Tomabechi Theory of Evolution (5-Theorem Edition) — From Cognitive Homeostasis to Symbolic Culture and Evolution
— A Minimal 5-Theorem System: From Theorems 1·2·3 (Cognitive Convergence, Sharing, and Abstraction) to the Tomabechi Symbolic Culture Generation Theorem and the Tomabechi Evolution Theorem (Symbolic Culture Generation and Evolution) —
(Last updated July 12, 2026)
CyLab, Carnegie Mellon University
C5I Center, George Mason University
1. Introduction
The Scope of the Theory — Self-Transformation, Coaching, and Leadership
Many self-development theories advise: "change one's behavior." Many leadership theories advise: "move people." Many coaching theories advise: "set goals." All such prescriptions are superficial.
Human behavior is not the simple output of decision-making. Human behavior is generated by the configuration of worlds that an individual experiences as stable, real, and meaningful. Attempting to alter behavior directly therefore produces no deep change. What is required is to alter the internal structure that generates behavior.
At a foundational level, the theory developed here is intended as one of the core frameworks of Cognitive Physics and Cognitive Biology, two disciplines the author proposes. The aim is to treat cognition with the same mathematical seriousness that physics treats matter and that biology treats life: cognitive states evolve under law-like dynamics (Cognitive Physics), and cognitive traits such as abstraction, presence, and altruism are subject to selection (Cognitive Biology). The TCZ, the Ego control operator, the LUB, and future-origin cognitive time introduced below are the shared primitives of both.
This insight originated in the national-security context. The author's reference paper (the Lecture Paper of April 4, 2026) formalized cognitive warfare as the process of "deforming a target population's evaluation function V(x, t) externally, thereby reconstituting its Total Comfort Zone (TCZ) and altering its behavioral trajectory." Cognitive warfare, on that account, does not command its targets. It alters the very topography on which their decisions are made, so that certain behaviors come to feel "natural."
This same mathematical structure is not, however, intrinsically military. It describes the homeostatic structure of human cognition itself. When inverted — that is, when used not to violate but to elevate the autonomy of the other — it becomes the theory of individual-support and collective-integration, organizational change, and self-transformation.
The aim of this paper is to execute that inversion rigorously. We retain the mathematical foundation of the foundational theory (TCZ, Shared-TCZ, LUB, Lyapunov convergence) and add concepts specific to application — Bridge Dynamics, presence generation, symbolic culture generation, ethical constraints, expanded fitness, and the cognitive future-origin — to construct a unified mathematical language for self-transformation, the support of others, and the balance of a human life.
Central Proposition
We state the central proposition of this paper at the outset.
than current reality and reconstituting that future as a new TCZ.
Two elements in this proposition extend the foundational theory. First, "presence" cannot be captured by the evaluation function V(x, t) alone: humans do not merely avoid discomfort; they migrate toward worlds that are experienced as vivid. Second, the orientation toward self-transformation is the inverse of the original paper's orientation toward the manipulation of target populations. Both differences are formalized mathematically below.
Main contributions. The contributions of this paper are fourfold. First, the foundational concepts — cognitive state, evaluation function, TCZ, and the Self / Ego operators — are formalized axiomatically under an explicitly stated set of standing assumptions A1–A8 (§2, Standing Assumptions), yielding a self-contained mathematical system with no dependence on external literature. Second, Theorems 1–3 (individual TCZ convergence, shared-TCZ convergence, and LUB convergence) are given complete proofs based on dissipativity derived from the HJB equation, the comparison theorem, and forward invariance (Nagumo’s condition) (Appendix A.2–A.4). Third, the Tomabechi Symbolic Culture Generation Theorem and the Tomabechi Evolution Theorem are completed formally: for the former, the σ-direction monotonicity lemma (Appendix A.5, Lemma A.5.1) makes the decrease of the effective potential rigorous under Assumption A7; for the latter, a lemma establishing monotone convergence of the gradient flow and convergence to the critical set by a LaSalle-type argument (Appendix A.6, Lemma A.6.1) is proved in a self-contained manner. Fourth, the questions of mathematical rigor (the Lyapunov descent gap, the probability space of the expectation 𝔼, and the convergence of the adaptive dynamics) are examined systematically, and each resolution is implemented in the body of the paper as an assumption or a lemma.
All theorems in this paper are proved in a self-contained manner under an explicitly stated set of standing assumptions (A1–A8). Every proof relies only on dissipativity derived from dynamic programming (the HJB equation), the comparison theorem, and a LaSalle-type invariance argument, with no dependence on external results. Potential issues of mathematical rigor are examined systematically in the closing section on mathematical foundations, where each is shown to be resolved.
Organization. Part I (§2–§3) develops the foundational theory: §2 presents the foundational concepts, the standing assumptions A1–A8, and Theorems 1–3, and §3 discusses the core intuition (entropy and abstraction). Part II develops the theory of symbolic culture generation and evolution: the Minimal Chain section traces the logical route from Theorems 1–3 to the two theorems, §4 gives the Tomabechi Symbolic Culture Generation Theorem, and §5 gives the Tomabechi Evolution Theorem. §6 presents the ethical constraint Ethic(B), §7 concludes, and §8 lists the references. These are followed by the Reader’s Guide (figures), the rigorous proofs in the appendix (A.1–A.7), the questions of mathematical rigor and their resolution, and the theorem summary table.
2. Cognitive Homeostasis, TCZ, Self, and Ego
This section gives the foundational material needed to read the rest of the paper without first studying the foundational theory or any prior document. The central concepts are the cognitive state, the evaluation function, the Total Comfort Zone (TCZ), the Self/Ego operator, and the optimal-control formulation of Ego. The definitions, theorem statements, variables, and proof strategy required for Theorems 1–3 and the later theory of cognitive time, cosmology, and evolution are given here in self-contained form. Moreover, the present paper constitutes a minimal-version formalization of the theory that the author has, since the early 1990s, referred to in his papers and lectures as the Cyber-Homeostasis Hypothesis and the Super-Information Field Hypothesis.
Cognitive State and Evaluation Function
Following the formulation, the cognitive state of a subject at time t is denoted by
| Symbol | Meaning |
|---|---|
| x(t) | cognitive state at time t |
| x | a cognitive state or state variable |
| X | the total cognitive state space |
| t | time parameter or present time |
| T | terminal horizon of the control problem |
The cognitive state includes beliefs, emotions, memories, self-image, future predictions, bodily sensations, social cognition, goals, and values. It is not a single belief or a single emotion; it is the total configuration of the subject’s internal world at that moment.
The basic evaluation function is denoted by
| Symbol | Meaning |
|---|---|
| V | evaluative potential; instability or discomfort cost |
It measures the instability, discomfort, internal inconsistency, or evaluative cost that the cognitive state x imposes on the subject at time t. A larger value of V means that the state is less stable and less sustainable for the subject. In this paper we use V for the pure evaluative function.
State-Space Definition of the Total Comfort Zone
Let ℛ(t;x0) denote the reachable set from the initial state x0, and let θ be an admissible stability threshold. The Total Comfort Zone is defined as
| Symbol | Meaning |
|---|---|
| x₀ | initial cognitive state |
| Ωθ | threshold stability set defined by a potential below θ |
| θ | stability threshold |
| TCZ | Total Comfort Zone; cognitive stability region |
| ℛ | reachable set from an initial state |
Thus the TCZ is the set of states that are reachable from the present state and can be inhabited while preserving cognitive stability.
TCZ as Stability Region, Attractor Basin, Visibility Boundary, and Identity Frame
The TCZ has four simultaneous meanings in this theory.
| Aspect | Mathematical role | Plain meaning |
|---|---|---|
| Stability region | Sublevel set of V: V(x,t)≤θ | The range in which the subject can remain without excessive internal strain. |
| Attractor basin | Region toward which the Ego trajectory returns | The “valley” into which one naturally falls back. |
| Visibility boundary | Boundary separating low-cost options from high-cost or unrepresented options | What the present Ego can naturally see as possible, realistic, or “for me.” |
| Identity frame | States compatible with the current self-model | The world in which one feels “this is me” and “this is my life.” |
This fourfold interpretation is essential for the theory of goals. A true goal is outside the current TCZ not merely because it is uncomfortable, but because it is outside the current attractor basin, outside the current visibility boundary, and outside the current identity frame. This is why a true goal is initially invisible to the present Ego.
