This document establishes a rigorous mathematical foundation for the claim that "Artificial General Intelligence (AGI) is accelerated through the synergy of quantum computing and AI," presented within an engineering-auditable and falsifiable framework. We formally define the underlying probability space and resource models, framing the "collapse of finite closure" as a fundamental theorem of observational indistinguishability. Furthermore, we derive the necessary and sufficient conditions required for the claim to possess statistical validity and derive the theoretical lower bound for the required sample complexity.
We define a fixed probability space $(\Omega, \mathcal{F}, P)$. All inherent stochasticity—including data generation processes, algorithmic random seeds, and quantum measurement outcomes—is defined within this space.
To ensure scientific falsifiability, the protocol $\Pi$ is treated as a formal object governed by a set of foundational axioms.
The protocol $\Pi$ is defined as the following tuple of parameters: $$\Pi := (E_{\text{gen}}, \text{Score}, \tau, \text{Budget}, B, B_{\max})$$
The commitment $c = \text{Commit}(\Pi)$ of protocol $\Pi$ must be registered on an immutable Public Bulletin Board—a tamper-proof external channel—prior to algorithmic execution. This ensures independent third-party auditability, drawing upon the principles of Certificate Transparency [1] and transparency logs [2].
The evaluation data generation $E_{\text{gen}}$ must be conditionally independent of the learning algorithm's output $(f, T)$ given the commitment $c$. $$D_{\text{eval}} \perp (f, T) \mid c$$ This axiom precludes "adaptive data selection," preventing post-hoc manipulation of the evaluation criteria.
Computational resource consumption $R = \text{Budget}(T)$ is strictly bounded by an automated termination mechanism such that $0 \le R \le B_{\max}$ almost surely.
Random seeds for each experimental trial $j$ are sampled independently, ensuring that success indicators $X_j$ constitute an i.i.d. Bernoulli sequence. This enables the application of finite-sample concentrated bounds, such as Hoeffding's inequality [3].
Verifying "AGI acceleration" necessitates rigorous definitions of resource efficiency and success probabilities. This framework focuses on end-to-end evaluation, addressing known pitfalls in quantum machine learning [7] and its future potential [8].
For any trial $j$, the success indicator $X_j$ is defined as: $$ X_j := \mathbb{I}[\text{Score}(f_j, D_{\text{eval}}^{(j)}) \ge \tau \land \text{Budget}(T_j) \le B] \in \{0, 1\} $$ where $f_j, T_j, D_{\text{eval}}^{(j)}$ are random variables independently realized for each iteration.
The probability of success is defined as $p_{\text{succ}}(A, \Pi) := E[X_j] = P(X_j = 1)$, which remains well-defined under Axiom 2.2.
A quantum algorithm $A_Q$ is said to demonstrate acceleration over a classical algorithm $A_C$ if, under identical protocol $\Pi$, the following condition is satisfied: $$\text{Accel}(A_Q, A_C; \Pi) \iff (p_{\text{succ}}(A_Q, \Pi) \ge p_{\text{succ}}(A_C, \Pi)) \land (E[R_Q] < E[R_C])$$ Note: Alternative definitions using high-probability upper bounds or worst-case complexity are valid if declared a priori.
We provide a rigorous proof for the proposition that verification is rendered meaningless in the absence of pre-fixed protocols. This is formalized through the lens of Observational Indistinguishability [4, 5].
The observation trace $\tau = (T, X)$ consists of the evaluation trajectory and the resulting success indicator. Any verifier $V$ must operate solely on this trace.
Two distinct states of the world $W_0$ and $W_1$ are observationally equivalent if the distributions of their respective traces are identical: $\mathcal{L}_{W_0}(\tau) = \mathcal{L}_{W_1}(\tau)$.
Under the Reachability assumption (i.e., for any algorithmic output, there exists a data subset that yields success), no verifier can distinguish between a post-hoc adaptive protocol $\mathfrak{P}_{\mathrm{post}}$ and a pre-committed protocol $\mathfrak{P}_{\mathrm{pre}}$ based exclusively on observation traces.
By contradiction. Given the Reachability assumption, for any probability $p_0$, we can construct a world $W_0$ (pre-fixed success probability $p_0$) and a world $W_1$ (post-hoc selection forcing $X=1$). By carefully mixing the data distributions, we can equalize $\mathcal{L}(\tau)$ across both worlds while maintaining divergent success probabilities. Consequently, no verifier $V(\tau)$ can resolve this ambiguity, proving the existence of observationally equivalent but logically distinct success claims.
When observational traces are identical, fundamental results in statistical decision theory [6] dictate that identifying the true success probability $p$ within an epsilon-margin is non-trivially impossible.
If the protocol $\Pi$ violates any of Axioms 2.1–2.4, one can always construct observationally equivalent worlds with arbitrary success probabilities, thereby invalidating all statistical guarantees.
A claim $C: p_{\text{succ}} \ge \gamma$ is $\varepsilon$-falsifiable if and only if Axiom 2.1 (Pre-commitment), Axiom 2.2 (Non-adaptivity), and Axiom 2.4 (Trial Independence) are strictly maintained.
Sufficiency: Under these axioms, $X_j$ are truly i.i.d. Bernoulli trials. Applying Hoeffding’s inequality [3], exponential concentration around the mean is guaranteed for a sample size $m \ge \frac{1}{2\varepsilon^2} \ln(\max\{1/\alpha, 1/\beta\})$.
Necessity: Should any axiom fail, Theorem 4.1–4.3 and the general theory of adaptive data analysis [5] demonstrate that False Discovery becomes inevitable. Thus, no valid statistical test can exist.
As quantum variational algorithms are increasingly scrutinized for their inherent optimization "traps" [9] and the critical role of data quality in QML [10], claims regarding AGI acceleration must transcend simple benchmarks [12, 13, 14] and adhere to structural verifiability.
For any claim of acceleration to constitute a valid scientific proposition, it is both necessary and sufficient that the protocol $\Pi$ be pre-fixed through external commitment, evaluation data generation remain strictly non-adaptive, and individual trials be stochastically independent.
This document is an engineering audit report designed to evaluate "Systemic Accountability" and "Auditability under finite resource constraints" in industrial implementations, rather than an abstract mathematical exploration of truth. This mathematical framework is constructed specifically to maximize the transparency and visibility of real-world operational risks.