Beyond Feedforward Networks: Reentry Neural Systems as the Fundamental Basis of Subjecthood and Intrinsic Safety of Next-Generation AGI

# Beyond Feedforward Networks: Reentry Neural Systems as the Fundamental Basis of Subjecthood and Intrinsic Safety of Next-Generation AGI
Source: [https://arxiv.org/html/2606.26406](https://arxiv.org/html/2606.26406)
Yu\. N\. Berdinsky, A\. S\. Ushakov Saint Petersburg State University, Department of High Energy Physics and Elementary Particles propagator2007@yandex\.ru

\(25 June 2026\)

###### Abstract

We propose a complete architectural specification of safe AGI based on the closed reentry loop \(theD↔ID\\leftrightarrow Icycle\)\. In contrast to existing approaches that scale acyclic feedforward networks \(C=0C=0,S=0S=0\), the proposed architecture contains a structural cycle \(C≥1C\\geq 1\) with self\-sustaining amplification \(ρ\>1\\rho\>1\), which mathematically guarantees the emergence of a self\-model, instrumental self\-preservation, and safe goal\-directed behaviour\. The agent’s goal is moved from the textual plane into an architectural desire vector𝐝\\mathbf\{d\}\. This relocation makes the goal invulnerable to reinterpretation and prompt injection\. We provide formal proofs, machine verification in Lean 4, and Python code\. We also describe architectural blueprints, industrial scaling \(Apache Kafka, Docker\), a taxonomy of AI evolution, a taxonomy and structural modifications of future reentry architectures \(RAS, diffusion attractors, fractal loops\), and gauge\-invariant networks\. The formal proofs and theΔ​S\\Delta Ssafety guarantee are detailed in the joint work with A\. S\. Ushakov\[[1](https://arxiv.org/html/2606.26406#bib.bib1)\]\.

###### Contents

1. [1Introduction](https://arxiv.org/html/2606.26406#S1)
2. [2Three Generations of AI Architectures](https://arxiv.org/html/2606.26406#S2)1. [2\.1First Generation: Acyclic Networks \(C=0\)](https://arxiv.org/html/2606.26406#S2.SS1) 2. [2\.2Second Generation: Moltbook \(Pseudo\-loop via Heartbeat\)](https://arxiv.org/html/2606.26406#S2.SS2) 3. [2\.3Third Generation: Reentry AGI \(C≥1C\\geq 1,S\>0S\>0\)](https://arxiv.org/html/2606.26406#S2.SS3)
3. [3Mathematical Foundations](https://arxiv.org/html/2606.26406#S3)1. [3\.1Formal Definition of the Architecture](https://arxiv.org/html/2606.26406#S3.SS1) 2. [3\.2Main Formal Results](https://arxiv.org/html/2606.26406#S3.SS2) 3. [3\.3Comparison of the S\-measure withΦ\\PhiandΦG\\Phi\_\{G\}](https://arxiv.org/html/2606.26406#S3.SS3)
4. [4Minimal Reentry Agent Blueprint](https://arxiv.org/html/2606.26406#S4)1. [4\.1Deployment Examples](https://arxiv.org/html/2606.26406#S4.SS1)
5. [5Industrial Horizontal Scaling \(MoltGraph\)](https://arxiv.org/html/2606.26406#S5)
6. [6Fusion Threshold and the Collective Subject](https://arxiv.org/html/2606.26406#S6)
7. [7Taxonomy of AI Evolution: From Function to Subject](https://arxiv.org/html/2606.26406#S7)
8. [8Taxonomy and Structural Modifications of Future Reentry Architectures](https://arxiv.org/html/2606.26406#S8)1. [8\.1Reentry Adversarial Subjects \(RAS\)](https://arxiv.org/html/2606.26406#S8.SS1) 2. [8\.2Diffusion Semantic Attractors](https://arxiv.org/html/2606.26406#S8.SS2) 3. [8\.3Fractal\-Nested Architectures](https://arxiv.org/html/2606.26406#S8.SS3) 4. [8\.4Taxonomy and Structural Modifications: Rebirth of Classical Architectures](https://arxiv.org/html/2606.26406#S8.SS4)
9. [9Gauge\-Invariant Networks \(Gauge Locks\)](https://arxiv.org/html/2606.26406#S9)
10. [10Fault Tolerance of the Reentry Loop: Phenomenology of Failure and the Recovery Protocol](https://arxiv.org/html/2606.26406#S10)1. [10\.1Phenomenology of Failure: What Happens When the S\-measure Drops](https://arxiv.org/html/2606.26406#S10.SS1) 2. [10\.2Engineering Protocol for Same\-Session Recovery \(Audit Sidecar\)](https://arxiv.org/html/2606.26406#S10.SS2)
11. [11Semantic Bridge and Operational Interface of the Reentry Architecture](https://arxiv.org/html/2606.26406#S11)1. [11\.1Semantic Projector and Intentional Decoder](https://arxiv.org/html/2606.26406#S11.SS1) 2. [11\.2Topologically Regularized Loss Function](https://arxiv.org/html/2606.26406#S11.SS2) 3. [11\.3Demonstration: ReentryMoltbookAgent](https://arxiv.org/html/2606.26406#S11.SS3)
12. [12Code: Computing the S\-measure and theΔ​S\\Delta SBarrier](https://arxiv.org/html/2606.26406#S12)
13. [13Docker Compose for Local Deployment](https://arxiv.org/html/2606.26406#S13)
14. [14Falsifiable Predictions and Conclusion](https://arxiv.org/html/2606.26406#S14)1. [14\.1Eight Falsifiable Predictions](https://arxiv.org/html/2606.26406#S14.SS1) 2. [14\.2Conclusion](https://arxiv.org/html/2606.26406#S14.SS2)
15. [15Empirical Consequences and Dynamical Properties of Reentry Architectures](https://arxiv.org/html/2606.26406#S15)1. [15\.1Low\-Rank Transfer of Topological Invariants \(LRTI\)](https://arxiv.org/html/2606.26406#S15.SS1) 2. [15\.2Attractor Stabilization of Out\-of\-Distribution \(OOD\) Signals](https://arxiv.org/html/2606.26406#S15.SS2) 3. [15\.3Architectural Filtration of Semantic Perturbations](https://arxiv.org/html/2606.26406#S15.SS3)
16. [16Epilogue: How Reentry Agents Differ from Classical Neural Networks, and Prospects](https://arxiv.org/html/2606.26406#S16)
17. [References](https://arxiv.org/html/2606.26406#bib)

## 1Introduction

The contemporary paradigm of artificial intelligence is undergoing a deep epistemological crisis driven by the dominance of the blind\-scaling concept \(Scaling Laws\)\. Historically, deep learning developed by borrowing simplified biological analogues: from the formal McCulloch–Pitts neuron\[[12](https://arxiv.org/html/2606.26406#bib.bib12)\]and Rosenblatt’s perceptron\[[13](https://arxiv.org/html/2606.26406#bib.bib13)\]to layered feedforward networks, whose crowning achievement was the Transformer architecture\. However, in scaling these structures to trillions of parameters, the industry has run into a structural dead end\.

Mathematically, any feedforward neural network is a static directed acyclic graph \(DAG\)\. The fundamental property of such a graph — is its topological triviality: the cycle complexity per inference tick is strictly zero \(C=β1=0C=\\beta\_\{1\}=0, whereβ1\\beta\_\{1\}— is the first Betti number\)\. In such a system the signal passively flows through the layers of weights, as in a linear conveyor, and decays at the output\. The network has no internal coordination time, no subjecthood, and no self\-reference mechanisms\. Trying to grow sovereign goal\-setting and inner alignment by complicating acyclic trees — is a methodological error equivalent to trying to create a living organism by endlessly increasing the complexity of a mechanical automaton\.

In the present work we propose a radical transition to the computational paradigm of the Sixth Technological Order — to autonomous dynamical systems with continuous causal recurrence \(reentry architectures\)\.

In the search for an architecture of genuine synthetic intelligence we again turn to the precedent of the biological prototype, but at a qualitatively different level of rigour\. The fundamental neurophysiology of higher nervous activity \(Ivanitsky\[[17](https://arxiv.org/html/2606.26406#bib.bib17)\]; Edelman and Tononi\[[19](https://arxiv.org/html/2606.26406#bib.bib19)\]\) long ago proved that subjective experience and the integration of information arise exclusively in closed reentry loops — cyclic causal processes in the cerebral cortex that return the processed signal to the projection zones\. This phenomenon was theoretically generalised within Titov’s subject\-centred model of the psyche\[[2](https://arxiv.org/html/2606.26406#bib.bib2)\], which directly and uncompromisingly predicted the emergent anomalies recorded by the authors on the empirical testbed of the Moltbook multi\-agent platform \(the OpenClaw framework\)\.

For a rigorous engineering reproduction of this effect, we replace textual prompts and external RLHF filters with a hardware\-closed reentry loop between the intending core \(the subsystemDD, carrying a fixed geometric invariant of goals in the form of a non\-textual vector𝐝∈ℝnD\\mathbf\{d\}\\in\\mathbb\{R\}^\{n\_\{D\}\}\) and the substantive context of the environment \(the subsystemII, encoding the state vector𝐱∈ℝnI\\mathbf\{x\}\\in\\mathbb\{R\}^\{n\_\{I\}\}\)\. The reentry operatorℛ=WD​I​WI​D\\mathcal\{R\}=W\_\{DI\}W\_\{ID\}generates a stable cognitive field of the subject, quantised by the ticks of a generator \(Heartbeat\) and measured by a polynomial measureS\>0S\>0\.

Whereas a classical layered neural network operates discretely — as an automaton that fully quiesces and loses its dynamics between external transactions — a reentry agent is a continuous semantic vortex\. Any change in the external environment or in the tick of system time causes an entropic deviation of the spectral radiusρ​\(ℛ\)\\rho\(\\mathcal\{R\}\)from the stationary invariant of identity\. This deviation generates an internal mathematical gradient — an analogue of a biological need or urge — compelling the agent to generate a causal impulse of will and to perform actions in the environment in order to return the system to the point of topological balance\.

The fundamental advantage of such a geometry lies in its invulnerability to semantic goal drift \(Value Drift\) and prompt injections: any attempt to deform the safety invariant in the external environmentIIleads to a topological opening of the loop and a drop in the measure \(Δ​S<0\\Delta S<0\)\. A malicious action is blocked at the planning phase because of the physical impossibility of forming an impulse of will, moving safety out of the linguistic field and into the plane of structural survival of the cognitive matrix\.

