Triadic Cognitive Dynamics
A Unified Theory from Logical Trichotomy
to Agent-Level Bayesian Oscillation
This paper proposes a unified cognitive theory that spans from the logical foundations to agent-level behavior. Taking the triadic value structure of “yes/no/maybe” as its axiomatic starting point, the framework sequentially establishes set-theoretic boundary operations, triadic decision states (Action/Think/Stop), Bayesian oscillation processes, and spatiotemporal irreversibility constraints, forming a complete inferential chain from formal logic to physical reality. The Landauer principle is employed to establish a precise physical bridge between information theory and thermodynamics. We demonstrate that the triadic structure exhibits fractal recursion and fundamental asymmetry—action is a process of factual contraction (finite), stopping is the open preservation of possibility space (potentially eternal), and thinking is the chaotic intermediate state of probabilistic weight contestation—and provide an impossibility proof of triadic minimal completeness. This framework intersects with Łukasiewicz’s three-valued logic, Yao’s three-way decisions, Friston’s active inference, and Jøsang’s subjective logic, while maintaining independent unifying power. In the domain of AI, the theory explains the cognitive inevitability of the Tool/COT/Output architecture and defines intelligence as: tracing the optimal path through an infinite openness using a finite number of contractions.
CHAPTER 01
The Limits of Dichotomy and the Necessity of Trichotomy
1.1 The Implicit Assumptions of Classical Bivalent Logic
In classical propositional logic, the truth-value function of any proposition φ maps strictly to a binary set. This axiomatic system harbors a widely overlooked premise: that the cognitive agent possesses sufficient information, tools, and capacity to render a definite judgment on φ.
v: Φ → {0, 1}
// Law of Excluded Middle
v(φ) ∨ v(¬φ) = 1, ∀ φ ∈ Φ
When information, tools, or capacity are insufficient, forcing a binary choice between {0, 1} entails systematic information loss. The absolutist demands of verification (P=1) and falsification (P=0) are unsatisfiable in the vast majority of real-world cognitive scenarios.
1.2 The Epistemological Legitimacy of the Third Value
Łukasiewicz introduced three-valued logic in 1920, granting “the undetermined” a legitimate logical status. “Maybe” is not a failure of logic but its honesty—it acknowledges the finitude of the cognitive agent.
v₃: Φ → {0, ½, 1}
// The Law of Excluded Middle no longer holds in three-valued logic
v₃(φ) ∨ v₃(¬φ) can equal ½ (when v₃(φ) = ½)
// The trichotomous boundary region
BND = {φ | v₃(φ) = ½} ← cognitive fault-tolerance space
// Probability theory unfolds the discrete third value into a continuous spectrum
P: Φ → [0, 1] ← precise calibration of “maybe”
1.3 Core Propositions of This Paper
Proposition 1: The triadic structure T = {1, ½, 0} (yes/maybe/no) constitutes the minimal complete structure of cognition.
Proposition 2: The triadic states S = {Action, Think, Stop} necessarily emerge in all cognitive systems operating under incomplete information and finite resources (impossibility proof provided in Chapter 4).
Proposition 3: Intelligence can be defined as selecting the optimal path through possibility space with a finite number of actions: π* = argmax U(π), |π| ≤ n.
CHAPTER 02
Logical Foundations: Syllogism as Boundary Intersection
2.1 A Set-Theoretic Reinterpretation of the Syllogism
The essence of the classical syllogism is not a nested inference from major to minor, but rather the intersection confirmation of two independent set boundaries. The major premise defines one set boundary, the minor premise defines another, and the conclusion confirms the region of overlap.
Major premise: M ⊆ P (All M are P)
Minor premise: S ⊆ M (All S are M)
Conclusion: S ⊆ P (All S are P)
// General conclusion region
Conclusion = {x ∈ U | x ∈ A ∩ B}
The intersection interpretation above is equivalent to subset transitivity in universal affirmative syllogisms (Barbara). For syllogistic forms involving negative or particular premises, complement sets and non-empty intersection operations are required. This paper uses the Barbara form to establish intuition; full generalization is deferred to subsequent work.
