THEOREM 27 · ENGLISH

Theorem 27: Ignorance-Conditioned Volitional Formations

Formula-Annotated Edition

Avijjāpaccayā saṅkhārā — From the Nonattainment of Nirvanic Tranquility to Directed Volitional Formation

Hideto Tomabechi · August 9, 2026

Cognitive Research Laboratories, Tokyo / CyLab, Carnegie Mellon University / C5I Center, George Mason University

Base: TomabechiFourDharmaSealsMini13

How to read this edition

This edition carries exactly the same content as the academic paper on Theorem 27, rewritten for two kinds of reader at once: those who are not at ease with mathematical notation, and those who have never studied Buddhist doctrine. Not a single formula has been dropped. Instead of dropping them, every formula is accompanied by an explanation of every symbol it contains, and by a restatement in pictures and ordinary words of what the formula is claiming.

You may read the chapters in any order, but a first reader is advised to begin with Chapter 0 (Theorem 27 in three minutes), then look through Chapter 1 (the Buddhist background) and Chapter 2 (the mathematical toolbox). Once those two are behind you, the formulas from Chapter 5 onward will look like combinations of tools you already know. If you are pressed for time, Chapter 0 together with Chapter 13 (the summary) will still give you an accurate grasp of what this theorem does claim and what it does not.

Four kinds of coloured box appear throughout. Blue explains the symbols in a formula; orange restates the formula in plain words; green gives the motivation for defining things that way; red warns against a reading that is easy to fall into but wrong. A reader who is uncomfortable with notation can follow the whole argument by reading only the orange boxes.

The theorem in one sentenceStarting from the positional fact that tranquility has not yet been attained, one first derives that the remaining quantity is necessarily decreasing; adding one further condition, one may then say that this decrease is the agent's own doing — that is the whole of it.

Contents

Ch. 0 Theorem 27 in three minutes
Ch. 1 Readable without prior Buddhist study
Ch. 2 The mathematical toolbox
Ch. 3 The two theorems underneath
Ch. 4 Full list of symbols
Ch. 5 Putting “ignorance” into a formula
Ch. 6 Putting “it is decreasing” into a formula
Ch. 7 Putting “volitional formation” into a formula
Ch. 8 Theorem 27 itself
Ch. 9 The proof, one step at a time
Ch. 10 Four readings to avoid
Ch. 11 What this theorem borrows from its foundation
Ch. 12 Frequently asked questions
Ch. 13 Summary
Appendix A All equations
Appendix B All symbols

Chapter 0 Theorem 27 in three minutes

The twelve links of dependent origination open with the line “with ignorance as condition, volitional formations come to be.” Theorem 27 translates that one line, and only that line, into mathematics. The translation consists of walking three boxes from left to right.

Fig. 17 The conclusion comes in two stages. Stage (2) follows from Theorem 26 alone; reaching stage (3) requires the extra hypothesis called Condition 27-A. This two-stage design is the single most important feature of the theorem.

(1) What ignorance is. In this system, ignorance is neither a mood nor a lack of cleverness. It is a fact about position: the place one currently occupies lies outside the set on which suffering is zero. It is the same kind of statement as “on the map, I am not yet inside the destination.”

(2) What follows from that. As long as one is outside the destination, the distance to it is not zero. The “height meter” supplied by Theorem 26 is tied to that distance and therefore necessarily takes a positive value. And since the same Theorem 26 guarantees that the meter must fall at a rate proportional to its own height, it follows that while ignorance holds, the meter is falling and cannot stall.

(3) May we call that a volitional formation? Here lies the crux of the theorem. A falling meter alone does not yet establish that the agent did anything: a boat also moves when it is merely carried by the current. Condition 27-A is therefore imposed, and it explicitly subtracts the drifting part from the rowing part. Only when that remainder is nonzero do we call it a volitional formation.

In one line: if tranquility has not been attained, something is necessarily being rowed. And the converse holds as well: once inside the room, that rowing becomes zero. But — and this is the point developed in Chapter 10 — what becomes zero is only the component directed out of the room. Living, thinking, and responding to others do not stop.

💡 In plain wordsThink of a climber halfway up a mountain. “Not yet at the hut” is ignorance; “the metres still showing on the altimeter” is the height meter; “the part actually produced by moving one's legs” is the volitional formation. If the altimeter falls because a cable car is carrying you, you are not walking. Condition 27-A is precisely the device that subtracts the cable car.

Chapter 1 Readable without prior Buddhist study

1.1 What the twelve links are

The twelve links of dependent origination are among the oldest doctrinal formulations in Buddhism, explaining how suffering arises as a chain of twelve items: ignorance, formations, consciousness, name-and-form, the six sense bases, contact, feeling, craving, clinging, becoming, birth, and aging-and-death. Each is joined to the next by a conditional relation of the form “because A is, B is,” and the whole is customarily drawn as a wheel.

This theorem addresses exactly one arrow of that wheel: “ignorance → formations,” in Pāli avijjāpaccayā saṅkhārā. About the remaining eleven relations the theorem says nothing whatever.

Fig. 1 The wheel of twelve links. Only the first arrow is turned into mathematics here; no claim is made about the other eleven.

1.2 Splitting the Pāli line into three parts

The Pāli formula has three parts: avijjā (ignorance), paccayā (with … as condition), and saṅkhārā (formations). Read in order, it says “with ignorance as condition, formations come to be.”

Saṅkhārā is plural. Following the widely used English rendering of Bhikkhu Bodhi, this paper reads it as volitional formations: roughly, the activity of shaping something under the direction of intention.

Fig. 19 The structure of the Pāli line. Only the conditional-implication form of this one sentence is formalized here.

⚠️ A common misreadingIn the Pāli originals, avijjā is explained as not knowing suffering, its arising, its cessation, and the path to that cessation; and SN 12.2 defines saṅkhārā as three kinds of formation — bodily, verbal, and mental. This theorem does not reduce any of those meanings to mere nonmembership in a set or to a single control component. The ignorance and formation that appear below are model-relative, operational notions defined inside the mathematical system of Theorem 26, and are not identical with the threefold classification of the Pāli originals.

Pāli source materials: Saṃyutta Nikāya 12.2 (Pāli text), together with the English translation of that discourse by Bhikkhu Bodhi.

1.3 “Nirvanic tranquility” in this system

One further term should be fixed in advance. In this system, nirvanic tranquility means the collection of states in which one is alive and the remaining suffering is exactly zero. The decisive point is that being alive is written into the definition from the outset; hence tranquility is not death. The collection is moreover allowed to move with time, so tranquility here is dynamic rather than static. Chapter 3 takes this up in detail.

Chapter 2 The mathematical toolbox

This chapter assembles the fifteen tools needed to read Theorem 27. Every one of them is obvious once drawn. Having passed through this chapter once, you will find that the formulas that follow are nothing but combinations of tools you already know.

Tool 1 Sets, and being in or out of one

A set is simply a collection of things. The only question we ask is whether a given point belongs to the collection or not; membership is written with the symbol for “is in,” nonmembership with the symbol for “is not in.” Throughout Theorem 27, the set is always the room where suffering is zero, and failing to be in it is what we call ignorance.

Fig. 3 Ignorance is the positional fact of being outside the room. It is not a matter of feeling or of ability.

Tool 2 The distance “dist” — length to the nearest wall

The distance from a point to a set is the length to the nearest point of that set, written dist(x, room). Inside the room the distance is zero; outside it is a positive number.

One mathematical caution is needed here. To assert that any outside point necessarily has positive distance, the room must be closed. For a set whose boundary is not quite included, one can have an outside point whose distance is nevertheless zero. Theorem 26 guarantees that this room is properly closed.

Fig. 4 Distance is the shortest length to the nearest wall. Because the room is closed, any point outside has positive distance.

