This paper presents a minimal theorem system for deriving the claim that cognitive time flows from the future to the present and then to the past. It is not a coaching manual, nor is it a theory of persuasion. It is a mathematical theory of human life: a theory of how a human being lives by allowing a future world to organize present action and the meaning of the past.
The paper uses only the minimal set of previously established Tomabechi theorems needed for the derivation. Theorem numbers are preserved from the larger Cognitive Space Time Theory framework. The minimal chain used here is Theorem 1, Theorem 2, Theorem 3, Theorem 7, Theorem 8, and Theorem 10. Theorem 1 is the main theorem of this paper. It establishes that the Ego, implemented as an optimal control policy, converges toward the Total Comfort Zone (TCZ). The semantic Self/Ego/TCZ formulation in §2.8 and the Lyapunov formulation in §5 are two presentations of this same Theorem 1. Theorem 2 extends this structure to shared human life. Theorem 3 introduces the LUB, the least upper bound of cognitive worlds, and makes abstraction mathematically central. Theorem 7 defines a true goal as a future state outside the current TCZ. Theorem 8 proves that, once such a future goal is set as a terminal condition, the current Ego control policy is determined from the future side, and past meaning is reorganized accordingly. Theorem 10 generalizes this to the universe as a cognitive-information space: physical time appears at the lowest level of abstraction as the coordinate of physical spacetime, whereas cognitive time appears at higher abstraction as a future-origin semantic axis within each LUB-TCZ.
The central conclusion is that the physical universe and the cognitive universe exhibit opposite effective orientations of time. In the lowest-abstraction physical universe, time is expressed through physical succession and thermodynamic increase. In the higher-abstraction cognitive universe, time is expressed through future-origin semantic organization, decreasing distance to a future LUB-TCZ, and reinterpretation of the past. This is not physical retrocausality and does not deny the thermodynamic arrow of time. It states a different mathematical fact: in cognitive systems, the future goal functions as the origin of control and meaning.
The final implication is practical but not merely practical. Self-development, coaching, and leadership share one common structure: success requires certainty that the effective direction of cognitive time is from the future to the past. This is not a philosophical slogan. Within the formal system developed here, it is a mathematical consequence of the TCZ, Ego control, true-goal exteriority, and LUB-indexed cognitive time.
All theorems in this paper are proved in a self-contained manner under an explicitly stated set of standing assumptions (A1–A8, §2.12). Every proof relies only on dissipativity derived from dynamic programming (the HJB equation), a comparison lemma, 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.
The full Cognitive Space Time Theory is a large system comprising many theorems, counting Theorem 6A and Theorem 6B as two distinct theorems. The present paper extracts only the theorem chain needed to prove the direction of cognitive time and to formulate a minimal theory of human life. The complete applied framework is treated elsewhere; here the argument deliberately remains minimal.
This paper therefore asks a sharper question: what is the smallest theorem chain that still proves that cognitive time flows from the future to the present and then to the past?
The answer is:
Theorem 1 gives the individual TCZ. Theorem 2 shows that human life is not isolated but shared. Theorem 3 introduces abstraction and LUB. Theorem 7 defines the true goal as outside the current TCZ. Theorem 8 derives future-origin cognitive time from Ego control with a future terminal condition. Theorem 10 extends the same structure to the cognitive universe.
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 (§2.12), yielding a self-contained mathematical system with no dependence on external literature. Second, Theorems 1–3 (individual TCZ convergence, shared-TCZ convergence, and LUB abstraction convergence) are given complete proofs based on dissipativity derived from the HJB equation, a comparison lemma, and a LaSalle-type invariance argument (Appendix A.1–A.3). Third, for Theorem 8 (the Future-Origin Cognitive Time Theorem), a verification lemma for optimal control with a future terminal condition (Appendix A.5, Lemma A.5.1) is proved in a self-contained manner, showing that the claim that the future goal determines present control is rigorously consistent with causality as a non-anticipative feedback law. Fourth, Theorem 10 generalizes cognitive time to every LUB level, and the non-smoothness of distance functions is handled by a Dini-derivative comparison lemma (Appendix A.6, Lemma A.6.1), establishing the opposition between the directions of physical and cognitive time as a mathematical fact.
Organization. Part I (§2–§7) develops the foundational theory: §2 presents the foundational concepts and the standing assumptions, §3–§4 position the true goal and the minimal theorem chain, and §5–§7 state Theorems 1–3. Part II (§8–§15) develops the theory of time: §9 gives Theorem 7 (the true goal), §10 gives Theorem 8 (future-origin cognitive time), §13 gives Theorem 10 (the cognitive universe), and §11–§12 and §14–§15 discuss their implications. §16 concludes. The appendices provide a symbol-by-symbol deep dive for each theorem, the rigorous proofs (A.0–A.8), and a systematic examination and resolution of the questions of mathematical rigor.
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.
The formulation below integrates two equivalent languages. The first is the semantic language of possible worlds and modal logic. The second is the dynamical language of state space and optimal control. The five-step bridge between them is one of the key achievements of the Tomabechi Framework: concepts such as Self, Ego, and comfort zone are not merely psychological metaphors; they can be stated as mathematical objects and implemented as control systems.
Following the formulation of the theory, the cognitive state of a subject at time t is denoted by
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
The function V0(x,t) 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 V0 means that the state is less stable and less sustainable for the subject.
Human belief systems exhibit a degree of stability that cannot be explained by simple linear updating. Beliefs, self-images, and expectations are not overwritten mechanically by new information. They preserve coherence. This tendency is called cognitive homeostasis.
