From Emptiness to Dependent Origination
A Synoptic Commentary on Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem), the Ignorance-Conditioned Volitional Formations Theorem — The Mathematics of Dynamic Tranquility across Theorems 24, 26, and 27
Cognitive Research Laboratories, Tokyo / CyLab, Carnegie Mellon University / C5I Center, George Mason University
Awakening does not transform the universe into emptiness. The world is already empty, whether or not it is apprehended as such. What changes is not the dependently arisen, non-self-subsistent character of phenomena, but the cognitive standpoint from which that character is apprehended. This commentary presents a Mahāyāna and Japanese Esoteric Buddhist reading from the standpoint of realization at ⊤ and an early Buddhist reading from conditioned human existence through the arising and cessation of suffering as two directions through one formal structure shared by Theorems 24, 26, and 27 in the present model.
Condition 24-A stipulates the impossibility of permanent zero suffering below emptiness. Given Lebesgue measurability of the trajectory evaluation, the existence of a finite-cost policy for every initial pair, and an optimal policy that attains the minimum, Theorem 24 (Universal Unsatisfactoriness Theorem) derives a positive, finite optimal residual value. Theorem 26 (Nirvanic Tranquility Theorem) proves exponential approach to the zero-residual family assumed at the highest abstraction level ⊤ (emptiness) in Condition 26-A, and shows that zero residual value is preserved within that family while vital activity continues. Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem) defines a state outside the family as operational ignorance and derives strict Lyapunov-residual descent. Under Condition 27-A, it further attributes that descent, Lebesgue-almost everywhere on the designated closed loop at the highest abstraction level, to a nonzero model-relative volitional formation through the actual actuator. Thus, the mathematics of dynamic tranquility at emptiness and the opening formula of dependent origination, avijjāpaccayā saṅkhārā, are connected within a common formal structure in the explicitly stated model.
1. The Precise Meaning of “Everything Is Empty”
Emptiness does not mean that only an awakened person sees the world as nothing. It means that every existent, experience, and relation lacks a fixed intrinsic nature independent of everything else and arises dependently.
An unawakened person may fail to apprehend the universe as empty. That cognitive failure does not exempt the observer from emptiness. The unawakened person is empty; life is empty; existence is empty; suffering is empty; and the activity by which suffering is brought to cessation is likewise empty. To say that suffering is empty does not deny experienced suffering or its causal effects. To say that cessation is empty does not deny the possibility of cessation. It means that neither suffering nor its cessation is to be reified as an independent and immutable entity.
What this commentary calls “viewing the universe from the Buddha’s standpoint” is an interpretive description, not a standard technical expression. In Prajñāpāramitā and Madhyamaka terms, it means apprehending all phenomena from the standpoint of awakening as dependently arisen and empty of self-subsistent essence, rather than inspecting emptiness as one object among others. The present system associates that standpoint with the greatest element ⊤ of the lattice of abstraction levels 𝕃 and interprets it as emptiness.
⊤ is a type—the highest abstraction level—whereas 𝒩⊤(t) is a time-indexed family of zero-residual-suffering states defined within that type. “Being at emptiness” and “having realized Nirvanic Tranquility” therefore cannot be identified without qualification. Theorem 26 (Nirvanic Tranquility Theorem) requires not only a=⊤, but also membership of the present state in 𝒩⊤(t).
2. Two Directions through One Dependently Arisen Universe
At the limited point of comparison developed here, the commentary does not treat a Mahāyāna—more specifically, Japanese Esoteric Buddhist—reading and an early Buddhist reading of dependent origination as mutually exclusive cosmologies. It places them as two interpretations of the same model from different standpoints and directions of description.
| Standpoint | Point of departure and direction | Expression in this system |
|---|---|---|
| Mahāyāna and Japanese Esoteric Buddhist reading adopted here | The universe and life are surveyed from the standpoint of realization at ⊤. Everything is already empty; suffering and its cessation are no exceptions. | A higher-order description from ⊤. Theorem 26 (Nirvanic Tranquility Theorem) models living, dynamic tranquility at emptiness. |
| Early Buddhist reading adopted here | One begins from conditioned human experience, contemplates impermanence, suffering, and non-self, and proceeds toward the cessation of suffering. | In the present system, Theorems 23 (Impermanence), 24 (Suffering), 25 (Non-Self), and 26 (Nirvanic Tranquility) may be read as an ascending route. |
| Connection | Arising contemplation follows the arising of suffering; cessation contemplation follows the cessation of its conditions. | Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem) connects the first relation of the twelve links to Theorems 24 (Universal Unsatisfactoriness Theorem) and 26 (Nirvanic Tranquility Theorem). |
This ascent is the formal organization of Theorems 23 (Impermanence Theorem) through 26 (Nirvanic Tranquility Theorem) in the present system; it is not a claim that early Buddhist sources historically systematized these teachings in this exact four-stage sequence.