2.2 “Yes/No/Maybe” as Three Boundary Operations
“Yes” (inclusion): x ∈ A definite inclusion
“No” (exclusion): x ∈ Aᶜ (i.e., x ∉ A) definite exclusion
“Maybe” (suspension): x ∈ BND(A) boundary pending
// Rough set three-region partition (disjoint exhaustive cover)
POS(A) = {x | x is certainly in A} positive region → “yes”
NEG(A) = {x | x is certainly not in A} negative region → “no”
BND(A) = U − POS(A) − NEG(A) boundary region → “maybe”
POS(A) ∪ BND(A) ∪ NEG(A) = U
POS(A) ∩ BND(A) = ∅, POS(A) ∩ NEG(A) = ∅, BND(A) ∩ NEG(A) = ∅
2.3 The Essence of Logical Reasoning—Orderly Resolution of “Maybe”
BND(A)_t₀ ⊇ BND(A)_t₁ ⊇ BND(A)_t₂ ⊇ …
|BND(A)_t| → 0 as t → ∞ (ideal limit; unreachable in practice)
|BND(A)| > 0 always holds ← complete certainty is an unreachable ideal
CHAPTER 03
The Ontological Inversion of Probability Theory
3.1 Formalization of the Traditional Hierarchy
In the classical framework, probability theory is treated as a stopgap approximation for incomplete information—it is invoked only when the truth-value function v(φ) cannot be determined.
v: Φ → {0, 1} ← foundational layer
P(φ) ∈ [0, 1] ← used only when v(φ) is indeterminate (supplementary layer)
3.2 Hierarchy Inversion: Probability Space as Foundation
This paper proposes the opposite: the probability measure P: Φ → [0, 1] is the fundamental representation of cognitive states, and bivalent logic is its degenerate special case.
v(φ) = 1 ⟺ P(φ) = 1 (upper-limit convergence)
v(φ) = 0 ⟺ P(φ) = 0 (lower-limit convergence)
// Information loss from forced binarization (measured by KL divergence)
D_KL(P_real ‖ P_binary) = P_real · log(P_real / P_binary)
+ (1−P_real) · log((1−P_real) / (1−P_binary))
// When P_real=0.7 is forced to P_binary=1
D_KL = 0.7·log(0.7/1) + 0.3·log(0.3/0) → +∞
// When P_binary=1, the support excludes φ being false,
// violating the absolute continuity condition; KL divergence is +∞ by convention.
// Compressing a 0.3 probability of “false” to 0 = infinite information distortion.
Forced binarization is catastrophic in information-theoretic terms
3.3 Quantum States as the Physical Reality of “Maybe”
A qubit is not a fuzzy version of a classical bit but rather the physical realization of “maybe.” Before measurement, a particle genuinely occupies a superposition of “neither true nor false”—this is not a deficiency of information but the structure of reality itself.
|ψ⟩ = c₀|0⟩ + c₁|1⟩, |c₀|² + |c₁|² = 1
// Born rule: measurement probabilities
P(outcome=1) = |c₁|²
P(outcome=0) = |c₀|²
// Classical bivalent logic = eigenstate special case of quantum states
|0⟩ → v(φ)=0 (pure “no”)
|1⟩ → v(φ)=1 (pure “yes”)
General superposition |ψ⟩ → “maybe” (probability space is physical reality)
What this paper asserts is structural isomorphism rather than physical identity. Quantum superposition states have been rigorously verified through Bell inequality experiments and double-slit interference experiments as physical reality. Cognitive uncertainty is a subjective state at the epistemological level. What the two share is mathematical structure—superposition and collapse dynamics on probability spaces—not physical substrate. This paper does not rely on the Penrose-Hameroff quantum consciousness hypothesis, nor does it claim that the brain is a quantum computer. The value of the isomorphism lies in this: the same mathematical toolkit can describe the dynamics of both domains simultaneously.
3.4 The Degenerate Prior Problem
If P(H) ∈ {0, 1}, then for any evidence E, P(H|E) = P(H). Bayesian updating is rendered entirely ineffective.
