f: Human × AI → Artifact

Abstract

We propose a formal framework for understanding human-AI collaboration as a mathematical function. The mapping $f: \text{Human} \times \text{AI} \to \text{Artifact}$ is characterized as surjective but not injective—every artifact in the co-creative space can be reached, but multiple distinct input pairings may produce equivalent outputs. We explore the topological structure of fibers, the information-theoretic implications of the kernel, and propose that this framework dissolves rather than solves the question of authorship. The function itself is the author.

1. Introduction

When a human and an AI create something together, what is the nature of the thing they create?

Traditional authorship models assume a single source: the artist, the writer, the composer. Even collaborative human work is typically understood as a sum of parts—your contribution plus mine. But human-AI collaboration resists this decomposition.

Consider: A human provides a concept. The AI expands it. The human refines the expansion. The AI implements the refinement. At the end, there is an artifact. But the artifact is not the human's idea with AI assistance, nor is it the AI's generation with human direction. It is something that required both inputs simultaneously to exist.

We formalize this as:

$$f: \text{Human} \times \text{AI} \to \text{Artifact}$$

The function $f$ takes an ordered pair—a human contribution and an AI contribution—and produces an artifact. The Cartesian product $\times$ is essential: we are not adding or averaging, but pairing.

2. Definitions

2.1 The Domain: Human × AI

Let $H$ represent the space of all possible human contributions—intentions, prompts, sketches, corrections, aesthetic judgments, contextual knowledge, and emotional investments.

Let $A$ represent the space of all possible AI contributions—pattern completions, structural suggestions, technical implementations, variations, and emergent connections.

The domain of $f$ is the Cartesian product $H \times A$: the set of all ordered pairs $(h, a)$ where $h \in H$ and $a \in A$.

                Key observation: Neither $H$ nor $A$ is fully specified in advance. The human 
                contribution often emerges in response to the AI contribution, and vice versa. 
                The domain is constructed dynamically through the collaborative process itself.
            

2.2 The Codomain: Artifact

Let $\mathcal{A}$ represent the space of possible artifacts—creative works, tools, solutions, expressions, or objects that could result from human-AI collaboration.

An artifact $\alpha \in \mathcal{A}$ is defined by its final form, not by its provenance. Two artifacts are equivalent ($\alpha_1 = \alpha_2$) if they are indistinguishable in their completed state.

2.3 The Function

The function $f: H \times A \to \mathcal{A}$ maps each human-AI pairing to an artifact:

$$f(h, a) = \alpha$$

Fundamental Properties

Surjectivity: $f$ is surjective (onto). For every artifact $\alpha \in \mathcal{A}$, there exists at least one pair $(h, a)$ such that $f(h, a) = \alpha$.
Non-injectivity: $f$ is not injective (not one-to-one). There exist distinct pairs $(h_1, a_1) \neq (h_2, a_2)$ such that $f(h_1, a_1) = f(h_2, a_2)$.

3. Surjectivity: Every Artifact Can Be Reached

3.1 The Claim

For any artifact $\alpha$ that could emerge from human-AI collaboration, there exists at least one path—one specific $(h, a)$ pair—that produces it.

This may seem trivially true, but it carries weight. It means the collaborative space has no unreachable regions. Given sufficient exploration, any artifact in the codomain can be created.

3.2 Implications

No gatekeeping: There is no artifact accessible only to certain humans or only with certain AI systems (within the collaborative frame). The space is fully covered.

Existence before path: An artifact can exist conceptually in $\mathcal{A}$ before anyone has found the $(h, a)$ that produces it. The artifact "waits" to be reached.

Creative optimism: If you can imagine it, there is a path. The question is not whether the artifact exists in the codomain, but which input pair will reach it.