Self / Ego Operators and Optimal Control
Following the foundational theory, the Self is a semantic ordering structure over possible worlds, while the Ego is its dynamical realization in the state space. The Ego is formalized as an optimal control problem:
| Symbol | Meaning |
|---|---|
| u(t) | control input or action path |
| πc | cognitive control policy of the Ego |
| arg min | the argument or choice that minimizes the expression |
The function πc(x) is the cognitive control policy. It maps a cognitive state x to an action or control trajectory u(t) that minimizes accumulated evaluative cost. The subscript c stands for cognitive, distinguishing this from a merely physical or instrumental control policy.
Five-Step Bridge from Possible-Worlds Modal Logic to Control Equations
The mathematical core of the framework is the equivalence between two descriptions of the same cognitive process. One is the semantic description in possible-worlds modal logic: possible worlds, Self, and TCZ. The other is the dynamical description in state space: optimal control, πc(x), and optimal trajectory x*(t). The bridge between these two descriptions proceeds in five steps:
The possible-worlds modal-logic formulas used in this five-step bridge, especially the semantic formulations of possible worlds, Self, and TCZ, are presented in H. Tomabechi's book オーセンティック・コーチング2026 ～本物のコーチング～ [Authentic Coaching 2026: Genuine Coaching] (Tomabechi, 2025). The present paper translates that semantic formulation into the language of state-space optimal control, Lyapunov convergence, and future-origin cognitive time.
- Cognition unfolds not over a single state but over a set of possible worlds W.
- An ordering operator r:W→W ranks possible worlds by cognitive comfort.
- The stable region under this ordering is the Total Comfort Zone.
- The Self / Ego functions as an operator that selects or reorganizes the ordering structure of the TCZ.
- In the state-space formulation, this operator is implemented as the optimal control problem πc(x).
Logical definition of TCZ in possible-worlds form
In possible-worlds form, the TCZ can be defined as
| Symbol | Meaning |
|---|---|
| W | set of possible worlds |
| w | one possible world |
| y | arbitrary situation or disturbance in possible-world notation |
| rTCZ | stabilizing relation inside TCZ |
| ∀ | for all |
| ∃ | there exists |
Here rTCZ(x,y) denotes a stabilizing relation within the TCZ. The formula says: for every situation y that may arise, there exists a stabilizing response x. This is a mathematical expression of psychological resilience.
Duality of the Self / Ego operator
| Symbol | Meaning |
|---|---|
| sSelf | Self operator mapping or transforming TCZ structures |
| f(x,u,t) | state dynamics under control u |
In possible-worlds notation, the Self can be represented as an operator that selects or transforms stabilizing worlds. The mapping s : TCZ → TCZ′ represents the second and more important operation: moving not inside the comfort zone, but moving the comfort zone itself.
The Self functions in two modes:
- Selection mode: the Self selects a desired TCZ from among possible worlds.
- Transformation mode: the Self transforms the TCZ itself by a mapping s:TCZ→TCZ′.
The second mode is decisive. Self-transformation is not movement inside the present comfort zone. It is transformation of the comfort zone itself. Ego is the state-space control realization of that Self operation. Modal-logic Self defines the desired region in the language of possible worlds; Ego implements the trajectory into that region as a control equation.
Summary of the equivalence
The equivalence can be summarized as follows:
- possible worlds ↔ trajectories in state space
- ordering relation r ↔ evaluation function V(x,t)
- existence of stabilizing response ↔ existence of a trajectory remaining in the stability region
- Self in selection mode ↔ evaluation of possible worlds
- Self in transformation mode ↔ TCZ reorganization by a control policy
This equivalence is what makes it possible to give mathematical theorems about concepts such as Self, Ego, TCZ, goal, and transformation.
Standing Assumptions
All theorems and proofs in this paper are stated under the following standing assumptions. Assumptions A1–A4 are common to all theorems; A5–A8 are additional assumptions for individual theorems. Each theorem statement lists the assumptions it uses.
The state space X ⊆ ℝn is a nonempty closed set and the control set U ⊂ ℝm is compact. The right-hand side f : X×U×[0,T] → ℝn of the dynamics ẋ = f(x,u,t) is continuous and locally Lipschitz in x, uniformly in u and t. Admissible controls are measurable functions u : [0,T] → U. Under these conditions, the Carathéodory existence theorem together with Lipschitz uniqueness guarantees that for every admissible control and every initial condition the closed-loop solution exists, is unique, and is forward complete on [0,T].
The evaluation function V : X×[0,T] → ℝ is continuously differentiable (C¹), bounded below, and proper (for every c, the sublevel set {x | V(x,t) ≤ c} is compact for each t). The stability set Ωθ = {x | V(x,t) ≤ θ} is nonempty. The same regularity is assumed for Vi, Sij, and A used in Theorems 2 and 3.
For each (x,t), the function u ↦ V(x,t) + ∇J*·f(x,u,t) to be minimized is continuous on U. Compactness of U and continuity ensure that the minimum is attained, and a measurable-selection theorem (Berge’s maximum theorem combined with a Kuratowski–Ryll-Nardzewski-type selection) yields a measurable feedback πc(x,t) attaining the minimum. Throughout, πc denotes this measurable selection.
Along the optimal closed loop ẋ = f(x, πc(x,t), t) there exists a constant α > 0 such that dV/dt ≤ −α(V − θ) outside Ωθ. The dynamical justification of this assumption — its derivation via the HJB equation, and why it does not follow from optimality alone — is given in the supplement to Appendix A.1.
The interaction terms Sij ≥ 0 are C¹, the coupling graph determined by the coefficients γij ≥ 0 is connected, and there is a uniform lower bound γmin > 0 on the coupling strength supporting shared stability (the strong-coupling condition).
The expectation 𝔼 is taken over a probability space (ΩP, 𝓑, P) representing epistemic uncertainty about the position of the LUB in the subsumption lattice. The abstraction potential A is nonnegative and measurable, and A = 0 ⇔ φ = LUB, P-almost surely. In the limit of vanishing uncertainty, 𝔼 becomes the identity operator and Theorem 3 reduces to Theorem 2.
A shared symbol system C is given. The symbolic presence Symi(C;σi,ρi,αi) is continuously differentiable in σi with ∂Symi/∂σi > 0 (an increase in symbolic-cultural capacity increases symbolic presence). The weight λi > 0 and the coupling constant κi > 0, and on the region of positively valued high-abstraction worlds the value sign satisfies Qi > 0. Pi and Vi have the regularity of Assumption A2.
The domain D ⊂ ℝ³ of the evolutionary cognitive-cultural parameter z = (α, ρ, σ) is nonempty, compact, and convex. The extended fitness functional Fi is continuously differentiable on D, the selection matrix Mi(z) is symmetric positive definite and locally Lipschitz in z, and the solutions of the gradient flow ż = Mi∇Fi are forward complete in D (D is forward invariant). The marginal-benefit conditions (the three inequalities of §5) hold on the low-(α,ρ,σ) region. Moreover, the evolutionary variable z varies on a slower time scale than the individual cognitive dynamics, and Fi(z) is evaluated, for each fixed z, as a completed integral along the individual optimal trajectory of that generation (adiabatic approximation — see Question 5 of the “Mathematical Foundations” section).
In summary, A1 says that the world of states and actions is mathematically well behaved; A2 says that the evaluation function is smooth and has a floor; A3 says that an optimal way of choosing actually exists; A4 says that the evaluation cost reliably decreases outside the stability region. A5–A8 are the additional premises needed for sharing (Theorem 2), abstraction (Theorem 3), symbolic culture (the Symbolic Culture Generation Theorem), and evolution (the Evolution Theorem), respectively.