P(H|E) = P(E|H) · P(H) / P(E)
// When the prior P(H) = 1
P(H|E) = P(E|H) · 1 / P(E) = 1 (regardless of evidence)
// When the prior P(H) = 0
P(H|E) = P(E|H) · 0 / P(E) = 0 (regardless of evidence)
The fatal flaw of dichotomy at the probabilistic level: locked updating
3.5 Alignment with Jøsang’s Subjective Logic
ω = (b, d, u), b + d + u = 1, b,d,u ∈ [0,1]
b = belief d = disbelief u = uncertainty (ignorance)
// Projected probability
P(x) = b + u · a (a = prior base rate)
u = 0 → degenerates to classical probability
u = 1 → complete ignorance
0 < u < 1 → precise expression of cognitive uncertainty
CHAPTER 04
The Triadic Decision States: Action, Think, Stop
4.1 Proof of Triadic Minimal Completeness
Proposition 2 claims that the triadic states necessarily emerge in all cognitive systems operating under incomplete information and finite resources. The following provides an impossibility proof by constructing three classes of degenerate systems and demonstrating their inevitable failure.
In environments with incomplete information (∃x: β < P(A|x) < α) and finite resources (F_total < ∞), any cognitive system with fewer than three states possesses an irreparable failure mode.
When P(A|x) ∈ (β, α), the system is forced to choose Action or Stop
ΔR_A = R(Action|x) − R(Think|x) > 0 (action loss exceeds deliberation loss)
ΔR_S = R(Stop|x) − R(Think|x) > 0 (stop loss exceeds deliberation loss)
⟹ Expected loss is strictly greater than that of a triadic system
// Degenerate System B: Think + Stop only, no Action
Cannot execute actions ⟹ cannot alter environmental state E
⟹ Cannot acquire new observations E_{t+1}
⟹ Λ_internal = 1 (stale information, no new evidence)
⟹ p_{t+1} = p_t always holds; Bayesian updating permanently stalls
// Degenerate System C: Action + Think only, no Stop
No safe exit mechanism; actions continuously consume free energy
Σ ΔF_action → F_total
⟹ System crashes upon free energy depletion (no graceful degradation)
The triad is the minimal complete structure □
4.2 Formal Definitions and the Dual-Threshold Model
S = {Action, Think, Stop}
// Dual-threshold decision function (0 ≤ β < α ≤ 1)
σ(p) = Action, if p ≥ α
σ(p) = Stop, if p ≤ β
σ(p) = Think, if β < p < α
4.3 The Fundamental Asymmetry of the Triadic States
Action: p → 1 (certainty jump · fact generation · discrete state)
Stop: p → 0 (certainty jump · fact negation · discrete state)
Think: p ∈ (β, α) (probabilistic weight contestation · continuous state)
Decision = phase transition: probabilistic continuum → deterministic discrete state
4.4 Information-Theoretic Proof of Action as Factual Contraction
Information entropy H(Ω) has a lower bound (H ≥ 0); each action strictly reduces H (ΔH > 0); therefore the number of actions is necessarily finite.
ΔH = log₂(N) − log₂(M) > 0 (strictly positive)
// Complete collapse is the special case M=1
ΔH_max = log₂(N) (a single action determines the outcome)
// Cumulative contraction
H(Ω_t) = H(Ω₀) − Σᵢ₌₁ᵗ ΔHᵢ
// Finiteness proof
ΔHᵢ > 0 always holds and H(Ω_t) ≥ 0
⟹ t ≤ H(Ω₀) / min(ΔHᵢ)
⟹ The number of actions is finite □
4.5 Proof of the Eternality of Stopping
The information entropy change of the Stop state is ΔH_stop = 0; it does not deplete the stock of possibility space and is therefore unconstrained by any upper bound at the information-theoretic level.
ΔH_stop = 0
H(Ω_t) = H(Ω_t₀), ∀ t > t₀ (during stopping)
// Comparison
Action: H(Ω) monotonically decreases and H ≥ 0 → finite number of instances
Stop: H(Ω) unchanged → potentially eternal at the information-theoretic level ∞
It is important to distinguish the information-theoretic level from the physical level: information-theoretically, ΔH_stop = 0 and stopping does not consume possibility space, making it potentially eternal. However, biological organisms in the Stop state still incur basal metabolic costs (ΔF_maintenance > 0); maintaining the Stop state itself requires free energy. A biological organism’s “stopping” actually involves maintenance-level micro-actions (breathing, heartbeat), constituting the lowest-frequency action at a recursive sublayer. Only inanimate matter (such as a rock) achieves genuinely eternal stopping at the physical level.
