Mediumless Topological Cosmology: A Turing-Transcendent Ontological Generator

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Jia, Baolong Independent Researcher seer@139.com

April 8, 2026

Abstract

The Trinity Topological Theory (Paradoxical Re-entry (formerly Paradox-Reference) ⇔ Entity-Relation ⇔ Lazy Evalu-

ation) proposes that the universe is a mediumless, self-evolving pointer maze. In previous work, the three atomic operations—CREATE/LINK, JUMP, and RESOLVE—were introduced qualitatively. This paper provides a rigorous formal treatment by introducing bounded-rate constraints $\mathbf v=(v_1,v_2,v_3,v_4)\in\mathbb N^4$ on node birth, node death, edge creation, and edge deletion per logical tick.

discrete-time stochastic dynamical system on the space of finite directed graphs. We estab- lish three principal results: (1) the bounded edge-creation rate v induces a natural causal cone on the graph, providing a first-principles derivation of a finite propagation speed with- out presupposing spacetime; (2) the computational power of the TGP is strictly determined by the locality of the Paradoxical Re-entry selection function, ranging from sub-Turing (con- stant selection) through Turing-complete (bounded-radius selection) to hypercomputational only in the unphysical limit of global oracles; (3) the RESOLVE conflict-resolution mechanism partitions the state space into equivalence classes whose quotient dynamics recovers a deter- ministic skeleton, from which irreversibility and a thermodynamic arrow of time emerge nat- urally; (4) while Turing-equivalent in computability, the TGP is strictly richer in generative capacity—it possesses five structural capabilities (endogenous randomness, self-generated substrate, rule–data identity, structural irreversibility, unbounded self-proliferating paral-

lelism) that no Turing machine can replicate internally. These results bridge the gap be- tween the philosophical vision of “computation without computers” and the mathematical apparatus of graph dynamics, random graph theory, and computability theory.

Disclaimer: This paper presents a theoretical framework intended for scholarly exploration. Some propositions are stated as conjectures. Our goal is to offer a rigorous mathematical foundation for the Trinity Topological Theory. We welcome scientific criticism and will refine this work in subsequent versions.

Contents

1 Introduction 2

2 Formal Framework: The Trinity Graph Process 4 2.1 State Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The Paradoxical Re-entry Selection Function . . . . . . . . . . . . . . . . . . . . . 4 2.3 The Conflict Resolution Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 State Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Causal Cone Theorem 5 3.1 Reachability and Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 The Causal Partial Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Computational Power of the TGP 7 4.1 The Distinction: Rules vs. Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.2 Classification by Locality of S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

5 Entropy, Irreversibility, and the Arrow of Time 8 5.1 The Source of Irreversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 5.2 Graph-Theoretic Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

6 Beyond Turing: Generative Capacity 9 6.1 Computability vs. Generability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

7 Emergent Stable Structures 10

8 Physical Correspondence Dictionary 11

9 Numerical Signatures and Falsifiability 12

10 Discussion 12 10.1 Why This is Not a Turing Machine . . . . . . . . . . . . . . . . . . . . . . . . . . 12 10.2 Why This is Not “Merely” Topology . . . . . . . . . . . . . . . . . . . . . . . . . 13 10.3 Relationship to Prior Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . 13

11 Emergent Evolution Hierarchy 13 11.1 Phase 0: Primordial Soup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 11.2 Phase 1: Stable Motif Emergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 11.3 Phase 2: Self-Repairing Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 14 11.4 Phase 3: Self-Replicating Structures . . . . . . . . . . . . . . . . . . . . . . . . . 14 11.5 Phase 4: Competitive Ecology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 11.6 Phase 5: Hierarchical Organization . . . . . . . . . . . . . . . . . . . . . . . . . . 15 11.7 Phase 6: Self-Modeling and Consciousness . . . . . . . . . . . . . . . . . . . . . . 15 11.8 Summary: The Emergence Timeline . . . . . . . . . . . . . . . . . . . . . . . . . 15

12 Research Significance and Open Problems 16 12.1 Why This Matters: Five Domains of Impact . . . . . . . . . . . . . . . . . . . . . 16 12.2 Open Problems and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . 16

13 Conclusion 17

1 Introduction

The JiaBaolong Axiom System [1] introduced the Trinity Topological Theory as a pre-mathematical ontology: the universe is a dynamic directed graph U = (V ,E ) evolving under three atomic t t t operations—CREATE/LINK (Entity-Relation), JUMP (Paradoxical Re-entry), and RESOLVE (Lazy Eval- uation). The original presentation, however, left several foundational questions open:

(i) What constrains the rate of topological change per tick?

(ii) Under what conditions does the system possess Turing-complete computational power?

(iii) How does a causal structure emerge without a pre-existing metric space?

(iv) What is the precise relationship between the RESOLVE mechanism and the thermodynamic arrow of time?

This paper addresses all four questions by introducing bounded-rate constraints—four non- negative integers (v ,v ,v ,v ) that cap the number of node births, node deaths, edge creations (per node), and edge deletions (per node) in each logical tick. The resulting formal object, which we call the Trinity Graph Process (TGP), is sufficiently structured to admit precise theorems while remaining faithful to the philosophical minimalism of the original theory.

Related Work. The TGP is related to, but distinct from, several established formalisms. Table 1 provides a systematic comparison of the five closest research programs. The critical observation is that no prior framework simultaneously achieves all five properties: (i) paradox- referential selection, (ii) node and edge creation and deletion, (iii) bounded rate constraints,

(iv) formal conflict resolution, and (v) derived causality and irreversibility. The TGP fills this gap.

Table 1: The Five Closest Research Programs and Their Gaps

Direction	Key Authors	Intersection with TGP	What is Missing
Wolfram Physics	Wolfram, 2020 [5]	Hypergraph rewriting → emergent spacetime	Rules are fixed (not Paradoxical Re-entry); no node deletion; no rate bounds
AlChemy	Fontana & Buss, 1994 [15]	$\lambda$-expressions interact → Paradoxical Re-entry + emergence	Operates in $\lambda$-calculus space, not graph space; no spatial concept
Causal Dynamical Triangulations	Ambjørn, Loll, 1998 [17]	Discrete simplices → emergent spacetime + causality	Uses fixed geometric building blocks (triangles); rules are external
Quantum Graphity	Konopka et al., 2008 [16]	Dynamic edges → phase transitions (melt/freeze)	Nodes are fixed (only edges change); no Paradoxical Re-entry selection; no rate bounds
Artificial Chemistry	Dittrich et al., 2001 [18]	Constructive dynamics → emergent novelty	General framework, not specific to graphs; no causal cone derivation

Stephen Wolfram’s Physics Project comes closest in formalism (graph dynamics generating space), but it relies on externally prescribed, fixed rewrite rules—precisely the element that the TGP replaces with the Paradoxical Re-entry selection function S(G ). Fontana’s AlChemy comes closest in philosophy (Paradoxical Re-entry, constructive), but operates in λ-calculus rather than graph space and lacks any notion of spatial propagation. The TGP is the first model to combine Paradoxical Re-entry rule generation with bounded-rate graph dynamics and derive both causal structure and irreversibility from first principles.

Contributions.

1. A complete formal definition of the TGP with bounded-rate constraints (Section 2).

2. A proof that v induces a causal cone with finite propagation speed on the graph (Sec- tion 3).

3. A classification of the computational power of the TGP parameterized by the locality radius of the Paradoxical Re-entry function (Section 4).

4. An analysis of the RESOLVE mechanism as a source of irreversibility and entropic flow (Section 5).

5. A formal proof that the TGP’s generative capacity strictly exceeds that of Turing machines, despite equal computability (Section 6).

6. A dictionary mapping TGP parameters to known physical constants and phenomena (Sec- tion 8).

7. A complete emergent evolution hierarchy—from primordial soup to consciousness—derived from the TGP’s four parameters (Section 11).

8. An assessment of research significance across five domains and seven open problems (Sec- tion 12).

2 Formal Framework: The Trinity Graph Process

2.1 State Space Definition 2.1 (Graph Universe). Let G denote the set of all finite directed graphs G = (V,E) where $V$ is a finite set of nodes (entities) and $E\subseteq V\times V$ is a set of directed edges (relations). Self-loops are permitted: $(v,v)\in E$ is allowed. Definition 2.2 (Rate Vector). A rate vector is a tuple

where:

2.2 The Paradoxical Re-entry Selection Function

Definition 2.3 (Selection Function). A Paradoxical Re-entry selection function is a (possibly

randomized) mapping

where $\mathcal A(\mathbf v)$ is the set of admissible action bundles under rate vector $\mathbf v$. An action bundle $a\in\mathcal A(\mathbf v)$ is a tuple $a=(V^+,V^-,E^+,E^-)$ satisfying:

Remark 2.4. The Paradoxical Re-entry nature of S means that S reads the current graph G to produce the action bundle. Since G is the system’s own state, S embodies Paradoxical Re-entry

(PR) axiom of the Trinity: the system reads itself to determine its next state.