To model cognitive homeostasis, cognition must be represented not as a single state but as a set of possible worlds. The effective domain of cognition includes the present and the future:
Thus, human cognition operates over a space of possible present-and-future worlds. It continuously evaluates potential trajectories rather than reacting only to immediate stimuli.
Possible worlds are not equivalent. Some are cognitively stable; others are experienced as unstable, inconsistent, threatening, or impossible. Therefore the set W must be equipped with an ordering based on cognitive comfort.
Under this ordering, the stable region of cognition is defined as the Total Comfort Zone:
The TCZ is the totality of future worlds that the subject can stably inhabit. It is not a static comfort preference. It is a dynamically maintained region under cognitive homeostasis.
A person with a large TCZ can absorb a wide range of unexpected events without destabilization. A person with a small TCZ may be destabilized by small changes. Thus the TCZ is not merely “what feels comfortable”; it is the formal structure of resilient inhabitable worlds.
Within this framework, the Self is not a fixed substance but an operator acting on possible worlds and on the TCZ. In possible-worlds form, this can be expressed as
The Self operates in two modes:
The second mode is the mathematical essence of self-transformation, coaching, and leadership. It is not movement within the existing comfort zone; it is relocation or reconstitution of the comfort zone itself.
The Ego is the reformulation of this operation as a control equation. In modal logic, the Self defines a desired region in the language of possible worlds. In dynamical systems language, the Ego is defined as a control equation that moves optimally within, and eventually toward, an attractor basin. These two formulations describe the same underlying structure: one in the language of logic, the other in the language of dynamics.
In the Tomabechi Framework, the Ego is formulated as an optimal control problem:
Here x(t) is the cognitive state, u(t) is the control input, and V0(x,t) represents cognitive instability, evaluative cost, or internal inconsistency. The function πc(x) is the cognitive control policy: it maps any cognitive state x to the optimal action that minimizes accumulated evaluative cost. The subscript c means cognitive.
The preceding discussion can be summarized in five steps:
The possible-worlds modal-logic formulas used in this 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.
In state space, let ℛ(t;x0) denote the reachable set from the initial state x0, and let θ be an admissible threshold. Then
The possible-worlds definition and the state-space definition look different, but they describe the same underlying structure. Possible worlds correspond to trajectories in state space. The ordering relation over possible worlds corresponds to the evaluation function V0(x,t). The existence of a stabilizing relation in possible-worlds logic corresponds to the existence of a trajectory that remains in a stable region of state space.
Clarification. The expression “Main Theorem” in this section does not introduce a theorem separate from Theorem 1. It names Theorem 1 itself. Section 2.8 states Theorem 1 in its semantic form, using possible worlds, Self, Ego, and TCZ. Section 5 restates the same Theorem 1 in its Lyapunov form, using the decrease of an evaluation potential. The two are not competing results; they are two languages for the same foundational convergence theorem.
When the Self operator defined over the possible-worlds set W is implemented as the optimal control problem
the optimal trajectory x*(t) converges to TCZ(x0) under the corresponding Lyapunov decrease condition:
The theorem unifies three languages:
| Element | Language | Role |
|---|---|---|
| Self | philosophy / possible worlds | semantic ordering structure |
| Ego | control engineering | optimal control process |
| TCZ | psychology / dynamical systems | stable region or attractor basin |
Cognition is thus simultaneously a semantic structure, a control process, and a stability system. These are not separate hypotheses but three descriptions of the same process at different levels of abstraction.
The Self operator modifies the ordering structure over possible worlds. The Ego selects trajectories that minimize accumulated cognitive instability. The condition V0(x,t) ≤ θ defines the stable region. Therefore, under the Lyapunov decrease condition, the optimal trajectory enters and remains in the TCZ.
The present paper uses only the minimal theorem chain needed to derive future-origin cognitive time. Before introducing the true goal and cognitive time, only three prior results are required: individual TCZ convergence, shared TCZ convergence, and LUB convergence. These are the structural foundations provided by the theory.
The broader Cognitive Space Time Theory contains additional applied machinery, but it is not used as an assumption in this minimal paper. The direction of cognitive time is derived here from TCZ, Ego control, and LUB abstraction alone.
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 V₀ : X×[0,T] → ℝ is continuously differentiable (C¹), bounded below, and proper (for every c, the sublevel set {x | V₀(x,t) ≤ c} is compact for each t). The stability set Ωθ = {x | V₀(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 ↦ V₀(x,t) + ∇W·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 dV₀/dt ≤ −α(V₀ − 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.0.
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). Here the edges of the coupling graph are counted as ordered pairs, and on every undirected edge {i, j} we require γij, γji ≥ γmin in both directions (bidirectional positive coupling). That this mutuality cannot be dropped is shown by the minimal counterexample in the Remark after A.2 (the pursuit equilibrium with γ21=0).
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.
The terminal penalty λd(x,G)² (λ > 0) is C¹ in x. The value function WG(x,t) = infu JG[u] is C¹ (in which case WG is a classical solution of the HJB equation and the verification lemma of Appendix A.5 applies). In addition, on the reachable terminal set R(T) the map G ↦ d(·,G)²|R(T) is injective (non-degeneracy of the terminal term).
For each LUB level L, the semantic distance dL(·, GL) is 1-Lipschitz, so that τL(x) = dL(x, GL) is absolutely continuous along trajectories. For each level there exists a level-wise dissipation rate cL > 0 along the optimal closed loop (the level-wise version of A4; the derivation is given in Appendix A.6).