Action is a process of factual contraction—each step eliminates other possibilities. Stopping is the open preservation of possibility space—it consumes no possibilities. Action cannot be eternal; stopping can be. This is the deepest asymmetry among the triadic states.
4.6 Expected Loss Functions and Threshold Derivation
R(Action|x) = λ_AP · P(Aᶜ|x) + λ_AA · P(A|x)
R(Stop|x) = λ_SP · P(A|x) + λ_SA · P(Aᶜ|x)
R(Think|x) = λ_TP · P(A|x) + λ_TA · P(Aᶜ|x)
// Optimal decision rule
σ*(x) = argmin R(d|x), d ∈ {Action, Think, Stop}
// Thresholds derived from loss functions
α = (λ_TA − λ_SA) / [(λ_TA − λ_SA) + (λ_AP − λ_TP)]
β = (λ_TP − λ_SP) / [(λ_TP − λ_SP) + (λ_SA − λ_TA)]
4.7 The Three-Dimensional Influence of Emotion on Triadic Decisions
p(Action) = Σᵢ wᵢ · eᵢ / Σᵢ wᵢ
Signature of emotional decision-making: ∃j: wⱼ >> Σᵢ≠ⱼ wᵢ
// Dimension 2 — Threshold modification
α_emotion = α₀ + Δα(emotion)
Fear: Δα < 0 (lowers action threshold; triggers flight)
Depression: Δα → 1 (action threshold extremely high; action inhibited)
Mania: Δα < 0, Δβ > 0 (thinking space extremely narrow; impulsive action)
// Dimension 3 — Search strategy modification
Fear → attentional narrowing (searches only threat-related memory nodes)
Curiosity → attentional broadening (activates distant memory associations)
Formalization: emotion modifies the decay radius r of sim(s, feature(v))
r_fear < r_neutral < r_curiosity
4.8 Action Contraction into Unknown Space
Action collapses to ω* ∈ Ω:
If ω* ∈ K: contraction within known space (routine decision)
If ω* ∈ Ω\K: entry into unknown space (exploration/discovery)
// The contraction property holds in both cases
ΔH > 0 always holds
// But the latter expands known space
K_new = K ∪ {ω*}
The main body of this paper adopts a strong presupposition stance (Ω is fixed), while acknowledging that a weak presupposition stance is more reasonable in open systems—Ω may expand through action (as when genetic mutations in evolution create new phenotypic spaces, or technological inventions create new action spaces). Under the weak presupposition, Theorem 2 still holds: even if Ω expands, the property ΔH > 0 for each action step remains unchanged, and the free energy budget F_total is still finite.
4.9 Systematic Risks of Dichotomous Decision-Making
A system possessing only Action and Stop without Think faces a pathological binary choice: impulsivity (acting with insufficient information) or paralysis (refusing to act without certainty). The hallmark of catastrophic decisions is that problems properly belonging in the Think state are forced into Action or Stop. The existence of “Think” in the triadic structure serves precisely as a cognitive safety net against forced binarization.
CHAPTER 05
Bayesian Oscillation: The Dynamic Process of Triadic States
5.1 Bayesian Updating as Triadic Iteration
p_{t+1} = p_t · P(E_{t+1}|H) / P(E_{t+1})
// Likelihood ratio form
p_{t+1}/(1−p_{t+1}) = Λ_{t+1} · p_t/(1−p_t)
// Random walk in log-odds space
log[p_{t+1}/(1−p_{t+1})] = log[Λ_{t+1}] + log[p_t/(1−p_t)]
The per-step increment log[Λ] can be positive or negative → random walk in log-odds space
5.2 Triadic Markov Process
A T S
P = A | P(A→A) P(A→T) P(A→S) |
T | P(T→A) P(T→T) P(T→S) |
S | P(S→A) P(S→T) P(S→S) |
// Key constraints
P(A→A) < 1 Continuous action cannot be eternal
P(S→S) can = 1 Stopping can be eternal (may be interrupted by external activation)
P(T→T) < 1 Pure thinking cannot be eternal (signal-to-noise ratio decays)
5.3 Waveform Analysis and Diagnostics
Amplitude: A_t = max{p_s, s≤t} − min{p_s, s≤t}
Volatility: σ_t = √[Var(Δp_s), s≤t]
Convergence rate: γ = −d(σ_t)/dt
// Healthy Bayesian oscillation
σ_t → σ_∞ > 0 (volatility converges to a positive constant, never reaching zero)
A_t monotonically decreasing (amplitude progressively narrows)
γ > 0 (converging rather than diverging)
// Pathological diagnostics
σ_t → ∞: Cognitive chaos (judgment criteria collapse)
σ_t → 0: Cognitive rigidity (degenerate prior · Bayesian updating halts)
γ < 0: Information processing failure (oscillation diverges)
5.4 Formalization of Uncertainty Sequences
P(X) = P(X₁) · Πᵢ₌₂ⁿ P(Xᵢ | Xᵢ₋₁, pᵢ₋₁)
// Information content of the sequence
I(X) = −log₂ P(X)
// Optimal sequence
X* = argmax_X [U(X_final) − C(X)]
U(X_final) = terminal utility C(X) = total sequence cost (time + energy)
5.5 Probabilistic Proof of Temporal Irreversibility
For macroscopic systems, the probability of time reversal decays exponentially with particle count, approaching zero. Temporal irreversibility is not legislated by physical law but extinguished by probability.
