The Tomabechi Four Dharma Seals Theorem System — Minimal Thirteen-Theorem Version

Abstract

This paper preserves the serial numbering of the Tomabechi theorem system while taking Theorems 1–4, 16, and 19–22 as its minimal foundation. On that basis, it formulates and proves Theorem 23 (Impermanence Theorem), Theorem 24 (All Conditioned Existence Is Suffering Theorem), Theorem 25 (Non-Self of All Dharmas Theorem), and Theorem 26 (Nirvanic Tranquility Theorem).

The mathematical core has four parts. First, a complete life–environment state subject to persistent strict dissipation admits no exact recurrence; and, so long as infinite, non-Zeno, nonterminal subsumption updates are repeated below emptiness, the cognitive TCZ neither terminates nor becomes fixed at any finite stage. Second, if a permanently zero-discomfort trajectory is impossible below emptiness, a positive residual evaluative cost remains for every initial condition even after optimal control. Third, a self-consciousness fixed point is a functional fixed point relative to an input history, not a fixed intrinsic nature common to every history. Fourth, if at the highest abstraction level ⊤ there exists a nonempty, forward-invariant family of zero-suffering sets and that family is Lyapunov stable, then the state distance to it can be made to decay exponentially while biological activity is preserved.

Every proof in this paper is conditional mathematics: the conclusion follows when the stated assumptions hold. The Buddhist terms are interpretive names for mathematical structures; the paper does not purport to deduce empirical or religious truth without assumptions.

Normalization of Theorem Numbers and the Minimal System
Common Definitions and the Unified Convergence Lemma
Theorem 1 — Tomabechi Main Theorem
Theorem 2 — Shared-TCZ Convergence Theorem
Theorem 3 — Abstract Shared-TCZ Convergence Theorem
Theorem 4 — Tomabechi Presence-Weighted Theorem
Theorem 16 — Tomabechi Self-Consciousness Existence and Emergence Theorem
Theorem 19 — The Tomabechi Free-Will Theorem
Theorem 20 — Tomabechi Symbolic-Presence Directionality Theorem
Theorem 21 — Tomabechi Subsumption-Poset Presence Directionality Theorem
Theorem 22 — Tomabechi Higher-LUB Presence Theorem
Theorem 23 — Impermanence Theorem
Theorem 24 — All Conditioned Existence Is Suffering Theorem
Theorem 25 — Non-Self of All Dharmas Theorem
Theorem 26 — Nirvanic Tranquility Theorem
Four Consistency Checks
Scope, Conclusions, and Canonical Sources

1. Normalization of Theorem Numbers and the Minimal System

The thirteen theorems adopted in this paper are listed below. Their numbers are not renumbered consecutively, so as to preserve their positions within the existing system of twenty-six theorems. Any content from omitted theorems that is required here is restated explicitly as an independent “common assumption.”

Theorem numbers, normalized official names, and roles adopted in this paper
No.	Normalized official name	Role in this paper
1	Tomabechi Main Theorem	Convergence to an individual TCZ; typed unification of Self, Ego, and TCZ
2	Shared-TCZ Convergence Theorem	Shared stable set for multiple subjects
3	Abstract Shared-TCZ Convergence Theorem	Convergence to a high-abstraction set representing the LUB
4	Tomabechi Presence-Weighted Theorem	Transformation of the effective potential by presence
16	Tomabechi Self-Consciousness Existence and Emergence Theorem	Inverse limit of multilayer TCZs and a history-relative fixed point
19	The Tomabechi Free-Will Theorem	Goal-conditioned control capacity and monotonicity in the level of abstraction
20	Tomabechi Symbolic-Presence Directionality Theorem	Directional component toward a specified LUB
21	Tomabechi Subsumption-Poset Presence Directionality Theorem	Threshold at which partial-information bias creates a local valley
22	Tomabechi Higher-LUB Presence Theorem	Stepwise ascent through subsumption rather than elimination
23	Impermanence Theorem	Nonrecurrence of the complete state and nonfixity of stagewise TCZs at every finite stage
24	All Conditioned Existence Is Suffering Theorem	Impossibility of permanently zero suffering below emptiness
25	Non-Self of All Dharmas Theorem	Nonexistence of a fixed point common to all histories, and of a fixed intrinsic nature at any abstraction layer (dependent origination)
26	Nirvanic Tranquility Theorem	Stability of the zero-suffering invariant set at emptiness

1 → 2 → 3 → 4 → 16 → 19 → 20 → 21 → 22 → 23 → 24 → 25 → 26

The arrows indicate the order of exposition in this paper. They do not mean that every theorem depends only on the immediately preceding theorem. The precise minimal dependencies are given in the table at the end.

2. Common Definitions and the Unified Convergence Lemma

2.1 State, Control, Evaluation, and TCZ

Let the cognitive state space be X⊂ℝⁿ, and let the control system be

ẋ = f(x,u,t)

Assume that f is locally Lipschitz continuous in the state and that admissible controls generate unique forward solutions. The baseline evaluation function

V₀(x,t) ≥ 0

represents discomfort, instability, internal inconsistency, or evaluation cost. Let ℛ(t;x₀) denote the reachable set and θ a threshold, and define

Ω_θ(t):={x∈X | V₀(x,t)≤θ}

The canonical TCZ is defined by pairing reachability and the evaluation condition at the same time:

TCZ(x₀):=⋃_τ≥0[ℛ(τ;x₀)∩Ω_θ(τ)]

To state convergence in distance to a time-varying set, however, for a fixed closed-loop policy π we separately use

K_π(x₀):=cl⋃_τ≥0ℛ_π(τ;x₀),　TCZ_π^cl(t;x₀):=K_π(x₀)∩Ω_θ(t)

which we call the “closed-reachable TCZ slice.” It is not the canonical TCZ itself, but rather a same-time convergence target restricted to the closed-loop reachable component. Whenever it serves as the target of distance convergence in this paper, we assume that the corresponding K_π(x₀) is forward invariant and that the slice at every time is nonempty. The time argument is omitted for autonomous systems.

2.2 Rigorous Interpretation of Finite-Horizon Control

Convergence as t→∞ cannot be derived from a single finite-horizon optimization. In this paper, π_c is the receding-horizon policy updated at each time by

u^*_t,x(·) ∈ arg min_{u(·)∈𝒰[t,t+T], x^u(t)=x} ∫_t^t+T Ψ(x^u(s),s)ds,　π_c(t,x):=u^*_t,x(t+)

We fix a Borel-measurable selection (t,x)↦u^*_t,x(·) from the set of optimal solutions and use the right limit u^*_t,x(t+) of a right-continuous representative. For existence of an optimal solution, we assume standard conditions such as compactness of the admissible-control set or coercivity of the control cost, together with lower semicontinuity; for the induced vector field, we require the Carathéodory conditions and forward completeness. Separately from optimality, we further require that the induced closed loop satisfy the following descent condition.

Lemma 0 (Unified Lyapunov Convergence Lemma)

Let K be a forward-invariant set and Φ:K×[0,∞)→[0,∞), and assume that each target set Ω(t):={z∈K | Φ(z,t)=0} is nonempty. Suppose that, along every closed-loop trajectory z(t), the map t↦Φ(z(t),t) is absolutely continuous and that, for some c,C>0,

D⁺_tΦ(z(t),t) ≤ −2cΦ(z(t),t)　(a.e. t≥t₀),　dist(z,Ω(t))² ≤ CΦ(z,t)　(∀(z,t)∈K×[0,∞))

hold for all closed-loop trajectories and every initial time t₀≥0. Then, for t≥t₀,

Φ(z(t),t) ≤ Φ(z(t₀),t₀)e^{−2c(t−t₀)},　dist(z(t),Ω(t)) ≤ √(CΦ(z(t₀),t₀))e^{−c(t−t₀)} → 0.

Proof

Apply Grönwall’s inequality on the interval [t₀,t] to the inequality for the total Dini derivative along the trajectory. Substituting the resulting estimate into the error bound for the same-time slice Ω(t) and taking the square root yields the distance estimate. ∎

2.3 Level of Abstraction, Lattice, and Emptiness

Let the totality of abstract concepts form a complete lattice

𝕃=(𝕃,≼,∨,∧,⊤,⊥)

Here a≼b means that b subsumes a and is at least as abstract. We interpret the least element ⊥ as the physical layer 0 and the greatest element ⊤, philosophically, as “emptiness.” From Theorem 19 onward, the abstraction variable ranges over all of 𝕃. For the inverse limit in Theorem 16, by contrast, to preserve a nontrivial multilayer construction, we exclude ⊤ and use an upward-directed subset 𝔄₁₆⊂𝕃∖{⊤} having no greatest element. This terminology is interpretive; the proofs use only the order-theoretic properties.

2.4 Typed Self, Ego, and TCZ

Self, Ego, and TCZ are not of the same mathematical type.

Types and roles of Self, Ego, and TCZ
Representation	Type	Role
Self	Semantic selection and transformation operator	Evaluates possible worlds and, when necessary, reconstructs the set of worlds
Ego	Control policy π_c	Implements the closed-loop trajectory on the evaluation landscape
TCZ	Stable set in state space	Specifies the region in which the trajectory remains or to which it converges

The three are “differently typed representations of the same cognitive process.” Theorem 25 states non-self while preserving this distinction.

3. Theorem 1 — Tomabechi Main Theorem

Assume that the closed-loop reachable component K₁:=cl⋃_τ≥0ℛ_{π_c}(τ;x₀) is forward invariant, and that

TCZ₁^cl(t;x₀):=K₁∩{x | V₀(x,t)≤θ}

is nonempty. Define the zero-residual Lyapunov function by

Φ₁(x,t):=[V₀(x,t)−θ]₊,　[r]₊:=max{r,0}

Here and below, Φ₁ is restricted to K₁.

