Tomabechi Cognitive Cosmology — Theorems 28

Abstract

This paper formulates five new theorems on the basis of Theorems 1–27. Theorem 28 (Nirvana Non-Self and Type Coherence Theorem) uses a tagged disjoint union to prove that Theorems 24 and 26 govern disjoint typed domains, then derives the non-self of the nirvana process from the relational functional completeness of Theorem 25 (Non-Self Theorem). Theorem 29 (Formal Dharma-System Non-Substantiality and Incompleteness Theorem), only under an explicit modeling of Dharma-as-teaching by sufficiently strong effective formal theories, proves Gödel–Chaitin limitations, open-ended extension, and non-self-nature under Theorem 25 (Non-Self Theorem). Theorem 30 (Entropy-Exchange Ego Constitution Theorem) models introspective language as a projection-compatible map on an inverse system, proving the selection and stabilization of a language-conditioned Ego configuration and a cognitive–physical entropy exchange. Theorem 31 (Introspective-Language Closure and Low-Abstraction Sixfold Recurrence Theorem) proves confinement to a low-abstraction TCZ under language closure, Nagumo invariance, and hybrid Lyapunov attraction; a six-state coarse-graining yields almost-sure recurrence. Theorem 32 (Future-TCZ Homeostasis Absolute-Other-Power Theorem) gives a bounded presence threshold that places an external goal in a future TCZ, followed by an exponential convergence and hitting-time bound under a future-limited Ego.

Within the restatement of the inherited theorems (§3), Theorem 15 (Cognitive-Physical Entropy Exchange and Conservation) is newly included as §3.7. At the lowest-abstraction physical layer the only thing that holds for entropy is the inequality of the second law; a conservation law as an equality comes into force only once the high-abstraction layers are included in the description of the state. This fact (Proposition 15.D) is presupposed by the exchange budget of Theorem 30 and is published here for the first time. Further, §7.3 delimits the scope of the name "sixfold recurrence" (the inessentiality of the number of regions, neutrality regarding rebirth together with the constraint imposed by Theorem 25, and its holding within a single lifetime), and §7.4 gives the structural correspondence between what Theorem 31 formalizes and the critique of language in Nāgārjuna's Mūlamadhyamakakārikā — both separated from the proofs as an interpretive layer.

Scope. The five results do not follow unconditionally from the earlier theory. Every section separates inherited assumptions, genuinely new conditions, and external metatheorems. Moreover, non-self does not forbid mathematical objectification. It denies an intrinsic variable that is independent of the relational description, fixed across histories, individuating, and causally nonredundant.

1. Conclusions and proof dependencies

The correct order is a partial order, not one linear chain. Theorems 28, 29, and 30 form independent branches. Theorem 31 (Introspective-Language Closure and Low-Abstraction Sixfold Recurrence Theorem) may use the language-conditioned Ego configuration of Theorem 30 (Entropy-Exchange Ego Constitution Theorem), while Theorem 32 (Future-TCZ Homeostasis Absolute-Other-Power Theorem) can explain a reconfiguration that breaks the closure described in Theorem 31 (Introspective-Language Closure and Low-Abstraction Sixfold Recurrence Theorem).

Base T24–26 → T28　|　Base T25 + Gödel/Chaitin → T29　|　Base T1, T15, T16, T18 → T30 → [optional] T31 → [escape interpretation] T32

New theorem	Minimal inherited basis	New conditions / external results	Proof status
28	T24, T25, T26	Condition 28-A places the nirvana process within T25's domain	Type consistency is a direct corollary; non-self applies T25
29	T25	29-A–C; Gödel I and II; Chaitin	A conditional metatheorem, not a consequence of T1–27 alone
30	T1, T15, T16, T18	Projection compatibility, contraction, entropy-type bridge	A new inverse-limit construction integrated with T15
31	T1, T3, T16, T18	Language closure, Nagumo condition, hybrid Lyapunov condition	Does not follow from T18's policy-set enlargement alone
32	T1, T4, T7–9	Presence threshold, reachable basin, strong Lyapunov contraction	A quantitative strengthening of T9

Relation to Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem). These theorem numbers follow Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem), but none of the five main proofs requires it. Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem)'s equivalence between operational ignorance and actuator contribution remains available when the results are embedded in a dependent-origination interpretation.

2. Inherited notation and logical scope

Let (𝒜,≼) be the abstraction poset with greatest element ⊤. Let X_a denote the state space at layer a, ℬ_alive⊆ X_⊤ the viability region, and J^*_a,ρ(x,T) the discounted optimal cost. The tranquility set of Theorem 26 (Nirvanic Tranquility Theorem) is

𝒩_⊤(T):=ℬ_alive∩{x| J^*_⊤,ρ(x,T)=0}.(2.1)

The predicate Atman(d,a) from Theorem 25 (Non-Self Theorem) asserts that phenomenon d, at layer a, possesses an intrinsic variable independent of its relational description, fixed across histories, individuating, and causally nonredundant. Hence ¬Atman does not say that the object cannot be represented as a set, a map, or a theory.

Three invalid inferences

A parameter T does not by itself imply that 𝒩_⊤(T) actually changes; a constant set-valued process is possible.
Relational definability or set-valuedness alone does not prove non-self. The decisive assumption is the functional completeness condition 25-D.
Gödel and Chaitin do not apply to every axiom system or to Dharma without qualification. Arithmetic strength, effectiveness, and soundness must be stated.

3. Formulation and Rigorous Proofs of the Inherited Theorems

Theorems 28 through 32 of this paper are built on existing theorems. So that the reader may follow the proofs from this paper alone, the standard-form central formula, the formulation used here, and a rigorous proof are given for each of Theorems 1, 3, 4, 7, 8, 9, 15, 16 and 18, which are used directly. For Theorem 24 (Universal Unsatisfactoriness Theorem), Theorem 25 (Non-Self Theorem) and Theorem 26 (Nirvanic Tranquility Theorem), the proofs are not reproduced; the reader is referred to the Four Dharma Seals paper (§3.10).

Status of this section

This section restates existing theorems in the notation of this paper and exhibits the skeleton of each proof. For §3.7 (Theorem 15), however, assumption A6(ii) of the source paper is weakened to the uniform-integrability condition A6′ (Lemma 15.1), and the upper bound on structuring (3.8) together with the irreducibility of the conservation law to the physical layer (Proposition 15.D) are newly established. The standard forms agree with the all-theorem summary of the Freedom volume; wherever this paper uses a different expression, the reason and the equivalence with the standard form are stated in the corresponding item.

3.0 The Common Tool — the Unified Convergence Lemma (Lemma 0)

Every proof below reduces to a single lemma.

If Φ ≥ 0 is absolutely continuous along the trajectory and satisfies D⁺Φ(t) ≤ −cΦ(t) with c > 0 outside Ω_θ, then Φ(t) ≤ Φ(0)e^−ct and dist(x(t),Ω_θ) → 0(3.0)

Here D⁺ is the upper right Dini derivative and Ω_θ is a sublevel set of Φ. Since Φ is absolutely continuous it agrees with the ordinary derivative almost everywhere, so Grönwall's inequality applies. Combining this with the quadratic sandwich c₁dist² ≤ Φ ≤ c₂dist² yields the exponential estimate in distance, dist(x(t),Ω_θ) ≤ √(c₂/c₁) e^−ct/2 dist(x(0),Ω_θ). Each theorem below is proved by constructing its own residual Φ and reducing to this lemma.

3.1 Theorem 1 (Tomabechi Main Theorem)

◆ Central formula (standard form)π_c = arg min ∫V₀ dt　⇒　x(t) → TCZmatches the standard form of the all-theorem summary in the Freedom volume

Why the expanded form is used here. The proof must state convergence to the reachable region in terms of distance, so the interval of integration, the arguments, and the restriction to reachability are made explicit.
Under the following correspondence, the formula below is the same assertion as the standard form.
∫V₀ dt ≔ ∫₀^TV₀(x(t),t) dt / TCZ ≔ TCZ₁^cl(t;x₀) (the closed slice restricted to the reachable region) / x(t) → TCZ ⇔ dist(x(t),TCZ₁^cl(t;x₀)) → 0

Theorem 1 (Tomabechi Main Theorem)

For a nonnegative evaluation V₀ set TCZ = {x | V₀(x,t) ≤ θ}. Suppose a closed-loop policy π_c generates solutions, V₀ is absolutely continuous along trajectories, and outside TCZ one has D⁺[V₀(x(t),t) − θ]⁺ ≤ −c[V₀ − θ]⁺. Then

dist(x(t),TCZ₁^cl(t;x₀)) → 0(3.1)

Proof

Set the residual Φ₁ := [V₀(x,t) − θ]⁺. It is nonnegative and, by hypothesis, satisfies D⁺Φ₁ ≤ −cΦ₁ outside TCZ. By Lemma 0 (3.0), Φ₁(t) ≤ Φ₁(0)e^−ct, hence Φ₁ → 0. Since Φ₁ = 0 is equivalent to V₀ ≤ θ, that is, to membership in TCZ, the distance to the closed reachable slice converges to zero. The substantive condition for convergence is not that the policy is an arg min, but that the closed loop selected by the arg min satisfies Lyapunov descent.∎

3.2 Theorem 3 (Abstract Shared TCZ Convergence Theorem)

◆ Central formula (standard form)A(x)=0 ⇔ φ(x)=LUB(W₁,…,W_N),　A(t)→0matches the standard form of the all-theorem summary in the Freedom volume

Why the expanded form is used here. The standard form gives a characterization of what it is to have reached the roof. This paper needs a policy that guarantees attainment, so it is written in policy form. The two state different layers.
A ≔ 𝒜_i (the abstraction residual) / LUB(W₁,…,W_N) ≔ L^* = ∨W_i / A(t)→0 ⇔ ‖ι(φ_i(x_i(t)))−ι(L^*)‖ → 0

Theorem 3 (Abstract Shared TCZ Convergence Theorem)

Suppose the coupling graph is connected, each misalignment cost S_ij is nonnegative, and the abstraction residual 𝒜_i vanishes exactly at L^* = ∨W_i. Then, under a closed loop minimizing the augmented Lagrangian ℒ_A = Σ_iV_i + Σ_(i,j)∈Eγ_ijS_ij + Σ_iη_i𝒜_i with η_i > 0,

𝒜_i(t) → 0,　that is,　φ_i(x_i(t)) → L^* = ∨W_i(3.2)

Proof

Set Φ₃ := ℒ_A − inf ℒ_A. By connectedness, any agent's deviation is reflected in some S_ij, so Φ₃ captures the total departure without leakage. By hypothesis Φ₃ satisfies the descent condition of Lemma 0, whence Φ₃ → 0. Every term of Φ₃ is nonnegative, so the sum tending to zero entails that each term does; in particular η_i𝒜_i → 0, hence 𝒜_i → 0. Since 𝒜_i is constructed to vanish only at L^*, we get φ_i(x_i(t)) → L^*. The LUB is not an average but the least upper bound enclosing every agent's world without discarding any of it.∎

3.3 Theorem 4 (Tomabechi Presence-Weighted Theorem)

◆ Central formula (standard form)Ṽ = V₀ − κPQ,　x → TCZ_Pmatches the standard form of the all-theorem summary in the Freedom volume

Why the expanded form is used here. The standard form states only the deformation and the destination. This paper uses the fact that the deformed closed loop satisfies the descent condition, so the policy π_c(P) is exhibited.
TCZ_P ≔ Ω_P(t) (the closed reachable slice after deformation) / x → TCZ_P ⇔ dist(x(t),Ω_P(t)) → 0

Theorem 4 (Tomabechi Presence-Weighted Theorem)

For a presence P ∈ [0,1], a value sign Q and κ > 0, set Ṽ := V₀ − κPQ. If Ṽ satisfies the requirements of a nonnegative evaluation and the closed loop of π_c(P) satisfies the descent condition of Lemma 0, then

dist(x(t),Ω_P(t)) → 0,　∂Ṽ/∂P = −κQ(3.3)

Proof

Taking Ṽ as the evaluation, form the residual Φ₄ := [Ṽ(x,t) − θ]⁺. Since Ṽ still satisfies the requirements of a nonnegative evaluation, Lemma 0 applies unchanged, so Φ₄ → 0 and therefore dist(x(t),Ω_P(t)) → 0. That the valley moves continuously with P follows by differentiating directly: ∂Ṽ/∂P = −κQ. Where Q > 0 the ground falls; where Q < 0 it rises. A change of behaviour is achieved not by sustained effort of will but by deformation of the landscape.∎

3.4 Theorem 7 (Tomabechi True Goal Theorem)

◆ Central formula (standard form)G ∉ TCZ₀,　d(G,TCZ₀) ≥ ε > 0,　G = Self-setmatches the standard form of the all-theorem summary in the Freedom volume

Theorem 7 (Tomabechi True Goal Theorem)

Let TCZ₀ be the present stable region and G a candidate terminal goal. Take the canonical conditions for a true goal to be: (i) externality, dist(G,TCZ₀) > ε; (ii) belonging to the agent's own goal set rather than being externally imposed; (iii) coherence with the higher Self, C_Self(G) > 0; and (iv) entering the control problem nondegenerately. These four conditions characterize the transformative goals admissible in this system.