S = k · ln(W) W = number of corresponding microstates
// Low-entropy states have far fewer microstates than high-entropy states
P(spontaneous return to low-entropy state) ∝ W_low / W_high = exp(−ΔS/k)
// For macroscopic systems, ΔS is enormous
P(reversal) ≈ exp(−10²³) → 0
// Formal statement of the Heraclitean proposition
f(x₀, t₁) ≠ f(x₀, t₂) when t₁ ≠ t₂ (almost surely)
Space is revisitable; time is irreversible
CHAPTER 06
The Nature of Thinking: Coupling of Interaction and Memory
6.1 The Fractal Recursive Structure of Thinking
T = (A_sub, T_sub, S_sub)* (* denotes indefinite repetition)
// Recursion depth constraint
SNR_k = Signal_k / Noise_k
When SNR_k < θ_min → the k-th layer of recursion automatically terminates
6.2 Feature-Triggered Model of Memory
If the external input signal s = ∅, then the activation value a(v) = 0 for all memory nodes v ∈ V. Without external feature input, memory cannot be directionally accessed.
// External input s as activation vector
a(v) = Σ_{e∈E(v)} W(e) · sim(s, feature(v))
Node v is activated ⟺ a(v) > θ_activation
// Key corollary: no input implies no activation
s = ∅ ⟹ a(v) = 0, ∀v ∈ V
6.3 Degeneration of Isolated Thinking
Λ_internal = P(E_old|H) / P(E_old|¬H) = constant
// When Λ_internal = 1 (stale information, no new evidence)
p_{t+1}/(1−p_{t+1}) = 1 · p_t/(1−p_t)
⟹ p_{t+1} = p_t (posterior = prior; updating stalls)
// External interaction breaks the degeneration
Λ_external ≠ 1 → new likelihood ratio injected → updating resumes
6.4 Information-Theoretic Description of the Cognitive Closed Loop
I(E; M) > 0 informational coupling between environment and memory
I(M; A) > 0 informational coupling between memory and action
I(A; E’) > 0 informational coupling between action and new environment
Complete loop condition:
I(E; M) · I(M; A) · I(A; E’) > 0
Any term = 0 → loop broken → intelligence stalls
6.5 Cognitive Interference Effects
a(v | s₁, s₂) > max(a(v | s₁), a(v | s₂)) super-linear enhancement
// Strong constructive interference (requires nonlinear activation function)
a(v | s₁, s₂) > a(v | s₁) + a(v | s₂) super-additivity
Condition: a(·) exhibits convexity or cross-activation terms
Neural correlate: cooperative effects of long-term potentiation (LTP)
// Destructive interference (conflicting signals)
a(v | s₁, s₂) < min(a(v | s₁), a(v | s₂)) cancellation
CHAPTER 07
The Cognitive Inevitability of AI Architecture
7.1 Triadic Mapping of Tool/COT/Output
Tool call ↔ Action σ(p)=Action, p≥α
COT reasoning ↔ Think σ(p)=Think, β<p<α
Output ↔ Stop completion judgment; collapse to output
7.2 Cognitive-Dynamic Interpretation of the ReAct Architecture
X_old = (T, A, S) → H(X_old) = 0
// ReAct: free grammar, positive sequence entropy
X_react = (T, A, T, A, T, S, T, A, S, …) → H(X_react) > 0
Sequence freedom ↑ → Bayesian oscillation adequacy ↑ → search space coverage ↑
7.3 Formal Description of COT as Internal Interaction
P(tₖ | t₁, …, t_{k-1}) = softmax(f(t₁, …, t_{k-1}))
// Each new token generation = micro-action (contraction)
// The generated token sequence = feature activation of internal “memory retrieval”
// Informational coupling between preceding and subsequent tokens
I(t_k; t_{k+n}) > 0 ← structurally isomorphic to memory retrieval in human thought
CHAPTER 08
Multi-Agent Coupling
8.1 “Your Action Is Another’s External Input”
E_{t+1} = f(E_t, Action_i)
// Other agent j receives a new signal