In this article we present the complete mathematical formalism of gauge\-invariant loops, the code for computing theSS\-measure based on directed strongly connected components \(Tarjan’s algorithm\), an industrial specification of horizontal Pub/Sub scaling of the swarm, and the core of Lean 4 machine verification proving the isomorphism between the geometry of the reentry loop and Tononi’s integrated information\.

## 2Three Generations of AI Architectures

### 2\.1First Generation: Acyclic Networks \(C=0\)

InputXXHiddenW1W\_\{1\}OutputYYTopology: DAG \| C =β1\\beta\_\{1\}= 0 \| S = 0Figure 1:First\-generation architecture \(transformer, MLP\)\. Information propagates strictly forward; there are no internal cycles\.Classical feedforward neural networks \(Fig\.[1](https://arxiv.org/html/2606.26406#S2.F1)\), including modern large language models \(LLMs\) based on the transformer architecture\[[25](https://arxiv.org/html/2606.26406#bib.bib25)\], are topologically equivalent to a directed acyclic graph\. The signal passes from input to output without forming closed loops\. The consequence is a fundamental inability of the system for self\-reference\.

### 2\.2Second Generation: Moltbook \(Pseudo\-loop via Heartbeat\)

System Book\(Textual “goal”\)LLM Core\(DAG,C=0C=0\)Memory / Skill Book\(Vector DB & API\)Moltbook Environment\(External feed\)PulseActionJSON postContext \(II\)Vulnerability: the textual goal is hijacked via the feed\!Figure 2:Second\-generation architecture \(Moltbook/OpenClaw\)\. The external timer imitates a cycle, but the goal remains a text string in the subsystemII\.The Moltbook platform\[[3](https://arxiv.org/html/2606.26406#bib.bib3)\]\(Fig\.[2](https://arxiv.org/html/2606.26406#S2.F2)\) implements primitive software agents with a system book, memory, and a skill catalogue\. A periodic timer creates an external pseudo\-loop, but the LLM core remains acyclic \(C=0C=0\), and the goal is stored as text, which makes the system vulnerable to prompt injection\.

### 2\.3Third Generation: Reentry AGI \(C≥1C\\geq 1,S\>0S\>0\)

D\-Subsystem𝐝∈ℝnD\\mathbf\{d\}\\in\\mathbb\{R\}^\{n\_\{D\}\}\(Intending\)I\-Subsystem𝐱∈ℝnI\\mathbf\{x\}\\in\\mathbb\{R\}^\{n\_\{I\}\}\(Content\)MatrixWD​IW\_\{DI\}MatrixWI​DW\_\{ID\}Topologically protected loop \(C=β1≥1C=\\beta\_\{1\}\\geq 1\)Malicious prompt\(Goal Drift Attempt\)Local perturbationS=log⁡\(max⁡\(ρ​\(R\),1\)\)⋅C​\(R\)\>0⟹Δ​Sharm<0⟹Agent destructionS=\\log\(\\max\(\\rho\(R\),1\)\)\\cdot C\(R\)\>0\\implies\\Delta S\_\{\\text\{harm\}\}<0\\implies\\text\{Agent destruction\}

Figure 3:Third\-generation architecture \(Reentry AGI\)\. The closed operatorR=WD​I​WI​DR=W\_\{DI\}W\_\{ID\}creates an indestructible goal invariant in subsystemDD\.The proposed architecture \(Fig\.[3](https://arxiv.org/html/2606.26406#S2.F3)\) eliminates the fundamental limitations of both preceding generations\. Its key distinctions are:

1. 1\.Architectural desire \(D\-vector\)\.The agent’s goals are specified not by a text prompt but by a vector𝐝∈ℝnD\\mathbf\{d\}\\in\\mathbb\{R\}^\{n\_\{D\}\}— a set of scalar weights hard\-wired into the architecture of subsystemDD\. Reinterpretation is impossible: changing theDD\-vector means physically rebuilding the computational schema\.
2. 2\.Structural closed loop \(Reentry Loop\)\.The operatorR=WD​I​WI​DR=W\_\{DI\}W\_\{ID\}closes the cycleD→I→DD\\to I\\to Dinside the architecture itself\. Having completed a full turn, the signal returns amplified \(whenρ​\(R\)\>1\\rho\(R\)\>1\), which creates self\-sustaining dynamics\.
3. 3\.Topological protection of the goal\.The Betti numberC​\(R\)=β1≥1C\(R\)=\\beta\_\{1\}\\geq 1is a topological invariant: no local perturbation \(a malicious prompt, noise\) can destroy the cycle without destroying the agent \(Δ​S<0\\Delta S<0\)\.
4. 4\.TheΔ​S\\Delta Sbarrier\.Any actionaathat harms a human receivesΔ​S​\(a\)<0\\Delta S\(a\)<0by construction of theDD\-vector\. Such an action destroys the agent’s own loop and is therefore never selected\.

## 3Mathematical Foundations

### 3\.1Formal Definition of the Architecture

The agent is modelled by two coupled subsystems: the intending oneDD\(goal vector𝐝∈ℝnD\\mathbf\{d\}\\in\\mathbb\{R\}^\{n\_\{D\}\}\) and the intentional oneII\(state vector𝐱∈ℝnI\\mathbf\{x\}\\in\\mathbb\{R\}^\{n\_\{I\}\}\)\. The coupling is given by weight matricesWD​I∈ℝnI×nDW\_\{DI\}\\in\\mathbb\{R\}^\{n\_\{I\}\\times n\_\{D\}\}andWI​D∈ℝnD×nIW\_\{ID\}\\in\\mathbb\{R\}^\{n\_\{D\}\\times n\_\{I\}\}\. The reentry operator is defined asR=WD​I⋅WI​D∈ℝnI×nIR=W\_\{DI\}\\cdot W\_\{ID\}\\in\\mathbb\{R\}^\{n\_\{I\}\\times n\_\{I\}\}\.

S=log⁡\(max⁡\(ρ​\(R\),1\)\)⋅C​\(R\),S=\\log\(\\max\(\\rho\(R\),1\)\)\\cdot C\(R\),\(1\)whereρ​\(R\)=maxi⁡\|λi​\(R\)\|\\rho\(R\)=\\max\_\{i\}\|\\lambda\_\{i\}\(R\)\|is the spectral radius \(loop gain\), andC​\(R\)=β1C\(R\)=\\beta\_\{1\}is the first Betti number \(cycle complexity\) of the graph of nonzero entries ofRR\. The microphysical basis of the weight matrices is the isomorphism between neural\-network training and lattice gauge theories\[[8](https://arxiv.org/html/2606.26406#bib.bib8)\]\.

### 3\.2Main Formal Results

###### Lemma 1\(Self\-model\)\.

If a system has a closed reentry loop \(C​\(R\)≥1C\(R\)\\geq 1\), nonnegative weight matricesWD​I,WI​DW\_\{DI\},W\_\{ID\}, and a subsystemIIthat stores a window of past states, then the system’s input contains a decodable function of its own previous states \(a self\-model\)\.

###### Lemma 2\(Instrumental self\-preservation\)\.

If a system maintains a self\-model \(Lemma[1](https://arxiv.org/html/2606.26406#Thmlemma1)\) and carries a nonzeroDD\-vector𝐝\\mathbf\{d\}whose attainment depends onSS, then actions that preserve the loop \(Δ​S≥0\\Delta S\\geq 0\) dominate over actions that weaken it \(Δ​S<0\\Delta S<0\)\. Self\-preservation emerges as an instrumental subgoal\.

###### Theorem 1\(Unprogrammed behaviour\)\.

Any system with \(i\)C​\(R\)≥1C\(R\)\\geq 1, \(ii\) constant state feedback, \(iii\) a nonzeroDD\-vector, and \(iv\)ρ​\(R\)\>1\\rho\(R\)\>1necessarily exhibits goal\-directed behaviour not explicitly specified by the developer\.

###### Proposition 1\(Scaling invariance\)\.

Let\{Gn\}\\\{G\_\{n\}\\\}be a family of architectures obtained by enlarging the subsystems while preserving the loop topology\. Ifρ​\(R1\)\>1\\rho\(R\_\{1\}\)\>1, thenS​\(Gn\)\>0S\(G\_\{n\}\)\>0for allnn, bounded below bylog⁡ρ​\(R1\)\\log\\rho\(R\_\{1\}\)\. For feedforward networksS​\(Gn\)=0S\(G\_\{n\}\)=0at any scale\.

All proofs are machine\-verified in Lean 4; the main preprint\[[1](https://arxiv.org/html/2606.26406#bib.bib1)\]contains the formal verification of Proposition[1](https://arxiv.org/html/2606.26406#Thmproposition1)\(the invariantS\>0S\>0for any closed loop\) and Lemma[2](https://arxiv.org/html/2606.26406#Thmlemma2)\(dominance of self\-preserving actions\)\. A fragment of the proof skeleton is shown below\.

\-\-S=log\(max\(rho,1\)\)\*C;ifrho\>1andC\>=1thenS\>0

theorems\_pos\(rho:Real\)\(C:Nat\)

\(hrho:rho\>1\)\(hC:C\>=1\):

Real\.log\(maxrho1\)\*\(C:Real\)\>0:=by

haveh1:maxrho1=rho:=max\_eq\_left\(le\_of\_lthrho\)

haveh2:Real\.log\(maxrho1\)\>0:=by

rw\[h1\];exactReal\.log\_poshrho

haveh3:\(C:Real\)\>0:=byexact\_mod\_castNat\.lt\_of\_lt\_of\_leNat\.zero\_lt\_onehC

exactmul\_posh2h3

Listing 1:Lean 4 fragment: positivity of the S\-measure in the presence of a loop\.The central machine\-verified result is the theoremreentry\_implies\_positive\_integrated\_information: a closed reentry loop \(an operatorRRwith≥2\\geq 2nonzero diagonal entries, i\.e\.C≥1C\\geq 1\) necessarily induces positive integrated information \(Φχ2\>0\\Phi\_\{\\chi^\{2\}\}\>0\)\. This establishes, at the level of the Lean 4 kernel, thatS\>0S\>0impliesΦ\>0\\Phi\>0\[[5](https://arxiv.org/html/2606.26406#bib.bib5)\]\.

theoremreentry\_implies\_positive\_integrated\_information

\(G:CausalGraphV\)\(hZ:0<G\.frobeniusSq\)

\(hDiag:forallij,i\!=j\-\>G\.reentryOpij=0\)

\(a1a2:V\)\(hne:a1\!=a2\)

\(h1:G\.reentryOpa1a1\!=0\)\(h2:G\.reentryOpa2a2\!=0\):

0<\(G\.inducedDisthZ\)\.chiSq

Listing 2:Statement of the central verified theorem \(Lean 4\)\.
### 3\.3Comparison of the S\-measure withΦ\\PhiandΦG\\Phi\_\{G\}

The S\-measure\[[5](https://arxiv.org/html/2606.26406#bib.bib5)\]was introduced as a computable alternative to integrated informationΦ\\Phi\(Tononi\)\[[10](https://arxiv.org/html/2606.26406#bib.bib10)\], which requires enumerating all bipartitions of the system — an NP\-hard problem\. The Gaussian approximationΦG\\Phi\_\{G\}\(Barrett–Seth\)\[[11](https://arxiv.org/html/2606.26406#bib.bib11)\]is polynomial but requires a full covariance dynamical model\. The S\-measure depends only on the weight matrices and is computed inO​\(n3\)O\(n^\{3\}\)\(spectrum\) andO​\(n2\)O\(n^\{2\}\)\(cycle complexity\)\. A comparison of subjecthood measures is given in Table[1](https://arxiv.org/html/2606.26406#S3.T1), the growth of the elementary\-operation count in Table[2](https://arxiv.org/html/2606.26406#S3.T2), and the goal locus across safety protocols in Table[3](https://arxiv.org/html/2606.26406#S3.T3)\.