The twelve links of dependent origination are more than a list—ignorance, volitional formations, consciousness, name-and-form, the six sense bases, contact, feeling, craving, clinging, becoming, birth, and aging-and-death. Arising contemplation observes how the entire mass of suffering arises when its conditions obtain. Cessation contemplation does not merely recite the factors backward: with the cessation of ignorance, formations cease; through the remaining conditional sequence, aging-and-death and the entire mass of suffering cease. The twelve links can therefore be understood both as a doctrinal formulation and as a practical, contemplative framework for observing the arising and cessation of suffering.
Udāna 1.1–1.3 depicts the Buddha, after awakening, attending respectively to the arising sequence of dependent origination, its cessation sequence, and both sequences. This canonical portrayal provides one basis for the contemplative reading adopted here; it does not reduce dependent origination to that reading alone.
3. Theorem 24 (Universal Unsatisfactoriness Theorem) — Structural Nonfulfillment below Emptiness
Let Va(x,t)≥0 be the nonnegative evaluation at abstraction level a∈𝕃. Define the discounted infinite-horizon optimal residual value by
minπ∈Pola(x,T) ∫T∞ e−ρ(t−T) Va(xx,Tπ(t),t) dt,
xx,Tπ(T)=x, ρ>0. (24.1)
Assume that the evaluation along each trajectory is Lebesgue measurable, that at least one finite-cost policy exists for every (a,x,T), and that an optimal policy exists and attains the minimum. Hence 0≤Ja,ρ*(x,T)<∞.
Condition 24-A requires that below emptiness no admissible policy can keep Va=0 Lebesgue-almost everywhere forever.
∄π∈Pola(x,T) such that
Va(xx,Tπ(t),t)=0 (a.e. t≥T). (24.A)
Under Condition 24-A and the stated measurability, finite-cost, and attainment assumptions, the fact that a nonnegative measurable function has zero integral only when it is zero almost everywhere yields
This does not assert that everyone consciously feels pain at every instant, nor does it establish one uniform positive lower bound shared by all states. It is structural nonfulfillment: from each fixed initial condition below emptiness, no amount of optimization realizes permanent zero suffering, and a positive residual cost remains.
4. Theorem 26 (Nirvanic Tranquility Theorem) — Dynamic Tranquility at Emptiness
At the highest abstraction level ⊤, let ℬalive⊂X⊤ be the forward-invariant set on which vital activity is sustained for the complete state, including brain and body. Under a single Borel-measurable Markov feedback π⊤0 that is simultaneously optimal for every initial pair (x,T)∈ℬalive×[0,∞), define the zero-residual-suffering family by
In the latest Mini13 formulation, Theorem 26 (Nirvanic Tranquility Theorem) does not have three separate Conditions 26-A, 26-B, and 26-C. It has one Condition 26-A (Zero-Suffering Sets and Stability). Taken together, the standing hypotheses of Theorem 26 (Nirvanic Tranquility Theorem) and Condition 26-A require forward completeness of the designated closed loop, forward invariance of ℬalive, and nonemptiness, closedness, and forward invariance of each 𝒩⊤(t). Condition 26-A also contains the following three displayed relations.