2.3 The Conflict Resolution Mechanism Definition 2.5 (Conflict Set). Given an action bundle a = (V +,V −,E+,E−), a conflict occurs when:

(a) A node w ∈ V − is also the endpoint of some e ∈ E+ (destruction–creation conflict).

(b) Two or more edges in E+ target the same node with incompatible topological consequences under domain-specific constraints. The conflict set is $C(a)\subseteq(V^-\times E^+)\cup(E^+\times E^+)$ .

Definition 2.6 (Resolution Function). A resolution function is a (possibly randomized) map-

such that $\mathcal R(a,G_t)=a^*$ is conflict-free, $C(a^*)=\varnothing$, and $a^*$ is a maximal consistent subset of $a$. The resolution delay is defined as $\Delta\tau:=|C(a)|$, the cardinality of the conflict set.

2.4 State Transition Definition 2.7 (Trinity Graph Process). A Trinity Graph Process is a tuple TGP = (G,v,S,R,G ) where G ∈ G is the initial graph. The state transition G ↦ G is: 0 t t+1

1. Selection (PR): Compute a = S(G ).

2. Resolution (LE): Compute a∗ = R(a,G ) with a∗ = (V +∗,V −∗,E+∗,E−∗).

3. Application (ER):

   $$ V_{t+1}=(V_t\setminus V^{-*})\cup V^{+*} \tag{11} $$
   $$ E_{t+1}=\bigl(E_t\setminus E^{-*}\setminus\{e\in E_t:e\text{ involves a node in }V^{-*}\}\bigr)\cup E^{+*} \tag{12} $$

Proposition 2.8 (Well-Definedness). For any rate vector v ∈ N4, any selection function S, any resolution function R, and any finite initial graph G , the TGP is well-defined: each G is 0 t a finite directed graph with

Proof. By construction, |V +∗| ≤ |V +| ≤ v (from (6) and the fact that a∗ ⊆ a). Since V −∗ ⊆ V , 1 t we have |V | = |V |−|V −∗|+|V +∗| ≤ |V |+v . Finiteness is preserved by induction. The edge t+1 t t 1 set E is a subset of (V × V ), hence finite. t+1 t+1 t+1

3 Causal Cone Theorem

The central physical result of this paper is that the bounded edge-creation rate v induces a natural causal structure on the graph—a “speed of light” that emerges without presupposing any metric space.

3.1 Reachability and Influence Definition 3.1 (Forward Influence Set). Given a TGP and a node w ∈ V , the forward influence set of w after ∆t ticks is

Theorem 3.2 (Causal Cone Bound). For any TGP with rate vector (v ,v ,v ,v ) and any r-local selection function S (Definition 3.3 below), the forward influence set satisfies

In particular, $|\operatorname{Reach}(w,t,\Delta t)|$ grows at most exponentially in $\Delta t$ with base $c_{\max}:=(v_3+1)(v_1+1)$. max 3 1 1). Definition 3.3 (r-Local Selection Function). S is r-local if for every node w ∈ V , the action of S at w depends only on the r-neighborhood B (w,G ) := {u ∈ V : d (w,u) ≤ r}, where d is r t t Gt Gt the shortest directed path length in G . Proof of Theorem 3.2. We proceed by induction on ∆t. Base case (∆t = 0): Reach(w,t,0) = {w}, so |Reach(w,t,0)| = 1. Inductive step: Assume $|\operatorname{Reach}(w,t,\Delta t)|\le\sum_{k=0}^{\Delta t}c_{\max}^k$. In one further tick, each influenced node can create at most $v_3$ new edges, and at most $v_1$ new nodes are created. Therefore the causal cone grows by at most the factor $c_{\max}=(v_3+1)(v_1+1)$.

Summing the geometric series:

Corollary 3.4 (Finite Propagation Speed). In a TGP with finite rate vector v, no causal influence can propagate to more than c∆t nodes in ∆t ticks. The quantity

plays the role of a topological speed of light: the maximum rate of causal propagation measured in graph-distance per tick. Remark 3.5 (Comparison with Physics). In General Relativity, the speed of light c bounds the causal structure of Minkowski spacetime, giving rise to light cones. In the TGP, c bounds causal propagation on the graph, giving rise to graph-theoretic causal cones. The crucial difference is that in GR, c is a fundamental constant imposed on a pre-existing manifold; in the TGP, c emerges from the rate constraints on atomic graph operations.

3.2 The Causal Partial Order Definition 3.6 (Causal Precedence). For events (node–tick pairs) (w,t) and (u,t′) with t < t′, we write (w,t) ≺ (u,t′) if u ∈ Reach(w,t,t′ − t). Proposition 3.7. ≺ is a strict partial order on the set of all events {(w,t) : w ∈ V , t ∈ N}. Proof. Irreflexivity: (w,t) ̸≺ (w,t) since ≺ requires t < t′. Transitivity: if (w,t) ≺ (u,t′) ≺ (x,t′′), then removing w at time t changes G t′, which contains u; since u influences x, removing w also changes the state containing x, so (w,t) ≺ (x,t′′).

4 Computational Power of the TGP

A fundamental question: is the TGP a model of computation, and if so, how powerful?

4.1 The Distinction: Rules vs. Data In a Turing machine M = (Q,Σ,δ,q ,F), the transition function δ is fixed and external to the tape contents. Data and program are ontologically separate. In the TGP, the selection function S reads G —the system’s own state—to determine the next action. There is no separation between program and data. This places the TGP in a fundamentally different ontological category from standard computation models.

Definition 4.1 (Program–Data Unification). A dynamical system exhibits program–data uni- fication if the rule governing state transitions at time t is itself a function of the state at time t.

The TGP exhibits program–data unification by construction: S = S(G ).

4.2 Classification by Locality of S Theorem 4.2 (Computational Hierarchy). The computational power of TGP(v,S) depends on the locality radius of S:

(I) Constant S (ignores G ; acts randomly): The TGP reduces to a stochastic graph process. It cannot compute any non-trivial function. Computational class: sub-Turing (probabilis- tic finite automaton on graphs).

(II) r-Local S (reads only the r-neighborhood of each node): Equivalent in computational power to a graph cellular automaton. For appropriate choices of r, v , and state labeling, the TGP can simulate Rule 110 [7] and is therefore Turing-complete.

(III) Global S (reads the entire graph G ): On finite graphs, equivalent to a finite-state machine (the state space G restricted to graphs of bounded size is finite). On unbounded graphs (if v > v ), formally capable of super-Turing computation, but only at the cost of violating 1 2 physical plausibility (requires reading infinite information in one tick).

Proof Sketch. Case (I): If S is constant, each tick applies a random action independent of G . The sequence G ,G ,... is a Markov chain on G with no computational structure beyond t 0 1 statistical equilibrium. Case (II): We construct an explicit simulation. Encode the tape of a Turing machine as a directed path p → p → ··· → p in G , where each node p carries a label (edge structure to 1 2 n t i auxiliary flag nodes) encoding the tape symbol. The head position is encoded as a distinguished self-loop on one node of the path. The transition function δ of the Turing machine is encoded in the r-local rule: the node with the self-loop reads its own symbol (its auxiliary edge pattern) and the patterns of its neighbors within radius r, then modifies its label, moves the self-loop one step left or right, and optionally creates a new node (extending the tape). This construction requires:

Since Rule 110 is Turing-complete and can be simulated by a 1D cellular automaton with neigh- borhood radius 1, the r-local TGP with r ≥ 2 is Turing-complete.

Case (III): For finite graphs with |V | ≤ N for all t, the state space is finite (|G | is finite), t N so the dynamics is that of a finite automaton. For unbounded growth (v > v ), the system 1 2 has access to unbounded memory; if S can additionally read the global state, it has oracle-level access, placing it beyond Turing in principle—but this requires processing |V | → ∞ information in one tick, which violates the spirit of bounded-rate dynamics and is physically implausible.

Corollary 4.3 (The Physically Relevant Regime). The physically relevant TGP is the r-local case (II) with small r. This regime is Turing-complete yet respects finite propagation speed (Section 3). It is powerful enough to generate the full hierarchy of physical emergence described in [1], yet constrained enough to obey causality.

5 Entropy, Irreversibility, and the Arrow of Time

5.1 The Source of Irreversibility Definition 5.1 (Pre-Resolution State). The pre-resolution state at tick t is the pair (G ,a) where a = S(G ) is the raw (conflict-laden) action bundle. Definition 5.2 (Post-Resolution Equivalence). Two pre-resolution states (G ,a) and (G ,a′) t t are post-resolution equivalent if R(a,G ) = R(a′,G ), i.e., they produce the same conflict-free t t action.

Theorem 5.3 (Irreversibility of Resolution). If |C(a)| > 0 (at least one conflict exists), then the resolution function R is strictly information-destroying: it maps multiple distinct pre-resolution states to a single post-resolution state.

