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October 24, 2025
─────────────────────────────────────────────── Contents 1 Introduction: Motivation and Observation 2 Theoretical Framework: Static Semantic Network Model 3 Hallucination as Semantic Drift 4 Emergent Self-Learning as Multi-Path Convergence 5 Greedy Decoding and the Local–Global Gap 6 Discussion and Implications ───────────────────────────────────────────────
1 Introduction: Motivation and Observation ─────────────────────────────────────────────── In the course of long-term and high-frequency interaction with large language models (LLMs)—averaging more than eight hours per day—it became increasingly evident that their behavior reveals several regular, almost physical, properties. Two empirical phenomena stand out:
1. Probabilistic path adaptation under question reformulation. When an LLM refuses to answer a question—for example, due to safety filters or internal constraints—it can often be induced to respond by slightly reformulating the same query. This suggests that each token-generation step represents a probabilistic traversal through an underlying semantic field: changing the wording of the prompt modifies the local probability distribution P(token_{t+1}|context_t), thereby shifting the model’s path toward a region of higher probability for “answer-allowed” continuations.
2. Structural invariance across semantically similar queries. For a given class of questions, the model’s answers tend to exhibit a stable syntactic and structural pattern. Even when explicitly instructed to respond in a different format, the model quickly “forgets” this deviation and reverts to its default response structure. This persistence implies that, once the user’s temporary contextual influence decays, the model’s generation process returns to its intrinsic probability field—its most stable path through the underlying semantic space.
Taken together, these observations motivate the following hypothesis: After training, an LLM does not operate as a dynamically changing reasoning system, but rather as a static semantic network in which each token exists as a node connected by probabilistically weighted directions. During inference, the model dynamically collapses this network by traversing one of many potential paths, guided by contextual probability gradients, until a specific semantic trajectory—or “Road”—emerges as the output sequence.
This interpretation reframes generation not as symbolic reasoning, but as probabilistic path selection over a frozen network, in which user prompts merely perturb the local probability field.
─────────────────────────────────────────────── 2 Theoretical Framework: Static Semantic Network Model ─────────────────────────────────────────────── During training, the attention-based backward correction mechanism ensures that the direction and endpoint of each token’s projection vector gradually evolve from a discrete and uncertain region into a set of concrete, learnable vectors. This implies that, once training is completed, the set of possible next tokens linked to a given token becomes fixed and static. Inference then only assigns dynamic probabilities to these pre-existing connections.
Definitions: T — set of token vectors representing the current state. W — total weighting matrix (all learned attention + feed-forward layers). D — collection of token-to-token directional vectors. P_i — probability of following direction D_i during inference. Road — path formed by successively selected D_i during generation. V — the static semantic network of all nodes and directed edges.
During inference: T_next = EWT The operation EWT assigns probabilities over all D_i in D. The model selects the direction with the highest P_i and moves to the next token. All selected directions form a path: R = {D_1, D_2, …, D_n}, R ⊂ V
Thus, R represents a semantic trajectory collapsed from the global probability field. Each potential path already exists within the trained network V; inference merely chooses among them.
─────────────────────────────────────────────── 3 Hallucination as Semantic Drift ─────────────────────────────────────────────── Within this framework, hallucination appears as deviation of the model’s traversal path from high-density regions of the learned semantic network V. Two structural cases arise:
Type I. Missing-Path Hallucination. When the computed vector from current token T does not match any existing D in V, the model interpolates to the nearest vector D′ = argmin_{D_i∈D} ||E(T)W – E(D_i)W||, producing an output that locally minimizes deviation but globally drifts away from the semantic manifold.
Type II. Disconnected-Path Hallucination. When the computed D set exists in V but points to tokens semantically unrelated to context, the LLM still assigns probabilities P_i = e^{s_i}/Σ e^{s_j}, generating fluent but factually wrong outputs.
In both cases, hallucination is not random noise but deterministic traversal in a static graph lacking proper constraints. Geometrically, hallucination = semantic drift: E(T_{t+1}) ∉ 𝒩(R_train) It unifies both factual and reasoning inconsistencies observed in LLMs.
─────────────────────────────────────────────── 4 Emergent Self-Learning as Multi-Path Convergence ─────────────────────────────────────────────── Human language and writing naturally encode knowledge, forming an implicit large-scale network—the Maximum Semantic Network. All knowledge items are connected: software → programming language → OS → hardware → instruction set → circuit implementation Explicit links = human-traceable paths; implicit links = real but unnoticed connections.
LLM self-learning arises from exploiting these implicit links. Through multiple rounds of probabilistic direction selection: D_i ∈ D, P_i = P(D_i | T, W) the model can traverse weak intermediate paths yet arrive at a correct token.