Theorem 1 (Tomabechi Main Theorem / Individual TCZ Convergence)

π_c = arg min_u(·) ∫ V₀(x^u(s),s)ds　⇒　dist(x(t),TCZ₁^cl(t;x₀)) → 0

Canonical central formula (the π_c continuity form of the system). The receding-horizon implementation below, with the descent conditions of Lemma 0, is its rigorous reading.

Suppose that the receding-horizon optimal control

u^*_t,x(·)∈arg min_{u(·)∈𝒰[t,t+T], x^u(t)=x}∫_t^t+TV₀(x^u(s),s)ds,　π_c(t,x):=u^*_t,x(t+)

generates a forward-complete closed loop and satisfies the descent condition and error bound of Lemma 0 for Φ₁. Then

dist(x(t),TCZ₁^cl(t;x₀)) → 0.

Proof

On K₁, Φ₁(x,t)=0 is equivalent to x∈TCZ₁^cl(t;x₀). Applying Lemma 0 to the assumed descent condition and error bound yields the conclusion. ∎

Rigor

Neither the existence of the arg min nor finite-horizon optimality alone implies convergence. The HJB equation ordinarily gives descent of the value function; descent of the residual in this theorem additionally requires dissipativity, detectability, or an independent control-Lyapunov condition. Self, Ego, and TCZ are not the same mathematical object, but differently typed interpretations—semantic, control-theoretic, and set-theoretic—of the same closed loop.

4. Theorem 2 — Shared-TCZ Convergence Theorem

Let the set of subjects be I={1,…,N}, and let the undirected coupling graph be G=(I,E). Introduce maps h_i:X_i→Y into a common representation space, and suppose that symmetric mismatch functions S_ij≥0 satisfy

S_ij(x_i,x_j)=0 ⇔ h_i(x_i)=h_j(x_j)

Let V_0,i be the baseline evaluation of each subject, and define the shared zero residual by

Φ₂(𝐱,t):=Σ_iw_i[V_0,i(x_i,t)−θ_i]₊ + Σ_{i,j}∈Eγ_ijS_ij(x_i,x_j)

For every undirected edge, let γ_ij=γ_ji>0, and let w_i>0. Fix a joint policy π₂ and define

K₂:=cl⋃_τ≥0ℛ_π₂(τ;𝐱₀),　Ω₂(t):={𝐱∈K₂ | Φ₂(𝐱,t)=0}

Assume that K₂ is forward invariant and that the shared zero set Ω₂(t) is nonempty at every time.

Theorem 2 (Shared-TCZ Convergence Theorem)

π_i = arg min_{u_i(·)} ∫ (V_0,i+Σ_jγ_ijS_ij)ds　⇒　dist(𝐱(t),TCZ^shared,cl(t)) → 0

Canonical central formula (the π_c continuity form). The shared zero residual Φ₂ below, with Lemma 0, is its rigorous reading.

Suppose that the coupling graph is connected and has positive bidirectional couplings, and that π₂, as either a joint policy or a distributed policy for which Φ₂ is an exact potential, satisfies the descent condition and error bound of Lemma 0. Then

dist(𝐱(t),Ω₂(t)) → 0.

Moreover, each subject’s TCZ residual and the mismatch measure on each coupling edge converge to zero. On the zero set, connectedness and the zero-equivalence condition above imply consistency among the representations of all subjects. We call this closed-reachable family of zero sets TCZ^shared,cl(t).

Proof

Because every term is nonnegative, Φ₂=0 is equivalent to simultaneous satisfaction of the individual TCZ conditions and the edgewise consistency conditions. Lemma 0 gives dist(𝐱(t),Ω₂(t))→0. Moreover,

0≤[V_0,i−θ_i]₊≤Φ₂/w_i→0,　0≤S_ij≤Φ₂/γ_ij→0.

In a connected graph, equality of edge representations on the zero set propagates transitively to all subjects. What is proved directly for the present trajectory is asymptotic convergence of both the distance to the family of shared zero sets and the mismatch measures to zero; neither finite-time entry into the set nor convergence of the representation values themselves is claimed. ∎

Necessary Additional Conditions

Bidirectional coupling alone does not imply shared convergence. If the valleys of the individual potentials differ, counterexamples exist in which a positive mismatch remains under finite coupling. Nonemptiness of the shared zero set and a strict descent condition that detects it are therefore explicitly required.

5. Theorem 3 — Abstract Shared-TCZ Convergence Theorem

Let φ_i:X_i→𝕃 map cognitive states to the lattice, and define the least upper bound of the multiple worlds represented by W_i∈𝕃 as

L^*:=∨_i=1^NW_i

Assume that 𝕃 admits a fixed injective order embedding ι:𝕃↪ℝ^m. Define the abstraction residual by

𝒜_i(x_i):=‖ι(φ_i(x_i))−ι(L^*)‖²

and let

Φ₃(𝐱,t):=Φ₂(𝐱,t)+Σ_iη_i𝒜_i(x_i),　η_i>0

Let π₃ be an LUB-weighted policy, and define

K₃:=cl⋃_τ≥0ℛ_π₃(τ;𝐱₀),　Ω₃(t):={𝐱∈K₃ | Φ₃(𝐱,t)=0}

Assume that K₃ is forward invariant and that Ω₃(t) is nonempty at every time.

Theorem 3 (Abstract Shared-TCZ Convergence Theorem)

π_i = arg min_{u_i(·)} ∫ (V_0,i+Σ_jγ_ijS_ij+η_i𝒜_i)ds　⇒　‖ι(φ_i(x_i(t)))−ι(L^*)‖ → 0

Canonical central formula (the π_c continuity form). The residual Φ₃ below, with Lemma 0, is its rigorous reading.

If the LUB-weighted closed loop π₃ satisfies the descent condition and error bound of Lemma 0, then

dist(𝐱(t),Ω₃(t))→0,　‖ι(φ_i(x_i(t)))−ι(L^*)‖→0.

Proof

Apply Lemma 0 to Φ₃. Every term is nonnegative, and 0≤𝒜_i(x_i(t))≤Φ₃(𝐱(t),t)/η_i→0. By definition, the distance to the LUB representation in the embedding space converges to zero. ∎

Type Correction

A state point x(t)∈X and a lattice element L^*∈𝕃 must not be compared directly. The conclusion must always be stated through the abstraction map and the order embedding.

6. Theorem 4 — Tomabechi Presence-Weighted Theorem

For presence P∈[0,1], value sign Q∈[−1,1], and weight κ>0, define

Ṽ(x,t):=V₀(x,t)−κP(x,t)Q(x,t) ≥ −κ

Let K be the forward-invariant closed-reachable component of the corresponding closed loop, and assume that the presence-weighted TCZ slice

Ω_P(t):={x∈K | Ṽ(x,t)≤θ_P}

is nonempty at every time. Define the zero residual as

Φ₄(x,t):=[Ṽ(x,t)−θ_P]₊

Theorem 4 (Tomabechi Presence-Weighted Theorem)

π_c^P = arg min_u(·) ∫ Ṽ(x^u(s),s)ds,　Ṽ = V₀−κPQ　⇒　dist(x(t),Ω_P(t)) → 0

Canonical central formula (the π_c continuity form). The residual Φ₄ below, with Lemma 0, is its rigorous reading.

Suppose that the receding-horizon optimal control

u^P,*_t,x(·)∈arg min_{u(·)∈𝒰[t,t+T], x^u(t)=x}∫_t^t+TṼ(x^u(s),s)ds,　π_c^P(t,x):=u^P,*_t,x(t+)

satisfies the conditions of Lemma 0 for Φ₄. Then

dist(x(t),Ω_P(t))→0.

Moreover, as a pointwise comparison with Q held fixed,

∂Ṽ/∂P=−κQ

Thus, all else being equal, an increase in presence lowers the effective cost when Q>0 and raises it when Q<0.

Proof

Convergence follows from Lemma 0. For the independent variables p,q, differentiating Ṽ=V₀−κpq with q fixed yields the partial-derivative formula. If both variables change simultaneously through an intervention variable r, then dṼ/dr=−κ(QdP/dr+PdQ/dr), and additional conditions are required to determine the sign of the total change. ∎

What Theorem 4 Does Not Yet Prove

∂Ṽ/∂P<0 shows only that the value decreases. For a specified future g to become a local attractor, one additionally needs ∇_xṼ(g)=0 and ∇_x²Ṽ(g)≻0, or equivalent valley-separation and descent conditions. Theorems 20 and 21 supply this directionality and valley formation.

7. Theorem 16 — Tomabechi Self-Consciousness Existence and Emergence Theorem

To interpret canonical Theorem 16 relative to history, this paper explicitly introduces the input-history and context index h. Fix a subject i and a history h, and associate a nonempty compact convex stable set with each layer of the upward-directed abstraction-level set (𝔄₁₆,≼), which has no greatest element:

K_i,α(h):=TCZ_i,α(h) ⊂ E_i,α

The space E_i,α is a locally convex Hausdorff space, and the common ambient space used to compare states under different histories is E_i:=∏_{α∈𝔄₁₆}E_i,α. The interlayer projections

p_βα:K_i,α(h)→K_i,β(h),　β≼α

are continuous and affine, satisfy p_γβ∘p_βα=p_γα and p_αα=id, and are assumed to admit a compatible point for every finite family of layers.

In addition, suppose that the layerwise feedback maps f_i,h,α:K_i,α(h)→K_i,α(h) satisfy

p_βα∘f_i,h,α=f_i,h,β∘p_βα

and induce a continuous self-map F_i,h on the inverse limit.

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

The inverse-limit space

SC_i,h:=lim←_{α∈𝔄₁₆}K_i,α(h) = {(x_α)_{α∈𝔄₁₆}∈∏_{α∈𝔄₁₆}K_i,α(h) | p_βα(x_α)=x_β} ⊂ E_i

is nonempty, compact, and convex. If there exists a continuous self-map commuting with the layerwise feedback,

F_i,h:SC_i,h→SC_i,h

then

∃S_i,h^*∈SC_i,h:　F_i,h(S_i,h^*)=S_i,h^*.