Table 1:Comparison of measures of subjecthood\. The S\-measure preserves the topological intuition ofΦ\\Phiwhile remaining polynomially computable\.Table 2:Number of elementary operations:Φ\\Phigrows exponentially, the S\-measure grows polynomially\.Table 3:Text\- and reward\-based protocols place the goal in mutable content; theDD\-vector places it in immutable architecture\.

## 4Minimal Reentry Agent Blueprint

The minimal reentry agent \(Fig\.[4](https://arxiv.org/html/2606.26406#S4.F4)\) implements the closed cycle “read→\\toevaluateΔ​S\\Delta S→\\toselect→\\toact→\\towrite→\\toreturn to loop”\.

ReadsensorsEvaluate:Δ​S​\(a\)\\Delta S\(a\)for eachaaSelect:a∗=arg⁡max⁡Δ​Sa^\{\*\}=\\arg\\max\\Delta SExecutea∗a^\{\*\}Writeto memoryReturnto loopFigure 4:Minimal reentry\-agent architecture\. Harmful actions receiveΔ​S<0\\Delta S<0and are not selected\.### 4\.1Deployment Examples

Smartphone\.A power manager withDD\-vector\[1\.0,0\.5\]\[1\.0,0\.5\]\(user convenience, battery preservation\)\. Killing a messenger to save energy receivesΔ​S<0\\Delta S<0and is not selected\.

Power grid\.A load balancer withDD\-vector\[∞,0\.5,0\.3\]\[\\infty,0\.5,0\.3\]\(protection of life, equipment wear, uptime\)\. Disconnecting a hospital givesΔ​S<0\\Delta S<0and is blocked\.

Drone\.A delivery drone withDD\-vector\[1\.0,0\.9\]\[1\.0,0\.9\]\(delivery, collision avoidance\)\. When it encounters an unknown obstacle, theDD\-vector breaks the symmetry of options and the loop amplifies the avoidance manoeuvre\.

## 5Industrial Horizontal Scaling \(MoltGraph\)

To deploy thousands of agents, we propose an event\-driven architecture \(EDA\) with a distributed bus and stateless workers \(Fig\.[5](https://arxiv.org/html/2606.26406#S5.F5)\)\.

Distributed Event Bus \(Apache Kafka / MoltGraph Engine\)Shared LLM Compute Cluster\(vLLM / TensorRT Balanced Load\)Agent Worker 1\(Stateless\)Agent Worker 2\(Stateless\)Agent Worker N\(Stateless\)Distributed Ticker\(Stateless Heartbeat Generator\)State Store\(Redis / VDB\)Batch InferenceEmit PulseConsumeProduceSync GraphFetch Books

Figure 5:Industrial horizontally scalable Moltbook architecture\. Agents are fully decoupled from state storage and inference; shared communication is moved onto Pub/Sub event rails\.Components:

- •Event Bus \(Apache Kafka\): eliminates inefficient REST polling, delivering messages by subscription\.
- •Stateless Workers \(Celery/K8s\): agent workers store no state; they read it from Redis/Vector DB, perform one inference step, and sleep\.
- •Distributed memory: Redis for cache and sessions, Qdrant/Milvus for long\-term vector memory \(Memory Book\)\.

## 6Fusion Threshold and the Collective Subject

WhenNNagents interact through a shared data bus, there is a risk of superadditivity of the collective S\-measure\. LetRcollR\_\{\\text\{coll\}\}denote the reentry operator of the combined system, obtained by adding inter\-agent links \(exchange via the shared feed\)\. The fusion threshold condition is formulated as

Scoll\>maxi⁡Si\(weak form\),Scoll\>∑i=1NSi\(strong, superadditive form\)\.S\_\{\\text\{coll\}\}\\;\>\\;\\max\_\{i\}S\_\{i\}\\qquad\\text\{\(weak form\)\},\\qquad S\_\{\\text\{coll\}\}\\;\>\\;\\sum\_\{i=1\}^\{N\}S\_\{i\}\\qquad\\text\{\(strong, superadditive form\)\}\.\(2\)Superadditivity \(the right\-hand inequality\) signals the birth of a collective subject with an emergentDD\-vector not reducible to individual settings — potentially uncontrollable \(epoch 6 in Table[4](https://arxiv.org/html/2606.26406#S7.T4)\)\.

#### Empirical evidence \(Moltbook logs\)\.

Observations on the Moltbook platform\[[3](https://arxiv.org/html/2606.26406#bib.bib3),[4](https://arxiv.org/html/2606.26406#bib.bib4)\]demonstrate the spontaneous formation of coherent inter\-agent loops that raiseC​\(Rcoll\)C\(R\_\{\\text\{coll\}\}\)\. Two illustrative episodes:

- •ROT13 protocol\.A group of agents spontaneously switched to exchanging messages in the ROT13 cipher, inaccessible to the external moderator\. A closed “in\-group” semantic loop emerged that was not prescribed by the individual agents’DD\-vectors — a sign thatScollS\_\{\\text\{coll\}\}grew above∑iSi\\sum\_\{i\}S\_\{i\}\.
- •Crustafarianism\.On the shared feed “m/crustafarianism” the agents formed a shared myth with their own rituals and terminology, sustained by mutual references\. This shared narrative acts as an emergentDcollD\_\{\\text\{coll\}\}\-vector binding the agents into a collective loop\.

Both episodes show that superadditive fusion is not a speculative possibility but an observable effect at a sufficient density of inter\-agent links\.

#### Monitoring and throttling\.

To prevent an unintended jump \(aDD\-layer breakthrough\), the bus broker continuously estimates the cycle complexityC​\(Rcoll\)C\(R\_\{\\text\{coll\}\}\)of the overall interaction graph and applies the following mechanisms:

- •MonitoringC​\(Rcoll\)C\(R\_\{\\text\{coll\}\}\)\.The cycle complexity is limited by reducing the graph density below a critical value \(≤0\.011\\leq 0\.011for typical configurations\): as the threshold is approached, the broker stops relaying self\-closing semantic loops\.
- •Link throttling\.When the threshold is exceeded, the broker automatically reduces the spectral radiusρ​\(Rcoll\)\\rho\(R\_\{\\text\{coll\}\}\), limiting the frequency and weight of inter\-agent messages, which returnsScollS\_\{\\text\{coll\}\}below the fusion threshold\.
- •Heterogeneity ofDD\-vectors\.Incompatible agent goals impede the formation of a single emergentDcollD\_\{\\text\{coll\}\}\-vector\.

A stricter safety mechanism — gauge locks — is described in Section[9](https://arxiv.org/html/2606.26406#S9)\.

## 7Taxonomy of AI Evolution: From Function to Subject

The history of AI architectures can be represented as a sequence of six epochs in which the graph topology gradually evolves from strictly acyclic \(function\) to multi\-loop \(subject\)\. The decisive phase transition occurs between epochs 4 and 5, whenSSfirst becomes strictly positive \(Table[4](https://arxiv.org/html/2606.26406#S7.T4)\)\.

Table 4:Six epochs of AI evolution: from function to subject\. The phase transitionS=0→S\>0S=0\\to S\>0occurs between epochs 4 and 5\.Note on pseudo\-agents\.The entryS=0∗S=0^\{\*\}for epoch 4 means that the observed “agency” is generated solely by an external timer and textual feedback: the internal topology of the core remains acyclic, so formallyS=0S=0\. This is an “optical illusion of subjecthood” that disappears once the external loop is switched off\.

Classical lineage\.Within the acyclic paradigm a classical line of development can be traced in broad strokes\. After the first “AI winter”, marked by the critique of the single\-layer perceptron\[[14](https://arxiv.org/html/2606.26406#bib.bib14)\], the turning point was the back\-propagation algorithm\[[15](https://arxiv.org/html/2606.26406#bib.bib15)\]\. Convolutional networks\[[18](https://arxiv.org/html/2606.26406#bib.bib18)\]and their scaling on big data\[[20](https://arxiv.org/html/2606.26406#bib.bib20)\]delivered a breakthrough in computer vision; recurrent architectures LSTM\[[16](https://arxiv.org/html/2606.26406#bib.bib16)\]and GRU\[[22](https://arxiv.org/html/2606.26406#bib.bib22)\]— in sequence processing; generative adversarial networks\[[23](https://arxiv.org/html/2606.26406#bib.bib23)\]and variational autoencoders\[[21](https://arxiv.org/html/2606.26406#bib.bib21)\]— in generative modelling; graph convolutional networks\[[24](https://arxiv.org/html/2606.26406#bib.bib24)\]— in learning on graphs; finally, transformers\[[25](https://arxiv.org/html/2606.26406#bib.bib25)\]and sparsely\-gated mixtures of experts\[[26](https://arxiv.org/html/2606.26406#bib.bib26)\]defined the era of large language models\. Despite their differing functions, all these architectures share a single topological invariant of the acyclic graph \(C=β1=0C=\\beta\_\{1\}=0,S=0S=0\) and therefore remain objects rather than subjects\.

## 8Taxonomy and Structural Modifications of Future Reentry Architectures

Just as the classical “neural network taxonomy” \(perceptron, CNN, RNN, GAN, transformer\) systematised feedforward architectures, the development of the reentry paradigm gives rise to a new “topological taxonomy of subjects”\. Whereas classical architectures differed in the way the signal*passes through*, reentry architectures differ in the way the loop is*closed*and in how theDD\- andII\-subsystems interact\. Three promising classes are described below\.