Formally: ∃ R_a, R_b ⊂ V, R_a ∘ R_b → T_k and accumulated probability: P(T_k) = Σ_{R_i→T_k} Π_{(u,v)∈R_i} P(D_{uv}) If P(T_k) surpasses baseline, the model “discovers” a relation never explicitly trained—manifesting self-learning.
Thus, self-learning = natural multi-path convergence in a static network, mirroring implicit interconnectivity of human language.
─────────────────────────────────────────────── 5 Greedy Decoding and the Local–Global Gap ─────────────────────────────────────────────── Greedy decoding chooses at each step the direction D_i with highest local probability. This guarantees local optimality, not global coherence.
Example: “eat soft cloud.” Each step (eat→soft, soft→cloud) fits local logic, but the entire path R is semantically wrong.
Global path probability: P(R) = Π_{(i,j)∈R} P(D_{ij}) Greedy decoding instead maximizes each step: D*_t = argmax_{D_i} P(D_i | T_t, W) Since max Π P ≠ Π max P, local optima ≠ global optimum. Hence, long-tail semantic drift occurs: plausible beginnings degrade into incoherence.
─────────────────────────────────────────────── 6 Discussion and Implications ─────────────────────────────────────────────── If the Maximum Semantic Network already exists, training is the act of reconstructing it mathematically. Discrepancies between learned and true structures are what must be corrected.
Human reasoning adds classification and correction beyond token prediction. When asked “What does a computer eat?”, humans classify “computer” as non-biological and infer that “eat” is invalid or metaphorical (“computers eat electricity”). This continuous factual correction prevents semantic drift.
Future LLMs should therefore include: 1. Hierarchical token classification grounded in linguistic/conceptual categories. 2. Real-time correction mechanisms adjusting P_i to maintain global coherence. 3. Modular functional segmentation—like the brain’s distributed regions for perception, generation, reasoning, and correction—sharing one semantic substrate.
Thus, the human brain serves as the optimal inference architecture. Future LLMs adopting similar modular and corrective structures could bridge statistical text generation and genuine reasoning, aligning token paths not only with probable continuations but with the coherent structure of reality itself. ───────────────────────────────────────────────
October 24, 2025
───────────────────────────────────────────────
Contents
1 Introduction: Motivation and Observation
2 Theoretical Framework: Static Semantic Network Model
3 Hallucination as Semantic Drift
4 Emergent Self-Learning as Multi-Path Convergence
5 Greedy Decoding and the Local–Global Gap
6 Discussion and Implications
───────────────────────────────────────────────
1 Introduction: Motivation and Observation
───────────────────────────────────────────────
In the course of long-term and high-frequency interaction with large language models (LLMs)—averaging more than eight hours per day—it became increasingly evident that their behavior reveals several regular, almost physical, properties. Two empirical phenomena stand out:
1. Probabilistic path adaptation under question reformulation.
When an LLM refuses to answer a question—for example, due to safety filters or internal constraints—it can often be induced to respond by slightly reformulating the same query. This suggests that each token-generation step represents a probabilistic traversal through an underlying semantic field: changing the wording of the prompt modifies the local probability distribution P(token_{t+1}|context_t), thereby shifting the model’s path toward a region of higher probability for “answer-allowed” continuations.
2. Structural invariance across semantically similar queries.
For a given class of questions, the model’s answers tend to exhibit a stable syntactic and structural pattern. Even when explicitly instructed to respond in a different format, the model quickly “forgets” this deviation and reverts to its default response structure. This persistence implies that, once the user’s temporary contextual influence decays, the model’s generation process returns to its intrinsic probability field—its most stable path through the underlying semantic space.
Taken together, these observations motivate the following hypothesis:
After training, an LLM does not operate as a dynamically changing reasoning system, but rather as a static semantic network in which each token exists as a node connected by probabilistically weighted directions. During inference, the model dynamically collapses this network by traversing one of many potential paths, guided by contextual probability gradients, until a specific semantic trajectory—or “Road”—emerges as the output sequence.
This interpretation reframes generation not as symbolic reasoning, but as probabilistic path selection over a frozen network, in which user prompts merely perturb the local probability field.
───────────────────────────────────────────────
2 Theoretical Framework: Static Semantic Network Model
───────────────────────────────────────────────
During training, the attention-based backward correction mechanism ensures that the direction and endpoint of each token’s projection vector gradually evolve from a discrete and uncertain region into a set of concrete, learnable vectors. This implies that, once training is completed, the set of possible next tokens linked to a given token becomes fixed and static. Inference then only assigns dynamic probabilities to these pre-existing connections.
