A synthesis of Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem) — the mathematics of dynamic tranquility running through Theorems 24, 26 and 27
Hideto Tomabechi · August 10, 2026
Cognitive Research Laboratories, Tokyo / CyLab, Carnegie Mellon University / C5I Center, George Mason University
How to read this edition
This edition carries exactly the same content as the academic synthesis, rewritten for two kinds of reader at once: those not at ease with mathematical notation, and those who have never studied Buddhist doctrine. Not a single formula has been dropped; instead, every formula is accompanied by an explanation of every symbol it contains, and by a restatement in pictures and ordinary words.
This synthesis addresses exactly one question. Is the world seen by one who has awakened a different universe from the world seen by one who has not? The answer is that it is one and the same universe, read from a different height and in a different direction. Confirming that mathematically is what this document does.
Five kinds of coloured box appear throughout. Blue explains the symbols; orange restates a formula in plain words; green gives the motivation; red warns against an easy misreading; gold shows the mathematical skeleton. A reader uncomfortable with notation can follow the whole argument by reading only the orange boxes.
🔍 The mathematical skeletonThe synthesis in brief. Theorem 24 (Universal Unsatisfactoriness Theorem): below emptiness, a remainder survives however well one optimizes. Theorem 26 (Nirvanic Tranquility Theorem): at emptiness there is a room, entered while alive, where the remainder is zero. Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem): outside that room, something is necessarily being rowed. These three are not stacked in series; they divide the territory by type and state.
Chapter 1 The synthesis in three minutes
Buddhism has, broadly, two registers of speech. One surveys the whole world from the point where things are already complete. The other traces, from the side of a human being still bound by conditions, how suffering arises and how it ends. The former is prominent in Mahāyāna and Japanese esoteric Buddhism, the latter in the early-Buddhist teaching of dependent origination.
The two are often treated as rival cosmologies. What this synthesis shows is that both describe one and the same formal structure, from different vantages and in different directions.
Fig. 1 The two readings capture one and the same formal structure from different vantages and directions; Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem) joins them.
1.1 What the two registers say
The register from the point of completion — the Mahāyāna vantage. Here the universe and life are seen from the Buddha's side. The world is already empty. To one who has not awakened the universe does not look empty; yet that unawakened person is themselves empty, existents are empty, everything is empty. Suffering is empty, and so is the cessation of suffering. There are no exceptions.
The register from the conditioned side — the early-Buddhist vantage. Here one climbs from the human side, one rung at a time, and the four seals are that path: first impermanence (all things pass), then universal unsatisfactoriness (a remainder survives even optimization), then non-self (no fixed substance called a self), and so toward nirvanic tranquility. It is an ascent from below.
This order has textual warrant. Dhammapada 277–279 lists impermanence, unsatisfactoriness and non-self in that sequence, and the Anattalakkhaṇa Sutta (Saṃyutta Nikāya 22.59) states the chain of reasoning itself: what is impermanent is unsatisfactory, and what is unsatisfactory is not self. The arrangement of this system — Theorem 23 (Impermanence Theorem) → Theorem 24 (Universal Unsatisfactoriness Theorem) → Theorem 25 (Non-Self Theorem) → Theorem 26 (Nirvanic Tranquility Theorem) — corresponds to that sequence. It also agrees with the internal dependencies: Theorem 26 (Nirvanic Tranquility Theorem) depends on Theorem 24 (Universal Unsatisfactoriness Theorem), so 24 must precede 26.
1.2 The twelve links are not a list of doctrines but the Buddha's method of meditation
Mistake this and the whole synthesis becomes unreadable. The twelve links must be understood as the method of meditation of Śākyamuni Buddha beneath the Bodhi tree, not as a mere inventory of twelve items.
The meditation runs in two directions. Forward contemplation: with ignorance as condition there are formations, with formations as condition consciousness, and so on down the order of arising. Reverse contemplation: where does aging-and-death come from, traced back step by step until one arrives at ignorance. It is this round trip that generated the early-Buddhist ascent.
What Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem) rigorously proves is only the first of those twelve links — “with ignorance as condition, volitional formations come to be.” About the remaining eleven it says nothing. And the places it connects to, in order to prove that one link, are Theorem 24 (Universal Unsatisfactoriness Theorem) and Theorem 26 (Nirvanic Tranquility Theorem) of the four seals.
💡 In plain wordsThink of one mountain seen as a map looking down from the summit and as an itinerary drawn step by step from the trailhead. Both depict the same mountain, but from different vantages and directions. Neither is the correct one and the other mistaken. Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem) is the cross-reference showing that the two drawings point at the same mountain.
Chapter 2 What “everything is empty” precisely means
Begin with the word most easily misunderstood. “Emptiness” does not mean “nothing exists.”
Emptiness is the structure whereby every existent, experience and relation lacks a fixed content independent of everything else and comes about within relation. That a desk is empty does not mean the desk does not exist; it means the desk has no independent core that would remain a desk once its relations to timber, working, use and observer were all removed.
What matters here is that the world is already empty for those who have not awakened. Awakening does not newly turn the universe into emptiness. What changes is not how the world is, but the cognitive vantage from which it is grasped.
⚠️ A common misreading“Suffering is empty” does not mean “there is no suffering.” Neither the experience of suffering nor its causal effects is denied. What is denied is only positing suffering as an independent, unchanging substance. Likewise “the cessation of suffering is empty” does not deny that cessation is possible. Confusing these two makes the whole system look like nihilism.
In this system the vantage of awakening is mapped to the top of the lattice of abstraction levels, and that top is read as “emptiness.” It is written ⊤ — a symbol for the top of an order, with no mystical import.
Fig. The ladder of abstraction: a remainder at every rung, with the zero-suffering room opening only at the top.
Chapter 3 Emptiness and the zero-suffering room are not the same
This is one of the most important distinctions in the whole synthesis.
“Being at emptiness” and “tranquility is realized” are different claims. The first is about a type — the height from which the world is viewed. The second is about whether one is presently inside a set defined within that type.
𝒩⊤(T) := ℬalive ∩ { x | J*⊤,ρ(x,T) = 0 }(26.1)
🔢 The symbols in this formula𝒩⊤(T) = the zero-residual-suffering family at time T, called “the room” in this volume / ℬalive = the set of states on which life is sustained / J*⊤,ρ(x,T) = the discounted total of suffering still remaining after doing one's best from x / ρ = the discount rate / ∩ = the common part
💡 In plain wordsIt means the room collecting exactly those states satisfying both “alive” and “zero remaining suffering.” Because being alive is written into the definition from the outset, this room is not death.
Fig. 4 Emptiness is a type; the room is a set defined inside that type. One can be at the top level and still be outside the room.
⚠️ A common misreadingWhat Theorem 26 (Nirvanic Tranquility Theorem) requires is that two conditions hold at once: that one is at the highest abstraction level, and that the present state belongs to that room. A persistent vegetative state, a disorder of consciousness in which all sensation and awareness have been lost, is not nirvanic tranquility. And the highest abstraction level means, by definition, an intellectual state that subsumes any and every item of knowledge in this world; it does not mean that one has no body, so the body still takes injury. In fact, after his awakening the Buddha died of food poisoning.
3.1 Is the room itself a substance?
🌱 Why it is set up this wayIt is not. Look at the definition of the room again. The room is defined only within a web of relations — to the evaluation V, the discount rate ρ, the set of living states, and the optimization structure. Strip these away and the very notion of the room disappears.
Moreover the room is a family of sets moving with time, not one fixed object. And it is a set, not a point.
So the room itself is not a substance cut off from relation either — and that is exactly what Theorem 25 (Non-Self Theorem) denies. Nirvāṇa too is non-self: this is not an interpretation but follows directly from the definition of the room.
⚠️ A common misreadingPrecisely for this reason, one cannot grasp nirvāṇa as “the one real substance.” The moment it is grasped, it becomes a new form of attachment.
