定理27を作った。仏教の十二支縁起（十二因縁）の最初は「無明に縁りて行が起こり」（無明によりて行があり）である。これを定理27とし、厳密証明した。四法印に続く、初の仏説の厳密完全証明である。「行」をパーリ語の意味合いに即した「志向的形成作用」とし、無明起行（Avijjāpaccayā saṅkhārā）を、涅槃寂静の未達によって生じる志向的形成作用として数理化した。十二支縁起は、無明起行が理解されれば、後は自然な帰結であり、別個の証明を必要としない。
定理27 無明起行定理
(Avijjāpaccayā saṅkhārā)
Avijjāpaccayā saṅkhārā――涅槃寂静の未達から志向的形成作用へ
十二支縁起の冒頭句「無明に縁りて行が起こる」を、定理26の動的寂静に接続して数理化する。定理26が成立する体系内で、現在状態が、時変集合族の一要素である零残余苦価値集合 𝒩⊤(t) に属さないことを、本体系における「操作的無明」と定義する。
行は生命活動一般ではなく、実際の制御アクチュエータを介して未達残差を厳密に減少させる、モデル相対的な志向的形成作用として定義する。
条件26-B（Mini13）のLyapunov不等式だけから直接従うのは、寂静未達時の正のLyapunov残差下降率である。それを主体の行へ帰属させるには、自然ドリフトと実制御入力を分離する条件27-Aがさらに必要である。
この条件のもとでは、無明時に方策が選択する志向信号と、その実効状態作用が非零となる。寂静集合内では、無明条件付きの正の横断寄与は消える一方、身体・脳・評価・フィードバックの接線方向の活動は存続できる。
1. パーリ語原典句と本定理の射程
パーリ語の定型句は avijjāpaccayā saṅkhārā であり、「無明を条件として、行が生じる」と読まれる。saṅkhārā は複数形であり、本稿ではBhikkhu Bodhiによる広く用いられる英訳 volitional formations に対応させて、「志向的形成作用」と読む。
ただし、パーリ語原典における avijjā は、苦・苦の生起・苦の滅・滅に至る道を知らないこととして説明される。また、SN 12.2は saṅkhārā を身体的・言語的・心的な行の三種として定義する。本定理は、これらの全語義を単なる集合非所属や一つの制御成分へ還元しない。
以下の Avijjā₂₇ は、定理26の数理体系におけるモデル相対的・操作的な無明であり、Saṅkhārā₂₇ は、パーリ語原典上の三種全体と同一ではなく、指定閉ループにおけるモデル相対的な無明条件付きアクチュエータ寄与である。
パーリ語原典資料：Saṃyutta Nikāya 12.2（パーリ本文） https://suttacentral.net/sn12.2/pli/ms?lang=jpn&layout=plain&reference=none¬es=asterisk&highlight=false&script=latin
同・Bhikkhu Bodhi英訳 https://suttacentral.net/sn12.2/en/bodhi?lang=jpn&reference=none&highlight=false
2. 定理24・26から継承する構造
抽象度束を 𝕃、最高抽象度を ⊤（空）とする。定理24の型付き永久零苦命題 PZS(a,x,T) は、「ある許容方策が、x(T)=x から非負評価 Vₐ=0 をLebesgueほとんど至る所で永久に保つ」ことをいう。
最小依存関係
依存 定理27で用いる内容
定理24・条件24-A (27.2)の空未満の枝：永久零苦不能と正の最適残余価値
定理26の常設仮定・条件26-A／26-B（26-Cは不要） 最高抽象度の指定方策、零残余苦価値集合、閉性・前向き不変性、Lyapunov挟み込みと厳密下降
3.2の正則性 Dini微分を有限の残差下降率として定義
条件27-A 残差下降を、実アクチュエータを介する指定入力と基準入力の差へ帰属
結論
本体系では、涅槃寂静の未達を、型付き操作的無明として表すことができる。最高抽象度で零苦不変集合の外にある状態は、条件26-A・26-Bにより、正のLyapunov無明残差と厳密に正の残差下降率をもつ。条件27-Aを加えると、その下降へ正に寄与する実アクチュエータ上の志向信号は非零である。したがって、明示したモデル内で、「無明に縁りて行が起こる」に対応する条件付き含意が成立する。
同時に、寂静は物理的静止でも全行為の消滅でもない。消えるのは無明に条件づけられた正の残差寄与であり、生命と自由の接線的・関係的活動は存続しうる。本定理は、パーリ語原典句のモデル理論的対応を与えるものであり、パーリ語原典上の無明・行の全語義を証明または尽くすものではない。これが、定理26の動的寂静と両立する定理27の最小形式である。
定理27 無明起行定理 （Avijjāpaccayā saṅkhārā）学術版日本語 https://tomabechi.jp/TomabechiAvijjaSankharaJA.html
定理27 無明起行定理 （Avijjāpaccayā saṅkhārā）学術版英語 https://tomabechi.jp/TomabechiAvijjaSankharaEN.html
ーーーー
I have formulated Theorem 27. The twelve links of dependent origination—the twelve nidānas—begin with “with ignorance as condition, volitional formations arise” (“with ignorance, there are volitional formations”).
