Appendix B
Working Figures
Nine results from Appendix B, each built as a small machine whose motion is the result. Every figure runs on its own; move the pointer over it, or focus it and use the arrow and number keys, to take over. The numbers on the figures come from the same models; those the book prints are checked against it, and values it does not give are marked illustrative in each caption.
Each figure opens with a few everyday questions; after it come plain words, the reasoning and some thoughts on living, with the mathematics folded away at the end for when you want it. The models are simplified on purpose, and the thoughts on living are directions drawn from them, offered for you to try on. They are not medical, psychological, legal or financial advice. Your life is richer than any model, and the final judgment is yours.
Fig 1 · Appendix B.5
Feedback loop
Start here
- Have you ever held a view so long that everything you saw afterward seemed to confirm it?
- When someone points out a mistake of yours, do you usually wave it off, take it in, or swing straight to the opposite view?
- Can correcting yourself too hard ever make things worse?
Bias as a loop on a chart recorder: the correction lever γ keeps κ = α + β − γ between −1 and 1; too little and the bias runs to the stop, too much and it swings apart.
In plain words
Today's bias carries into tomorrow, what you already believe adds to it, and correction pulls it back. Add the three together and you get one number, κ. Above 1, the bias grows every day; between −1 and 1, it fades back to zero; below −1, you overcorrect and it swings back and forth wider each time. Lucidity here is the correction, and it helps only in the right amount. The numbers are illustrative: the book gives none.
Why it holds
Each round multiplies the last round's bias by the same κ. Multiplying again and again shrinks a number only when its size is below 1, so |κ| < 1 is the whole condition for the bias to die out, and the sign of κ decides whether it fades straight back or swings. Correction γ enters κ with a minus sign, which is why lucidity can pull κ into that band and can also push it through the far side. The machine is a chart recorder because the result is about a path in time: the paper shows every round at once.
For thinking and living
Bias needs no villain. When yesterday's view and today's beliefs feed it more than you correct it, it grows on its own. The figure suggests two habits. Correct a little every day, the way a driver keeps adjusting the wheel, instead of waiting for a crisis. And watch for overcorrection: if each fix lands you further out on the other side, a drift has become a swing. In a quarrel, an investment or a diet, the useful question is the same: am I damping this, or feeding it?
A gentle note
A real mind is more than one number, and κ here is illustrative. Use the picture as a mirror for noticing patterns, and be gentle with yourself: everyone carries some drift, and noticing it is where correction starts.
For advanced readers: the mathematics
- Why is |κ| < 1 both necessary and sufficient for the bias to die out? What changes if κ varies from round to round?
- Correction enters κ with a minus sign. For given persistence and reinforcement, which range of correction keeps the bias stable, and what happens at exactly κ = −1?
- Add a random shock each round. How does the long-run spread of the bias behave as |κ| approaches 1?
Fig 2 · Appendix B.6
Obscuration sieve
Start here
- Do all wines taste the same to you, or all jazz sound alike? What would you gain by learning to tell them apart?
- Have you ever lumped two different people, or two different problems, together under one label?
- Once a distinction is lost, can it be recovered from what is left?
Eight signals pour through swinging chutes into glass tubes; every merge the agent cannot undo lowers the entropy it can reach, and the scale reads 𝒪 = 1 − H(P_t)/H(P_𝒳) from 0 to 1.
In plain words
Reality sends eight kinds of signal. If you cannot tell two of them apart, you have lost a distinction. Each merge loses one more, and the scale shows what share of the original information, measured in bits, you can no longer resolve. All eight kept apart reads 0, nothing hidden; all poured into one tube reads 1, everything blurred together.
Why it holds
Pouring two signals into one tube discards the difference between them, and no later step can bring it back. A coarser sorting of the same signals never carries more information, so each merge lowers the entropy the agent can reach and 𝒪 only moves up. The machine uses liquid and chutes for exactly this property: what has been poured together cannot be separated again.