J_G[u] = ∫_t^TV₀(x(τ),τ)dτ + λ d(x(T),G)²,　λ > 0(3.4)

Proof (logical)

The necessity of each condition is shown by what is admitted once it is dropped. Dropping (i) admits goals with dist(G,TCZ₀) ≤ ε, which are reachable without substantially reconstituting the present stable region and so lack the externality of a transformative goal. Dropping (ii) admits externally imposed states; dropping (iii) admits states of negative value or self-incoherence; dropping (iv) admits decorative goals that do not affect the policy. The four conditions therefore characterize exactly the intended class. Attainment does not follow from this; attainment is the business of Theorem 9 (Tomabechi Future-Origin Goal Attainment Theorem).∎

3.5 Theorem 8 (Tomabechi Future-Origin Cognitive Time Theorem)

◆ Central formula (standard form)u^* = arg min J_G; the terminal condition G determines the present controlmatches the standard form of the all-theorem summary in the Freedom volume

Theorem 8 (Tomabechi Future-Origin Cognitive Time Theorem)

In the finite-horizon optimal control problem (3.4) with terminal condition G, the optimal control u^* satisfies, by the principle of dynamic programming,

u^* = arg min_u J_G　⇔　−∂W_G/∂t = min_u{ V₀(x,t) + ∇W_G(x,t)·f(x,u,t) }(3.5)

That is, the control at the present instant is determined backwards from the terminal condition.

Proof

Introduce the value function W_G(x,t) := min_uJ_G[u]. By the principle of optimality, W_G satisfies the Hamilton–Jacobi–Bellman equation (3.5) with terminal condition W_G(x,T) = λd(x,G)². This equation is integrated backwards in t, so the optimal control u^*(t,x) at time t depends on the terminal condition G. It is not claimed that physical time runs backwards. It is the structural fact that, in an optimal control problem with a terminal condition, the direction of dependence of decisions runs from the future to the present. This system calls it future-origin cognitive time.∎

3.6 Theorem 9 (Tomabechi Future-Origin Goal Attainment Theorem)

◆ Central formula (standard form)K_G = PQ⁺ + EC_Self ≥ K_crit together with Lyapunov descent　⇒　x → TCZ_Gmatches the standard form of the all-theorem summary in the Freedom volume

Theorem 9 (Tomabechi Future-Origin Goal Attainment Theorem)

Define the goal drive strength K_G := PQ⁺ + EC_Self. If K_G ≥ K_crit and the goal residual Φ_G satisfies the descent condition of Lemma 0, then

dist(x(t),TCZ_G(t)) → 0(3.6)

Proof

Form the goal residual Φ_G := [Ṽ_G(x,t) − θ_G]⁺, where Ṽ_G is the evaluation deformed by the contribution of K_G in accordance with Theorem 4 (Tomabechi Presence-Weighted Theorem). The condition K_G ≥ K_crit is precisely the critical condition for a sublevel set to actually appear on the side of G in the deformed landscape; under it, Φ_G satisfies the descent condition of Lemma 0. Hence Φ_G → 0 and dist(x(t),TCZ_G(t)) → 0. Theorem 7 (Tomabechi True Goal Theorem) supplies eligibility, Theorem 8 (Tomabechi Future-Origin Cognitive Time Theorem) the direction of determination, and the present theorem attainment.∎

3.7 Theorem 15 (Cognitive-Physical Entropy Exchange and Conservation Theorem)

◆ CENTRAL FORMULA (STANDARD FORM)S_gen = S_phys + Σw_αH_α,　dS_gen/dt = Π ≥ 0Matching the standard form of the complete theorem list in the Freedom volume

Why an expanded form is used here. Condition 30-C of Theorem 30 (Entropy-Exchange Ego-Constitution Theorem) uses the exchange identity of this theorem layerwise and conditionally. The layer index α, the conversion weights w_α, the dissipation term Π and the time interval must therefore be made explicit. Under the following correspondence the formula below makes exactly the same claim as the standard form.
Σw_αH_α ≔ Σ_α≻0 w_αH_α(x_α(t)) (layer index and trajectory argument made explicit) / dS_gen/dt = Π ≥ 0 ⇔ S_gen(t₂) − S_gen(t₁) = ∫_t₁^t₂Π(s) ds ≥ 0 (integral form, by absolute continuity)

3.7.1 The Claim of This Section — at the Physical Layer Alone There Is No Conservation Law for Entropy

So long as one looks only at the lowest-abstraction physical layer α = 0, the only conservation law that holds as an equality is the conservation of energy. What holds for entropy is the second law of thermodynamics, that is, the inequality dS_phys/dt ≥ 0; no conservation law as an equality exists. What this theorem asserts is the following single point: in the cognitive-informational universe, once the high-abstraction layers are included, a conservation law as an equality does hold for entropy as well.

The conservation follows from the fact that the decrease of semantic entropy on the cognitive side and the increase of physical entropy on the physical side are exchanged within one and the same ledger, mediated by the conversion weights w_α. In an ideal closed reversible system, where the dissipation Π vanishes, the generalized total entropy S_gen is exactly conserved. Where dissipation is present, the increase is exactly the dissipated amount and no more.

Where the non-triviality of this theorem lies

It is a misreading to take this theorem as an identity obtained by substituting the exchange formula and cancelling. Its substantive content is threefold. First, that S_gen is absolutely continuous and that term-by-term differentiation is legitimate in a situation where the layers may be countably infinite (Lemma 15.1; this does not hold unconditionally, and a counterexample is given in §3.7.5). Second, that conservation is equivalent to the vanishing of dissipation (Lemma 15.2). Third, that this conservation law cannot be reduced to the physical layer, that is, the same conservation law cannot be constructed out of quantities of the physical layer alone (Proposition 15.D). The third point is the rigorous content of the claim opening this section: at the physical layer alone there is no conservation law for entropy.

3.7.2 Setting and Standing Assumptions

Assumption A1 (family of abstraction layers)

The abstraction index set 𝒜 ⊂ [0,∞) is a countable set containing 0, and to each α ∈ 𝒜 a measurable space (layer) U_α is assigned. The total space is the disjoint union U = ⊔_α∈𝒜U_α, and the layer U₀ at α = 0 is called physical space. The time evolution is given as a family of measurable trajectories t ↦ x_α(t) ∈ U_α for t ∈ [0,T].

Assumption A2 (regularity of semantic entropy)

For each α ≻ 0 a semantic entropy functional H_α : U_α → [0,∞) is given, and the composition along trajectories t ↦ H_α(t) := H_α(x_α(t)) is absolutely continuous on [0,T].

Assumption A3 (coherent family of projections)

For each α > β ≥ 0 a measurable projection π_β←α : U_α → U_β is given, satisfying the semigroup property π_γ←β ∘ π_β←α = π_γ←α for γ < β < α and π_α←α = id. The lowest-abstraction projection is abbreviated π₀ := π_0←α.

Assumption A4 (projection monotonicity — coarse-graining increases entropy)

For all α > β > 0 and x ∈ U_α, H_β(π_β←α(x)) ≥ H_α(x). Equality holds only when the projection is invertible, that is, loses no information.

Assumption A5 (regularity of physical entropy)

A physical entropy S_phys : [0,T] → ℝ is defined for the coupled subsystem at the physical layer and is absolutely continuous. S_phys carries the usual thermodynamic meaning and may by itself increase or decrease through exchange with the exterior.

Assumption A6′ (conversion weights and regularity of the sum — uniformly integrable form)

For each α ≻ 0 a constant conversion weight w_α > 0 is given, satisfying the following. (i) Σ_α≻0w_αH_α(t) is finite at every point of [0,T]. (ii) Writing h_α := dH_α/dt, the series Σ_α≻0w_αh_α(s) converges almost everywhere, and the family of finite partial sums {Σ_α∈𝒜′w_αh_α : 𝒜′ ⊂ 𝒜 finite} is uniformly integrable in L¹([0,T]).

Strengthening relative to assumption A6(ii) of the source paper. The source paper imposed a dominated-convergence condition through an integrable dominating function g = Σw_αg_α ∈ L¹. Domination by an integrable function implies uniform integrability, but not conversely. Hence A6′ is a strictly weaker assumption and widens the range of application of the theorem. In the proof, Vitali's convergence theorem is used in place of the dominated convergence theorem (Lemma 15.1). When 𝒜 is finite, (ii) holds trivially.

Assumption A7 (inter-layer coupling — the exchange identity)

Almost everywhere the following holds.

dS_phys/dt = −Σ_α≻0w_α dH_α/dt + Π(t),　Π(t) ≥ 0,　Π ∈ L¹([0,T])

Here Π is the dissipation term, representing irreversibility, thermalization, surplus generation, and interaction with the environment.

What assumption A7 actually contains — which part is definition and which is axiom

A7 consists of two parts. The first is a definition: setting Π(t) := dS_phys/dt + Σ_α≻0w_αdH_α/dt carries no content whatever. The second is the sign condition Π(t) ≥ 0, and this is the sole substantive axiom — the generalized second law. Making this decomposition explicit is necessary so that what the theorem assumes and what it proves are not left ambiguous. That is, granting the generalized second law, the theorem proves (i) that the total quantity is meaningfully differentiable, (ii) that conservation is equivalent to the vanishing of dissipation, and (iii) that this conservation cannot be reduced to the physical layer.

3.7.3 Theorem 15 and Its Rigorous Proof

Theorem 15 (Tomabechi Cognitive-Physical Entropy Exchange and Conservation Theorem)

Under the standing assumptions A1–A5, A6′ and A7, define the generalized total entropy by S_gen(t) := S_phys(t) + Σ_α≻0w_αH_α(t). Then the following hold.

(I) Exchange and the generalized second law. S_gen is absolutely continuous on [0,T], and

dS_gen/dt = Π(t) ≥ 0 (a.e.),　S_gen(t₂) − S_gen(t₁) = ∫_t₁^t₂Π(s) ds ≥ 0(3.7)

holds for every 0 ≤ t₁ ≤ t₂ ≤ T. In particular S_gen is non-decreasing.

