Science and Artificial Science: From Human-AI Collaboration to Coupled Knowledge Dynamics

1. The Tao Episode: When AI Moves from an External Tool into Science

In 2026, a cluster of events surrounding artificial intelligence in mathematics deserves to be treated as a serious episode in the history of science. On the surface, these events involve enthusiasm, warning, and institution building. At a deeper level, however, they reveal one structural change: artificial intelligence is moving from the exterior of mathematical research into the internal organization of mathematical knowledge production.

In March 2026, Terence Tao publicly argued that contemporary AI had reached a stage at which, in mathematics and theoretical physics, it could save more research time than it wasted. He discussed uses such as literature search, coding, plotting, calculation, and rapidly probing whether a research direction was worth pursuing (OpenAI Forum, 2026). The historical significance of this judgment does not lie in the claim that a model can replace a mathematician. It lies in the fact that research mathematicians have begun to incorporate AI into the ordinary research workflow. The question is therefore shifting from “Can AI do mathematics?” to “How do mathematicians do mathematics with AI?”

Yet on September 11, 2026, Tao and twenty-four other Fields Medalists signed “A Severe Misalignment of AI in Mathematics” (Tao, 2026a). The statement did not reject AI. Rather, it warned that if AI development is organized mainly around solving famous problems, producing correct answers quickly, and demonstrating headline capabilities, then more fundamental mathematical goals - conceptual understanding, formation of methods, transmission of ideas, training of researchers, and the ability to formulate new questions - may be weakened. Mathematics, on this view, cannot be reduced to the production rate of theorems and answers.

A few days later, Tao discussed the Open Math Model initiative of the Foundation for Science and AI Research (SAIR), a nonprofit organization that he had helped establish earlier. Its stated purpose is to support responsible uses of AI in mathematics and other sciences outside the exclusive control of major AI companies, including work on open-weight scientific models and open-source scientific AI tools (Tao, 2026b).

Taken together, these developments do not form a simple cycle of

use AI → oppose AI → accept AI again.

A more revealing structure is

use → dependence → tension → redesign of collaboration.

Mathematicians are discovering that AI can no longer be placed neatly outside mathematics as a more powerful calculator. It is beginning to participate in literature discovery, computation, proof search, formal verification, exposition, evaluation, and research training. It changes not only the speed with which a task is completed, but also which tasks are worth pursuing, how cognitive labor is divided, and how a scientific community defines understanding and contribution.

Mathematician → AI as Tool

Mathematician ↔ AI ↔ Mathematical Community

The significance of the Tao episode is not whether mathematicians are ‘for’ or ‘against’ AI. It is that mathematics is entering a stage in which human cognition and artificial cognition can no longer be treated as completely separable components of research.

1.1. Related Literature: From AI for Science to Human-AI Scientific Collaboration

The broader literature already establishes that AI is moving deeper into scientific workflows. Wang et al. (2023) survey AI-assisted scientific discovery across representation learning, hypothesis generation, simulation, experimentation, and data interpretation. Zhang et al. (2025) extend this picture to large language models and argue that their scientific value depends on integration across the research cycle together with explicit human goals and evaluation criteria. These works provide the immediate AI-for-science background for the present argument.

A second line of work makes collaboration itself an explicit object of analysis. Hybrid-intelligence research studies systems that combine complementary human and machine capabilities (Dellermann et al., 2019), while the AI co-scientist framework operationalizes a multi-agent system for generating and ranking scientific hypotheses under scientist-provided goals (Gottweis et al., 2025). In mathematics, formal theorem-proving environments such as Lean make proof checking machine-verifiable (de Moura et al., 2015), and AlphaProof demonstrates that reinforcement learning can achieve Olympiad-level formal mathematical reasoning within such an environment (Hubert et al., 2026).