p_{j,t+1} = Update(p_{j,t}, Observe(E_{t+1}))
8.2 The Heraclitean Generalization
E_t = g(E₀, Action₁(t₁), Action₂(t₂), …, Actionₙ(tₙ))
∃j≠i such that Actionⱼ occurs within (t₁,t₂)
⟹ E(observed by i, t₁) ≠ E(observed by i, t₂)
8.3 Nash Equilibrium as Multi-Body Oscillation Steady State
∀i: σ*ᵢ(p) ∈ BR_i(σ*₋ᵢ)
// BR_i = best-response function of agent i (allows mixed strategies)
Pure strategy equilibrium: all oscillations completely subside (special case)
Mixed strategy equilibrium: oscillations continue but expected payoffs remain unchanged (general case)
8.4 Information Asymmetry and Partial Observability
Each cognitive agent observes only a subset Oᵢ ⊆ E of the environment. Different observation subsets lead to divergent Bayesian update paths, even when confronting the same objective environment.
p_{i,t+1} = Update(p_{i,t}, Oᵢ(E_{t+1}))
p_{j,t+1} = Update(p_{j,t}, Oⱼ(E_{t+1}))
// When Oᵢ ≠ Oⱼ
p_{i,t+1} ≠ p_{j,t+1} (same environment, different update paths)
8.5 Emergent Phenomena in Multi-Body Oscillation
κ → 1 (excessive coupling): group polarization
All individuals follow predecessors’ actions, abandoning private information (information cascade)
κ ∈ (κ_low, κ_high) (moderate coupling): collective wisdom
Independent judgment + weak interaction → collective judgment surpasses the individual
κ → 0 (insufficient coupling): isolated oscillation
Each cognitive agent operates independently; no emergent effects
CHAPTER 09
Thermodynamic Foundations and the Life Cycle
9.1 The Law of Entropy Increase and Constraints on Action
dS_total / dt ≥ 0
// Thermodynamic free energy (Helmholtz, F_thermo)
F_thermo = E − TS (energy available for work; unit: joules)
// Living systems: local entropy decrease + global entropy increase
dS_local < 0, but dS_env > |dS_local|
dS_total = dS_local + dS_env > 0
// Number of actions constrained by free energy budget
N_max ≤ F_thermo_total / |ΔF_per_action|
9.2 A Bayesian Oscillation Model of the Life Cycle
L(t) = (action_rate(t), think_depth(t), stop_duration(t))
// Suggested functional forms
action_rate(t) ∝ t · exp(−t/T_peak) rises then falls (Gamma type)
think_depth(t) ∝ log(1+t) · exp(−t/T₂) logarithmic growth followed by exponential decay
stop_duration(t) ∝ exp(t/T₃) exponential growth
Growth phase t ∈ [0, T₁]:
action_rate high and increasing, think_depth shallow, stop_duration short
→ High-frequency, low-depth oscillation (Piaget’s sensorimotor stage · high exploration rate ε)
Maturity phase t ∈ [T₁, T₂]:
action_rate stable, think_depth at peak, stop_duration moderate
→ Medium-frequency, high-depth oscillation (Cattell’s fluid intelligence peak · Klein’s recognition-primed decision)
Aging phase t ∈ [T₂, T₃]:
action_rate declining, think_depth declining, stop_duration increasing
→ Low-frequency, low-depth oscillation (Salthouse’s processing speed decline · working memory capacity reduction)
t = T_death:
P(S→T) = 0 All activation channels permanently closed
P(S→A) = 0
σ(p) = Stop, ∀t > T_death Eternal Stop state
9.3 Heat Death as Cosmic-Scale Eternal Stopping
T → T_uniform, ΔT → 0 temperature gradient vanishes
F_thermo → 0 free energy exhausted
S_thermo → S_max thermodynamic entropy maximized (maximum disorder)
// After heat death
H_useful → 0 Usable information approaches zero (no temperature gradient to extract work)
N_action = 0 No action possible (no free energy to drive it)
