Chapter map

1. 1. Lattice
2. 2. Cell A
3. 3. Cell B
4. 4. Mass
5. 5. Stream to A
6. 6. Stream to B
7. 7. Conservation
8. 8. Momentum
9. 9. Collision
10. 10. Viscosity
11. 11. Boundary
12. 12. Mach check
13. 13. Refinement
14. 14. Conclusion

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

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1. Lattice

This is where modelling choices begin to affect evidence. Use D1Q3 velocities -1,0,+1.

A concrete situation

Weights 1/6,2/3,1/6.

What the mathematics does not license

State order. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

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

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2. Cell A

The central mathematical idea is concrete. Set rho=1 at equilibrium.

A concrete situation

Distributions=(1/6,2/3,1/6).

What the mathematics does not license

Zero velocity. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

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

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3. Cell B

Here the geometry becomes operational. Set rho=2 at equilibrium.

A concrete situation

Distributions=(1/3,4/3,1/3).

What the mathematics does not license

Zero velocity. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

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

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4. Mass

This step deserves a careful pause. Sum both triples.

A concrete situation

Total=3.

What the mathematics does not license

Check weights. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

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

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5. Stream to A

The useful question is not only how to calculate, but what the calculation assumes. Take both movers from B in two-cell periodic grid.

A concrete situation

A new rho=4/3.

What the mathematics does not license

Rest remains local. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

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

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6. Stream to B

A small example makes the mechanism visible. Take both movers from A.

A concrete situation

B new rho=5/3.

What the mathematics does not license

Rest remains local. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

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

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7. Conservation

This is where modelling choices begin to affect evidence. Add new densities.

A concrete situation

Total=3.

What the mathematics does not license

Periodic streaming. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

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

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8. Momentum

The central mathematical idea is concrete. Calculate first moment.

A concrete situation

Local momentum may change during transport.

What the mathematics does not license

Track sign. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record 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.

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9. Collision

Here the geometry becomes operational. Relax toward local equilibrium.

A concrete situation

Conserved moments unchanged.

What the mathematics does not license

Specify tau. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

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

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10. Viscosity

This step deserves a careful pause. Map tau to kinematic viscosity.

A concrete situation

Formula depends on lattice units.

What the mathematics does not license

State scaling. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

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

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11. Boundary

The useful question is not only how to calculate, but what the calculation assumes. Replace periodic wrap with wall rule.

A concrete situation

Mass and momentum effects change.

What the mathematics does not license

Define wall location. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

Record 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.

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12. Mach check

A small example makes the mechanism visible. Increase velocity.

A concrete situation

Compressibility error can grow.

What the mathematics does not license

Stay in validated range. The same reasoning helps students move from an exercise to a real dataset.

Try it, check it, transfer it

Record 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.

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13. Refinement

This is where modelling choices begin to affect evidence. Change dx and dt consistently.

A concrete situation

Physical scaling changes.

What the mathematics does not license

Recalculate parameters. That distinction matters because a correct formula can still answer the wrong operational question.

Try it, check it, transfer it

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

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14. Conclusion

The central mathematical idea is concrete. State populations,collision,streaming,boundaries and validation.

A concrete situation

Worked densities become 4/3 and 5/3.

What the mathematics does not license

Mass alone is insufficient. The habit of checking assumptions is one of the practical benefits of learning mathematics.

Try it, check it, transfer it

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

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A seven-day practice lab

1. Day 1: define the decision, observation unit, X variables and Y or reference condition.
2. Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
3. Day 3: build a transparent baseline and write what it assumes.
4. Day 4: fit the main method inside a training-only pipeline; record every parameter.
5. Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
6. Day 6: compare validation metrics with the practical workload created by errors.
7. Day 7: present one supported conclusion, one unresolved uncertainty and one safe next experiment.

Questions for a student, parent or mentor

• Which part of the model is a mathematical definition, and which part is a choice?
• What small calculation would catch a sign, scale or dimension error?
• What evidence comes from training, validation and the untouched test?
• What benign condition could look like a fault or prediction failure?
• How would the answer change for another response, asset, school project or time period?
• What should a human review before acting on the score?

A method-specific audit before deployment

The lattice Boltzmann method evolves discrete particle distribution functions on a lattice through local collision and exact streaming steps. Macroscopic density and momentum are moments of those distributions.

Use the one-dimensional D1Q3 lattice with velocities -1, 0 and +1 and equilibrium weights 1/6, 2/3 and 1/6 at zero velocity. The weights add to one.

Let periodic cell A have density 1 and cell B density 2, each initially at its zero-velocity equilibrium. Their distribution triples are (1/6,2/3,1/6) and (1/3,4/3,1/3), ordered left, rest, right.

After streaming on a two-cell periodic lattice, both moving populations entering A come from B. Cell A receives 1/3+1/3 around its unchanged rest population 2/3, so its new density is 4/3.