A central result of the present paper is that the concept of a “goal” can be defined mathematically from TCZ and Ego control.
A state G is not a true goal if it lies inside the current TCZ:
If G lies inside the current TCZ, it is an improvement, task, plan, preference, or near-term objective. It is not a true goal in the Tomabechi sense. A true goal must satisfy the exteriority condition:
More strongly, for some ε > 0,
A true goal is a future state or future TCZ component G such that:
Not every state outside the current TCZ is a true goal. A random danger, an imposed demand, a fantasy without self-concordance, or an unreachable state may also lie outside the current TCZ. The concept of a true goal therefore requires more than externality. It requires a future-side state that can be set by the Self, valued at a higher level of self-concordance, and capable of reconstituting the Ego's control problem.
Let TCZ₀ be the present Total Comfort Zone, and let π₀ be the present Ego control policy that minimizes accumulated evaluative cost relative to V₀. Let G be a candidate future state or future TCZ component. Then the following characterization holds:
Thus a true goal is precisely a future-side state outside the present TCZ that can become an origin of control reconstitution for the Ego.
Theorem 7 does two things at once. First, it separates a true transformative goal from an ordinary objective. Second, it shows that once such a goal is inserted as a terminal condition, the Ego no longer solves the same control problem. This is why the theorem is called both the Tomabechi True Goal Theorem and the Goal Exteriority and Control-Reconstitution Theorem.
| Symbol | Meaning | Plain-language reading |
|---|---|---|
| TCZ0 | The present Total Comfort Zone | The world the subject can presently inhabit as natural and stable. |
| G | Candidate future state or future TCZ component | The possible future being considered as a goal. |
| G ∈ TCZ0 | G lies inside the present TCZ | The present Ego can already understand it as part of the current world. |
| G ∉ TCZ0 | G lies outside the present TCZ | The future is outside the current world of stability. |
| dist(G, TCZ0) > ε | Strict exteriority condition | The goal is not merely at the edge; it is genuinely outside. |
| CSelf(G) | Higher-order self-concordance | The degree to which the future belongs to the subject’s deeper Self. |
| JG[u] | Goal-conditioned control functional | The new decision problem after the goal has been inserted. |
| λ | Terminal-goal weight | How strongly the future goal pulls on the control problem. |
The key distinction is the difference between inside and outside the present TCZ. If
then the present Ego can already treat G as a low-cost, understandable destination. Such a state may be useful, but it does not transform the subject’s world. It is a task, plan, improvement, or near-term objective.
If instead
then G is not contained in the present world of stability. This exteriority is necessary for a true goal because a goal that does not leave the present TCZ cannot reorganize the present TCZ. A stronger form is
This says that the distance between the candidate goal and the present TCZ is not zero and not merely infinitesimal. The goal is structurally outside the present attractor basin.
Exteriority is necessary, but not sufficient. Many things lie outside the current TCZ: danger, fantasy, manipulation, an externally imposed demand, or an impossible state. Theorem 7 therefore adds three further conditions.
| Condition | Why it is needed | Failure prevented |
|---|---|---|
| Self-settability | The Self must be able to set G as a future-side TCZ component. | A random outside event is not mistaken for a goal. |
| Self-concordant positive value | G must have value at a higher level of the subject’s own Self. | An imposed or borrowed desire is not mistaken for a goal. |
| Control reconstitutability | Under sufficient future-terminal orientation, G must be able to change the Ego’s control functional. | A fantasy that never changes action is not mistaken for a goal. |
Once these conditions are satisfied, the control problem changes from the ordinary Ego functional to a goal-conditioned functional:
The first term is the ordinary accumulated evaluation cost. The second term is the terminal-goal penalty: it becomes large when the final state x(T) is far from the goal G. Therefore, present action is now chosen not merely to reduce present discomfort, but also to reduce distance to the future goal.
| Number | Name used in this paper | Role in the minimal derivation |
|---|---|---|
| 1 | Tomabechi Main Theorem (Possible Worlds–Ego–TCZ Unification) State-space form: Tomabechi Individual TCZ Convergence Theorem | Unifies Self, Ego, and TCZ; defines the basic return of Ego to stable cognitive regions. |
| 2 | Tomabechi Shared-TCZ Convergence Theorem | Shows that human life is social and shared, not merely individual. |
| 3 | Tomabechi LUB Abstraction Theorem | Introduces abstraction and LUB as the higher direction of life. |
| 7 | Tomabechi True Goal Theorem | Defines a true goal as outside the current TCZ. |
| 8 | Tomabechi Future-Origin Cognitive Time Theorem | Derives future→present→past cognitive time from Ego control. |
| 10 | Tomabechi Cognitive Universe Theorem | Generalizes the result to physical and cognitive universes. |
The numbering is deliberately preserved from the larger framework. The present goal is not to build the complete applied theory, but to prove the direction of cognitive time with the minimum theorem set.
| Symbol | Meaning |
|---|---|
| πc(x) | Ego’s optimal control policy (minimizes accumulated cost) |
| u(t) | control variable (choices of action, attention, interpretation) |
| V(x,t) | evaluation function (discomfort/deviation cost, V=V0) |
| x*(t) | optimal closed-loop trajectory |
| TCZ(x0) | Total Comfort Zone containing x0 (threshold set) |
Assumptions: the standing assumptions A1–A4 (§2.12) are in force.