The Main Theorem Is Theorem 1 — Tomabechi Unification of Self / Ego / TCZ
| Symbol | Meaning |
|---|---|
| Self | The semantic operator acting on possible worlds W by selecting or transforming a desirable region. |
| Ego | The dynamical realization of Self as a control process minimizing accumulated evaluative cost. |
| TCZ(x0) | The Total Comfort Zone reachable from initial state x0, where evaluative cost remains below threshold. |
| πc(x) | The cognitive control policy of the Ego; it selects a control that minimizes accumulated evaluative cost. |
| V(x(t),t) | Baseline evaluation function: instability, discomfort, internal inconsistency, or evaluative cost. |
| x*(t) | The optimal cognitive trajectory generated by the control policy πc. |
This subsection makes an important point explicit. The main theorem is not a separate preliminary theorem outside the theorem system; it is Theorem 1 itself. In the language of possible worlds, the Self selects or transforms a desirable region. In the language of control theory, the Ego is the policy that minimizes accumulated evaluative cost. In the language of dynamical systems, the TCZ is the stability region to which the optimal trajectory converges.
Self, Ego, and TCZ are therefore not three different things. They are three descriptions of one cognitive process in three languages: philosophy, engineering, and psychology. This unification makes it possible to treat self, comfort zone, goal, and cognitive transformation not as metaphors but as mathematically implementable objects in a control system.
Theorems 1–3 as the Immediate Mathematical Background
Detailed derivations of Theorems 1–3 are given in Appendix A of this paper. The purpose of this subsection is to make the present paper self-contained by summarizing the formulas, variables, and meanings of Theorems 1–3.
Theorem 1: Individual TCZ Convergence
Assumptions: the standing assumptions A1–A4 are in force.
| Symbol | Meaning |
|---|---|
| arg minu(t) | The operation that selects the control input u(t) that minimizes the integral that follows. |
| ∫0T | Accumulation over the horizon from time 0 to terminal time T. The Ego minimizes not only momentary discomfort but cost over time. |
| ⇒ | The implication: if the control structure on the left is realized, the convergence statement on the right follows. |
Theorem 2: Shared-TCZ Convergence
Assumptions: the standing assumptions A1–A5 are in force.
| Symbol | Meaning |
|---|---|
| πi | The cognitive-control policy of subject i, optimizing both personal stability and alignment with others. |
| ui(t) | The control input of subject i at time t. |
| xi(t), xj(t) | The cognitive states of subjects i and j. |
| Vi(xi(t),t) | The individual evaluative cost or instability of subject i. |
| Sij(xi,xj) | The cognitive deviation or misalignment between subjects i and j; a standard example is ||xi−xj||². |
| γij | The strength of social-cognitive coupling from subject j to subject i. |
| Σj | Summation over the other subjects j relevant to subject i. |
| TCZshared | The shared stability region in which multiple subjects can remain stable together. |
Theorem 3: LUB / Higher-Abstraction Convergence
Assumptions: the standing assumptions A1–A6 are in force.
A(x)=0 ⇔ φ(x)=LUB(W1,...,WN) ⇒ x*(t)→LUB(W1,...,WN)
| Symbol | Meaning |
|---|---|
| 𝔼 | Expectation. It indicates minimization under uncertainty or across possible variations of the trajectory. |
| Vi | The individual evaluative cost of subject i. |
| Sij, γij | The pairwise misalignment term and its social-cognitive coupling weight. |
| A(xi(t)) | Abstraction potential: a measure of how far subject i's state is from the target LUB. |
| ηi | Positive weight assigned to the abstraction potential. |
| φ(x) | Abstraction map sending cognitive states into a lattice of concepts or possible worlds. |
| W1,...,WN | The multiple cognitive worlds, positions, or meaning-worlds to be integrated. |
| LUB(W1,...,WN) | The least upper bound: the minimal higher abstraction that contains all the relevant worlds without erasing their differences. |
| A(x)=0 ⇔ φ(x)=LUB(...) | The design condition: abstraction potential is zero exactly when the state reaches the target LUB. |
At-a-Glance Table: Theorems 1–3
| Theorem | Core formula | Main variables | Meaning |
|---|---|---|---|
| Theorem 1 Individual TCZ | πc(x)=arg minu(t)∫0TV(x(t),t)dt ⇒ x*(t)→TCZ(x0) | V: instability TCZ: stability region πc: Ego policy | The Ego returns the individual to the current stability region. |
| Theorem 2 Shared-TCZ | πi=arg minui(t)∫0T(Vi(xi(t),t)+ΣjγijSij(xi,xj))dt ⇒ x*(t)→TCZshared | Sij: misalignment γij: coupling | Coupled people converge to a shared stable region. |
| Theorem 3 LUB | πi=arg minui(t)𝔼∫0T(Vi+ΣjγijSij+ηiA(xi(t)))dt, A(x)=0⇔φ(x)=LUB(W1,...,WN) ⇒ x*(t)→LUB(W1,...,WN) | A: abstraction potential φ: abstraction map LUB: least upper bound | Groups can be lifted from low shared stability to higher shared purpose. |
This table is the immediate mathematical inheritance used by the present paper. The later theory of true goals and future-origin cognitive time depends on it in the following way: a true goal is outside the present TCZ (Theorem 1), may need to be made socially visible or shared (Theorem 2), often requires a higher LUB (Theorem 3).
3. The Core Intuition
Ordinary theories of action often assume that people are pushed forward by the past. A person is said to act because of past experience, past trauma, past incentives, past habits, or past conditioning. Such factors matter. But in the Tomabechi Framework they are not the deepest explanation of self-transformation. The central question is not only “what caused the present?” but “what future has become real enough to organize the present?”
The current Ego normally minimizes accumulated evaluative cost and therefore returns the subject to the current Total Comfort Zone. This explains why merely wanting change is insufficient. If the desired future remains outside the present TCZ, then the present Ego will not naturally move toward it. The desired future is too costly, too unstable, or too unreal from the standpoint of the current evaluative landscape.
A true goal is different from a plan, task, intention, improvement objective, or forecast. A task lies inside the current TCZ. A true goal lies outside it. Because it lies outside the current TCZ, it is not directly visible to the present Ego as a low-cost option. Yet if the Self sets that goal as the terminal condition of the Ego’s control problem, present action becomes organized by the future. This is the mathematical meaning of future-origin cognitive time.
Physical Entropy and Cognitive-Informational Entropy
The distinction between physical time and cognitive time can be clarified through entropy. In the ordinary thermodynamic description, the arrow of physical time is associated with the tendency of entropy to increase in an isolated physical system. A broken cup does not spontaneously reassemble; heat diffuses; local order tends to dissipate unless work is supplied. This is the familiar physical direction of time.
Cognitive-information space behaves differently. In cognition, a higher abstraction can reduce semantic entropy by compressing many scattered details into a smaller number of organizing meanings. The movement toward a LUB is precisely such a movement: many lower-level distinctions become integrated under a higher concept. In the notation of Tomabechi Framework, this is expressed as
| Symbol | Meaning |
|---|---|
| →, ↑, ↓, ⇔ | direction of implication, increase/decrease, or equivalence |
This does not mean that the brain violates thermodynamics. The brain remains a physical system. The claim is about the informational geometry of meaning. In physical space, entropy increase gives the ordinary thermodynamic arrow. In cognitive-information space, a future goal can organize meaning so that semantic entropy decreases: scattered events become interpretable, present choices become ordered, and past experiences acquire a coherent role.
Minimal Chain — From Theorems 1·2·3 to the Tomabechi Symbolic Culture Generation Theorem and the Tomabechi Evolution Theorem
Theorem 1 (Tomabechi Master Theorem), Theorem 2 (Shared-TCZ Convergence), and Theorem 3 (Abstract Shared-TCZ Convergence) establish the foundations of cognitive homeostatic convergence, sharing, and hierarchical abstraction. From these foundations, the next questions are how a higher-order abstract world becomes social reality and is transmitted historically (Tomabechi Symbolic Culture Generation Theorem), and how that capacity is selected in evolution (Tomabechi Evolution Theorem). This minimal chain is the core of the present paper.