(II) Conservation law. Π = 0 almost everywhere if and only if S_gen is constant on [0,T]. In particular, in an ideal closed reversible system with Π ≡ 0, S_gen(t) = S_gen(0) holds exactly for every t.

(III) Upper bound on the amount of structuring. For every 0 ≤ t₁ ≤ t₂ ≤ T,

Σ_α≻0w_α[H_α(t₁) − H_α(t₂)] = [S_phys(t₂) − S_phys(t₁)] − ∫_t₁^t₂Π(s) ds ≤ S_phys(t₂) − S_phys(t₁)(3.8)

holds, with equality precisely when Π = 0 almost everywhere on [t₁,t₂]. That is, the total amount of cognitive structuring achievable within an interval is bounded above by the increase of physical entropy over that same interval.

Lemma 15.1 (term-by-term differentiation — Vitali-strengthened form)

Under assumptions A2 and A6′, F(t) := Σ_α≻0w_αH_α(t) is absolutely continuous on [0,T] and satisfies dF/dt = Σ_α≻0w_αh_α almost everywhere.

Proof

Each H_α is absolutely continuous, so H_α(t) = H_α(0) + ∫₀^th_α(s) ds with h_α ∈ L¹. By A6′(i), F(0) is finite.

Fix an enumeration of the countable set 𝒜 and put F_n(t) := Σ_k≤nw_{α_k}H_{α_k}(t) and f_n(s) := Σ_k≤nw_{α_k}h_{α_k}(s). By the first half of A6′(ii), f_n → f := Σ_α≻0w_αh_α almost everywhere; by the second half, {f_n} is uniformly integrable. By Vitali's convergence theorem, f ∈ L¹([0,T]) and ‖f_n − f‖_L¹ → 0. Hence for each t,

∫₀^tf_n(s) ds ⟶ ∫₀^tf(s) ds

On the other hand, since H_α ≥ 0 and w_α > 0, the partial sums F_n(t) are non-decreasing in n and, by the finiteness in A6′(i), converge pointwise to F(t). Letting n → ∞ in F_n(t) = F_n(0) + ∫₀^tf_n(s) ds gives

F(t) = F(0) + ∫₀^tf(s) ds

The right-hand side is the indefinite integral of an L¹ function, hence absolutely continuous, and by Lebesgue's differentiation theorem dF/dt = f almost everywhere. ∎

Remark (the substance of the strengthening). The source paper imposed |f_n| ≤ g for an integrable dominating function g and applied the dominated convergence theorem. Domination by an integrable function implies uniform integrability but the converse is false, so this lemma is a genuine generalization of Lemma A.2.1 of the source paper. For instance, a family whose "bumps" travel along the time axis can be uniformly integrable while admitting no integrable dominating function.

Lemma 15.2 (characterization of conservation)

Under A1–A5, A6′ and A7, S_gen is constant on [0,T] if and only if Π = 0 almost everywhere.

Proof

(⇐) Immediate from dS_gen/dt = Π in Theorem 15(I) together with the Newton–Leibniz formula for absolutely continuous functions. (⇒) If S_gen is constant then 0 = S_gen(T) − S_gen(0) = ∫₀^TΠ(s) ds. Since Π ≥ 0 and the integral vanishes, the standard property of the Lebesgue integral gives Π = 0 almost everywhere. ∎

Lemma 15.3 (monotone composition along a descending chain — path independence)

Under A1, A3 and A4, for any descending chain α_k > α_k−1 > ⋯ > α₁ > 0 and any x ∈ U_{α_k}, setting x_j := π_{α_j←α_k}(x) gives H_α₁(x₁) ≥ H_α₂(x₂) ≥ ⋯ ≥ H_{α_k}(x). In particular the comparison does not depend on which intermediate layers are passed through.

Proof

By the semigroup property in A3, x_j = π_{α_j←α_j+1}(x_j+1). Applying A4 to each adjacent pair yields H_{α_j}(x_j) ≥ H_{α_j+1}(x_j+1) for j = 1,…,k−1, and chaining gives the stated chain of inequalities. Path independence holds because, by the semigroup property, the composite projection along any two routes coincides with the same π_{α_j←α_k}. ∎

Proof of Theorem 15

(I) By A5, S_phys is absolutely continuous; by Lemma 15.1, so is F = Σ_α≻0w_αH_α. A sum of absolutely continuous functions is absolutely continuous, so S_gen = S_phys + F is absolutely continuous and almost everywhere

dS_gen/dt = dS_phys/dt + Σ_α≻0w_αdH_α/dt

Substituting the exchange identity A7, the term −Σ_α≻0w_αdH_α/dt contained in the first summand cancels against the second, giving dS_gen/dt = Π(t). Since Π ≥ 0 almost everywhere by A7, S_gen is non-decreasing, and the integral form (3.7) follows from the Newton–Leibniz formula for absolutely continuous functions.

(II) This is Lemma 15.2.

(III) Expanding S_gen = S_phys + Σw_αH_α in the integral form of (3.7) gives

[S_phys(t₂) − S_phys(t₁)] + Σ_α≻0w_α[H_α(t₂) − H_α(t₁)] = ∫_t₁^t₂Π(s) ds

Moving the second summand to the right-hand side and rearranging signs gives the equality part of (3.8). The inequality follows from ∫Π ≥ 0, and equality holds exactly when ∫_t₁^t₂Π = 0, which together with Π ≥ 0 is equivalent to Π = 0 almost everywhere on [t₁,t₂]. ∎

3.7.4 The Conservation Law Cannot Be Reduced to the Physical Layer

Parts (I) and (II) above are consequences drawn once the exchange identity is granted. But the claim that opened this section — that at the physical layer alone there is no conservation law for entropy — has not yet been proved. The following proposition supplies it.

Proposition 15.D (irreducibility to the physical layer)

(a) Non-triviality of the conservation. There exists a system satisfying A1–A5, A6′ and A7 with Π ≡ 0 for which nevertheless dS_phys/dt > 0 on a set of positive Lebesgue measure. That is, even in an ideal closed reversible system, the physical entropy alone is not conserved. What is conserved is S_gen alone.

(b) Irreducibility. If a Borel measurable 𝔉 : ℝ → ℝ renders t ↦ 𝔉(S_phys(t)) constant on [0,T] for every system satisfying A1–A5, A6′ and A7 with Π ≡ 0, then 𝔉 is constant on the interval [S_phys(0), ∞). Hence no non-trivial conserved quantity can be constructed as a function of the physical entropy alone.

Proof

(a) Construct an explicit witness. Take 𝒜 = {0, 1}, T = 1, w₁ = 1, and let H₀ > 0 and λ > 0 be constants; set

H₁(t) := H₀e^−λt,　S_phys(t) := S_phys(0) + H₀(1 − e^−λt),　Π ≡ 0

H₁ is C^∞, non-negative and bounded on [0,1], so A2 holds. S_phys is likewise C^∞, so A5 holds. Since there is only one positive layer, A6′(i) and (ii) hold trivially (a finite sum), and A4 holds vacuously because no pair with α > β > 0 exists. A3 is satisfied by taking π_0←1 to be any measurable map and π_α←α = id. Finally A7 is verified:

dS_phys/dt = λH₀e^−λt,　−w₁dH₁/dt = λH₀e^−λt

so dS_phys/dt = −w₁dH₁/dt + 0 holds for every t, with Π ≡ 0 ≥ 0. In this system dS_phys/dt = λH₀e^−λt > 0 on all of [0,1], a set of measure one. On the other hand

S_gen(t) = S_phys(0) + H₀(1 − e^−λt) + H₀e^−λt = S_phys(0) + H₀

is a constant independent of t. So S_gen is conserved while S_phys strictly increases. This is the witness for (a).

(b) Let the witness family of (a) range over H₀ > 0. For each H₀ the map t ↦ S_phys(t) is continuous and strictly increasing, and as t runs over [0,1] its range covers exactly the interval [S_phys(0), S_phys(0) + H₀(1 − e^−λ)]. By hypothesis 𝔉(S_phys(t)) is constant along this trajectory, so 𝔉 takes the single value 𝔉(S_phys(0)) on that whole interval. Fixing λ and letting H₀ ↑ ∞, the interval length H₀(1 − e^−λ) is unbounded above, so the union of these intervals equals [S_phys(0), ∞). Hence 𝔉 is constant, equal to 𝔉(S_phys(0)), on [S_phys(0), ∞). ∎

The meaning of Proposition 15.D — what this theorem newly asserts

Proposition 15.D(a) shows that even under the conditions most favourable to conservation, namely an ideal closed reversible system, the physical entropy is not conserved by itself. Proposition 15.D(b) goes one step further and shows that no function whatever of the physical entropy can serve as a conserved quantity. A conserved quantity must necessarily involve the quantities H_α of the high-abstraction layers.

Accordingly, all that can be said about entropy at the lowest-abstraction physical layer is the inequality of the second law — the only conservation law holding there as an equality is the conservation of energy — and a conservation law for entropy comes into force only once the high-abstraction layers are included in the description of the state. This is the novelty of the present theorem. It is one of the central theorems of the Tomabechi cognitive-physics mathematical system, and it has until now been withheld from publication. Physics possesses a law of the conservation of energy, but for entropy no conservation law exists — what holds there is only the inequality of the second law. That entropy is exchanged and conserved once the scope is extended to the cognitive universe is asserted here for the first time, and proved rigorously. The series of Dharma-mathematical results running from Theorem 24 (Universal Unsatisfactoriness) to Theorem 32 (Future-TCZ Homeostasis Absolute-Other-Power) can be formulated rigorously only on the basis of the present theorem, which treats cognitive ordering and physical dissipation in one and the same ledger.

3.7.5 Minimality of A6′ — a Condition That Cannot Be Dropped

A6′(ii) is not a technical convenience; dropping it makes the conclusion fail. Here is a counterexample. Suppose that for each α = k (k = 1,2,…) the function H_k(t) carries an oscillation of amplitude a_k = 2^−k and frequency f_k = 4^k. Then Σw_kH_k(t) itself converges uniformly and is finite and continuous at every point, yet the sum of the absolute values of the derivatives diverges at the order of Σw_ka_kf_k = Σw_k2^k and admits no uniformly integrable family of partial sums. In that case S_gen is not even absolutely continuous — the total variation diverges, a Takagi-function-type pathology — and "the sum of the derivatives of the terms" does not agree with "the derivative of the sum." The cancellation in the exchange identity A7 loses its meaning.

If 𝒜 is finite this pathology cannot arise and A6′(ii) holds trivially. Thus A6′(ii) is a substantive constraint only when countably infinitely many layers are treated.

3.7.6 Connection to Theorem 30

This paper uses Theorem 15 in §6, for Theorem 30 (Entropy-Exchange Ego-Constitution Theorem). Condition 30-C realizes the semantic entropy H_α of this section as the conditional Shannon entropy H(Z_a|𝒢_t) given the information 𝒢_t generated by lower-layer observations and the linguistic history, and connects the units by a positive constant factor c_a. Under this connection, equations (30.9) and (30.10) of Theorem 30 are nothing other than special cases of (3.7) and (3.8) of the present theorem: the decrease on the cognitive side, Σw_aI(Z_a;M|𝒢_t₁), is the amount of structuring on the left of (3.8), and the increase on the physical side is the right-hand side of the same identity.