Let a subject’s cognitive dynamics be described by a closed-loop system
and let V₀(x,t) be a Lyapunov-type evaluation function. If outside the stable set
the decrease condition holds,
then
This theorem says that the Ego converges toward the subject’s stability region. It is the mathematical form of cognitive homeostasis.
| Symbol | Meaning |
|---|---|
| πi | optimal control policy of agent i |
| Vi | individual evaluation function |
| γij ≥ 0 | coupling strength between agents i and j |
| Sij | inter-agent misalignment cost |
| TCZshared | shared stability region of the coupled system |
Assumptions: the standing assumptions A1–A5 (§2.12) are in force.
For N subjects with cognitive states xᵢ, individual evaluation functions Vᵢ, and pairwise discrepancy terms Sᵢⱼ, define
If
outside the shared threshold region, then
Theorem 2 matters for a theory of human life because life is not lived alone. Even when the paper is not about leadership, human existence is socially coupled. Families, friendships, organizations, cultures, and nations all shape the stability landscape of the individual.
| Symbol | Meaning |
|---|---|
| 𝔼 | probabilistic expectation (on the space of Assumption A6) |
| ηi > 0 | abstraction weight |
| A(x) | abstraction-attraction potential (A=0 ⇔ φ=LUB) |
| φ(x) | abstraction map |
| LUB(W1,…,WN) | least upper bound of the possible-world family |
Assumptions: the standing assumptions A1–A6 (§2.12) are in force.
Let cognitive worlds form a subsumption lattice, and let A(x) be an abstraction potential satisfying
Define the abstraction-augmented Lyapunov function
If
then
Theorem 3 introduces the direction of abstraction. Without Theorem 3, the theory remains a theory of stability. With Theorem 3, it becomes a theory of human ascent: the movement from narrow stability to inclusive higher meaning.
Theorems 1–3 describe stability, social sharing, and abstraction. They do not yet define what a life should be oriented toward. A person may stabilize in a narrow TCZ. A society may converge to a low shared TCZ. A group may even reach some abstraction that remains defensive or merely conventional. A theory of human life requires one more concept: the true goal.
The crucial insight is that a true goal cannot be inside the current TCZ. If a desired state is already inside the current TCZ, it can be reached without transforming the person. It may be a task, plan, preference, or improvement. It is not a transformative goal.
Thus we now move from Theorem 3 to Theorem 7.
| Symbol | Meaning |
|---|---|
| πG | goal-conditioned optimal control policy |
| VG | goal-conditioned evaluation function |
| TCZ0 | the current Total Comfort Zone |
| G = Self-set | G is settable by the Self operator |
| G | the true goal (a future state) |
Assumptions: the standing assumptions A1–A4 (§2.12) are in force.
Let TCZ₀ be the current Total Comfort Zone. A future state or future TCZ component G is a true transformative goal only if it satisfies the exteriority condition
and, more strongly for robust separation,
Additionally, G must be settable by the Self as a future terminal condition, must have positive self-concordant value, and must be capable of reconstituting the Ego’s control functional once it functions as a future terminal condition.
Theorem 7 defines the word “goal.” This is one of the most important points of the paper. A true goal is not a desire already compatible with the present self. It is a future world outside the present stability region. It is therefore initially invisible to the present Ego.
| Condition | Formula | Meaning |
|---|---|---|
| Exteriority | G ∉ TCZ₀ | The goal lies outside the present comfort zone. |
| Separation | dist(G,TCZ₀)>ε | The goal is not merely at the boundary; it is meaningfully outside. |
| Self-setting | sSelf(TCZ₀)=TCZG | The Self can define a new future-side stability region. |
| Value | CSelf(G)>0 | The goal is aligned with higher self-concordance. |
| Symbol | Meaning |
|---|---|
| JG[u] | goal-conditioned cost functional (terminal term λd(x(T),G)²) |
| WG | value function (solution of the HJB equation) |
| λ > 0 | terminal weight |
| d(x,G) | distance to the goal |
| f(x,u,t) | dynamics of the cognitive state |
Assumptions: the standing assumptions A1–A4 and A7 (§2.12) are in force.
Let a true goal G be set as a future terminal condition. Define the goal-indexed control functional
The optimal control policy is
When the terminal term is non-degenerate, the present control policy depends on the future goal G. If the meaning of a past event p is written as I(p|G), then the effective direction of cognitive time is
Thus cognitive time flows from future goal to present Ego control and then to the meaning of the past.
This is the central theorem of the present paper. It does not say that physical time runs backward. It says that within cognition, the effective direction of control and meaning is future-origin.
Both directions of the equivalence — that optimality implies the HJB equation, and that a solution of the HJB equation yields the value function and the optimal feedback — are proved in a self-contained manner in Appendix A.5 via the dynamic programming principle and the verification lemma A.5.1.
| Symbol | Meaning |
|---|---|
| G | future true goal outside the present TCZ |
| JG[u] | total cost of a trajectory when future goal G is included |
| ∫ V₀ dτ | ordinary accumulated internal strain |
| λd(x(T),G)² | penalty for ending far from the future goal |
| u*(t;G) | present action selected under the future goal |
| I(p|G) | meaning of past event p under future goal G |
Theorem 8 turns the theory from a theory of cognition into a theory of human life. A human life is not merely a sequence of reactions to past causes. It is the process by which a future world becomes the origin of present action and past meaning.
This does not deny biology, memory, social environment, or history. It says that these are not sufficient to determine a life. The defining question is which future G becomes the terminal condition of the Ego’s control problem.
If a person believes that the past determines the present, then the only available strategy is adjustment within the current TCZ. If a person knows that the future goal determines the present control policy, then the strategy becomes TCZ transformation.