Tool 3 A square is never negative

Any real number squared is at least zero. This obvious property does heavy work below as the guarantee that a sign cannot flip. In particular, “the square of the distance” has the convenient feature that it vanishes exactly when the distance vanishes.

Tool 4 A Lyapunov function — the height meter

Control theory routinely uses a device that expresses “how much is left to the destination” as a single number: a Lyapunov function. The name is forbidding; the thing is a fuel gauge run backwards. It must satisfy only two properties.

It is zero at the destination and positive everywhere else.
It necessarily decreases as time passes.

Given these two, one may conclude that as long as the meter keeps falling, the destination is being approached. The height meter used in Theorem 27 is written W_⊤.

Fig. 13 The height meter W_⊤: a single gauge for how much is still left. The instant it reaches zero is tranquility.

Tool 5 The sandwich — meter and distance move together

It would be useless if meter and distance were unrelated. Two positive constants are therefore introduced, trapping the meter between two multiples of the squared distance.

c₁ dist(x,𝒩_⊤(t))² ≤ W_⊤(x,t) ≤ c₂ dist(x,𝒩_⊤(t))²(26.A)

🔢 The symbols in this formulac₁, c₂ = positive constants (the lower and upper multipliers) / dist = distance to the room / 𝒩_⊤(t) = the zero-suffering room at time t / W_⊤ = the height meter / x = the present state / t = time

💡 In plain wordsIt says that the meter's value moves at roughly the same magnitude as the squared distance to the room. Hence positive distance means positive meter, and zero meter means zero distance. Being able to travel in both directions is what matters later.

Fig. 5 The sandwich. The meter is necessarily trapped between two multiples of the squared distance.

Tool 6 The derivative — a slope

A derivative is the slope of a graph. A negative slope means a downhill, that is, the quantity is decreasing; the size of the slope is the speed of that decrease.

Tool 7 The upper right Dini derivative — slope even at corners

The ordinary derivative has a weakness: where a graph has a corner, no slope is defined, because approaching from the left and from the right give different answers.

So one adopts a measurement that looks only at the approach from the right and takes the largest available value. This is the upper right Dini derivative, written D⁺_t. It is always defined, even at corners, and at ordinary points it agrees exactly with the ordinary derivative. It is an extension of the usual notion, not a different one.

Fig. 7 The upper right Dini derivative. By looking only at the slope on the right, a value is always defined, even at a corner.

🌱 Why it is set up this wayWhy is such a tool needed? In real systems the meter is often built from “the distance to the nearest wall,” and such a quantity is not necessarily smooth everywhere: a corner appears at the instant when which part of the wall is nearest switches over. The Dini derivative is the tool that faces such corners head-on rather than detouring around them.

Tool 8 Strict descent — decreasing without stalling

There are weak and strong senses of “decreasing.” A decrease that slows down and eventually stalls partway is of no use. So a strong condition is imposed: the speed of decrease is proportional to the amount still remaining.

D⁺_tW_⊤(x(t),t) ≤ −λ W_⊤(x(t),t)(26.B)

🔢 The symbols in this formulaD⁺_t = the upper right Dini derivative (Tool 7) / λ = a positive constant fixing the rate of decay / W_⊤ = the height meter / x(t) = the state at time t

💡 In plain words“The meter must fall at a rate proportional to its present height.” The more that remains, the faster it falls; as little remains, it eases off. What matters is that while the remainder is positive, the falling rate is never zero. It therefore cannot stall partway.

Fig. 6 Strict descent. While height remains, the rate of fall never reaches zero.

Tool 9 Vectors and the inner product — how well two directions agree

A vector is an arrow with a direction and a length. The inner product of two arrows is a single number saying how far they point the same way: positive if they agree, negative if they oppose, zero if they are perpendicular.

Here the inner product is written by transposing one factor (for example ∇W_⊤^Tv), read as “the inner product of ∇W_⊤ with v.”

Tool 10 The gradient — the steepest uphill direction

Stand on a hillside and point in the steepest uphill direction: that is the gradient. It is written with the nabla symbol, so the gradient of the meter is ∇W_⊤.

Taking the inner product of the gradient with the direction of actual motion tells you whether that motion raises or lowers the meter. A negative inner product means there is a downhill component, so the meter falls.

Fig. 8 The gradient points steepest uphill. If the inner product with the direction of motion is negative, the meter falls.

Tool 11 The chain rule — splitting a compound effect

The meter's value changes both because time itself passes and because the state moves. The chain rule adds these two effects together, thus.

D_tW_⊤ = ∂W_⊤/∂t + ∇W_⊤^T ẋ

🔢 The symbols in this formula∂W_⊤/∂t = the change with the state held still and only time advancing / ∇W_⊤ = the gradient (Tool 10) / ẋ = the velocity vector of the state / D_t = the time derivative along the trajectory

💡 In plain words“The meter moved for two reasons: the clock advanced, and I moved.” In Chapter 7 the right-hand side is split further into four parts, three of which cancel.

Tool 12 The Cauchy–Schwarz inequality — extracting a lower bound on length

The inner product of two arrows a and b can be written thus.

a^Tb = ‖a‖ ‖b‖ cos θ

🔢 The symbols in this formula‖a‖, ‖b‖ = the lengths of the two arrows / θ (theta) = the angle between them / cos θ = a quantity equal to 1 at zero degrees, 0 at a right angle, and −1 at 180 degrees

However hard it tries, cos θ cannot exceed 1. Hence the size of the inner product cannot exceed the product of the two lengths. That is the Cauchy–Schwarz inequality.

A picture makes it obvious. Shine light on a pole: the length of its shadow is ‖b‖cos θ. A shadow is never longer than the pole that casts it. The moment the light moves off vertical, the shadow gets shorter.

Fig. 24 The geometry of Cauchy–Schwarz. A shadow never exceeds its pole; turned around, the shadow's length bounds the pole's length from below.

💡 In plain wordsThe use of this inequality lies in dividing by it. Dividing both sides by ‖a‖ gives ‖b‖ ≥ |a^Tb| / ‖a‖ — that is, “since the shadow is this long, the pole must be at least this long”. A length one cannot measure directly gets a lower bound.

🔍 The mathematical skeletonThe assignment here is: a = G^T∇W_⊤ (the actuator's leverage), b = η₂₇ (the size of the volitional signal), and a^Tb = the work actually done. The work is known to be at least λW_⊤ by (27.6), and the leverage is known to be at most L₂₇. Dividing the one by the other yields the lower bound λW_⊤/L₂₇ on the size of the effort. That is equation (27.8).

🌱 Why it is set up this wayEquality holds only when the two arrows point in exactly the same direction. So the bound in (27.8) is the most generous possible estimate: even rowing in the most efficient way imaginable, at least this much is required. In practice more is needed.

Fig. 12 The same point as a lever. If the work required is pinned down and the leverage has a ceiling, the size of the force is bounded below.

Tool 13 “Almost every instant” — points have no width

Begin with the notion of length. The interval [a, b] has length b − a. What, then, is the length of a single point? Zero. A point has no width.

Does gathering many points produce length? For finitely many, plainly not. Strikingly, even for countably many — an infinity that can be labelled 1, 2, 3, … — the gathered length is still zero. For instance the fractions in the interval [0, 1] are infinite in number, yet all of them together have length zero.

The name for the measuring device that extends this notion of length from intervals to complicated sets is the Lebesgue measure. And the phrase “there may be exceptions, but all of them together have measure zero” is rendered as almost everywhere, or “at almost every instant,” abbreviated a.e.

Fig. 25 Exceptional instants appear as points with no width. No number of them adds any length of time.

🌱 Why it is set up this wayIn this paper the exceptions are the instants at which the meter's graph has a corner. A corner arises at the moment when which part of the wall is nearest switches over. Such switches occur only at isolated moments, so gathering all of them gives length zero. That is why “at almost every instant” suffices.

