GhostDrift Limit Theorem:
Mathematical Boundaries of Decision and Responsibility

Mathematical Necessity of Exploration Intervals for Responsible Decision Making
GhostDrift Research Project
Date: January 4, 2026
Version 4.2 (Final Polished Edition)
Abstract This paper rigorously formalizes the structural conditions required for "Responsibility" in decision-making systems within the framework of Zermelo-Fraenkel set theory without Choice (ZF). We define "Responsibility Fixability" as the existence of a left inverse on the image of a decision mapping and establish its equivalence to injectivity (Lemma 1). Furthermore, we define "Context Richness" as the existence of a surjection from the context space to the power set of the trace space. Utilizing Cantor's theorem, we derive the GhostDrift Limit Theorem, which proves the fundamental impossibility of fixing responsibility in systems lacking an Exploration Interval. Finally, we axiomatize the Exploration Interval as a sequence of events with temporal structure and provide mathematical sufficient conditions to restore Responsibility Fixability. These results provide theoretical boundary conditions for current discussions on accountability, provenance, and algorithmic recourse found in [Kroll2017], [Cheney2009], and [Karimi2022].

1. Introduction

In response to the phenomenon of "Evaporation of Responsibility" in automated and bureaucratic decision-making, this paper provides a structural impossibility proof rather than an ethical critique. We proceed within the axiomatic system of ZF set theory to ensure a constructive proof with minimal assumptions, explicitly excluding dependence on the Axiom of Choice.

1.1 Related Work and Positioning

The claims of this paper intersect with the following research domains: (i) Accountability gaps in algorithmic decision-making (e.g., [Matthias2004], [Kroll2017]). (ii) Data and Decision Provenance frameworks ensuring auditability (e.g., [W3CPROV], [Cheney2009], [Singh2019]). (iii) Robustness of algorithmic recourse and post-hoc explanations (e.g., [Karimi2022], [Upadhyay2021], [Rudin2019]).

While most existing studies discuss "what should be explained or recorded" as a matter of policy or design, this paper is distinct in providing a necessary condition (impossibility) from a set-theoretic perspective: without an Exploration Interval (Exploration Log), Responsibility Fixability (the existence of a left inverse) cannot exist mathematically.

2. Definitions and Basic Lemmas

2.0 Set-Theoretic Preliminaries (ZF)

We proceed within the framework of ZF set theory. We adopt the standard Kuratowski definition for ordered pairs, \((a,b):=\{\{a\},\{a,b\}\}\), and treat relations as sets of ordered pairs. A function (or mapping) \(f\) is defined as a relation such that for any \(x\) in the domain \(\mathrm{Dom}(f)\), there exists a unique \(y\) satisfying \((x,y)\in f\); this unique \(y\) is denoted as \(f(x)\). The notation \(f:A\to B\) implies \(\mathrm{Dom}(f)=A\) and \(\mathrm{Im}(f)\subseteq B\). The image is defined as \(\mathrm{Im}(f):=\{y\in B\mid \exists x\in A\,((x,y)\in f)\}\) (via the Axiom of Separation), and the restriction mapping is \(f\upharpoonright S:=\{(x,y)\in f\mid x\in S\}\). A function is injective if \(\forall x_1,x_2\in\mathrm{Dom}(f)\,(f(x_1)=f(x_2)\Rightarrow x_1=x_2)\), and surjective if \(\mathrm{Im}(f)=B\).

2.1 Mathematical Model and Responsibility Fixability

We assume the following entities are sets within ZF:

Note: The concept of "Responsibility Fixability" defined below is isomorphic to "identifiability" in statistical causal inference (Compare: [Halpern2016], [Pearl2009]). It ensures the causal origin can be uniquely determined from the effect.
Definition 1 (Responsibility Fixability) A mapping \(\Phi: \mathcal{C} \to \mathcal{X}\) is said to be Responsibility Fixable if there exists a mapping \(R: \mathrm{Im}(\Phi) \to \mathcal{C}\) such that for any \(c \in \mathcal{C}\): $$ R(\Phi(c)) = c $$

2.2 Fundamental Equivalence

Lemma 1 (Equivalence of Fixability and Injectivity) For any mapping \(\Phi: \mathcal{C} \to \mathcal{X}\), the following are equivalent:
  1. \(\Phi\) is Responsibility Fixable.
  2. \(\Phi\) is Injective.
Proof

(1 \(\Rightarrow\) 2)
Assume \(\Phi\) is Responsibility Fixable with left inverse \(R\). Let \(\Phi(c_1) = \Phi(c_2)\). Since this value is in \(\mathrm{Im}(\Phi)\), applying \(R\) yields: $$ c_1 = R(\Phi(c_1)) = R(\Phi(c_2)) = c_2 $$ Thus, \(\Phi\) is injective.