Dt+W⊤(x(t),t)≤−λW⊤(x(t),t), λ>0 (26.B)
0≤J⊤,ρ*(x,t)≤ω(dist(x,𝒩⊤(t))),
ω:[0,∞)→[0,∞) continuous and increasing, ω(0)=0. (26.C)
Equations (26.A) and (26.C) hold for every x∈ℬalive, t≥0; equation (26.B) holds along every trajectory of the designated closed loop π⊤0. Equation (26.A) sandwiches the Lyapunov residual W⊤ between positive multiples of the squared distance to 𝒩⊤(t). Equation (26.B) gives an exponential upper bound and strict descent whenever W⊤>0. Equation (26.C) ensures that the optimal residual value approaches zero as the distance does. Thus, for every T≥0, x(T)=xT∈ℬalive, and t≥T, Grönwall’s inequality yields
dist(x(t),𝒩⊤(t))≤√(W⊤(x(T),T)/c1)e−λ(t−T)/2→0. (26.2)
This conclusion is generally asymptotic and does not imply finite-time arrival. If the initial state already belongs to 𝒩⊤(T), however, forward invariance maintains zero residual value at every future time while vital activity continues. Under Condition 24-A, the standing hypotheses of Theorem 26 (Nirvanic Tranquility Theorem), and Condition 26-A, the connection between Theorems 24 (Universal Unsatisfactoriness Theorem) and 26 (Nirvanic Tranquility Theorem) is expressed in one formula using the typed permanent-zero-suffering predicate PZS(a,x,T):
[∀a≺⊤, ∀x∈Xa: ¬PZS(a,x,T)] ∧
[∀x∈ℬalive: PZS(⊤,x,T)⇔x∈𝒩⊤(T)]. (26.3)
On the typed domain 𝔛27:=(⊔a≺⊤({a}×Xa))⊔({⊤}×ℬalive),
PZS(a,x,T) ⇔ [a=⊤ ∧ x∈𝒩⊤(T)]
This is the minimal typed form of “permanent cessation of suffering” on the typed domain of (26.3), under Condition 24-A, the standing hypotheses of Theorem 26 (Nirvanic Tranquility Theorem), and Condition 26-A.
The inclusion 𝒩⊤(T)⊂ℬalive is built into the definition. Writing x∈ℬalive again usefully emphasizes “while alive,” but is logically redundant. The nonemptiness of 𝒩⊤(T) is also an explicit assumption of Condition 26-A; it does not follow from equation (26.A) alone.
5. The “Extinction of Defilements,” V, J, and Buddhahood in This Very Body
Nirvanic Tranquility is often described as a quiet and peaceful state in which the flames of the defilements have been extinguished. But the mathematics is misconstrued if this metaphor is identified with making the evaluative map globally and identically zero, V(·,·)≡0, or, more strongly, with stopping or deleting the evaluative mechanism itself.
| Expression | Meaning | Role in this system |
|---|---|---|
| V(·,·)≡0 | The evaluative map is made identically zero on its entire domain, or the evaluative mechanism itself is removed. | Theorem 26 (Nirvanic Tranquility Theorem) does not require this. Deleting evaluation would remove the V-driven closed-loop architecture of Theorems 1 (Tomabechi Main Theorem) through 4 (Tomabechi Presence-Weighted Theorem). |
| V⊤(x(t),t)=0 a.e. | An operating evaluative map returns zero almost everywhere along the designated zero-cost trajectory. | Given V⊤≥0 and attainment of the optimum, this holds on the designated trajectory within 𝒩⊤. |
| J⊤,ρ*(x,T)=0 | The discounted optimal residual value over the future is zero. | This is central to Theorem 26 (Nirvanic Tranquility Theorem) and defines membership in 𝒩⊤(T). |
What vanishes in Theorem 26 (Nirvanic Tranquility Theorem) is not the existence of the evaluative mechanism, but the optimal residual value. More precisely, on the trajectory of the designated policy π⊤0 beginning from x(T)∈𝒩⊤(T), nonnegativity and attainment of the optimum give
V⊤(x(t),t)=0 (for Lebesgue-almost every t≥T).
For a general trajectory beginning outside the family, Theorem 26 (Nirvanic Tranquility Theorem) establishes only J⊤,ρ*(x(t),t)→0; it does not assert that V⊤=0 almost everywhere from some intermediate time onward. The evaluative map V⊤ and the optimal feedback π⊤0 nevertheless remain defined and may continue to operate. Dynamic tranquility is the preservation, under an active closed loop, of forward-invariant membership in the zero-residual family.
At the same time, the body, brain, environment, and physical entropy may continue to change, consistently with the impermanence formalized in Theorem 23 (Impermanence Theorem). Nirvanic Tranquility is therefore not physical stasis. The minimal mathematical possibility condition that this commentary associates with sokushin jōbutsu (“attaining Buddhahood in this very body”) is
This is not the extinction of the evaluative flame through termination of living control. It is the continued membership of a complete living state in the zero-residual-value invariant family at emptiness. The present commentary interprets this living dynamic tranquility as a mathematical model of emptiness that can be read as formally consistent with the ideal of attaining Buddhahood in this very body in Japanese Esoteric Buddhism.