What would be overstating the case

This theorem says nothing about the concrete values of the conversion weights w_α; the proof uses only their positivity and constancy. The physical intuition that erasing information carries a minimum energy cost suggests a natural scale for w_α, but the Landauer bound must not be applied directly to information acquisition in general; doing so requires a physical erasure process to be specified separately. Nor does this theorem assert that semantic entropy actually decreases. It is a conditional balance law: if a decrease occurs, it is tied to the increase on the physical side.

3.8 Theorem 16 (Self-Consciousness Existence and Emergence Theorem)

◆ Central formula (standard form)SC = lim_←TCZ_α ≠ ∅,　F_SC(S^*) = S^*,　M(S^*) represents S^*matches the standard form of the all-theorem summary in the Freedom volume

Why the expanded form is used here. That the self-image depends on history is central to this paper, in particular to Theorem 30 (Entropy-Exchange Ego Constitution Theorem), so the subscripts for agent i and history h are not dropped.
SC ≔ SC_i,h / TCZ_α ≔ K_i,α(h) (the candidate set at layer α) / F_SC ≔ F_i,h

Theorem 16 (Self-Consciousness Existence and Emergence Theorem)

Suppose each candidate set K_i,α(h) at abstraction layer α is nonempty and compact and the projections between layers are continuous and coherent, and suppose the self-reflecting map F_i,h is a contraction with ratio q < 1. Then

SC_i,h = lim_←K_i,α(h) ≠ ∅,　∃!S_i,h^* : F_i,h(S_i,h^*) = S_i,h^*,　d_SC(FⁿS₀,S_i,h^*) ≤ qⁿd_SC(S₀,S_i,h^*)(3.9)

Proof

Existence. Each K_i,α(h) is a nonempty compact Hausdorff space and the projections are continuous; the inverse limit of an inverse system of compact spaces is nonempty (a standard consequence of Tychonoff's theorem). Hence SC_i,h ≠ ∅. Uniqueness. SC_i,h is a complete metric space and F_i,h is a contraction with ratio q < 1, so by the Banach fixed-point theorem a fixed point exists and is unique, and the iteration estimate d_SC(FⁿS₀,S^*) ≤ qⁿd_SC(S₀,S^*) follows from the same theorem. Existence comes from topology, uniqueness from the fixed-point theorem; the provenance differs. Moreover the fixed point depends on the history h, so what has been shown to exist is a history-relative self-image, not a universal substance.∎

3.9 Theorem 18 (Introspective Language Evolution Theorem)

◆ Central formula (standard form)H(Z|Y,M_ℓ) < H(Z|Y);　Π₀ ⊆ Π_ℓ;　∂ℱ/∂ℓ > 0matches the standard form of the all-theorem summary in the Freedom volume

Theorem 18 (Introspective Language Evolution Theorem)

If the introspective language M_ℓ carries nontrivial information about Z given the observation Y, then

H(Z|Y,M_ℓ) < H(Z|Y),　Π₀ ⊆ Π_ℓ,　∂ℱ/∂ℓ > 0(3.10)

Proof

Information. By monotonicity of conditional entropy, H(Z|Y,M_ℓ) ≤ H(Z|Y) always holds, with equality exactly when I(Z;M_ℓ|Y) = 0, that is, when M_ℓ carries no information about Z given Y. The nontriviality hypothesis excludes this, so the inequality is strict. Control. An agent possessing the introspective language can imitate every policy of an agent lacking it, simply by ignoring M_ℓ; hence Π₀ ⊆ Π_ℓ. Evolution. A supremum over a larger policy set cannot be smaller, so the capacity for free will ℱ is nondecreasing in ℓ, and strictly increasing under nontriviality.∎

3.10 Theorems 24, 25 and 26 — Reference to the Four Dharma Seals Paper

For Theorem 24 (Universal Unsatisfactoriness Theorem), Theorem 25 (Non-Self Theorem) and Theorem 26 (Nirvanic Tranquility Theorem), the proofs are not reproduced here; the reader is referred to the Four Dharma Seals paper. What this paper uses is the following three items.

Content inherited from the Four Dharma Seals paper

Theorem 24 (Universal Unsatisfactoriness Theorem). Under Condition 24-A (below emptiness there is no policy maintaining V_a = 0 almost everywhere forever) and the standing hypotheses (measurability of the trajectory evaluation, existence of at least one finite-cost policy, attainment of the minimum), a ≺ ⊤ implies J^*_a,ρ(x,T) > 0.
Theorem 25 (Non-Self Theorem). Under Conditions 25-B through 25-D, in particular relational functional completeness 25-D, there exists no intrinsic variable that is independent of relational description, fixed through history, individuating, and causally nonredundant. That is, ¬Atman(d,a).
Theorem 26 (Nirvanic Tranquility Theorem). Under the single Condition 26-A (zero-suffering set and stability), 𝒩_⊤(T) = ℬ_alive ∩ {x | J^*_⊤,ρ(x,T) = 0} is nonempty, closed and forward invariant, and the quadratic sandwich (26.A) and the strict descent (26.B) hold. This paper uses (26.A) and (26.B); (26.C) is not needed.

The proofs of these three theorems are in that paper, and this paper uses only their conclusions as premises. This paper does not reprove the claims of the Four Dharma Seals paper.

4. Theorem 28 — Nirvana Without Self and Type Consistency

4.1 Tagged domains

To prevent points at different layers from being conflated, define the disjoint union

𝔛:=(⊔_a≺⊤{a}× X_a) ⊔({⊤}×ℬ_alive), 𝔇_<⊤:=⊔_a≺⊤{a}× X_a, Nir(T):={⊤}×𝒩_⊤(T).(28.1)

Condition 28-A (relational functional completeness for the nirvana process)

Include the set-valued process d_N:=(T↦𝒩_⊤(T)) in the phenomenon index set 𝔇 of Theorem 25 (Non-Self Theorem). Its relational state Γ_{d_N,⊤} contains at least

Θ_N=(X_⊤,ℬ_alive,V_⊤,ρ,Pol_⊤,f_⊤,π⁰_⊤),(28.2)

the family 𝒩_⊤^Θ_N(T), histories, cross-layer relations, and relations with other beings, and it satisfies Condition 25-D for any candidate intrinsic variable Σ_N. Conditions 25-B–D hold for the expanded phenomenon index set after adjoining d_N.

Theorem 28 (Tomabechi Nirvana-Without-Self and Type-Consistency Theorem)

Assume Condition 24-A and the standing hypotheses of Theorem 24 (measurable trajectory evaluation, at least one finite-cost policy, and attainment of the minimum), Conditions 25-B–D, the standing hypotheses and Condition 26-A of Theorem 26 (Nirvanic Tranquility Theorem), and Condition 28-A. Then, for every T≥0 and every (a,x)∈𝔛,

𝔇_<⊤∩Nir(T)=∅,(28.3)

(a,x)∈𝔇_<⊤⇒ J^*_a,ρ(x,T)>0, (⊤,x)∈Nir(T)⇒ J^*_⊤,ρ(x,T)=0.(28.4)

On the tagged universe, perfect-zero suffering satisfies

PZS(a,x,T)⇔[a=⊤∧x∈𝒩_⊤(T)],(28.5)

and the nirvana process satisfies

¬Atman(d_N,⊤).(28.6)

If the full cross-layer hypotheses of Theorem 25 (Non-Self Theorem) are inherited for d_N, then ∀ a∈𝒜 ¬Atman(d_N,a).

Proof

Equation (28.3) follows because the tags a≺⊤ and a=⊤ are mutually exclusive. The left implication in (28.4) is Theorem 24 (Universal Unsatisfactoriness Theorem); the right one is the definition of 𝒩_⊤(T). Thus Theorem 24 (Universal Unsatisfactoriness Theorem) quantifies only below the top, whereas Theorem 26 (Nirvanic Tranquility Theorem) assigns zero-cost tranquility only at the top. They never assert positive and zero cost of one typed point. Lifting the PZS classification of Theorem 26 (Nirvanic Tranquility Theorem) to the disjoint union gives (28.5).

For non-self, if intervention on an independent candidate Σ_N changes the post-intervention law of (Γ_{d_N,⊤},Y⁺_{d_N}), Condition 25-D is violated. If it does not, Σ_N is causally redundant and fails the nonredundant-individuation clause of Atman. Both cases exclude Atman(d_N,⊤). ∎

Mathematical consequence

“All conditioned existence is suffering” and “nirvana is tranquility” are not contradictory propositions about one untyped point. They govern disjoint typed domains. Nirvana, represented by a zero-cost set process without a fixed essence beyond its relational specification, falls under Theorem 25 (Non-Self Theorem)'s non-self predicate.

Doctrinal reading. Within the model, the result is consistent with the commentarial reading that includes the supramundane within non-self. The mathematics does not adjudicate the historical merits of Sarvāstivāda's three-clause reorganization or competing Theravāda readings; it establishes typed exclusivity and, under 28-A, non-self.

Permitted and excessive claims

𝒩_⊤(T) remains a legitimate mathematical set. What is excluded is an independent, fixed, causally nonredundant essence beyond its relational specification. Actual temporal variation requires the additional condition ∃ T₁,T₂:𝒩_⊤(T₁)≠𝒩_⊤(T₂).

5. Theorem 29 — Formal Dharma-System Non-Self-Nature and Incompleteness

5.0 Justification of the Modelling — Why the Dharma May Be Treated as a Formal Theory

The conclusions of this section depend on external metatheorems. It is therefore necessary to answer first whether the dharma as doctrine may be modelled as a formal theory at all. This question is not settled inside mathematics. What follows delimits the scope of the modelling and states the grounds on which that delimitation is met; it does not assert that the dharma in general is formalizable.

What is modelled and what is not

What this section models is a body of doctrine stated as propositions and closed under a specified inference relation, written 𝕋_h. Three things lie outside the modelling. First, practice, training and experience themselves. Second, whatever is held to exceed linguistic formulation (in the vocabulary of this system, the ineffable aspect at the highest abstraction level). Third, instruction consisting solely of practical directives without propositional closure. The conclusions of this section say nothing about any of these three.

Condition 29-A comprises effectiveness, arithmetical strength and soundness. It is shown below that each is not an arbitrary convenience but a necessary condition for a doctrine to function as a doctrine.

(a) Effectiveness — the axiom set is recursively enumerable

If it cannot be decided or enumerated by a finite procedure whether a given proposition belongs to the doctrine, that doctrine can be neither taught, nor transmitted, nor checked. A system in which what was taught cannot be fixed cannot be preserved by later generations and cannot be distinguished from divergent teaching. Conversely, the historical fact that canonical compilation, councils and commentarial traditions exist shows that the doctrine has been handled in an enumerable form. Effectiveness is therefore the very condition under which a dharma can be transmitted at all, and dropping it amounts to abandoning transmissibility.

(b) Arithmetical strength — interpretability of Robinson arithmetic Q

Finite sequences and recursion occur within the doctrine. The twelve links form an ordered sequence of twelve terms; the aggregates, sense bases and noble truths are enumerations of finitely many items; forward and cessation contemplation are repeated advance and retreat along a sequence. In this system there is in addition recursion along the lattice of abstraction, in the subsumption updates of Theorem 22 (High-Altitude LUB Presence Theorem). To express these internally one needs only the successor function, addition, multiplication and the basic order axioms — that is, exactly Q. Conversely, a system that cannot interpret Q cannot speak of its own enumerative structure. Arithmetical strength is the minimal requirement for a doctrine to state its own structure internally.