Furthermore, suppose there exist a compact Hausdorff self-representation space Rep_i,h, a closed representation relation ℜ_i,h⊂Rep_i,h×SC_i,h, and continuous maps M_i,h:SC_i,h→Rep_i,h and F_i,h^Rep:Rep_i,h→Rep_i,h such that

(M_i,h(S),S)∈ℜ_i,h　(∀S∈SC_i,h),　M_i,h∘F_i,h=F_i,h^Rep∘M_i,h

Then M_i,h is called a faithful and equivariant self-representation. Injectivity is an optional additional condition that strengthens the discrimination of self-states; it is not a defining requirement of faithfulness. When a faithful M_i,h represents S_i,h^*, the latter is an operationally defined functional and structural state of self-consciousness, and equivariance also yields F_i,h^Rep(M_i,h(S_i,h^*))=M_i,h(S_i,h^*). If F_i,h is a contraction with contraction constant 0≤L<1 with respect to a complete metric, then the fixed point is unique and

d(S_n,S_i,h^*)≤Lⁿd(S₀,S_i,h^*)

holds. This iterative convergence is called the constructive emergence of self-consciousness.

Proof

The product space ∏_{α∈𝔄₁₆}K_i,α(h) is compact by Tychonoff’s theorem. The projection-coherence conditions define closed sets, and finite coherence gives them the finite-intersection property; hence their total intersection, the inverse limit, is nonempty. Because the projections are affine, the inverse limit is convex. The Schauder–Tychonoff fixed-point theorem gives a fixed point for a continuous self-map of a nonempty compact convex set. Adding contractivity yields uniqueness and geometric convergence by the Banach fixed-point theorem. A faithful self-representation makes that fixed point self-consciousness by the operational definition. Substituting the fixed point into the equivariance equation shows that its representation is also a fixed point of the lifted dynamics. ∎

Types and Scope

SC_i,h is a “space,” whereas S_i,h^* is a “point” in that space; they must not be identified. The fixed point is also relative to the fixed history h and does not denote an invariant entity common to all inputs and all times. The theorem proves functional self-consciousness, but it does not prove the existence of qualia or any particular neural implementation.

8. Theorem 19 — The Tomabechi Free-Will Theorem

Fix the subject S_i,h^* and history h of Theorem 16 as reformulated in this paper with a history index, and suppress the index h below. At each abstraction level α∈𝕃, let the nonempty set of admissible goal-conditioned control pairs be

𝔠_i,α:={(d,π) | d∈𝔇_i,α,　π∈Pol_i,α(d)}

Each problem d has a current-state variable X_d, a discrete goal random variable G_d that is represented within the subject and takes values in a finite set, and an action output Y_d^π, with H(G_d|X_d)<∞.

Theorem 19 (The Tomabechi Free-Will Theorem)

Define free-will capacity by taking the supremum over both the problem family and the policy family:

ℱ_i(α):=sup_{d∈𝔇_i,α} sup_{π∈Pol_i,α(d)}I_d(G_d;Y_d^π|X_d) = sup_{(d,π)∈𝔠_i,α}I_d(G_d;Y_d^π|X_d) ∈ [0,∞)

and assume that this supremum is finite at every abstraction level. If, whenever α≼β, there exists an injection preserving the joint distribution and evaluation value,

ι_αβ:𝔠_i,α↪𝔠_i,β

then

α≼β ⇒ ℱ_i(α)≤ℱ_i(β).

Consequently, for the least element 0 and greatest element ⊤,

ℱ_i(0)=min_α∈𝕃ℱ_i(α),　ℱ_i(⊤)=max_α∈𝕃ℱ_i(α).

If H(G_d|X_d)=0 for every problem at the physical layer, then ℱ_i(0)=0. Moreover, if ℱ_i(⊤)>ℱ_i(0), then

f_i(α):=[ℱ_i(α)−ℱ_i(0)]/[ℱ_i(⊤)−ℱ_i(0)]

is the unique increasing affine normalization that fixes both endpoints, and it satisfies f_i(0)=0 and f_i(⊤)=1.

Furthermore, a goal-independent random output Y⊥G|X has zero capacity. By contrast, under a measurable deterministic policy Y=φ(X,G), if, for P_X-almost every x, the map g↦φ(x,g) is injective, then that problem attains

I(G;Y|X)=H(G|X)

Free-will capacity does not require randomness and is compatible with deterministic implementation.

Proof

By the evaluation-value-preserving injection, the set of mutual-information values attainable at a lower abstraction level is contained in the evaluation set at a higher abstraction level. Set inclusion cannot decrease a supremum, so monotonicity follows. The endpoint minimum and maximum follow from 0≼α≼⊤. At the physical layer, I(G;Y|X)≤H(G|X)=0. The denominator in the normalization formula is positive, and solving the two linear endpoint equations shows that the increasing affine map sending the endpoints to 0 and 1 is unique. Conditional independence gives zero mutual information. Under a deterministic injective policy, H(G|Y,X)=0, and therefore I(G;Y|X)=H(G|X). ∎

The Precise Meaning of “Proportional”

What is proved is monotone nondecrease with respect to abstraction level, not linear proportionality between raw capacity and an abstraction index. “Proportionality” is made precise as the uniqueness of an affine normalization fixing the two endpoint capacity values at 0 and 1. Positive capacity also requires the existence of a problem for which H(G|X)>0 and goal differences are mapped to differences in action.

9. Theorem 20 — Tomabechi Symbolic-Presence Directionality Theorem

Let the information available to subject i be the sublattice 𝕃_i(t)⊊𝕃, let the set of representations denoted by a symbol σ be W_σ⊂𝕃_i(t), and define its target address by

u_σ:=∨W_σ

Let the representational set Z_u⊂X be nonempty and closed, and define its distance-type function D_u≥0 by

D_u(x)=0 ⇔ x∈Z_u

Let P be baseline presence, let the symbol σ produce amplification Sym_σ, and let λ≥0 be the coupling coefficient. Define symbolic presence by

P_σ(x,t):=P(x,t)+λSym_σ(x,t),　q_σ∈[−1,1]

Within a neighborhood during a single application interval, suppose D_u,V₀,P_σ∈C¹, s∈C¹, and s′<0, and let S_u(x):=s(D_u(x)). Let the mobility matrix be continuous and symmetric, with M(x)≽γI and γ>0, and set

Ṽ_σ=V₀−κq_σP_σS_{u_σ},　ẋ=−M∇Ṽ_σ,　d_u:=−M∇D_u

Assume that the closed-loop vector field is locally Lipschitz, and define ‖ξ‖_M²:=ξ^TMξ. Within any one application interval, hold u_σ,Z_u,D_u fixed.

Theorem 20 (Tomabechi Symbolic-Presence Directionality Theorem)

Let q_σ>0. Suppose that, off the target within a compact forward-invariant set K, there exist b,c>0 such that the following inequalities hold uniformly:

−⟨∇D,M∇V₀⟩ + κq_σS⟨∇D,M∇P_σ⟩ ≤ b‖∇D‖_M²　(20.A)

κq_σP_σ(−s′(D)) ≥ b+c　(20.B)

and suppose that on K∖Z_{u_σ}, ∇D≠0. Then

Ḋ ≤ −c‖∇D‖_M²<0,　⟨ẋ,d_{u_σ}⟩_M⁻¹=−Ḋ>0.　(20.1)

If, in addition, ‖∇D‖_M²≥2μD and dist(x,Z_{u_σ})≤C√D, then

D(t)≤D(0)e^−2μct,　dist(x(t),Z_{u_σ})≤C√D(0)e^−μct→0.　(20.2)

Proof

Since ∇S=s′(D)∇D,

Ḋ=−⟨∇D,M∇V₀⟩+κq_σS⟨∇D,M∇P_σ⟩+κq_σP_σs′(D)‖∇D‖_M².

Substitution of (20.A) and (20.B) gives the first conclusion. Moreover,

⟨ẋ,d_u⟩_M⁻¹=ẋ^TM⁻¹(−M∇D)=−⟨∇D,ẋ⟩=−Ḋ.

With the additional PL-type bound, Ḋ≤−2μcD, and exponential convergence follows from Grönwall’s inequality. ∎

What Is Not Proved

This theorem gives a direction toward a particular LUB u_σ within partial information. It does not entail u_σ=⊤, the truth or goodness of the symbolic content, or comparability with another LUB. Proving directionality does not prove that the direction is correct.

10. Theorem 21 — Tomabechi Subsumption-Poset Presence Directionality Theorem

Let the distribution of symbols to which the subject is exposed be the measure μ_i, and define, from the reconstruction kernel K(x,a),

S_{μ_i}(x):=∫_{𝕃_i}K(x,a)μ_i(da),　b:=∨supp μ_i.　(21.1)

Assume b∈𝕃_i. This is guaranteed, for example, if supp μ_i is finite or if 𝕃_i is a complete sublattice. Let the representational center be x_b, and let the closed ball be U_b=B̄_r(x_b)⊂X, where r>0.

Here p>0 is not itself the normalized presence P∈[0,1] of Theorem 4; it is an unbounded effective presence gain that combines symbolic amplification and coupling strength.

Definition 21.1 ((r,m)-Directional Bias)

When S_{μ_i}∈C²(U_b) satisfies

∇S_{μ_i}(x_b)=0,　−∇²S_{μ_i}(x)≽mI　(x∈U_b),　m>0　(21.2)

the measure μ_i is said to be biased toward b with parameters (r,m).

Theorem 21 (Tomabechi Subsumption-Poset Presence Directionality Theorem)

Let 𝕃_i⊊𝕃 and ⊤∉𝕃_i, and suppose that μ_i is biased in the sense above. Suppose V₀,S_{μ_i}∈C²(U_b) and that on U_b there exist constants B,β≥0 such that

‖∇V₀‖≤B,　∇²V₀≽−βI　(21.3)

and let

Ṽ_μ,p:=V₀−κpS_{μ_i},　p>p_crit:=(1/κm)max{β,B/r}.　(21.4)

Then the following statements hold.