(2 \(\Rightarrow\) 1)
Assume \(\Phi\) is injective. Consider the class of ordered pairs defined by \(c\mapsto(\Phi(c),c)\) for all \(c\in\mathcal{C}\). By the Axiom of Replacement, the set $$ R := \{(\Phi(c),c)\mid c\in\mathcal{C}\} $$ exists. We show \(R\) is a function. If \((x,c_1)\in R\) and \((x,c_2)\in R\), then \(x=\Phi(c_1)=\Phi(c_2)\). By injectivity, \(c_1=c_2\). Thus, \(R\) maps each \(x\) to a unique \(c\). The domain of \(R\) is \(\mathrm{Im}(\Phi)\), and for any \(c\in\mathcal{C}\), since \((\Phi(c),c)\in R\), it follows that \(R(\Phi(c))=c\).

Q.E.D.

3. GhostDrift Limit Theorem

We now formulate the condition where the context space \(\mathcal{C}\) is "sufficiently complex" relative to the trace space \(\mathcal{X}_0\). We term this "Context Richness."

3.1 Context Richness and Cantor's Theorem

Definition 2 (Context Richness) The set \(\mathcal{C}\) is said to be Rich relative to the trace space \(\mathcal{X}_0\) if there exists a surjection from \(\mathcal{C}\) onto the power set of \(\mathcal{X}_0\): $$ \exists h:\mathcal{C}\twoheadrightarrow \mathcal{P}(\mathcal{X}_0). $$
Lemma 3 (Cantor / Non-existence of Surjection to Power Set) For any set \(X\), there exists no surjection from \(X\) onto \(\mathcal{P}(X)\): $$ \neg\exists f:X\twoheadrightarrow \mathcal{P}(X). $$
Proof (Lemma 3)

Let \(f:X\to\mathcal{P}(X)\) be an arbitrary mapping. By the Axiom of Separation, the set $$ D:=\{x\in X\mid x\notin f(x)\}\subseteq X $$ exists. For any \(x\in X\), if \(D=f(x)\), then \(x\in D \iff x\notin f(x) \iff x\notin D\), a contradiction. Thus, \(\forall x\in X, D\neq f(x)\). Consequently, \(f\) cannot be surjective.

Q.E.D.

3.2 Proof of the Limit Theorem

Theorem 1 (GhostDrift Limit Theorem / DR-Limit Theorem)

Assumption: \(\mathcal{C}\) is Rich relative to \(\mathcal{X}_0\) (Definition 2).

Claim: No decision mapping \(\Phi_0: \mathcal{C} \to \mathcal{X}_0\) is Responsibility Fixable.

Proof

We proceed by contradiction. Assume there exists a mapping \(\Phi_0: \mathcal{C}\to\mathcal{X}_0\) that is Responsibility Fixable.
1. By Lemma 1 (1 \(\Rightarrow\) 2), \(\Phi_0\) is injective.
2. Following the construction in Lemma 1 (2 \(\Rightarrow\) 1), there exists a left inverse \(R:\mathrm{Im}(\Phi_0)\to\mathcal{C}\) such that \(R(\Phi_0(c))=c\) for all \(c\).
3. By the Assumption (Definition 2), there exists a surjection \(h:\mathcal{C}\twoheadrightarrow\mathcal{P}(\mathcal{X}_0)\).
Now, consider the mapping \(g:\mathcal{X}_0\to\mathcal{P}(\mathcal{X}_0)\) defined as: $$ g(x):=\begin{cases} (h\circ R)(x) & \text{if } x\in \mathrm{Im}(\Phi_0),\\ \emptyset & \text{if } x\notin \mathrm{Im}(\Phi_0). \end{cases} $$ (Note: The graph of \(g\) is constructible in ZF via Separation and Union axioms.)
For any subset \(S\in\mathcal{P}(\mathcal{X}_0)\), since \(h\) is surjective, there exists \(c\in\mathcal{C}\) such that \(h(c)=S\). Let \(x:=\Phi_0(c)\). Then \(x\in\mathrm{Im}(\Phi_0)\), and: $$ g(x) = h(R(\Phi_0(c))) = h(c) = S. $$ Thus, \(g\) is surjective onto \(\mathcal{P}(\mathcal{X}_0)\). However, by Lemma 3 (Cantor's Theorem), no such surjection exists. This is a contradiction. Therefore, \(\Phi_0\) cannot be Responsibility Fixable.

Q.E.D.

Note: Theorem 1 demonstrates impossibility specifically under the assumption of Richness (\(\exists h:\mathcal{C}\twoheadrightarrow\mathcal{P}(\mathcal{X}_0)\)). If \(\mathcal{C}\) lacks this richness (i.e., the context is constrained), this specific impossibility proof does not apply.