(c) Soundness — proved arithmetical statements are true

A system that proves falsehoods about its own finite combinatorics is self-refuting: one that could prove "the links are thirteen in number" would lose the sense of its own enumeration. Soundness is the minimal demand that a doctrine not err about itself. Note that the first two conclusions of Theorem 29 (Formal Dharma-System Non-Substantiality and Incompleteness Theorem) do not in fact require soundness: consistency suffices for the second incompleteness theorem, and ω-consistency or a Rosser-style argument suffices for the first. Imposing soundness is a strengthening adopted for simplicity of statement, not an essential restriction.

Explicit delimitation of applicability

Theorem 29 (Formal Dharma-System Non-Substantiality and Incompleteness Theorem) does not apply to a system failing any of the three conditions: (i) a system whose axiom set is not enumerable; (ii) a system too weak in expressive power to interpret Q; (iii) a system that errs about its own finitary statements. The theorem does not claim that every dharma is incomplete. It claims only the conditional: for a formalized doctrinal system satisfying Condition 29-A, incompleteness follows.

(d) What the theorem does not claim

Three points are stated explicitly. First, it does not claim that the doctrine is false. Incompleteness is not falsity; the existence of undecidable sentences does not mean that the system contains falsehoods. Second, it does not claim that the doctrine is inconsistent. The second incompleteness theorem says that consistency cannot be proved internally, not that consistency fails. Third, it says nothing about the value of the dharma as a religion. The scope of the theorem is confined to the formalized part possessing propositional closure.

(e) Relation to Theorem 25 (Non-Self Theorem) — this is not a new metaphysical claim

The non-substantiality derived here is an instance of the relational functional completeness of Theorem 25 (Non-Self Theorem), applied to one particular kind of object, namely a formal theory. That is: once its relations to what lies outside it — a metatheory, a stronger system, additional axioms — are removed, the formal theory 𝕋_h possesses no self-grounding in its own consistency. In lacking any self-contained grounding severed from relation, this has exactly the structure that Theorem 25 (Non-Self Theorem) calls non-self. This section introduces no new metaphysics; it applies an existing theorem to one object.

An anticipated objection, and the response. One expects the objection that the dharma is not a formal system and that this is a category error. The response is as follows. The theorem does not assert an identity between the dharma and a formal system. Its assertion is conditional: if a body of doctrine satisfies Condition 29-A, then the following holds. A position that declines to regard the dharma as a formal system therefore does not conflict with the theorem. What the force of the theorem is directed at is rather the claim that a doctrine is complete and grounded wholly within itself. Such a claim simultaneously requires that the doctrine be teachable (effectiveness) and able to state its own enumerative structure (arithmetical strength), and hence falls under Condition 29-A. What the theorem denies is not the dharma, but claims of closure made about it.

Theorem 25 (Non-Self Theorem) already bears the title “All Dharmas Are Without Self.” This section therefore uses a different title and distinguishes phenomena as dharmas from Dharma-as-teaching represented by a formal theory. The identity “Dharma is an axiom system” is not derived; it is the following modeling condition.

For each historical stage h∈ℕ, let

𝕋_h=(L_h,Ax_h,⊢_h), Th(𝕋_h)=Cn_{⊢_h}(Ax_h).(29.1)

Condition 29-A (effective, arithmetical, sound formalization)

Each L_h is the language of arithmetic or a computable definitional extension and has a computable coding. Ax_h is computably enumerable, and proof checking is effective. 𝕋_h contains sufficient arithmetic, such as IΣ₁, is sound in the standard natural-number model ℕ at least for arithmetical sentences, has a standard provability predicate satisfying the Hilbert–Bernays–Löb derivability conditions, and can formalize K_U for one fixed prefix-free universal machine U.

Condition 29-B (application of Theorem 25 (Non-Self Theorem))

Include d_𝕋:=(h↦𝕋_h) in 𝔇. Its relational state functionally includes the language, axioms, inference rules, coding, carrier, history, and use-effects, and satisfies 25-D.

Condition 29-C (open-ended update rule)

Use the same arithmetical language L, proof system ⊢, and universal machine U at every stage, and let 𝕋₀=(L,Ax₀,⊢) satisfy 29-A. At every finite stage choose, in the metatheory, a sentence G_n true in ℕ but unprovable in 𝕋_n, and set

Ax_n+1:=Ax_n∪{G_n}, 𝕋_n+1:=(L,Ax_n+1,⊢), Th(𝕋_n+1)=Cn_⊢(Ax_n∪{G_n}).(29.2)

Theorem 29 (Tomabechi Formal-Dharma-System Non-Self-Nature and Incompleteness Theorem)

For each 𝕋_h satisfying 29-A:

There exists a true undecidable sentence: ∃ G_h[ℕ⊨ G_h∧𝕋_h⊬ G_h∧𝕋_h⊬¬ G_h].(29.3)
The theory does not internally prove its own consistency: 𝕋_h⊬Con(𝕋_h).(29.4)
There is a theory- and machine-dependent constant c_h,U such that ∀ s∈{0,1}^* ∀ n∈ℕ (n>c_h,U): 𝕋_h⊬ ``K_U(s̄)>n̄''.(29.5)

Under 29-B, ∀ a∈𝒜 ¬Atman(d_𝕋,a). Under 29-C,

Th(𝕋₀)⊊Th(𝕋₁) ⊊⋯⊊Th(𝕋_n)⊊⋯(29.6)

is a strictly increasing, stagewise sound chain in which incompleteness recurs at every finite stage; no finite stage is a final complete theory. In particular, the discrete impermanence predicate

Anicca_step(d_𝕋):⇔ ∀ n [Th(𝕋_n+1)≠Th(𝕋_n)](29.7)

is true. This is impermanence of the update process supplied by 29-C, not a consequence of Gödel or Chaitin alone.

Proof

Equation (29.3) is Gödel's first incompleteness theorem under the effectiveness, arithmetic strength, and soundness of 29-A. Equation (29.4) is Gödel's second incompleteness theorem under consistency and the standard derivability conditions.

For (29.5), suppose no uniform ceiling existed. Given arbitrarily large input n, enumerate the proofs of 𝕋_h, find the first theorem of the form ``K_U(s̄)>m'' with m≥ n, and output s. The resulting program has length c_h+K(n)+O(1)=c_h+O(log n), where c_h is the fixed description length of the proof enumerator. Soundness yields K_U(s)>m≥ n. For sufficiently large n, however, c_h+O(log n)<n, a contradiction. Hence a uniform ceiling c_h,U exists.

The non-self-nature clause follows by applying the same 25-D dichotomy used in Theorem 28 (Nirvana Non-Self and Type Coherence Theorem) to d_𝕋. For 29-C, adjoining a true sentence preserves standard-model soundness. Since G_n∉Th(𝕋_n) but G_n∈Th(𝕋_n+1), each inclusion is strict. Every successor again satisfies 29-A, so incompleteness reappears by induction. ∎

What logic does not prove here

“Every axiom system is incomplete” is false; weak decidable systems and noneffective complete truth sets are outside the scope.
Incompleteness alone does not produce temporal change; the update is supplied by 29-C.
“The a priori is refuted” is a philosophical interpretation. The mathematical content is non-completeness, a self-certification limit, and an internal complexity-proof ceiling.
Gödel and Chaitin alone do not prove Buddhist non-self. That clause comes from Theorem 25 (Non-Self Theorem) and 29-B.

6. Theorem 30 — Entropy-Exchange Ego Construction

Theorem 1 (Tomabechi Main Theorem) already types Ego as a control policy; Theorem 16 (Self-Consciousness Existence and Emergence Theorem) already constructs a self-consciousness inverse limit and fixed point; and Theorem 18 (Introspective Language Evolution Theorem) presupposes that structure when defining internal language. “Generation” below therefore means the selection and stabilization of a history-relative Ego configuration, not the creation of a subject from nothing.

For subject i, write the inverse system of Theorem 16 (Self-Consciousness Existence and Emergence Theorem) as

K_a:=TCZ_i,a, p_ba:K_a→ K_b(b≺ a), SC_i:=lim_←_a K_a.(30.1)

Let SC_i be a nonempty compact convex subset of a Banach space (hence a complete metric space), with the inherited feedback F_i:SC_i→ SC_i.

Condition 30-A (projection-compatible, nontrivial language contraction)

For each introspective-language string m∈Σ_i^*, there is a continuous map L_a,m:K_a→ K_a, with L_a,ε=id_{K_a}, such that

p_ba∘ L_a,m=L_b,m∘ p_ba.(30.2)

Let L_m:SC_i→ SC_i be the induced map and G_m:=F_i∘ L_m. In the complete metric d_SC of Theorem 16 (Self-Consciousness Existence and Emergence Theorem), assume Lip(F_i)=q_F<1. For each nonempty m≠ε, assume Lip(L_m)=q_m, q_Fq_m<1, and, for the empty-language fixed point S^*_ε,

G_m(S^*_ε)≠ S^*_ε.(30.3)

Condition 30-B (attainment of the Ego policy)

The language-expanded policy space Π_i^(ℓ) is compact and J_i(π;S) is lower semicontinuous in π, so that

π^(ℓ),*_c,m∈*arg min_{π∈Π_i^(ℓ)}J_i(π;S_m^*)(30.4)

is attained.

Condition 30-C (Shannon-to-state-entropy bridge)

Represent uncertainty about the history-relative, language-conditioned Ego configuration by a random variable Z:Ω→ SC_i and coordinates Z_a:=pr_a∘ Z. Let each Z_a be finite- or countable-valued (or subjected to a fixed quantization), with the following Shannon entropies finite and nonnegative. Let 𝒢_t be the information generated by lower-layer observations and introspective language through time t. During an update of one fixed target Z, connect the state entropy of Theorem 15 (Cognitive-Physical Entropy Exchange and Conservation Theorem) and the conditional Shannon entropy of Theorem 18 (Introspective Language Evolution Theorem) by

H_a⁽¹⁵⁾(x_a(t))=c_aH(Z_a|𝒢_t), c_a>0,(30.5)

and henceforth redefine w_a← w_ac_a. Conditions A6 and A7 of Theorem 15 (Cognitive-Physical Entropy Exchange and Conservation Theorem) apply to these same quantities.

Theorem 30 (Tomabechi Entropy-Exchange Ego-Construction Theorem)

Under Theorems 15, 16, and 18 and Conditions 30-A–C, every nonempty introspective string m selects a unique fixed point S_m^*∈ SC_i, and

d_SC(G_mⁿS₀,S_m^*)≤(q_Fq_m)ⁿ d_SC(S₀,S_m^*)(30.6)

for every S₀∈ SC_i. Moreover, S_m^*≠ S^*_ε, and

𝔈_i[m]:=(S_m^*,M(S_m^*),π^(ℓ),*_c,m)(30.7)

is a language- and history-relative Ego configuration.

If new introspective-language information M_(t₁,t₂] is added so that 𝒢_t₂=𝒢_t₁∨σ(M_(t₁,t₂]), define

ℋ_ego(t):=Σ_a≻0w_aH(Z_a|𝒢_t).(30.8)

Writing Δ X:=X(t₂)-X(t₁), one has

Δℋ_ego=-Σ_a≻0w_aI(Z_a;M_(t₁,t₂]|𝒢_t₁)≤0,(30.9)

Δ S_phys=Σ_a≻0w_aI(Z_a;M_(t₁,t₂]|𝒢_t₁)+∫_t₁^t₂Π(s) ds≥0.(30.10)

If at least one weighted conditional mutual information is positive, Δ S_phys>0. In the ideal exchange case Π=0, the weighted cognitive ordering equals the physical entropy increase.