I designated this as Theorem 27 and gave it a rigorous proof. Following the Four Dharma Seals, this is the first rigorous and complete proof of a teaching of the Buddha.
Taking saṅkhārā in its Pāli sense as “volitional formations,” the mathematics formulates Avijjāpaccayā saṅkhārā as directed volitional formation arising from the nonattainment of Nirvanic Tranquility. Once Avijjāpaccayā saṅkhārā is understood, the remaining links of dependent origination follow naturally and require no separate proof.
Theorem 27: Ignorance-Conditioned Volitional Formations (Avijjāpaccayā saṅkhārā)
Avijjāpaccayā saṅkhārā—From the Nonattainment of Nirvanic Tranquility to Directed Volitional Formation
The opening formula of the twelve links of dependent origination—“with ignorance as condition, volitional formations arise”—is formalized by connecting it to the dynamic tranquility of Theorem 26.
Within a system in which Theorem 26 holds, the present state’s nonmembership in the zero-residual-suffering value set 𝒩⊤(t), a member of a time-indexed family of sets, is defined as “operational ignorance” in the present system. Volitional formation is not identified with biological activity in general; it is defined as a model-relative, directed formative operation that strictly decreases the nonattainment residual through the actual control actuator.
What follows directly from the Lyapunov inequality in Condition 26-B of Mini13 is only a positive Lyapunov-residual descent rate whenever tranquility remains unrealized. Attributing that descent to the agent’s volitional formation additionally requires Condition 27-A, which separates natural drift from the actual control input.
Under this condition, both the policy-selected volitional signal and its effective state action are nonzero whenever operational ignorance holds. Within the tranquility set, the positive ignorance-conditioned transverse contribution vanishes, while tangential bodily, neural, evaluative, and feedback activity may continue.
1. The Canonical Pāli Formula and the Scope of This Theorem
The standard Pāli formula is avijjāpaccayā saṅkhārā, read as “with ignorance as condition, volitional formations come to be.” Saṅkhārā is plural. This paper interprets it as volitional formative activity, corresponding to Bhikkhu Bodhi’s widely used English rendering, volitional formations.
In the Pāli canon, however, avijjā is explained as not knowing suffering, the origin of suffering, the cessation of suffering, and the way leading to its cessation. SN 12.2 also defines saṅkhārā as three kinds of volitional formation: bodily, verbal, and mental. The present theorem does not reduce the full meanings of these terms to mere set nonmembership or to a single control component.
The following Avijjā₂₇ is a model-relative, operational notion of ignorance within the mathematical system of Theorem 26. Saṅkhārā₂₇ is not claimed to be coextensive with the entire canonical triad; it is a model-relative, ignorance-conditioned actuator contribution within the designated closed loop.