For thinking and living
Labels are useful, and every label pours several things into one tube. The figure says that what is poured together stays together: no amount of later reasoning recovers a distinction you never kept. So the cheapest moment to stay lucid is the moment of perception. Before you file a person, a feeling or an event under a familiar heading, ask what the merge would lose. Learning a craft, a language or a person well is largely learning to keep apart what you used to lump together.
A gentle note
Lumping things together is how anyone gets through a day; a mind that kept every distinction would never act. The figure measures what is lost. Which distinctions matter to you is still your call.
For advanced readers: the mathematics
- Why can merging never lower 𝒪? Relate this to the data-processing inequality.
- Two different merges can reach the same 𝒪. What does that say about measuring obscuration with a single number?
- If the eight signals are not equally likely, which merge costs the least information, and why?
Fig 3 · Appendix B.11
Tipping track
Start here
- Have you ever stayed quiet in a meeting, then learned afterward that most people agreed with you?
- How many others would need to speak up before you felt safe to?
- Can a group that is mostly right still lose its nerve?
A ball rolls away from a ridge set by the perceived threshold: with enough fear, a group already past the real threshold still falls silent.
In plain words
Being lucid pays off for each person only once enough of the group is lucid too. The real tipping point is fixed by costs and payoffs (40% here, our choice); people act on the tipping point they believe in. Fear pushes that believed point up, courage pulls it down. So a group already past the real point can still slide back into silence if it is afraid, and a courageous group can rise before reaching it.
Why it holds
Acting lucid pays more the more others already do, so the net advantage g(p) rises with the lucid share p and crosses zero at one point, the threshold. Below it the share shrinks, above it the share grows, so the threshold is a balance point nothing can rest on, a ridge, and the two ends, all silent and all lucid, are where the ball comes to rest. Fear and courage move only where agents believe the ridge stands; the ball rolls by the believed ridge. That is why perception alone can decide which end a group reaches.
For thinking and living
Silence often survives on a miscount. Each person estimates the tipping point from what they can see, and fear pushes the estimate up. The figure suggests two things. Your guess about how many others agree is a variable worth checking: ask around, or speak first on small matters. And courage works in public: one person who lowers the believed threshold makes it cheaper for everyone who comes after. Being early has a cost; the figure shows why it can also be the step that decides the outcome.
A gentle note
Speaking up has real costs, and sometimes keeping yourself safe is the wise choice. The model knows nothing about your workplace, your family or where you live. Please weigh your own situation; no figure should talk you into a risk you cannot afford.
For advanced readers: the mathematics
- The threshold is an unstable equilibrium. Linearize the dynamics near it: what sets how fast a group moves away from it?
- Model fear as a shift added to the perceived threshold. With the real threshold at 40%, which starting shares end lucid as the shift varies?
- Let the perceived threshold differ from person to person, a distribution instead of one number. Compare the result with Granovetter's threshold model of collective behaviour.
Fig 4 · Appendix B.12
Lucidity shutter
Start here
- Do you know someone who has an explanation for everything? And someone who says nothing can really be known?
- Which of the two are you closer to?
- If your attention is limited, is it better to split it evenly or to go all in on one side?
Two roller blinds geared so that λ + ξ = 0.90; the light let through is 𝓜 = λ·ξ, largest when the window is square.
In plain words
λ is how much of the world's Pattern you understand; ξ is your awareness that reality goes beyond what you understand. Here they share a fixed total of 0.90, so raising one lowers the other. Lucidity is their product, so it is largest when the two are equal (0.45 each, 𝓜 ≈ 0.203) and small when either side dominates: the Logonaut and the Mystient both get 0.080.
Why it holds
With λ + ξ fixed at S, the product λξ is largest when the two are equal: by the inequality of arithmetic and geometric means, λξ ≤ (S/2)² = S²/4. Moving a distance d off the middle costs exactly d², whichever side gains. Two geared blinds build the constraint into the machine, since opening one closes the other, and the light through the window is its area, a product.