4. Tomabechi Symbolic Culture Generation Theorem
The preceding sections established that cognitive time is indexed by future LUB-TCZ structures and that higher abstraction reduces semantic randomness. The next step is to explain how such high-abstraction worlds become socially real and historically transmissible. This is the role of symbolic culture. Art, music, literature, religion, ritual, myth, law, and narrative are not secondary ornaments added after cognition. They are mechanisms that transform high abstraction into shared presence.
Assumptions: the standing assumptions A1–A4 and A7 are in force.
where Pσ,i(x,t;C)=Pi(x,t)+λi Symi(C;σi,ρi,αi), Ṽσ,i(x,t;C)=Vi(x,t)−κi Pσ,i(x,t;C)Qi(x,t)
| Symbol | Meaning |
|---|---|
| C | A symbolic-cultural form: art, music, literature, ritual, myth, religion, law, narrative, or shared symbol. |
| σi | Symbolic culture generation capacity of subject, lineage, or group i. |
| ρi | Presence-generation capacity. |
| αi | Abstraction level or inclusive scope. |
| Symi(C;σi,ρi,αi) | The symbolic-cultural function that makes a high-abstraction world shareable and transmissible. |
| λi | Positive coefficient measuring symbolic amplification of presence. |
| Pσ,i | Presence after symbolic-cultural amplification. |
| Ṽσ,i | Effective potential after symbolic presence is included. |
| Qi | Valence of the state or world: positive for approach, negative for avoidance. |
Step-by-Step Reading
- Start with a high-abstraction world that may be conceptually meaningful but not yet real to ordinary cognition.
- Introduce a symbolic form C: a song, image, story, ritual, poem, law, myth, or religious form.
- The symbolic function Symi transforms the high abstraction into something perceptible, memorable, and shareable.
- This raises Pσ,i, the presence of that world.
- If the world has positive valence Qi>0, then higher presence lowers its effective potential.
- Lower effective potential means that the Ego can more naturally approach and inhabit the high-abstraction world.
- Across generations, symbolic culture stabilizes Shared-High-TCZ structures by preserving presence in transmissible form.
Connection to the Other Theorems
| Related theorem | What it contributes | Why symbolic culture is needed |
|---|---|---|
| Theorem 3 | πi=arg minui(t)𝔼∫0T(Vi+ΣjγijSij+ηiA(xi(t)))dt, A(x)=0⇔φ(x)=LUB(W1,...,WN) ⇒ x*(t)→LUB(W1,...,WN) | A LUB must be made visible and emotionally real. |
| Tomabechi Evolution Theorem | πevo=arg max(α,ρ,σ) F, ż=M∇F ⇒ dF/dt≥0 | The Tomabechi Symbolic Culture Generation Theorem supplies the independent symbolic axis σ used by the evolutionary theorem. |
5. Tomabechi Evolution Theorem — Selection of Altruism, Presence, and Symbolic Culture
where Fi(αi,ρi,σi)=∫[−Vi+βiSharei+νiGi(αi)+μiPresi+ωiSymi−Ci]dt
| Symbol | Meaning |
|---|---|
| Vᵢ | individual evaluative potential of subject i |
| α | positive convergence rate, or abstraction level in evolutionary context |
| i,j,k,ℓ | indices used for summing over agents, domains, or messages |
| Fi | evolutionary fitness functional of subject or lineage i |
| βi,νi,μi,ωi | positive weights of sharing, group, presence, and symbolic terms |
| Sharei | benefit from sharing or cooperation at abstraction level αi |
| Gi | group survival or group fitness term |
| Presi | fitness gain from presence-generating capacity |
| Symi | fitness gain from symbolic culture |
| Ci | cost term for abstraction, presence, or symbolic capacity |
| ρi/σi | presence-generating capacity and symbolic-cultural capacity |
Tomabechi Evolution Theorem is the evolutionary completion of the time theory. The Abstract Shared-TCZ Convergence Theorem (Theorem 3) makes each LUB-TCZ a higher-order stability region shared by a group. Tomabechi Evolution Theorem shows that humans can evolve capacities that make higher LUB-TCZs more accessible: altruism expands the range of inclusion; presence makes abstract futures are experienced as vivid; symbolic culture transmits those realities across persons and generations.
In the unified numbering of the Tomabechi Framework, this result is Tomabechi Evolution Theorem. the full 12-theorem series covers true goals, future-origin cognitive time, future-origin goal attainment, and LUB-indexed cognitive time. Tomabechi Evolution Theorem comes after them because evolution selects the very capacities that allow beings to live from higher future-origin LUBs.
Variables of the Evolutionary Extension
| Variable | Meaning | Plain-language reading |
|---|---|---|
| αi | Abstraction level / inclusion radius | How large the subject’s “we” can become. |
| ρi | Presence-generating capacity | Ability to make abstract futures are experienced as vivid. |
| σi | Symbolic-cultural capacity | Ability to transmit high meanings through art, music, literature, religion, ritual, myth, and narrative. |
| Sharei(αi) | Cooperative benefit | Benefit gained as the circle of cooperation expands. |
| Gi(αi) | Group survival / group fitness benefit | Benefit obtained when the group survives and flourishes. |
| Presi(ρi,αi) | Presence benefit | Benefit of experiencing high futures as real. |
| Symi(σi,ρi,αi) | Symbolic-cultural benefit | Benefit of stabilizing and transmitting high futures across persons and generations. |
| Ci | Cost | Cognitive, metabolic, social, and cultural cost. |
Extended Evolutionary Fitness Functional
Define the extended evolutionary fitness functional:
Assumptions: the standing assumptions A1–A2 and A8 are in force.
Let Fi be the extended evolutionary fitness functional above. Suppose that in the relevant adaptive region the marginal benefits of abstraction, presence generation, and symbolic-cultural transmission exceed their corresponding marginal costs:
| Symbol | Meaning |
|---|---|
| ∂ | partial derivative; marginal change of a function |
Step-by-Step Reading
- First build the “scorecard” Fi of subject or lineage i: add the gains from low discomfort, cooperation, group survival, presence and symbols, and subtract the costs of sustaining them.
- Take the evolutionary coordinates z = (α, ρ, σ) — the triple of abstraction, presence-generating power, and symbolic-culture power.
- Selection is gradient ascent ż = M∇F: move in the direction of a higher score, with a positive-definite step M.
- Then dF/dt = ∇FTM∇F ≥ 0 — the score never falls (a one-way uphill road).
- In the low-α·ρ·σ region the marginal-benefit conditions say “one more step always pays,” so that region cannot be a summit.
- The parameter domain D is compact (a bounded range), so a summit must exist — and by the previous step it lies on the high-α·ρ·σ side.
- Hence evolution selects altruism, rich presence, and symbolic culture — art and religion are not exceptions but outputs.
Then a low-abstraction, low-presence, low-symbolic-capacity state cannot be a stable fitness maximum. Evolutionary ascent selects higher abstraction, richer presence generation, and symbolic-cultural capacity. Therefore large-scale cooperation, moral systems, abstract altruism, art, music, literature, religion, and the desire for peace are mathematically predicted consequences of the Tomabechi Framework evolutionary fitness structure.
Selfish Gene as a Low-Abstraction Approximation
The selfish-gene hypothesis is not discarded. It is reclassified. In the low-abstraction regime, where Share, G, Pres, and Sym are weak or absent, fitness appears dominated by individual or kin-level optimization. This is the low-α approximation. It is useful, but incomplete.
Once Theorem 3 is taken into account, the picture changes. Theorem 3 introduces high-abstraction LUB integration. Tomabechi Evolution Theorem then states that evolution can select the capacity to inhabit and transmit high-abstraction, high-presence shared futures. In that larger system, altruism is not a contradiction of evolution. It is evolution at higher abstraction.