4. Sufficient Conditions for Resolution via Exploration Intervals

4.0 Mathematical Axiomatization of Exploration Intervals

We axiomatize the Exploration Log as a finite sequence of events. Let \(\mathcal{E}\) be the set of events (observations, branches, etc.), and let \(n\in\omega\) be a finite ordinal. A log of length \(n\) is a function \(\gamma:n\to\mathcal{E}\). We define the log space as: $$ \mathcal{L}:=\bigcup_{n\in\omega} \mathcal{E}^n $$ The generation rule is a relation \(P\subseteq \mathcal{C}\times\mathcal{L}\) assigning a unique \(\gamma\) to each \(c\), defining the function \(\ell: \mathcal{C} \to \mathcal{L}\). (Existence: \(\mathcal{L}\) exists in ZF via Power Set, Separation, Replacement, and Union axioms).

4.1 Fiber Separation and Resolution

Consider the extended mapping \(\Phi_1(c) = (\Phi_0(c), \ell(c))\). Let \(F_x = \{ c \in \mathcal{C} \mid \Phi_0(c) = x \}\) be the fiber of \(\Phi_0\) over \(x\).

Lemma 2 (Injectivity Condition for Extended Mapping) \(\Phi_1\) is injective if and only if for all \(x \in \mathcal{X}_0\), the restriction \(\ell|_{F_x}: F_x \to \mathcal{L}\) is injective.
Proof

(\(\Rightarrow\)) Assume \(\Phi_1\) is injective. For any \(x\in\mathcal{X}_0\) and \(c_1,c_2\in F_x\) with \(\ell(c_1)=\ell(c_2)\), we have \(\Phi_1(c_1)=(\Phi_0(c_1),\ell(c_1))=(x,\ell(c_2))=\Phi_1(c_2)\). Thus \(c_1=c_2\).
(\(\Leftarrow\)) Conversely, if \(\Phi_1(c_1)=\Phi_1(c_2)\), then \(\Phi_0(c_1)=\Phi_0(c_2)=:x\) and \(\ell(c_1)=\ell(c_2)\). Thus \(c_1,c_2 \in F_x\). Since \(\ell|_{F_x}\) is injective, \(c_1=c_2\).

Q.E.D.

Corollary 1 (Sufficient Condition for Resolution) If the exploration log \(\ell\) is fiber-separating—that is, \(\forall x\in\mathcal{X}_0, \ell\upharpoonright F_x\) is injective—then \(\Phi_1\) is Responsibility Fixable.
Implementation Example: The condition of Corollary 1 is not tautological; it provides a design requirement. For instance, if \(\ell(c)\) captures a sufficient statistic of the context or a unique commit hash generated during the exploration of \(c\), then \(\ell\) distinguishes all elements within the same fiber \(F_x\), rendering \(\Phi_1\) Responsibility Fixable.

5. Conclusion and Implications

5.1 Post-hoc Impossibility

Theorem 1 implies that under rich contexts, any mapping \(\Phi_0\) based solely on the trace space \(\mathcal{X}_0\) is necessarily non-injective. Consequently, there exist distinct contexts \(c_1, c_2\) yielding the same result \(x\). For such \(x\), no function \(R\) exists to uniquely identify the cause. Thus, any attempt to assign responsibility becomes a Post-hoc Impossibility.

5.2 Conclusion

We have established the "necessity of Exploration Intervals" as a theorem within ZF set theory. Responsibility Fixability requires injectivity (Lemma 1), and injectivity fails for rich contexts without exploration data (Theorem 1). The condition for restoring responsibility is precisely the fiber separation property of the log (Lemma 2).

Appendix A. Humanistic Background: Conditions of Responsibility

This paper presents a limitative claim: without specific structural conditions (exploration intervals), the attribution of responsibility is mathematically undefined. This form parallels Gödel's incompleteness theorems or Arrow's impossibility theorem, where internal mathematical limits inform external philosophical frameworks ([Godel1931], [Arrow1951]).

A.1 "Giving Reasons" and Responsibility

Philosophically, responsibility is linked to the agent's ability to offer reasons ([Strawson1962], [Anscombe1957]). The Exploration Interval Log serves as the minimal structural condition to preserve these reasons (branches, rejections) for verification.

A.2 "Evaporation of Responsibility" in Modern Institutions

In bureaucracy and automation, the link between decision and reason often "evaporates" ([Weber1946], [Arendt1963]). We formalize this not as a moral failure, but as a mapping-theoretic impossibility when information is lossily compressed.

A.3 Return to Engineering

This connects directly to Decision Provenance ([Singh2019]). The Exploration Log is the implementation of the fiber-separating function required to make systems mathematically accountable.

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