Canonical Pāli source: Saṃyutta Nikāya 12.2, Pāli text https://suttacentral.net/sn12.2/pli/ms?lang=jpn&layout=plain&reference=none¬es=asterisk&highlight=false&script=latin
Bhikkhu Bodhi’s English translation https://suttacentral.net/sn12.2/en/bodhi?lang=jpn&reference=none&highlight=false
2. Structures Inherited from Theorems 24 and 26
Let 𝕃 be the lattice of abstraction levels, with greatest element ⊤, interpreted as emptiness. The typed permanent-zero-suffering proposition PZS(a,x,T) from Theorem 24 means that some admissible policy, starting from x(T)=x, keeps the nonnegative evaluation Vₐ=0 Lebesgue-almost everywhere forever.
Minimal Dependencies
Dependency Content used in Theorem 27
Theorem 24 and Condition 24-A The below-⊤ branch of (27.2): the impossibility of permanent zero suffering and a positive optimal residual value
The standing assumptions and Conditions 26-A and 26-B of Theorem 26; Condition 26-C is not required The designated policy at the highest abstraction level, the zero-residual-suffering set, closedness, forward invariance, the Lyapunov sandwich bounds, and strict descent
Regularity assumptions in Section 3.2 Definition of the Dini derivative as a finite residual-descent rate
Condition 27-A Attribution of residual descent to the difference between the designated and reference inputs through the actual actuator
Conclusion
Within the present system, the 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 a positive Lyapunov ignorance residual and a strictly positive residual-descent rate.
When Condition 27-A is added, the volitional signal on the actual actuator that positively contributes to that descent is nonzero. Thus, within the explicitly specified model, the conditional implication corresponding to “with ignorance as condition, volitional formations arise” is established.
At the same time, tranquility is neither physical stasis nor the cessation of all activity. What vanishes is the positive residual contribution conditioned by operational ignorance; the tangential and relational activities of life and freedom may continue.
This theorem provides a model-theoretic counterpart to the canonical Pāli formula. It neither proves nor exhausts the full canonical meanings of avijjā and saṅkhārā. This is the minimal form of Theorem 27 that remains consistent with the dynamic tranquility of Theorem 26.
Theorem 27: Ignorance-Conditioned Volitional Formations (Avijjāpaccayā saṅkhārā)—Japanese Academic Edition https://tomabechi.jp/TomabechiAvijjaSankharaJA.html
Theorem 27: Ignorance-Conditioned Volitional Formations (Avijjāpaccayā saṅkhārā)—English Academic Edition https://tomabechi.jp/TomabechiAvijjaSankharaEN.html
定理27 無明起行定理
(Avijjāpaccayā saṅkhārā)
Avijjāpaccayā saṅkhārā――涅槃寂静の未達から志向的形成作用へ
十二支縁起の冒頭句「無明に縁りて行が起こる」を、定理26の動的寂静に接続して数理化する。定理26が成立する体系内で、現在状態が、時変集合族の一要素である零残余苦価値集合 𝒩⊤(t) に属さないことを、本体系における「操作的無明」と定義する。
行は生命活動一般ではなく、実際の制御アクチュエータを介して未達残差を厳密に減少させる、モデル相対的な志向的形成作用として定義する。
条件26-B（Mini13）のLyapunov不等式だけから直接従うのは、寂静未達時の正のLyapunov残差下降率である。それを主体の行へ帰属させるには、自然ドリフトと実制御入力を分離する条件27-Aがさらに必要である。
この条件のもとでは、無明時に方策が選択する志向信号と、その実効状態作用が非零となる。寂静集合内では、無明条件付きの正の横断寄与は消える一方、身体・脳・評価・フィードバックの接線方向の活動は存続できる。