Ṽ_μ,p has, in the interior of U_b, a unique minimizer x_b^*.
With c:=κpm−β>0, one has ‖x_b^*−x_b‖≤B/c.
Suppose A_b=A_b^T∈C¹, A_b(x)≽γI, and γ>0. If the closed loop ẋ=−A_b∇Ṽ_μ,p has a unique solution and starts in a nonempty forward-invariant sublevel set that contains x_b^* and whose closure is contained in int U_b—denote this set by 𝒞_b—then it converges exponentially to x_b^*.
Even if the support of a finite goal variable G lies within the biased branch, if H(G|X)>0 and there exists a measurable π such that, for P_X-almost every x, the map g↦π(x,g) is injective, then the deterministic output Y_π:=π(X,G) satisfies I(G;Y_π|X)=H(G|X)>0. Convergence to a biased well is compatible with positive free-will capacity.

Proof

First,

∇²Ṽ_μ,p=∇²V₀−κp∇²S_{μ_i}≽(κpm−β)I=cI

so the function is strongly convex. At a boundary point x∈∂U_b, with d=x−x_b and ‖d‖=r,

⟨∇Ṽ_μ,p,d⟩≥−Br+κpmr²>0.

Hence the minimizer exists in the interior rather than on the boundary, and strong convexity makes it unique. Strong monotonicity gives c‖x_b^*−x_b‖≤‖∇V₀(x_b)‖≤B. Setting W=Ṽ−Ṽ(x_b^*) yields

Ẇ≤−γ‖∇Ṽ‖²≤−2γcW

and therefore exponential convergence. Finally, injectivity gives H(G|Y_π,X)=0, so the mutual information equals H(G|X). ∎

The Distinction Between Freedom and Autonomy

Even if the support of the alternatives is confined to a biased branch, capacity may be positive when that branch contains multiple goals that map to actions. Thus, “making a choice” and “having an unmanipulated space of alternatives” are distinct questions. If an external reference LUB u_ref and reference admissible set 𝒜_ref⊂𝕃 are specified separately, and if b is incomparable with u_ref or b∉𝒜_ref, the theorem’s local well cannot be identified with the globally maximal direction of subsumption.

11. Theorem 22 — Tomabechi Higher-LUB Presence Theorem

Starting from an initial LUB u₀∈𝕃, accumulate new information as a join without deleting prior information:

u_n+1:=u_n∨v_n+1　(22.1)

This is equivalent to updating the information set by join closure. Let the representational center, presence field, and background landscape at each stage be x_{u_n},S_{u_n},R_n, respectively, and let the local region be U_n:=B̄_{r_n}(x_{u_n})⊂X, with r_n>0. Each p_n>0 is likewise not normalized presence but an unbounded effective presence gain.

Theorem 22 (Tomabechi Higher-LUB Presence Theorem)

At every stage, suppose R_n,S_{u_n}∈C²(U_n), m_n>0, and B_n,β_n≥0, and suppose that on U_n,

∇S_{u_n}(x_{u_n})=0,　−∇²S_{u_n}≽m_nI,　‖∇R_n‖≤B_n,　∇²R_n≽−β_nI

Ṽ_n:=R_n−κp_nS_{u_n},　p_n>(1/κm_n)max{β_n,B_n/r_n}　(22.2)

hold. Let A_n=A_n^T∈C¹, A_n≽γ_nI, and γ_n>0, and apply the closed loop ẋ=−A_n∇Ṽ_n from the stage-start time t_n. Assume the same unique-solution and invariant-sublevel-set conditions as in Theorem 21, reachability in the sense that the switching state from the preceding stage lies in the basin of attraction of the next stage, and time-scale separation sufficient to stabilize each stage before switching. Then

u_n≼u_n+1≼⊤,　v_n+1≰u_n ⇒ u_n≺u_n+1.　(22.3)

Each stage has a unique local minimizer x_n^*; let c_n:=κp_nm_n−β_n>0. If the stage-n vector field is continued without switching, denote the resulting hypothetical trajectory by x_n^fr(t_n+τ). Then

Ṽ_n(x_n^fr(t_n+τ))−Ṽ_n(x_n^*)≤e^{−2γ_nc_nτ}[Ṽ_n(x(t_n))−Ṽ_n(x_n^*)],　τ≥0.　(22.4)

The actual switched trajectory coincides with this hypothetical trajectory for 0≤τ≤T_n. If the distance estimate is ‖x_n^fr(t_n+τ)−x_n^*‖≤C_ne^−γ_nc_nτ and 0<ε_n<C_n, then a sufficient waiting time to attain error ε_n is

T_n≥max{0,(1/γ_nc_n)log(C_n/ε_n)}.　(22.5)

If the admissible problem families of Theorem 19 embed into higher stages while preserving their joint distributions, then, because the abstraction variable ranges over 𝕃,

ℱ_i(u_n)≤ℱ_i(u_n+1).　(22.6)

If 𝕃 is directed-complete, then u_∞:=∨_nu_n exists and u_∞≼⊤. In order-theoretic terms, the sequence can be said to “approach emptiness” only when the additional condition ∨_nu_n=⊤ holds.

Proof

By the definition of join, u_n≼u_n∨v_n+1=u_n+1. If v_n+1≰u_n but u_n+1=u_n, the upper-bound property of the join would give v_n+1≼u_n, a contradiction. The unique well and exponential convergence at each stage follow by applying the strong-convexity proof of Theorem 21 with stage indices. Capacity monotonicity follows from the evaluation-value-preserving embedding of Theorem 19. The increasing sequence is a directed set, so directed completeness gives the existence of its supremum. ∎

On Reaching Emptiness

The join update alone does not imply u_n≠⊤. If nonattainment of emptiness at any finite stage is needed, ∀n<∞,u_n≺⊤ must be assumed independently. Nor does Theorem 22 say that new nonsubsumed information emerges automatically.

12. Theorem 23 — Impermanence Theorem

Let 𝒵 be the complete state space encompassing life, cognition, and the environment, and define the complete state on the interval of survival I_alive by

z(t):=(x₀(t),(x_α(t))_α>0,e(t))

Environmental, memory, and historical variables required to make 𝒮 single-valued are included among the components of z; the clock coordinate itself, however, is not added merely to trivialize the nonrecurrence conclusion. Here, the balance law of the omitted Theorem 15 (Tomabechi Cognitive–Physical Entropy Exchange and Conservation Theorem) is rendered as a function of the complete state, with 𝒮(z(t))=S_gen(t) and Π_gen(t)=Π(t) identified along the trajectory. Define the generalized total entropy as the time-independent, single-valued state functional

𝒮:𝒵→ℝ,　𝒮(z):=Ŝ_phys(z)+Σ_α>0w_αĤ_α(z)

and assume, as in the canonical formulation, that w_α>0 and, if the set of layers is infinite, that the series converges absolutely. Assume that t↦𝒮(z(t)) is absolutely continuous and satisfies, almost everywhere,

d𝒮(z(t))/dt=Π_gen(t)≥0

Because mere nondecrease does not exclude equilibrium, strict dissipation during life is imposed as an independent condition.

Condition 23-A (Persistent Strict Dissipation)

∀t₁,t₂∈I_alive,　t₂>t₁ ⇒ ∫_t₁^t₂Π_gen(s)ds>0.　(23.A)

For cognitive updating below emptiness, impose the following additional condition.

Condition 23-B (Infinite, Non-Zeno, Nonterminal Subsumption Updating)

At every finite stage, u_n≺⊤, and some new information v_n+1≰u_n is supplied.
Every update satisfies the presence threshold, reachability, and time-scale separation conditions of Theorem 22, with 0<T_n<∞, t_n+1=t_n+T_n, and Σ_n=0^∞T_n=∞.
The representation is faithful, successive unique minimizers satisfy δ_n:=‖x_n+1^*−x_n^*‖>0, and the line segment [x_n^*,x_n+1^*]⊂U_n+1.

Define the sublevel set, the closed-loop reachable component, and the TCZ at each stage by

Λ_n(θ_n):={x∈U_n | Ṽ_n(x)−Ṽ_n(x_n^*)≤θ_n}

K_n:=cl⋃_τ≥0ℛ_n^fr(τ;x(t_n)),　TCZ_n^cl:=Λ_n(θ_n)∩K_n

and let θ₀≥0 and, for every n≥0, let 0≤θ_n+1<(c_n+1/2)δ_n². Here ℛ_n^fr is the reachable set of the hypothetical closed loop obtained by continuing, without switching, the stage-n vector field of Theorem 22.

Theorem 23 (Impermanence Theorem)

Under Condition 23-A, for any t₁,t₂∈I_alive with t₂>t₁,

z(t₂)≠z(t₁).　(23.1)

That is, a complete life–environment process with persistent dissipation never returns to an exactly identical complete state over any nonzero time interval.

Furthermore, under Condition 23-B, u_n≺u_n+1 and

TCZ_n+1^cl≠TCZ_n^cl.　(23.2)

Consequently, no finite stage below emptiness is a terminal TCZ, and the sequence of stagewise TCZs does not become fixed at any finite stage. If “operation” is defined as the successive implementation of the goal-directed closed loop at each stage, then the full sequence of stages does not accumulate at any finite physical time, and there is no time at which operation terminates permanently. This does not deny the possible existence of a limiting TCZ for the sequence of sets.

