular. Complete the evidence step: Plot supported sites. 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: Transcript field

Begin with this mechanism: One gene supplies a nonnegative value at every supported location. Work through the concrete case: A layered marker starts concentrated at one edge. Make the limitation visible: Normalization changes relative mass. Complete the evidence step: Version preprocessing. 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: Neighbour sets

Begin with this mechanism: Platform geometry defines which sites exchange simulated mass. Work through the concrete case: A square site has four orthogonal neighbours. Make the limitation visible: Hexagonal layouts require another stencil. Complete the evidence step: Use the documented grid. 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: Inner and boundary points

Begin with this mechanism: Saturated inner sites have the full expected neighbour count. Work through the concrete case: An edge site has fewer neighbours. Make the limitation visible: Boundary treatment changes diffusion speed. Complete the evidence step: Report boundary rules. 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: Continuous heat equation

Begin with this mechanism: The time derivative equals diffusion coefficient times the spatial Laplacian. Work through the concrete case: A peak spreads and flattens. Make the limitation visible: This is an analytical metaphor. Complete the evidence step: Separate model from biology. 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: Discrete Laplacian

Begin with this mechanism: Finite differences approximate curvature from neighbour values. Work through the concrete case: A centre of ten surrounded by zeros has negative curvature. Make the limitation visible: Grid spacing enters the formula. Complete the evidence step: Keep units consistent. 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: Explicit time step

Begin with this mechanism: The next field equals the current field plus D times Laplacian times dt. Work through the concrete case: One step lowers a sharp peak. Make the limitation visible: Large dt can become unstable or negative. Complete the evidence step: Check the stability condition. 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: Mass behaviour

Begin with this mechanism: A well-posed update should preserve total concentration under appropriate boundaries. Work through the concrete case: Five units remain five after a step. Make the limitation visible: Numerical error can drift. Complete the evidence step: Track totals. 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: Normalized proportions

Begin with this mechanism: Divide every field value by the current total. Work through the concrete case: Values sum to one. Make the limitation visible: Zeros require careful logarithms. Complete the evidence step: Use the zero-log-zero limit. 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: Shannon entropy

Begin with this mechanism: Minus the sum of p log p measures spatial evenness. Work through the concrete case: A uniform field has higher entropy than one peak. Make the limitation visible: Entropy ignores where values occur. Complete the evidence step: Pair it with maps. 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: Entropy trajectory

Begin with this mechanism: Entropy is recorded after every update. Work through the concrete case: A smooth gradient approaches uniformity slowly. Make the limitation visible: Discretization can create nonmonotone artifacts. Complete the evidence step: Plot the full trace. 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: Convergence threshold

Begin with this mechanism: Stop when average entropy change over inner sites is below epsilon. Work through the concrete case: Two consecutive steps differ negligibly. Make the limitation visible: Epsilon changes the stopping time. Complete the evidence step: Sweep thresholds. 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: Diffusion-time metric

Begin with this mechanism: The convergence time ranks spatial organization. Work through the concrete case: A structured stripe outlasts random speckles. Make the limitation visible: Scale and mask affect time. Complete the evidence step: Compare only defensible cohorts. 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: Min-max ranking

Begin with this mechanism: Times are rescaled within a sample between zero and one. Work through the concrete case: The slowest gene receives one. Make the limitation visible: The scale is relative, not universal. Complete the evidence step: Preserve raw times. 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: Parallel computation

Begin with this mechanism: Genes diffuse independently and can use several cores. Work through the concrete case: Four profiles run simultaneously. Make the limitation visible: Hardware changes speed, not the metric. Complete the evidence step: Record software and tolerances. 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 grid, stencil, D, dt, epsilon and normalization. Work through the concrete case: A student diffuses a five-site line. Make the limitation visible: Long diffusion time is a ranking signal. Complete the evidence step: Write a sepal numerical 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.