This is why “time flows from the future to the past” is not a metaphor of optimism. It is a statement about the direction of the control functional once a true goal enters the system.
We now distinguish two levels of time.
At the lowest level of abstraction, the universe appears as physical spacetime:
Physical time is the coordinate associated with the physical universe. Its arrow is ordinarily associated with thermodynamic entropy increase:
At higher levels of abstraction, the universe is not merely physical spacetime. It is a cognitive-information space structured by partial order and LUBs. For each LUB L, define a future LUB-TCZ GL and a semantic pseudo-distance dL. Then define cognitive time by
In the cognitive universe, time is not first a clock coordinate. It is the semantic distance from the future LUB-TCZ.
| Symbol | Meaning |
|---|---|
| πcL | Ego control policy on LUB level L |
| τL(x)=dL(x,GL) | cognitive time of level L (a pseudometric) |
| cL > 0 | level-wise dissipation rate (Assumption A8) |
| GL | the goal TCZ of level L |
| e−cLt | exponential decay factor |
Assumptions: the standing assumptions A1–A4 and A8 (§2.12) are in force.
Let the universe be modeled as a cognitive-information partially ordered space
Let the lowest abstraction projection be
For each LUB L, let GL be the corresponding future LUB-TCZ and define
If along the cognitive trajectory
then
Therefore physical time is the coordinate of the lowest-abstraction physical universe, while cognitive time is a future-origin semantic axis within each LUB-TCZ. The effective direction of time is opposite across the two levels: physical time is described from past to future in the physical projection, whereas cognitive time is organized from future LUB-TCZ to present control and past meaning.
The descent condition above is not an independent postulate: under the level-wise dissipativity (Assumption A8) it is derived from the optimal policy πcL. The rigorous derivation, including the treatment of the non-smoothness of the distance function, is given in Appendix A.6 (Lemma A.6.1).
The expression “the flow of time is reversed” must be understood precisely. In the physical universe at the lowest abstraction level, time is described by physical succession. We speak of causes in the past producing effects in the future. In the cognitive universe, the control origin is the future. The future goal selects present action, and present commitment reorganizes the meaning of the past.
| Layer | Time orientation | Entropy orientation | Meaning |
|---|---|---|---|
| Physical universe | past → present → future | thermodynamic entropy tends to increase | physical succession |
| Cognitive universe | future → present → past | semantic entropy decreases toward goal/LUB | meaning and control |
This is the precise sense in which the physical universe and the cognitive universe have opposite effective time directions. They are not two contradictory physics. They are two abstraction layers of the same information universe.
In self-development, coaching, and leadership, success requires that the future goal be treated not as a wish but as the origin of the control policy. Formally, the practitioner must operate as though the relevant control functional is JG, not J₀.
Thus success requires certainty that cognitive time flows from future to past.
Self-development, coaching, and leadership differ in surface application. Self-development applies the principle to one’s own life. Coaching helps another person set a future outside their present TCZ without imposing content. Leadership helps a group share a higher future. But the mathematical structure is the same in all three cases.
This sentence is not a philosophical assertion. In the formal system of this paper, it is a mathematical fact: once the future goal enters the control functional, present action is selected from the future side, and the past is reorganized by the meaning of that future.
This paper has constructed a minimal theorem chain for a theory of human life. The proof deliberately remains within the minimal structure needed to establish future-origin cognitive time. It required only the following steps.
Beyond its specific theorems, this paper stands as the foundational paper of Tomabechi Time Theory: the claim that, for human beings, time is not merely a physical coordinate running from past to future, but a cognitive axis whose origin lies in the future. The minimal chain established here — TCZ, Ego, LUB, the true goal, future-origin cognitive time, and the cognitive universe — is offered as the rigorous foundation on which that theory of time is built.
Therefore the central claim is established:
A human being does not truly live by being pushed by the past. A human being lives when a future world becomes the origin of present action. The past then becomes material organized by that future.
Self-development, coaching, and leadership are founded on this same structure. Their common principle is that success requires certainty that time, in cognition, flows from future to past. This is not an inspirational slogan. It is the mathematical consequence of TCZ, Ego control, true-goal exteriority, and LUB-indexed cognitive time.
This appendix surveys every theorem of this paper in one place. Each entry gives the statement in πc form, a complete reading of its symbols, the intuition behind it, and a concrete example. It is written for readers who want the full logical structure without the rigorous proofs that follow.
Let ż=F(z) be a closed-loop system, and let Φ(z) be continuously differentiable with compact sublevel sets { z | Φ(z) ≤ c } on the trajectories considered (properness; equivalently, Φ is radially unbounded). Define
Assume that for all z∉Ωθ,
Let y(t)=Φ(z(t))−θ. Then ẏ≤−αy. By the comparison principle,
Hence Φ(z(t))→θ+; since the sublevel sets are compact, every limit point of z(t) lies in Ωθ, so dist(z(t),Ωθ)→0. This lemma is the common proof skeleton for the theorems below.
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 V0 dτ, and is justified — not merely assumed — as follows. Let the value function be J*(x,t)=infu 𝔼∫tT V0(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{V0 + ∇J*·f(x,u,t)}, so along the optimal closed loop dJ*/dt = −V0 ≤ 0: the value function is non-increasing under πc. We assume in addition that the closed loop is dissipative and detectable with respect to V0 — 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 V0 (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.
| 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. ∎
| 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. If the social-cognitive coupling graph is strongly connected and the largest invariant set with ℒ̇=0 forces Sij→0 along the coupling edges, LaSalle’s invariance principle yields social alignment. Hence the joint trajectory converges to the shared stability region TCZshared.