Symbols of Theorem 23
Symbol	Meaning
𝒵,　z(t)=(x₀,(x_α)_α>0,e)	the complete state space, and the complete state bundling the physical layer, all abstraction layers, and the environment
𝒮(z)	the generalized total-entropy state functional (the balance of omitted Theorem 15 promoted to a function of the complete state)
Π_gen(t)	the total entropy production rate; d𝒮/dt=Π_gen≥0
I_alive	the interval of being alive; Condition 23-A is sustained strict dissipation on it
u_n,　v_n+1	the stage-n LUB and the non-subsumed new information (Condition 23-B)
x_n^*,　δ_n,　θ_n	the unique stage minimizer, the inter-stage distance ‖x_n+1^−x_n^‖, and the sublevel threshold
Λ_n(θ_n),　K_n,　TCZ_n^cl	the sublevel set, the closed reachable part of the frozen loop, and their intersection: the stage TCZ

Proof

From the balance equation and (23.A),

𝒮(z(t₂))−𝒮(z(t₁))=∫_t₁^t₂Π_gen(s)ds>0.

If z(t₂)=z(t₁), then the state functional 𝒮 would also have equal values at the two times, a contradiction. Thus (23.1) holds.

Next, Condition 23-B(1) and Theorem 22 give u_n≺u_n+1. By the exponential convergence of Theorem 22, x_n^* belongs to the closure of the reachable set at stage n; because its potential difference is zero, x_n^*∈TCZ_n^cl. Since the line segment lies within U_n+1, c_n+1-strong convexity on that region yields

Ṽ_n+1(x_n^*)−Ṽ_n+1(x_n+1^*)≥(c_n+1/2)δ_n²>θ_n+1.

Hence x_n^*∉Λ_n+1(θ_n+1), and therefore x_n^*∉TCZ_n+1^cl; the two TCZs are thus distinct. Because this argument repeats at every finite stage, there is no terminal or fixed TCZ at any finite stage. Finally, ΣT_n=∞ ensures that the switching times do not exhibit Zeno accumulation at a finite time. ∎

The Precise Meaning of “Always Changing”

The second-law condition Π≥0 alone neither excludes equilibrium nor implies that every individual observable changes at every instant. The strong nonrecurrence conclusion of this theorem requires (23.A). Moreover, because π_c may continue to operate even as a time-invariant function, the theorem does not claim that “the policy function is continually transformed.” What changes is the complete state and, under Condition 23-B, the goal TCZ including its reachability component. The physical impermanence of the body does not cease even upon attaining emptiness.

13. Theorem 24 — All Conditioned Existence Is Suffering Theorem

Fix the same subject as in Theorem 19 and suppress the subject and history indices. Stratify the baseline evaluation V₀ of Theorems 1–4 by abstraction level, and, at a∈𝕃, let V_a(x,t)≥0 denote the resulting nonnegative evaluation of discomfort and inconsistency. Define its discounted infinite-horizon optimal value by

J_a,ρ^*(x,T):=min_{π∈Pol_a(x,T)}∫_T^∞e^−ρ(t−T)V_a(x_x,T^π(t),t)dt,　x_x,T^π(T)=x,　ρ>0.　(24.1)

Assume that the evaluation along each trajectory is Lebesgue measurable, that at least one finite-cost policy exists for every (a,x,T), and that an optimal policy attaining the minimum exists. Thus 0≤J_a,ρ^*(x,T)<∞.

Condition 24-A (Impossibility of Permanently Zero Suffering below Emptiness)

∀a≺⊤, ∀x∈X_a, ∀T≥0:　∄π∈Pol_a(x,T): V_a(x_x,T^π(t),t)=0　(for Lebesgue-almost every t≥T).　(24.A)

Theorem 24 (All Conditioned Existence Is Suffering Theorem)

Under Condition 24-A and the existence of an optimal policy, for every abstraction level below emptiness a≺⊤, initial state x∈X_a, and starting time T≥0,

J_a,ρ^*(x,T)>0.　(24.2)

That is, in any below-emptiness model satisfying Condition 24-A, no optimal control can realize a trajectory of permanently zero suffering.

Symbols of Theorem 24
Symbol	Meaning
a≺⊤	an abstraction level below emptiness; ⊤ is the top (emptiness)
V_a(x,t)≥0	the nonnegative discomfort/instability/incoherence evaluation at level a
Pol_a(x,T)	the set of admissible policies for the initial pair (x,T)
ρ>0	the discount rate
J_a,ρ^*(x,T)	the discounted infinite-horizon optimal residual value (Eq. 24.1)
x_x,T^π(t)	the closed-loop trajectory under policy π with x_x,T^π(T)=x
Condition 24-A	no-permanent-zero-suffering below emptiness: no policy keeps V_a=0 almost everywhere forever

Proof

Fix (a,x,T) and suppose that J_a,ρ^*(x,T)=0. The integrand e^−ρ(t−T)V_a is nonnegative, and a policy attaining the minimum exists. If the integral of a nonnegative measurable function is zero, the function is zero Lebesgue-almost everywhere. It follows that V_a=0 almost everywhere along the optimal trajectory. This contradicts (24.A). Hence the optimal value is positive. ∎

Operational Meaning of All Conditioned Existence Is Suffering

This is not the proposition that “one subjectively feels pain at every instant.” Under Condition 24-A, it is a structural nonfulfillment: below emptiness, the evaluation-minimization problem itself cannot be eliminated permanently, and a positive residual cost remains after optimization for every fixed initial condition. The theorem does not assert a uniform positive lower bound common to all states. Moreover, because the Ṽ of Theorem 4 may take negative values, this theorem measures suffering using the nonnegative baseline evaluation V_a.

14. Theorem 25 — Non-Self of All Dharmas Theorem

In this theorem, “all dharmas” means every entity, phenomenon, relation, memory, and symbol treated by the present model. “Non-self” does not deny functional, historical, or relational existence. Rather, it means that there is no immutable intrinsic nature that exists independently of other dharmas, input histories, and interlayer relations, individuates a dharma in a fixed manner, and exerts an additional causal effect.

25.1 The Current Self-Process and Fixed Points

Let ℋ be the set of input histories, and suppose that, for each h∈ℋ, there is a contraction operator F_i,h from Theorem 16 with a unique fixed point S_i,h^*∈SC_i,h⊂E_i. Define the three typed representations of the self-process by

R_i[h]:=(Self_i[h],Ego_i[h],TCZ_i[h])

Let H be the random global history of the entire model, including admissible inputs, interlayer representations, and histories of relations with other entities. Let the random variable corresponding to the self-process be R_i:=R_i[H]. Self is a semantic selection-and-transformation operator, Ego is a control policy, and TCZ is a stable set in state space. The three do not have the same mathematical type; they are differently typed representations of the same cognitive process.

Condition 25-A (History Sensitivity and Functional Completeness of the Self-Process)

For some h,h′∈ℋ, within the common ambient space E_i, S_i,h^*≠S_i,h′^*.
R_i provides a functionally complete description of the future causal outputs Y_i⁺. For any candidate additional self-variable Σ_i independent of the input history, and for every history-intervention value h and candidate-variable-intervention value s,

ℒ(R_i,Y_i⁺ | do(H=h),do(Σ_i=s))=ℒ(R_i,Y_i⁺ | do(H=h))　(∀h,∀s).　(25.A)

25.2 Layerwise Existence Profiles of All Dharmas

Let the totality of abstract concepts be the same complete lattice used in the existing system:

𝕃=(𝕃,≼,∨,∧,⊤,⊥),　⊥=0 (physical layer),　⊤=emptiness

α≼β means that β subsumes α and is at least as abstract. Let 𝔇 be the index set of all dharmas treated by this model. Here, d∈𝔇 is a descriptive index and is not itself presupposed to be an entity independent of relations.

For each abstraction level α∈𝕃, introduce a typed representation space E_α and adjoin an absence symbol ∂_α∉E_α indicating that the dharma is not directly manifested at that layer.

E_α^∂:=E_α⊔{∂_α},　Z_d[h]:=(z_d,α[h])_α∈𝕃∈∏_α∈𝕃E_α^∂.　(25.B1)

Call Z_d[h] the layerwise existence profile of dharma d, and define its manifestation support by

Supp_h(d):={α∈𝕃 | z_d,α[h]≠∂_α}

Condition 25-B (Existence Across the Subsumption Partial Order)

For every d∈𝔇 and h∈ℋ, the manifestation support is nonempty, upward closed, and contains the greatest element.

Supp_h(d)≠∅,　α∈Supp_h(d) ∧ α≼β ⇒ β∈Supp_h(d),　⊤∈Supp_h(d).　(25.B2)

Identify the representation space at the highest abstraction level with the singleton E_⊤:={★_⊤}, and set

z_d,⊤[h]=★_⊤　(∀d∈𝔇,∀h∈ℋ).　(25.B3)

★_⊤ is the common non-individuating representation assigned to all dharmas at “emptiness”; it is not a repository that preserves a separate fixed entity for each dharma. Moreover, ★_⊤ is a non-individuation marker in the representation space, not an entity belonging to 𝔇, a universal subject, or an independent causal variable.

The Type of “Existing Across the Layers”

This means that every dharma is situated as a profile indexed by all elements of 𝕃; it does not mean that every coordinate is always non-absent. Because ∂_α is assigned at a layer where there is no direct manifestation, the absence of a living self-regulatory realization from the physical layer after death is consistent with relational persistence at higher abstraction layers.

25.3 Vertical Subsumption and Bidirectional Relations with Other Dharmas

For a history h, define the layerwise dharma nodes and vertical subsumption edges by

V[h]:={(d,α) | d∈𝔇, α∈Supp_h(d)},
E_≼[h]:={((d,α),(d,β)) | α≼β, α,β∈Supp_h(d)}

Furthermore, let Rel be the set of relation labels, and represent causal, constitutive, role-based, and symbolic relations with other dharmas by the set of labeled directed edges E_rel[h].