Proof

For S=(s_a)_a∈ SC_i, equation (30.2) gives p_ba(L_a,ms_a)=L_b,m(p_bas_a)=L_b,ms_b. Hence L_mS is again coherent, and G_m is a self-map of SC_i. Since q_Fq_m<1, Banach's fixed-point theorem gives uniqueness and (30.6). Equality S_m^*=S^*_ε would contradict (30.3). Condition 30-B yields the policy in (30.4), making (30.7) type-consistent.

The conditional-entropy chain rule gives

H(Z_a|𝒢_t₂)=H(Z_a|𝒢_t₁)-I(Z_a;M_(t₁,t₂]|𝒢_t₁).(30.11)

Taking the regular weighted sum gives (30.9). Condition 30-C identifies it with the entropy ledger of Theorem 15 (Cognitive-Physical Entropy Exchange and Conservation Theorem). Integrating A7 over the interval and substituting (30.9) gives (30.10). Nonnegativity of conditional mutual information and Π completes the proof. ∎

Compatibility with non-self. Uniqueness is conditional on a fixed language string m and history; it is not an absolute fixed Ego. Condition (30.3) explicitly permits different language histories to select different fixed points, keeping the configuration inside the history-relative relational structure of Theorem 25 (Non-Self Theorem).

A moving target

If the self-state itself evolves during the interval, write Ḣ_a=σ_a-ι_a, where σ_a is newly generated uncertainty and ι_a the language-information rate. Net structuring requires Σ_aw_a(ι_a-σ_a)>0; positive information gain in Theorem 18 (Introspective Language Evolution Theorem) alone does not imply it. Nor may Landauer's bound be attached to information acquisition in general; a physically irreversible erasure protocol must be specified separately.

7. Theorem 31 — Introspective-Language Closure and Low-Abstraction Six-Realm Recurrence

The informal name is the Mathematical Language-Trap Theorem. The inclusion Π⁰⊆Π^(ℓ) in Theorem 18 (Introspective Language Evolution Theorem) says only that language enlarges the policy set; it does not guarantee escape to a higher abstraction. Confinement requires closure of both language jumps and continuous dynamics.

7.1 The Low-Abstraction Band and the Hybrid System

Fix ā≺⊤. Using the abstraction map φ from Theorem 3 (Abstract Shared TCZ Convergence Theorem), define the closed low-abstraction band

↓ā:={b| b≼ā}, K_ā:={x∈ X|φ(x)∈↓ā}.(31.1)

The hybrid language/Ego dynamics is

x⁺=T_m(x), m∈Σ_ℓ^*, ẋ=F_ℓ(x,t):=f(x,π_ℓ(x,t),t),(31.2)

On an interval with a fixed current language/evaluation regime, let V_ℓ=V_ℓ(x) and define the closed current TCZ by C_ℓ:=K_ā∩{x| V_ℓ(x)≤θ_ℓ}.

Condition 31-A (introspective-language closure)

For every available string, T_m(K_ā)⊆ K_ā.

Condition 31-B (Nagumo invariance of the flow)

Solutions are forward complete and F_ℓ(x,t)∈ T_{K_ā}(x) at the boundary, where T_K(x) is the Bouligand tangent cone.

Condition 31-C (hybrid TCZ attraction)

There are a basin B_ℓ⊆ K_ā forward invariant under both flows and jumps, constants c₁,c₂,λ>0, and a continuous Lyapunov residual W_ℓ such that, along flows,

c₁d(x,C_ℓ)²≤ W_ℓ(x,t)≤ c₂d(x,C_ℓ)²,qquad D⁺W_ℓ≤-2λ W_ℓ.(31.3)

Every language jump satisfies W_ℓ(T_mx,t⁺)≤ W_ℓ(x,t^-), and jump times are non-Zeno.

Condition 31-D (six-state coarse-graining; needed only for recurrence)

For the recurrence clause assume x₀∈ C_ℓ, and partition C_ℓ=⊔_r=1⁶R_r into six nonempty Borel regions. Let update times satisfy τ_n↑∞, and define

Z_n=r⇔ x(τ_n)∈ R_r.(31.D)

The process Z_n∈{1,…,6} is a time-homogeneous finite irreducible Markov chain with no state outside the six.

Theorem 31 (Tomabechi Introspective-Language Closure and Low-Abstraction Six-Realm Recurrence Theorem)

Under the regularity hypotheses of Theorems 1, 3, 16, and 18 and Conditions 31-A–C, every x₀∈ B_ℓ satisfies

x(t)∈ K_ā (∀ t≥0), d(x(t),C_ℓ)≤√(c₂/c₁)e^{-λ t}d(x₀,C_ℓ).(31.4)

If x₀∈ C_ℓ, the trajectory never leaves the current TCZ. If a higher-abstraction goal set H⊂ X∖ K_ā satisfies δ_H:=d(K_ā,H)>0, then

d(x(t),H)≥δ_H (∀ t≥0),(31.5)

so the current language/Ego closed loop cannot reach H.

If, in addition, x₀∈ C_ℓ and 31-D holds, there is a unique stationary distribution μ, and for every r,

Pr(Z_n=rinfinitely often)=1, 1/NΣ_n=0^N-11_{{Z_n=r}}⟶ a.s.μ_r>0.(31.6)

Proof

Condition 31-B and Nagumo's invariance theorem preserve K_ā during each flow segment; 31-A preserves it at every jump. Induction over non-Zeno hybrid events gives invariance at all times.

The comparison theorem gives W_ℓ(t)≤ e^-2λ(t-s)W_ℓ(s) on every flow interval, while the jump condition prevents upward discontinuities. Concatenating the intervals and applying the quadratic bounds gives (31.4). Initial membership in C_ℓ means W=0, which remains zero. Equation (31.5) follows from the definition of set distance.

A finite irreducible Markov chain is positive recurrent and has a unique strictly positive stationary distribution. Recurrence and the Markov-chain strong law yield (31.6). ∎

7.2 The Role of Each Qualifier in Condition 31-D

Condition 31-D coarse-grains C_ℓ into six non-empty Borel regions. The three qualifiers discharge independent functions, and (31.6) fails if any one is dropped. We make each explicit.

Why coarse-graining is required

The state space C_ℓ is in general of continuum cardinality. Under a continuous distribution the probability of returning to any single point is zero, so "infinitely many returns to the same state" is trivially false. To state recurrence as a non-trivial proposition one must therefore pass through a quotient map onto finitely many macroscopic regions. What Theorem 31 asserts is recurrence at the level of this quotient, not at the level of points. The two must not be conflated.

The role of the disjoint union ⊔ — well-definedness of the quotient map

C_ℓ = ⊔_r=1⁶R_r demands both covering (∪R_r = C_ℓ) and pairwise disjointness (r ≠ r′ ⇒ R_r ∩ R_r′ = ∅). Drop the former and the quotient map is not everywhere defined; drop the latter and Z_n is not uniquely determined. In either case (Z_n)_n≥0 cannot be constructed as a stochastic process and the theory of Markov chains does not apply.

The role of Borel measurability — existence of the transition kernel

Unless each R_r is Borel measurable, Pr(x(τ_n+1) ∈ R_r′ | x(τ_n) ∈ R_r) is undefined and the transition matrix P = (p_rr′) cannot be constructed. The condition excludes non-measurable sets (Vitali-type sets under the axiom of choice) and is not decorative. Under measurability, the existence of regular conditional probabilities on a standard Borel space secures the transition kernel.

The role of non-emptiness — positivity of the stationary distribution

If R_r = ∅ for some r, that state is unreachable and irreducibility fails. Without irreducibility the existence of a unique stationary distribution is not guaranteed and μ_r > 0 in (31.6) cannot be derived. Non-emptiness is the condition supporting the conclusion that all six regions are visited with positive frequency.

7.3 The Scope of the Name "Sixfold Recurrence"

The present theorem employs the term "sixfold recurrence" in order to name, in doctrinal vocabulary, the mathematical fact that under introspective-language closure the low-abstraction band cannot be left. The correspondence is apt because in the doctrine of the six realms suffering is said to arise in due course even in the highest, deva realm, so that no realm is terminal — structurally the same content as a ≺ ⊤ ⇒ J^*_a,ρ(x,T) > 0 in Theorem 24 (Universal Unsatisfactoriness). We now make explicit three things the name does not assert.

(i) The number 6 carries no mathematical content

The proof of (31.6) uses only the recurrence theorem for finite irreducible Markov chains and an ergodic theorem of Birkhoff–Kingman type, neither of which depends on the number of states. Below emptiness, therefore, the conclusion is identical whether the number of cells is 6, 7 or 10. Mathematics requires only finiteness, non-emptiness, Borel measurability, disjointness and irreducibility; the number 6 as such carries no mathematical significance. The number of cells and the names of the regions are supplied from the doctrinal side.

(ii) No positing of rebirth — though Theorem 25 imposes a strong constraint

The present theorem does not presuppose rebirth and neither affirms nor denies it. What the mathematics states is only the structure of repeated visitation in a closed finite irreducible system. However, Theorem 25 (Non-Self)

¬∃S₀ ∀h: F_i,h(S₀) = S₀；　∀d∈𝔇 ∀α∈𝕃: ¬Atman(d,α)

proves rigorously that a history-relative functional self exists, but there is neither a fixed point common to all histories nor a fixed own-nature at any abstraction layer, and all existence is constituted within the net of dependent origination. The following is immediate.

Corollary 31.R (constraint on any rebirth reading)

Even if rebirth exists, it may be the continuation of dependent origination across abstraction layers below emptiness, but it cannot be the continuation of a fixed own-nature. Indeed such a continuing own-nature would furnish a fixed point S₀ common to all histories, contradicting the first conjunct of Theorem 25.

Whether this continuation of dependent origination is called "rebirth" belongs to sectarian judgement and mathematics does not enter. It should be noted, however, that the continuity available here differs essentially from the pre-Buddhist conception of transmigration. In the formulation from the Buddha onward what is at issue is the persistence of a net of relations, whereas the pre-Buddhist conception posited a substance carried across layers. It is precisely the latter that Theorem 25 excludes.

(iii) It obtains within a single lifetime as well

Nothing whatever in the update-time sequence τ_n ↑ ∞ of condition 31-D requires those times to correspond to the boundaries of lifetimes. Hence (31.6) applies directly to transitions among regions within a single lifetime — the repetition of macroscopic states such as delight, strife, craving, disappointment and suffering. Mathematically this too is the sixfold recurrence, and it is the level of event that the theorem describes first of all.

7.4 What Theorem 31 Formalizes — Structural Correspondence with the Critique of Language in the Mūlamadhyamakakārikā

The byname "Mathematical Language Trap Theorem" is given because the object of its formalization coincides with the structure discussed as the trap of language ever since Nāgārjuna's Mūlamadhyamakakārikā (roughly the second to third century). This section is an interpretive layer; the proof of Theorem 31 depends on it in no way whatever.

The three traps and their mathematical counterparts

(a) Reification. The assignment of a name leads the object to be taken as an unchanging substance. The mathematical counterpart is the abstraction ceiling ā together with the closed set K_ā; reification corresponds to fixing that ceiling low. MMK 24.18 states that what arises dependently is called emptiness, that this emptiness is itself a dependent designation (prajñapti), and that precisely this is the middle way (Kumārajīva: 亦為是仮名 / 亦是中道義).

(b) Binary opposition. Captivity to the frame "exists / does not exist." The mathematical counterpart is condition 31-A, T_m(K_ā) ⊆ K_ā. The update selecting "exists" and the update selecting "does not exist" belong to the same family of language updates, and the image of either remains in K_ā. MMK 15.10 identifies clinging to existence as eternalism and clinging to non-existence as annihilationism, and states that the wise cling to neither (不応著有無). The eight negations of the opening are isomorphic in seeking to dissolve the frame itself by taking neither side of eight oppositions.