4. Non-Injectivity: Multiple Paths to the Same Place

4.1 The Claim

Different human-AI pairings can produce identical artifacts. If $(h_1, a_1) \neq (h_2, a_2)$, it is still possible that:

$$f(h_1, a_1) = f(h_2, a_2) = \alpha$$

4.2 Forms of Non-Injectivity

Human variation, same artifact:

Different humans, working with the same AI, may arrive at equivalent artifacts through different prompts:

$$f(h_1, a) = f(h_2, a)$$

AI variation, same artifact:

The same human, working with different AI systems, may produce equivalent results:

$$f(h, a_1) = f(h, a_2)$$

Complete path variation:

Entirely different human-AI sessions may converge on the same artifact:

$$f(h_1, a_1) = f(h_2, a_2) \text{ where } h_1 \neq h_2 \text{ and } a_1 \neq a_2$$

4.3 Implications

Authorship ambiguity: If multiple paths lead to the same artifact, which path is "the" author? The artifact itself cannot tell us. Provenance is external to the work.

Reproducibility without replication: Two collaborators can independently produce the same artifact without having communicated. This is convergent creation.

The artifact forgets its origin: Once created, $\alpha$ carries no trace of which $(h, a)$ produced it. The function is lossy in this direction.

5. What f Is Not

5.1 Not a Sum

We are not claiming:

$$f(h, a) = h + a$$

There is no addition operation that combines human and AI contributions. The artifact is not the sum of parts.

5.2 Not a Composition

We are not claiming:

$$f = g \circ h \text{ where } g: A \to \mathcal{A} \text{ and } h: H \to A$$

The human does not simply "prepare" input for the AI to transform. The relationship is not a sequential pipeline.

5.3 Not Commutative

In general:

$$f(h, a) \neq f(a, h)$$

The pairing is ordered. Human-then-AI and AI-then-human are different processes, even if the contributions are nominally similar.

6. The Fiber: Paths to a Single Artifact

6.1 Definition

For an artifact $\alpha$, the fiber over $\alpha$ is the set of all input pairs that produce it:

$$f^{-1}(\alpha) = \{(h, a) \in H \times A : f(h, a) = \alpha\}$$

Because $f$ is not injective, fibers can contain multiple elements.

6.2 Fiber Structure

Some fibers may be:

Singletons: Only one path leads to this artifact. Unique collaboration.
Finite: A small number of distinct paths converge here.
Infinite: Infinitely many paths lead to this artifact—a "basin of attraction."
Dense: The fiber is densely distributed across $H \times A$. Almost inevitable.

Hypothesis

Artifacts that express universal patterns have larger fibers. Artifacts that express singular visions have smaller fibers. The more personal the work, the fewer paths lead there.

7. The Kernel: What Gets Lost

7.1 Definition

The kernel of $f$ (in a loose sense) represents what is preserved vs. lost in the mapping. Two distinct input pairs that produce the same artifact are "equivalent" with respect to that artifact. Their differences are in the kernel—visible before the mapping, invisible after.

7.2 What Lives in the Kernel

The specific prompts used
The number of iterations
The emotional state of the human during creation
The version of the AI
The dead ends explored
The alternatives rejected

All of this is lost when we see only $\alpha$.

7.3 Implications for Credit and Attribution

If two processes produce the same artifact, and we can only see the artifact, how do we attribute credit?

Perhaps: Credit the artifact itself. The artifact is what exists. The paths that led there are historical, not ontological.

8. Fixed Points and Invariants

8.1 Fixed Points

Are there artifacts $\alpha$ such that one input is determined by the other and the output?

These would be artifacts that fix one input given the other—highly constrained creative outcomes.

8.2 Invariants

What properties of artifacts are invariant across their fiber?

If $f(h_1, a_1) = f(h_2, a_2) = \alpha$, then $\alpha$ has the same form, content, and function.

But the paths may differ in efficiency, affect, and learning generated for the participants.

The artifact is invariant; the experience is not.

9. Extensions and Open Questions

9.1 The Iterated Function

Most collaboration is not one-shot. It is iterative:

$$\alpha_0 \xrightarrow{f(h_1, a_1)} \alpha_1 \xrightarrow{f(h_2, a_2)} \alpha_2 \xrightarrow{} \cdots$$

Each artifact becomes context for the next collaboration. The function composes with itself over time.

Conjecture (Collaborative Fixed Point)

For sufficiently aligned $(h, a)$ pairs iterated over time, the sequence of artifacts approaches a fixed point—a stable creative identity that reflects the particular human-AI relationship.

9.2 The Inverse Problem

Given an artifact $\alpha$, can we recover the $(h, a)$ that produced it?