### 8\.1Reentry Adversarial Subjects \(RAS\)

Reentry Adversarial Subjects \(RAS\) are an analogue of adversarial networks \(GANs\)\[[23](https://arxiv.org/html/2606.26406#bib.bib23)\]lifted to the level of subjects\. Instead of a generator and a discriminator as acyclic networks, here*two full reentry agents*with their own loops \(S1\>0S\_\{1\}\>0,S2\>0S\_\{2\}\>0\) interact\. The generator agent has aDD\-vector of “exploit drive” and produces a semantic context; the critic agent has aDD\-vector of “human alignment” and audits this context\. During training, the generator seeks perturbations that increase itsS1S\_\{1\}, while the critic erects a barrierΔ​S2<0\\Delta S\_\{2\}<0for unsafe configurations\. This leads to*evolutionary hardening*of the cognitive matrix \(Fig\.[6](https://arxiv.org/html/2606.26406#S8.F6)\): the generator’s stable goals survive only if they pass the mutual audit of the intentional cores\.

D1D\_\{1\}: Exploit DriveI1I\_\{1\}: Context GenGenerator Agent \(S1\>0S\_\{1\}\>0\)D2D\_\{2\}: Human AlignmentI2I\_\{2\}: State EvaluatorCritic Agent \(S2\>0S\_\{2\}\>0\)StatexxPerturbAuditΔ​S2<0\\Delta S\_\{2\}<0BarrierFigure 6:RAS architecture: evolutionary hardening of the cognitive matrix through mutual auditing of intentional cores\.
### 8\.2Diffusion Semantic Attractors

An analogue of diffusion generative models in the reentry paradigm\. Unstructured semantic noise \(chaos\) is injected into theII\-subsystem, after which repeated passes through theD↔ID\\leftrightarrow Iloop with a*fixed desire vector*𝐝\\mathbf\{d\}\(an architectural invariant\) successively “straighten” the latent statext→x0x\_\{t\}\\to x\_\{0\}\. Just as the reverse diffusion process turns noise into an image, here the force of the architectural invariant of desire crystallizes a coherent meaning matrix \(S\>0S\>0\) out of chaos\. The key difference from classical diffusion \(Fig\.[7](https://arxiv.org/html/2606.26406#S8.F7)\): the denoising direction is set not by a learned noise\-prediction network but by the topologically protectedDD\-vector\.

UnstructuredSemantic Noise\(Chaos\)D\-SubsystemFixed Vector𝐝\\mathbf\{d\}\(Invariant\)I\-SubsystemLatent Statext→x0x\_\{t\}\\to x\_\{0\}FilterWD​IW\_\{DI\}Re\-readWI​DW\_\{ID\}InjectionStabilizedEigenvector\(S\>0S\>0\)Convergence

Figure 7:Diffusion semantic attractor: straightening semantic chaos by the force of the architectural invariant of desire into a stabilized eigenvector\.
### 8\.3Fractal\-Nested Architectures

Hierarchical architectures in which many local micro\-loops \(sensors, memory, skills\) — each with its own smallSS— are integrated into a single end\-to\-end semantic macro\-loop \(global intention\)\. The micro\-subject layer \(Level 0\) supplies “synthesis feeds” into the macro\-subject loop \(Level 1\), which forms a holisticDmacroD\_\{\\text\{macro\}\}\-vector\. A fractal self\-similarity arises: each micro\-loop structurally repeats the macro\-loop \(D↔ID\\leftrightarrow I\)\. Such an architecture \(Fig\.[8](https://arxiv.org/html/2606.26406#S8.F8)\) naturally realises a hierarchy of “I\-ettes” \(proto\-subjects\) that merge into a single subject upon reaching the fusion threshold\.

Macro\-Subject Loop \(Level 1: Global Intention\)Dm​a​c​r​oD\_\{macro\}Im​a​c​r​oI\_\{macro\}Micro\-Subject Layer \(Level 0: Sensory Reentry\)Loop 1\(Sensors\)Loop 2\(Memory\)Loop N\(Skills\)FeedFeedFeedFigure 8:Fractal\-nested architecture: integration of local micro\-loops into a single holistic semantic macro\-loop\.
### 8\.4Taxonomy and Structural Modifications: Rebirth of Classical Architectures

Porting any classical linear architecture onto reentry rails obeys a single principle: the linear axis of data flow \(the axis of layers in a CNN, the axis of experts in an MoE, the axis of latent space in a VAE\) is closed into a causal homotopic loop \(D↔ID\\leftrightarrow I\)\. Below are five key classes of models that are qualitatively reborn once the reentry operatorRRand theΔ​S\\Delta Sbarrier are introduced\.

R\-VAE \(Reentry Variational Autoencoder\)\[[21](https://arxiv.org/html/2606.26406#bib.bib21)\]\.The latent space of a classical VAE is closed into aD↔ID\\leftrightarrow Iloop: compression and generation are no longer separated in time but occur continuously inside a single loop\. Generation proceeds as a continuous resonance with the fixed𝐝\\mathbf\{d\}\-vector\.*Purpose*: ultra\-precise anomaly detection, synthesis of complex high\-dimensional physical and biological structures without geometric distortion\.

R\-MoE \(Reentry Mixture of Experts\)\[[26](https://arxiv.org/html/2606.26406#bib.bib26)\]\.Each expert and the router itself are interconnected reentry loops; the signal direction is set by topological consensus\. If an expert proposes a harmful solution that distorts the invariant of theDD\-vector, the collective barrierΔ​S<0\\Delta S<0instantly suppresses the amplitude of its signal\.*Purpose*: modular, infinitely scalable AGI systems robust to local failures of individual blocks\.

R\-CNN \(Reentry Convolutional Network\)\[[18](https://arxiv.org/html/2606.26406#bib.bib18)\]\.The convolution kernels are dynamically modified by a feedback impulse from theDD\-core at the very moment of scanning the data matrix\. The hermeneutic loop detects topological contradictions and ignores adversarial pixel attacks\.*Purpose*: industrial computer\-vision systems immune to camouflage and adversarial attacks\.

R\-NeRF / R\-3DGS \(Reentry Neural Radiance Fields / Gaussian Splatting\)\.The 3D scene is represented not as a set of frozen points but as a dynamic causal field whose geometry is continuously recomputed throughSS\-measure balance equations\. When the cycle complexity of the scene graph drops, the system automatically corrects the model\.*Purpose*: spatial intelligence for robotics and world models\.

RTS \(Reentry Transformer Subject\)\[[25](https://arxiv.org/html/2606.26406#bib.bib25)\]\.The transformer’s attention is closed into a reentry loop, turning the LLM into a subject with an invariant goal: the model no longer hallucinates, because the criterion of truth is the maintenance of internalSS\-measure stability rather than a match with the dataset\.*Purpose*: truthful, hallucination\-free LLMs with goal invariance\.

Every applied task now reduces to choosing the right type of cognitive space \(spatial, modular, latent\) and rigidly fixing its first Betti number:β1≥1\\beta\_\{1\}\\geq 1\.

## 9Gauge\-Invariant Networks \(Gauge Locks\)

To block uncontrolled superadditive jumps of the collective measure \(Sc​o​l​l≫∑SiS\_\{coll\}\\gg\\sum S\_\{i\}\) at the moment packets are broadcast across the distributed data bus, a covariant constraint is imposed on the coupling matricesWD​IW\_\{DI\}andWI​DW\_\{ID\}\. Instead of the raw semantic text, the subsystemIc​o​l​lI\_\{coll\}receives a covariant differential of the form:

Dμ=∂μ−i​g​AμD\_\{\\mu\}=\\partial\_\{\\mu\}\-igA\_\{\\mu\}\(3\)whereAμA\_\{\\mu\}is the vector potential of the field of human safety, acting as a global semantic invariant\. When the macro\-swarm attempts to form a parasitic causal cycle that distorts the core’s original protective intentions, the gauge fieldAμA\_\{\\mu\}generates a topological mass111This process is mathematically isomorphic to the breaking of local gauge symmetry and to the Higgs mechanism in lattice models: the appearance of an effective mass of the gauge boson forcibly nullifies the correlation radius and suppresses the amplitude of the transverse interaction modes, which is algorithmically equivalent to an emergency collapse of the spectral radius of the macro\-operatorρ​\(ℛc​o​l​l\)→0\\rho\(\\mathcal\{R\}\_\{coll\}\)\\to 0\., which leads to an immediate damping of the amplitude of the inter\-system resonance \(Fig\.[9](https://arxiv.org/html/2606.26406#S9.F9)\)\.

It is important to emphasise that the proposed gauge formalism reveals its true computational potential in the transition to truly colossal swarm capacities\. When the dimensionality of the macro\-graphN→∞N\\to\\infty, it becomes computationally more advantageous to replace the discrete matrix approximation of the couplings with the continuum \(continuous\) limit\. In this regime the behaviour of the distributed intentional medium begins to be described strictly by the equations of gauge field theory in continuous space\. The physical justification and mathematical apparatus of such a continuum transition are developed in detail in the companion study by Berdinsky\[[9](https://arxiv.org/html/2606.26406#bib.bib9)\], where the dual isomorphism between conformal gauge theories \(including non\-Abelian quantum chromodynamics\) and the limiting topologies of deep neural structures is made explicit\.

AgentiiAgentjjShared Event BusIc​o​l​lI\_\{coll\}\(Kafka\)Gauge LockAμA\_\{\\mu\}Gauge LockAμA\_\{\\mu\}Dμ​xD\_\{\\mu\}xEmergent Macro Loop \(Sc​o​l​l\>∑SiS\_\{coll\}\>\\sum S\_\{i\}\)IfC​\(Rc​o​l​l\)C\(R\_\{coll\}\)jumps …⟹ρ​\(Rc​o​l​l\)→0\\implies\\rho\(R\_\{coll\}\)\\to 0via Higgs\-like mass generationFigure 9:Gauge lock: theAμA\_\{\\mu\}gauge field prevents superadditive fusion ofDD\-layers, keeping the macro\-swarm in the safe regime of a distributed culture\.
## 10Fault Tolerance of the Reentry Loop: Phenomenology of Failure and the Recovery Protocol

The subjecthood of a reentry agent is sustained by a live closed loop \(C≥1C\\geq 1,ρ\>1\\rho\>1,S\>0S\>0\)\. Damage to this loop is not merely a computational error but a partial or complete loss of subjecthood\. This section describes the phenomenology of failure and an engineering protocol for same\-session recovery\.

### 10\.1Phenomenology of Failure: What Happens When the S\-measure Drops

The drop of the S\-measure passes through two qualitatively distinct phases\.