For thinking and living
Understanding, and a sense of what lies beyond understanding, draw on the same budget of attention, and in this model lucidity is their product. The cost of leaning grows with the square: small imbalances cost little, large ones cost a lot. So the practical advice is modest. Perfect balance is unnecessary; what matters is noticing when you have drifted far to one side. A scientist who never says "we don't know" and a mystic who never checks anything have made the same mistake from opposite ends.
A gentle note
λ and ξ are not scores you can look up for yourself, and people rightly spend whole seasons of life leaning one way, in study or in grief. The shutter describes a tendency over years; there is no need to grade your today by it.
For advanced readers: the mathematics
- Prove λξ ≤ S²/4, and show that moving a distance d off the middle costs exactly d². Why does it not matter which side gains?
- If lucidity were λᵃξᵇ instead of λξ, where would the best split sit? What could justify unequal exponents?
- Drop the fixed total and let λ + ξ ≤ 1, with δ = 1 − λ − ξ unaware. How does the picture change?
Fig 5 · Appendix B.13
Face tube
Start here
- Is it better to know a little about many things, or a lot about a few?
- When you try to keep every side of a question in view at once, do you feel clearer or more scattered?
- Why might two perspectives held well beat five held thinly?
Light crosses one plate per face of Reality; with the total fixed, the product of the openings peaks at two faces and falls to 0.192, 0.063, 0.018 as faces are added.
In plain words
Suppose reality has several faces and your total awareness is fixed. Lucidity multiplies your awareness of every face, and each one is below 1, so every extra face multiplies in another fraction. With the same total, two faces let through the most light (0.500); three give 0.192, four 0.063, five 0.018. Leaning your awareness toward one face makes the product smaller still.
Why it holds
The total richness fixes the sum of squares, x₁² + … + xₙ² = r², and lucidity is the product of the x's. By the same mean inequality the product is largest when every face gets an equal share, giving (r/√n)ⁿ, and that falls as n grows because each added face multiplies in a smaller fraction. One face is ruled out: it would need x₁ = r = 1, and awareness of any face stays below 1 (T1). So at this richness two faces are the best an agent can do. The tube puts one plate per face in the light's path, so what comes through is the product.
For thinking and living
With a fixed budget of attention, each extra face you try to hold multiplies in another fraction, so in this model spreading thin is expensive. The best count here is two faces, held evenly. In practice: before adding another lens to a problem, ask whether you have really held the first two. And when one face starts to crowd out the rest, the product drops faster than it feels.
A gentle note
Reality's faces cannot be counted in any simple way, and the figure fixes the total richness only to keep the arithmetic clean. Curiosity about many things is a joy worth keeping; read this as a note about depth, and leave your breadth alone.
For advanced readers: the mathematics
- Show that under x₁² + … + xₙ² = r² the product peaks at equal shares, at (r/√n)ⁿ. Is there any r for which three faces beat two, given every xᵢ < 1?
- Why is the one-face case excluded by T1 instead of by the arithmetic?
- Replace the sum-of-squares budget with a linear one, x₁ + … + xₙ = S. Does the conclusion survive?
Fig 6 · Appendix B.14
Lucidity reservoir
Start here
- Have you noticed that clarity is easier to keep once you already have some?
- What quietly drains your clarity in an ordinary week?
- Can a habit that leans too far one way empty the tank, however full it started?
The four-mode master equation as a tank: a feedback pump, a ceiling collar, a balance lever and a drain. Past 15° or 75° the tank empties.
In plain words
The water level is lucidity. The pump is growth: the more lucidity you have, the faster it builds, slowing as it nears the ceiling. The drain is a steady loss at rate γ. The lever is the balance between understanding and Mystery-awareness; it sets the ceiling, which is highest at 45°. At the book's values, push the lever past 15° or 75° and the ceiling drops so low that the drain wins: the tank empties however full it started.