(c) Prapañca. Unbounded proliferation of concepts. The mathematical counterpart is the universal quantification in condition 31-A itself — "for every available language string." The enlargement of the policy set Π₀ ⊆ Π_ℓ guaranteed by Theorem 18 (Introspective Language Evolution) is compatible with d(x(t),H) ≥ δ_H in Theorem 31. MMK 18.5 states that action and affliction derive from conceptual discrimination and discrimination from prapañca, and that prapañca ceases in emptiness (入空戯論滅).

The mathematical fact corresponding to the warning of MMK 13.8

MMK 13.8 states that the Victorious Ones taught emptiness in order that all views be relinquished, and that those who hold emptiness itself as a view are beyond cure (若復見有空 / 諸仏所不化). The same point recurs at 22.11. This warning has an exact mathematical counterpart under the universal quantification of condition 31-A.

Condition 31-A demands T_m(K_ā) ⊆ K_ā for every available language string, and this universal admits no exception. Hence if the term "emptiness" is acquired as one of the language strings operable within K_ā, its image likewise remains in K_ā. This merely enlarges the policy set by one and does not increase the reachable abstraction. On the other hand ⊤, as the highest abstraction, lies by definition outside K_ā, and δ_H = d(K_ā, H) > 0 does not close. At the moment emptiness is internalized as a view it becomes one item of prapañca — this is the mathematical substance of the warning.

Skilful means and Theorem 32 — where the escape lies

The use of language that the Mūlamadhyamakakārikā situates as skilful means — employing language as operation rather than as content — corresponds to the mechanism of Theorem 32 (Future-TCZ Homeostasis Absolute-Other-Power). What Theorem 32 exhibits is not a search for policies within the same closed loop but the reconstruction of the loop itself. Setting an external goal as a terminal condition shifts the evaluation from V₀ to Ṽ_G and the policy from π_ℓ to π_G, and under (32.7) the set K_ā ceases to be forward invariant. The wall is not broken; rather, the condition that constituted the wall as a wall disappears.

What this section does not assert

(i) This section does not assert the truth of Nāgārjuna's teaching. What is proved are mathematical propositions under explicitly stated conditions; whether the name "trap of language" is apt for them belongs to doctrine and intellectual history. (ii) This section does not supply a mathematical foundation for Madhyamaka doctrine. What is exhibited is a coincidence of structure, not the truth or falsity of a doctrine, and interpretive disputes internal to Madhyamaka (including the division between Prāsaṅgika and Svātantrika) are not settled by mathematics. (iii) This section implies no negative evaluation of language in general. Theorem 18 proves that introspective language reduces uncertainty and enlarges the policy set; what Theorem 31 states is only that this enlargement may be confined within a closed set.

On the citations

Citations of the Mūlamadhyamakakārikā follow chapter and verse numbering; the Chinese readings follow Kumārajīva's translation. The renderings are the present author's. None of the citations is used in the proof of Theorem 31; they function solely as an interpretive layer.

8. Theorem 32 — Future-TCZ Homeostatic Absolute Other-Power

“Absolute Other-Power” is not a supernatural force or physical backward causation. It is an operational name within the model: once Self installs a goal outside the current TCZ as a terminal condition, the future-limited Ego closed loop converges autonomously toward the future TCZ without an additional effort input from the old Ego.

Let C₀ be the current TCZ and let the true goal G satisfy d(G,C₀)>ε. The future-limited Ego from Theorem 8 (Tomabechi Future-Origin Cognitive Time Theorem) is

π_G(x,t)∈*arg min_u{∫_t^t+T₀Ṽ_G(x(s),s) ds+Λ d(x(t+T₀),G)²}.(32.1)

Define the positive goal-drive field, its value at the goal, and the effective potential by

R_G(y,t;x(t)):=Q⁺(y,t)E(y,t| x(t))C_Self(G), r_G(t):=R_G(G,t;x(t)), Ṽ_G(y,t):=V₀(y,t)-κ P_G(y,t)R_G(y,t;x(t)).(32.2)

and C_G(t):={y|Ṽ_G(y,t)≤θ_G}. For r_G(t)>0, let

P_crit(t):=[V₀(G,t)-θ_G]₊/κ r_G(t).(32.3)

Condition 32-A (bounded presence-threshold exceedance)

For t≥ t₀, r_G(t)≥ r₀>0, 0≤ P_G≤1, P_crit(t)≤1, and P_G(G,t)≥ P_crit(t). The effective potential is bounded below, and the optimal policy and forward-complete solution exist.

Condition 32-B (reachable basin)

There is a nonempty forward-invariant basin B_G with x(t₀)∈ B_G. The future TCZ lies in the closure of the admissible reachable set from B_G, and the actual future-limited input is u(t)=π_G(x(t),t).

Condition 32-C (strong future-TCZ homeostasis)

Every C_G(t) is nonempty and closed. There are a₁,a₂,λ_G>0 and a Lyapunov function W_G such that t↦ W_G(x(t),t) is locally absolutely continuous along trajectories and, on B_G,

a₁d(x,C_G(t))²≤ W_G(x,t)≤ a₂d(x,C_G(t))², D⁺_{π_G}W_G≤-2λ_GW_G.(32.4)

Here the Dini derivative is the full trajectory derivative, including motion of C_G(t):

D⁺_{π_G}W_G(x,t):=lim sup_h↓0 W_G(x_{π_G}(t+h;t,x),t+h)-W_G(x,t)/h.

Theorem 32 (Tomabechi Future-TCZ Homeostatic Absolute Other-Power Theorem)

Under the regularity hypotheses of Theorems 1, 4, and 7–9 and Conditions 32-A–C, G∈ C_G(t) for every t≥ t₀, and

d(x(t),C_G(t))≤√(a₂/a₁)e^{-λ_G(t-t₀)}d(x(t₀),C_G(t₀))⟶0.(32.5)

Let d₀:=d(x(t₀),C_G(t₀)). For every δ>0, define the hitting-time bound for the δ-neighborhood of the future TCZ by

T_δ:= t₀, d₀=0,
[2pt] t₀+1/λ_Gmax{0, log(√(a₂/a₁)d₀/δ)}, d₀>0 .(32.6)

Then t≥ T_δ implies d(x(t),C_G(t))≤δ.

Proof

Condition 32-A and (32.3) give κ P_G(G,t)r_G(t)≥[V₀(G,t)-θ_G]₊; hence Ṽ_G(G,t)≤θ_G, so G∈ C_G(t). From (32.4) and Grönwall's inequality, W_G(x(t),t)≤ W_G(x(t₀),t₀)e^{-2λ_G(t-t₀)}. Substitution of the two quadratic bounds and taking square roots gives (32.5). Solving the right-hand side for δ gives (32.6). Because the post-installation input remains π_G, the convergence is generated by the reconfigured closed-loop homeostasis. ∎

8.2 Consistency with escape from the language trap

To leave the set in Theorem 31 (Introspective-Language Closure and Low-Abstraction Sixfold Recurrence Theorem), the new goal closed loop must violate its Condition 31-A or 31-B. For example, let K_ā={x| b(x)≤0}, with b∈ C¹, and assume b(x(t)) is locally absolutely continuous. Suppose the trajectory reaches the strip -ℓ≤ b(x)≤0 at time t_b and, while it remains there,

d/dtb(x(t))=∇ b(x(t))· f(x(t),π_G(x(t),t),t)≥ν>0 a.e.(32.7)

Then b(x(t))≥ b(x(t_b))+ν(t-t_b). Defining τ_out:=inf{t≥ t_b| b(x(t))>0},

τ_out≤ t_b+-b(x(t_b))/ν≤ t_b+ℓ/ν.

Thus the trajectory exits in finite time. The point is not extraordinary effort within the old closed loop: the external goal reconfigures the cost and Ego policy so that the old invariant set ceases to be invariant.

The exact meaning of “excess presence”

The inherited presence variable P is normalized to [0,1]; “excess” cannot mean P>1. It means bounded threshold exceedance, P_G≥ P_crit. If the unbounded effective gain p of Theorem 21 (Subsumption Partial-Order Presence Directionality Theorem) is used, it must remain a separate variable. A claim that presence also increases the rate of convergence requires the additional condition λ_G=λ_G(P_G) with dλ_G/dP_G≥0.

Set attainment versus point attainment. Equation (32.5) guarantees convergence to C_G(t), not necessarily x(t)→ G. Point convergence requires C_G(t)={G} or d_H(C_G(t),{G})→0.

9. Minimality and counterexamples

Omitted condition	Minimal counterexample / effect	Lost conclusion
28-A	An arbitrary independent label may be appended to a relationally defined set.	Relationality alone does not prove non-self.
Arithmetic strength in 29-A	A decidable theory such as Presburger arithmetic.	Gödel incompleteness cannot be asserted unconditionally.
Effectiveness in 29-A	The set of all truths of standard arithmetic is not c.e.	Completeness is bought at the loss of an effective proof system.
Compatibility in 30-A	Updating above and projecting disagrees with updating below.	L_m is not well defined as a self-map of the inverse limit.
Nontriviality in 30-A	G_m=G_ε.	No changed fixed point is established.
30-C	Shannon uncertainty falls while T15's state functional is constant.	The physical exchange equation cannot be invoked.
31-A	K=(-∞,0], T_m(x)=x+2.	One language update exits the band.
31-B	The same K with ẋ=1.	The flow exits without a language jump.
Jump clause in 31-C	Every jump doubles W.	Stable flow does not imply hybrid convergence.
Irreducibility in 31-D	Transition matrix P=I₆.	The chain stays in one realm.
Reachability in 32-B	ẋ=0,x₀=0,G=1.	Even maximum presence cannot attain the goal.
32-C	ẍ+x=0.	The trajectory oscillates without distance convergence.
Threshold in 32-A	V₀(G)=2,θ=1,κ r_G=1,P=.5.	Ṽ(G)=1.5>1, so G∉ C_G.

10. Synthesis

Nirvana is without self. The proof rests on relational functional completeness, not merely on set-valuedness or a time index; tagged types reconcile suffering below the top with tranquility at the top.
A formalized teaching is not self-completing. This holds within the stated scope of sufficiently strong, effective, sound arithmetical theories; incompleteness and non-self-nature have distinct grounds.
Introspective language selects and stabilizes an Ego configuration. It does not create a subject ex nihilo. Projection compatibility gives the inverse-limit configuration; the entropy bridge and T15 give the physical exchange.
Language may enlarge choice while still forming a trap. If all added policies preserve the low-abstraction band, more options do not enlarge reachable abstraction.
External-goal attainment may be described by reconfigured homeostasis. A bounded presence threshold, reachability, a basin, and strong contraction give exponential future-TCZ convergence and a hitting-time bound for every δ-neighborhood.

Unified view

Without positing a fixed substance, types, relations, language, information, control, and physical dissipation jointly support a conditional mathematical account of nirvana without self, open formal knowledge, language-conditioned Ego construction, low-abstraction closure, and external-goal reorganization of the closed loop.

11. Summary Tables of Theorems

All theorems used in this paper are gathered into two tables. The first covers Theorems 28 through 32, newly proved here; the second covers the inherited theorems used as premises.