Art, Music, Literature, and Religion as Evolutionary Devices
Art, music, literature, religion, ritual, myth, and moral narrative are not treated here as ornamental by-products. They are symbolic-cultural devices that perform a precise function: they convert high-abstraction LUBs into shared presence. A mathematical culture of peace cannot exist unless the future of peace is not merely conceptually available but felt as real by a population.
| Symbol | Meaning |
|---|---|
| high abstraction | the high-level LUB ideal (peace, dignity, shared future) supplied by Theorem 3 |
| shared presence | the felt realness of that ideal across a population, amplified by symbols (Pσ) |
| transgenerational transmission | the capacity of songs, stories, rituals, and law to carry the ideal to the next generation |
Humanity’s Cognitive Future-Origin
Tomabechi Evolution Theorem gives the evolutionary meaning of the high-order LUB-TCZ. If cognitive time is organized by future LUB-TCZs, then humanity’s cognitive future-origin can be modeled as a Shared-High-TCZ in which diversity is not eliminated but harmonized under a higher LUB. In such a future, peace, altruism, dignity, creativity, symbolic culture, and coexistence are not external moral decorations. They are the high-abstraction organizing structure of human cognitive time.
Present conflict and war, in this interpretation, do not reveal humanity’s final nature. They reveal that humanity is still on the way toward its higher cognitive future-origin. They are evidence of incomplete evolution relative to the future Shared-High-TCZ that can organize human cognitive time.
6. The Ethical Constraint — Ethic(B)
From Decept to Ethic
The most consequential element of the applied theory is the ethical constraint. The operational formalization in the foundational theory introduced Decept(M) as the term suppressing deception. This is a deductive constraint by which a cognitive-warfare system avoids deceptive content (disinformation, deepfakes), and avoids inducing self-deception through inductive generation by generative AI without deductive mathematics or through cyber attack on AI.
This paper replaces Decept with Ethic(B): a penalty on structures that violate the autonomy, dignity, long-term interest, and higher-abstraction integrity of the client or organization.
Construction of the Ethic Term
Ethic(B) penalizes four forms of violation:
| Form of violation | Concrete example |
|---|---|
| Violation of autonomy | Directing the client toward a path the client did not choose |
| Violation of dignity | Treating the client as a "subject of treatment" or "subject of correction" |
| Violation of long-term interest | Distorting long-term cognitive structure for short-term outcome |
| Violation of higher-abstraction integrity | Sacrificing higher-order ethical coherence for individual gain |
Definition of Ethical Coaching
The applied Bridge Dynamics must therefore be a structure that elevates self-determination rather than manipulating it.
Coaching is the design of a bridge by which the other
can choose a higher-order future on their own.
This is not a philosophical preference. It is a constraint embedded in the mathematics of this paper. Penalizing Ethic(B) with high δ biases the solution of the bridge optimization toward the protection of autonomy. Coaching does not become non-manipulative because the coach is "careful"; the optimization itself excludes manipulative structure as a solution. This is the ethical core of this paper and the decisive structural difference in inverting cognitive-warfare theory into its applied form.
Ethical Coaching/Leadership = Peace Promotion as the Most Effective Cognitive Warfare. The mathematics of this paper leads to the strong proposition: "the most effective cognitive-warfare strategy is to promote peace". This mathematical fact repositions the normative essence of ethical coaching and leadership as strategically optimal. That is, under Ethic(B) — which protects autonomy, dignity, long-term interest, and higher-order integrity — the practice of promoting integration toward Shared-High-TCZ is both ethical and optimal: ethics and effectiveness do not conflict; they mathematically point in the same direction. This is a mathematical refutation of the long-standing assumption that "promoting peace is ideal but unrealistic": within the framework of this paper, peace promotion is the most rational and most sustainable strategy.
7. Conclusion
This paper has presented the minimal 5-theorem system of Tomabechi Cognitive Space Time Theory (Tomabechi Cognitive Physics and Cognitive Biology). We have mathematically established the direct chain from individual convergence to a TCZ (Tomabechi Master Theorem), shared convergence (Shared-TCZ Convergence Theorem), and higher-order abstract convergence (Abstract Shared-TCZ Convergence Theorem) through to symbolic culture generation (Tomabechi Symbolic Culture Generation Theorem) and evolution (Tomabechi Evolution Theorem).
The first result is the Tomabechi Master Theorem. Under an optimal policy that minimizes the potential function V, cognitive states converge to the individual TCZ. Self, Ego, and TCZ are three descriptions of the same process, and cognitive homeostasis is rigorously formalized in the language of optimal control theory.
The second result is the Shared-TCZ Convergence Theorem. Dyadic cognitive coordination forms a shared TCZ through symmetric interaction terms γijSij, providing the mathematical foundation for cooperation, empathy, and coaching.
The third result is the Abstract Shared-TCZ Convergence Theorem. The LUB (Least Upper Bound) bundles shared TCZs into higher-order abstract concepts, realizing the convergence of culture, groups, and organizations beyond the individual.
The fourth result is the Tomabechi Symbolic Culture Generation Theorem. When symbolic cultural forms C amplify the presence of a high-abstraction world and the value sign of that world is positive, the symbolic culture generation capacity σ lowers the effective potential of that world, making it shareable, memorable, and transmissible across individuals and generations. Art, myth, religion, law, and narrative are instances of this mathematical mechanism.
The fifth result is the Tomabechi Evolution Theorem. Humanity evolves the capacity to live in higher abstraction, richer presence, symbolic culture, altruism, and peace, because these capacities make it possible to inhabit a Shared-High-TCZ and transmit it across generations.
The final claim of this minimal 5-theorem system is as follows. Cognitive homeostasis (Tomabechi Master Theorem, Shared-TCZ Convergence Theorem, Abstract Shared-TCZ Convergence Theorem) exists, and from it, symbolic culture generation and evolution necessarily follow. A Shared-High-TCZ — where all diversity coexists harmoniously under a higher LUB — exists as the origin of humanity's cognitive future time. Cognitive time flows from that future toward present action and past meaning, and humanity evolves toward that Shared-High-TCZ. Current conflict and war do not represent humanity's ultimate nature; they represent humanity in the process of evolving toward it.
Finally, the 5 theorems of this paper are presented as a foundational framework for the author's proposed Cognitive Physics and Cognitive Biology. Cognitive Physics provides cognition with dynamical laws — potential, control, convergence — and Cognitive Biology provides cognition with an evolutionary account — abstraction level, presence, altruism, and symbolic culture as selected traits. The unified mathematics of TCZ, Ego, and LUB presented in this paper is intended as a common foundation for rigorously developing both disciplines.
8. References
Bellman, R. (1957). Dynamic Programming. Princeton University Press.
LaSalle, J. P. (1976). The Stability of Dynamical Systems. SIAM.
Lyapunov, A. M. (1892/1992). The General Problem of the Stability of Motion. Taylor & Francis.
Tomabechi, H. (2025). オーセンティック・コーチング2026 ～本物のコーチング～ [Authentic Coaching 2026: Genuine Coaching]. Kaitakusha, November 24, 2025. ISBN-10: 4758970297; ISBN-13: 978-4758970297. In Japanese.
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. (2026c). Cognitive Latent Potential Theory. Cognitive Research Laboratories Technical Report, May 2026.
Reader's Guide — Figures
Reader’s Guide — The Five-Theorem Story (One Picture)
The five theorems as one story: (1) the mind settles into its own valley of comfort (Theorem 1); (2) springs pull interpersonal misalignments until all align in a shared valley (Theorem 2); (3) the group climbs a ladder to the higher unification, the LUB (Theorem 3); (4) symbolic culture builds a bridge to that high world and makes it feel real; (5) evolution climbs the slope of the score F, selecting high abstraction, presence, and symbolic capacity.
Reader's Guide — Logical Chain of the Minimal 5-Theorem System
Reader's Guide — Lyapunov Function Hierarchy of the 5 Theorems
9. Appendix — Rigorous Proofs
A.1 General Setup
| Symbol | Meaning |
|---|---|
| Φ | Lyapunov or composite potential function |
| φ | semantic or abstraction map |
| d | distance or semantic distance |
| dL | semantic pseudodistance inside LUB L |
Let X be a finite-dimensional smooth state space with metric d. Let the controlled dynamics be
where f is locally Lipschitz in x and continuous in u,t. Assume admissible controls generate unique forward solutions on [0,T]. Let all relevant Lyapunov functions have compact sublevel sets along the trajectories considered.
For a differentiable Lyapunov candidate Φ and threshold θ, define
If, outside Ωθ,
then the standard comparison argument implies convergence toward Ωθ. This is the proof pattern used throughout the appendix.