Condition 25-C (Dependent-Origination Relations with Converse Roles and Connectedness)

For every relation label r∈Rel, there is a converse-role label r^⌣, and

((d,α),r,(e,β))∈E_rel[h] ⇔ ((e,β),r^⌣,(d,α))∈E_rel[h],　(r^⌣)^⌣=r.　(25.C1)

Every dharma has, at one or more layers, a relational edge to another dharma.
Let the horizontal relation graph obtained by projecting the layerwise relations onto the dharma indices be

G_rel[h]:=(𝔇,{{d,e} | d≠e, ∃α,β,r: ((d,α),r,(e,β))∈E_rel[h]})　(25.C2)

The underlying undirected graph is connected. Thus, “everything is connected” does not mean a complete graph having a direct edge between every pair of dharmas. It means that any dharma can be reached from any other through a finite path composed of relational edges to other dharmas. Connectedness is not trivialized by relying solely on subsumption into the common greatest element ⊤.

Let Inc_d,α[h] be the collection of incoming and outgoing relational edges of dharma d at layer α, and let Vert_d,α[h] be its vertical subsumption neighborhood. Define its relational state by

Γ_d,α[h]:=(Z_d[h],Vert_d,α[h],Inc_d,α[h]).　(25.C3)

Let the random variable corresponding to the global history H be Γ_d,α:=Γ_d,α[H]. Thus, a dharma is described not as an isolated point, but as a typed state comprising its layerwise profile, vertical subsumption relations, relations with other dharmas, and history.

Example of father, mother, and child.

Let 𝔇_born⊂𝔇 be the set of entities that have a biological birth, and let ℋ_gene⊂ℋ be the set of histories in which genealogical relations are defined. In a simplified biological genealogy model, the roles of father, mother, and child can be represented by the following converse-role pairs.

∀h∈ℋ_gene　∀f,c∈𝔇: FatherOf(f,c;h) ⇔ HasFather(c,f;h),
∀h∈ℋ_gene　∀m,c∈𝔇: MotherOf(m,c;h) ⇔ HasMother(c,m;h),
∀h∈ℋ_gene　∀c∈𝔇_born　∃f,m∈𝔇: FatherOf(f,c;h) ∧ MotherOf(m,c;h).　(25.C4)

A father is a father in relation to a child (son or daughter), and a mother is likewise a mother in relation to a child. At the same time, a child relationally exists as someone who has a father and who has a mother. FatherOf(f,c;h)⇔HasFather(c,f;h) is the converse-role description of the same relation; it means neither the symmetry FatherOf(f,c;h)⇔FatherOf(c,f;h) nor symmetry of causal action. Nor does it mean that the father, mother, and child are all simultaneously alive or physically present at the current physical layer.

After death and memory.

Let h_t be the complete history up to time t, let τ_d be the time of death of person d, and let AliveRealization₀(d,t) indicate whether the living self-regulating process is directly implemented at the physical layer. In the present model, in which the complete history retains the causal history of past bodily existence, after death the person is no longer directly implemented at the physical layer as a living process but is represented in at least one higher historical layer.

t≥τ_d ⇒ AliveRealization₀(d,t)=0 ∧ ∃α_hist: 0≺α_hist≺⊤ ∧ z_{d,α_hist}[h_t]≠∂_{α_hist}.　(25.C5)

Furthermore, insofar as memories, records, social roles, or causal effects of actions are retained by carriers, they also constitute relational representations at higher layers. For example, the memory content Mem_j,t(d)∈E_α of another person j is part of a higher abstraction layer, while the corresponding brain state, document, or medium serves as a physical-layer carrier. What remains after death is not the same fixed subject but representations and traces that depend on other persons, media, and histories. If the carriers of particular memories or records are lost, the persistence of those specific representations is not guaranteed. A corpse or material trace is distinguished as a physical entity separate from the person’s living self-regulating process.

25.4 Functional Completeness for All Dharmas

For arbitrary d∈𝔇 and α∈𝕃, let Y_d,α⁺ be the future causal output at that layer. When the dharma is not directly manifested at that layer, let the output take values in the type 𝒴_α^∂:=𝒴_α⊔{∂_α^Y}, formed by adjoining the absence-output symbol ∂_α^Y, thereby defining its type at every layer. A candidate intrinsic-nature variable Σ_d,α means an additional variable hypothesized to individuate d in a fixed manner independently of history, layerwise profile, subsumption relations, and relations with other dharmas.

Condition 25-D (Relational Functional Completeness for All Dharmas and All Abstraction Layers)

Γ_d,α contains a functionally complete representation of the causally effective layerwise and relational state of the dharma. Therefore, for every admissible d,α,h,s,

ℒ(Γ_d,α,Y_d,α⁺ | do(H=h),do(Σ_d,α=s))=ℒ(Γ_d,α,Y_d,α⁺ | do(H=h)).　(25.D)

Define Atman(d,α) as the proposition that “there exists a candidate intrinsic-nature variable Σ_d,α that individuates d in a fixed manner independently of history, interlayer relations, and relations with other dharmas, and that has a nonredundant causal effect beyond Γ_d,α.”

Theorem 25 (Non-Self of All Dharmas Theorem)

Under Condition 25-A, there exists no fixed self-consciousness state common to all histories for the subject of Theorem 16.

¬∃S₀∈⋂_h∈ℋSC_i,h　∀h∈ℋ: F_i,h(S₀)=S₀.　(25.1)

Furthermore, under Conditions 25-B, 25-C, and 25-D, for every dharma d treated by this model and every abstraction layer α—including the physical layer 0 and emptiness ⊤—there exists no fixed intrinsic nature that is independent of the relational state and has a nonredundant causal effect beyond that state.

∀d∈𝔇　∀α∈𝕃:　¬Atman(d,α).　(25.2)

Accordingly, all dharmas are situated as layerwise implementations, representations, and traces across the subsumption partial-order system whose least element is the physical layer and whose greatest element is emptiness. At the same time, they arise within a web of dependent origination that includes converse-role relations with other dharmas. Yet at no abstraction layer does a “self,” construed as a fixed entity independent of other dharmas and histories, arise within the minimal causal description of this model.

Symbols of Theorem 25 (extended)
Symbol	Meaning
ℋ,　h,　H	the set of input histories, an individual history, and the global random history
R_i[h]	the three typed representations of the self-process (Self, Ego, TCZ)
𝕃,　⊥=0,　⊤	the complete lattice of abstraction; bottom = the physical layer, top = emptiness
𝔇,　d	the index set of all dharmas treated by the model, and an index
E_α^∂,　∂_α	the representation space at layer α, with the absence symbol "not directly present at this layer"
Z_d[h],　Supp_h(d)	the layerwise existence profile and its presence support: nonempty, upward closed, containing ⊤ (25.B2)
★_⊤	the non-individuating representation of emptiness, common to all dharmas; it preserves no individuals (25.B3)
Rel,　r^⌣	relation labels and converse-role labels (the FatherOf ⇔ HasFather pattern, 25.C1)
G_rel[h]	the horizontal relation graph projected to dharma indices; connected (25.C2)
Γ_d,α	the relational state: layerwise profile, vertical subsumption neighborhood, incident relation edges (25.C3)
Y_d,α⁺,　∂_α^Y	the future causal output at layer α, and the absent-output symbol
Σ_d,α,　Atman(d,α)	a candidate intrinsic nature, and the proposition "independent, rigidly individuating, with non-redundant causal effect"

Proof

Stage 1 — Existing self-process fixed-point proof. Suppose that there exists a common fixed point S₀ belonging to the intersection. Because the fixed point of each F_i,h is unique by contractivity, for the two histories h,h′ in Condition 25-A(1), S₀=S_i,h^*=S_i,h′^*. This contradicts S_i,h^*≠S_i,h′^*. Hence (25.1).

Stage 2 — Generalization to all dharmas and all layers. Fix arbitrary d∈𝔇 and α∈𝕃, and suppose that Atman(d,α) holds. If an intervention on the candidate intrinsic-nature variable Σ_d,α changes the joint interventional law of (Γ_d,α,Y_d,α⁺), either directly or through Γ_d,α, it contradicts the functional completeness in Condition 25-D. If it does not change that law, then Σ_d,α has no nonredundant causal effect and fails to satisfy the definition of Atman(d,α). Either case is a contradiction.

Because d and α were arbitrary, (25.2) holds at every abstraction layer from the physical layer to emptiness. At the greatest abstraction level in particular, Condition 25-B makes the greatest-layer coordinate z_d,⊤[h]=★_⊤ itself common to all dharmas and devoid of fixed content that individuates one dharma from another. Any difference that may remain in Γ_d,⊤ is a non-intrinsic difference inherited from lower-layer profiles, histories, and relations, not a fixed intrinsic nature preserved in emptiness. Even after the living physical implementation ends at death, higher-layer components may persist relationally as memories, records, roles, and causal histories; this is not the persistence of an independent fixed subject. ∎

Conclusion as Dependent Origination

Existence = Layerwise Profile + Subsumption Relations + Converse-Role Relations with Other Dharmas + History

Functional and relational selves and forms of existence remain well-defined. However, in the minimal causal description of this model, no fixed intrinsic nature subsisting independently of and beyond those relational descriptions exists at any abstraction layer. This is what the present system calls the non-self of all dharmas.

Consistency with Theorem 16 and Scope of the Proof

Theorem 16 proves F_i,h(S_i,h^*)=S_i,h^* relative to each fixed history. What Theorem 25 denies is a fixed point common to all histories, together with an independent, fixed, causally nonredundant intrinsic nature beyond the relational state. The existence of a coherent self-process at each point in time is compatible with the nonexistence of a fixed intrinsic nature.

Moreover, non-self does not follow merely from the existence of the subsumption graph, the connectedness of the graph, or the father–mother–child relations. It is mathematically possible to assign a unique fixed attribute to every vertex of a connected graph. The decisive condition for deriving non-self across all dharmas and all layers is the relational functional completeness of Condition 25-D. This theorem is relative to the explicitly stated functional and relational model; it does not claim that mathematics alone unconditionally excludes every metaphysical entity of every kind.

Emptiness ⊤ is not a place that preserves the substantial personhood of the dead. Memories, records, and social roles are higher-layer representations dependent on carriers and histories; the permanent persistence of their specific representations does not follow from this theorem.

