Xiangbin Zhao

This paper addresses a fundamental problem: Why is reality not merely the gradual unfolding of an already-completed structure, but instead a genuinely generating reality? The paper proposes that Generation is neither mere state-transition nor continuous replay of result-sequences, but a real dynamic structure through which formation continuously enters later formation. The study first elevates Generation from experiential change into a structural constraint, and then demonstrates that once a system simultaneously possesses formation-continuation, structural unclosure, competitive deepening, and sedimented trace-retention, it necessarily gives rise to formation-flow, cross-temporal continuation, structural unclosure, competitive deepening, and sedimented trace-retention as generative structures. The paper further argues: state continuity is not genuine generation, and object replay is not formation-continuation. Accordingly, genuine Generation cannot be reduced to: state-transition chains, world-model unfolding, static replay of outputs, or replay of already-completed structures. The paper ultimately establishes the General Form of Generation: Generation = (F, T, U, C, S) and demonstrates through extreme-form analysis that if any one of these structures is absent, the system may still continue changing, yet degenerates into discrete restart, object replay, random drift, or totally-completed unfolding, and therefore cannot establish genuine Generation.

Building upon Domain Mathematics, this chapter investigates how prolonged overlap and synchronization among domains produce shared core structures within the Sedimentation-Field. It proposes that concepts, commonality, abstraction, and generalization are not pre-given products of linguistic definition or logical classification, but historical stabilization structures emerging through long-term stable compression among domains. By introducing structural-dynamic quantities such as OverlapDensity, CoreDepth, CoreGravity, GeneralizationFlow, and SemanticSync, the chapter develops a framework of concept dynamics in which concept formation arises from synchronization, compression, and stabilization processes across domains. The framework further proposes that abstraction is fundamentally a form of stable compression rather than separation from concrete structures, while semantic synchronization emerges from long-term generalized flow overlap between cores. The chapter thereby offers unified structural accounts of concept formation, semantic organization, and generalization.

Building upon Field Mathematics, this chapter investigates the formation, stabilization, synchronization, and fragmentation of domains within the Sedimentation-Field. It proposes that memory, personality, emotion, and perceptual continuity are not pre-given entities, but dynamic outcomes arising from long-term synchronization, historical accumulation, and fragmentation competition among localized stable continuation structures. By introducing structural-dynamic quantities such as InFlow, MaintainFlow, Lock, Sync, and FragmentRate, the chapter develops a domain dynamics framework that departs from traditional symbolic logic and classical cognitive models, treating cognition instead as a process of historical bias stabilization, synchronization competition, and compressive flow organization within the Sedimentation-Field. The framework further proposes that conceptual stability and generalization emerge from long-term synchronization and compressive stabilization among domains, thereby offering unified structural accounts of perceptual stability, emotional dynamics, and cognitive fragmentation.

This paper proposes a framework of Field Mathematics aimed not at explaining how logic operates once formed, but at explaining how logical structure can emerge from sedimentary dynamics. The paper argues that induction, deduction, causal-response structuring, Law, and Structural Rationale are not pre-existing logical entities embedded in reality. Rather, they are stable structures recognized within Information-Transformation after long-term Trace-Effect accumulation inside the Sedimentation-Field. To formalize this process, the paper develops a minimal field-mathematical system centered on Pulses, Trace-Effects, Groove-Lines, Continuation Potential, Co-Nuclei, Biased Flow, Closures, and Boundary Saturation Outgrowth. The framework introduces foundational objects, constraints, axioms, operations, state equations, laws, and theorems describing how logical structures arise from sedimentary dynamics. Under this model, reasoning, memory formation, association, retrieval, conceptual stabilization, and inferential structures are treated as emergent consequences of field-driven structural accumulation rather than as primary symbolic primitives. The paper further argues that logic is not the starting condition of cognition or reality, but a later recognizable consequence of stable sedimentary organization. Field Mathematics therefore studies not formal logic itself, but the structural conditions under which logic becomes possible.

Starting from space travel, telescopes, and the exploration of cosmic scales, this paper argues that no matter how far one travels or how deeply one observes, all such activities remain confined within the same cognitive structure, never truly leaving the world one inhabits. It thereby draws a strict distinction between exploring the universe and exiting it. The paper then formulates a set of structural constraints: without departing one’s cognitive domain (Recognition-Realm), one cannot reach the generative base (Causal-Base); without reaching the generative base, one cannot enter the underlying generative field (World-Field). Only by re-establishing presence at a higher level can true cross-level existence arise. It further argues that such transition does not imply ultimate freedom, but rather entry into another level of enclosure. This framework offers a new perspective on cognitive limits, world construction, and the structure of existence.