💡 In plain wordsThe practical import is this: so long as one integrates over time or takes time averages, exceptions of measure zero make no difference whatever. It is the same reason that adding lines of no width to a region does not change its area.

⚠️ A common misreadingStill, “almost every” and “every” are strictly different. The question “does it hold at the particular instant t = 5?” cannot be answered, since the possibility that this instant is one of the exceptions cannot be excluded. Hence in (27.10) of Chapter 8 the first equivalence is stated for every instant and the second for almost every instant; the two must not be collapsed.

Tool 14 Lipschitz regularity — a speed limit on steepness

Smoothness of a function comes in degrees. What Theorem 27 requires is the intermediate one, Lipschitz continuity, defined in a single line.

| f(x) − f(y) | ≤ L ‖ x − y ‖

🔢 The symbols in this formulaf = the function in question (here the height meter W_⊤) / x, y = any two points / ‖x − y‖ = the distance between them / L = the Lipschitz constant, a positive number giving the ceiling on the slope

💡 In plain words“The difference in values is at most L times the distance between the points.” On a graph: nowhere is the slope steeper than L. Steep hills are allowed; sheer cliffs are not.

Fig. 21 Corners are allowed; cliffs are excluded. Lipschitz sits exactly between continuity and differentiability.

The three degrees of smoothness compared
Property	Meaning	Example
continuous	no breaks	√x (infinite slope at the origin)
Lipschitz	no breaks, and a ceiling on the slope	\|x\| (a corner at the origin, but slope at most 1)
differentiable	a unique tangent line at each point	x²

🔍 The mathematical skeletonThis property does three jobs here.
(1) It is the source of “almost every instant.” By Rademacher's theorem, a locally Lipschitz function is differentiable almost everywhere; corners can occupy only a set of measure zero. That is what licenses the a.e. of Tool 13, and with it the chain rule. Without Lipschitz, one could not even say a.e.
(2) It guarantees that the descent rate is finite. With a ceiling on the slope, −D⁺_tW_⊤ cannot blow up to infinity. Des₂₇(t) is meaningful as a finite quantity thanks to this regularity.
(3) It explains why “locally” suffices. No single L need work across the whole space; it is enough that some L works on a neighbourhood of the trajectory under study. For real systems that is all one needs.

🌱 Why it is set up this wayWhy not demand differentiability outright? Because W_⊤ is typically built from dist(x, 𝒩_⊤(t)), and a corner necessarily appears at the moment when which part of the wall is nearest switches over. Demanding differentiability would exclude almost every realistic system. Lipschitz is the minimum strength that lets one face those corners rather than detour around them.

Tool 15 The actuator — where intention meets reality

In engineering, an actuator is a motor, a hydraulic cylinder, a rudder — a device that turns a control signal into physical force, the exact converse of a sensor, which turns the world into information. In the formulas it is the matrix G(x,t), multiplying the input u to produce the state action Gu.

In the boat metaphor: u is how hard one pulls, G is the relation between oar and water, and Gu is how far the boat actually moved.

Fig. 22 The actuator G maps the input space into the state space. An input lying in its kernel produces no state effect, however hard one strains.

Three properties must be kept in view.

(1) G is neither square nor invertible. The input dimension m and the state dimension n do not agree. The plain fact that one cannot do everything one might wish is written into the shape of the matrix from the outset.

(2) It has a kernel. There can exist v ≠ 0 with Gv = 0 — an input direction along which one strains while nothing happens in the world. A feathered oar sliding flat along the surface is exactly this. It is the existence of such components that allows (27.9) to say that η₂₇ itself need not become zero. Entering tranquility does not extinguish intention as such.

(3) G must be physically fixed. Here lies the crux of the design. The same ẋ can be written in many ways by re-partitioning it between f₀ and G. If that partition could be redrawn after the fact to suit us, one could simply rename the natural drift and call it a formation. So G must be the actual actuator — the route that really moves the body — and Condition 27-A compares u⁰ with u^tr under the same f₀ and the same G.

🔍 The mathematical skeletonThe real role of G is to type the attribution. Looked at alone, η₂₇ is merely a difference inside the head. Only through G can one say that the volition acted, along a real route, in the direction that lowers the meter. That is why (27.5) is written as −∇W_⊤^TGη₂₇ > 0 rather than simply as η₂₇ ≠ 0.

💡 In plain wordsThe condition ‖G^T∇W_⊤‖ ≤ L₂₇ appearing in (27.8) says that the actuator's effectiveness has a ceiling — the leverage is not infinite. That is what permits the conclusion “to do this much work, at least this much input is required.”

Tools 14 and 15 contrastedLocal Lipschitz regularity is a condition on W_⊤, guaranteeing that things can be measured. The actuator G is a condition on the structure of the system, guaranteeing that things can be attributed. Theorem 27 stands only when both are in place.

Chapter 3 The two theorems underneath

Theorem 27 is not built from nothing. It adds exactly one storey on top of two theorems already proved. This chapter presents those two without formulas.

Fig. 2 The overall map: the two boxes on the left are the foundation, the one on the right is this paper.

3.1 Theorem 24 (universal unsatisfactoriness) — below emptiness, suffering does not reach zero

The system regards cognitive abstraction as layered. The topmost layer is written ⊤ and called emptiness; layers below it are called “below emptiness” and written a ≺ ⊤.

Theorem 24 says: at any layer below emptiness, no matter how well one optimizes, the total remaining suffering cannot be brought to zero. Suffering here does not mean a moment-by-moment sting. It means the structural unsatisfactoriness that a positive remainder survives even after optimization has been carried through.

Fig. 16 The ladder of abstraction. At every rung below the top the remainder is positive; only at the top does the zero-suffering room open.

3.2 Theorem 26 (nirvanic tranquility) — a zero-suffering room, while alive

At the highest abstraction level ⊤, Theorem 26 guarantees three things.

There is a room. Inside the set ℬ_alive on which life is sustained, there exists a nonempty collection 𝒩_⊤(t) of states whose remaining suffering is exactly zero.
The room is closed, and once entered it is not left. It is a closed set containing its boundary, and it is forward invariant: once inside, one stays inside thereafter.
There is a height meter. There exists a W_⊤ satisfying the sandwich of Tool 5 and the strict descent of Tool 8.

Note that the room is allowed to move from instant to instant; that is why it is written 𝒩_⊤(t), with the t attached. Tranquility is therefore not a frozen point but a safe zone that may itself move.

𝒩_⊤(T) := ℬ_alive ∩ { x | J^*_⊤,ρ(x,T) = 0 }

🔢 The symbols in this formula𝒩_⊤(T) = the zero-suffering room at time T / ℬ_alive = the set of states on which life is sustained / J^*_⊤,ρ(x,T) = the total suffering that still remains after doing one's best from state x, discounted / ρ = the discount rate, weighting the far future slightly less / ∩ = the common part of two collections

💡 In plain wordsIt means the room collecting exactly those states that satisfy both “alive” and “remaining suffering is zero.” Because being alive is a condition from the outset, this room is not death.

The two inequalities borrowed from the foundation

c₁ dist(x,𝒩_⊤(t))² ≤ W_⊤(x,t) ≤ c₂ dist(x,𝒩_⊤(t))²(26.A)

D⁺_tW_⊤(x(t),t) ≤ −λ W_⊤(x(t),t)(26.B)

These two lines are all that the proof of Theorem 27 actually uses.

Note on the base

Note on the base (Mini13). This edition is built on TomabechiFourDharmaSealsMini13. There, Theorem 26 is stated as a single Condition 26-A (zero-suffering set and stability), whose content comprises three equations: the quadratic sandwich (26.A), the strict descent (26.B), and a continuity bound on the residual value in terms of distance (26.C). This volume uses only (26.A) and (26.B); (26.C) is not needed. Closedness of the room likewise follows from the zero-set representation of the continuous W_⊤, namely 𝒩_⊤(t) = ℬ_alive ∩ {x | W_⊤(x,t)=0}. The proposition “permanent zero suffering can be maintained” is given the name PZS, and the composition of Theorems 24 and 26 is cited as (26.3).