Why it holds
The level settles where growth equals loss: α M (sin 2θ − M) = γ M, so the resting level is M* = sin 2θ − γ/α. That is positive only while sin 2θ > γ/α. At the book's values γ/α = 0.5, and sin 2θ = 0.5 at exactly 15° and 75°. Outside that band no positive level can hold, so the tank drains from any start. The tank splits the equation into parts you can watch one at a time: the pump is the growth term, the collar the ceiling sin 2θ, the lever θ, the drain γ.
For thinking and living
The tank holds two lessons. Lucidity compounds, so small daily practice matters most when you have little; and loss is steady, so it has to be met steadily. The harder lesson is the band: at the book's values, lean too far toward pure analysis or pure wonder and no starting level survives. A good start cannot save a lopsided habit. Check the lever before you worry about the level.
A gentle note
γ and α are the book's illustrative values, and a real life has seasons when the drain runs fast: illness, loss, exhaustion. A falling level in those seasons says nothing about your worth. Rest is one way to slow the drain.
For advanced readers: the mathematics
- Derive M* = sin 2θ − γ/α and check its stability from the sign of the derivative at M*.
- What happens to the band of viable angles as γ/α falls toward 0, and as it rises toward 1?
- At sin 2θ = γ/α the two equilibria meet and exchange stability. What kind of bifurcation is this?
Fig 7 · Appendix B.15
Coupled vessels
Start here
- Has someone close to you ever made you clearer just by being around?
- Do you become more like the people you spend time with?
- If two friends are very close, should they end up exactly alike?
Two agents with different growth rates joined by a valve. Opening it narrows the gap like 1/β without closing it.
In plain words
Two people grow in lucidity at different rates, so on their own they settle at different levels, 0.30 apart. Open the valve, that is, let them influence each other with strength β, and the higher one lifts the lower. The stronger the link, the smaller the gap (0.19, 0.11, 0.05, 0.02), but it never closes, because each still grows at their own rate.
Why it holds
Each level feels two pulls: its own growth and loss, which alone would hold it at 1 − γ/αᵢ (0.70 and 0.40 here), and the coupling β(M̄ − Mᵢ), which pulls it toward the pair's mean. At rest the pulls cancel. The two growth rates differ, so balancing them takes a gap, and the coupling supplies a pull proportional to β times that gap; the stronger β, the smaller the gap that suffices, roughly as 1/β, but a zero gap would supply no pull at all. A valve between two tanks is this coupling made literal: the flow through it follows the difference in level.
For thinking and living
Influence can lift. In the model, the more lucid partner raises the other and gives up a little of their own level in doing so; the stronger the bond, the closer the two come. The gap never closes, because each keeps their own growth rate, and the figure treats that as normal: a friendship or a marriage can bring two people near while keeping their difference. Choose your close company with care, and when you are the steadier one, know that your steadiness travels.
A gentle note
People are far more than tanks and valves, and no one has to raise another person's level alone. If a bond keeps pulling you down, the same arithmetic is running in reverse; protecting your own level and asking for help are both fair.
For advanced readers: the mathematics
- Solve for the two-person equilibrium with coupling β(M̄ − Mᵢ) and show that the gap shrinks roughly as 1/β. Where does the approximation fail?
- Does coupling raise the pair's average level, or only move lucidity from one to the other? Check against the logistic growth terms.
- Extend the model to n people on a network. What role does the graph Laplacian play?
Fig 8 · Appendix B.16
Collective pool
Start here
- Have you been on a team that was smarter than any of its members? Or one that was dumber?
- What made the difference: the people, or how they talked to each other?
- Can one person drag a whole group down without changing anyone's mind?
Five agents whose lucidity never changes feed one pool; the pattern of their links alone lifts the group above its mean or drags it below.
In plain words
A group's lucidity is its members' average plus a term set by how they are linked. Here the five members keep their own levels throughout; only the links change. A cooperative link adds a bonus, largest between two members who are both lucid and close in level; an adversarial link subtracts one. With the right links the group reaches 0.725, 0.165 above the members' mean: that surplus is the emergence gap. The weight of the link term, η = 0.5, is our choice.