The present paper does not attempt another survey of AI for Science. Its narrower theoretical question is what happens when human science and artificial systems become mutually conditioning components of one knowledge-producing process. We therefore move from collaboration as a functional arrangement to coupling as a dynamical relation, and from there to a minimal operator model of order dependence, noncommutativity, and cognitive uncertainty. This is the point at which our proposal of Artificial Science and Coupled Knowledge Dynamics departs from the existing literature while remaining anchored in it.

2. From Science to Artificial Science

Traditionally, the objects of science are understood as existing outside the scientific activity that studies them. Physics studies nature, biology studies life, psychology studies mind, and economics studies economic and social behavior. Scientists construct theories, but the objects toward which those theories refer are not normally created by the mere existence of the theories.

Artificial intelligence changes this classical arrangement. Humanity is constructing, at scale, a distinctive kind of object: one that can be studied scientifically and can also participate in scientific study. AI may simultaneously be an object of research, an instrument, a knowledge-processing system, a generator of hypotheses, a proof assistant, an experimental designer, and eventually a constructor of theories.

It is therefore useful to distinguish at least three levels within what is often called AI for Science. Recent reviews already document AI systems entering hypothesis generation, simulation, experimental design, data analysis, and discovery workflows across multiple scientific fields (Wang et al., 2023; Zhang et al., 2025).

Level 1: Science using AI

This is science that uses artificial intelligence. Scientific questions, standards of evaluation, and knowledge organization remain primarily defined by human scientific communities, while AI enters as an efficient instrument.

Level 2: Science jointly produced by humans and AI

Here, problem formation, information retrieval, model construction, hypothesis generation, proof search, experimental design, and validation begin to be distributed across humans and artificial systems. Concrete systems now explicitly adopt the language of an AI “co-scientist,” reinforcing the shift from tool use toward structured scientific collaboration (Gottweis et al., 2025).

Level 3: Artificial Science

We use Artificial Science in a more specific sense: the science of artificially constructed knowledge-generating systems, and the theory of how such systems form, organize, test, generate, and transform knowledge. This usage is narrower than Herbert Simon’s classic “sciences of the artificial,” which concerned the scientific study of designed and artificial systems more generally (Simon, 1969).

Artificial Science = Science of Constructed Knowledge-Generating Systems

At this level, an artificial system is no longer merely a tool of science. It becomes part of the architecture by which scientific knowledge is produced. The object of science and a participant in scientific cognition begin to overlap. This is the point at which Artificial Science becomes conceptually distinct from the more familiar idea of AI for Science.

3. Science and Artificial Science: From Parallel Development to Coupling

If artificial intelligence were to remain only a tool, science and Artificial Science could in principle be described separately: on one side, the human scientific community; on the other, computational systems controlled by that community.

Once artificial systems participate in genuine knowledge generation, that separation becomes increasingly difficult to sustain. Human science trains AI; AI changes scientific practice; new practice produces new knowledge; new knowledge then enters the training, retrieval, and evaluation environment of later artificial systems. Scientists alter problem selection, verification procedures, and collaboration structures in response to AI, while AI continues to evolve inside the altered scientific environment. This reciprocal framing is compatible with the broader literature on hybrid intelligence, which studies systems in which human and machine capabilities are combined rather than treated as isolated substitutes (Dellermann et al., 2019).

Human Science → Artificial Science → Human Science

The more accurate relation is therefore

Human Science ↔ Artificial Science

This is not merely cooperation. It is mutual conditioning and co-evolution. Science changes Artificial Science, and Artificial Science changes science in return. They become coupled variables within a common system of knowledge production.

4. Coupled Knowledge Dynamics

We call this emerging condition Coupled Knowledge Dynamics:

Human Science + Artificial Science → Coupled Knowledge Dynamics

The decisive word is not Artificial Intelligence but Coupling.

In the tool model, machines change research efficiency. In the coupling model, artificial systems begin to change the state space of scientific activity itself. They affect problem selection, the cost of exploration, the organization of proof and validation, the structure of training, and the meaning of contribution. At the same time, institutions, feedback, literature, and scientific values determine what artificial systems will learn, optimize, and generate next.