This has been a point of debate in the texts as well: whether “all dhammas are non-self” at Dhammapada 279 includes nirvāṇa. The Theravāda commentary Dhammapada-aṭṭhakathā places the supramundane — the level that includes nirvāṇa — within the scope of non-self. A variant reading, “all formations are non-self,” has also been proposed, but the standard critical editions all adopt “all dhammas are non-self.” The answer given by this system is the same under either reading, because the three points above — definable only relationally, moving with time, a set rather than a point — yield the nonsubstantiality of the room independently.
A parallel form exists in the northern transmission. In the Saṃyukta Āgama (Taishō T99) the triad reads “all formations are impermanent; all dhammas are non-self; nirvāṇa is quiescence,” and nirvāṇa alone is stated in positive rather than negative terms. The present system does the same: Theorem 26 (Nirvanic Tranquility Theorem) is written not as the absence of suffering but as the positive condition of belonging to the room.
First the “total amount of suffering” must be defined properly. “I am suffering now” and “I will suffer from here on” are different, so the future must be summed too.
🔢 The symbols in this formulaVa(x,t) ≥ 0 = a nonnegative evaluation at level a: the degree of present strain / e−ρ(t−T) = a weight counting the far future less / ρ > 0 = the discount rate / minπ = the smallest value over all admissible policies / J* = the total remaining even after doing one's best
Fig. The discounted total: the summed bar area is the total, and it vanishes only when every bar does.
🌱 Why it is set up this wayWhy discount? Counting the far future with the same weight as the present makes the sum diverge, and nothing can then be compared. Discounting lets the total hardship of a life from this state be expressed as a single finite number.
Next, one hypothesis is placed about levels below emptiness. This is Condition 24-A.
🔢 The symbols in this formula∀ = for all / ∄ = there does not exist / a ≺ ⊤ = an abstraction level below emptiness / a.e. = at almost every instant (all exceptions together have length zero)
💡 In plain words“Below emptiness, no policy however skilful can keep suffering at zero forever” — that is Condition 24-A. It is not a proved fact but a hypothesis. Theorem 24 (Universal Unsatisfactoriness Theorem) states what follows under it.
Combining this hypothesis with the nonnegativity of the evaluation yields the conclusion.
a ≺ ⊤ ∧ 条件24-A ⇒ 0 < J*a,ρ(x,T) < ∞(24.2)
Fig. If the discounted total of a nonnegative quantity is zero, the quantity is zero almost everywhere; any surviving bump makes the area positive.
🔍 The mathematical skeletonThe proof is a single push of reductio. (1) Suppose the total were zero. (2) Since the evaluation is nonnegative, a vanishing integral forces it to be zero almost everywhere. (3) That policy would then keep suffering at zero forever. (4) But Condition 24-A says no such policy exists. Contradiction, so the total is positive. ∎
⚠️ A common misreadingUniversal unsatisfactoriness is not “everyone stings every moment.” Nor is a uniform positive lower bound across all states asserted. The claim is the structural unsatisfactoriness that, for each fixed initial condition below emptiness, no amount of optimization achieves permanent zero suffering and a positive residual cost survives.
Chapter 5 Theorem 26 (Nirvanic Tranquility Theorem) — dynamic tranquility at emptiness, while alive
At the highest level the story changes: there is a zero-suffering room, and it can be entered while alive. That is Theorem 26 (Nirvanic Tranquility Theorem).
In the current edition, the condition of Theorem 26 (Nirvanic Tranquility Theorem) is not three separate conditions but a single Condition 26-A (zero-suffering set and stability), and that one condition contains three equations.
🔢 The symbols in this formulaW⊤ = the height meter (Lyapunov residual): zero inside the room, positive outside / dist = the shortest distance to the room / c1, c2 = positive sandwich constants / D+t = the upper right Dini derivative, a slope measurable even at corners / λ = the decay rate / ω = a continuous increasing function with ω(0)=0
Fig. The sandwich of (26.A): the meter is trapped between two multiples of the squared distance.Fig. The strict descent of (26.B): while height remains, the falling rate never reaches zero.