The interpretation also connects naturally to the preceding theorems. The evaluation V is a control core of Theorems 1 (Tomabechi Main Theorem) through 4 (Tomabechi Presence-Weighted Theorem) and the source of the residual in Theorem 24 (Universal Unsatisfactoriness Theorem). By contrast, J⊤,ρ*=0 is consistent with fi(⊤)=1 in Theorem 19 (Tomabechi Free Will Theorem), the inclusion ladder open toward emptiness in Theorem 22 (High-Altitude LUB Presence Theorem), the positive below-emptiness residual of Theorem 24 (Universal Unsatisfactoriness Theorem), and de-individuation by ★⊤ in Theorem 25 (Non-Self Theorem), while remaining compatible with the impermanence of Theorem 23 (Impermanence Theorem).
6. Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem) — From Dynamic Tranquility to avijjāpaccayā saṅkhārā
What Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem) rigorously proves is not the exhaustive canonical meaning of ignorance and volitional formations themselves, but their conditional relation after both terms have been mapped into an explicitly specified model. To avoid conflating types, first fix the domain.
In SN 12.2, avijjā is explained as not knowing suffering, its origin, its cessation, and the way leading to its cessation. The discourse also enumerates three kinds of formation: bodily formation, verbal formation, and mental formation. The theorem does not reduce the full canonical meaning of either term to set nonmembership or to one control component. Avijjā27 is the restricted predicate of typed operational ignorance, and Saṅkhārā27 is the restricted predicate of an ignorance-conditioned contribution through the actual actuator on the designated closed loop.
For every (a,x)∈𝔛27 and T≥0, typed operational ignorance is defined as the nonexistence of an admissible policy that maintains permanent zero suffering.
Combining Theorems 24 (Universal Unsatisfactoriness Theorem) and 26 (Nirvanic Tranquility Theorem) gives
Theorem 24 (Universal Unsatisfactoriness Theorem) and Condition 24-A support the below-⊤ branch of this equivalence. At the highest abstraction level, writing the Lyapunov ignorance residual as (27.3) and the residual descent rate as (27.4), set Ign27:=W⊤ and Des27:=−Dt+W⊤. The standing hypotheses of Theorem 26 (Nirvanic Tranquility Theorem), the local Lipschitz regularity of t↦W⊤(x(t),t) along the trajectory, and equations (26.A) and (26.B), contained in the single Condition 26-A, yield
⇒Des27(t)≥λIgn27(x(t),t)
≥λc1dist(x(t),𝒩⊤(t))2>0. (27.6)
What is directly established here is strict Lyapunov-residual descent while tranquility remains unrealized. That descent alone cannot distinguish natural drift from the agent’s input, because a system may reduce its residual through natural dynamics alone.
Condition 27-A therefore separates the designated input u0 from an independently fixed reference input utr under the same natural drift f0 and the same actual actuator G. The reference input is an admissible feedback taking values in the same input space as u0. The condition also assumes C1 regularity of W⊤, absolute continuity of the trajectory, satisfaction of the closed-loop equation almost everywhere by that trajectory, and measurable, finite-valued Gη27.
η27(t,x):=u0(t,x)−utr(t,x),
F⊤tr(x,t):=f0(x,t)+G(x,t)utr(t,x),
∂tW⊤+∇xW⊤TF⊤tr=0. (27.A2)
The reference-field identity (27.A2) is assumed to hold on a neighborhood containing the trajectory of interest.
Model-relative directed volitional formation is then defined as the component acting through the actual actuator that contributes positively to descent of the ignorance residual.
−∇xW⊤(x(t),t)TG(x(t),t)η27(t,x(t))>0. (27.5)
On the designated closed loop at the highest abstraction level, the first equivalence then holds at every time, and the second holds Lebesgue-almost everywhere under Condition 27-A.
Avijjā27(⊤,x(t),t)⇔Saṅkhārā27(x(t),t) (a.e., under Condition 27-A). (27.10)
The theorem does not prove that ignorance drives the state away from tranquility. Under the designated optimal policy, it proves that the residual of unrealized tranquility descends toward zero and that a nonzero directed signal through the actual actuator contributes to that descent. Within the tranquility family, what vanishes is the positive transverse contribution conditioned by operational ignorance. Life maintenance, evaluation, thought, responsiveness to others, free-will capacity, motion tangent to the family, and input components in the kernel of G need not vanish.
