Setup
- Treat the system as coupled to an infinite set of diameters on the unit circle.
- Watch it over time: take one thousand snapshots in a fixed window.
- At each snapshot, for each idea, either both poles adhere (balanced) or exactly one pole fails to adhere (a “missing pole”). The idea itself is still present; only one pole is absent.
Negative photograph rule
- When a pole at angle “theta” is missing, draw a unit arrow at the opposite angle “theta plus one hundred eighty degrees.”
- Sum all such unit arrows tip-to-tail across the one thousand snapshots.
- The long arrow you get is the resultant. Its length is “C.” Its direction is the system’s bias direction under the chosen template.
- If you want comparability, divide by one thousand at the end to get a mean resultant per snapshot.
What lights up
- A pole that often fails to adhere produces a bright line at that angle in the negative photograph.
- The resultant points toward the antipole of what is missing. That is why a “just” system shows a bright missing line at “injustice,” and the arrow points toward “justice.”
Worked fairness example (just vs typical)
Template
- Fairness idea has two poles: “justice” at eighty degrees and “injustice” at two hundred sixty degrees.
Perfectly just system (in a vacuum)
- Across one thousand snapshots, “injustice” fails to adhere one thousand times; “justice” never fails.
- Negative photograph: a razor-bright line at two hundred sixty degrees.
- Resultant: one thousand unit arrows pointing to eighty degrees, summed tip-to-tail, then divided by one thousand if you normalize.
- Direction says “justice.”
- Realization part equals magnitude times cosine of the direction.
- Imaginary part, the contribution to the denominator, equals magnitude times sine of the direction. With a direction near ninety degrees, the imaginary part is large.
Typical meso system
- Say “injustice” is missing six hundred times and “justice” is missing one hundred times; everything else balances out.
- Net contribution toward “justice” is five hundred unit arrows (six hundred minus one hundred) at eighty degrees.
- Normalize by one thousand if you want a mean: zero point five toward eighty degrees.
- Realization part and imaginary part follow from that arrow: realization equals magnitude times cosine of eighty degrees (small), imaginary equals magnitude times sine of eighty degrees (large).
- Interpretation: strong actualization pressure from fairness, little realized pattern yet.
Read-off rules (fast grading)
- Where does the arrow point? That is the bias direction under your template.
- How long is the arrow? That is the bias strength: larger means a steadier, more persistent missing pole.
- Imaginary part “k” (what you pass to the denominator) Larger absolute “k” means stronger tightening and lower realized ratio when you keep actual and predictor fixed at one.
- Realization part “j” Large “j” means you see many realized instances, but the idea’s mark is fuzzier. Small “j” with large “k” means few realized instances but a crisp mark.
Macro vs meso vs micro expectations
- Macro (eternal now, very large window): many faint lines that mostly cancel; resultant near zero.
- Meso (cultures, institutions): a stable fingerprint of a few bright missing poles; clear nonzero arrow.
- Micro (persons, projects): sharp gaps and larger swings; arrow can be quite long.
Classroom cadence (five minutes to run)
- Draw the unit circle. Explain “infinite diameters present; we only see what fails to adhere.”
- Overlay the negative photograph: tick marks grow at angles where poles go missing.
- Every ten snapshots, update the tip-to-tail sum of antipole arrows so students watch the arrow drift and settle.
- At the end, state the three numbers you care about: direction, magnitude, and the imaginary contribution “k.”
- If you want, finish by plugging “k” into “predictor plus i times k” with actual equals one and predictor equals one, so they see how ideas alone change the ratio.
One-line takeaway for students
A system’s bias is not “what it loves,” it is “what persistently fails to adhere.” Plot the missing poles, add the opposite arrows, and the average arrow you get is the bias; its sine projection gives the imaginary part you pass to the denominator.