⚠️ A common misreadingTheorem 27 does not treat the case in which Condition 26-A fails. Within a system where it does hold, the case in which the present state has not yet entered the room is what we call nonattainment of tranquility. If the foundational hypotheses themselves break down, none of the conclusions below can be drawn.

Finally, combining Theorems 24 and 26 gives: below emptiness, permanent zero suffering cannot be maintained; and at the highest abstraction level ⊤, being able to maintain permanent zero suffering is the same as being inside the room. This combined result is cited here as (26.3).

Chapter 4 Full list of symbols

Every symbol used below is listed here in advance. Whenever an unfamiliar symbol appears, return to this table. There is nothing to memorize.

Every symbol appearing in Theorem 27
Symbol	Reading and meaning
⊤	the highest abstraction level, read “emptiness”; a symbol for the top of an order, with no mystical import
a ≺ ⊤	an abstraction level below emptiness
𝕃	the lattice of all abstraction levels
X_a	the state space at abstraction level a
ℬ_alive	the set of states on which life is sustained
𝒩_⊤(t)	the set of states at time t that are alive and have zero remaining suffering (the room)
x(t)	the state at time t; one's present position on the map
dist(x,𝒩_⊤(t))	the shortest distance from the present position to the room
W_⊤(x,t)	the height meter (Lyapunov function): zero inside the room, positive outside
c₁, c₂	the positive lower and upper constants of the sandwich
λ	the decay rate, a positive constant fixing how fast the meter falls
D⁺_t	the upper right Dini derivative: a slope measurable even at corners
PZS(a,x,T)	the permanent-zero-suffering proposition: some admissible policy starting from x at time T keeps suffering at zero forever after
Avijjā₂₇	operational ignorance in this system: the failure of PZS
Ign₂₇	the Lyapunov ignorance residual; at the highest level it is W_⊤ itself
Des₂₇	the residual descent rate, i.e. how fast the meter falls; on its own it does not mean formation
f₀(x,t)	the natural drift: how the state moves when nobody does anything (the current)
G(x,t)	the actual actuator map: the route by which an input really acts on the state (oar against water)
u	the control input: the quantity the agent may choose
u⁰(t,x)	the designated optimal input, chosen by the policy π⁰_⊤ of Theorem 26
u^tr(t,x)	the reference input: an independently fixed baseline input covering life-support and tangential motion
η₂₇(t,x)	the volitional signal: the designated input minus the reference input
Gη₂₇	the effective state action of the volitional signal through the actual actuator
Saṅkhārā₂₇	volitional formation in this system, defined by −∇W_⊤^TGη₂₇ > 0
L₂₇	a positive constant bounding ‖G^T∇W_⊤‖ from above
∇	the gradient: the steepest uphill direction
‖ · ‖	the length (norm) of a vector
π⁰_⊤	the designated optimal policy at the highest level, simultaneously optimal for all initial pairs
L (Lipschitz constant)	an upper bound on slope; locally Lipschitz means such an L exists near the trajectory (Tool 14)
ker G	the kernel of G: input directions v with Gv = 0, producing no state effect however hard one strains (Tool 15)
a.e.	at almost every instant (the exceptional instants together have zero total length)

Chapter 5 Putting “ignorance” into a formula

5.1 Preparation: keeping the types apart

First the range of states under discussion is fixed. At each layer below emptiness we take the whole state space of that layer; at the highest level ⊤ we take only the living states. Laying these side by side without mixing them gives 𝔛₂₇.

𝔛₂₇ := ( ⊔_a≺⊤ ( {a} × X_a ) ) ⊔ ( {⊤} × ℬ_alive )

🔢 The symbols in this formula⊔ = putting collections side by side without overlap (disjoint union) / {a} × X_a = a state tagged with the label “at layer a” / ℬ_alive = the set of living states

💡 In plain wordsThe convention is that every state carries a tag saying which layer it belongs to. Comparing states across layers without tags leads to false conclusions, so the possibility is shut off at the start.

5.2 The definition of ignorance

Now to the point. Ignorance is defined as follows.

Avijjā₂₇(a,x,T) :⇔ ¬ PZS(a,x,T)(27.1)

🔢 The symbols in this formula:⇔ = the left side is defined by the right side / ¬ = “not” / PZS(a,x,T) = the proposition that there exists an admissible policy which, starting from state x at time T, keeps suffering at zero forever after / a = abstraction level / x = state / T = starting time

💡 In plain words“From where I now stand, there is no way at all to keep suffering at zero forever” — that is ignorance. It is not a psychological matter of lacking knowledge or being agitated; it is a statement about what is possible: no such move exists.

🌱 Why it is set up this wayWhy define it by what cannot be done rather than by what is not known? Because inside a mathematical model the notion must be checkable. Whether someone knows something lies outside the model, whereas whether a policy exists can be decided within it. Hence the term operational ignorance.

Applying (26.3) to this definition splits ignorance into two cases.

Avijjā₂₇(a,x,T) ⇔ [ a ≺ ⊤ ] ∨ [ a = ⊤ ∧ x ∉ 𝒩_⊤(T) ](27.2)

🔢 The symbols in this formula⇔ = the two sides are equivalent / ∨ = “or” / ∧ = “and” / ∉ = “is not a member of”

💡 In plain wordsRead out: to be in ignorance is either (i) to be at a layer below emptiness, or (ii) to be at the highest level but not yet inside the room. Case (i) is underwritten by Theorem 24, case (ii) by Theorem 26.

⚠️ A common misreadingAlmost everything that follows concerns case (ii), that is, the designated closed loop at the highest abstraction level. Branch (i) is used only in this single line of (27.2), and only there is Condition 24-A required.

5.3 Measuring the amount of ignorance

The ignorance defined in (27.1) is a truth value: yes or no. But if one can obtain a number saying how much ignorance there is, it becomes possible to place it in a chain of inequalities, and far stronger conclusions follow. The amount of ignorance at the highest level is therefore defined as follows.

Ign₂₇(x,t) := W_⊤(x,t)(27.3)

🔢 The symbols in this formula:= = the left side is defined by the right side / Ign₂₇ = the Lyapunov ignorance residual, i.e. the amount of ignorance / W_⊤ = the height meter (Tool 4) / x = the state / t = time

💡 In plain wordsAs a formula, the two sides are the same thing. No new quantity has been created; a second name has been given to the height meter that Theorem 26 already supplied.

🌱 Why it is set up this wayWhy bother with a second name at all? For three reasons.
(1) It types the role. W_⊤ is a quantity of control theory; Ign₂₇ is a name carrying the interpretation "amount of ignorance." Though the number is the same, the part each plays in the argument differs. Keeping them distinct on the page lets the reader see where the mathematics ends and the interpretation begins.
(2) It converts a yes-or-no into a quantity. (27.1) is a truth value, (27.3) a real number. This bridge is what allows the step from "ignorance holds" to the quantitative conclusion "the descent rate is at least λ times as large."
(3) The zeros coincide. What entitles us to the second name is the sandwich (26.A). By it, Ign₂₇ = 0 ⇔ the distance to the room is 0 ⇔ one is inside the room ⇔ ignorance does not hold. The point where the quantity vanishes and the point where the predicate turns false are exactly the same point. Were they not, this quantity could not be called the amount of ignorance.

⚠️ A common misreadingThe absolute value of Ign₂₇ carries no meaning in itself. The sandwich constants c₁ and c₂ leave a free multiplier, so the meter's scale can be rescaled at will. Only two things are meaningful: whether it is zero or positive, and whether it is decreasing. The statement "the amount of ignorance is 3.7" has no content.
Nor is this a psychological quantity. It does not measure intensity of pain or degree of anxiety; it measures position — how far one is from the room.