1. パーリ語原典句と本定理の射程
パーリ語の定型句は avijjāpaccayā saṅkhārā であり、「無明を条件として、行が生じる」と読まれる。saṅkhārā は複数形であり、本稿ではBhikkhu Bodhiによる広く用いられる英訳 volitional formations に対応させて、「志向的形成作用」と読む。
ただし、パーリ語原典における avijjā は、苦・苦の生起・苦の滅・滅に至る道を知らないこととして説明される。また、SN 12.2は saṅkhārā を身体的・言語的・心的な行の三種として定義する。本定理は、これらの全語義を単なる集合非所属や一つの制御成分へ還元しない。
以下の Avijjā₂₇ は、定理26の数理体系におけるモデル相対的・操作的な無明であり、Saṅkhārā₂₇ は、パーリ語原典上の三種全体と同一ではなく、指定閉ループにおけるモデル相対的な無明条件付きアクチュエータ寄与である。
パーリ語原典資料：Saṃyutta Nikāya 12.2（パーリ本文） https://suttacentral.net/sn12.2/pli/ms?lang=jpn&layout=plain&reference=none¬es=asterisk&highlight=false&script=latin
同・Bhikkhu Bodhi英訳 https://suttacentral.net/sn12.2/en/bodhi?lang=jpn&reference=none&highlight=false
2. 定理24・26から継承する構造
抽象度束を 𝕃、最高抽象度を ⊤（空）とする。定理24の型付き永久零苦命題 PZS(a,x,T) は、「ある許容方策が、x(T)=x から非負評価 Vₐ=0 をLebesgueほとんど至る所で永久に保つ」ことをいう。
最小依存関係
依存 定理27で用いる内容
定理24・条件24-A (27.2)の空未満の枝：永久零苦不能と正の最適残余価値
定理26の常設仮定・条件26-A／26-B（26-Cは不要） 最高抽象度の指定方策、零残余苦価値集合、閉性・前向き不変性、Lyapunov挟み込みと厳密下降
3.2の正則性 Dini微分を有限の残差下降率として定義
条件27-A 残差下降を、実アクチュエータを介する指定入力と基準入力の差へ帰属
結論
本体系では、涅槃寂静の未達を、型付き操作的無明として表すことができる。最高抽象度で零苦不変集合の外にある状態は、条件26-A・26-Bにより、正のLyapunov無明残差と厳密に正の残差下降率をもつ。条件27-Aを加えると、その下降へ正に寄与する実アクチュエータ上の志向信号は非零である。したがって、明示したモデル内で、「無明に縁りて行が起こる」に対応する条件付き含意が成立する。
同時に、寂静は物理的静止でも全行為の消滅でもない。消えるのは無明に条件づけられた正の残差寄与であり、生命と自由の接線的・関係的活動は存続しうる。本定理は、パーリ語原典句のモデル理論的対応を与えるものであり、パーリ語原典上の無明・行の全語義を証明または尽くすものではない。これが、定理26の動的寂静と両立する定理27の最小形式である。
定理27 無明起行定理 （Avijjāpaccayā saṅkhārā）学術版日本語 https://tomabechi.jp/TomabechiAvijjaSankharaJA.html
定理27 無明起行定理 （Avijjāpaccayā saṅkhārā）学術版英語 https://tomabechi.jp/TomabechiAvijjaSankharaEN.html
ーーーー
I have formulated Theorem 27. The twelve links of dependent origination—the twelve nidānas—begin with “with ignorance as condition, volitional formations arise” (“with ignorance, there are volitional formations”).
I designated this as Theorem 27 and gave it a rigorous proof. Following the Four Dharma Seals, this is the first rigorous and complete proof of a teaching of the Buddha.
Taking saṅkhārā in its Pāli sense as “volitional formations,” the mathematics formulates Avijjāpaccayā saṅkhārā as directed volitional formation arising from the nonattainment of Nirvanic Tranquility. Once Avijjāpaccayā saṅkhārā is understood, the remaining links of dependent origination follow naturally and require no separate proof.
Theorem 27: Ignorance-Conditioned Volitional Formations (Avijjāpaccayā saṅkhārā)
Avijjāpaccayā saṅkhārā—From the Nonattainment of Nirvanic Tranquility to Directed Volitional Formation
The opening formula of the twelve links of dependent origination—“with ignorance as condition, volitional formations arise”—is formalized by connecting it to the dynamic tranquility of Theorem 26.
Within a system in which Theorem 26 holds, the present state’s nonmembership in the zero-residual-suffering value set 𝒩⊤(t), a member of a time-indexed family of sets, is defined as “operational ignorance” in the present system. Volitional formation is not identified with biological activity in general; it is defined as a model-relative, directed formative operation that strictly decreases the nonattainment residual through the actual control actuator.