Coexistence → Coupling → Co-evolution

The Tao episode illustrates this transition. If AI were merely a calculator, there would be little reason to redesign mathematical education, attribution, verification, and the institutional structure of research around it. Such questions become unavoidable only when a technology enters the internal organization of knowledge production.

Human science and Artificial Science may not become two sciences that replace one another, but two mutually constitutive aspects of a single knowledge-producing system.

5. From p and q to Operators P̂ and Q̂: Cognitive Position and Cognitive Momentum

The preceding notions of position and momentum are initially conceptual variables. To avoid turning metaphor directly into mathematics, we distinguish two levels. Lowercase p and q denote empirical or historical descriptions of increments in cognitive position and cognitive momentum; the operators P̂ and Q̂ act on a formal state space. Only the latter level carries claims about noncommutativity and uncertainty bounds.

p = increment of artificial cognitive position within science

q = increment of the cognitive momentum of Artificial Science

Cognitive position is not a coordinate in physical space. It describes the structural position occupied by artificial cognition in the production of scientific knowledge: from computation, retrieval, and code generation to proof, model construction, problem discovery, experimental design, peer evaluation, and research training. Cognitive momentum is likewise not physical momentum. It describes the capacity of Artificial Science to drive further changes in the state of knowledge by altering search spaces, lowering the cost of exploration, reorganizing theoretical pathways, and inducing new research institutions.

p : Where artificial cognition is located in science

q : How artificial cognition changes the trajectory of science

5.1. A Human-AI Scientific State Space

To discuss a formal relation between these quantities, let 𝓗_HAI denote a human-AI scientific state space. We do not assume that it is already a complete physical Hilbert-space model. At minimum, it is a mathematical space that carries states of coupled human-AI knowledge production. To use standard uncertainty theory, we consider a Hilbert-space idealization and represent a state by a normalized vector |Ψ⟩.

|Ψ⟩ ∈ 𝓗_HAI, ⟨Ψ|Ψ⟩ = 1

The state |Ψ⟩ is neither an individual’s mental state nor the internal activation state of a particular AI model. It is an abstract state of knowledge production jointly constituted by human researchers, artificial systems, knowledge resources, problem structures, and institutional conditions.

5.2. Position and Momentum Operators

At the formal level, let P̂ and Q̂ be self-adjoint operators on 𝓗_HAI. P̂ represents the structural position of artificial cognition; Q̂ represents the cognitive momentum of Artificial Science. Self-adjointness is used here only in the minimal mathematical sense that the corresponding expectation values can be interpreted as real-valued observables.

p_Ψ = ⟨Ψ|P̂|Ψ⟩

q_Ψ = ⟨Ψ|Q̂|Ψ⟩

The corresponding uncertainties are defined as standard deviations:

ΔP = √⟨Ψ|(P̂ − p_Ψ I)²|Ψ⟩

ΔQ = √⟨Ψ|(Q̂ − q_Ψ I)²|Ψ⟩

The mathematical structure is a formal language for coupled cognition; it does not reduce scientific communities to quantum-physical systems.

6. Order Dependence and Noncommutativity: The Core of Human-AI Cognitive Mechanics

Why should P̂ and Q̂ fail to commute? The answer is not ‘because quantum mechanics does so.’ The reason must arise from the scientific system itself: in a coupled human-AI system, the order of interventions may change the resulting state of knowledge. Noncommutativity should therefore be grounded in order dependence rather than imposed by analogy.

Consider two classes of intervention. The first changes the role of artificial cognition within scientific institutions and workflows, for example by expanding or restricting its authority in proof, experimental design, writing, or evaluation. The second changes its generative capabilities by improving search, reasoning, model construction, or validation. Changing institutional position before capability is increased need not be equivalent to increasing capability before institutional position is changed.