15. Theorem 26 — Nirvanic Tranquility Theorem

Let ⊤ be the highest abstraction level. Here x∈X_⊤ is a complete state that includes the brain and body, and ℬ_alive⊂X_⊤ is the set on which vital activity is maintained. Assume that there exists a single Borel-measurable Markov feedback π_⊤⁰:[0,∞)×X_⊤→U that is simultaneously optimal for every initial pair and satisfies

J_⊤,ρ^*(x,T)=∫_T^∞e^−ρ(t−T)V_⊤(x_x,T^π_⊤⁰(t),t)dt　(∀(x,T)∈ℬ_alive×[0,∞))

Under this single feedback, define the family of zero-suffering sets at the highest abstraction level by

𝒩_⊤(T):=ℬ_alive∩{x | J_⊤,ρ^*(x,T)=0}.　(26.1)

Condition 26-A (Zero-Suffering Sets and Stability)

The closed loop under π_⊤⁰ is forward complete and renders ℬ_alive forward invariant.
For every T≥0, 𝒩_⊤(T) is nonempty and closed, and the family of sets is forward invariant under π_⊤⁰; that is, x(T)∈𝒩_⊤(T)⇒∀t≥T, x(t)∈𝒩_⊤(t).
There exist c₁,c₂,λ>0 and a time-varying Lyapunov function W_⊤:ℬ_alive×[0,∞)→[0,∞) such that

c₁dist(x,𝒩_⊤(t))²≤W_⊤(x,t)≤c₂dist(x,𝒩_⊤(t))²　(26.A)

D⁺_tW_⊤(x(t),t)≤−λW_⊤(x(t),t)　(26.B)

where (26.A) holds for every x∈ℬ_alive and t≥0, and (26.B) holds along every trajectory of the specified closed loop. In addition, let ω:[0,∞)→[0,∞) be continuous and increasing, with ω(0)=0, and suppose that for every x∈ℬ_alive and t≥0,

0≤J_⊤,ρ^*(x,t)≤ω(dist(x,𝒩_⊤(t))).　(26.C)

Theorem 26 (Nirvanic Tranquility Theorem)

Under Condition 26-A, for every starting time T≥0 and every initial state x_T∈ℬ_alive, if x(T)=x_T, then

W_⊤(x(t),t)≤W_⊤(x(T),T)e^−λ(t−T),　dist(x(t),𝒩_⊤(t))≤√(W_⊤(x(T),T)/c₁)e^{−λ(t−T)/2}→0.　(26.2)

Consequently, J_⊤,ρ^*(x(t),t)→0. If the starting state satisfies x(T)∈𝒩_⊤(T), then J_⊤,ρ^*(x(t),t)=0 is maintained at all times while vital activity continues, and, under the optimal policy π_⊤⁰, V_⊤(x(t),t)=0 holds Lebesgue-almost everywhere. This forward-invariant state of zero suffering is called tranquility.

Define the typed proposition PZS(a,x,T) (with x∈X_a) to mean that “some admissible policy, starting from x(T)=x, permanently maintains V_a=0 almost everywhere.” Together with Theorem 24, the following holds without conflating types:

∀T≥0:　[∀a≺⊤, ∀x∈X_a: ¬PZS(a,x,T)] ∧ [∀x∈ℬ_alive: PZS(⊤,x,T) ⇔ x∈𝒩_⊤(T)].　(26.3)

Symbols of Theorem 26
Symbol	Meaning
⊤	the top abstraction level; philosophically read as emptiness
ℬ_alive	the forward-invariant set on which life processes are sustained
π_⊤⁰	the single Borel-measurable Markov feedback simultaneously optimal for all initial pairs
𝒩_⊤(T)	the zero-residual-suffering family ℬ_alive∩{x\|J_⊤,ρ^*(x,T)=0}
W_⊤,　c₁,c₂,λ	the time-varying Lyapunov function and its quadratic sandwich / decay constants (26.A, 26.B)
ω	the continuous increasing modulus bounding the optimal residual value by the distance (26.C); ω(0)=0
PZS(a,x,T)	the typed permanent-zero-suffering proposition: existence of a policy keeping V_a=0 almost everywhere forever

Proof

Applying Grönwall’s inequality to (26.B) gives the first bound. The distance bound follows from the left-hand inequality in (26.A), and (26.C), together with ω(0)=0, implies that the optimal residual value also converges to zero. By the forward invariance of the family of sets, a trajectory starting within the set remains in it. In that case, the minimum of the nonnegative integral is zero and is attained by π_⊤⁰; hence the instantaneous evaluation is zero almost everywhere.

Finally, Condition 24-A excludes PZS(a,x,T) below emptiness. If PZS(⊤,x,T) holds at the highest abstraction level, the existence of a zero-cost policy implies J_⊤,ρ^*(x,T)=0, and hence x∈𝒩_⊤(T). Conversely, if the state belongs to the set, the minimizing policy π_⊤⁰ yields a trajectory of permanently zero suffering. ∎

Dynamic Tranquility

For a general set Ω_θ(t)={x|V(x,t)≤θ} with θ>0, the result is bounded suffering, not its extinction. Strict extinction of suffering requires a zero-evaluation trajectory. Moreover, (26.2) generally describes asymptotic approach and does not imply finite-time arrival. Exponential convergence is proved for the state’s distance to the set; the convergence rate of the optimal residual value depends on the modulus ω. Tranquility is not physical stasis but dynamic stability: the living process may continue to move within the invariant zero-suffering set.

The "Extinction of Defilements" Reading, V(x,t), and the Mathematics of Living Buddhahood (Sokushin Jōbutsu)

Nirvanic tranquility is commonly interpreted as "a quiet and peaceful state of mind because the flames of the defilements have been extinguished." Taken literally, this popular reading would mathematize as the extinction of the evaluation function itself, V(x,t)≡0 — or the halting of the evaluative process. Within the present system this reading is unrealizable while alive. First, V₀ is a constitutive component of the closed-loop control of Theorems 1–4: evaluation and its gradient drive the homeostatic policy π_c. Identically halting the evaluative process halts the control loop, which is incompatible with the life-maintenance condition of ℬ_alive. Second, what can vanish is not the function V but the discounted optimal residual value J_⊤,ρ^*. Theorem 24 shows J_a,ρ^*>0 below emptiness — a lower bound on the residual that presupposes the persistence of V, not a statement of its possible extinction. Under the "the value of V(x,t) disappears" reading, therefore, living nirvana — sokushin jōbutsu, attaining Buddhahood in this very body — would be impossible, and the reading is incomplete from the Mahāyāna-esoteric standpoint.

The mathematics of Theorem 26 resolves exactly this difficulty. On 𝒩_⊤(T)=ℬ_alive∩{x | J_⊤,ρ^*(x,T)=0}, the evaluation function and the optimal feedback π_⊤⁰ keep operating, and body, brain, and physical entropy keep changing impermanently as in Theorem 23; yet the residual suffering value is preserved at exactly zero (dynamic tranquility). The precise mathematics of nirvanic tranquility is thus not "the extinction of the evaluative flame" but "membership in the zero-residual-value invariant family at emptiness," and this holds while alive. The mathematical possibility condition of living Buddhahood is

a=⊤　∧　x∈𝒩_⊤　∧　x∈ℬ_alive

whose nonemptiness is guaranteed by Condition 26.A. This interpretation is also the natural one from the mathematical linkage of all the theorems. V is the driving core of Theorems 1–4 and the source of the residual in Theorem 24; its extinction is incompatible with the system’s very definition of being alive. By contrast, J_⊤,ρ^*=0 is consistent with f_i(⊤)=1 of Theorem 19, with the inclusion ladder open toward emptiness of Theorem 22, with the below-emptiness lower bound of Theorem 24, and with the de-individuation by ★_⊤ of Theorem 25 — while remaining compatible with the impermanence of Theorem 23. Hence taking the mathematics of Theorem 26 as the interpretation of the Buddha’s teaching of nirvanic tranquility follows naturally from the linkage of the entire system.

16. Four Consistency Checks

Apparent Tensions among the New Theorems and Their Resolution
Apparent tension	Distinction	Conclusion
“Existence of self-consciousness” in Theorem 16 versus “non-self” in Theorem 25	A history-relative functional fixed point S_i,h^* versus a fixed entity S₀ common to all histories	A functional self exists; a fixed intrinsic self-nature does not. There is no contradiction.
“Impermanence” in Theorem 23 versus “tranquility” in Theorem 26	Change in the complete physical state versus membership in an invariant zero-suffering set	The body and environment may change while the evaluation cost remains zero. Tranquility is not stasis.
“Freedom is maximal at emptiness” in Theorem 19 versus “suffering” in Theorem 24	Goal-conditioned control capacity versus nonnegative evaluation cost	Maximal capacity alone does not imply zero suffering. The zero-set assumption of Theorem 26 is separately required.
“Ascent” in Theorem 22 versus “no finite-stage fixation” in Theorem 23	Monotonicity of a given subsumption update versus the nonterminal condition that new information is supplied repeatedly	Theorem 23 requires Condition 23-B. Automatic updating does not follow from Theorems 20–22 alone.