Why it holds
The group's lucidity is Ψ = M̄ + η·S_W. No member changes, so the mean M̄ cannot move, and everything the group gains or loses beyond it comes from the link term S_W. Each link adds g(a, b) = min(a, b)(1 − |a − b|): the less lucid member caps it and a wide gap between the two discounts it, which is why a link pays most between two lucid members of similar level. A negative weight turns the same quantity into a loss, which is how one heavy adversarial link or a propagandist drags the group below its mean. The pool shows the mean as a fixed mark and the link term as what lifts the water above it or lowers it below.
For thinking and living
The pool shows that a group is more than a count of good people. With every member fixed, the links alone lifted the group 0.165 above its mean, and one adversarial link can sink it below. Two habits follow. Invest in links between people who are both clear and close enough to hear each other, and help the weaker side of each link, since that side caps what the link can give. And watch for the one hostile tie that costs more than any member adds.
A gentle note
η = 0.5 is our choice, and five members make a toy. Real groups have history, power and care that no formula holds. If you feel like the weak side of a link, remember that in this model your level is what the link is built on, so being helped is part of how the whole group rises.
For advanced readers: the mathematics
- For fixed a, show that g(a, b) = min(a, b)(1 − |a − b|) is largest at b = a. What is that largest value?
- How should S_W scale with group size so that Ψ stays comparable between a group of five and a group of fifty?
- For given members and weights, which arrangement of signed links minimizes Ψ? How few adversarial links suffice to push Ψ below the mean?
Fig 9 · Appendix B.17
Trust dishes
Start here
- Why is it hard to trust someone who seems certain about everything?
- Would you rather negotiate with someone who admits what they don't know, or someone who never does?
- Can more contact cure distrust, or does something else have to change first?
Two dishes trade beams; trust holds while each side's gain beats its dissipation, and as the other's Mystery-awareness falls toward 0 the threshold β† runs off the dial: the Chain of Suspicion.
In plain words
Two civilizations can cooperate only if each gains more from the other than it loses on its own. What one gains from the other grows with how strongly they interact (β) and with the other's Mystery-awareness, its awareness that reality goes beyond what it understands. If either side's Mystery-awareness falls toward zero, no amount of interaction is enough: the β needed runs off the dial and cooperation cannot hold. That is the Chain of Suspicion. The values are illustrative.
Why it holds
Cooperation is an equilibrium only when both conditions hold: β ξⱼ > γᵢ and β ξᵢ > γⱼ. Solved for β, each side needs β above its own dissipation divided by the other's ξ, so the smaller Mystery-awareness sets the bar, β† = γ_max / min(ξᵢ, ξⱼ). As that ξ falls toward 0, γ/ξ grows without bound and no finite β meets it. Two dishes make the symmetry visible: each beam's strength is set by the side it comes from, so either dish going blind breaks the link for both.
For thinking and living
Trust in this model rests on the other side's humility. What each side gains depends on the other's sense that reality exceeds its understanding, and if either side loses that sense entirely, no amount of contact makes cooperation stable. That turns a familiar instinct around. Before pushing for more meetings, more deals or more messages, look at whether both sides can still say "we might be wrong". Your own Mystery-awareness is the half of the bargain you control.
A gentle note
This is a model of civilizations, borrowed here as a lens on people, and its values are illustrative. Trust between real people also rests on safety. If someone has shown they will hurt you, protecting yourself is the lucid choice, and no figure asks you to keep the channel open.
For advanced readers: the mathematics
- Show that the exact threshold is max(γᵢ/ξⱼ, γⱼ/ξᵢ). When does it equal γ_max / min(ξᵢ, ξⱼ), and when is the latter only a sufficient bound?
- Liu Cixin's Chain of Suspicion assumes neither side can verify the other's intentions. Which part of that assumption does ξ → 0 formalize, and which part does it leave out?
- Let each side estimate the other's ξ with noise. How does misperception change the region where cooperation is stable?