What follows directly from the Lyapunov inequality in Condition 26-B of Mini13 is only a positive Lyapunov-residual descent rate whenever tranquility remains unrealized. Attributing that descent to the agent’s volitional formation additionally requires Condition 27-A, which separates natural drift from the actual control input.
Under this condition, both the policy-selected volitional signal and its effective state action are nonzero whenever operational ignorance holds. Within the tranquility set, the positive ignorance-conditioned transverse contribution vanishes, while tangential bodily, neural, evaluative, and feedback activity may continue.
1. The Canonical Pāli Formula and the Scope of This Theorem
The standard Pāli formula is avijjāpaccayā saṅkhārā, read as “with ignorance as condition, volitional formations come to be.” Saṅkhārā is plural. This paper interprets it as volitional formative activity, corresponding to Bhikkhu Bodhi’s widely used English rendering, volitional formations.
In the Pāli canon, however, avijjā is explained as not knowing suffering, the origin of suffering, the cessation of suffering, and the way leading to its cessation. SN 12.2 also defines saṅkhārā as three kinds of volitional formation: bodily, verbal, and mental. The present theorem does not reduce the full meanings of these terms to mere set nonmembership or to a single control component.
The following Avijjā₂₇ is a model-relative, operational notion of ignorance within the mathematical system of Theorem 26. Saṅkhārā₂₇ is not claimed to be coextensive with the entire canonical triad; it is a model-relative, ignorance-conditioned actuator contribution within the designated closed loop.
Canonical Pāli source: Saṃyutta Nikāya 12.2, Pāli text https://suttacentral.net/sn12.2/pli/ms?lang=jpn&layout=plain&reference=none¬es=asterisk&highlight=false&script=latin
Bhikkhu Bodhi’s English translation https://suttacentral.net/sn12.2/en/bodhi?lang=jpn&reference=none&highlight=false
2. Structures Inherited from Theorems 24 and 26
Let 𝕃 be the lattice of abstraction levels, with greatest element ⊤, interpreted as emptiness. The typed permanent-zero-suffering proposition PZS(a,x,T) from Theorem 24 means that some admissible policy, starting from x(T)=x, keeps the nonnegative evaluation Vₐ=0 Lebesgue-almost everywhere forever.
Minimal Dependencies
Dependency Content used in Theorem 27
Theorem 24 and Condition 24-A The below-⊤ branch of (27.2): the impossibility of permanent zero suffering and a positive optimal residual value
The standing assumptions and Conditions 26-A and 26-B of Theorem 26; Condition 26-C is not required The designated policy at the highest abstraction level, the zero-residual-suffering set, closedness, forward invariance, the Lyapunov sandwich bounds, and strict descent
Regularity assumptions in Section 3.2 Definition of the Dini derivative as a finite residual-descent rate
Condition 27-A Attribution of residual descent to the difference between the designated and reference inputs through the actual actuator
Conclusion
Within the present system, the 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 a positive Lyapunov ignorance residual and a strictly positive residual-descent rate.
When Condition 27-A is added, the volitional signal on the actual actuator that positively contributes to that descent is nonzero. Thus, within the explicitly specified model, the conditional implication corresponding to “with ignorance as condition, volitional formations arise” is established.
At the same time, tranquility is neither physical stasis nor the cessation of all activity. What vanishes is the positive residual contribution conditioned by operational ignorance; the tangential and relational activities of life and freedom may continue.
This theorem provides a model-theoretic counterpart to the canonical Pāli formula. It neither proves nor exhausts the full canonical meanings of avijjā and saṅkhārā. This is the minimal form of Theorem 27 that remains consistent with the dynamic tranquility of Theorem 26.
Theorem 27: Ignorance-Conditioned Volitional Formations (Avijjāpaccayā saṅkhārā)—Japanese Academic Edition https://tomabechi.jp/TomabechiAvijjaSankharaJA.html
Theorem 27: Ignorance-Conditioned Volitional Formations (Avijjāpaccayā saṅkhārā)—English Academic Edition https://tomabechi.jp/TomabechiAvijjaSankharaEN.html