Chapter 6 Putting “it is decreasing” into a formula

Next, a name is given to the speed at which the amount of ignorance falls. One caution is needed. The meter's value is often built from “the distance to the nearest wall,” and such a quantity is not necessarily smooth everywhere: a corner appears in the graph at the instant when which part of the wall is nearest switches over.

Tool 7, the upper right Dini derivative, handles this: it is always defined, even at corners. As regularity we assume that W_⊤ is continuous and locally Lipschitz on a neighbourhood containing the trajectory, and that the trajectory x(·) is absolutely continuous. A locally Lipschitz function is differentiable almost everywhere, and at such points the Dini derivative coincides with the ordinary one. Under this regularity the Dini derivative takes finite values.

Des₂₇(t) := − D⁺_t W_⊤(x(t),t)(27.4)

🔢 The symbols in this formulaDes₂₇(t) = the residual descent rate, i.e. how fast the meter falls / D⁺_t = the upper right Dini derivative / the minus sign = a sign flip so that “falling” is recorded as a positive number / x(t) = the state at time t

💡 In plain wordsThe minus sign is there simply so that “the amount by which it is falling” comes out positive. Des₂₇(t) > 0 means that the trajectory does not stay at the same level but genuinely decreases.

⚠️ A common misreadingAt this stage the decrease may not yet be attributed to the agent. That the meter falls is a fact, but who made it fall is not settled: it may be falling by the current alone. Making that distinction is the subject of the next chapter.

Chapter 7 Putting “volitional formation” into a formula

This chapter is the heart of Theorem 27. There is only one question: who is responsible for the meter's fall?

Fig. 9 Did it fall by the current, or by one's own rowing? Condition 27-A is what separates the two.

7.1 Splitting the mechanism of motion in two

First, the mechanism of motion is written as a control-affine system. The name is forbidding; the content is simple: write "what moves by itself" and "what the input does" as a sum.

ẋ = f₀(x,t) + G(x,t) u ,　　u⁰(t,x) := π⁰_⊤(t,x)(27.A1)

🔢 The symbols in this formulaẋ = the velocity of the state (the dot means a time derivative) / f₀(x,t) = the natural drift, what moves without anyone doing anything (the current) / G(x,t) = the actual actuator map, the real route by which an input acts on the state (Tool 15) / u = the control input / u⁰ = the optimal input selected by the designated policy π⁰_⊤ of Theorem 26

💡 In plain wordsIn the boat metaphor: f₀ is the current, G is how a stroke of the oar translates into motion through the water, and u is how hard one pulls. The boat's velocity is the sum of the drifting part and the rowed part.

Why it is called affine

Affine means a linear function with a constant added. The line y = ax passes through the origin; y = ax + b does not. It is the presence of that "+ b" that makes a relation affine rather than linear.

Read the equation above as a function of u. Doubling u doubles Gu — that part is linear. But setting u = 0 does not make ẋ zero, because f₀ remains. That is the "+ b", and it is why the system is affine rather than linear.

Fig. 23 A control-affine system. Varying the input makes the tips fall on a straight line, but that line misses the origin; the offset is the natural drift.

🔍 The mathematical skeletonWhy insist on this particular form? Here lies the decisive point.
A general control system is written ẋ = f(x, u, t). In that form, how the input acts on the state is dissolved into f and cannot be extracted: splitting the input does not split the action.
In a control-affine system, by contrast, the moment the input is split as u⁰ = u^tr + η₂₇, the action splits automatically too: G u⁰ = G u^tr + G η₂₇. It is precisely this separability that allows Step 3 of the proof to decompose the meter's change into four terms, cancel three of them, and isolate the extra rowing. The control-affine form is not decoration: it is the structure that makes the proof of Theorem 27 possible at all.

7.2 Introducing the reference input

Now the decisive move. We assume that there exists an independently fixed, admissible reference feedback u^tr(t,x), taking values in the same input space as the designated input u⁰. It corresponds to the “bare” business of living: life-support, change over time, and tangential motion within the tranquility set. The difference of the two inputs is then called the volitional signal.

η₂₇(t,x) := u⁰(t,x) − u^tr(t,x) ,　　F^tr_⊤(x,t) := f₀(x,t) + G(x,t) u^tr(t,x)

🔢 The symbols in this formulaη₂₇ (eta) = the volitional signal, the chosen input minus the reference input / u^tr = the reference input / F^tr_⊤ = the closed-loop motion driven by the reference input alone (the reference field)

Fig. 10 Volitional formation is a subtraction: the chosen input minus the staying-alive part leaves the volitional signal η₂₇.

🌱 Why it is set up this wayWhy must the reference input be fixed independently? Because if it could be chosen after the fact to suit us, any quantity whatever could be declared to be the formation. Fixing the reference in advance blocks the move of simply renaming the natural drift and calling it a formation.

7.3 Condition 27-A — the reference alone does not change the height

Condition 27-A (attribution of volitional action to the actual actuator)

The control system on the relevant region has the form (27.A1); W_⊤ is of class C¹ in time and state on an open neighbourhood containing the trajectory; the trajectory x(·) is absolutely continuous and satisfies the closed-loop equation almost everywhere; and Gη₂₇ is measurable and finite-valued. In addition, the reference closed loop alone does not change the residual level: on a neighbourhood of the trajectory,

∂W_⊤/∂t + ∇W_⊤^T F^tr_⊤ = 0(27.A2)

🔢 The symbols in this formula∂W_⊤/∂t = the meter's change when only time advances / ∇W_⊤ = the gradient of the meter / F^tr_⊤ = the motion driven by the reference input alone / class C¹ = once differentiable with continuous derivative

💡 In plain words(27.A2) says exactly one thing: “left to itself, the meter's height does not change.” Hence if the meter does fall, the situation was not left to itself — someone put in an extra stroke.

🌱 Why it is set up this wayThanks to this condition, Gη₂₇ ceases to be “the natural drift under a new name.” It acquires a definite type: the effective state action obtained by comparing the designated input with the reference input under the same drift and the same actual actuator.

7.4 The definition of volitional formation

Everything is now in place. Under Condition 27-A, volitional formation in this model, at almost every instant, is defined as follows.

Saṅkhārā₂₇(x(t),t) :⇔ − ∇_xW_⊤(x(t),t)^T G(x(t),t) η₂₇(t,x(t)) > 0(27.5)

🔢 The symbols in this formulaSaṅkhārā₂₇ = volitional formation in this model / ∇_xW_⊤ = the gradient of the meter with respect to the state / G η₂₇ = the effective state action of the volitional signal through the actuator / the superscript T = transpose, the notation that forms an inner product / > 0 = this inner product is positive

💡 In plain words“The extra stroke really does act in the direction that lowers the meter” — that is a volitional formation. Because of the leading minus sign, this expression being positive means that the gradient (uphill) and the effective action point opposite ways: the push is downhill.

⚠️ A common misreadingThis is not volitional formation in general as the Pāli originals speak of it. It expresses the restricted state of affairs that there exists an actual-actuator component positively contributing to the decrease of the ignorance residual. The threefold bodily, verbal and mental classification is not being reduced to this single inner product.

Chapter 8 Theorem 27 itself

Theorem 27 (Ignorance-Conditioned Volitional Formations / Avijjāpaccayā saṅkhārā)

Under the standing hypotheses of Theorem 26 (in particular the existence of π⁰_⊤ and its simultaneous optimality for all initial pairs), Condition 26-A, and the local Lipschitz regularity of Chapter 6, the following hold along any closed-loop trajectory of π⁰_⊤ and at any time t.