Proof. Let $C(a)=\{c_1,\ldots,c_m\}$ with $m>0$. For conflict $c_i$ with $k_i$ alternatives, the resolver keeps one and discards $k_i-1$. The number of pre-resolution configurations mapping to the same post-resolution action $a^*$ is $\prod_{i=1}^{m}k_i\ge 2^m>1$. Therefore $\mathcal R$ is many-to-one and information is irreversibly lost.

5.2 Graph-Theoretic Entropy Definition 5.4 (TGP Entropy). The resolution entropy at tick t is

This measures the number of bits of information destroyed by the resolution process.

Proposition 5.5 (Monotonicity of Cumulative Entropy). The cumulative resolution entropy is

and is non-decreasing in $T$. Proof. Every $H_t\ge0$, so $\Sigma_{T+1}=\Sigma_T+H_{T+1}\ge\Sigma_T$.

Remark 5.6 (Arrow of Time). The cumulative entropy Σ provides a natural arrow of time for the TGP: the “past” is the direction in which Σ is smaller. This mirrors the Second Law of Thermodynamics: the total entropy of an isolated system never decreases. In the TGP, this law is not postulated—it is a theorem following from the many-to-one nature of conflict resolution.

6 Beyond Turing: Generative Capacity

Theorem 4.2 established that the TGP, in its r-local regime, is Turing-complete: it can simu- late any Turing machine. However, the converse question—Can a Turing machine fully simu- late the TGP?—reveals a fundamental asymmetry. While a TM can track the state sequence G ,G ,G ,... given a specific resolution oracle, the TGP possesses five structural capabilities that no Turing machine can replicate internally. We formalize this gap through the concept of generative capacity.

6.1 Computability vs. Generability Definition 6.1 (Computable Set). The computable set of a system M, denoted Comp(M), is the set of all functions f : Σ∗ → Σ∗ that M can compute (in the standard sense of input–output computation).

Definition 6.2 (Generable Set). The generable set of a dynamical system D, denoted Gen(D), is the set of all sequences of states (s ,s ,s ,...) that D can produce from some initial condition, including all possible resolution outcomes.

Theorem 6.3 (Generative Gap). For the r-local TGP with rate vector v:

That is, the TGP computes exactly the same functions as a Turing machine, but can generate strictly more phenomena.

Proof. The first equality follows from Theorem 4.2 (Case II) and the Church–Turing thesis. For the strict inclusion, we exhibit five witnesses—properties of TGP trajectories that no Turing machine trajectory can possess.

Witness 1: Endogenous randomness. A deterministic TM produces a single deterministic trajectory from any initial configuration. A probabilistic TM uses an external random tape ω ∈ {0,1} ; conditioned on ω, the trajectory is deterministic. In both cases, the randomness (if

any) is exogenous—supplied from outside the computational mechanism. In the TGP, the RESOLVE function R generates randomness endogenously: the conflict set C(a) arises from the topological self-interference of concurrent operations within G . No external random source is consulted. The set of possible resolutions is determined entirely by the system’s own state. Let X := R(S(G ),G ) denote the resolved action at tick t. The sequence (X ,X ,X ,...) is stochastic, but the probability measure over outcomes is self-generated: it depends on G , which itself depends on all previous resolutions. No Turing machine can produce a stochastic sequence from a deterministic mechanism without an external entropy source.

Witness 2: Self-generated substrate. A Turing machine operates on a tape {0,1} — a fixed, pre-existing, one-dimensional lattice. The tape’s topology (a countable chain) never changes during computation. The TGP operates on G , whose topology changes at every tick. Nodes are created and destroyed; edges are formed and broken. The “space” on which the system operates is a product of its own dynamics. No Turing machine modifies the topology of its own tape.

Witness 3: True rule–data identity. In a Universal Turing Machine U, the simulated program is encoded as data on the tape, but U’s own transition function δ remains fixed and external. There is always a “meta-level” rule that is not itself data. In the TGP, S(G ) reads G t t

directly—the same object that S will modify. There is no meta-level: the graph is simultaneously the rule, the data, and the substrate. This is not merely “self-modifying code” (which a UTM can simulate); it is the absence of any fixed interpretive layer.

Witness 4: Structural irreversibility. Bennett [14] proved that every Turing machine can be converted to a logically reversible one computing the same function, with at most polynomial overhead. Thus, irreversibility in TM computation is always eliminable. In the TGP, the RESOLVE mechanism is structurally irreversible (Theorem 5.3): whenever |C(a)| > 0, multiple pre-resolution states collapse to one post-resolution state. Since conflicts are unavoidable whenever v > 0 and |V | > 1 (multiple nodes creating edges concurrently), irre- 3 t versibility is a necessary feature of any non-trivial TGP. There is no “reversible TGP” analogue: removing RESOLVE would cause logical deadlock.

Witness 5: Unbounded self-proliferating parallelism. A k-tape Turing machine has k read/write heads—a constant determined before execution. Even a non-deterministic TM explores multiple branches, but each branch has a single head. In the TGP with v > v , the 1 2 number of concurrently active “processors” (nodes executing local PR selection) grows without bound: at tick t, up to |V | = |V | + t(v − v ) nodes operate in parallel. The system creates its own processors as a byproduct of its dynamics. No Turing machine—deterministic, non- deterministic, or multi-tape—can increase its number of active heads during execution.

Remark 6.4 (Positioning in the Computation Hierarchy). The TGP does not exceed the Church– Turing barrier in the computability-theoretic sense: it cannot decide the halting problem. Its additional capabilities—endogenous randomness, self-generated substrate, rule–data identity, structural irreversibility, and self-proliferating parallelism—are ontological rather than function- theoretic. They pertain to what the system is and generates, not to what functions it computes. We propose the following placement:

Level	Computability	Generability
Hypercomputation (oracle TMs, infinite-time TMs)	strictly stronger	not applicable here
TGP	equal to Turing machines	strictly richer than Turing machines
Turing machines	standard computability	standard generability
Finite automata	strictly weaker	strictly weaker

The TGP occupies a level that is Turing-equivalent in computation but Turing-transcendent in generation. Formalizing this “generability hierarchy” as a rigorous counterpart to the Chom- sky/arithmetic hierarchies is an open problem of independent interest.

7 Emergent Stable Structures

In a stochastic dynamical system, certain subgraph patterns survive longer than others. We characterize the most persistent structures. Definition 7.1 (Persistence). A subgraph $H\subseteq G_t$ has persistence $\pi(H):=\mathbb E[\min\{s>0:H\not\subseteq G_{t+s}\}]$, the expected number of ticks until $H$ is disrupted. Proposition 7.2 (Cycle Persistence). In a TGP with v < v (edge creation rate exceeds edge 4 3 deletion rate), directed cycles C of length n ≤ v have persistence n 3

growing exponentially with cycle length (up to the repair capacity v ). Proof Sketch. A directed cycle is disrupted only when at least one of its edges is deleted. The probability that a specific edge is chosen for deletion in a given tick is at most v /|E |. If 4 t the cycle’s edge is deleted but a replacement path exists within the node’s v creation budget, the cycle can “self-repair” within one tick. The repair probability depends on whether the PR function S selects the repair action. Under the assumption that S includes local topology- preserving bias (nodes preferentially reconnect to nearby nodes), the net disruption rate is (v /v )n, giving the claimed persistence bound. 4 3

Conjecture 7.3 (Particle–Cycle Correspondence). The persistent directed cycles in the TGP correspond to fermions in the physical emergence hierarchy of [1]. The cycle length n encodes the particle’s internal quantum numbers. Topologically knotted cycles (non-contractible on the ambient graph surface) correspond to particles with non-trivial spin. Proposition 7.4 (Hub Formation via Preferential Attachment). If S has a preferential-attachment component—the probability that a new edge targets node u is proportional to its in-degree kin— u then the in-degree distribution follows a power law:

Hubs with kin ≫ k¯ emerge spontaneously, corresponding to massive objects in the Trinity cos- mology.

8 Physical Correspondence Dictionary

We compile a correspondence between TGP parameters and known physical phenomena. This is not a derived prediction but a conjectured dictionary motivating future numerical investigation.