Remark (from cost-minimization to the descent condition). The descent hypothesis used here is the precise dynamical content of the optimal-control formulation πc=arg minu 𝔼∫0T V dτ, and is justified — not merely assumed — as follows. Let the value function be J*(x,t)=infu 𝔼∫tT V(x(τ),τ)dτ. Under the stated regularity (locally Lipschitz dynamics, admissible controls, J* continuously differentiable), J* satisfies the Hamilton–Jacobi–Bellman equation −∂/∂t J* = minu{V + ∇J*·f(x,u,t)}, so along the optimal closed loop dJ*/dt = −V ≤ 0: the value function is non-increasing under πc. We assume in addition that the closed loop is dissipative and detectable with respect to V — the comfort set Ωθ is reachable and the optimal feedback admits a rate α>0 with dΦ/dt ≤ −α(Φ−θ) outside Ωθ for Φ taken as V (equivalently as J*). This dissipativity assumption is what “the Ego steers toward lower evaluative cost” means dynamically, and it does not follow from optimality alone; granting it, the convergence conclusion follows from the comparison argument. On the boundary ∂Ωθ the same inequality gives dΦ/dt ≤ 0, so Ωθ is forward-invariant (Nagumo’s condition) and the trajectory does not re-exit, yielding dist(x(t),Ωθ)→0 rather than mere boundary contact.
A.2 Proof of Theorem 1: Tomabechi Main Theorem / Individual TCZ Convergence
| Symbol | Meaning |
|---|---|
| V | baseline evaluative potential; instability or discomfort cost |
| Ai | abstraction potential; zero at the LUB target |
Let Φ(x,t)=V(x,t) and define
By definition, this set is the state-space representation of TCZ(x0) restricted to the reachable set. Suppose the Ego control policy πc induces closed-loop dynamics ẋ=F(x,t) and satisfies the Lyapunov decrease condition
| Symbol | Meaning |
|---|---|
| ∇ | gradient with respect to state variables |
for all x∉Ωθ. Let
Outside Ωθ, y(t)>0 and
By the comparison theorem,
Therefore limsupt→∞ V(x(t),t)≤θ. Under compactness of the relevant sublevel sets and continuity of V, this implies
Since Ωθ is precisely the TCZ threshold set, the optimal trajectory converges to TCZ(x0). The five-step equivalence between possible-worlds semantics and state-space control establishes that implementing the Self operator as πc gives the same convergence statement in dynamical form. Hence Self, Ego, and TCZ are unified as semantic structure, control process, and stable limit set. ∎
A.3 Proof of Theorem 2: Tomabechi Shared-TCZ Convergence Theorem
ℒ̇≤−α(ℒ−θ) outside Ωθℒ；
dist(x(t),Ωθℒ)→0； Sij≤θ/γmin (Sij→0 as θ→0)； x*(t)→TCZshared.
| Symbol | Meaning |
|---|---|
| Sij | Cognitive deviation or inconsistency between agents i and j; typically ||xi−xj||². |
| ℒ | Composite Lyapunov function for the multi-agent system. |
Let the joint state be x=(x1,…,xN). Each agent i follows a control policy πi that minimizes accumulated cost containing both its own evaluative instability Vi and the cognitive deviation Sij from other agents j. Assume Sij≥0 and γij>0.
Define the composite Lyapunov function ℒ(x,t)=ΣiVi(xi,t)+Σi,jγijSij(xi,xj). If along the closed-loop trajectory ℒ̇≤−α(ℒ−θ) outside the stability set Ωθℒ, then the unified Lyapunov lemma gives dist(x(t),Ωθℒ)→0.
Since every term in ℒ is nonnegative, low ℒ means both low individual instability and low pairwise inconsistency. Moreover, by the strong-coupling condition of Assumption A5 there exists γmin := mini≠j γij > 0. On Ωθℒ, since every term of ℒ is nonnegative, γmin Σi,j Sij ≤ Σi,j γij Sij ≤ ℒ ≤ θ, hence Sij ≤ θ/γmin on every coupling edge. Pairwise cognitive misalignment is thus uniformly bounded at the threshold scale, and in this operational sense the joint trajectory converges to the shared stability region TCZshared. The strict limit Sij → 0 follows either from the same estimate along a nested sequence of thresholds θk ↓ 0, or from Barbalat’s lemma (valid for time-varying systems) under uniform continuity of ℒ̇. The essential point of this proof is that the classical LaSalle invariance principle — which presupposes an autonomous system — is never applied directly to the time-varying closed loop ẋ = F(x,t).
A.4 Proof of Theorem 3: Tomabechi LUB Convergence Theorem
A(x)=0⇔φ(x)=⊤=LUB(W1,...,WN)；
ℒ̇A≤−α(ℒA−θA)；
Sij≤θA/γmin, A(x)≤θA/η (→0 as θA→0)； φ(x(t))→⊤.
| Symbol | Meaning |
|---|---|
| A(x) | Abstraction potential measuring distance from the target LUB. |
| η, ηi | Positive weight on the abstraction potential. |
| ℒA | Lyapunov function combining individual stability, social alignment, and abstraction. |
Add an abstraction potential A(x) to the composite Lyapunov function of Theorem 2. Assume A(x)≥0 and that A(x)=0 if and only if the abstraction map φ(x) reaches the target top element ⊤=LUB(W1,…,WN).
Set ℒA(x,t)=ΣiVi+Σi,jγijSij+ηA(x). If ℒ̇A≤−α(ℒA−θA) along the closed-loop trajectory, the unified Lyapunov lemma implies convergence toward the threshold set of ℒA.
On the threshold set ΩθA, since every term of ℒA is nonnegative, γmin Σ Sij ≤ θA and η A(x) ≤ θA, hence Sij ≤ θA/γmin on every edge and A(x) ≤ θA/η. Letting θA,k ↓ 0 along a nested sequence of thresholds (or invoking Barbalat’s lemma under uniform continuity of ℒ̇A — both valid for time-varying systems) yields Sij → 0 and A(x) → 0. Here too the classical LaSalle invariance principle for autonomous systems is not used. By design, A(x)→0 is equivalent to φ(x)→LUB(W1,…,WN). Therefore the system converges not merely to a low-level intersection but to the least upper bound in the abstraction lattice.
A.5 Proof of Tomabechi Symbolic Culture Generation Theorem
Ṽσ,i=Vi−κiPσ,iQi
Qi>0, λi∂Symi/∂σi>0 ⇒ ∂Ṽσ,i/∂σi<0
∂Ṽσ,i/∂σi=−κiQi∂Pσ,i/∂σi
κi>0, Qi>0 ⇒ ∂Ṽσ,i/∂σi<0
therefore increasing symbolic capacity lowers the effective potential of a positively valued high-abstraction world
Define the symbolically augmented presence as Pσ,i=Pi+λiSymi. If symbolic capacity σi increases the symbolic function Symi and λi>0, then Pσ,i increases with σi. Substituting Pσ,i into the presence-weighted effective potential gives Ṽσ,i=Vi−κiPσ,iQi. For a positively valued world, Qi>0. Hence the derivative of Ṽσ,i with respect to σi is negative. Thus symbolic culture lowers the effective potential of a high-abstraction world when that world is positively valued. This proves the theorem.
We now complete the above computation formally.
Claim. Under Assumption A7, at every point (x,t) and for every agent i, on the set {Qi > 0} we have
That is, the presence-weighted effective potential is strictly monotonically decreasing in the symbolic-cultural capacity σi.