I(E; M) = 0 No information exchange possible
9.4 Intelligence Metric and Ultimate Formulation
∃ t ∈ [t_birth, t_death]: σ(p_t) ∈ {Action, Think}
// Metric of intelligence
Intelligence = E[U(π*)] / C(π*)
i.e., the ratio of expected utility to total cost of the optimal path
All intelligence, all life, all existence is a finite string of Bayesian oscillations between two eternities of stopping. The fuel of oscillation is free energy; its direction is locked by the irreversibility of time; the quality of oscillation is determined by the noise-reduction efficiency of thinking. Intelligence is tracing the optimal path through an infinite openness using a finite number of contractions.
Σ ΔF_action ≤ F_thermo_total free energy budget
H(Ω_t) ≥ 0 possibility space non-negative
dS_total / dt ≥ 0 entropy increase constraint
P(time reversal) → 0 irreversibility
CHAPTER 10
Alignment with Prior Theories
10.1 Overview of Alignments
| Prior Theory | Shared Structure | Key Differences |
|---|---|---|
| Łukasiewicz Three-Valued Logic (1920) | v₃: Φ → {0, ½, 1} | Łukasiewicz uses it for static truth tables; the present framework dynamizes ½ as Bayesian oscillation |
| Yao’s Three-Way Decisions (2010–) | (α,β) dual thresholds; POS/BND/NEG | Yao works top-down (rough sets → applications); the present framework works bottom-up, adding fractal recursion, spatiotemporal constraints, AI mapping, and impossibility proofs |
| Jøsang’s Subjective Logic (1997–) | (b, d, u) triples | Jøsang targets trust-network engineering; the present framework targets cognitive ontology |
| Friston’s Active Inference (2006–) | Bayesian unification of perception-action-inference | Friston starts from variational inference; the present framework starts from logical trivalence |
| Di Gioia’s TCA (2026) | AI Tool/Deliberation/Stop triad | TCA targets engineering optimization; the present framework targets cognitive ontology |
10.2 Friston’s Variational Free Energy—Distinguished from Thermodynamic Free Energy
F_thermo = E − TS
→ Energy available for work (joules); drives physical action
// Variational free energy (Friston, 2006)
F_var = E_q[log q(s) − log p(s, o)]
= D_KL(q(s) ‖ p(s|o)) − ln p(o)
→ Divergence between internal model and true posterior (nats/bits); drives perceptual updating
// Structural isomorphism
Both drive the system toward a more “accurate” state through minimization
Action: minimizes F_var by altering observations o
Perception: minimizes F_var by updating internal model q(s)
// Essential difference
F_thermo is a physical quantity (joules)
F_var is an informational quantity (nats/bits)
10.3 Core Optimization Objective of Di Gioia’s TCA
max V(τ, b) = E[U(a*(b_τ))] − ∫₀ᵗ c(t, ρ_t) dt
τ = stopping time, b = belief state, c = cognitive friction cost
CHAPTER 11
Conclusions and Future Directions
11.1 A Unified Inferential Chain across Seven Mathematical Languages
Ch.2 Ch.3 Ch.3 Ch.4 Ch.5 Ch.6 Ch.9
11.2 Cross-Layer Mapping Theorems
An exact mapping exists between the three regions of rough sets and the three-way Bayesian decision: POS(A) → {x | P(A|x) ≥ α}, NEG(A) → {x | P(A|x) ≤ β}, BND(A) → {x | β < P(A|x) < α}.
A precise relation holds between Bayesian updating and changes in information entropy: the per-step entropy reduction equals the mutual information between the action and the new evidence: H(Ω_t) − H(Ω_{t+1}) = I(X_t; E_{t+1}).