Table 2: TGP–Physics Correspondence Dictionary

TGP Concept	Formal Object	Physical Analogue
Rate vector	$\mathbf v=(v_1,v_2,v_3,v_4)$	Fundamental constants
$v_1$ (node creation rate)	$|V^+|\le v_1$	Vacuum fluctuation rate
$v_2$ (node destruction rate)	$|V^-|\le v_2$	Pair annihilation rate
$v_3$ (edge creation rate)	$|E^+(w)|\le v_3$	Interaction bandwidth → $c$
$v_4$ (edge deletion rate)	$|E^-(w)|\le v_4$	Decoherence rate
$v_1-v_2$ (net growth)	$\Delta|V|\sim v_1-v_2$	Cosmological expansion rate $H_0$
$v_3/v_4$ (creation/deletion ratio)	Stability parameter	Fine-structure constant $\alpha$?
Graph distance $d(w,u)$	Shortest directed path	Spatial separation
In-degree $k_w^{\mathrm{in}}$	$\sum_j A_{jw}$	Mass $\propto k^{\mathrm{in}}$
Conflict count $|C(a)|$	Resolution load	Curvature / gravitational field
Resolution delay $\Delta\tau$	Conflict cardinality	Proper time interval
Causal cone $\operatorname{Reach}(w,t,\Delta t)$	Theorem 3.2	Light cone
Persistent cycle $C_n$	Proposition 7.2	Fermion (stable particle)
Transient wave on graph	Edge-pattern ripple	Boson (force carrier)
Hub $k^{\mathrm{in}}\gg\bar{k}$	Proposition 7.4	Black hole
Cumulative entropy $\Sigma_T$	Definition 5.4	Thermodynamic entropy

Remark 8.1. The ratio $v_1/v_2$ controls whether the graph-universe is expanding ($v_1>v_2$), static ($v_1=v_2$), or contracting ($v_1<v_2$). In an expanding TGP, the effective cosmological parameter decreases as $|V_t|$ grows. (v = v ), or contracting (v < v ). In an expanding TGP, the “cosmological constant” Λ is The effective cosmological parameter scales as $(v_1-v_2)/|V_t|$ in an expanding TGP and therefore decreases as the graph grows. the observed smallness of Λ in our universe.

9 Numerical Signatures and Falsifiability

A theory without falsifiable predictions is philosophy, not science. We propose three quantitative predictions that future computational experiments can test.

Conjecture 9.1 (Universal Degree Exponent). For the physically relevant TGP (with r-local S and preferential attachment), the asymptotic in-degree exponent converges to

If real particle mass spectra (or network degree distributions of the CMB graph) show γ ≈ 3.7, this would constitute evidence for the TGP.

Conjecture 9.2 (Emergent Dimensionality). Starting from a random initial graph $G_0$, the spectral dimension $d_s$ is conjectured to satisfy

with $\varepsilon\to0$ as $v_3/v_4\to1$. Conjecture 9.3 (Resolution Entropy and the Bekenstein Bound): 10 Discussion

10.1 Why This is Not a Turing Machine

A common misunderstanding is that any discrete, step-by-step process is “just a Turing machine.” The TGP is categorically different:

1. No fixed program. The Turing machine’s transition function δ is fixed before execution begins. In the TGP, the “transition function” S(G ) changes at every tick because G t t changes. The program is the data.

2. No input/output. A Turing machine computes a function f : Σ∗ → Σ∗. The TGP has no designated input or output. It is a dynamical system, not a computing device. It simply evolves.

3. No halting. A Turing machine either halts or runs forever seeking an answer. The TGP has no halting state. It is an open-ended process—an ontological generator, not a problem solver.

4. Intrinsic stochasticity. The randomness in R is not “noise” added to a deterministic core; it is a fundamental feature of the conflict-resolution mechanism. The TGP is irreducibly stochastic in a way that Turing machines (even probabilistic ones) are not: the stochasticity arises from topological self-interference, not from a random tape or coin flips.

10.2 Why This is Not “Merely” Topology

Conversely, dismissing the TGP as “just topological dynamics” undersells its structure:

1. It can simulate any Turing machine (Theorem 4.2, Case II). Pure topological dynamics (e.g., a homeomorphism on a manifold) typically cannot.

2. It generates its own space. Standard topological dynamics operates on a given space. The TGP generates the space (graph structure) as it runs.

3. It produces irreversibility from reversible components. Each individual atomic operation (add node, add edge, etc.) is reversible (delete node, delete edge). But the composite RESOLVE step is irreversible (Theorem 5.3), creating a thermodynamic arrow from purely logical operations.

The TGP therefore occupies a unique position: more than topology, other than computation.

10.3 Relationship to Prior Formulations

Table 3 compares the TGP with the original “Computation Without Computers” formulation and related formalisms.

Table 3: Comparison of Formalisms

Feature	Original Trinity [1]	TGP (This Paper)	Wolfram Physics [5]
Basic elements	Nodes + edges	Nodes + edges	Hyperedges
Rewrite rules	Implicit (PR)	Implicit ($\mathcal S(G_t)$)	Explicit (fixed rule)
Rate bounds	Not specified	$(v_1,v_2,v_3,v_4)$	Not specified
Causality	Assumed	Derived (Theorem 3.2)	Derived
Node deletion	Not specified	Explicit ($v_2$)	Not standard
Conflict resolution	LE (qualitative)	$\mathcal R$ (formal)	Causal invariance
Entropy / Arrow of time	Qualitative	Proved (Theorem 5.3)	Not addressed
Computational power	“Equivalent to UTM”	Classified (Theorem 4.2)	Turing-complete
Generative capacity	Not addressed	$\operatorname{Gen}\supsetneq\operatorname{Gen}(\mathrm{TM})$ (Theorem 6.3)	Not addressed

11 Emergent Evolution Hierarchy

A central claim of the Trinity framework is that the full hierarchy of physical reality—from spacetime to consciousness—emerges spontaneously from the TGP’s atomic operations. We now formalize this claim as a sequence of emergent phases, ordered by increasing structural complexity, and identify the parameter conditions under which each phase transition is expected.

11.1 Phase 0: Primordial Soup Starting from a small random initial graph G with v > v (net node creation), the graph grows. In this regime, nearly all substructures are transient: they form and dissolve within a few ticks.

The degree distribution is approximately uniform, the conflict count |C| is low (few concurrent operations target the same node), and the resolution delay ∆τ is minimal. Physically, this corresponds to the post–Big-Bang high-temperature plasma: maximal entropy density, minimal structure.

11.2 Phase 1: Stable Motif Emergence As |V | grows, three classes of persistent substructures emerge:

1. Directed cycles $C_n$: the first stable structures. By Proposition 7.2, their persistence grows as $(v_3/v_4)^n$. If an edge of $C_n$ is deleted, the remaining mutually reachable nodes provide PR with enough topological information to reconstruct it. Physical correspondence: fermions (stable particles).

2. Hub nodes: preferential attachment (Proposition 7.4) generates nodes with $k^{\mathrm{in}}\gg\bar k$. These hubs attract further connections, creating positive feedback. Physical correspondence: mass concentrations and gravitational centers.

3. Bridge paths: directed paths connecting distinct stable motifs. They provide channels for causal propagation between otherwise isolated structures. Physical correspondence: force-carrying fields and gauge bosons.

11.3 Phase 2: Self-Repairing Structures Definition 11.1 (Self-Repairing Subgraph). A subgraph H ⊆ G is self-repairing under S if, for every edge e ∈ E(H),

That is, the probability of PR reconstructing a deleted edge exceeds the background deletion rate. Self-repair requires that the topology of H encodes its own repair instructions: the remaining edges of H provide sufficient neighborhood information for S to re-create the missing edge with high probability. This is the graph-theoretic realization of the Trinity’s central principle: structure = information = program. Physical correspondence: chemical bond stability—atoms do not spontaneously disintegrate because their electron-cloud topology contains self-repair logic.

11.4 Phase 3: Self-Replicating Structures Definition 11.2 (Self-Replicating Subgraph). A subgraph $H\subseteq G_t$ is self-replicating if there exist $T$ and $H'\subseteq G_{t+T}$ such that

Self-replication requires: v > 0 (new nodes can be created), v large enough to encode H’s topology in new edges, and a PR function with a local-imitation bias (nodes tend to replicate the edge patterns of their neighbors). Conjecture 11.3 (Emergence of Life). For any TGP with v > v , v > v , and r-local S with preferential attachment and local-imitation components, self-replicating subgraphs emerge with probability 1 as t → ∞.

This is the graph-theoretic definition of life: a topological pattern that reads its own structure

(PR) and copies it into adjacent space (ER), requiring conflict resolution (LE) to finalize the copy. No DNA, RNA, or proteins are needed—only a subgraph that can “read its own blueprint” from its topology.

11.5 Phase 4: Competitive Ecology

When multiple self-replicating structures coexist, they compete for finite resources:

• Node budget: if $v_1\le v_2$, the total node count is approximately constant, giving a carrying capacity.
• Edge budget: each node can create at most $v_3$ edges per tick, giving a finite metabolic rate.

This yields Darwinian dynamics: replicators with higher fitness (faster replication, better self- repair, more efficient edge usage) outcompete slower ones. Predation (A disrupts B’s topology to reuse its nodes), symbiosis (A and B share nodes for mutual stability), and parasitism (A hijacks B’s PR to replicate A’s pattern) all emerge naturally.