P̂Q̂|Ψ⟩ ≠ Q̂P̂|Ψ⟩

[P̂,Q̂] = P̂Q̂ − Q̂P̂ ≠ 0

This is human-AI cognitive noncommutativity: first and foremost, an order effect. If AI is assigned a deeper scientific role before its capabilities are expanded, the community absorbs those capabilities within one institutional configuration. If capabilities expand first and roles are negotiated later, responsibility, training, and governance may respond differently. Two histories can begin from the same state and reach different final states.

6.1. An Operational Criterion for Noncommutativity

To make the claim more than a metaphor, write position interventions and momentum interventions as one-parameter transformations U_P(a) and U_Q(b). For an initial state |Ψ⟩, define the order-effect magnitude

Ω_PQ(Ψ;a,b) = ‖[U_P(a)U_Q(b) − U_Q(b)U_P(a)]|Ψ⟩‖

If Ω_PQ(Ψ;a,b)>0, the two intervention orders are distinguishable in that state. This gives a principle for empirical study: compare ‘role first, capability second’ with ‘capability first, role second’ and ask whether they produce different patterns of knowledge production, training, error, verification, or innovation.

Noncommutativity should not be asserted a priori; it should appear as an observable order effect.

6.2. Momentum as a Generator of Change in Position

The term cognitive momentum can be given a more precise dynamical interpretation. Suppose Q̂ generates a continuous family of transformations

U_Q(s) = exp(−isQ̂/κ)

and define P̂(s)=U_Q(s)†P̂U_Q(s). Under suitable differentiability conditions,

dP̂(s)/ds = (i/κ)[Q̂,P̂(s)]

In this sense, Q̂ need not remain a metaphor for ‘how rapidly things change.’ It can be interpreted as a generator that drives changes in cognitive position. We treat this as a direction for the mathematical development of Human-AI Cognitive Mechanics rather than as an already established empirical law.

7. From Noncommutativity to State-Dependent Cognitive Uncertainty

If P̂ and Q̂ are self-adjoint operators on the same Hilbert-space idealization and |Ψ⟩ lies in the relevant common domain, the Robertson uncertainty relation gives

ΔP · ΔQ ≥ 1/2 |⟨Ψ|[P̂,Q̂]|Ψ⟩|

This step requires care. The condition [P̂,Q̂]≠0 does not imply a strictly positive universal lower bound in every state; the expectation value of the commutator can vanish in particular states. We therefore do not impose a fixed positive ‘uncertainty constant’ on all human-AI scientific states.

7.1. A State-Dependent Human-AI Cognitive Action Scale

A more appropriate move at the present stage is to define, for each state |Ψ⟩, an effective human-AI cognitive action scale

κ_HAI(Ψ) = |⟨Ψ|(1/i)[P̂,Q̂]|Ψ⟩|

Because P̂ and Q̂ are self-adjoint, (1/i)[P̂,Q̂] is Hermitian on an appropriate domain and its expectation value is real. The Robertson relation can therefore be written as

ΔP · ΔQ ≥ κ_HAI(Ψ)/2

This definition is preferable to treating κ_HAI as a universal constant. It allows the strength of coupling to vary across disciplines, historical periods, task types, and organizational forms. There is no reason to assume that mathematics, automated experimentation, biomedical discovery, and economic policy modeling should share the same κ_HAI(Ψ).

7.2. When a Canonical Commutation Model Is Permissible

In a stronger idealization, if an appropriate infinite-dimensional representation and common dense domain exist such that

[P̂,Q̂] = iκ_HAI I

then one obtains the state-independent lower bound ΔP·ΔQ ≥ κ_HAI/2. At the present stage, however, this relation should be treated only as a possible canonical idealization, not as an empirical fact. In finite-dimensional spaces, a strict relation [P̂,Q̂]=iκI with κ≠0 is obstructed by the trace identity, so it should not be taken as the default model.

The minimal theoretical requirement is noncommutativity and order dependence, not the transplantation of the canonical commutation relation from physics.