Avijjā₂₇(⊤,x(t),t) ⇒ Des₂₇(t) ≥ λ Ign₂₇(x(t),t) ≥ λ c₁ dist(x(t),𝒩_⊤(t))² > 0(27.6)

Hence, so long as nirvanic tranquility remains unattained, the ignorance residual has a strict descent rate along the designated closed loop. Condition 24-A is not needed for this conclusion at the highest level; it is needed only for the below-emptiness branch of (27.2).

If in addition Condition 27-A holds, then at Lebesgue-almost every instant the upper right Dini derivative agrees with the ordinary trajectory derivative, and the following hold.

Des₂₇(t) = − ∇W_⊤^TGη₂₇ ≥ λ W_⊤ ≥ 0 ,
Avijjā₂₇(⊤,x(t),t) ⇒ − ∇W_⊤^TGη₂₇ > 0
⇒ Saṅkhārā₂₇(x(t),t) ∧ η₂₇ ≠ 0 ∧ Gη₂₇ ≠ 0(27.7)

Moreover, at instants where ignorance holds, if ‖G^T∇_xW_⊤‖ ≤ L₂₇ for some constant L₂₇ > 0, then

‖η₂₇(t)‖ ≥ λ W_⊤(x(t),t) / L₂₇ ≥ ( λ c₁ / L₂₇ ) dist(x(t),𝒩_⊤(t))² > 0(27.8)

Conversely, if x(T) ∈ 𝒩_⊤(T) for some T, then

Des₂₇(t) = 0 　(for all t ≥ T),
− ∇W_⊤^TGη₂₇ = 0 　(for Lebesgue-almost every t ≥ T)(27.9)

What vanishes here is the positive actuator contribution conditioned on ignorance — not η₂₇ itself, nor the whole of biological activity, evaluation, optimal feedback, or tangential motion within the tranquility set. A tangential component, or a component in the kernel of G, may survive in η₂₇.

Hence, along the designated closed loop at the highest abstraction level, the first equivalence holds at every instant and the second, under Condition 27-A, at almost every instant.

Avijjā₂₇(⊤,x(t),t) ⇔ Des₂₇(t) > 0 ,
Avijjā₂₇(⊤,x(t),t) ⇔ Saṅkhārā₂₇(x(t),t) 　(a.e., Condition 27-A)(27.10)

8.1 Reading the four equations one line at a time

(27.6) If in ignorance, it is necessarily decreasing

🔢 The symbols in this formula⇒ = “implies” / Des₂₇(t) = the speed at which the meter falls / λ = the decay rate / Ign₂₇ = the amount of ignorance (= W_⊤) / c₁ = the lower constant of the sandwich / dist(…)² = the squared distance to the room

💡 In plain wordsThree inequality signs in a row, but the reading is easy: the falling speed is at least λ times the amount of ignorance; that amount is at least c₁ times the squared distance; and the distance is positive, so the whole chain is positive. In other words, while ignorance holds the descent cannot stall, and a minimum amount proportional to how far away one is is guaranteed.

(27.7) That decrease is the agent’s own rowing

💡 In plain wordsOnce Condition 27-A is in force, the cause of the descent is pinned down: the speed of descent coincides exactly with the effect of the extra rowing. Hence in ignorance that inner product must be positive, a formation obtains, and neither the volitional signal nor its effective action can be zero.

(27.8) The size of the formation is bounded below too

🔢 The symbols in this formula‖η₂₇(t)‖ = the length (size) of the volitional signal / L₂₇ = a positive constant bounding ‖G^T∇_xW_⊤‖ from above / ‖ · ‖ = the length of a vector

💡 In plain wordsThis is a step stronger than “nonzero.” The farther one is from the room, the larger the formation must be. A lower bound proportional to the squared distance appears. As with a lever: if the work required is fixed, the size of the force is bounded from below.

(27.9) Inside the room, that contribution becomes zero

💡 In plain wordsOnce inside, forward invariance keeps one inside thereafter. Inside, the meter is identically zero, so the descent rate is zero as well; combining with Condition 27-A, the actual-actuator residual contribution is zero almost everywhere too.

⚠️ A common misreadingOnly the positive contribution conditioned on ignorance becomes zero. It is not claimed that η₂₇ itself becomes zero. A tangential component moving within the room, or a kernel component that produces no state effect through G, may survive in η₂₇. This distinction is the subject of Chapter 10.

(27.10) The two equivalences

🔍 The mathematical skeletonGathering (27.6)–(27.9) yields two equivalences. The first (ignorance ⇔ positive descent) holds at every instant. The converse direction is available because, once inside the room, forward invariance makes W_⊤ identically zero, so the descent rate is zero at every instant. The second (ignorance ⇔ formation) holds, under Condition 27-A, at almost every instant. The qualifier appears because the Dini and ordinary derivatives agree only almost everywhere.

Chapter 9 The proof, one step at a time

The proof has just five steps. After each, what the step accomplished is restated in ordinary language. The only tools used are those assembled in Chapter 2; no new mathematics enters.

Proof

Step 1. Assume Avijjā₂₇(⊤,x(t),t). By (27.2), x(t) ∉ 𝒩_⊤(t). The set is closed, so the distance is positive, and (26.A) gives

W_⊤(x(t),t) ≥ c₁ dist(x(t),𝒩_⊤(t))² > 0

Step 2. Multiplying both sides of (26.B) by minus one,

Des₂₇(t) = − D⁺_tW_⊤(x(t),t) ≥ λ W_⊤(x(t),t) > 0

This is (27.6). At this stage the descent may still be due to natural drift, so it is not yet called a formation.

Step 3. Under Condition 27-A, using u⁰ = u^tr + η₂₇ and the chain rule, at almost every instant

D_tW_⊤ = ∂W_⊤/∂t + ∇W_⊤^T[ f₀ + Gu^tr + Gη₂₇ ] = ∇W_⊤^TGη₂₇

A locally Lipschitz function is differentiable almost everywhere, and at such points the upper right Dini derivative agrees with the ordinary derivative. Hence in ignorance − ∇W_⊤^TGη₂₇ ≥ λW_⊤ > 0. By the definition of formation, Saṅkhārā₂₇ holds, and η₂₇ ≠ 0 and Gη₂₇ ≠ 0.

Step 4. Furthermore, by the Cauchy–Schwarz inequality,

λW_⊤ ≤ − ∇W_⊤^TGη₂₇ ≤ ‖G^T∇W_⊤‖ ‖η₂₇‖ ≤ L₂₇‖η₂₇‖

Hence (27.8) follows.

Step 5. Finally, if x(T) ∈ 𝒩_⊤(T), forward invariance gives x(t) ∈ 𝒩_⊤(t) for all t ≥ T. By (26.A), W_⊤ is identically zero along the trajectory, so its upper right Dini derivative is zero at every instant. Adding the chain rule of Condition 27-A, the actual-actuator residual contribution is zero almost everywhere as well. This is (27.9), and combining the two cases yields (27.10). ∎

9.1 The five steps in ordinary language

💡 In plain wordsStep 1: If you are outside the room, the distance to the wall is not zero — and because the room is closed, this can be said with certainty. Positive distance forces, via the sandwich, a positive meter value.
Step 2: Theorem 26 says the meter falls at a speed proportional to its height. Flipping the sign of both sides turns this into “the falling speed is at least λ times the height.” Since the height is positive, so is the falling speed. That completes (27.6).
Step 3: Now Condition 27-A enters. Decomposing the meter's change into four parts, three of them (the effect of time itself, the current, and the staying-alive rowing) cancel exactly by (27.A2). Only the extra-rowing term survives. So if it is falling, the extra rowing is not zero.
Step 4: The size of an inner product cannot exceed the product of the two lengths. Since the work done is pinned at λW_⊤ or more, the length of the arrow is bounded from below.
Step 5: The converse. Once inside, one never leaves, so the meter stays at zero; staying at zero, its slope is zero. Hence the descent rate is zero, and so is the actuator contribution.

Fig. 11 The decomposition in Step 3. Three of the four terms vanish by (27.A2), leaving only the extra rowing.