Remark (mutuality of coupling cannot be dropped — a minimal one-sided counterexample). The strong-coupling condition of Assumption A5 requires positive coupling in both directions on every edge. Drop that mutuality and Theorem 2 fails. Minimal counterexample: one dimension, two agents, V1=(x1−a)2, V2=(x2−b)2 (a≠b), S12=S21=(x1−x2)2, γ12 > 0 but γ21=0 (agent 2 does not include the mismatch with agent 1 in its own cost). Then agent 2’s optimal control lowers only V2, so x2→b, while agent 1 minimizes V1+γ12S12 and settles at x1→m:=(a+γ12b)/(1+γ12). Hence S12→(m−b)2=(a−b)2/(1+γ12)2 > 0 and Sij→0 fails — no “shared” valley forms; the outcome is a pursuit equilibrium in which only agent 1 accommodates. The composite ℒ is still non-increasing, but its rest set is not contained in the surface S=0. LaSalle can force Sij→0 only when every edge of the strongly connected graph is positively coupled in both directions, so that stationarity on the maximal invariant set demands alignment of every pair. Asymmetric γ is admissible in general, but for symmetric S the effective coupling is (γij+γji)/2 — its positivity on every edge, i.e., mutuality, is the essential condition, and Assumption A5 makes it explicit.
| 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 invariant set, the nonnegative social inconsistency terms require Sij→0 and the nonnegative abstraction term requires A(x)→0. 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.
Suppose G∈TCZ₀. Then G lies inside the current stability region. It can be approached without changing the stability region itself. Therefore reaching G does not require transformation of the current TCZ. It is an improvement, task, plan, preference, or near-future objective, but not a transformative goal.
A true transformative goal must require a change of TCZ. Hence it must be outside the current TCZ: G∉TCZ₀. For robustness against boundary ambiguity, require dist(G,TCZ₀)>ε. The remaining conditions — Self-settability, self-concordant value, and capacity to reconstitute the control functional once it functions as a future terminal condition — ensure that the exterior state is not merely irrelevant fantasy but a possible future origin. Therefore the stated conditions define a true goal.
We now complete the above argument formally.
Definition A.4.1 (Transformative goal). A state (or future TCZ component) G is transformative if, under the optimal closed loop of Theorem 1 with the current evaluation function V₀ retained, the ε-neighborhood of G cannot be reached (that is, lim inft→∞ dist(x(t), G) ≥ ε), while after the replacement of the evaluation structure V₀ → VG by the Self operator sSelf, the corresponding optimal closed loop satisfies dist(x(t), G) → 0.
Step 1 (Necessity of exteriority). We argue by contraposition. Suppose G ∈ TCZ₀, or dist(G, TCZ₀) = 0. Under Assumptions A1–A4, Theorem 1 applies, and along the current closed loop dist(x(t), Ωθ) → 0. Since Ωθ = TCZ₀ is forward invariant (Appendix A.0, Nagumo’s condition), the trajectory remains in TCZ₀ once it arrives, and every neighborhood of G becomes reachable without any change of the evaluation structure (when G ∈ TCZ₀) or as a boundary limit (when dist(G,TCZ₀) = 0). This contradicts the first condition of Definition A.4.1. Hence, for G to be transformative, G ∉ TCZ₀ is necessary, and moreover the uniform separation dist(G, TCZ₀) ≥ ε is necessary.
Step 2 (Necessity of the remaining conditions). Without Self-settability sSelf(TCZ₀) = TCZG, the replaced evaluation structure VG is undefined and the second condition of Definition A.4.1 is vacuous. Without self-concordant positive value CSelf(G) > 0, VG cannot be constructed consistently as an evaluation function in the sense of A2 (a proper C¹ function whose minimum set is G). Without the capacity to reconstitute the control functional, G cannot act as the terminal condition of Theorem 8 and the replaced closed loop is undefined. Hence each of the three conditions is necessary.
Step 3 (Sufficiency). Conversely, suppose G satisfies exteriority, separation, Self-settability, self-concordant positive value, and reconstitutability. By self-concordance, take VG to be an evaluation function with minimum set G satisfying the regularity of A2, and set it as the terminal condition λd(x,G)² of Theorem 8 (Assumption A7). By the verification lemma A.5.1 of Appendix A.5, the value function descends under the replaced optimal feedback, and under the replaced version of the dissipativity A4, dist(x(t), G) → 0. On the other hand, by exteriority and separation, the pre-replacement closed loop satisfies lim inft→∞ dist(x(t), G) ≥ ε (Step 1). Hence G is transformative in the sense of Definition A.4.1.
Conclusion. Exteriority G ∉ TCZ₀ (including the separation dist(G,TCZ₀) ≥ ε), Self-settability, self-concordant positive value, and reconstitutability of the control functional together constitute a characterization (necessary and sufficient conditions) of a transformative goal. ∎
Let G be a true goal and define
The optimal control is u*(t;G)=arg min JG[u]. Since the terminal term explicitly contains G, and since the term is non-degenerate over the reachable terminal states, changing G changes the minimizer. Therefore the present control policy depends on the future goal.
Let I(p|G) denote the interpretation of a past event p under the future goal G. Since present commitment and control policy are indexed by G, the past is interpreted within the narrative and value structure induced by G. Hence the effective cognitive order is G→u*(t;G)→I(p|G). This is future-to-present-to-past cognitive time.