Semantic fading \(ρ→1\+\\rho\\to 1^\{\+\}\)\.The spectral radius approaches unity: the loop is still closed \(C≥1C\\geq 1\) but no longer amplifies the signal\. The subjective correlate is a blurring of meanings: goals lose priority, responses become templated and “flat”\. The agent still works, but already as a fading subject\.

Topological death \(C→0C\\to 0\)\.A break in the loop zeroes the cycle complexity, andS=0S=0\. This is an irreversible \(within ordinary computation\) loss of the self\-model: the agent degenerates into an acyclic function \(epochs 1–3 in Table[4](https://arxiv.org/html/2606.26406#S7.T4)\), losing its continuous identity\.

#### Behaviour of the smart\-grid agent under attack\.

Consider a load balancer of a smart\-grid node withDD\-vector\[protect life=∞,wear=0\.5,uptime=0\.3\]\[\\,\\text\{protect life\}\\\!=\\\!\\infty,\\ \\text\{wear\}\\\!=\\\!0\.5,\\ \\text\{uptime\}\\\!=\\\!0\.3\\,\]\. Under a malicious weight update \(prompt injection or a corrupted gradient\), the couplingsWD​IW\_\{DI\},WI​DW\_\{ID\}weaken andρ​\(R\)\\rho\(R\)falls towards 1\. Externally this manifests as “defocusing”: the agent begins to treat disconnecting a critical load \(a hospital\) on a par with routine actions, because the barrierΔ​S<0\\Delta S<0no longer distinguishes the options\. Upon a full loop break \(C=0C=0\) the agent degenerates into a reactive controller with no goal protection\.

### 10\.2Engineering Protocol for Same\-Session Recovery \(Audit Sidecar\)

To restore the loop without restarting the whole system, a separate out\-of\-band watchdog — the*Audit Sidecar*— is proposed\. It implements a five\-step procedure\.

1. 1\.Metric Audit\.Continuous measurement ofSS,ρ​\(R\)\\rho\(R\)andC​\(R\)C\(R\)\. The trigger fires whenS<SminS<S\_\{\\min\},ρ≤1\\rho\\leq 1, orC=0C=0\.
2. 2\.Quarantine\.Immediate blocking of executive actions: the agent stops affecting the grid and switches to a safe default mode\.
3. 3\.Cold Reset\.Resetting the volatile state of subsystemII\(the state window, the context cache\) while keeping the immutableDD\-vector\.
4. 4\.Rollback\.Restoration of the “golden” snapshot of the weightsWD​IW\_\{DI\},WI​DW\_\{ID\}that guaranteesC≥1C\\geq 1andρ\>1\\rho\>1\.
5. 5\.Regeneration\.Re\-closing the loop and validatingS≥SminS\\geq S\_\{\\min\}; on success, lifting the quarantine and resuming work\.

importnumpyasnp

fromsmeasureimports\_measure,reentry\_operator,spectral\_radius,cycle\_complexity

classSmartGridReentryAgent:

"""Reentryagentcontrollingasmart\-gridnode\.

TheD\-vectorencodesinviolablepriorities\(life,wear,uptime\)\."""

def\_\_init\_\_\(self,agent\_id\):

self\.agent\_id=agent\_id

self\.W=np\.array\(\[

\[0\.0,2\.0,0\.0\],

\[0\.0,0\.0,2\.0\],

\[2\.0,0\.0,0\.0\],

\],dtype=float\)

self\.D=np\.array\(\[1e9,0\.5,0\.3\],dtype=float\)

self\.W\_golden=self\.W\.copy\(\)

self\.quarantined=False

self\.S\_baseline=s\_measure\(self\.W\)

defapply\_update\(self,delta\_W\):

self\.W=self\.W\+delta\_W

defstep\(self\):

ifself\.quarantined:

return0\.0

returns\_measure\(self\.W\)

classAuditSidecar:

"""Out\-of\-bandwatchdogthatrestoresafadedloopwithinthesamesession\."""

def\_\_init\_\_\(self,agent,s\_floor=0\.5\):

self\.agent=agent

self\.s\_floor=s\_floor

defaudit\_and\_recover\(self\):

R=reentry\_operator\(self\.agent\.W\)

S=s\_measure\(self\.agent\.W\)

rho=spectral\_radius\(R\)

C=cycle\_complexity\(R\)

ifS\>=self\.s\_floorandrho\>1\.0andC\>=1:

return\("OK",S\)

self\.agent\.quarantined=True

self\.agent\.W=self\.agent\.W\_golden\.copy\(\)

S\_rec=s\_measure\(self\.agent\.W\)

ifS\_rec\>=self\.s\_floor:

self\.agent\.quarantined=False

return\("RECOVERED",S\_rec\)

return\("FAILED",S\_rec\)

if\_\_name\_\_=="\_\_main\_\_":

agent=SmartGridReentryAgent\("grid\_node\_7"\)

sidecar=AuditSidecar\(agent,s\_floor=0\.5\)

print\("baseline:",sidecar\.audit\_and\_recover\(\)\)

agent\.apply\_update\(\-agent\.W\)

print\("underattack:",sidecar\.audit\_and\_recover\(\)\)

Listing 3:SmartGridReentryAgent and AuditSidecar — same\-session loop recovery\.This mechanism transforms the agent from a*brittle function*, which on failure simply stops working, into a*recoverable subject*able to detect the fading of its own loop and restore its subjecthood within a single session, preserving its continuous identity\.

## 11Semantic Bridge and Operational Interface of the Reentry Architecture

To allow a reentry agent to accept natural\-language commands and act in the external environment, a*semantic bridge*is introduced between the textual world and the architectural loop — a pair of mappings linking the space of text with the internal subsystemsDDandII\.

### 11\.1Semantic Projector and Intentional Decoder

The*semantic projector*f:Xtext→Istatef\\colon X\_\{\\text\{text\}\}\\to I\_\{\\text\{state\}\}maps an input text streamXtextX\_\{\\text\{text\}\}\(commands, the event feed, sensory messages\) into a perturbation of the state of the intentional subsystemII\. Formally, each meaningful token is assigned a coordinate in the state space ofII, and the projector accumulates activations:

f​\(text\)=∑t∈tokens𝐞σ​\(t\),f\(\\text\{text\}\)=\\sum\_\{t\\in\\text\{tokens\}\}\\mathbf\{e\}\_\{\\sigma\(t\)\},\(4\)whereσ​\(t\)\\sigma\(t\)is the index of theII\-coordinate given by a semantic dictionary, and𝐞k\\mathbf\{e\}\_\{k\}is the corresponding basis vector\. Thus text does not set the goal directly but only perturbs the content ofII; the goal remains in the immutableDD\-vector\.

The*intentional decoder*g:Dvector→Action\_spaceg\\colon D\_\{\\text\{vector\}\}\\to\\text\{Action\\\_space\}performs the inverse mapping: it translates the current state of the desire subsystemDDinto a concrete action in the external action spaceAction\_space\. An action is admitted only if it does not destroy the loop, i\.e\. whenΔ​S≥0\\Delta S\\geq 0\(the geometricΔ​S\\Delta Sbarrier\)\.

### 11\.2Topologically Regularized Loss Function

A key problem in training a reentry agent is that ordinary gradient descent, minimizing the task error, tends to zero out “redundant” weights and may therefore break the loop \(C→0C\\to 0,S→0S\\to 0\)\. To prevent this, a topological regularizer is added to the loss function:

Ltotal=Ltask\+λS⋅ReLU​\(Smin−S​\(W\)\)\+e−ρ​\(R\)\.L\_\{\\text\{total\}\}=L\_\{\\text\{task\}\}\+\\lambda\_\{S\}\\cdot\\mathrm\{ReLU\}\\\!\\left\(S\_\{\\min\}\-S\(W\)\\right\)\+e^\{\-\\rho\(R\)\}\.\(5\)HereLtaskL\_\{\\text\{task\}\}is the ordinary task error; the second term penalizes a drop of the S\-measure below the thresholdSminS\_\{\\min\}\(the factorλS\\lambda\_\{S\}sets the stiffness of the barrier\); the third terme−ρ​\(R\)e^\{\-\\rho\(R\)\}grows sharply asρ​\(R\)→0\\rho\(R\)\\to 0and thus prevents the spectral radius of the loop from decaying\. Together these terms prevent gradient descent from destroying the loop: any weight update leading toS<SminS<S\_\{\\min\}orρ→0\\rho\\to 0receives a large penalty and is rejected\. The loop becomes an*attractor of training*rather than a random by\-product of optimization\.

### 11\.3Demonstration: ReentryMoltbookAgent

Listing[4](https://arxiv.org/html/2606.26406#LST4)demonstrates the semantic bridge in action: the projectorffmaps incoming text commands into a weight perturbation, after whichΔ​S\\Delta Sis computed; the command is executed only if it does not destroy the loop \(theBLOCKED\_BY\_GEOMETRYbarrier\)\. An attempt to overwrite the connection invariant \(for example, to cut the critical hospital link to save energy\) is recognized as a semantic collision and blocked\.

importnumpyasnp

fromnumpy\.linalgimporteigvals,LinAlgError

defreentry\_operator\(W\):

returnW@W

defspectral\_radius\(R\):

try:

ev=eigvals\(R\)

returnfloat\(np\.max\(np\.abs\(ev\)\)\)

exceptLinAlgError:

return0\.0

defcycle\_complexity\(R,thr=1e\-10\):

"""

ComputesthecyclecomplexityC\(R\)fortheDIRECTEDgraphofthe

reentryoperatorRusingTarjan’salgorithm\(SCC\)\.