7. The Typed Synthesis of Theorems 24, 26, and 27
a≺⊤∧24-A⇒¬PZS ∧ 0<J*<∞
x∉𝒩⊤⇒−D+W⊤>0
Avijjā27(⊤,x(t),t)⇔Saṅkhārā27(x(t),t) a.e.
This is a synthesis of the full typed architecture, not a serial proof-dependency diagram. Theorem 24 (Universal Unsatisfactoriness Theorem) supplies the below-⊤ branch; the highest-level equivalence follows from Theorem 26 (Nirvanic Tranquility Theorem), the stated regularity, and Condition 27-A.
| Type and state | Formal conclusion | Interpretation |
|---|---|---|
| a≺⊤ | Condition 24-A gives ¬PZS; the measurability, finite-cost, and attainment assumptions further give 0<Ja,ρ*<∞. | Structural nonfulfillment below emptiness; the below-emptiness branch of typed operational ignorance. |
| a=⊤, x∉𝒩⊤(t) | Avijjā27 and positive residual descent; almost everywhere under Condition 27-A on the designated highest-level closed loop, Saṅkhārā27. | From unrealized tranquility to nonzero directed volitional formation that reduces the residual. |
| a=⊤, x∈𝒩⊤(t) | Membership in the family is forward invariant, so J⊤,ρ*=0 is preserved. Under Condition 27-A, the positive ignorance-conditioned contribution through the actual actuator is zero a.e. | Dynamic tranquility with vital activity, impermanence, and relational action intact. |
7.1 Nonsubstantiality of the Zero-Residual-Suffering Family
One philological issue bearing on the typed synthesis deserves treatment from the mathematical side: whether sabbe dhammā anattā at Dhammapada 279 includes nirvāṇa within the scope of non-self. The Theravāda commentary Dhammapada-aṭṭhakathā glosses “all dhammas” in that verse as the dhammas of four levels — sense-sphere, form, formless and supramundane — and thereby places the supramundane, which includes nirvāṇa, within the scope of non-self. A variant reading sabbe saṅkhārā anattā has occasionally been proposed, restricting the scope to formations; it is not, however, the mainstream of the surviving manuscript and commentarial tradition, and standard critical editions, including that of the Pali Text Society, adopt sabbe dhammā anattā.
Within the present system the question is settled by definition rather than by interpretation. The zero-residual-suffering family is introduced by (26.1), and that definition resists substantialization on three counts.
First, relational dependence. 𝒩⊤ is specified only through its relation to the evaluation V, the discount rate ρ, the life-sustaining set ℬalive, and the optimization structure; strip these away and the definition itself is lost. Second, time dependence. 𝒩⊤(T) is a family indexed by time, not a single fixed object. Third, nonindividuality. 𝒩⊤ is a set and not a point, so it does not even take the form of the “substance existing as a point cut off from relation” that Theorem 25 (Non-Self Theorem) denies.
Consequently, within this system nirvanic tranquility too falls within the scope of Theorem 25 (Non-Self Theorem); the system agrees with the commentarial reading cited above. The conclusion does not, however, depend on which reading is adopted. Even if the variant were taken and the scope of non-self restricted to formations, the nonsubstantiality of 𝒩⊤ would still follow independently from the three counts above. This subsection states a property internal to the model; it does not adjudicate a textual-critical dispute over the Pāli canon.
Correspondence with the Sarvāstivāda parallel. The triad found in the Saṃyukta Āgama (Taishō T99) — “all formations are impermanent; all dhammas are non-self; nirvāṇa is quiescence” — assigns impermanence to formations and non-self to dhammas, and states nirvāṇa through a positive property rather than a negation. The articulation of the present system corresponds to this. Theorem 23 (Impermanence Theorem) treats the level of formations as nonrecurrence of the trajectory; Theorem 25 (Non-Self Theorem) treats the level of dhammas as relational structure; and Theorem 26 (Nirvanic Tranquility Theorem) is stated not as an absence of suffering but as the positive condition of membership in a zero-residual invariant set. Moreover, the difficulty of speaking of nirvāṇa within the framework of suffering is dissolved here by typing: the hypothesis of Theorem 24 (Universal Unsatisfactoriness Theorem) is a ≺ ⊤ and does not apply to nirvāṇa, while at a = ⊤ with x ∈ 𝒩⊤(T) one has J*⊤,ρ = 0. The two are not competing claims about one object but cover regions partitioned exclusively by type and state.