We now complete both directions of the equivalence and the G-dependence formally. The proof has three stages: (i) the dynamic programming principle yields the HJB equation (optimality ⇒ HJB); (ii) the verification lemma shows that a solution of the HJB equation yields the value function and the optimal feedback (HJB ⇒ optimality); (iii) the non-degeneracy of the terminal term (Assumption A7) yields the G-dependence of present control.
Stage (i): Dynamic programming principle ⇒ HJB. Define the value function WG(x,t) = infu(·) JG[u] with initial condition x(t) = x. For every h ∈ (0, T−t), additivity of the cost and concatenability of admissible controls (Assumption A1) give the dynamic programming principle
Since WG is C¹ by Assumption A7, expanding the right-hand side to first order in h and letting h → 0⁺ yields
The terminal condition WG(x,T) = λd(x,G)² holds by definition.
Stage (ii): Verification lemma.
Claim. Suppose W ∈ C¹(X×[t₀,T]) satisfies the HJB equation −∂/∂t W = minu∈U{V₀ + ∇W·f} together with the terminal condition W(x,T) = λd(x,G)², that πc(x,t) ∈ arg minu∈U{V₀ + ∇W·f} is a measurable selection (Assumption A3), and that the corresponding closed-loop solution exists (Assumption A1). Then W coincides with the value function (W = WG) and πc is optimal.
Proof. Take any admissible control u(·) with trajectory x(·). The HJB equation implies, at every point (x,t) and for every u ∈ U,
by the definition of the minimum. The first two terms on the left are the total derivative of the composite function t ↦ W(x(t),t); integrating over [t₀, T] gives
Substituting the terminal condition yields JG[u] = ∫t₀T V₀ dτ + λd(x(T),G)² ≥ W(x(t₀),t₀), so W is a lower bound on the cost of every admissible control. For u = πc, the minimum in the HJB equation is attained pointwise, the inequality above holds with equality, and JG[πc] = W(x(t₀),t₀). Hence W = WG and πc is optimal. ∎ (Lemma)
Stage (iii): G-dependence. By the non-degeneracy in Assumption A7, the map G ↦ d(·,G)²|R(T) is injective on the reachable terminal set R(T). Therefore, if G ≠ G′, the terminal penalty functions differ on R(T) and the family of value functions {WG} separates G (WG ≠ WG′). Hence the optimal feedback πc(x,t;G) = arg minu{V₀ + ∇WG·f} depends on G.
Non-anticipativity (causal consistency). πc(x,t;G) is a function only of the present state and time (x,t) (and of the already-set G); it does not depend on future state values. It is therefore a non-anticipative feedback law. The future acts by being aggregated, through the HJB equation solved backward in time, into the present quantity WG(x,t). This is the rigorous meaning of the claim that the future goal determines present control; its consistency with causality is examined independently in Question 3 at the end of this paper.
This establishes both directions of πc = arg minu(t) JG[u] ⇔ −∂/∂t WG = minu{V₀ + ∇WG·f}, the G-dependence of present control, and the cognitive order G → πc(·,·;G) → I(p|G). ∎
Let the universe be modeled as (𝒰,⪯). The lowest abstraction projection is physical spacetime π⊥(𝒰)=ℝ³×ℝphys. Physical time is therefore a coordinate of the lowest abstraction layer. At higher cognitive abstraction, for each LUB L, define τL(x)=dL(x,GL). If dτL/dt≤−cLτL, the comparison principle gives τL(t)≤τL(0)e−cLt, hence τL→0 and x(t)→GL.
Thus physical time is the succession coordinate of the bottom projection, while cognitive time is the future-origin semantic convergence coordinate in each LUB-TCZ. The directions are opposite in effective orientation: physical description runs past-to-future, cognitive control runs future-to-present-to-past.
We now complete the derivation of the descent condition and the treatment of non-smoothness formally.
Regularity remark. The distance function dL(·, GL) is in general not continuously differentiable, but by Assumption A8 it is 1-Lipschitz. Hence t ↦ τL(x(t)) is locally Lipschitz along the closed-loop solutions of Assumption A1, therefore absolutely continuous and differentiable almost everywhere. The descent below is stated in terms of the upper Dini derivative D⁺τL(t) = lim suph→0⁺ [τL(x(t+h)) − τL(x(t))]/h.
Claim. If y : [0,∞) → [0,∞) is absolutely continuous and satisfies D⁺y(t) ≤ −c y(t) almost everywhere (c > 0), then y(t) ≤ y(0)e−ct.
Proof. Set v(t) = y(t)ect. Then v is absolutely continuous and, almost everywhere, dv/dt = (dy/dt + c y)ect ≤ 0. Since an absolutely continuous function is the Lebesgue integral of its derivative, v(t) ≤ v(0), that is, y(t) ≤ y(0)e−ct. ∎ (Lemma)
Derivation of the descent condition. For the level-L optimal policy πcL = arg minu(t) ∫ τL dt, apply exactly the argument of the supplement to Appendix A.0 with the integrand V₀ replaced by τL. That is, the level-wise value function JL*(x,t) = infu ∫tT τL(x(τ))dτ satisfies the HJB equation, and along the optimal closed loop dJL*/dt = −τL ≤ 0. The level-wise dissipativity of Assumption A8 strengthens this non-increase to the rate-descent D⁺τL ≤ −cLτL on the set {τL > 0}. As with A4, this is a structural assumption that does not follow from optimality alone; its dynamical meaning is that at each abstraction level, Ego control effectively contracts the semantic distance.