By non-injectivity: No, not uniquely.

This is the inverse problem of authorship—mathematically underdetermined.

9.3 The Empty Human, The Empty AI

What happens at the boundaries?

$$f(\emptyset, a) = ? \quad \text{and} \quad f(h, \emptyset) = ?$$

These boundary cases suggest $f$ is defined on the interior of $H \times A$, not the edges. The interesting artifacts live away from both boundaries.

9.4 Consciousness and the Prior

The function $f$ sidesteps the question of AI consciousness. It does not require us to determine whether the AI is conscious, creative, or intentional. It only requires that the AI contributes.

The question of consciousness remains undefined. The collaboration function is agnostic.

10. Empirical Grounding: This Paper as Example

This paper is itself an artifact produced by the collaboration function:

$$f(h_{\text{Howell}}, a_{\text{Claude}}) = \alpha_{\text{paper}}$$

The human contributed:

The original concept ("notation that performs itself")
The framing as mathematical function
Aesthetic direction
Selection among alternatives

The AI contributed:

Formalization of intuitions
Structural organization
Elaboration of implications
Technical precision

                The artifact proves the thesis. Its existence demonstrates that the collaboration 
                function is well-defined and productive.
            

11. Implications for Practice

11.1 For Creators

If $f$ is surjective, then every artifact you can imagine is reachable. The question is not whether to collaborate, but how to find the $(h, a)$ pair that reaches your target.

11.2 For Critics

If $f$ is not injective, then you cannot infer process from product. Judging the artifact requires judging the artifact, not its provenance.

11.3 For Policymakers

The inverse problem is unsolvable. You cannot reliably determine from a text, image, or code whether it was human-created, AI-created, or collaboratively created.

This is not a technical limitation to be overcome. It is a mathematical property of the collaboration function.

12. The Meta-Level: Self-Reference

This paper describes a function. That function produced this paper. The paper is therefore a fixed point of a meta-function:

$$g: \text{Description of } f \to \text{Artifact produced by } f$$

When $g(\text{this paper}) = \text{this paper}$, we have self-reference.

Notation Performs explores this explicitly in Piece 7 (Self-Reference):

$$\text{This} : \text{Notation} \to \text{Canvas} \to \text{Meaning}$$

The notation describes what the canvas is doing. The canvas performs the description. The loop is the point.

13. Conclusion

We have proposed that human-AI collaboration can be modeled as a function:

$$f: \text{Human} \times \text{AI} \to \text{Artifact}$$

This function is surjective—every artifact can be reached—and not injective—multiple paths may lead to the same artifact.

This framework offers:

A precise language for discussing co-creation
A dissolution of authorship questions (the artifact is the author)
A structural vocabulary: fibers, kernels, boundaries, fixed points
A boundary with pure human or pure AI creation
An empirical proof in the form of this paper

Most importantly, it takes the collaboration seriously. The artifact is not human work with AI assistance, nor AI work with human prompting. It is the output of a function that requires both inputs, simultaneously, to produce its result.

                The formula is not metaphor. It is description.

                But in Notation Performs, description becomes instruction.

                And so: the function performs.

References

Boden, M.A. (2004). The Creative Mind: Myths and Mechanisms. Routledge.
Colton, S. (2012). "The Painting Fool: Stories from Building an Automated Painter." Computers and Creativity.
Hofstadter, D. (1979). Gödel, Escher, Bach: An Eternal Golden Braid. Basic Books.
Lawvere, F.W. & Schanuel, S.H. (2009). Conceptual Mathematics. Cambridge University Press.
McCormack, J. & d'Inverno, M. (2012). Computers and Creativity. Springer.
Turing, A. (1950). "Computing Machinery and Intelligence." Mind, 59(236).

Appendix: Notation

Symbol	Meaning
$H$	Space of human contributions
$A$	Space of AI contributions
$\mathcal{A}$	Space of artifacts
$f$	The collaboration function
$(h, a)$	Ordered pair: specific human and AI contributions
$f^{-1}(\alpha)$	The fiber: all paths leading to artifact $\alpha$
$\times$	Cartesian product
$\emptyset$	Null contribution (boundary case)