🔍 The mathematical skeletonWhat is used where. Positivity of the distance to a closed set (Step 1); the sandwich (26.A) (Steps 1 and 5); the strict descent (26.B) (Step 2); the chain rule together with (27.A2) (Step 3); almost-everywhere differentiability of a locally Lipschitz function (Step 3, Tool 14); the Cauchy–Schwarz inequality (Step 4); forward invariance (Step 5). That is the complete list. Not one new hypothesis has been added.

Chapter 10 Four readings to avoid

The conclusions so far are strong when read correctly, but a small misstep makes them look like something quite different. Four misreadings are dismantled in turn.

10.1 “Ignorance drives one away from tranquility” — false

Under the designated policy π⁰_⊤, inheriting (26.B), W_⊤ decreases. What this theorem directly proves is therefore a Lyapunov descent toward zero residual while tranquility is unattained. It does not prove that ignorance necessarily carries one away from tranquility.

The possibility that a policy under partial information heads toward a mistaken least upper bound falls within the scope of Theorems 20 and 21 and requires a separate observation-and-belief model.

⚠️ A common misreadingDo not reverse the direction. Theorem 27 says “if in ignorance, a descent toward tranquility is under way” — not “if in ignorance, one falls away.”

10.2 “Zero formation means all activity stops” — also false

By Theorem 23, even in tranquility the full life-and-environment state may change. By Theorem 26, the evaluation function and π⁰_⊤ keep operating. What (27.9) sets to zero is only the Lyapunov residual descent rate and the positive actuator contribution conditioned on ignorance. It does not erase tangential motion within the tranquility set, life-support, responsiveness to others, or the capacity for free will.

Fig. 18 What Theorem 27 says becomes zero, and what it does not. Equation (27.9) zeroes only the left-hand box.

Fig. 14 Tranquility is not physical stasis. Only the component directed out of the room vanishes; tangential motion within the room remains.

10.3 “Theorem 26 alone already gives the formation” — it does not

The inequality − D⁺_tW_⊤ > 0 shows that the residual strictly decreases. But that decrease cannot immediately be attributed to the agent's control input. Condition 27-A compares the designated input u⁰ with the reference input u^tr under the same natural drift f₀ and the same actual actuator G, thereby typing the conclusion about the volitional signal η₂₇ as a difference of real inputs.

The natural-drift counterexample

Take ẋ = −x, admissible input set U = {0}, and V_⊤(x) = x². Then J^*_⊤,ρ(x) = x²/(ρ+2), 𝒩_⊤ = {0}, W_⊤ = x², and Ẇ_⊤ = −2W_⊤, so a strict descent of the Theorem 26 type holds. Yet the real input is identically zero and the descent is due to natural drift alone. Under Condition 27-A we have u⁰ = u^tr = 0, hence Gη₂₇ = 0, and the requirement (27.A2) on the reference field fails off a null set. Therefore no formation is concluded from this system.

💡 In plain wordsThe counterexample makes exactly one point: “the meter is falling” and “the agent rowed” are different things. In the system above the passenger has no oar at all (U = {0}); the destination is nevertheless approached by the current alone. Without Condition 27-A, one could have made the error of calling that current a volitional formation.

Fig. 15 The counterexample. The meter genuinely falls, yet the input is always zero. “It falls, therefore someone rowed” does not follow.

10.4 “Below emptiness one may also conclude a nonzero input at every instant” — one may not

By (27.2), every state below emptiness satisfying Condition 24-A belongs to typed operational ignorance. However, to conclude an instantaneous nonzero control at a given layer, one needs that layer's own Lyapunov function together with an actual-actuator attribution corresponding to Condition 27-A. A nonzero control input at every instant does not follow from the positive residual value of Theorem 24 alone.

Chapter 11 What this theorem borrows from its foundation

Theorem 27 has almost no new hypotheses of its own. It uses only theorems already proved and conditions explicitly stated. Here everything it borrows is listed without exception. With such a list in hand, the reader can check for themselves how far the theorem holds and which premise, if it failed, would bring it down.

Fig. 26 What Theorem 27 borrows from its foundation. It has almost no new hypotheses of its own.

Minimal dependencies — what is borrowed, from where, and where it is used
Borrowed from	What is borrowed	Where it is used
standing hypotheses of Theorem 26	a designated policy π⁰_⊤ exists at the highest level and is simultaneously optimal for all initial pairs	the whole statement of Chapter 8
Condition 26-A (zero-suffering set and stability)	the room 𝒩_⊤(t) is nonempty, closed, and never left once entered (forward invariance)	Steps 1 and 5 of the proof
two equations inside that same Condition 26-A	the sandwich (26.A) and the strict descent (26.B)	Steps 1 and 2 of the proof
Theorem 24 and Condition 24-A	permanent zero suffering cannot be maintained below emptiness; used only for the below-emptiness branch of (27.2)	equation (27.2) in Chapter 5
the regularity of Chapter 6	local Lipschitz continuity of W_⊤ and absolute continuity of the trajectory, so that the Dini derivative defines a finite residual descent rate	Chapter 6 and Step 3 of the proof
Condition 27-A	attribution of the residual descent to the difference between the designated and reference inputs through the actual actuator	Chapter 7 and Step 3 of the proof

11.1 What is not used — equation (26.C)

Condition 26-A contains one further equation beyond the two quoted above: (26.C), a continuity bound on the residual value in terms of distance. This theorem never uses it.

The reason is simple. Theorem 27 needs only two things: (i) the lower estimate “positive distance forces a positive meter,” and (ii) the descent “the meter necessarily decreases.” Continuity of the residual value itself appears at no step of the proof. Nor need closedness of the room be assumed separately: since W_⊤ is continuous, writing the room as the collection of points where W_⊤ vanishes makes closedness follow automatically.

💡 In plain wordsListing an equation one does not use is not a boast but a matter of accuracy. Knowing that it was unnecessary means Theorem 27 applies unchanged to systems where (26.C) fails. The fewer hypotheses a theorem uses, the further its reach.

11.2 The list of borrowings is also a map for refutation

This table is also a map for anyone who doubts Theorem 27. Exhibiting a single system that satisfies all six items above while the conclusion nevertheless fails would refute the theorem. Producing a system in which one of the six fails, by contrast, would not — such a system lies outside the theorem's scope from the start.

The natural-drift counterexample of Chapter 10 (ẋ = −x, U = {0}) was exactly an instance of this distinction. That system fails (27.A2) of Condition 27-A off a null set, so it lies outside the scope and no formation is concluded. The counterexample itself shows that the hypothesis is not idling.

Chapter 12 Frequently asked questions

Q1. Is ignorance not simply “not knowing”?
That is how the Pāli originals explain it. But to handle it inside a mathematical model one needs a decidable form. This paper therefore defines it as a statement about possibility: no policy exists that can maintain permanent zero suffering. That is what is called operational ignorance. The canonical meaning has not been replaced; one counterpart within the model has been fixed.

Q2. Is “a formation occurring” a bad thing?
The theorem passes no value judgement. Formation here is the actuator component positively contributing to a decrease of the ignorance residual. If anything, it describes the structure “not yet arrived, therefore rowing in order to arrive.”

Q3. Once in tranquility, does one stop doing anything?
No. What (27.9) sets to zero is only the transverse contribution conditioned on ignorance. Tangential motion within the room, life-support, responsiveness to others, and the capacity for free will all remain. See Figures 14 and 18 in Chapter 10.

Q4. Is tranquility not just death?
It is not. The definition of the room 𝒩_⊤(t) intersects ℬ_alive, the set on which life is sustained, from the very beginning. The definition covers only cessation of suffering while alive.

Q5. If the meter is falling, may we not already call that a formation?
That is precisely the error the theorem guards against most carefully. In the counterexample of Chapter 10 (ẋ = −x, U = {0}) the meter falls although there is no oar at all. Condition 27-A is the device that subtracts the drifted part.