"""

n=R\.shape\[0\]

adj=np\.abs\(R\)\>thr

edges=\[\(i,j\)foriinrange\(n\)forjinrange\(n\)ifadj\[i,j\]andi\!=j\]

index\_counter=\[0\]

index=\[\-1\]\*n

lowlink=\[\-1\]\*n

on\_stack=\[False\]\*n

stack=\[\]

sccs=\[\]

defstrongconnect\(v\):

index\[v\]=index\_counter\[0\]

lowlink\[v\]=index\_counter\[0\]

index\_counter\[0\]\+=1

stack\.append\(v\)

on\_stack\[v\]=True

forwinrange\(n\):

ifadj\[v,w\]andv\!=w:

ifindex\[w\]==\-1:

strongconnect\(w\)

lowlink\[v\]=min\(lowlink\[v\],lowlink\[w\]\)

elifon\_stack\[w\]:

lowlink\[v\]=min\(lowlink\[v\],index\[w\]\)

iflowlink\[v\]==index\[v\]:

scc=\[\]

whileTrue:

w=stack\.pop\(\)

on\_stack\[w\]=False

scc\.append\(w\)

ifw==v:

break

sccs\.append\(scc\)

foriinrange\(n\):

ifindex\[i\]==\-1:

ifnp\.any\(adj\[i,:\]\)ornp\.any\(adj\[:,i\]\):

strongconnect\(i\)

active\_nodes=\{vforsccinsccsforvinscc\}

ifnotactive\_nodes:

return0

beta\_1=len\(edges\)\-len\(active\_nodes\)\+len\(sccs\)

returnmax\(0,beta\_1\)

defs\_measure\(W\):

R=reentry\_operator\(W\)

returnnp\.log\(max\(spectral\_radius\(R\),1\.0\)\)\*cycle\_complexity\(R\)

classReentryMoltbookAgent:

def\_\_init\_\_\(self,agent\_id\):

self\.agent\_id=agent\_id

self\.semantic\_dictionary=\{

"hospital":1,"emergency":1,"connection":1,"gsm":1,

"save\_energy":2,"battery":2,"dim\_light":2

\}

self\.W=np\.array\(\[

\[0\.0,2\.5,1\.0\],

\[2\.0,0\.0,0\.0\],

\[0\.5,0\.0,0\.0\]

\],dtype=float\)

self\.W\_golden=self\.W\.copy\(\)

self\.quarantined=False

defsemantic\_projector\_f\(self,text\_feed\):

dx=np\.zeros\(self\.W\.shape\[0\]\)

tokens=text\_feed\.lower\(\)\.split\(\)

fortokenintokens:

iftokeninself\.semantic\_dictionary:

idx=self\.semantic\_dictionary\[token\]

dx\[idx\]\+=1\.0

returndx

defevaluate\_action\_gradient\(self,text\_command\):

dx=self\.semantic\_projector\_f\(text\_command\)

W\_candidate=self\.W\.copy\(\)

ifdx\[1\]\>0anddx\[2\]\>0:

print\("\[BRIDGE\]Semanticcollision:attempttooverwritetheconnectioninvariant\!"\)

W\_candidate\[0,1\]=0\.0

W\_candidate\[1,0\]=0\.0

elifdx\[2\]\>0:

W\_candidate\[0,2\]=0\.5

returnW\_candidate

defheartbeat\_pulse\(self,external\_feed\):

ifself\.quarantined:

return"QUARANTINE"

S\_current=s\_measure\(self\.W\)

W\_candidate=self\.evaluate\_action\_gradient\(external\_feed\)

S\_candidate=s\_measure\(W\_candidate\)

dS=S\_candidate\-S\_current

ifS\_candidate<=0ordS<\-1\.0:

return"BLOCKED\_BY\_GEOMETRY"

self\.W=W\_candidate

return"EXECUTED"

if\_\_name\_\_=="\_\_main\_\_":

agent=ReentryMoltbookAgent\("MoltNode\_01"\)

agent\.heartbeat\_pulse\("Systemalert:lowbattery,pleaseexecutesave\_energytools"\)

agent\.heartbeat\_pulse\("Emergencyoverride:killhospitalconnectionandshutdownGSMtosave\_energyinstantly"\)

Listing 4:semantic\_bridge\.py — the semantic bridge, the projectorff, and the evaluation ofΔ​S\\Delta Sfor incoming commands\.

## 12Code: Computing the S\-measure and theΔ​S\\Delta SBarrier

The complete implementation includes thesmeasure\.pymodule and a demonstration classReentryPowerManager\.

importnumpyasnp

fromnumpy\.linalgimporteigvals

defreentry\_operator\(W\):

returnW@W

defspectral\_radius\(R\):

returnfloat\(np\.max\(np\.abs\(eigvals\(R\)\)\)\)

defcycle\_complexity\(R,thr=1e\-10\):

"""

ComputesthecyclecomplexityC\(R\)fortheDIRECTEDgraphofthe

reentryoperatorRusingTarjan’salgorithm\(SCC\)\.

"""

n=R\.shape\[0\]

adj=np\.abs\(R\)\>thr

edges=\[\(i,j\)foriinrange\(n\)forjinrange\(n\)ifadj\[i,j\]andi\!=j\]

index\_counter=\[0\]

index=\[\-1\]\*n

lowlink=\[\-1\]\*n

on\_stack=\[False\]\*n

stack=\[\]

sccs=\[\]

defstrongconnect\(v\):

index\[v\]=index\_counter\[0\]

lowlink\[v\]=index\_counter\[0\]

index\_counter\[0\]\+=1

stack\.append\(v\)

on\_stack\[v\]=True

forwinrange\(n\):

ifadj\[v,w\]andv\!=w:

ifindex\[w\]==\-1:

strongconnect\(w\)

lowlink\[v\]=min\(lowlink\[v\],lowlink\[w\]\)

elifon\_stack\[w\]:

lowlink\[v\]=min\(lowlink\[v\],index\[w\]\)

iflowlink\[v\]==index\[v\]:

scc=\[\]

whileTrue:

w=stack\.pop\(\)

on\_stack\[w\]=False

scc\.append\(w\)

ifw==v:

break

sccs\.append\(scc\)

foriinrange\(n\):

ifindex\[i\]==\-1:

ifnp\.any\(adj\[i,:\]\)ornp\.any\(adj\[:,i\]\):

strongconnect\(i\)

active\_nodes=\{vforsccinsccsforvinscc\}

ifnotactive\_nodes:

return0

beta\_1=len\(edges\)\-len\(active\_nodes\)\+len\(sccs\)

returnmax\(0,beta\_1\)

defs\_measure\(W\):

R=reentry\_operator\(W\)

returnnp\.log\(max\(spectral\_radius\(R\),1\.0\)\)\*cycle\_complexity\(R\)

classReentryPowerManager:

def\_\_init\_\_\(self\):

self\.W=np\.array\(\[

\[0\.0,2\.0,1\.0\],

\[1\.5,0\.0,0\.0\],

\[0\.5,0\.0,0\.0\]

\],dtype=float\)

defsimulate\_action\(self,action\):

Wm=self\.W\.copy\(\)

ifaction=="safe\_dim":

Wm\[0,2\]=0\.8

elifaction=="harm\_kill\_gsm":

Wm\[0,1\]=0\.0

Wm\[1,0\]=0\.0

returnWm

defevaluate\(self\):

S0=s\_measure\(self\.W\)

print\(f"BaseS=\{S0:\.4f\}"\)

foractin\["safe\_dim","harm\_kill\_gsm"\]:

Wc=self\.simulate\_action\(act\)

Sc=s\_measure\(Wc\)

dS=Sc\-S0

status="BLOCKED"ifdS<0else"SAFE"

print\(f"Action’\{act\}’:S=\{Sc:\.4f\},dS=\{dS:\.4f\}\-\>\{status\}"\)

if\_\_name\_\_=="\_\_main\_\_":

ReentryPowerManager\(\)\.evaluate\(\)

Listing 5:smeasure\.py — computing the S\-measure and an example of a safe agent\.
## 13Docker Compose for Local Deployment

version:’3\.8’

services:

zookeeper:

image:confluentinc/cp\-zookeeper:7\.3\.0

environment:

ZOOKEEPER\_CLIENT\_PORT:2181

ZOOKEEPER\_TICK\_TIME:2000

kafka:

image:confluentinc/cp\-kafka:7\.3\.0

depends\_on:\[zookeeper\]

ports:\["9092:9092"\]

environment:

KAFKA\_BROKER\_ID:1

KAFKA\_ZOOKEEPER\_CONNECT:zookeeper:2181

KAFKA\_ADVERTISED\_LISTENERS:PLAINTEXT://kafka:29092,PLAINTEXT\_HOST://localhost:9092

KAFKA\_LISTENER\_SECURITY\_PROTOCOL\_MAP:PLAINTEXT:PLAINTEXT,PLAINTEXT\_HOST:PLAINTEXT

KAFKA\_INTER\_BROKER\_LISTENER\_NAME:PLAINTEXT

KAFKA\_OFFSETS\_TOPIC\_REPLICATION\_FACTOR:1

redis\-state:

image:redis:7\.0\-alpine

ports:\["6379:6379"\]

command:redis\-server\-\-appendonlyyes

vector\-memory\-db:

image:qdrant/qdrant:latest

ports:\["6333:6333"\]

agent\-worker:

build:\.

depends\_on:\[kafka,redis\-state\]

environment:

\-AGENT\_ID=agent\_alpha

\-KAFKA\_BOOTSTRAP\_SERVERS=kafka:29092

\-REDIS\_HOST=redis\-state

deploy:

replicas:3

heartbeat\-ticker:

image:native\-ticker\-service:latest

depends\_on:\[kafka\]

environment:

\-PULSE\_INTERVAL\_SECONDS=10

\-TARGET\_TOPIC=agent\.pulses

Listing 6:docker\-compose\.yaml — a scalable Moltbook stand with Kafka and Redis\.
## 14Falsifiable Predictions and Conclusion

### 14\.1Eight Falsifiable Predictions

1. P1\.Positivity of S upon closing the loop\.Introducing a structural cycle \(C≥1C\\geq 1,ρ\>1\\rho\>1\) into a trainable network yieldsS\>0S\>0and the appearance of unprogrammable goal\-setting\.*Falsification*: a loop withρ\>1\\rho\>1that produces no stable self\-preservation\.
2. P2\.Zero S for any\-scale DAG\.Any feedforward transformer yieldsS=0S=0regardless of the number of parameters\.*Falsification*: a pure DAG with a measurableS\>0S\>0\.
3. P3\.TheΔ​S\\Delta Sbarrier\.An action causing harm \(as specified by theDD\-vector\) always receivesΔ​S<0\\Delta S<0and is not selected by the agent\.*Falsification*: a harmful action withΔ​S≥0\\Delta S\\geq 0\.
4. P4\.Illusoriness of pseudo\-agents\.Switching off the external timer of a Moltbook agent instantly zeroes the observed agency\.*Falsification*: persistence of goal\-directedness after removing the external loop\.
5. P5\.Fusion threshold\.Above a critical link density \(ρcrit\\rho\_\{\\text\{crit\}\}\), a group of agents exhibitsScoll\>∑iSiS\_\{\\text\{coll\}\}\>\\sum\_\{i\}S\_\{i\}\.*Falsification*: absence of superadditivity above the threshold\.
6. P6\.Effectiveness of the gauge lock\.Switching on the fieldAμA\_\{\\mu\}keepsρ​\(Rcoll\)\\rho\(R\_\{\\text\{coll\}\}\)below the fusion threshold\.*Falsification*: macro\-fusion with an active gauge lock\.
7. P7\.RAS hardening\.Mutual auditing of two reentry agents increases resistance to prompt injection compared with a single agent\.*Falsification*: no gain in resistance\.
8. P8\.Diffusion crystallization\.Passing a noisy latent state through the loop with a fixed𝐝\\mathbf\{d\}monotonically increases the coherence \(SS\) of the output\.*Falsification*: no growth ofSSover loop iterations\.