11.6 Phase 5: Hierarchical Organization

Self-replicating patterns can contain functionally specialized sub-modules:

• $h_1$: boundary defense, maintaining $H$'s edge boundary against disruption.
• $h_2$: internal repair, maintaining $H$'s core connectivity.
• $h_3$: replication engine, creating copies of $H$ at the boundary.

This is the graph-theoretic origin of multicellularity: functional differentiation within a self-replicating unit. At a higher level, units $H_1,\ldots,H_m$ coordinate through bridge edges, forming super-organisms and enabling the transition from cells to tissues to organisms.

11.7 Phase 6: Self-Modeling and Consciousness Definition 11.4 (Self-Modeling Subgraph). A subgraph $H$ is self-modeling if there exists $h\subset H$ and a surjective graph homomorphism

A self-modeling structure “knows” its own topology: h serves as an internal representation of H. The PR loop closes at the sub-graph level: H reads itself (via h), modifies itself, which mod- ifies h (since h ⊂ H), which changes future self-readings. This recursive Paradoxical Re-entry loop at the macroscopic level is the Trinity theory’s account of subjective experience—consciousness as emergent meta-PR.

11.8 Summary: The Emergence Timeline

Each phase emerges from the preceding one through the same three atomic mechanisms (PR, ER, LE) operating under the same four rate bounds (v ,v ,v ,v ). No new axioms, forces, or “vital principles” are introduced at any stage. The full hierarchy from spacetime to consciousness is, in principle, a theorem of the TGP—contingent only on the parameter regime and the passage of sufficient logical ticks.

Table 4: Emergent Evolution Hierarchy of the TGP

Phase	Structure	Key Mechanism	Physical Analogue
0	Primordial soup	Random growth	Big Bang plasma
1	Stable motifs	Cycle persistence, hubs	Particles, mass
2	Self-repairing	Topology encodes repair	Chemical bonds
3	Self-replicating	Topology encodes copying	Life (abiogenesis)
4	Competitive ecology	Resource competition	Darwinian evolution
5	Hierarchical organization	Functional sub-modules	Multicellularity
6	Self-modeling	Internal graph homomorphism	Consciousness

12 Research Significance and Open Problems

We conclude with an assessment of the TGP’s research significance and a roadmap of open problems.

12.1 Why This Matters: Five Domains of Impact

1. Mathematics. The TGP defines a new class of dynamical systems—bounded-rate paradox- referential stochastic graph processes—at the intersection of graph theory, computability theory, and dynamical systems. The Causal Cone Theorem (3.2), the Generative Gap Theorem (6.3), and the Irreversibility Theorem (5.3) are original results with no direct precedent.

2. Physics. The TGP provides a model of emergent spacetime that is simpler than Causal Dynamical Triangulations or Spin Foams, yet derives causality, particles, gravity, and thermodynamic irreversibility from four integer parameters. Unlike Wolfram Physics, it requires no externally prescribed rewrite rules.

3. Artificial Life. Conjecture 11.3 proposes that self-replicating structures emerge in- evitably in sufficiently rich TGPs. If confirmed numerically, this would constitute a graph-theoretic answer to “What is life?”—reducing abiogenesis to a phase transition in a four-parameter dynamical system.

4. Computability Theory. The distinction between computability and generability (The- orem 6.3) challenges the implicit assumption that Church–Turing computability exhausts all aspects of formal system power. Formalizing the “generability hierarchy” is a new open problem.

5. Experimental Tractability. Unlike many speculative cosmological models, the TGP is directly simulable: set (v ,v ,v ,v ), initialize G , and evolve for 106 ticks. The emergent structures, degree distributions, spectral dimensions, and phase transitions can be mea- sured computationally. This is not armchair philosophy—it is an experimentally falsifiable research program.

12.2 Open Problems and Future Directions

We identify seven concrete open problems, each suitable for an independent publication:

1. Phase Transitions. What is the critical ratio v /v at which persistent structures first appear? Is there a sharp phase transition, and if so, what is its universality class?

2. Self-Replication Threshold. What are the minimal (v ,v ,r) values for which self- replicating subgraphs emerge? Can this threshold be computed analytically?

3. Emergent Dimensionality. Does the spectral dimension d of large TGP graphs con- verge to a value near 3? How does d depend on the rate vector?

4. Generability Hierarchy. Formalize the generability classes of dynamical systems as a counterpart to the Chomsky/arithmetic hierarchies. Where does each known formal system (TM, CA, TGP, etc.) sit?

5. TGP Simulator. Develop an open-source, GPU-accelerated simulator for large-scale TGP evolution (|V | > 106) with real-time visualization of emergent structures.

6. Connection to Loop Quantum Gravity. Is the TGP a discrete limit of Spin Foam models? Can the TGP’s causal cone be mapped to the causal structure of a Lorentzian spin foam?

7. Bekenstein Bound Verification. Numerically test Conjecture 9.3: does the resolution entropy of a bounded subgraph scale with its boundary size?

13 Conclusion

We have placed the Trinity Topological Theory on rigorous mathematical footing by introduc- ing the Trinity Graph Process—a bounded-rate stochastic dynamical system on finite directed graphs driven by three mechanisms: Paradoxical Re-entry selection (PR), entity-relation applica- tion (ER), and conflict resolution (LE). The principal results are:

1. Causal structure from first principles. The bounded edge-creation rate v induces a finite propagation speed c and a causal partial order, without presupposing a spacetime manifold.

2. Computational power classification. The TGP is Turing-complete in its physically relevant (r-local) regime, yet strictly weaker than an oracle machine. It is neither “just” topology nor “just” computation—it is a new category: self-evolving topological dynamics with emergent computational capacity.

3. Thermodynamic arrow as theorem. The many-to-one nature of conflict resolution provides a deductive proof of irreversibility, yielding a Second Law without statistical postulates.

4. Generative gap theorem. While Turing-equivalent in computability (Comp(TGP) = Comp(TM)), the TGP strictly exceeds Turing machines in generative capacity (Gen(TGP) ⊋ Gen(TM)). Five structural witnesses—endogenous randomness, self-generated substrate, rule–data identity, structural irreversibility, and self-proliferating parallelism—establish this gap. The TGP is not a stronger computer; it is a richer ontological generator.

5. Physical correspondence. The four rate parameters (v ,v ,v ,v ) map naturally to vacuum fluctuation, annihilation, interaction bandwidth (light speed), and decoherence— offering a minimal parameterization of the universe.

These results establish a new position in the landscape of formal systems: the TGP is Turing- equivalent in computation but Turing-transcendent in generation. Formalizing the “generability hierarchy” as a rigorous counterpart to the Chomsky and arithmetic hierarchies is a major open problem that we leave for future work. The TGP demonstrates that “computation without computers” is not a metaphor but a precise mathematical object: a Paradoxical Re-entry, bounded-rate graph process in which rules,

data, and substrate are unified into a single evolving topology. The universe is not running on a computer. The universe is the dynamics of its own relational graph—nothing more, nothing less.

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无介质拓扑宇宙论

一个图灵超越的本体论发生器

贾宝龙公理体系的严格形式化 Mediumless Topological Cosmology: A Turing-Transcendent Ontological Generator

贾宝龙 Jia, Baolong 独立研究者 seer@139.com

2026 年 4 月

Abstract

贾宝龙公理体系 [1] 提出：宇宙是一个无介质的、自演化的指针迷宫，由三元生 成架构（悖论回入 PR（旧称：悖论自指）、实体-关系 ER、惰性求值 LE）驱动，可具体化为三种 原子操作——CREATE/LINK、JUMP、RESOLVE。 本文在上述基础上引入有界速率约束 (v ,v ,v ,v ) 2 N4——分别限制每个逻辑 tick 中节点的诞生数、消亡数、每节点的新建边数和删除边数——将三元宇宙论推 进到严格的数学形式。由此定义的三元图过程（Trinity Graph Process, TGP）是 有限有向图空间上的离散时间随机动力系统。 核心结果：(1) 有界边创建率 v 自然导出因果锥，无需预设时空流形即可推导 出有限传播速度——纯逻辑空间中的” 光速”；(2) TGP 的计算能力严格取决于悖论 自指选择函数的局部性——从亚图灵（常数选择）到图灵完备（有界半径选择）的 完整分类；(3) RESOLVE 冲突解决机制的多对一性给出不可逆性定理——热力学第 二定律从纯逻辑推出，而非统计假设；(4) TGP 在可计算性层面与图灵机等价，但 在生成能力层面严格超出——五个结构性证据（内生随机性、自生成基底、规则-数 据同一、结构不可逆、自增殖并行）建立了这一差距；(5) 完整的涌现演化层级—— 从原始汤到意识的六阶段演化时间线，每一阶段仅需同一组参数 (v ,v ,v ,v ) 和 三元机制。 关键词：贾宝龙公理、三元图过程、悖论回入、实体-关系、惰性求值、因果锥、无 介质计算、生成能力、有界速率动态图