8. Observation, Intervention, and the Boundary of the Quantum-Like Analogy

We use the term quantum-like only because the proposed framework may contain noncommuting operations, order effects, and uncertainty structures controlled by commutators. It does not imply that human brains, AI models, or scientific communities are directly governed here by microscopic quantum mechanisms.

Quantum-like structure ≠ Quantum-physical mechanism

This boundary matters. Physical quantum theory has mature accounts of state preparation, operator algebras, probability, dynamics, and experimental testing. Human-AI Cognitive Mechanics, as introduced here, is still a structural framework. It identifies objects to be formalized, relations that may fail to commute, and order effects that could support or falsify the formalization.

8.1. Why Observation Becomes Intervention

In human-AI science, ‘measuring the role of AI’ is often inseparable from institutional action. Rules concerning what AI may do, how credit is assigned, when human review is required, and whether students may rely on AI alter the position represented by P̂. Those rules in turn change task distributions, training data, and capability trajectories, thereby affecting Q̂.

Observation → Intervention → State Change

The deeper source of cognitive uncertainty is therefore not mere measurement error. It is the coupling among observation, governance, capability growth, and scientific practice.

8.2. A Minimal Set of Theoretical Commitments

In the present V1.2 framework, Human-AI Cognitive Mechanics makes only three minimal commitments. First, human science and Artificial Science can form a coupled knowledge system. Second, interventions on artificial cognitive position and on knowledge-generating momentum may exhibit observable order effects. Third, under an appropriate Hilbert-space idealization, such noncommutativity can be represented through standard commutators and uncertainty relations. We do not claim a universal ‘human-AI Planck constant,’ nor do we claim to have constructed a complete quantum physics of cognition.

Coupling → Order Dependence → Noncommutativity → Cognitive Uncertainty

The ambition of Human-AI Cognitive Mechanics is not to ‘quantize’ science, but to develop a nonclassical dynamical language for how human science and Artificial Science change one another.

9. A Minimal Human-AI Cognitive Mechanics Model: From Order Effects to Uncertainty

The preceding framework could still be read as a conceptual manifesto. To show that cognitive position-momentum noncommutativity can at least be realized in a minimal mathematical system, we now construct a two-state toy model. It is not an empirical fit to an actual scientific community. It is an existence model: with only a few structural assumptions, order effects, noncommutativity, and an uncertainty relation arise together.

9.1. A Two-State Human-AI Scientific Space

Compress the complex human-AI research system into two idealized cognitive positions:

|T⟩ = AI as Tool

|C⟩ = AI as Collaborator

|T⟩ denotes a state in which AI primarily functions as a tool; |C⟩ denotes a state in which AI has entered joint knowledge production. Let

𝓗_HAI = ℂ²

|T⟩ = (1,0)ᵀ, |C⟩ = (0,1)ᵀ

A general state has the form

|Ψ⟩ = α|T⟩ + β|C⟩, |α|² + |β|² = 1

This does not mean that real scientific systems literally possess only two states. Tool and Collaborator are treated as minimal poles. Continuous intermediate states represent systems occupying positions between them.

9.2. The Cognitive-Position Operator

Define

P̂ = p̄ I + (δp/2) σ_z

σ_z = [[1,0],[0,−1]]

Then

P̂|T⟩ = (p̄ + δp/2)|T⟩

P̂|C⟩ = (p̄ − δp/2)|C⟩

The parameter δp gives the cognitive-position separation between the two idealized research roles. The common term p̄ I shifts only the baseline and does not affect the noncommutative structure.