Sources: Dhammapada 277–279 (Pāli text); Dhammapada-aṭṭhakathā on verse 279; Saṃyukta Āgama, Taishō T99. The four-seal formulation derives from a confluence of the triad of impermanence, unsatisfactoriness and non-self with the Sarvāstivāda emphasis on the quiescence of nirvāṇa, and does not itself appear in the Pāli triad.
8. Synthesis — In What Sense Is the Mathematics One?
The dynamic tranquility supplied by Theorem 26 (Nirvanic Tranquility Theorem) at the highest abstraction level ⊤ forms the foundation of the highest-level part of Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem). Theorem 24 (Universal Unsatisfactoriness Theorem) supplies the below-⊤ branch of (27.2) through the impossibility of permanent zero suffering and thereby completes the account of typed operational ignorance across the lattice. By contrast, the highest-level equivalence Avijjā27⇔Saṅkhārā27 follows from the standing hypotheses of Theorem 26 (Nirvanic Tranquility Theorem), equations (26.A) and (26.B), the stated regularity, and Condition 27-A. Theorem 24 (Universal Unsatisfactoriness Theorem) is therefore indispensable to the typed synthesis, but is not a direct premise of the highest-level ignorance–formation equivalence.
Within the range formalized here, the opening formula of early Buddhist dependent origination can therefore be placed within a common formal structure in the present model together with living dynamic tranquility interpreted in Mahāyāna and Japanese Esoteric Buddhist terms. A description that views the whole from the standpoint of realization at ⊤ and a description that begins from the below-⊤ human standpoint and follows the arising and cessation of suffering differ in direction, yet are mathematically compatible. For this restricted correspondence, their difference can be represented primarily as one of standpoint and direction of description rather than a discontinuity in the mathematical mechanism.
Within the interpretive mapping proposed here, emptiness viewed from the standpoint of awakening and dependent origination traced from the human standpoint of nonattainment need not be modeled as descriptions of two different universes. At the specific point of connection formalized in this system, they can be represented as two perspectives on a common formal architecture: the greatest element ⊤, the zero-residual-suffering family, and the Lyapunov dynamics governing approach to that family.
This is a claim of formal compatibility within the present model, not a claim of historical or doctrinal identity among early Buddhism, Mahāyāna Buddhism, and Esoteric Buddhism. Nor does Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem) exhaust the canonical meanings of avijjā and saṅkhārā. What is rigorously proved is the conditional relation that holds under the stated assumptions after the two terms are mapped to the model-relative, operational notions Avijjā27 and Saṅkhārā27. Moreover, the theorem formalizes only the first relation among the twelve links; it does not prove the remaining eleven conditional relations.
On the base edition. The direct base for Theorems 24 (Universal Unsatisfactoriness Theorem) and 26 (Nirvanic Tranquility Theorem) and for Condition 26-A is TomabechiFourDharmaSealsMini13, but Theorem 26 (Nirvanic Tranquility Theorem) is completely identical in Mini13 and Mini14. The two editions differ only in whether Theorem 15 (the Tomabechi Cognitive-Physical Entropy Exchange and Conservation Theorem) appears in the main text or the same budget is stated as an explicit common assumption, which does not affect the apparatus used here. The present account therefore applies unchanged to readers working from Mini14.
Sources and Editions
- Hideto Tomabechi, The Tomabechi Four Dharma Seals Theorem System—Minimal Thirteen-Theorem Version (the direct basis for Theorems 24 (Universal Unsatisfactoriness Theorem) and 26 (Nirvanic Tranquility Theorem) and Condition 26-A)
- Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem) — Japanese Academic Edition
- Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem) — English Academic Edition
- Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem) — Japanese Mathematics Explained Edition
- Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem) — English Mathematics Explained Edition
- Saṃyutta Nikāya 12.2 — Pāli text / Bhikkhu Bodhi’s English translation
- Udāna 1.1 — arising sequence / Udāna 1.2 — cessation sequence / Udāna 1.3 — arising and cessation sequences
- Prajñāpāramitāhṛdaya (Heart Sutra) / Nāgārjuna, Mūlamadhyamakakārikā 24.18 (emptiness and dependent origination)
- Kūkai, Sokushin jōbutsu gi (Buddhahood in This Very Body)
Mathematical note: in the latest formulation of Theorem 26 (Nirvanic Tranquility Theorem), the single condition is “Condition 26-A,” which contains equations (26.A), (26.B), and (26.C). The highest-level proof in Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem) directly uses equations (26.A) and (26.B); equation (26.C) is not required for that proof.