Definitions:
T — set of token vectors representing the current state.
W — total weighting matrix (all learned attention + feed-forward layers).
D — collection of token-to-token directional vectors.
P_i — probability of following direction D_i during inference.
Road — path formed by successively selected D_i during generation.
V — the static semantic network of all nodes and directed edges.
During inference:
T_next = EWT
The operation EWT assigns probabilities over all D_i in D.
The model selects the direction with the highest P_i and moves to the next token.
All selected directions form a path:
R = {D_1, D_2, …, D_n}, R ⊂ V
Thus, R represents a semantic trajectory collapsed from the global probability field.
Each potential path already exists within the trained network V; inference merely chooses among them.
───────────────────────────────────────────────
3 Hallucination as Semantic Drift
───────────────────────────────────────────────
Within this framework, hallucination appears as deviation of the model’s traversal path from high-density regions of the learned semantic network V.
Two structural cases arise:
Type I. Missing-Path Hallucination.
When the computed vector from current token T does not match any existing D in V, the model interpolates to the nearest vector D′ = argmin_{D_i∈D} ||E(T)W – E(D_i)W||, producing an output that locally minimizes deviation but globally drifts away from the semantic manifold.
Type II. Disconnected-Path Hallucination.
When the computed D set exists in V but points to tokens semantically unrelated to context, the LLM still assigns probabilities P_i = e^{s_i}/Σ e^{s_j}, generating fluent but factually wrong outputs.
In both cases, hallucination is not random noise but deterministic traversal in a static graph lacking proper constraints.
Geometrically, hallucination = semantic drift:
E(T_{t+1}) ∉ 𝒩(R_train)
It unifies both factual and reasoning inconsistencies observed in LLMs.
───────────────────────────────────────────────
4 Emergent Self-Learning as Multi-Path Convergence
───────────────────────────────────────────────
Human language and writing naturally encode knowledge, forming an implicit large-scale network—the Maximum Semantic Network.
All knowledge items are connected:
software → programming language → OS → hardware → instruction set → circuit implementation
Explicit links = human-traceable paths; implicit links = real but unnoticed connections.
LLM self-learning arises from exploiting these implicit links.
Through multiple rounds of probabilistic direction selection:
D_i ∈ D, P_i = P(D_i | T, W)
the model can traverse weak intermediate paths yet arrive at a correct token.
Formally:
∃ R_a, R_b ⊂ V, R_a ∘ R_b → T_k
and accumulated probability:
P(T_k) = Σ_{R_i→T_k} Π_{(u,v)∈R_i} P(D_{uv})
If P(T_k) surpasses baseline, the model “discovers” a relation never explicitly trained—manifesting self-learning.
Thus, self-learning = natural multi-path convergence in a static network, mirroring implicit interconnectivity of human language.
───────────────────────────────────────────────
5 Greedy Decoding and the Local–Global Gap
───────────────────────────────────────────────
Greedy decoding chooses at each step the direction D_i with highest local probability.
This guarantees local optimality, not global coherence.
Example: “eat soft cloud.”
Each step (eat→soft, soft→cloud) fits local logic, but the entire path R is semantically wrong.
Global path probability:
P(R) = Π_{(i,j)∈R} P(D_{ij})
Greedy decoding instead maximizes each step:
D*_t = argmax_{D_i} P(D_i | T_t, W)
Since max Π P ≠ Π max P, local optima ≠ global optimum.
Hence, long-tail semantic drift occurs: plausible beginnings degrade into incoherence.
───────────────────────────────────────────────
6 Discussion and Implications
───────────────────────────────────────────────
If the Maximum Semantic Network already exists, training is the act of reconstructing it mathematically.
Discrepancies between learned and true structures are what must be corrected.
Human reasoning adds classification and correction beyond token prediction.
When asked “What does a computer eat?”, humans classify “computer” as non-biological and infer that “eat” is invalid or metaphorical (“computers eat electricity”).
This continuous factual correction prevents semantic drift.
Future LLMs should therefore include:
1. Hierarchical token classification grounded in linguistic/conceptual categories.
2. Real-time correction mechanisms adjusting P_i to maintain global coherence.
3. Modular functional segmentation—like the brain’s distributed regions for perception, generation, reasoning, and correction—sharing one semantic substrate.
Thus, the human brain serves as the optimal inference architecture.
Future LLMs adopting similar modular and corrective structures could bridge statistical text generation and genuine reasoning, aligning token paths not only with probable continuations but with the coherent structure of reality itself.
───────────────────────────────────────────────
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