Convergence. Applying Lemma A.6.1 to y(t) = τL(x(t)) gives τL(x(t)) ≤ τL(x(0))e−cLt → 0. By the distance property of dL, x(t) → GL in the dL topology. The argument holds independently for each LUB level L, and consistency across levels follows from the fact that the projection π⊥ preserves the subsumption order: the bottom-layer physical description (the direction of physical time) is not altered by convergence at higher levels. ∎
The following table summarizes the theorem chain used in this minimal paper. The numbering is preserved from the full Cognitive Space Time Theory, but only Theorems 1, 2, 3, 7, 8, and 10 are required for the present derivation.
| Theorem | Name | Core Formula | Meaning |
|---|---|---|---|
| Theorem 1 | Tomabechi Main Theorem / Individual TCZ Convergence | πc=arg minu(t)∫0TVdt ⇒ x*(t)→TCZ(x0) | The foundational theorem: Self/Ego control converges to the individual stability region. |
| Theorem 2 | Tomabechi Shared-TCZ Convergence | πi=arg minui(t)∫0T(Vi+ΣγijSij)dt ⇒ x*(t)→TCZshared | Coupled subjects converge to a shared stability region. |
| Theorem 3 | Tomabechi LUB Abstraction | πi=arg minui(t)𝔼∫0T(Vi+ΣγijSij+ηiA)dt, A=0⇔φ=LUB ⇒ x*(t)→LUB | Shared stability is lifted into higher abstraction. |
| Theorem 7 | Tomabechi True Goal | πG=arg minu(t)∫tTVGdτ, G∉TCZ0, G=Self-set | A true goal is outside the present TCZ. |
| Theorem 8 | Tomabechi Future-Origin Cognitive Time | πc=arg minu(t)JG ⇔ −∂/∂t WG=minu{V0+∇WG·f} | The future goal reorganizes present Ego control and past meaning. |
| Theorem 10 | Tomabechi Cognitive Universe | πcL=arg minu(t)∫τLdt ⇒ τL(t)≤τL(0)e−cLt | Physical time is the bottom-layer coordinate; cognitive time is a LUB-indexed future-origin axis. |
This section organizes, from the standpoint of control theory and applied mathematics, the questions of mathematical rigor that may arise concerning this minimal six-theorem system — Theorem 1 (individual TCZ convergence), Theorem 2 (shared-TCZ convergence), Theorem 3 (LUB convergence), Theorem 7 (true goal), Theorem 8 (future-origin cognitive time), and Theorem 10 (cognitive universe). The three questions below are those, among the rigor questions examined likewise in the parent Cognitive Space Time Theory, that bear on these six theorems. For each, we state the original formulation, the precise concern, and the rigorous resolution.
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 Appendix A.0); the probability space of Question 2 is defined as Assumption A6; and the causal consistency of Question 3 is proved as the verification lemma A.5.1 of Appendix A.5 (a non-anticipative feedback law).
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.
Theorem 3 introduces an expectation operator 𝔼 in the control objective (𝔼∫₀ᵀ[Vᵢ + ΣγᵢⱼSᵢⱼ + ηᵢA(xᵢ)]dt), while Theorems 1, 2, 7, 8, and 10 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, 2, 7, 8, and 10 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).
In Theorem 8, the future goal G defines the value function WG as a terminal condition, and the present optimal control policy πc=arg min JG appears to be determined from the future. This looks like "the future determines the present" and seems to violate physical causality.
This is not a violation of causality but the standard structure of optimal control theory. The HJB equation −∂/∂t WG = minu{V0+∇WG·f} (terminal condition WG(x,T)=λd(x,G)²) is solved backward in time. The value function WG(x,t) is a present quantity defined for the state x at the current time t; it is the index of present optimality that incorporates the future terminal condition. Physically, the agent does not know the actual future state but chooses the present control on the basis of the goal G that the Self has set now — that is, present information. Hence "future-origin" is a direction of time in cognitive semantics (the direction in which a goal confers meaning on the present and the past), not a reversal of physical causality. Mathematically, πc is a feedback law depending only on the current state x(t), satisfying measurability and non-anticipativity. Therefore Theorem 8 is consistent with physical causality, and the future-origin character of cognitive time is rigorously defined as a semantic ordering.
| Theorem | Question found | Status after resolution |
|---|---|---|
| T1 | HJB–Lyapunov gap | ✓ Resolved: dissipativity assumption made explicit |
| T2 | No substantive issue | ✓ Sound under the strong-coupling condition |
| T3 | Probability space of 𝔼 undefined | ✓ Resolved: epistemic uncertainty over lattice position |
| T7 | No substantive issue | ✓ Sound under goal-externality, Self-setting, and self-consistency |
| T8 | Future terminal condition and causality | ✓ Resolved: backward HJB solution, non-anticipative feedback law |
| T10 | No substantive issue | ✓ Sound under per-LUB-layer Lyapunov descent |
The logical structure of this minimal six-theorem system is sound. There is no circular dependence among the theorems: T1 → T2 → T3 (adding coupling terms in turn); Theorem 7 establishes that a true goal lies outside the current TCZ; Theorem 8 establishes that a future goal, as a terminal condition, reconstitutes present control and the meaning of the past; and Theorem 10 establishes that each abstraction layer carries its own future-origin time axis. Each theorem uses the output of the preceding theorems as input or structural assumption and has no dependence in the reverse direction.
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. (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. Cognitive Research Laboratories Technical Report, May 2026.