💡 In plain wordsIn one breath: (26.A) the meter tracks the distance to the room; (26.B) the meter must fall at a rate proportional to its own height; (26.C) as the distance shrinks so does the remaining suffering. Given these three, while outside the room the meter cannot stall and the room is continually approached.
🔍 The mathematical skeletonFrom these three, Grönwall's inequality yields exponential approach. But this is asymptotic approach, not arrival in finite time. If, however, the starting point is already inside the room, forward invariance keeps the residual value at zero for all time, with life sustained.
Fig. Theorem 26 (Nirvanic Tranquility Theorem): tranquility is the room where one is alive and remaining suffering is exactly zero.
5.1 The mathematical possibility condition for Buddhahood in this very body
Gathering what has been said into a single condition:
a = ⊤ ∧ x ∈ 𝒩⊤(t) ∧ x ∈ ℬalive
🔢 The symbols in this formulaa = ⊤ = one is presently viewing the world from the highest abstraction level (emptiness) / x ∈ 𝒩⊤(t) = the present state belongs to the zero-residual-suffering set / x ∈ ℬalive = life is being sustained
💡 In plain wordsAll three holding at once is the mathematical possibility condition for Buddhahood in this very body. That it is while alive rather than after death is written into the third conjunct. And that this set is nonempty is guaranteed by Condition 26-A.
⚠️ A common misreadingTaking the popular reading literally into mathematics makes tranquility-while-alive impossible. Reading the phrase “the flames of defilement go out” as V(x,t) ≡ 0 runs aground twice. First, V0 is a constituent of the closed-loop control of Theorems 1 through 4. The evaluation and its gradient drive homeostatic control, so halting the evaluative process identically means halting the control loop, which is incompatible with the life-sustaining condition of ℬalive. Second, what can vanish is not the function V itself but the discounted optimal residual value. That Theorem 24 (Universal Unsatisfactoriness Theorem) exhibits a positive lower bound below emptiness is a statement about that residual, presupposing that V persists; it does not say V could vanish. The mathematics of Theorem 26 (Nirvanic Tranquility Theorem) resolves exactly this difficulty. The evaluation function and the optimal feedback keep operating, and body, brain and physical entropy keep changing, impermanent as Theorem 23 (Impermanence Theorem) says. Yet the residual suffering value is held strictly at zero. That is dynamic tranquility.
🌱 Why it is set up this wayWhy build “while alive” into the definition? Because then “tranquility is not death” follows immediately from the definition and needs no proof. And the time index on the room permits the room itself to move, making tranquility dynamic rather than static.
Chapter 6 Three kinds of zero that must be distinguished
Tranquility is often described as “a still mind in which the flames of defilement have gone out.” The metaphor is beautiful, but carried straight into the mathematics it produces a mistake. What exactly becomes zero must be split three ways.
Fig. 2 Three kinds of zero. What vanishes in Theorem 26 (Nirvanic Tranquility Theorem) is the rightmost, not the leftmost.
The three zeros distinguished
Expression
Meaning
Status in this system
V(·,·) ≡ 0
the evaluation map is identically zero, or the machinery itself is deleted
Theorem 26 (Nirvanic Tranquility Theorem) does not require this. Deleting the evaluation process would lose the very model structure of Theorems 1–4.
V⊤(x(t),t) = 0 a.e.
the operating evaluation returns zero almost everywhere along the designated zero-cost trajectory
holds on designated trajectories inside the room, given nonnegativity and attainment of the optimum
J*⊤,ρ(x,T) = 0
the discounted optimal residual value over the future is zero
the core of Theorem 26 (Nirvanic Tranquility Theorem). It defines membership in the room.
⚠️ A common misreadingWhat vanishes in Theorem 26 (Nirvanic Tranquility Theorem) is not the existence of the evaluation machinery but the optimal residual value. Delete the evaluative activity itself and the foundation of the system — the mind rolling into valleys, Theorems 1–4 — collapses. Dynamic tranquility is not the removal of the evaluative system but the maintained membership in the zero-residual set under an operating closed loop.