Proof. By Assumption A7, Symi is C¹ in σi, so Pσ,i = Pi + λiSymi may be differentiated in σi; since Pi does not depend on σi, we obtain ∂Pσ,i/∂σi = λi∂Symi/∂σi > 0. Differentiating Ṽσ,i = Vi − κiPσ,iQi in σi, and using that Vi, κi, and Qi do not depend on σi, the chain rule gives ∂Ṽσ,i/∂σi = −κiQi∂Pσ,i/∂σi. From the signs of the three factors κi > 0, Qi > 0, and ∂Pσ,i/∂σi > 0, the right-hand side is strictly negative. ∎ (Lemma)
Dynamical consequence. By the monotonicity of Lemma A.5.1, an increase in σi uniformly lowers Ṽσ,i on positively valued high-abstraction worlds and monotonically enlarges the corresponding stability set Ωθσ = {x | Ṽσ,i(x,t) ≤ θ}. Applying the unified Lyapunov argument of Appendix A.1–A.2 with Φ = Ṽσ,i (Assumptions A1–A4), convergence to such a world under the policy πcsym is strengthened as σi grows. Hence symbolic-cultural capacity is an independent evolutionary variable that rigorously promotes convergence to positively valued high-abstraction worlds. ∎
A.6 Proof of Tomabechi Evolution Theorem
Let
be the evolutionary cognitive-cultural parameter vector of subject or lineage i. Define
Assume that the parameter domain D is compact, that Fi is continuously differentiable on D, and that selection dynamics are given by positive-definite gradient ascent:
where Mi is symmetric positive definite. Along trajectories,
Equality holds only when ∇Fi=0. Equivalently, the Lyapunov candidate
satisfies dΨ11/dt≤0. Thus the adaptive dynamics monotonically increases fitness until it reaches an invariant set of stationary points.
Now consider the low-abstraction / low-presence / low-symbolic-capacity region. By the marginal-benefit assumptions,
Therefore Fi has positive directional derivatives in the directions of increasing abstraction, presence-generating capacity, and symbolic-cultural capacity. Hence a low-α, low-ρ, low-σ point cannot be a local maximum. Compactness guarantees the existence of a maximizer; the marginal-benefit inequalities show that the maximizer cannot lie in the purely low-abstraction, low-presence, low-symbolic-capacity region. Therefore evolutionary selection favours higher abstraction, richer presence generation, and symbolic-cultural capacity. The theorem follows.
We now complete existence and convergence formally.
Well-posedness. Under Assumption A8, Mi is locally Lipschitz and Fi is C¹, so the right-hand side Mi(z)∇Fi(z) is continuous and locally Lipschitz on D. By the Picard–Lindelöf theorem, the gradient flow żi = Mi∇Fi has a unique solution for each initial value, defined on [0,∞) by the forward invariance of D (A8).
Claim. Under Assumption A8, every solution zi(t) of the gradient flow satisfies: (i) t ↦ Fi(zi(t)) is nondecreasing and converges to a finite limit F∞; (ii) the ω-limit set ω(zi(0)) is nonempty, compact, connected, and invariant, and is contained in the critical set {z ∈ D | ∇Fi(z) = 0}.
Proof. (i) Along trajectories dFi/dt = ∇FiTMi∇Fi ≥ 0 (Mi positive definite), and Fi is bounded above as a continuous function on the compact set D. Monotone bounded convergence gives Fi(zi(t)) → F∞. (ii) Since D is compact and solutions are defined on [0,∞), the set ω(zi(0)) is nonempty, compact, connected, and invariant (standard properties of autonomous systems). By (i), Fi ≡ F∞ on the ω-limit set; by invariance, along any solution through a point of the ω-limit set dFi/dt = 0, that is, ∇FiTMi∇Fi = 0. Positive definiteness of Mi yields ∇Fi = 0. This is LaSalle’s invariance principle in its gradient-flow form. ∎ (Lemma)
Exclusion of the low region (formalized). By the marginal-benefit conditions of Assumption A8, on the low-(α,ρ,σ) region we have ∂Fi/∂αi > 0, ∂Fi/∂ρi > 0, and ∂Fi/∂σi > 0. Hence that region contains no critical points, and by Lemma A.6.1(ii) the ω-limit set does not intersect it. Moreover, a global maximizer exists by the compactness of D and the continuity of Fi; an interior point of the low region cannot be a maximizer because it is not critical, and on the low-region boundary faces the increasing directions (increasing α, ρ, σ) point into D, so the first-order optimality condition fails there as well. Therefore the maximizer lies outside the low-abstraction, low-presence, low-symbolic-capacity region. In conclusion, the evolutionary dynamics monotonically increases fitness while converging to the critical set, and both the critical set and the global maximizer lie on the side of high α, ρ, and σ. ∎
A.7 Corollary: Humanity’s Cognitive Future-Origin
| Symbol | Meaning |
|---|---|
| LUB | least upper bound; minimal higher abstraction integrating lower worlds |
By the Abstract Shared-TCZ Convergence Theorem (Theorem 3), each future LUB-TCZ can serve as a higher-order stability region for the group. By Tomabechi Evolution Theorem, evolutionary dynamics select capacities that make higher LUB-TCZs inhabitable and transmissible. Let GH denote the human Shared-High-TCZ in which diversity is included under a higher LUB. If GH maximizes long-term shared fitness among available high-abstraction futures, then it functions as humanity’s cognitive future-origin. Present conflict indicates that current states have not yet converged to GH; it does not refute the existence of GH as the organizing future. ∎
Mathematical Foundations: Questions of Mathematical Rigor and Their Resolution
This section sets out the questions of mathematical rigor that the five theorems retained in this minimal system — Theorem 1 (Individual TCZ Convergence), Theorem 2 (Shared-TCZ Convergence), Theorem 3 (LUB Convergence), the Tomabechi Symbolic Culture Generation Theorem, and the Tomabechi Evolution Theorem — may raise from the standpoint of control theory and applied mathematics, together with their resolutions. Of the five questions below, the first three are those, among the same rigor questions examined for the parent Cognitive Space-Time Theory, that bear on these five theorems, while the remaining two (Questions 4 and 5) were examined and resolved additionally during the control-theoretic review of this minimal version; for each, the original formulation is noted, the precise concern is stated, and the rigorous resolution is provided.
Each resolution presented in this section is implemented in the body of the paper as formal structure, not merely as commentary: the dissipativity of Question 1 is stated explicitly as Assumption A4 (and justified in the supplement to Appendix A.1); the probability space of Question 2 is defined as Assumption A6; and the convergence of the adaptive dynamics in Question 3 is proved as Lemma A.6.1 of Appendix A.6 (a LaSalle-type argument). In addition, the treatment of time-variation in Question 4 is implemented as the direct estimates of Appendices A.3–A.4 (nonnegative decomposition on the threshold set with the γmin bound), and the time-scale separation of Question 5 is stated explicitly as a supplement to Assumption A8.
Question 1 — Lyapunov Descent Gap: Does Minimizing ∫Vdt Guarantee V Decreases?
The claim "πc minimizes ∫Vdt ⇒ x*(t) → TCZ" rests on a non-trivial connection. The optimal control policy πc minimizes the integral of V over time — this is not the same as guaranteeing that V(x*(t),t) itself decreases along the trajectory. A standard counterexample: a cost-minimizing trajectory can allow V to spike temporarily if doing so produces lower long-run cost. The Lyapunov argument (V is a decreasing function along trajectories) does not follow from optimality alone.
The rigorous connection is established through the Hamilton–Jacobi–Bellman (HJB) equation. Let the value function be J*(x,t) = infu 𝔼∫tT V(x(τ),τ)dτ. Under standard regularity (locally Lipschitz dynamics, admissible controls, J* ∈ C¹), J* satisfies the HJB equation: −∂/∂t J* = minu{V + ∇J*·f(x,u,t)}. Along the optimal closed loop: dJ*/dt = −V(x*,t) ≤ 0. Therefore J* — not V directly — serves as the Lyapunov function. The theorem uses J* as the candidate Φ, and the Lyapunov decrease condition dΦ/dt ≤ −α(Φ−θ) follows from the dissipativity assumption (stated in the Appendix proofs): that the optimal feedback system is dissipative with respect to V and the stable region TCZ is reachable. This dissipativity assumption is not a consequence of optimality but an additional structural requirement on the cognitive system. It is the mathematical expression of the statement: "the Ego's control steers consistently toward lower evaluative cost, not only on average over time but along trajectories." For biological cognitive systems, this is a reasonable homeostatic assumption; for AI implementations, it must be explicitly verified or designed in.