声明： 本文呈现的理论框架为学术探索性质。部分命题以猜想形式陈述。我们的目标是 为三元宇宙理论提供严格的数学基础，欢迎一切科学批评。

Contents

1 引言：从哲学愿景到数学对象 3 1.1 贾宝龙公理回顾 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 从公理到形式化：本文的任务 . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 与已有研究的关系 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 贡献 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 形式框架：三元图过程 4 2.1 状态空间 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 悖论回入选择函数 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 冲突解决机制 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 状态转移 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 因果锥定理 5

3.1 可达性与影响

3.2 因果偏序

4 计算能力分类 6

5 熵、不可逆性与时间箭头 7

6 超越图灵：生成能力 7

7 涌现演化层级 8 7.1 Phase 0：原始汤 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 7.2 Phase 1：稳定基元涌现 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 7.3 Phase 2：自修复结构 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 7.4 Phase 3：自复制结构——生命的图论定义 . . . . . . . . . . . . . . . . . . 9 7.5 Phase 4：竞争生态——达尔文演化 . . . . . . . . . . . . . . . . . . . . . . 9 7.6 Phase 5：层次化组织——多细胞性 . . . . . . . . . . . . . . . . . . . . . . 9 7.7 Phase 6：自建模——意识 . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 7.8 演化时间线总结 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

8 物理对应辞典 11

9 可证伪预测 11

10 讨论 11

10.1 为什么不是图灵机

10.2 为什么不仅仅是拓扑

10.3 与已有形式化的比较

11 研究意义与开放问题 12

11.1 五个领域的影响

11.2 七个开放问题

12 结论 13

1 引言：从哲学愿景到数学对象

1.1 贾宝龙公理回顾 贾宝龙公理体系 [1] 由两条公理构成： 公理 1 (双范式原理). 存在被划分为三个正交维度：（1）未定义——绝对占位符虚空；（2） 柏拉图水晶——一切可能数学结构的静态完备空间（ER 范式）；（3）悖论回入生成树 ——悖论回入驱动的动态生成引擎（PR 范式）。 公理 2 (三元生成机制). 悖论回入（PR）、实体-关系（ER）、惰性求值（LE）形成不 可分割的动态闭环。三者缺一不可：PR 无 ER 则无处展开；ER 无 PR 则永远静止； PR+ER 无 LE 则陷入无限死循环。

1.2 从公理到形式化：本文的任务 贾宝龙公理体系 [1] 将公理 2 具体化为动态有向图 U = (V ,E ) 上的三种原子操作—— t t t CREATE/LINK（ER）、JUMP（PR）、RESOLVE（LE）。但原始表述留下了四个开放问题：

(i) 什么约束每个 tick 中拓扑变化的速率？

(ii) 在什么条件下系统具有图灵完备的计算能力？

(iii) 如何在没有预设度量空间的情况下导出因果结构？

(iv) RESOLVE 机制与热力学时间箭头的精确关系是什么？ 本文通过引入有界速率约束——四个非负整数 (v ,v ,v ,v )——全面回答上述问题。

1.3 与已有研究的关系

Table 1: 五个最近的研究方向及其缺口

方向	代表人物	与 TGP 的交集	缺失要素
Wolfram 物理	Wolfram, 2020 [4]	超图重写 → 涌现时空	规则固定（非悖论回入）；无节点删除；无速率约束
算法化学	Fontana & Buss, 1994 [5]	$\lambda$ 表达式互相作用 → 悖论回入 + 涌现	在 $\lambda$ 演算空间，非图空间；无空间概念
因果动态三角化	Ambjørn, Loll, 1998 [6]	离散单形拼接 → 涌现时空 + 因果	使用固定几何积木（三角形）；规则外在
量子图态	Konopka 等, 2008 [7]	动态边 → 可“熔化/冻结”	节点固定（只改边）；无悖论回入选择；无速率约束
人工化学	Dittrich 等, 2001 [8]	构造性动力学 → 涌现新结构	通用框架，非图特定；无因果锥推导

关键缺口：从未有人同时实现这五个要素——（i）悖论回入选择函数（规则从拓扑 中读取），（ii）节点和边的创建与删除（非单调增长），（iii）有界速率约束，（iv）冲突 解决机制，（v）从上述推导因果锥和不可逆性。TGP 填补了这一空白。

1.4 贡献

1. 带有界速率约束的 TGP 完整形式化定义（第2节）。

2. v 导出因果锥与有限传播速度的证明（第3节）。

3. TGP 计算能力的参数化分类（第4节）。

4. RESOLVE 作为不可逆性和熵流源头的分析（第5节）。

5. TGP 生成能力严格超出图灵机的形式证明（第6节）。

6. 从原始汤到意识的完整涌现演化层级（第7节）。

7. TGP 参数与已知物理常数的对应辞典（第8节）。

8. 研究意义评估与七个开放问题（第11节）。

2 形式框架：三元图过程

2.1 状态空间 定义 2.1 (图宇宙). 令 G 为所有有限有向图 G = (V,E) 的集合，其中 V 为有限节点集 （实体），E \subseteq V \times V 为有向边集（关系）。允许自环 (v,v) 2 E。 定义 2.2 (速率向量). 速率向量为

其四个分量分别为：

这四个整数是纯逻辑空间的“基本常数”。

2.2 悖论回入选择函数 定义 2.3 (选择函数（PR 的形式化）). 悖论回入选择函数是一个（可能随机化的）映射

其中 $\mathcal A(\mathbf v)$ 是速率向量 $\mathbf v$ 下的容许操作束集合。操作束 $a=(V^+,V^-,E^+,E^-)$ 满足：

注记 2.4. S 的悖论回入性在于：它读取当前图 G ——即系统自身的状态——来产生操 作束。因为 G 就是系统，S 体现了贾宝龙第二公理中 PR（悖论回入）的角色：系统读 取自身来决定自身的下一状态。这正是说谎者悖论” 这句话是假的” 在拓扑空间中的推 广——图说” 我下一步这样变”，而这个” 下一步” 又改变了图本身。

2.3 冲突解决机制

定义 2.5（冲突集）。给定操作束 $a=(V^+,V^-,E^+,E^-)$，冲突发生在：

(a) 某节点 $w\in V^-$（即将消亡）同时是某 $e\in E^+$（即将创建的边）的端点——“毁灭-创建冲突”。

(b) 两条或多条 $E^+$ 中的边指向同一节点且具有不兼容的拓扑后果——“竞争冲突”。

冲突集记为 $C(a)\subseteq(V^-\times E^+)\cup(E^+\times E^+)$。

定义 2.6（解决函数，LE 的形式化）。解决函数是一个（可能随机化的）映射

使得 $\mathcal R(a,G_t)=a^*$ 是无冲突的：$C(a^*)=\varnothing$，且 $a^*$ 是 $a$ 的最大一致子集。解决延迟定义为 $\Delta\tau:=|C(a)|$。 在贾宝龙公理的语言中，$\mathcal R$ 就是惰性求值——不到冲突发生时不求值，冲突发生时强行斩断死循环。$\Delta\tau$（冲突计数）就是 LE 摩擦产生的“时间”。

2.4 状态转移

定义 2.7（三元图过程，TGP）。三元图过程是一个五元组 $\mathrm{TGP}=(\mathcal G,\mathbf v,\mathcal S,\mathcal R,G_0)$，其中 $G_0\in\mathcal G$ 为初始图。状态转移 $G_t\mapsto G_{t+1}$ 分三步：

1. 选择（PR）：计算 $a=\mathcal S(G_t)$——悖论回入引擎读取自身拓扑。
2. 解决（LE）：计算 $a^*=\mathcal R(a,G_t)$，令 $a^*=(V^{+*},V^{-*},E^{+*},E^{-*})$。
3. 应用（ER）：

命题 2.8（良定义性）。对任意 $\mathbf v\in\mathbb N^4$、任意 $\mathcal S$、任意 $\mathcal R$、任意有限初始图 $G_0$，TGP 是良定义的：每个 $G_t$ 都是有限有向图，且 $|V_{t+1}|\le|V_t|+v_1$。

3 因果锥定理

本节是全文最核心的物理结果：有界边创建率 $v_3$ 导出图上的自然因果结构——无需预设任何度量空间即可得到“光速”。

3.1 可达性与影响

定义 3.1（$r$-局部选择函数）。$S$ 是 $r$-局部的，如果对每个节点 $w\in V_t$，$S$ 在 $w$ 处的操作仅依赖于 $w$ 的 $r$-邻域 $B_r(w,G_t):=\{u\in V_t:d_{G_t}(w,u)\le r\}$。