Q6. Is Condition 27-A not just a convenient hypothesis added after the fact?
It is designed so as not to be. The reference input u^tr is fixed independently and is moreover required to satisfy the checkable identity (27.A2). In the counterexample above it is exactly (27.A2) that fails off a null set, which is why no formation is concluded. The counterexample thus shows that the hypothesis is doing real work.

Q7. Is “at almost every instant” not a weak conclusion?
Collecting all exceptional instants gives zero total duration. Moreover the first equivalence of (27.10) holds at every instant. Only the second is qualified, because the Dini and ordinary derivatives agree only almost everywhere. The distinction is written out rather than collapsed.

Q8. Does this theorem prove a Buddhist doctrine?
It does not. What it supplies is a model-theoretic analogue of one line of the Pāli originals. It neither proves nor exhausts the meanings of ignorance and formations in those originals. That the threefold classification of SN 12.2 must not be reduced to a single inner product is a position maintained throughout.

Q9. Do the same conclusions hold at layers below emptiness?
By (27.2), all states below emptiness belong to operational ignorance. But to conclude an instantaneous nonzero control one additionally needs that layer's Lyapunov function and an actual-actuator attribution corresponding to Condition 27-A (Chapter 10, §10.4).

Q10. Is the theorem falsifiable?
Yes. All hypotheses are stated explicitly: the standing hypotheses of Theorem 26, the 26-series conditions, Condition 24-A, and Condition 27-A. Exhibiting one system that satisfies them while the conclusion fails would refute it. Producing a system in which one of the hypotheses fails, however, would not.

Chapter 13 Summary

Within this system, nonattainment of nirvanic tranquility can be represented as typed operational ignorance. At the highest abstraction level, a state outside the zero-suffering invariant set has, by Conditions 26-A and 26-B, a positive Lyapunov ignorance residual and a strictly positive residual descent rate. Adding Condition 27-A, the actual-actuator volitional signal that contributes positively to that descent is nonzero. The stated model therefore establishes a conditional implication corresponding to “with ignorance as condition, volitional formations come to be.”

At the same time, tranquility is neither physical stasis nor the extinction of every action. What ceases is the positive residual contribution conditioned by ignorance; the tangential and relational activities of life and freedom may continue. The theorem supplies a model-theoretic analogue of a line of the Pāli originals; it neither proves nor exhausts the meanings of ignorance and formations there. This is the minimal form of Theorem 27 consistent with the dynamic tranquility of Theorem 26.

The opening of the twelve links, in the form this system gives it

nonrealization of tranquility (operational ignorance)
⇒ positive Lyapunov residual descent rate
+ actual-actuator attribution (Condition 27-A)
⇒ nonzero volitional formation

Avijjā₂₇(⊤,x(t),t) ⇒ Saṅkhārā₂₇(x(t),t)
(along the designated closed loop at the highest level, a.e. under Condition 27-A)

💡 In plain wordsOnce more, in plain words: “If you have not yet arrived, you are necessarily rowing something. Once you arrive, that rowing ceases — but what ceases is only the rowing directed outward; living, thinking, and answering to another do not stop.” That is the whole of Theorem 27.

Appendix A All equations

Every equation appearing in this volume
Label	Equation	In one line
(26.A)	c₁dist² ≤ W_⊤ ≤ c₂dist²	the sandwich: the meter tracks the squared distance
(26.B)	D⁺_tW_⊤ ≤ −λW_⊤	strict descent: the meter must fall at a rate proportional to its height
(26.3)	PZS fails below ⊤; at ⊤, PZS ⇔ inside the room	the composition of Theorems 24 and 26
(27.1)	Avijjā₂₇ :⇔ ¬PZS	definition of ignorance: permanent zero suffering cannot be maintained
(27.2)	Avijjā₂₇ ⇔ [a≺⊤] ∨ [a=⊤ ∧ x∉𝒩_⊤]	the two cases of ignorance
(27.3)	Ign₂₇ := W_⊤	the amount of ignorance is the height meter
(27.4)	Des₂₇ := −D⁺_tW_⊤	the residual descent rate, recorded as a positive number
(27.A1)	ẋ = f₀ + Gu,　u⁰ := π⁰_⊤	control-affine system: drift and rowing separated as a sum
(27.A2)	∂W_⊤/∂t + ∇W_⊤^TF^tr_⊤ = 0	the reference closed loop alone does not change the height
(27.5)	Saṅkhārā₂₇ :⇔ −∇W_⊤^TGη₂₇ > 0	definition of formation: the extra rowing acts positively on the descent
(27.6)	Des₂₇ ≥ λIgn₂₇ ≥ λc₁dist² > 0	in ignorance the descent cannot stall
(27.7)	Des₂₇ = −∇W_⊤^TGη₂₇ ≥ λW_⊤ ≥ 0	the descent equals the agent's own rowing
(27.8)	‖η₂₇‖ ≥ λW_⊤/L₂₇ > 0	a lower bound on the size of the formation
(27.9)	inside the room Des₂₇ = 0, and the contribution is zero a.e.	in tranquility the transverse contribution vanishes
(27.10)	ignorance ⇔ descent > 0 (all t); ignorance ⇔ formation (a.e.)	the two equivalences

Appendix B All symbols

The table of Chapter 4 is reproduced here so that it can be consulted from the end of the volume while reading.

Every symbol appearing in Theorem 27
Symbol	Reading and meaning
⊤	the highest abstraction level, read “emptiness”; a symbol for the top of an order, with no mystical import
a ≺ ⊤	an abstraction level below emptiness
𝕃	the lattice of all abstraction levels
X_a	the state space at abstraction level a
ℬ_alive	the set of states on which life is sustained
𝒩_⊤(t)	the set of states at time t that are alive and have zero remaining suffering (the room)
x(t)	the state at time t; one's present position on the map
dist(x,𝒩_⊤(t))	the shortest distance from the present position to the room
W_⊤(x,t)	the height meter (Lyapunov function): zero inside the room, positive outside
c₁, c₂	the positive lower and upper constants of the sandwich
λ	the decay rate, a positive constant fixing how fast the meter falls
D⁺_t	the upper right Dini derivative: a slope measurable even at corners
PZS(a,x,T)	the permanent-zero-suffering proposition: some admissible policy starting from x at time T keeps suffering at zero forever after
Avijjā₂₇	operational ignorance in this system: the failure of PZS
Ign₂₇	the Lyapunov ignorance residual; at the highest level it is W_⊤ itself
Des₂₇	the residual descent rate, i.e. how fast the meter falls; on its own it does not mean formation
f₀(x,t)	the natural drift: how the state moves when nobody does anything (the current)
G(x,t)	the actual actuator map: the route by which an input really acts on the state (oar against water)
u	the control input: the quantity the agent may choose
u⁰(t,x)	the designated optimal input, chosen by the policy π⁰_⊤ of Theorem 26
u^tr(t,x)	the reference input: an independently fixed baseline input covering life-support and tangential motion
η₂₇(t,x)	the volitional signal: the designated input minus the reference input
Gη₂₇	the effective state action of the volitional signal through the actual actuator
Saṅkhārā₂₇	volitional formation in this system, defined by −∇W_⊤^TGη₂₇ > 0
L₂₇	a positive constant bounding ‖G^T∇W_⊤‖ from above
∇	the gradient: the steepest uphill direction
‖ · ‖	the length (norm) of a vector
π⁰_⊤	the designated optimal policy at the highest level, simultaneously optimal for all initial pairs
L (Lipschitz constant)	an upper bound on slope; locally Lipschitz means such an L exists near the trajectory (Tool 14)
ker G	the kernel of G: input directions v with Gv = 0, producing no state effect however hard one strains (Tool 15)
a.e.	at almost every instant (the exceptional instants together have zero total length)