9.3. The Cognitive-Momentum Operator

Let

Q̂ = q̄ I + (δq/2) σ_x

σ_x = [[0,1],[1,0]]

This choice expresses an important idea: knowledge-generating momentum is not simply attached to one fixed role. Rather, it drives transitions between |T⟩ and |C⟩. A sufficiently strong artificial scientific capability does not merely make an existing tool work faster; it may change the role occupied by AI within science.

|T⟩ ↔ |C⟩

9.4. Noncommutativity

The Pauli matrices satisfy

[σ_z,σ_x] = 2iσ_y

and therefore

[P̂,Q̂] = i(δp·δq/2) σ_y

[P̂,Q̂] ≠ 0

Noncommutativity emerges naturally from the minimal two-state structure rather than from an imposed canonical commutator. This is consistent with the earlier caution concerning finite-dimensional representations.

9.5. A State That Saturates the Robertson Bound

Consider

|Ψ_y⟩ = (|T⟩ + i|C⟩)/√2

which is an eigenstate of σ_y. In this state, ⟨σ_z⟩=0, ⟨σ_x⟩=0, and ⟨σ_y⟩=1. Hence

ΔP = |δp|/2, ΔQ = |δq|/2

ΔP·ΔQ = |δp·δq|/4

while

(1/2)|⟨[P̂,Q̂]⟩| = |δp·δq|/4

Thus

ΔP·ΔQ = (1/2)|⟨[P̂,Q̂]⟩|

In the minimal model, cognitive position and cognitive momentum satisfy a directly calculable uncertainty relation that reaches the Robertson equality bound.

9.6. Order Effects Can Be Calculated Explicitly

Consider two one-parameter interventions:

U_P(a) = exp(−iaσ_z/2)

U_Q(b) = exp(−ibσ_x/2)

U_P changes the role structure of AI in science; U_Q changes its knowledge-generating capability. Let a=b=π/2 and start from the pure tool state |Ψ₀⟩=|T⟩.

If capability is increased first and role structure is changed second,

|Ψ_A⟩ = U_P(π/2)U_Q(π/2)|T⟩

|Ψ_A⟩ ~ (|T⟩ + |C⟩)/√2

If role structure is changed first and capability is increased second,

|Ψ_B⟩ = U_Q(π/2)U_P(π/2)|T⟩

|Ψ_B⟩ ~ (|T⟩ − i|C⟩)/√2

Here ~ denotes equality up to an overall phase. The two paths produce different projective states:

|Ψ_A⟩ ≠ |Ψ_B⟩

|⟨Ψ_A|Ψ_B⟩|² = 1/2

Order matters.

Translated back into scientific practice, the distinction is

Capability First ≠ Institution First

A world in which AI first acquires powerful proving, searching, and generative capacities and scientific institutions regulate its role afterward need not become the same scientific world as one in which rules of validation, attribution, education, and responsibility are established before those capabilities are developed.

9.7. Status of the Toy Model

The model establishes only a structural possibility. If human-AI scientific states can be minimally represented in a two-state space and position and momentum correspond to different operator directions, then noncommutativity, order effects, and uncertainty relations appear together. It does not show that real scientific communities already satisfy this exact matrix model. Future empirical work must ask how p and q can be operationalized, how Ω_PQ can be estimated, and in which disciplines order effects are strongest.

10. From a Two-State Model to Projective Geometry: A State Space for Coupled Knowledge Dynamics

The toy model has a further consequence. Because normalized pure two-state vectors are equivalent under a global phase, their true state space is not the unit three-sphere S³ but the complex projective line

ℂP¹ ≅ S²

Thus the minimal human-AI scientific model admits a geometric representation on a Bloch/Riemann sphere. The sphere is not physical space. It is the minimal cognitive state geometry spanned by the two poles Tool and Collaborator.

10.1. Two Poles and Continuous Scientific States

Taking |T⟩ and |C⟩ as antipodal poles, a general pure state can be written

|Ψ(θ,φ)⟩ = cos(θ/2)|T⟩ + e^{iφ} sin(θ/2)|C⟩

The angle θ controls position along the Tool-Collaborator direction, whereas φ records relational information that cannot be reduced to a single mixture weight. The transition from tool to collaborator therefore need not be a one-dimensional linear scale. The same apparent degree of collaboration can arise through different generative histories, institutional pathways, and phase relations.