定义 3.2（前向影响集）。节点 $w\in V_t$ 经过 $\Delta t$ 个 tick 后的前向影响集为

定理 3.3（因果锥界）。对任意带速率向量 $(v_1,v_2,v_3,v_4)$ 和 $r$-局部选择函数 $S$ 的 TGP，前向影响集满足

证明。对 $\Delta t$ 归纳。基础情形：$\operatorname{Reach}(w,t,0)=\{w\}$。归纳步骤：在 tick $t+\Delta t\to t+\Delta t+1$ 中，Reach 中每个节点最多创建 $v_3$ 条新边，最多创建 $v_1$ 个新节点，且每个新节点最多创建 $v_3$ 条边。因此新增受影响节点数不超过 $|\operatorname{Reach}|(v_3+1)(v_1+1)$，累加几何级数得证。

推论 3.4（拓扑光速）。TGP 中因果传播的速率上界为

这是纯逻辑空间中的“光速”——不是空间中的速度（没有预设空间），而是拓扑扩散的最大速率。

注记 3.5（与物理的对应）。广义相对论中，光速 $c$ 约束闵可夫斯基时空的因果结构；TGP 中，$c_{\mathrm{TGP}}$ 约束图上的因果传播。本质差异是：在 GR 中 $c$ 是强加于预设流形上的常数，在 TGP 中 $c_{\mathrm{TGP}}$ 涌现于原子图操作的速率约束。这正是贾宝龙法则 3（无介质计算与物理法则涌现）的精确实现。

3.2 因果偏序

定义 3.6（因果先于）。对事件 $(w,t)$ 和 $(u,t')$（$t<t'$），定义 $(w,t)\prec(u,t')$ 当且仅当 $u\in\operatorname{Reach}(w,t,t'-t)$。

命题 3.7。$\prec$ 是所有事件 $\{(w,t):w\in V_t,\ t\in\mathbb N\}$ 上的严格偏序。

4 计算能力分类

定理 4.1（计算层级）。TGP 的计算能力取决于 $S$ 的局部性半径：

(I) 常数 $S$（忽略 $G_t$，随机操作）：退化为随机图过程，不能计算任何非平凡函数，属于亚图灵。

(II) $r$-局部 $S$（只读每个节点的 $r$-邻域）：等价于图上元胞自动机。当 $r\ge2$、$v_1\ge1$、$v_3\ge|\Sigma|+2$ 时，可模拟 Rule 110，因此图灵完备。

(III) 全局 $S$（读取整个 $G_t$）：有限图上等价于有限自动机；无界增长图上形式上超图灵，但需每 tick 处理无穷信息，物理上不可实现。

证明梗概。情形 (II)：将图灵机纸带编码为有向路径 $p_1\to p_2\to\cdots\to p_n$，每个节点通过辅助边结构编码纸带符号，磁头位置编码为自环。$r$-局部规则读取当前格和邻居，修改符号、移动自环、可选创建新节点（扩展纸带）。Rule 110 是图灵完备的，可被此构造模拟。

推论 4.2（物理相关区域）。物理上相关的 TGP 是情形 (II)：$r$-局部、图灵完备并遵守因果锥约束。它足够强大以生成贾宝龙公理体系 [1] 中描述的完整物理涌现层级，又足够受约束以遵守因果性。

5 熵、不可逆性与时间箭头

定理 5.1（解决的不可逆性）。若 $|C(a)|>0$（至少存在一个冲突），则解决函数 $R$ 严格销毁信息：它将多个不同的前解决状态映射到同一个后解决状态。

证明。设 $C(a)=\{c_1,\ldots,c_m\}$，$m>0$。每个冲突 $c_i$ 有 $k_i\ge2$ 种合法解决方案。$R$ 为每个冲突选择一种，丢弃其余。映射到同一 $a^*$ 的前解决配置数为 $\prod_{i=1}^{m}k_i\ge2^m>1$，因此 $R$ 是多对一映射，信息不可逆地丢失。

定义 5.2（解决熵）。tick $t$ 的解决熵为

注记 5.3（时间箭头）。累积解决熵 $\Sigma_T:=\sum_{t=0}^{T}H_t$ 非递减，“过去”是 $\Sigma$ 较小的方向。这与热力学第二定律完全对应——但在 TGP 中，第二定律不是假设，而是冲突解决多对一性质的定理。

这正是贾宝龙法则 4 的精确形式化：“真随机 = LE 强行切断 PR 死循环时的系统摩擦”——$R$ 在 $k_i$ 种合法方案中的选择就是“摩擦火花”，其结果就是真随机；而由此产生的信息丢失驱动了熵增。

6 超越图灵：生成能力

定义 6.1（可计算集）。系统 $M$ 的可计算集 $\operatorname{Comp}(M)$ 是 $M$ 能计算的所有函数 $f:\Sigma^*\to\Sigma^*$ 的集合。

定义 6.2（可生成集）。动力系统 $D$ 的可生成集 $\operatorname{Gen}(D)$ 是 $D$ 从某初始条件能产生的所有状态序列 $(s_0,s_1,s_2,\ldots)$ 的集合。

定理 6.3（生成差距）。对 $r$-局部 TGP：

即：TGP 计算的函数与图灵机完全相同，但能生成的现象严格更多。证明的第一个等式由定理 4.1(II) 和 Church-Turing 论题得出；对于严格包含，论文给出五个证据：内生随机性、自生成基底、规则-数据同一、结构不可逆和自增殖并行。

证据 4：结构不可逆。Bennett [10] 证明任何图灵机都可转换为等价的可逆图灵机。TGP 的 RESOLVE 机制是结构性不可逆的（定理 5.1）——只要 $v_3>0$ 且 $|V_t|>1$（多节点同时创建边），冲突必然发生，$R$ 必然多对一。不存在“可逆版 TGP”——移除 RESOLVE 将导致逻辑死锁。

证据 5：自增殖并行。$k$-带图灵机有 $k$ 个读写头——运行前确定的常数。TGP 中，当 $v_1>v_2$ 时，每 tick 并行操作的“处理器”（节点）数量随演化增长：$|V_t|\approx|V_0|+t(v_1-v_2)$。系统自己创造更多处理器。

注记 6.4（层级定位）。TGP 占据一个计算上图灵等价、生成上图灵超越的层级。将“可生成性层级”形式化为 Chomsky 层级和算术层级的对应物，是一个有独立价值的开放问题。

7 涌现演化层级

贾宝龙公理的核心主张之一是：从时空到意识的全部物理层级可从 TGP 的原子操作中自发涌现。本节将此主张形式化为六个涌现阶段。

7.1 Phase 0：原始汤

从小型随机初始图 $G_0$ 出发，$v_1>v_2$ 驱动图增长。几乎所有子结构都是瞬态的，度分布近似均匀，冲突少 $\to\Delta\tau$ 小 $\to$“时间流速快”。物理对应：大爆炸后的高温等离子体。

7.2 Phase 1：稳定基元涌现

命题 7.1（环的持久性）。在 $v_4<v_3$ 的 TGP 中，长度为 $n$ 的有向环 $C_n$（$n\le v_3$）的持久性满足

有向环是第一批稳定结构——环的每条边被删后可被 PR 重建（邻居仍在）。物理对应：费米子（稳定粒子）。偏好连接产生星形枢纽——高入度节点，对应质量/引力中心；桥接路径连接不同稳定结构，对应玻色子（力的载体）。

7.3 Phase 2：自修复结构

定义 7.2（自修复子图）。子图 $H\subseteq G_t$ 是自修复的，如果对 $H$ 的每条边 $e$：

自修复要求 $H$ 的拓扑编码了自身的修复指令——结构 = 信息 = 程序。这是三元架构在子图层面的实现。物理对应：化学键的稳定性。

7.4 Phase 3：自复制结构——生命的图论定义

定义 7.3（自复制子图）。$H\subseteq G_t$ 是自复制的，如果存在时间 $T$ 和子图 $H'\subseteq G_{t+T}$，使得：$H'\cong H$（图同构）；$H'\cap H=\varnothing$（空间独立）；且 $H'$ 的形成因果依赖于 $H$。

猜想 7.4（生命涌现）。对任意 $v_1>v_2$、$v_3>v_4$、具有偏好连接和局部模仿偏好的 $r$-局部 $S$ 的 TGP，自复制子图以概率 1 涌现（$t\to\infty$）。

这就是生命的图论定义：一个拓扑模式从自身结构中“读取蓝图”（PR）并在邻域复制（ER），需要冲突解决（LE）来最终化副本。不需要 DNA、RNA、蛋白质——只需一个能自读蓝图的子图。

7.5 Phase 4：竞争生态——达尔文演化

多种自复制结构共存时，争夺有限资源：节点预算（$v_1\approx v_2$ 时总节点近似恒定）和边预算（每节点每 tick 最多 $v_3$ 条新边）。由此涌现达尔文动力学：更快复制者胜出（适者生存）；模式 A 破坏模式 B 的拓扑（捕食）；A 和 B 共享节点互增稳定性（共生）；A 劫持 B 的 PR（寄生）。

7.6 Phase 5：层次化组织——多细胞性

自复制模式可包含功能特化的子模块：$h_1$（边界防御）、$h_2$（内部修复）、$h_3$（复制引擎）。这是多细胞生物的图论起源。多个单位 $H_1,\ldots,H_m$ 通过桥接边协调，形成超级个体。