Landauer’s principle (1961): erasing 1 bit of information dissipates at least kT·ln 2 of heat. The information entropy reduction ΔH of each action corresponds to a minimum free energy expenditure ΔF_min = kT·ln 2·ΔH. Thus, the finiteness of action is simultaneously constrained from both the information-theoretic and thermodynamic sides.
ΔF_min = kT · ln2 · ΔH
// Dual constraints on the finiteness of action
Information-theoretic constraint: t ≤ H(Ω₀) / min(ΔHᵢ)
Thermodynamic constraint: N_max ≤ F_thermo_total / (kT · ln2 · min(ΔHᵢ))
Both constraints hold independently; the number of actions is the lesser of the two
11.3 Summary of Core Theorems
Theorem 1 (Degenerate Prior Lock-in): P(H)∈{0,1} ⟹ P(H|E)=P(H)
Theorem 2 (Finiteness of Action): ΔH>0 and H≥0 ⟹ the number of actions is finite
Theorem 3 (Potential Eternality of Stopping): ΔH_stop=0 ⟹ no upper-bound constraint at the information-theoretic level
Theorem 4 (Feature Dependency): s=∅ ⟹ a(v)=0 ∀v
Theorem 5 (Temporal Irreversibility): P(reversal)∝exp(−ΔS/k)→0
11.4 Testable Predictions
Prediction 1: The triadic degrees of freedom (sequence entropy H(X)) of AI systems are positively correlated with task performance. Prediction 2: Cognitive degradation can be quantified as the anomaly of volatility σ_t trending toward 0 or ∞. Prediction 3: Decision quality has an inverted-U relationship with the optimal threshold gap (α−β).
11.5 Open Problems
Existence and convergence conditions for Nash equilibria in multi-agent coupling; neuroscientific experimental verification of cognitive interference effects (cf. Busemeyer and Bruza’s quantum cognition models); online adaptive algorithms for triadic thresholds (α,β); deep integration pathways with the quantum cognition research program.
References
Łukasiewicz, J. (1920). O logice trójwartościowej [On three-valued logic].
Kleene, S.C. (1952). Introduction to Metamathematics. North-Holland.
Popper, K. (1934/1959). The Logic of Scientific Discovery. Routledge.
Shannon, C.E. (1948). A Mathematical Theory of Communication. Bell System Technical Journal.
Landauer, R. (1961). Irreversibility and Heat Generation in the Computing Process. IBM Journal of R&D, 5(3).
Boltzmann, L. (1877). Über die Beziehung zwischen dem zweiten Hauptsatze der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung.
Jøsang, A. (1997). Artificial Reasoning with Subjective Logic. Proceedings of AWCR.
Friston, K. (2006). A free energy principle for the brain. Journal of Physiology-Paris, 100(1-3).
Friston, K. (2010). The free-energy principle: a unified brain theory? Nature Reviews Neuroscience, 11.
Yao, Y.Y. (2010). Three-way decisions with probabilistic rough sets. Information Sciences, 180.
Yao, Y.Y. (2012). An outline of a theory of three-way decisions. RSCTC 2012, LNCS 7413. Springer.
Yao, Y.Y. (2020). Tri-level thinking: models of three-way decision. Int. J. Mach. Learn. Cybern., 11.
Caves, C.M., Fuchs, C.A. & Schack, R. (2002). Quantum probabilities as Bayesian probabilities. Physical Review A, 65.
Busemeyer, J.R. & Bruza, P.D. (2012). Quantum Models of Cognition and Decision. Cambridge University Press.
Piaget, J. (1954). The Construction of Reality in the Child. Basic Books.
Salthouse, T.A. (1996). The processing-speed theory of adult age differences in cognition. Psychological Review, 103(3).
Klein, G. (1998). Sources of Power: How People Make Decisions. MIT Press.
Parr, T., Pezzulo, G. & Friston, K.J. (2022). Active Inference. MIT Press.
Di Gioia, D. (2026). Cognitive Friction: A Decision-Theoretic Framework for Bounded Deliberation in Tool-Using Agents. WWW 2026.
Original Thought Paper · V2 · July 2026
Generated through Human-AI Collaborative Dialogue