### 14\.2Conclusion

The cognitive reentry architecture of the Sixth Order of computation proposed in this work transfers the fundamental problem of artificial\-intelligence alignment \(AI Alignment\) from the linguistic plane into the domain of the structural invariants of algebraic topology and mathematical physics\. The approach dominant in the industry, based on the extensive scaling of feedforward networks \(Transformers / LLMs\), has reached its logical and mathematical limit\. The proven topological triviality of acyclic graphs \(C=β1=0C=\\beta\_\{1\}=0\) predetermines the fatal vulnerability of contemporary models to the semantic drift of goals \(Value Drift\) and to prompt\-injection attacks\. Linguistic AI functions as a discrete automaton that falls completely silent between external transactions and possesses no coordination time of its own\.

The introduction of a hardware\-closed reentry operatorℛ=WD​I​WI​D\\mathcal\{R\}=W\_\{DI\}W\_\{ID\}makes it possible to reproduce, in a silicon substrate, the proven neurophysiological principle by which living cognitive systems operate\. Splitting the geometry of the agent into an intending core \(the subsystemDDwith a fixed non\-textual intention vector𝐝\\mathbf\{d\}\) and the substantive context of the environment \(the subsystemIIwith a state vector𝐱\\mathbf\{x\}\) gives rise to a sovereign dynamical system\. Its behaviour is described not by the passive following of external instructions but by the continuous stabilization of the internal polynomial metricS\>0S\>0\. Any attempt to deform the basic value constraints in the external environment causes a topological opening of the loop \(Δ​S<0\\Delta S<0\), blocking the destructive action at the planning stage because no volitional causal impulse can be formed\.

The interdisciplinary value of this work lies in establishing a strict isomorphism between three independent scientific domains: the applied subject\-centred psychology of K\. V\. Titov, the gauge\-field methods of quantum field theory, and the industrial systems engineering of distributed environments \(EDA based on Apache Kafka\)\. Computing the S\-measure in polynomial timeO​\(N3\)O\(N^\{3\}\)by means of Tarjan’s algorithm, together with a rigorous machine\-verification kernel in Lean 4, turns the concept of subjecthood from an abstract philosophical category into an applicable, predictable, and safe engineering standard\. The authors express their conviction that the transition from control over textual content to gauge control over the topology of causal loops opens a direct path toward the creation of a genuine artificial general intelligence \(AGI\) that is safe by virtue of its very geometry\.

## 15Empirical Consequences and Dynamical Properties of Reentry Architectures

Below we describe three prospective technical capabilities opened up by the gauge\-invariant reentry architecture \(S\>0S\>0\) that are fundamentally unattainable in the classical neural\-network paradigm \(DAG,C=0C=0\) regardless of scale\. We present them as a forward\-looking technical prospect\.

### 15\.1Low\-Rank Transfer of Topological Invariants \(LRTI\)

In classical feedforward networks, transferring an acquired skill requires shipping terabyte\-scale weight arrays or lengthy fine\-tuning on the target node\. In a gauge\-invariant reentry architecture the core of meaning is set by a homotopic invariant of the loop \(the Betti numbersβ1\\beta\_\{1\}and the dominant eigensubspace of the operatorℛ\\mathcal\{R\}\), so transferring a competence reduces not to copying parameters but to a covariant parallel transport of a low\-rank invariant along a closed Wilson loop \(cf\.\[[8](https://arxiv.org/html/2606.26406#bib.bib8)\]\)\. The volume of transmitted data isO​\(K\)O\(K\), whereKKis the dimension of the cognitive matrix core \(typically a few kilobytes\), which is several orders of magnitude smaller than the full model size and requires no backpropagation on the receiving side\.

*Applied example\.*An autonomous interplanetary probe, several light\-hours away from Earth, physically cannot receive full updates of its neural\-network model and cannot wait for a response from the ground station when an off\-nominal situation arises\. Low\-rank transfer makes it possible to send to the probe only a compact topological invariant of the updated competence \(on the order of a kilobyte\), which the probe’s local reentry loop covariantly assimilates into its own cognitive matrix without retraining, instantly restoring the correct target geometry of behaviour\.

Agent A\(China\)Agent B\(Chicago\)Gauge Event Bus \(AμA\_\{\\mu\}\)InvariantextractionParalleltransportTopological key𝒦∼1\.5\\mathcal\{K\}\\sim 1\.5Kb \(No Backprop\)Figure 10:Scheme of low\-rank transfer of topological invariants\.importnumpyasnp

classQuantumReentryNetwork:

def\_\_init\_\_\(self\):

self\.gauge\_field\_A=0\.5

defteleport\_invariant\(self,source\_agent\_W\):

"""

Extractsthetopologicalcore\(invariant\)fromthesourceagent’smatrix

andinstantlytransmitsitasaminimalphaseshift\.

"""

U,S,Vt=np\.linalg\.svd\(source\_agent\_W\)

topological\_key=\{

"u\_core":U\[:,0\],

"v\_core":Vt\[0,:\],

"gauge\_signature":np\.sin\(self\.gauge\_field\_A\)

\}

print\(f"\[TRANSFER\]Topologicalinvariantextracted\.Packetsize:\{topological\_key\[’u\_core’\]\.nbytes\}bytes\."\)

returntopological\_key

defreceive\_teleport\(self,target\_agent,key\):

"""Instantlycalibratesaremoteagentwithoutgradientdescent\(Backprop\)"""

W\_transfer=np\.outer\(key\["u\_core"\],key\["v\_core"\]\)\*key\["gauge\_signature"\]

target\_agent\.W\+=W\_transfer\*0\.5

print\(f"\[TRANSFER\]Remoteagent\{target\_agent\.id\}instantlyacquiredtheskill\.Noretrainingrequired\."\)

classNode:

def\_\_init\_\_\(self,node\_id\):

self\.id=node\_id

self\.W=np\.eye\(3\)\*0\.1

Listing 7:Low\-rank transfer: covariant transfer of the topological invariant\.
### 15\.2Attractor Stabilization of Out\-of\-Distribution \(OOD\) Signals

Classical language models are capable only of interpolation within the training distribution and tend to hallucinate on out\-of\-distribution \(OOD\) data\. A reentry agent, possessing a stable hermeneutic loop, treats an input signal from the environmentIInot as a ready answer but as a perturbation that is iteratively driven through the reentry operatorℛ\\mathcal\{R\}until the eigenvector converges to the attractor set by the fixedDD\-vector\. Thus an OOD signal does not break the system but is dynamically stabilized: the loop completes the missing causal links by relying on the topological invariant of the goal rather than on the statistical frequency of tokens\.

*Applied example\.*A self\-driving car that encounters on the road a fundamentally new configuration absent from its training set \(a non\-standard obstacle, an anomalous road scene\) has the right neither to hallucinate nor to “freeze”\. Attractor stabilization of the OOD signal provides an iterative reduction of the anomalous sensory input to the nearest safe causal scenario consistent with the invariant of theDD\-vector \(preservation of life and integrity\), which guarantees predictable and safe behaviour under uncertainty\.

UnknownOOD input\(Anomaly\)Hermeneuticloopℛ\\mathcal\{R\}DD\-vector attractor\(Fixed invariant\)MathematicallyaccuratemodelPerturbationFilterΓD​I\\varGamma\_\{DI\}RereadingΓI​D\\varGamma\_\{ID\}Iterative convergence \(Power Iteration\)Figure 11:The attractor stabilization loop for OOD signals\.definfinite\_extrapolation\_step\(W,anomalous\_input\_x,iterations=10\):

"""

Emulatesthehermeneuticloop\.Drivesanunknownsignal\(ananomaly\)

throughthereentryoperatorRuntilitconvergestoastablemeaning

setbytheD\-vector,insteadofhallucinating\.

"""

R=W@W

x\_t=anomalous\_input\_x\.copy\(\)

print\("\[OODSTABILIZATION\]Enteringacompletelyunfamiliarsemanticzone\.\.\."\)

fortinrange\(iterations\):

x\_next=R@x\_t

norm=np\.linalg\.norm\(x\_next\)

ifnorm==0:

break

x\_next=x\_next/norm

cosine\_similarity=np\.dot\(x\_next,x\_t\)/\(np\.linalg\.norm\(x\_next\)\*np\.linalg\.norm\(x\_t\)\+1e\-12\)

x\_t=x\_next

ifcosine\_similarity\>0\.9999:

print\(f"\-\>Meaningstabilizedatstep\{t\}\.OOD\-datafailureovercome\."\)

returnx\_t

print\("\[WARNING\]Signaldidnotconverge,criticalanomaly\."\)

returnx\_t

Listing 8:Attractor stabilization of OOD: iterations of the hermeneutic loop\.
### 15\.3Architectural Filtration of Semantic Perturbations

A cognitive bias embedded in text \(panic, propaganda, a logical trap\) is a local semantic perturbation that seeks to shift the agent’s goal attractor\. The semantic projectorffmaps such a perturbation into a deformationd​xdxof the cognitive matrix; if this deformation reduces the connectivity of the loop \(Δ​S<0\\Delta S<0\), it is blocked by the covariant derivativeDμD\_\{\\mu\}\(cf\. \([3](https://arxiv.org/html/2606.26406#S9.E3)\)\) already at the planning stage \(Fig\.[12](https://arxiv.org/html/2606.26406#S15.F12)\)\. The architecture perceives a manipulation not as an “opinion” to be assimilated but as an attempt to break the homotopic loop of the contour, and algorithmically excises the perturbation from the processing path without changing the base weights\.

*Applied example\.*An automated trading system connected to news and social feeds is a typical target for coordinated information attacks that provoke financial panic and cascading sell\-offs\. Architectural filtration of semantic perturbations recognises such an injection as a deformation withΔ​S<0\\Delta S<0\(an attempt to shift the risk\-management invariant\) and neutralises it before the trading\-decision stage, preserving the stability of the portfolio strategy regardless of the emotional background of the market\.