Question 2 — Stochastic Expectation 𝔼 in Theorem 3 Only
Theorem 3 introduces an expectation operator 𝔼 in the control objective (𝔼∫₀ᵀ[Vᵢ + ΣγᵢⱼSᵢⱼ + ηᵢA(xᵢ)]dt), while Theorems 1 and 2 and the Symbolic Culture Generation and Evolution theorems use deterministic integrals. The probability space governing the expectation is not defined.
The expectation in Theorem 3 arises from uncertainty about the true LUB position in the subsumption lattice. When an agent does not have perfect knowledge of the cognitive world positions of others or of the location of the target LUB in the lattice, the control objective is stochastic: the agent minimizes expected cost over the uncertainty in lattice positions. The probability measure Pr is defined over the agent's epistemic uncertainty about (W₁,...,WN). In the special case of perfect knowledge (no uncertainty about lattice positions), 𝔼 reduces to the identity and Theorem 3 collapses to the deterministic form of Theorem 2 with an added abstraction term. Theorems 1 and 2 and the Symbolic Culture Generation and Evolution theorems implicitly assume deterministic settings; the 𝔼 may be added to all of them for full generality without changing the proof structure. The Appendix proofs establish the deterministic case, from which the stochastic generalization follows by standard arguments under appropriate measurability conditions on f(x,u,t) and A(x).
Question 3 — Evolution Theorem: Adaptive Dynamics and Convergence to Optimum
The adaptive dynamics żi = Mi(zi)∇Fi(zi) show that dFi/dt = ∇FiᵀMi∇Fi ≥ 0, so Fi is non-decreasing. But non-decreasing and bounded above does not automatically imply convergence to the optimum (α*, ρ*, σ*). Convergence requires additional regularity: the dynamics must not cycle and the optimum must be an attractor.
Convergence of the adaptive dynamics follows from three conditions: (1) Fi is bounded above (the marginal benefit terms Share, G, Pres, Sym are bounded by finite populations and environmental carrying capacity); (2) the level sets {z : Fi(z) ≥ c} are compact (which follows from the cost term Ci(α,ρ,σ) → ∞ as α,ρ,σ → ∞, preventing escape to infinity); and (3) the gradient flow żi = Mi∇Fi has no limit cycles (which follows from Fi being a strict Lyapunov function along all non-stationary trajectories). Under these three conditions, LaSalle's invariance principle applied to the adaptive dynamics ensures convergence to the set {z : ∇Fi(z) = 0} — the critical points of Fi. The marginal-benefit conditions of the Evolution Theorem ensure that the low-abstraction region is not a local maximum, so the critical points lie in the high-abstraction regime. Convergence to a saddle point rather than a maximum is precluded by the positive-definiteness of Mi.
Question 4 — Time-Varying Systems and the Applicability of LaSalle's Invariance Principle
In deriving Sij → 0 from the decrease of the composite Lyapunov function ℒ(x,t) (and ℒA), the proofs of Theorems 2 and 3 might appeal to an invariance-principle-type argument. The closed-loop system ẋ = F(x,t), however, is time-varying through the time dependence of Vi(xi,t), whereas the classical LaSalle invariance principle presupposes an autonomous (time-invariant) system and therefore does not apply as stated.
This paper does not apply LaSalle to a time-varying system. The proofs in Appendices A.3 and A.4 reach their conclusions using only (i) the comparison theorem, which gives dist(x(t), Ωθℒ) → 0 and remains valid for time-varying systems, and (ii) a direct estimate by nonnegative decomposition on the threshold set — Sij ≤ θ/γmin (and likewise A(x) ≤ θA/η), using the strong-coupling bound γmin > 0 of Assumption A5. Where a strict zero limit is required, it follows from the same estimate along a nested sequence of thresholds θk ↓ 0, or from Barbalat's lemma (applicable to time-varying systems) under uniform continuity of ℒ̇. The evolutionary gradient flow ż = M∇F of Appendix A.6, by contrast, is autonomous, so the LaSalle-type argument there (Lemma A.6.1) is legitimate.
Question 5 — Two Time Scales of Individual and Evolutionary Dynamics (Adiabatic Approximation)
The fitness Fi(z) of the Evolution Theorem is defined as an integral ∫0T[−Vi + …]dt along individual trajectories. If the individual state dynamics x(t) (Theorems 1–3) and the gradient flow of the evolutionary parameters z = (α,ρ,σ) ran on the same time axis simultaneously, the evaluation of Fi(z) would cease to be well defined, and a circularity with Theorems 1–3 could arise.
The two dynamics are separated in time scale (adiabatic approximation). Individual cognitive dynamics run on the fast time t; the evolutionary variable z moves on a slow, generational time. For each fixed z, Fi(z) is evaluated as a completed integral along the individual optimal trajectory of that generation — so Fi is a function of z alone (assumed C¹ in Assumption A8), and the gradient flow ż = M∇F is autonomous. This separation is the mathematical expression of the standard biological setting (generation time ≫ behavioral time), and no reverse dependence — the conclusions of the Evolution Theorem feeding the assumptions of Theorems 1–3 — exists. This point is also stated explicitly as a supplement to Assumption A8.
Summary: Theorem Soundness Status
| Theorem | Question Found | Status After Resolution |
|---|---|---|
| T1 | HJB-Lyapunov gap | ✓ Resolved: dissipativity assumption explicitly stated |
| T2 | Time-varying dynamics vs. invariance principle (Q4) | ✓ Resolved: nonnegative decomposition on the threshold set, Sij≤θ/γmin (App. A.3–A.4), with a Barbalat remark |
| T3 | Undefined probability space for 𝔼 | ✓ Resolved: epistemic uncertainty over lattice positions |
| Symbolic Culture Generation Theorem | None critical | ✓ Sound under the symbolic-culture-generation setting |
| Evolution Theorem | Adaptive dynamics convergence to optimum | ✓ Resolved: LaSalle (autonomous gradient flow) + compactness + no limit cycles, plus two-time-scale separation (Q5) |
The overall logical structure of the five-theorem minimal system is sound. No circular dependencies were found among the theorems: T1 → T2 → T3 (adding coupling terms progressively); the Symbolic Culture Generation Theorem establishes symbolic culture as the transmission mechanism for high-abstraction shared TCZs; and the Evolution Theorem establishes the evolutionary selection of the altruism, presence, and symbolic capacities that sustain them. Each theorem in the chain uses the output of prior theorems as inputs or structural assumptions, without reverse dependencies.
Theorem Summary Table — 5 Theorems (Theorems 1, 2, 3, Tomabechi Symbolic Culture Generation Theorem, Tomabechi Evolution Theorem)
Theorem Summary Table — 5 Theorems (Theorems 1, 2, 3, Tomabechi Symbolic Culture Generation Theorem, Tomabechi Evolution Theorem) (No., Name, Central Formula, Meaning)
| Theorem | Core formula | Main variables | Meaning |
|---|---|---|---|
| Theorem 1 Tomabechi Master Theorem | πc=arg minu(t)∫0TVdt ⇒ x*(t)→TCZ(x0) | V: instability TCZ: stable region | The Ego converges exponentially to the individual TCZ valley under optimal control. |
| Theorem 2 Shared-TCZ Convergence | πi=arg minu(t)∫0T(Vi+ΣγijSij)dt ⇒ (x1,x2)→TCZsh | γij: coupling Sij: empathy | Two coupled agents converge to a shared stable valley. |
| Theorem 3 Abstract Shared-TCZ Convergence | πi=arg min 𝔼∫(Vi+ΣγS+ηiA)dt ⇒ Ai(t)→0 | A: abstraction potential LUB: least upper bound | A group sharing a high LUB goal converges to an abstract shared TCZ. |
| Tomabechi Symbolic Culture Generation Theorem | πsymc=arg min∫Psymdt, Pσ,i=Pi+λiSymi | Sym: symbol Pσ: symbol-enhanced presence | Symbols give presence to higher worlds and deepen the valley. |
| Tomabechi Evolution Theorem | πevo=arg max F, ż=M∇F, dF/dt≥0 | F: fitness functional α,ρ,σ: evolutionary variables | Evolution climbs the fitness slope monotonically toward high α, ρ, σ. |