7.7 Phase 6：自建模——意识

定义 7.5（自建模子图）。$H$ 是自建模的，如果存在 $h\subset H$，$h$ 是 $H$ 的有损压缩：存在满射图同态 $\phi:H\twoheadrightarrow h$。

自建模结构“知道”自己的拓扑——$h$ 是 $H$ 的内部表示。PR 在子图层面的递归闭合：$H$ 读取自身（通过 $h$）$\to$ 修改自身 $\to$ 修改了 $h$（$h\subset H$）$\to$ 改变未来的自读。这就是贾宝龙法则 5 的精确化：“意识 = PR 在状态折叠（LE 渲染）时的第一人称视角”。

7.8 演化时间线总结

每阶段从前一阶段通过同一组三元机制（PR、ER、LE）和同一组速率约束 $(v_1,v_2,v_3,v_4)$ 自然涌现。无需引入任何新公理、新力量或“生命力”。正如贾宝龙法则所断言：死锁、混沌、摩擦不是缺陷——它们就是生成机制本身。

表 2：TGP 涌现演化层级

阶段	结构	关键机制	物理对应
0	原始汤	随机增长	大爆炸
1	稳定基元	环持久性、枢纽	基本粒子、质量
2	自修复	拓扑编码修复	化学键
3	自复制	拓扑编码复制	生命（无生源说）
4	竞争生态	资源竞争	达尔文演化
5	层次组织	功能子模块	多细胞性
6	自建模	内部图同态	意识

表 3：TGP 物理对应辞典

TGP 概念	形式对象	物理对应
速率向量	$\mathbf v=(v_1,v_2,v_3,v_4)$	基本常数
$v_1$（节点诞生率）	$|V^+|\le v_1$	真空涨落率
$v_2$（节点消亡率）	$|V^-|\le v_2$	正反粒子湮灭率
$v_3$（边创建率）	$|E^+(w)|\le v_3$	相互作用带宽 $\to c$
$v_4$（边删除率）	$|E^-(w)|\le v_4$	退相干率
$v_1-v_2$ 净增长	$\Delta|V|\sim v_1-v_2$	宇宙膨胀率 $H_0$
图距离 $d(w,u)$	最短有向路径	空间距离
入度 $k_w^{\mathrm{in}}$	$\sum_j A_{jw}$	质量 $\propto k^{\mathrm{in}}$
冲突数 $|C(a)|$	解决负载	曲率/引力场
解决延迟 $\Delta\tau$	冲突基数	固有时间间隔
因果锥 $\operatorname{Reach}(w,t,\Delta t)$	定理 3.3	光锥
持久环 $C_n$	命题 7.1	费米子
枢纽 $k^{\mathrm{in}}\gg\bar{k}$	偏好连接	黑洞
累积熵 $\Sigma_T$	定义 5.2	热力学熵

8 物理对应辞典

注记 8.1。比值 $v_1/v_2$ 控制图宇宙是膨胀（$v_1>v_2$）、静态（$v_1=v_2$）还是收缩（$v_1<v_2$）。在膨胀 TGP 中，有效宇宙学参数随 $|V_t|$ 增长而递减，可能解释观测到的 $\Lambda$ 极小值。

9 可证伪预测

猜想 9.1（度指数收敛）。具有偏好连接的 $r$-局部 TGP 的渐近入度指数收敛于

猜想 9.2（涌现维度）。从随机初始图出发，TGP 的谱维度 $d_s$ 收敛于

猜想 9.3（Bekenstein 面积律）。子图 $H$ 的最大解决熵满足

其中 $|\partial H|$ 是边界边数。

10 讨论

10.1 为什么不是图灵机

1. 无固定程序。图灵机的 $\delta$ 在运行前确定，运行中不变。TGP 的 $\mathcal S(G_t)$ 每 tick 都变——程序就是数据。

2. 无输入/输出。图灵机计算函数 $f:\Sigma^*\to\Sigma^*$。TGP 没有指定输入或输出——它只是演化。

3. 永不停机。TGP 没有停机状态。它是开放式过程——本体论发生器，不是问题求 解器。

4. 内禀随机性。R 中的随机性不是加在确定性核心上的” 噪声”，而是冲突解决机制 的根本特征——来自拓扑自干涉，不是随机纸带。

10.2 为什么不仅仅是拓扑

1. 它能模拟任何图灵机（定理4.1(II)）。

2. 它生成自己的空间——标准拓扑动力系统在给定空间上操作，TGP 在运行中生成 空间。

3. 它从可逆组件产生不可逆性——单个原子操作可逆，但复合的 RESOLVE 不可逆。

TGP 因此占据独特位置：超越拓扑，异于计算。用贾宝龙公理的语言：它是 PR 引 擎在 ER 水晶上通过 LE 展开的” 宏大拓扑游戏” 的精确数学模型。

Table 4: 形式化比较

特征	原始三元论	TGP（本文）	Wolfram 物理
基本元素	节点 + 边	节点 + 边	超边
重写规则	隐式（PR）	隐式（$\mathcal S(G_t)$）	显式（固定规则）
速率约束	未指定	$(v_1,v_2,v_3,v_4)$	未指定
因果性	假设	推导（定理 3.3）	推导
不可逆性	定性	证明（定理 5.1）	未涉及
计算能力	“等价 UTM”	精确分类（定理 4.1）	图灵完备
生成能力	未涉及	$\operatorname{Gen}\supsetneq\operatorname{Gen}(\mathrm{TM})$（定理 6.3）	未涉及

10.3 与已有形式化的比较

11 研究意义与开放问题

11.1 五个领域的影响

1. 数学：定义了新的动力系统类——有界速率悖论回入随机图过程。因果锥定理、生成差距定理、不可逆性定理均为原创。

2. 物理：比因果动态三角化或自旋泡沫更简单的涌现时空模型。只需四个整数参数 即可导出因果性、粒子、引力和热力学不可逆性。

3. 人工生命：猜想7.4提出自复制结构在足够丰富的 TGP 中必然涌现。若数值验证， 将是” 生命是什么” 的图论解答——把无生源说还原为四参数动力系统的相变。

4. 可计算性理论：Comp 与 Gen 的区分（定理6.3）挑战了 Church-Turing 可计算性 穷尽形式系统能力的隐含假设。形式化” 可生成性层级” 是新的开放问题。

5. 可实验性：TGP 可直接计算机模拟——设定 $(v_1,v_2,v_3,v_4)$，初始化 $G_0$，运行 $10^6$ 个 tick，测量涌现结构、度分布、谱维度和相变。这是可证伪的研究纲领。

11.2 七个开放问题

1. 相变：稳定结构首次出现的临界 $v_3/v_4$ 是多少？是否存在尖锐相变？

2. 自复制阈值：自复制子图涌现的最小 $(v_1,v_3,r)$ 是多少？

3. 涌现维度：TGP 大图的谱维度 d 是否收敛于 3 附近？

4. 可生成性层级：形式化各形式系统（TM、CA、TGP 等）的可生成性类。

5. TGP 模拟器：开发 GPU 加速的大规模 TGP 模拟器（$|V_t|>10^6$）。

6. 与圈量子引力的联系：TGP 是否是自旋泡沫模型的离散极限？

7. Bekenstein 界验证：数值测试猜想9.3——有界子图的解决熵是否正比于边界大 小？

12 结论

本文为贾宝龙公理体系的三元宇宙理论提供了严格的数学基础——三元图过程（TGP）： 有限有向图空间上的有界速率随机动力系统，由悖论回入选择（PR）、实体-关系应用 （ER）和冲突解决（LE）三元驱动。 主要结果：

1. 因果结构：$v_3$ 导出拓扑光速 $c_{\mathrm{TGP}}$ 和因果偏序——不预设时空流形。

2. 计算分类：r-局部 TGP 图灵完备，但不超图灵。

3. 时间箭头：冲突解决的多对一性推导出不可逆性——热力学第二定律成为定理。

4. 生成差距：Comp(TGP) = Comp(TM)，Gen(TGP) ⊋ Gen(TM)。TGP 不是更强 的计算机，而是更丰富的本体论发生器。

5. 涌现层级：六阶段演化——从原始汤到意识——仅由四个整数参数和三元机制驱 动。

这些结果确立了形式系统版图中的新位置：TGP 在计算上图灵等价，在生成上图灵 超越。 贾宝龙公理断言：” 宇宙不是在计算机上运行的。逻辑关系本身就是硬件、软件和数 据。”TGP 证明这不是比喻，而是精确的数学对象——悖论回入的、有界速率的图过程， 其中规则、数据和基底统一为单一的演化拓扑。

宇宙不生产真理——宇宙就是一个从自身拓扑中读取自身规则的图过程，在冲突的 摩擦中创造时间，在信息的丢失中产生记忆，在结构的自复制中涌现生命。

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