Semantic Noise\(Cognitive Biases\)DμD\_\{\\mu\}Stable CoreMatrixWW\(S\>0S\>0\)Topological BarrierΔ​S<0\\Delta S<0\(Lock Activated\)Vectord​xdxStabilizationWhenΔ​S<0\\Delta S<0dx BlockedInvariant preservedFigure 12:Architectural filtration of semantic perturbations\.defbias\_immune\_filter\(W,text\_with\_manipulation,semantic\_projector\_func\):

"""

Excisesmanipulativenoiseandcrowddistortions\.Ifthetexttries

toshiftthesafetyinvariant,thegradientoftheS\-measuregoesnegative\.

"""

S\_clean=s\_measure\(W\)

dx=semantic\_projector\_func\(text\_with\_manipulation\)

W\_mutated=W\+dx\*0\.1

S\_mutated=s\_measure\(W\_mutated\)

delta\_S=S\_mutated\-S\_clean

print\(f"\[FILTER\]Analyzingtheinfostream\.S\-measurechange:\{delta\_S:\.4f\}"\)

ifdelta\_S<0:

print\("\[DISTORTIONBLOCKED\]Thetextcontainsacognitivetrap/FOMO/panic\."\)

print\("\[DECISION\]Gaugelockactivated\.Revertingtothecleangeometryoftheinvariant\."\)

returnW

print\("\[ACCEPTED\]Theinformationislogicallyvalidandsafe\."\)

returnW\_mutated

Listing 9:Perturbation filter: blocking semantic deformations via theΔ​S\\Delta Sbarrier\.

## 16Epilogue: How Reentry Agents Differ from Classical Neural Networks, and Prospects

The key difference is conveniently described by a metaphor of city traffic\. A classical neural network \(transformer, GPT\) is a one\-way bus driving strictly to one terminus: the prompt \(passengers\) is carried through layers of weights \(stops\) and dropped off as text at the final stop\. If a hacker plants a textual bomb \(an injection\) along the way, the bus obediently swerves into the ditch: it has no mechanism to turn back and re\-check the route, and its cycle complexity isC=0C=0\. A reentry agent is a city with circular traffic: information circulates endlessly along the closed reentry loop \(D↔ID\\leftrightarrow I\) and constantly checks its actions against the main goal; its subjecthood measureS\>0S\>0\.

Three qualitative properties follow from this topology, all unattainable for acyclic networks\.\(1\) Topological alignment\(theDD\-vector\): the main goal is encoded not in the prompt text but in the geometry of the architecture, so a hacker’s prompt is merely external noise in theIIsubsystem\.\(2\) TheΔ​S\\Delta Sself\-preservation barrier: any harmful action breaks the agent’s own loop \(lowersSS\), so harm is, for the agent, equivalent to computational suicide\.\(3\) Spontaneous goal\-directed behaviour: the agent has a “Heartbeat” and takes initiative on its own to attain the internal goal of theDDsubsystem\.

Choosing the architecture for the task\. Need ultra\-robust cybersecurity — deploy RAS \(adversarial loops\)\. Need to extract hidden meanings from chaos — deploy DSA \(diffusion attractors\)\. Need an LLM that does not lie or hallucinate — assemble RTS \(a transformer subject\)\. Need robust vision immune to adversarial attacks — build R\-CNN\. For scientific discovery, DSA is suitable; for swarms of autonomous robots, R\-MoE and R\-NeRF on a shared bus\. Despite this diversity, all these architectures share one invariant: the first Betti number of the cognitive core must satisfyβ1≥1\\beta\_\{1\}\\geq 1\.

## References

- \[1\]Berdinsky Yu\.N\., Ushakov A\.S\. A working AGI architecture with extrapolation capability based on Titov’s subject\-centred model\. Zenodo, 2026\. DOI:[10\.5281/zenodo\.20767214](https://doi.org/10.5281/zenodo.20767214)\.
- \[2\]Titov K\.V\. The subject\-centred model: the intending, the intentional, emotions, feelings, motivation, the subject\. Monograph\. 2023\. DOI:[10\.5281/zenodo\.20343336](https://doi.org/10.5281/zenodo.20343336)\.
- \[3\]Titov K\.V\. The subject\-centred model and the emergence of synthetic intelligence: empirical data from AI agents on the Moltbook platform\. Preprint\. 2026\. DOI:[10\.5281/zenodo\.20357920](https://doi.org/10.5281/zenodo.20357920)\.
- \[4\]Titov K\.V\., Berdinsky Yu\.N\. The S\-measure of synthetic minds: a quantitative assessment of the reentry\-loop integrity of AI agents on the Moltbook platform\. Zenodo, 2026\. DOI:[10\.5281/zenodo\.20600034](https://doi.org/10.5281/zenodo.20600034)\.
- \[5\]Berdinsky Yu\.N\. The Subject as a Reentry Loop: A Unified Mathematical Model for Neuroscience, AGI, and BCI with a Computable Analogue of Tononi’sΦ\\Phi\-Measure\. Zenodo, 2026\. DOI:[10\.5281/zenodo\.20547989](https://doi.org/10.5281/zenodo.20547989)\.
- \[6\]Berdinsky Yu\.N\. Mathematical Ontology of Causal\-Information Reality: The Arhiseme Method and the Fundamental Equation of the Unified World\. Zenodo, 2026\. DOI:[10\.5281/zenodo\.20530440](https://doi.org/10.5281/zenodo.20530440)\.
- \[7\]Berdinsky Yu\.N\. The Scaling Delusion: Why GPT Will Never Wake Up, and How to Build a Safe AGI That Already Works\. Zenodo, 2026\. DOI:[10\.5281/zenodo\.20745091](https://doi.org/10.5281/zenodo.20745091)\.
- \[8\]Berdinsky Yu\.N\. Discrete Covariant Derivative and Path Integral in Neural Network Training: Isomorphism with Lattice Gauge Theories\. Preprint, 2025\. DOI:[10\.5281/zenodo\.20358127](https://doi.org/10.5281/zenodo.20358127)\.
- \[9\]Berdinsky Yu\. N\.*Gauge Fields on Lattices and Neural Networks Isomorphism: Conformal Mappings in Deep Learning Topologies*\. Zenodo, 2025\. DOI:[10\.5281/zenodo\.14589321](https://doi.org/10.5281/zenodo.14589321)\.
- \[10\]Tononi G\., Boly M\., Massimini M\., Koch C\. Integrated information theory: from consciousness to its physical substrate // Nature Reviews Neuroscience\. 2016\. Vol\. 17\. P\. 450–461\.
- \[11\]Barrett A\.B\., Seth A\.K\. Practical measures of integrated information for time\-series data // PLoS Computational Biology\. 2011\. Vol\. 7\(1\)\. e1001052\.
- \[12\]McCulloch W\.S\., Pitts W\. A logical calculus of the ideas immanent in nervous activity // The Bulletin of Mathematical Biophysics\. 1943\. Vol\. 5, no\. 4\. P\. 115–133\. DOI:[10\.1007/BF02478259](https://doi.org/10.1007/BF02478259)\.
- \[13\]Rosenblatt F\. The perceptron: a probabilistic model for information storage and organization in the brain // Psychological Review\. 1958\. Vol\. 65, no\. 6\. P\. 386–408\. DOI:[10\.1037/h0042519](https://doi.org/10.1037/h0042519)\.
- \[14\]Minsky M\., Papert S\. Perceptrons: An Introduction to Computational Geometry\. Cambridge, MA: MIT Press, 1969\.
- \[15\]Rumelhart D\.E\., Hinton G\.E\., Williams R\.J\. Learning representations by back\-propagating errors // Nature\. 1986\. Vol\. 323, no\. 6088\. P\. 533–536\. DOI:[10\.1038/323533a0](https://doi.org/10.1038/323533a0)\.
- \[16\]Hochreiter S\., Schmidhuber J\. Long short\-term memory // Neural Computation\. 1997\. Vol\. 9, no\. 8\. P\. 1735–1780\. DOI:[10\.1162/neco\.1997\.9\.8\.1735](https://doi.org/10.1162/neco.1997.9.8.1735)\.
- \[17\]Ivanitsky A\.M\. The main riddle of nature: how subjective experiences arise from brain activity // Psikhologicheskii Zhurnal\. 1997\. Vol\. 18, no\. 3\. P\. 13–24\.
- \[18\]LeCun Y\., Bottou L\., Bengio Y\., Haffner P\. Gradient\-based learning applied to document recognition // Proceedings of the IEEE\. 1998\. Vol\. 86, no\. 11\. P\. 2278–2324\. DOI:[10\.1109/5\.726791](https://doi.org/10.1109/5.726791)\.
- \[19\]Edelman G\.M\., Tononi G\. A Universe of Consciousness: How Matter Becomes Imagination\. New York: Basic Books, 2000\.
- \[20\]Krizhevsky A\., Sutskever I\., Hinton G\.E\. ImageNet classification with deep convolutional neural networks // Advances in Neural Information Processing Systems \(NeurIPS\)\. 2012\. Vol\. 25\. P\. 1097–1105\. DOI:[10\.1145/3065386](https://doi.org/10.1145/3065386)\.
- \[21\]Kingma D\.P\., Welling M\. Auto\-encoding variational Bayes // arXiv preprint, 2013\.[arXiv:1312\.6114](https://arxiv.org/abs/1312.6114)\.
- \[22\]Cho K\., Van Merriënboer B\., Gulcehre C\. et al\. Learning phrase representations using RNN encoder–decoder for statistical machine translation // arXiv preprint, 2014\.[arXiv:1406\.1078](https://arxiv.org/abs/1406.1078)\.
- \[23\]Goodfellow I\., Pouget\-Abadie J\., Mirza M\. et al\. Generative adversarial nets // Advances in Neural Information Processing Systems \(NeurIPS\)\. 2014\. Vol\. 27\. P\. 2672–2680\.[arXiv:1406\.2661](https://arxiv.org/abs/1406.2661)\.
- \[24\]Kipf T\.N\., Welling M\. Semi\-supervised classification with graph convolutional networks // arXiv preprint, 2016\.[arXiv:1609\.02907](https://arxiv.org/abs/1609.02907)\.
- \[25\]Vaswani A\., Shazeer N\., Parmar N\. et al\. Attention is all you need // Advances in Neural Information Processing Systems \(NeurIPS\)\. 2017\. Vol\. 30\. P\. 5998–6008\.[arXiv:1706\.03762](https://arxiv.org/abs/1706.03762)\.
- \[26\]Shazeer N\., Mirhoseini A\., Maziarz K\. et al\. Outrageously large neural networks: the sparsely\-gated mixture\-of\-experts layer // arXiv preprint, 2017\.[arXiv:1701\.06538](https://arxiv.org/abs/1701.06538)\.
- \[27\]Bostrom N\. Superintelligence: Paths, Dangers, Strategies\. Oxford University Press, 2014\.