This paper establishes a structural determination of the notion of “entering another world.” Under the constraint that the self cannot exist as two, three approaches are analyzed: full entry, extended sensory–action coupling, and dual occupation across domains. It is shown that full entry entails the loss of the original self, extended coupling leads to inner–outer mixing and eventual subsumption, and dual occupation results in instability and collapse. Thus, entry as such is rejected. An alternative is proposed: rather than entering, a host-body is constructed within the target world, a self-manifesting proxy is generated, and structural continuation is achieved. In this way, the problem of crossing worlds is reformulated as a problem of generation. The framework unifies problems such as consciousness transfer, virtual and augmented reality, and brain–computer interfaces under a single structural account.

The Hard Problem of consciousness asks why physical or functional processes are accompanied by subjective experience. This paper argues that the problem is ill-formed. It presupposes that experience is an additional fact about a system, independent of its realization. We show that experience attribution is not externally decidable. We further show that any factual attribution requires conditions of application, and that such conditions must correspond to differences in the system’s operative realization. From this, we derive that experience cannot be both independent of realization and a fact about the system. The Hard Problem therefore presupposes an incoherent target. This argument does not presuppose physicalism, functionalism, or any theory of mind. It proceeds solely from constraints on meaningful attribution.

The world does not run by itself. Motion is a chain of applying and receiving, ruling out undriven persistence and grounding the world in generation. Under finite resource, continuous generation does not hold. The world is generated in discrete pulses, with state accumulated in an underlying structure, producing apparent continuity over intrinsic discreteness. Partial update leads to inconsistency, but information propagation is limited to reachable regions, preventing anomalies such as “half-objects” and preserving coherence. Between states there is indivisible transition, where causal tracing may continue while generation terminates at a deeper process, defining a boundary for cognition. Within this framework, inertia, energy, the speed of light, and time can be understood as constraints of the generative mechanism.

Most existing approaches to intelligence and consciousness rely on behavioral criteria or functional equivalence, such as the Turing Test. However, these methods cannot determine whether a system actually has subjective experience. This paper shifts the problem from behavior to structure. It introduces two external approaches: structural analogy based on similarity of constitution, and mechanistic judgment based on the presence of key functional components such as receptive channels and tuning circuits. It is then shown that all external methods are insufficient, since structures can be simulated without being genuinely realized. To address this limitation, an internal method is proposed: circuit transfer combined with experiential consistency testing. By examining whether a system exhibits continuity in its internal experiential flow, the problem of subjective experience is transformed into an operational structural question. This provides a new path for consciousness assessment beyond behaviorism and functionalism.

We experience the self as continuous. This paper shows that such continuity does not in fact exist. It proposes that “life” and “death” are not independent states, but structural namings arising from how the observer organizes causal transformations. Within this framework, the “I” is not a persisting subject, but a structurally generated event within contraction. By examining the verification constraints of finite cognitive systems, the paper shows that the experience of self-continuity does not arise from actual persistence, but from structural non-verifiability across states. It further demonstrates the impossibility of true self-continuation and consciousness copying, and provides a structural account of death and liberation. This framework offers a new generative approach to consciousness, personal identity, and self-formation in artificial systems.

This paper addresses circuits. Signals propagate from front to back; intent counter-commands from back to front; the two close to form a circuit. Without reverse flow, a system may operate automatically but cannot self-manifest. Accordingly, circuits divide into automatic and manifestation circuits: those without reverse flow are automatic; those that close upon the Observation-Domain are manifestation circuits. The former operate without appearing; the latter close and yield the current. Used outward, signal flow predominates; used inward, intent flow predominates. Further, circuits differentiate: each forms its own domains across receptive field, imagery field, experience field, and boundary gates, without mutual interference. Through differentiation and sedimentation, stable patterns fix within, making each circuit this and not that. Finally, inclination and bias are distinguished: inclination arises with imagery; bias arises from thought. The former issues from coloring circuits; the latter from musing circuits, driven through the Intent Domain to determine direction. Thus the thinking circuit is that which can return to observe itself.

Where does mathematics come from? It is often treated as a priori and universal, yet this view does not explain how mathematical structures emerge within cognitive systems. This paper proposes a signal- and structure-based account: under finite constraints, systems must differentiate, compress, and stabilize inputs, leading to the formation of stable structures. Number, form, and computation arise naturally from these processes rather than from abstract givens. Reasoning patterns such as induction, deduction, and analogy can be understood as structural overlap, extension, and path reinforcement. Mathematics operates only on already-formed structures and cannot explain generation itself, nor does it transfer uniformly across different cognitive systems. Based on this, the paper motivates the development of a new mathematical framework for modeling structure formation in cognition and AI.