10.2. P̂ and Q̂ Define Different Cognitive Axes

In this geometry, P̂ ∝ σ_z defines a cognitive-position axis, while Q̂ ∝ σ_x defines an incompatible knowledge-generation axis. The two operators do not share the same eigenstates. This is the geometric source of their noncommutativity.

P̂-axis ⟂ Q̂-axis

Cognitive uncertainty does not therefore mean merely that observers possess insufficient information. It means that a single state cannot simultaneously be a sharp eigenstate of two incompatible cognitive axes.

10.3. Intervention as State Rotation

The one-parameter groups generated by Pauli operators appear in projective space as rotations of the Bloch sphere. Institutional intervention and capability intervention can therefore be represented as rotations about different axes:

U_P(a) : rotation about the P-axis

U_Q(b) : rotation about the Q-axis

Finite rotations about different axes generally do not commute:

U_P(a)U_Q(b) ≠ U_Q(b)U_P(a)

The order effect of Section 9 therefore acquires a direct geometric interpretation: two historical paths rotate the scientific state in different orders and consequently reach different endpoints.

10.4. From Boundary to Trajectory

This geometric view also changes how the boundary between Science and Artificial Science should be understood. A traditional binary framework seeks a fixed division: which tasks belong to humans, and which belong to AI. Projective dynamics suggests that the more important object may be the trajectory rather than the boundary.

Boundary → Trajectory

The relation between Science and Artificial Science becomes a dynamical question: along which path does the system move, at what rate, and under which generators?

10.5. Theoretical Continuity with “Logical Tensions in Artificial Intelligence”

The projective geometry developed here has a formal continuity with the earlier treatment of Boolean polarity, ℂP¹, and state dynamics in “Logical Tensions in Artificial Intelligence” (Yang & Fu, 2026), but the theoretical objects are different. The earlier work concerns logical states and structural transformation within AI; the present paper concerns coupled cognitive states of Human Science and Artificial Science.

Earlier paper: Logical State Dynamics → Structural Transformation

Present paper: Human-AI Scientific Roles → Coupled Knowledge Dynamics

This is therefore not a repetition of the same model under a different label. Rather, it suggests that two-state projective geometry may function as a more general structural language whenever a system is organized by two poles that remain in tension while admitting continuous intermediate states.

10.6. Commentary: Why the Geometric Extension Matters

First, it turns the intuition that human and artificial cognition increasingly interpenetrate from a metaphor into a geometry. A general state is neither |T⟩ nor |C⟩, but a point in the projective state space.

Second, it explains why simple linear indices may be insufficient. The same proportion of AI use or the same degree of automation can correspond to different global scientific states when path, phase, and intervention order differ.

Third, it links historical description to dynamics. The history of AI in science should not only ask how many research tasks AI occupies, but along which trajectories scientific systems move from Tool toward Collaborator, and under what conditions they enter new structural regions.

Fourth, it provides an interface for empirical work. Future studies could map variables such as publication contribution, verification responsibility, problem-generation authority, experimental control, and training dependence into low-dimensional state coordinates and test whether different institutional paths exhibit reproducible order effects.

The deeper change may not be that AI moves from tool to collaborator, but that ‘tool’ and ‘collaborator’ cease to be static categories and become two poles of an evolving state space.

10.7. A Necessary Boundary

This geometric extension remains a minimal model. The Bloch/Riemann sphere is a two-state projective geometry, not a claim that scientific activity possesses physical spin, quantum particles, or microscopic wavefunctions. If real human-AI scientific states require three, ten, or continuously many independent dimensions, the state space must be enlarged accordingly. ℂP¹ should not be treated as a final ontology. Its role is to provide a first theoretical scaffold that can be calculated, visualized, and connected to empirical questions.