Cell B receives 1/6+1/6 around rest population 4/3, so its new density is 5/3. The total density remains 3, matching the initial total 1+2.

A collision step relaxes local distributions toward equilibrium while conserving selected moments. In the single-relaxation-time BGK model, the relaxation parameter controls viscosity within stated scaling assumptions.

Did You Know? Streaming moves each discrete population to an exact neighbouring lattice node. Much of the computational regularity comes from this simple shift combined with local arithmetic.

The popular D2Q9 lattice uses one rest velocity, four axial velocities and four diagonals. Its weights are chosen so low-order velocity moments reproduce the isotropy needed for the target hydrodynamics.

Macroscopic velocity is obtained from the first moment divided by density, with forcing corrections when used. The method evolves distributions, not velocity directly.

Low-Mach and near-incompressible assumptions matter for many standard formulations. Raising flow speed until compressibility errors appear is not a legitimate accuracy strategy.

Boundary rules such as bounce-back approximate no-slip walls, but their effective wall position and order depend on implementation and geometry.

Students can verify mass before and after collision and streaming separately. This catches index, periodic-wrap and direction-label mistakes.

Applications include porous flow, multiphase systems, microfluidics and complex moving geometries. Each extension introduces additional modelling assumptions and stability limits.

Grid spacing, time step, relaxation, forcing and physical-unit conversion must be consistent. Lattice units cannot be reported as physical predictions without a clear scaling map.

For parents and students, the transferable lesson is that a complex fluid picture can emerge from repeated local rules when the conserved totals and scaling are checked.

A responsible report states the lattice, populations, equilibrium, collision operator, relaxation, forcing, boundaries, unit conversion, Mach range, conservation, convergence and validation.

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Frequently asked questions

What does lattice Boltzmann evolve?

Discrete particle distribution functions associated with a finite set of lattice velocities.

What are collision and streaming?

Local relaxation of distributions followed by movement to neighbouring nodes.

How is density obtained?

By summing all distributions at a lattice node.

Does mass conservation guarantee an accurate flow?

No. Momentum, viscosity, boundaries, Mach range, resolution and validation also matter.

What are lattice units?

A nondimensional computational scaling that must be mapped consistently to physical units.

What must be reported?

Lattice, equilibrium, collision, relaxation, forcing, boundaries, scaling, stability, convergence and validation.

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Primary source and fact-check note

Guy R. McNamara and Gianluigi Zanetti, “Use of the Boltzmann Equation to Simulate Lattice-Gas Automata,” Physical Review Letters 61, 1988. 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

• Finite volume method
• Method of lines
• Computational mathematics hub
• How Mathematics Works
• Study and learning methods

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.

Method audit: representation, transfer and evidence

Moment audit

Verify density and momentum before and after collision.

Streaming audit

Check every direction, neighbour and periodic wrap.

Scaling audit

Map lattice viscosity, speed and time to physical units.

Boundary audit

Measure wall location, slip and conservation.

Mechanism workshop: calculate, challenge and transfer

Population ledger

List every distribution before collision.

Moment check

Compute density and momentum per cell.

Collision check

Confirm conserved moments remain fixed.

Streaming map

Draw each direction to its destination.

Periodic test

Verify wrap indices on two and three cells.

Wall test

Compare bounce-back placements.

Scaling

Map a lattice viscosity to physical units.

Mach sweep

Measure density variation as speed grows.

Student explanation

Explain local rules and global flow.

Parent explanation

Connect repeated rules with conserved totals.

Evidence notebook: reconstruct every step

Evidence notebook 1: Lattice

Step 1 isolates lattice. Use D1Q3 velocities -1,0,+1. The checked result is: Weights 1/6,2/3,1/6. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: State order. Record lattice. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 2: Cell A

Step 2 isolates cell a. Set rho=1 at equilibrium. The checked result is: Distributions=(1/6,2/3,1/6). Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Zero velocity. Record A. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 3: Cell B

Step 3 isolates cell b. Set rho=2 at equilibrium. The checked result is: Distributions=(1/3,4/3,1/3). Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Zero velocity. Record B. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 4: Mass

Step 4 isolates mass. Sum both triples. The checked result is: Total=3. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Check weights. Record total. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 5: Stream to A

Step 5 isolates stream to a. Take both movers from B in two-cell periodic grid. The checked result is: A new rho=4/3. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Rest remains local. Record A. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 6: Stream to B

Step 6 isolates stream to b. Take both movers from A. The checked result is: B new rho=5/3. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Rest remains local. Record B. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 7: Conservation

Step 7 isolates conservation. Add new densities. The checked result is: Total=3. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Periodic streaming. Record check. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 8: Momentum

Step 8 isolates momentum. Calculate first moment. The checked result is: Local momentum may change during transport. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Track sign. Record values. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 9: Collision

Step 9 isolates collision. Relax toward local equilibrium. The checked result is: Conserved moments unchanged. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Specify tau. Record step. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 10: Viscosity