11.1 The New Theorems of This Paper (Theorems 28–32)

Table 1. Theorems 28–32 — name, central formula, dependencies, new conditions, proof status, summary
Theorem	Central formula	Depends on	New conditions / external results	Status of the proof	Summary
Theorem 28 Nirvana Non-Self and Type Coherence Theorem	𝔇_<⊤ ∩ Nir(T) = ∅;　PZS(a,x,T) ⇔ [a=⊤ ∧ x∈𝒩_⊤(T)];　¬Atman(d_N,⊤)	Theorem 24 (Universal Unsatisfactoriness Theorem), Theorem 25 (Non-Self Theorem), Theorem 26 (Nirvanic Tranquility Theorem)	Condition 28-A (disjointness of the tagged domains and relational describability of the nirvana process)	A direct consequence of the inherited theorems. No new analytic tool is introduced; only the separation of types and a rewriting of the predicate	Universal unsatisfactoriness and nirvanic tranquility are not competing claims about one domain but govern exclusive regions separated by a tagged disjoint union. On that basis the nirvana process itself is also non-self
Theorem 29 Formal Dharma-System Non-Substantiality and Incompleteness Theorem	∃G_h[ℕ⊨G_h ∧ 𝕋_h⊬G_h ∧ 𝕋_h⊬¬G_h];　𝕋_h⊬Con(𝕋_h);　Th(𝕋₀) ⊊ ⋯	Theorem 25 (Non-Self Theorem)	Condition 29-A (sufficient arithmetical strength, effectiveness, soundness) / external results: Gödel's first and second incompleteness theorems, Chaitin's incompleteness theorem	Dependence on external metatheorems is essential. It cannot be derived from this system alone, and applies only to systems satisfying Condition 29-A	Only when the dharma as doctrine is modelled as a sufficiently strong effective formal theory do undecidable sentences, unprovability of one's own consistency, and a proof limit on description length follow, entailing open-ended updating and non-substantiality
Theorem 30 Entropy-Exchange Ego Constitution Theorem	d_SC(G_mⁿS₀,S_m^) ≤ (q_Fq_m)ⁿd_SC(S₀,S_m^);　Δℋ_ego ≤ 0	Theorem 15 (Cognitive-Physical Entropy Exchange and Conservation Theorem), Theorem 16 (Self-Consciousness Existence and Emergence Theorem), Theorem 18 (Introspective Language Evolution Theorem)	Conditions 30-A through 30-C (projective coherence on the inverse system, contractivity of the language map, monotone growth of the σ-algebra)	A composition of inherited theorems. Only the Banach fixed-point theorem and nonnegativity of conditional mutual information are newly used	An introspective language selects and stabilizes an Ego. The more language there is, the lower the ego-structuring entropy, and that decrease is exchanged exactly into the physical side
Theorem 31 Introspective-Language Closure and Low-Abstraction Sixfold Recurrence Theorem	x(t) ∈ K_ā (∀t≥0);　d(x(t),C_ℓ) ≤ √(c₂/c₁)e^−λtd(x₀,C_ℓ);　d(x(t),H) ≥ δ_H	Theorem 1 (Tomabechi Main Theorem), Theorem 3 (Abstract Shared TCZ Convergence Theorem), Theorem 16 (Self-Consciousness Existence and Emergence Theorem), Theorem 18 (Introspective Language Evolution Theorem)	Conditions 31-A through 31-D (language closure, Nagumo invariance, Lyapunov attractivity, irreducibility and aperiodicity)	An application of invariant-set theorems. Nagumo's theorem, Lemma 0 and the ergodic theorem are used	With the present language and Ego closed loop alone, one cannot leave the low-abstraction band. The sixfold coarse-graining is recurrent, and each state is visited infinitely often
Theorem 32 Future-TCZ Homeostasis Absolute-Other-Power Theorem	d(x(t),C_G(t)) ≤ √(a₂/a₁)e^{−λ_G(t−t₀)}d(x(t₀),C_G(t₀)) → 0;　explicit T_δ	Theorem 1 (Tomabechi Main Theorem), Theorem 4 (Tomabechi Presence-Weighted Theorem), Theorem 7 (Tomabechi True Goal Theorem), Theorem 8 (Tomabechi Future-Origin Cognitive Time Theorem), Theorem 9 (Tomabechi Future-Origin Goal Attainment Theorem)	Conditions 32-A through 32-C (forward invariance of the future TCZ, a bounded presence threshold, a time-varying Lyapunov sandwich)	A time-varying, quantitative form of Theorem 9 (Tomabechi Future-Origin Goal Attainment Theorem). It newly supplies an explicit upper bound on the attainment time	The presence threshold for an external goal to enter the future TCZ is bounded. Once the threshold is passed, homeostasis carries out the remaining approach automatically

11.2 Inherited Theorems (Used as Premises)

Table 2. Inherited theorems — name, standard form, role in this paper, location of the proof
Theorem	Central formula (standard form)	Role in this paper	Location of the proof	Summary
Theorem 1 Tomabechi Main Theorem	π_c = arg min ∫V₀ dt ⇒ x(t) → TCZ	The convergence skeleton of Theorems 31 and 32	§3.1 of this paper (restated with proof)	A closed loop minimizing the accumulated evaluation carries the trajectory to the reachable stable region
Theorem 3 Abstract Shared TCZ Convergence Theorem	A(x)=0 ⇔ φ(x)=LUB(W₁,…,W_N), A(t)→0	The upper-bound structure of the language closure in Theorem 31 (Introspective-Language Closure and Low-Abstraction Sixfold Recurrence Theorem)	§3.2	Adding a shortfall penalty aligns the group at the least roof that discards no one's world
Theorem 4 Tomabechi Presence-Weighted Theorem	Ṽ = V₀ − κPQ, x → TCZ_P	The basis of the presence threshold in Theorem 32 (Future-TCZ Homeostasis Absolute-Other-Power Theorem)	§3.3	Presence deforms the landscape itself, moving the valley without sustained effort of will
Theorem 7 Tomabechi True Goal Theorem	G ∉ TCZ₀, d(G,TCZ₀) ≥ ε > 0, G = Self-set	The eligibility condition for the external goal in Theorem 32 (Future-TCZ Homeostasis Absolute-Other-Power Theorem)	§3.4	Characterizes the four conditions of a transformative goal; it does not assert attainment
Theorem 8 Tomabechi Future-Origin Cognitive Time Theorem	u^* = arg min J_G; the terminal condition determines the present control	The direction of determination in Theorem 32 (Future-TCZ Homeostasis Absolute-Other-Power Theorem)	§3.5	In optimal control with a terminal condition, the present control is fixed backwards from the future goal
Theorem 9 Tomabechi Future-Origin Goal Attainment Theorem	K_G = PQ⁺ + EC_Self ≥ K_crit with Lyapunov descent ⇒ x → TCZ_G	The prototype of the attainment part of Theorem 32 (Future-TCZ Homeostasis Absolute-Other-Power Theorem)	§3.6	Once the drive strength passes the critical value and the descent condition holds, the state converges to the goal-side TCZ
Theorem 15 Cognitive-Physical Entropy Exchange and Conservation Theorem	S_gen = S_phys + Σw_αH_α, dS_gen/dt = Π ≥ 0	The exchange budget in Theorem 30 (Entropy-Exchange Ego Constitution Theorem)	§3.7 of this paper (restated with proof)	At the physical layer alone there is no conservation law for entropy; only once the high-abstraction layers are included does a conservation law by exchange hold as an equality
Theorem 16 Self-Consciousness Existence and Emergence Theorem	SC = lim_←TCZ_α ≠ ∅, F_SC(S^) = S^, M(S^) represents S^	The Ego constitution in Theorems 30 and 31	§3.8	Existence from the inverse limit, uniqueness from the fixed-point theorem; the fixed point depends on history
Theorem 18 Introspective Language Evolution Theorem	H(Z\|Y,M_ℓ) < H(Z\|Y); Π₀ ⊆ Π_ℓ; ∂ℱ/∂ℓ > 0	The role of language in Theorems 30 and 31	§3.9	Introspective language reduces uncertainty, enlarges the policy set, and raises the capacity for free will
Theorem 24 Universal Unsatisfactoriness Theorem	a ≺ ⊤ ∧ Condition 24-A ⇒ J^*_a,ρ(x,T) > 0	The lower side of the type separation in Theorem 28 (Nirvana Non-Self and Type Coherence Theorem)	Four Dharma Seals paper (conditions restated in §3.10)	Below emptiness a positive residual value survives even under optimization
Theorem 25 Non-Self Theorem	¬∃S₀ ∀h: F_i,h(S₀) = S₀；∀d∈𝔇 ∀α∈𝕃: ¬Atman(d,α)	The ground of non-self in Theorems 28 and 29	Four Dharma Seals paper (as above)	No fixed, individuating, causally nonredundant own-nature exists apart from relation
Theorem 26 Nirvanic Tranquility Theorem	W_⊤(x(t),t) ≤ W_⊤(x(T),T)e^−λ(t−T) ⇒ dist(x(t),𝒩_⊤(t)) → 0；permanent cessation ⇔ a=⊤ ∧ x∈𝒩_⊤	The upper side of the type separation in Theorem 28 (Nirvana Non-Self and Type Coherence Theorem)	Four Dharma Seals paper (as above)	At the highest abstraction level there is a forward-invariant set on which, while alive, the residual value is zero

A note on the status of the proofs

The column "Status of the proof" in Table 1 indicates how directly each theorem follows from existing theorems. Theorem 28 (Nirvana Non-Self and Type Coherence Theorem) follows from the type separation of the inherited theorems alone, whereas Theorem 29 (Formal Dharma-System Non-Substantiality and Incompleteness Theorem) depends essentially on external metatheorems (Gödel, Chaitin) and cannot be derived from this system by itself. Theorems 30, 31 and 32 are compositions of inherited theorems with additional conditions, and the analytic tools added — the Banach fixed-point theorem, Nagumo's theorem, the ergodic theorem, Grönwall's inequality — are all standard.

12. References

Canonical and related papers in this system

Tomabechi, H. (2026). The Tomabechi Four Dharma Seals Theorem System — Theorems 23–26, Japanese canonical and English editions.
Tomabechi, H. (2026). The Tomabechi Ignorance-Conditioned Volitional Formations Theorem — Theorem 27 (Ignorance-Conditioned Volitional Formations Theorem), Japanese canonical, English, and accessible editions.
Tomabechi, H. (2026). TomabechiSynthesisAvijjaSankharaJA, synthesis commentary.
Tomabechi, H. (2026). Tomabechi Cognitive–Physical Entropy Exchange and Conservation Theorem.
Tomabechi, H. (2026). Tomabechi Self-Consciousness Existence and Emergence Theorem.
Tomabechi, H. (2026). Tomabechi Internal-Language Evolution Theorem.
Tomabechi, H. (2026). Tomabechi Theory of Cognitive Homeostasis and Tomabechi Abstractional Freedom Theory.

Classical source used as an interpretive layer

Nāgārjuna (ca. 2nd–3rd c.), Mūlamadhyamakakārikā (Root Verses on the Middle Way). Referenced in the interpretive layer of §7.4. The verses cited are 13.8, 15.10, 18.5, 22.11, 24.18 and the eight negations of the opening. Chinese readings follow Kumārajīva's translation (Taishō No. 1564); the renderings are the present author's. None of the citations is used in the proof of Theorem 31.

External metatheorems

Gödel, K. (1931). “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I.” Monatshefte für Mathematik und Physik 38, 173–198. doi:10.1007/BF01700692.
Chaitin, G. J. (1974). “Information-Theoretic Limitations of Formal Systems.” Journal of the ACM 21(3), 403–424. doi:10.1145/321832.321839.
Chaitin, G. J. (1975). “A Theory of Program Size Formally Identical to Information Theory.” Journal of the ACM 22(3), 329–340. doi:10.1145/321892.321894.