Minimal Dependencies

Minimal Dependencies of Theorems 23–26
Theorem	Prior structure used directly	Newly stated indispensable conditions
23	The generalized-entropy balance of omitted Theorem 15; Theorem 22 (with Theorem 21 operating through Theorem 22)	Persistent strict dissipation; infinite, non-Zeno updating; overlap of basins of attraction; reachable TCZs
24	The shared framework of nonnegative evaluation and control (no direct proof dependence on a preceding theorem)	Impossibility of permanently zero suffering; existence of a finite-cost policy and an infinite-horizon optimal policy
25	The history-relative fixed point of Theorem 16; the typed unification of Self, Ego, and TCZ	History sensitivity and functional completeness (25-A); layerwise profiles with upward-closed presence support (25-B); bidirectional dependent-origination relations with connectedness (25-C); relational functional completeness for all dharmas and all layers (25-D)
26	The greatest element ⊤ of the common lattice; Theorem 24	Nonemptiness, forward invariance, and Lyapunov stability of the family of zero-suffering sets; a single designated policy optimal for every initial pair

17. Scope, Conclusions, and Canonical Sources

17.1 What This Paper Establishes

The convergence results of Theorems 1–4 have been reproved in a unified Lyapunov formulation that distinguishes the canonical TCZ from closed-reachable TCZ slices and separates finite-horizon optimization from infinite-time stability.
In Theorem 16, the inverse-limit space and the fixed-point subject have been separated by type, and a history index has been introduced to establish consistency with Theorem 25.
The free-will capacity of Theorem 19 has been normalized as the double supremum over a family of problems and a family of policies, consistently with the canonical appendix, rather than as the single supremum used in the main text.
Theorems 23–26 have been proved with all additional conditions needed to exclude counterexamples stated explicitly rather than left implicit.

17.2 What This Paper Does Not Claim

It does not claim, from the second law alone, that every individual phenomenon changes at every instant.
It does not claim, from the mere fact that an evaluation function is minimized, that residual suffering must be positive.
It does not infer the nonexistence of metaphysical entities in general merely from the fact that the self is a function.
It does not claim, from the fact that free-will capacity is maximal at emptiness alone, that suffering is zero.
It does not identify direction toward a particular LUB within partial information directly with direction toward truth, goodness, or emptiness.

17.3 Conclusion

Minimal Conclusion

Life is impermanent because, as a complete state satisfying Condition 23-A, it undergoes continuing dissipation. A below-emptiness control system satisfying Condition 24-A cannot realize permanently zero suffering and therefore has the residual structure expressed by “all conditioned existence is suffering.” Self-consciousness exists as a relational fixed point relative to history, but it is non-self because it is not a fixed entity common to all histories. Finally, if the forward-invariant family of zero-suffering sets in Condition 26-A exists at the highest abstraction level and is Lyapunov stable, then a dynamic tranquility in which suffering is extinguished without terminating vital activity is mathematically possible.

Impermanence ≠ chaotic inability to remain still; Suffering ≠ pain at every instant; Non-self ≠ nonexistence of a self-process; Tranquility ≠ physical cessation

17.4 Theorem List (All Thirteen Theorems)

Theorem name, central formula, and brief explanation
No.	Theorem	Central formula	Brief explanation
1	Tomabechi Main Theorem	π_c=arg min_u(·)∫V₀ds ⇒ dist(x(t),TCZ₁^cl(t;x₀))→0	Ego’s receding-horizon control converges exponentially to the individual closed-reachable TCZ slice.
2	Shared-TCZ Convergence Theorem	π_i=arg min_{u_i(·)}∫(V_0,i+Σγ_ijS_ij)ds ⇒ dist(𝐱(t),TCZ^shared,cl(t))→0	Connected coupled agents cancel individual residuals and misalignments together, converging to the shared zero set.
3	Abstract Shared-TCZ Convergence Theorem	π_i=arg min_{u_i(·)}∫(V_0,i+ΣγS+η_i𝒜_i)ds ⇒ ‖ι(φ_i(x_i(t)))−ι(L^*)‖→0	Adding the abstraction residual lifts the group to the representation of the LUB L^*=∨W_i.
4	Tomabechi Presence-Weighted Theorem	π_c^P=arg min_u(·)∫Ṽds,　Ṽ=V₀−κPQ ⇒ dist(x(t),Ω_P(t))→0	Presence P and value sign Q deform the effective terrain and shift the target TCZ slice.
16	Tomabechi Self-Consciousness Existence and Emergence Theorem	SC_i,h=lim←K_i,α(h)≠∅,　∃S_i,h^: F_i,h(S_i,h^)=S_i,h^*	On the inverse limit of layered TCZs there exists (uniquely, under contraction) a history-relative fixed point of self-consciousness.
19	Tomabechi Free-Will Theorem	ℱ_i(α)=sup_(d,π)I(G;Y\|X),　α≼β⇒ℱ_i(α)≤ℱ_i(β),　f_i(⊤)=1	Goal-conditioned control capacity is monotone in abstraction and maximal at emptiness; it is compatible with determinism.
20	Tomabechi Symbolic-Presence Directionality Theorem	κq_σP_σ(−s′(D))≥b+c ⇒ ⟨ẋ,d_{u_σ}⟩_M⁻¹=−Ḋ>0	Under dominance, the actual motion of the mind always carries a positive component toward the LUB the symbol points at.
21	Tomabechi Subsumption-Poset Presence Directionality Theorem	p>p_crit=(1/κm)max{β,B/r} ⇒ ∇²Ṽ_μ,p≽cI,　∃!x_b^,　‖x_b^−x_b‖≤B/c	Bias presence above the threshold defeats the old terrain, carving a unique local valley that captures exponentially.
22	Tomabechi Higher-LUB Presence Theorem	u_n+1=u_n∨v_n+1 ⇒ u_n≺u_n+1,　exp. stability,　ℱ_i(u_n)≤ℱ_i(u_n+1),　u_∞=∨u_n≼⊤	Staged updating by inclusion, not erasure, raises the order strictly, never lowers capacity, and its limit stays at or below the top.
23	Impermanence Theorem	∫_t₁^t₂Π_gends>0 ⇒ z(t₂)≠z(t₁); 23-B ⇒ TCZ_n+1^cl≠TCZ_n^cl	A living complete state under sustained dissipation never recurs, and below emptiness no stage TCZ is ever final.
24	All Conditioned Existence Is Suffering Theorem	a≺⊤ ∧ 24-A ⇒ J_a,ρ^*(x,T)>0	Below emptiness no optimal control achieves a permanent zero-suffering trajectory; a positive residual cost remains.
25	Non-Self of All Dharmas Theorem	¬∃S₀∀h: F_i,h(S₀)=S₀;　∀d∈𝔇 ∀α∈𝕃: ¬Atman(d,α)	A history-relative functional self exists, but neither a fixed point common to all histories nor a fixed intrinsic nature at any abstraction layer exists; every dharma stands within the web of dependent origination.
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 state approaches the zero-suffering invariant family of emptiness exponentially: dynamic tranquility with life processes intact.

17.5 Canonical Sources Inherited by This Paper

Hideto Tomabechi (2026), Tomabechi Theory of Cognitive Homeostasis — Extension to Cognitive Space by Presence and Goal Attainment.

Hideto Tomabechi (2026), Tomabechi Self-Consciousness Existence and Emergence Theorem.

Hideto Tomabechi (2026), Tomabechi Abstractional Freedom Theory — Cognitive Universe Theory, Theorem 19.

Hideto Tomabechi (2026), Directionality of Symbolic Presence in Partial-Information Spaces — Theorems 20, 21, and 22. Reference edition: Which Way Does a Symbol Turn the Mind? Theorems 20, 21, and 22 — Accessible Complete Edition (the authorized lay edition in one-to-one correspondence with the canonical text).

Hideto Tomabechi (2026), Tomabechi Cognitive–Physical Entropy Exchange and Conservation Theorem.

Tomabechi, H. (2026a). A Unified Theory of Latent Potentials: Homeostasis and Cognitive Warfare — Toward a Mathematical Foundation of Cognitive Control in Physical, Social, and AI Systems. National Defense University Lecture Paper, April 4, 2026, public revised edition. Available at: https://tomabechi.jp/TomabechiNDUpaperENpublic.pdf

Tomabechi, H. (2026b). “Cognitive Warfare as Control of Complex Cognitive Potential Landscapes: A Lyapunov-Based Framework for Stability, Abstraction, Presence, and Peace-Oriented Cognitive Operations.” In Complexity and Security: Theorizing Within and Beyond Borders. Routledge, in press.

Tomabechi, H. (2026c). Cognitive Latent Potential Theory: From Cognitive Warfare to Human and Organizational Transformation. Cognitive Research Laboratories Technical Report, May 2026.

Abstract

Contents

1. Normalization of Theorem Numbers and the Minimal System

2. Common Definitions and the Unified Convergence Lemma

2.1 State, Control, Evaluation, and TCZ

2.2 Rigorous Interpretation of Finite-Horizon Control

2.3 Level of Abstraction, Lattice, and Emptiness

2.4 Typed Self, Ego, and TCZ

3. Theorem 1 — Tomabechi Main Theorem

4. Theorem 2 — Shared-TCZ Convergence Theorem

5. Theorem 3 — Abstract Shared-TCZ Convergence Theorem

6. Theorem 4 — Tomabechi Presence-Weighted Theorem

7. Theorem 16 — Tomabechi Self-Consciousness Existence and Emergence Theorem

8. Theorem 19 — The Tomabechi Free-Will Theorem

9. Theorem 20 — Tomabechi Symbolic-Presence Directionality Theorem

10. Theorem 21 — Tomabechi Subsumption-Poset Presence Directionality Theorem

11. Theorem 22 — Tomabechi Higher-LUB Presence Theorem

12. Theorem 23 — Impermanence Theorem

13. Theorem 24 — All Conditioned Existence Is Suffering Theorem

14. Theorem 25 — Non-Self of All Dharmas Theorem

25.1 The Current Self-Process and Fixed Points

25.2 Layerwise Existence Profiles of All Dharmas

25.3 Vertical Subsumption and Bidirectional Relations with Other Dharmas

25.4 Functional Completeness for All Dharmas

15. Theorem 26 — Nirvanic Tranquility Theorem

16. Four Consistency Checks

Minimal Dependencies

17. Scope, Conclusions, and Canonical Sources

17.1 What This Paper Establishes

17.2 What This Paper Does Not Claim

17.3 Conclusion

17.4 Theorem List (All Thirteen Theorems)

17.5 Canonical Sources Inherited by This Paper