💡 In plain wordsBody, brain, environment and physical entropy keep changing, consistently with Theorem 23 (Impermanence Theorem). Tranquility is therefore not physical stasis. Nothing halts; the direction changes.
Chapter 7 Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem) — from dynamic tranquility to “with ignorance as condition, formations”
Here the link is made to the opening line of the twelve links: in Pāli avijjāpaccayā saṅkhārā, “with ignorance as condition, volitional formations come to be.”
First, ignorance is defined. In this system it is neither a mood nor a lack of knowledge but a fact about possibility: that no policy exists which can keep suffering at zero forever.
Avijjā27(a,x,T) :⇔ ¬ PZS(a,x,T)(27.1)
Avijjā27(a,x,T) ⇔ [ a ≺ ⊤ ] ∨ [ a = ⊤ ∧ x ∉ 𝒩⊤(T) ](27.2)
🔢 The symbols in this formula:⇔ = the left side is defined by the right / ¬ = not / PZS = the permanent-zero-suffering proposition: some policy from that state keeps suffering at zero forever after / ∨ = or / ∧ = and / ∉ = is not a member of
💡 In plain wordsRead out: to be in ignorance is either (i) to be at a level below emptiness, or (ii) to be at the highest level but not yet inside the room. Case (i) is underwritten by Theorem 24 (Universal Unsatisfactoriness Theorem), case (ii) by Theorem 26 (Nirvanic Tranquility Theorem).
Fig. Ignorance is the positional fact of being outside the room, not a matter of feeling or ability.
Next, names are given to the amount of ignorance and to the speed at which it falls.
Ign27(x,t) := W⊤(x,t)(27.3)
Des27(t) := − D+t W⊤(x(t),t)(27.4)
💡 In plain wordsNo new quantity is created. A second name is given to the height meter that Theorem 26 (Nirvanic Tranquility Theorem) already supplied. The minus sign is there so that “the amount by which it is falling” comes out positive.
Fig. The height meter; the instant it reaches zero is tranquility.
From (26.A) and (26.B) of Condition 26-A, the following now follows.
💡 In plain words“While outside the room, the meter is falling and cannot stall.” Moreover a minimum amount proportional to how far away one is is guaranteed.
⚠️ A common misreadingBut this descent alone does not establish that the agent did anything. It may be falling by the current alone; a counterexample where the natural dynamics alone lower the residual actually exists. This is the crucial fork in Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem).
Fig. Carried by the current, or rowing? Condition 27-A is what separates the two.
So Condition 27-A is imposed: under the same natural drift and the same actual actuator, the designated input minus an independently fixed reference input is taken.
ẋ = f0(x,t) + G(x,t) u , u0(t,x) := π0⊤(t,x)(27.A1)
η27 := u0 − utr , ∂W⊤/∂t + ∇xW⊤T Ftr⊤ = 0(27.A2)
🔢 The symbols in this formulaf0 = the natural drift, what moves without anyone doing anything / G = the actual actuator map, the real route by which an input acts / u0 = the optimal input chosen by the designated policy / utr = an independently fixed reference input, the staying-alive part / η27 = the volitional signal, the difference of the two inputs / Ftr⊤ = the closed loop driven by the reference input alone
Fig. Volitional formation is a subtraction: the chosen input minus the staying-alive part leaves the volitional signal.
💡 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.
Fig. A two-stage conclusion: stage 2 needs only Theorem 26 (Nirvanic Tranquility Theorem); stage 3 needs Condition 27-A.
⚠️ A common misreadingNote the direction that is proved. Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem) does not prove that ignorance carries one away from tranquility. It proves that under the designated optimal policy the residual descends toward zero and a nonzero volitional signal contributes to that descent. And what becomes zero inside the room is only the positive transverse contribution conditioned on ignorance — life-support, evaluation, thought, responsiveness to others, the capacity for free will and tangential motion within the room do not vanish.