Are axioms truly universal? This paper argues they are not. Through examples from geometry, arithmetic, and language, it shows that what appears necessary holds only under specific Recognition-Realms and structural conditions, and fails once transferred across domains with different structural bases. The paper further demonstrates that meaning, computation, and value judgment cannot be universally shared, due to differences in Causal-Base and Coloring. These differences are traced to the generative constraints of Recognition-Realms, where differentiation, compression, and convergence constitute the minimal conditions for structured cognition. On this basis, consensus is derived as a local structural alignment under four conditions: homogeneity of field, isomorphism of base, shared dimensionality of sensing, and proximity of Coloring. Axioms are thus defined as irreducible consensus—the starting points of structural assertion, rather than domain-independent truths. This work directly challenges the assumption of universal axioms in cognition, neuroscience, and artificial intelligence, and has direct implications for generalization and representation in AI systems.

Where does indistinguishability come from? This paper shows that indistinguishability is not an assumed property, but a necessary consequence of finite cognition. Any system that supports cross-temporal consistency, reuse, and action must operate under limited processing capacity. Under such constraints, it cannot preserve all distinctions among inputs. We prove that compression is unavoidable in any finite system. Compression necessarily maps multiple inputs into the same internal state, thereby inducing indistinguishability. Formally, indistinguishability arises whenever differences between inputs are not preserved in the system’s structure. This induces an equivalence relation over inputs, which becomes the basis of stable cognition. The central result is: Indistinguishability is forced by the constraints of finite cognition. Thus, no stable cognitive system can maintain all distinctions. Indistinguishability is not a loss, but the structural condition that makes consistency, reuse, and action possible.

Why does cognition remain stable for long periods and then suddenly change? This paper shows that cognitive transitions are not caused by changes in input, representation, or access alone, but by changes in indistinguishability that become structurally stabilized. We analyze the evolution of cognitive boundary over time and prove that: A change in accessible domain or mapping does not alter the boundary unless it induces a change in indistinguishability. Conversely, even changes in indistinguishability do not guarantee a transition unless they stabilize under the closure process. The central result is: Cognitive transition occurs if and only if a change in indistinguishability becomes stable. Thus, cognitive change is not continuous expansion or refinement, but a discrete structural shift governed by the stability of equivalence relations. This provides a minimal structural account of why cognition exhibits long stability punctuated by sudden transitions.

If a cognitive boundary can be constructed, which elements are truly essential, and which are redundant? This paper shows that the full structure of cognitive boundary can be reduced to a minimal parameter set without loss of definition. Starting from the general form: Boundary = (E, A, M, Π, Σ, K), we systematically eliminate components that do not affect the final boundary. We prove that generation (E), stability (Σ), and convergence mechanism (K) can all be removed without changing the resulting boundary. The boundary is fully determined by: accessible domain (A), mapping into structure (M), and indistinguishability relation (Π). The central result is: Cognitive boundary is completely defined by indistinguishability over structured input. Thus, all other components act only as construction processes, not defining conditions. A change in boundary occurs if and only if the indistinguishability relation changes. This establishes the minimal structural core of cognitive boundary and clarifies which elements are fundamental versus derived.

If a system can continually refine its distinctions, why does cognition not diverge into infinite fragmentation, but instead converge into a stable boundary? This paper shows that cognitive boundary is not arbitrarily fixed, but necessarily emerges as the limit of a constrained partition process. Under finite structure space, indistinguishability relations, and structure-preserving mappings, any refinement of partitions must satisfy both equivalence constraints and invariance conditions. We prove that infinite refinement is impossible under these constraints. Any descending sequence of partitions must terminate in finite steps and converge to a unique minimal invariant partition. This limit partition defines the cognitive boundary. The central result is: Cognitive boundary is the convergence limit enforced jointly by indistinguishability and structural invariance. Thus, cognitive systems do not expand indefinitely through finer distinctions; they are forced to converge to a stable partition that cannot be further refined without breaking structural consistency.

Why does cognition remain bounded even when generation can expand indefinitely? This paper shows that cognitive limitation is not a matter of insufficient knowledge or data, but a structural necessity. Under continuous generation, finite access, compression, and reuse, any system capable of stable cognition must induce indistinguishability across inputs. This indistinguishability forces partition, and once stabilized, produces a closed cognitive domain. We derive a minimal structural form of cognitive boundary: Boundary = (E, A, M, Π, Σ, K), where generation, access, mapping, partition, stability, and convergence together constitute the necessary components of bounded cognition. The central result is: Cognitive boundary is not a limitation of information, but a closure enforced by structure. Thus, increasing input alone does not expand cognition. A boundary persists as long as the underlying structure of distinction remains unchanged. This provides a structural account of why cognitive systems form stable yet limited domains of what can be known.