Step 10 isolates viscosity. Map tau to kinematic viscosity. The checked result is: Formula depends on lattice units. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: State scaling. Record map. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 11: Boundary

Step 11 isolates boundary. Replace periodic wrap with wall rule. The checked result is: Mass and momentum effects change. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Define wall location. Record rule. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 12: Mach check

Step 12 isolates mach check. Increase velocity. The checked result is: Compressibility error can grow. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Stay in validated range. Record limit. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 13: Refinement

Step 13 isolates refinement. Change dx and dt consistently. The checked result is: Physical scaling changes. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Recalculate parameters. Record table. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Evidence notebook 14: Conclusion

Step 14 isolates conclusion. State populations,collision,streaming,boundaries and validation. The checked result is: Worked densities become 4/3 and 5/3. Keep dimensions, kernel or lattice conventions, node and direction order, normals, sign, time level and precision beside every number. The safeguard is: Mass alone is insufficient. Write conclusion. Change one input, recompute the local rule and global invariant, and explain whether the difference comes from quadrature, collision, streaming, interface transport, force transfer, spatial resolution, time integration, rounding or a changed model. Ask a second reader to reproduce the arithmetic and finish with the narrow claim the evidence supports.

Failure cases and responsible recovery

Velocity-is-population

Macroscopic fields are moments. Build a small counterexample, restore the missing condition and recalculate before making the claim.

Collision changes mass

Conserved moments should survive eligible collisions. Build a small counterexample, restore the missing condition and recalculate before making the claim.

Direction index swapped

Streaming becomes wrong. Build a small counterexample, restore the missing condition and recalculate before making the claim.

Lattice units are physical

Scaling must be documented. Build a small counterexample, restore the missing condition and recalculate before making the claim.

High Mach shortcut

Compressibility error grows. Build a small counterexample, restore the missing condition and recalculate before making the claim.

Bounce-back-is-exact

Wall location and order depend on implementation. Build a small counterexample, restore the missing condition and recalculate before making the claim.

Tau copied

Viscosity and stability change. Build a small counterexample, restore the missing condition and recalculate before making the claim.

Mass-only proof

Flow accuracy needs more evidence. Build a small counterexample, restore the missing condition and recalculate before making the claim.

Verified case study: fourteen steps from model to decision

1. Define the model

Write the PDE or kinetic rule, variables, units, domain and boundary or initial data.

2. Identify the representation

Name boundary densities, lattice populations, a level-set field or Lagrangian markers.

3. State the transfer rule

Write the kernel, collision-streaming map, Hamilton-Jacobi speed or delta coupling.

4. Calculate a tiny case

Use the smallest nontrivial example with labelled indices.

5. Check a conserved quantity

Verify source weight, mass, volume, force or work as eligible.

6. Audit signs and geometry

Check normals, directions, level-set sign, marker coordinates and Jacobians.

7. Test a singular or sharp feature

Approach a kernel source, interface, wall or concentrated force.

8. Refine

Improve panels, lattice, interface grid and marker spacing consistently.

9. Separate errors

Distinguish modelling, discretisation, quadrature, iteration and rounding.

10. Compare a baseline

Use an exact, manufactured, direct or independently implemented reference.

11. Count cost

Record unknowns, nonzeros or dense entries, cell updates, memory and elapsed time.

12. Validate

Compare application outputs with independent measurements or accepted benchmarks.

13. Communicate limits

State what the result supports and what remains untested.

14. Preserve evidence

Save equations, data, code, meshes, parameters, logs and corrected arithmetic.

Learner lab: ten checks for transferable understanding

Lab question 1

Create a reproducible mini-investigation of populations, moments, collision, streaming, lattice units, walls and stability. For task 1, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 2

Create a reproducible mini-investigation of populations, moments, collision, streaming, lattice units, walls and stability. For task 2, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 3

Create a reproducible mini-investigation of populations, moments, collision, streaming, lattice units, walls and stability. For task 3, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 4

Create a reproducible mini-investigation of populations, moments, collision, streaming, lattice units, walls and stability. For task 4, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 5

Create a reproducible mini-investigation of populations, moments, collision, streaming, lattice units, walls and stability. For task 5, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 6

Create a reproducible mini-investigation of populations, moments, collision, streaming, lattice units, walls and stability. For task 6, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 7

Create a reproducible mini-investigation of populations, moments, collision, streaming, lattice units, walls and stability. For task 7, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 8

Create a reproducible mini-investigation of populations, moments, collision, streaming, lattice units, walls and stability. For task 8, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 9

Create a reproducible mini-investigation of populations, moments, collision, streaming, lattice units, walls and stability. For task 9, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.

Lab question 10

Create a reproducible mini-investigation of populations, moments, collision, streaming, lattice units, walls and stability. For task 10, state one prediction, calculate a transparent case, record units and conventions, compare an independent baseline, identify one failure mode, and explain the result once for a student and once for a parent. End with one practical next step and one claim the evidence does not support.