Fig. What becomes zero, and what does not.
Chapter 8 The threefold split by type — which theorem covers which region
Fig. 3 The threefold split by type and state; the three cases are exclusive and exactly cover the typed domain.
The threefold split by type and state
Type and state
Formal conclusion
Interpretation
a ≺ ⊤
permanent zero suffering is impossible by Condition 24-A; with measurability, finite cost and attainment, 0 < J* < ∞
structural unsatisfactoriness below emptiness; the below-top branch of typed operational ignorance
a = ⊤, x ∉ 𝒩⊤(t)
operational ignorance holds with positive residual descent; under Condition 27-A, formation holds at almost every instant
from nonattainment to a nonzero volitional formation that lowers the residual
a = ⊤, x ∈ 𝒩⊤(t)
membership is forward invariant and the optimal residual stays zero; the ignorance-conditioned positive actuator contribution is zero almost everywhere
dynamic tranquility retaining life, impermanence and relational action
💡 In plain wordsThe three boxes neither overlap nor leave gaps: every state falls into exactly one. This single table therefore says completely how far the system claims what.
Chapter 9 The dependency is not a serial chain
This is where misreading easily occurs. Theorems 24, 26 and 27 are not linked as a single chain 24→26→27.
Fig. 5 How Theorems 24, 26 and 27 depend on one another — a synthesis of the typed picture, not a serial proof graph.
🔍 The mathematical skeletonPrecisely: (1) Theorem 24 (Universal Unsatisfactoriness Theorem) supplies only the below-top branch of (27.2). (2) The top-level equivalence between ignorance and formation follows from the standing hypotheses of Theorem 26 (Nirvanic Tranquility Theorem), equations (26.A) and (26.B), the stated regularity, and Condition 27-A. (3) Theorem 24 (Universal Unsatisfactoriness Theorem) is therefore indispensable to the typed synthesis but is not used in the direct proof of the top-level correspondence.
🌱 Why it is set up this wayWhy insist on this? Because writing the dependencies precisely tells you what collapses when which premise fails. Someone doubting Condition 24-A does not thereby shake the top-level correspondence. Remove Condition 27-A, on the other hand, and (2) collapses: only “it is falling” survives.
Chapter 10 Summary — in what sense is the mathematics one?
Within what this system formalizes, the opening line of the early-Buddhist twelve links and the living dynamic tranquility read in Mahāyāna and Japanese esoteric terms connect on one and the same formal structure. A description surveying the whole from the vantage of awakening and one tracing arising and cessation from the human vantage below emptiness differ only in direction; mathematically they are compatible.
🔍 The mathematical skeletonThe shared formal structure is threefold: (1) the greatest element that is the highest abstraction level; (2) the zero-residual family of sets inside it; (3) the Lyapunov dynamics governing approach to that family. The two readings are these three grasped from different vantages and directions.
🔍 The mathematical skeletonThe conclusion in one line. The mathematics of the dynamic emptiness of Mahāyāna esoteric Buddhism is the core of Theorem 26 (Nirvanic Tranquility Theorem). And that same mathematics of emptiness is, unchanged, the skeleton of Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem), which rigorously proves the first link of the twelve — “with ignorance as condition, volitional formations come to be” — contemplated by Śākyamuni beneath the Bodhi tree.
Hence, at least mathematically, there is no difference between early Buddhism and Mahāyāna esoteric Buddhism. What differs is only the vantage, as said at the outset: looking down from the Buddha's side, or climbing from the human side.
⚠️ A common misreadingThis is a claim of formal compatibility within the model, not of historical or doctrinal identity between early Buddhism, Mahāyāna and esoteric Buddhism. Nor does Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem) exhaust the meanings of avijjā and saṅkhārā in the Pāli originals. What is strictly proved is only the conditional relation obtaining after both words are mapped to model-relative, operational notions. And the scope covers only the first of the twelve relations, not the remaining eleven.
💡 In plain wordsFinally, 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.” And from beginning to end, this is a story about one and the same universe.