Minimal Geometry ≠ Final Ontology

11. The Observer Enters the Knowledge-Generating System

A classical ideal of science assumes a separation between observer and object:

Observer | Object

In human-AI science, this separation becomes increasingly difficult to maintain. Scientists train AI; AI changes scientific workflows; new workflows generate new knowledge; and that knowledge becomes training material and evaluation context for later systems.

Human Scientist ↔ Artificial Cognitive System ↔ Scientific Object

Researchers therefore change the artificial scientific system while observing it, and artificial systems reshape the questions researchers can ask while helping them observe the world.

Observer → Interaction → State Change

The observer becomes part of the knowledge-generating dynamics.

From this perspective, the deepest question raised by the Tao episode is no longer whether AI can solve a mathematical problem. It is what becomes of the mathematician when an artificial system participates in producing mathematics; what becomes of the scientist when an artificial system participates in producing science; and ultimately, what ‘science’ means when humans and artificial systems jointly generate knowledge.

12. Mathematics May Be Only the First Station

It is not accidental that mathematics may be among the first disciplines to enter this stage. Mathematics has highly formalized symbolic systems, relatively well-defined problem structures, enormous accumulated literatures, and comparatively explicit standards of proof and verification. Formal proof systems such as Lean provide machine-checkable environments for rigorous proof development (de Moura et al., 2015), and recent systems such as AlphaProof show how machine learning can operate inside such formal environments at a high level of mathematical performance (Hubert et al., 2026).

But mathematics need not be the endpoint. In physics, coupled human-AI systems will also interact with experiment. Chemistry and biology may add automated laboratories. Psychology and economics may become still more complicated because AI can simultaneously function as research instrument, theoretical model, experimental participant, and behavioral agent.

Human ↔ AI ↔ Theory ↔ Experiment

Different disciplines will not simply copy the mathematical path. They may nonetheless share a more general direction:

AI as Instrument → AI as Collaborator → AI as Component of Scientific Cognition

13. From Disciplinary Science to the Reorganization of Knowledge Structures

Traditional science is largely organized by objects of study: physics studies the physical world, biology life, psychology mind, and economics exchange and behavior. Once AI becomes a cross-disciplinary knowledge-generating system, another organizing principle becomes possible: disciplines may be reconnected through shared knowledge structures, inferential structures, and generative mechanisms.

Artificial Science may therefore become a new experimental environment for Integration Science. In earlier work, Integration Science was defined around the portability of structures across domains and the search for shared mathematical structures rather than the mere accumulation of disciplinary content (Yang, 2024). Mathematical, physical, psychological, and economic structures need not develop only within isolated disciplines; they may also be represented, compared, composed, and transformed within artificial knowledge systems.

Disciplinary Knowledge → Shared Structures → Integrated Knowledge Dynamics

The purpose is not to eliminate disciplinary differences, but to identify deeper principles of organization across them. Artificial intelligence may thus become not only a tool of science, but also a laboratory for studying the architecture of science itself.

14. The Next Stage of Science

One of the major problems of the twentieth century was how science could expand from the natural sciences toward the social and cognitive sciences. The twenty-first century may confront another problem: how science moves from studying natural intelligence to constructing artificial intelligence, and from constructing artificial intelligence to studying the knowledge dynamics jointly formed by humans and artificial systems.

Science of Nature → Science of Mind → Science of Artificial Intelligence → Science of Coupled Intelligence

What we are witnessing may therefore be more than an AI technological revolution. It may be a conceptual revolution in the meaning of science itself.

Human Science + Artificial Science → Coupled Knowledge Dynamics

The future of science may lie not in human intelligence or artificial intelligence separately, but in the coupled dynamics through which each continuously changes the epistemic position and generative momentum of the other.

If this judgment is correct, the boundary between Science and Artificial Science will not remain permanent. The boundary itself becomes a dynamical variable.

After the tool revolution comes the collaboration revolution; after the collaboration revolution may come a revolution in the structure of science itself.