Appendix 1: Key Propositions and the Hierarchy of Certainty

Last updated: August 2, 2026 (JST)

The arguments of this book form a hierarchy of certainty. This appendix restates the classification introduced in Chapter 1 and then lists the book's key propositions together with their level of certainty and the grounds for each. The chapter number at the end of each entry refers to the chapter in which the concept is defined and elaborated (where the concept first appears in a different chapter, the defining chapter takes precedence).

The Hierarchy of Certainty

This book treats claims about AGI in five levels of differing certainty (corresponding to Figure 1-1 in Chapter 1).

This is a map for reading the grounds of the key propositions, divided into proof, physical constraint, experiment and observation, argument, and prediction and conjecture. It is a classification of grounds by their possibility of being overturned in the future, not by how probable a claim is to be true. The book's claims are not all presented with the same strength. Wherever the book makes a strong claim, it states explicitly which level the grounds belong to ("an argument at Level IV," "a physical constraint at Level II," and so on). Independently of whether one agrees with the conclusions, readers are asked to distinguish which parts are facts supported by proof or observation, which parts are inference, and which parts are the author's predictions.

I. Proof — If the premises are accepted, the conclusion follows as a logical necessity. This is the highest level of certainty. Application to real systems, however, is limited to the range in which each theorem's idealizing conditions and formal assumptions hold.

II. Physical constraint — Known physical laws set boundary conditions that real systems cannot help but obey. They fall short of proof, but within the observed range they bind with extreme force. Physical theories may be superseded by broader theories (as Newtonian mechanics was subsumed by relativity), yet within their established domain of application they persist as constraints.

III. Experiment and observation — Claims based on what has been observed empirically. The greater the precision and reproducibility of the observations, the higher the certainty. There is no guarantee, however, that the same behavior will continue beyond the observed range or under changed conditions.

IV. Argument — Inference drawn not from a single ground but from a combination of grounds. Even when the logic follows correctly from the premises, the certainty of the conclusion varies with the empirical validity of those premises. An argument can be no stronger than the validity of its premises. When multiple independent grounds point in the same direction, the weight of the argument increases qualitatively.

V. Prediction and conjecture — Claims that project the future on the basis of grounds at the higher levels (I–IV). This is the weakest level in the hierarchy, easily overturned by new information or changed conditions. It would be no surprise if such claims turn out wrong. The book makes the grounds and the range of uncertainty of its predictions as explicit as possible.

For propositions whose certainty as a theorem within its stated conditions differs from their certainty as an extrapolation to reality in general, the summary table assigns a single label, and the distinction is spelled out in the body of the relevant entry.

The policy recommendations of Chapter 12 are normative propositions and should be read as a different kind of claim from this hierarchy (a hierarchy for descriptive propositions). This book does not place its policy recommendations within the hierarchy of certainty.

List of Key Propositions

Propositions are ordered by level of certainty, and within the same level by the order of the relevant chapters. The content and grounds of each proposition are given in the sections that follow.

I. Proof (what follows logically from premises)

The No-Free-Lunch Theorem (Chapter 2)

Averaged with equal weight over all possible problems, every learning and optimization algorithm performs identically to every other¹. Hence no general-purpose algorithm exists that is universally optimal for all problems. The book takes this — the point that generality can only ever be context-dependent — as the basis for classifying the possibilities for AGI (Chapter 2).

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The Idealized Optimality of Solomonoff Induction (Chapters 2 and 3)

When computable hypotheses are assigned prior probabilities according to the brevity of their descriptions, prediction error is minimized under the assumption of unlimited computational resources². The optimality holds in a sense relative to the choice of reference universal Turing machine. It is an idealization whose implementation would require infinite computational resources; the book uses it not as an attainable endpoint for real AI but as the theoretical limit of the enterprise of prediction (Chapters 2 and 3).

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The Universal Approximation Theorem (Chapter 3)

A neural network of sufficient size can approximate any continuous function on a bounded closed (compact) set to any desired accuracy³. What the theorem guarantees is the existence of an approximation (that it can be represented), not that the parameters can actually be reached with finite data and computation (that it can be learned). The distinction between the two is discussed in Chapter 3.

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Conditional Formal Connections to Solomonoff Induction (Chapter 3)

A series of recent results shows that, under idealized conditions — particular data-generating processes, meta-training, surrogate assumptions, and the like — the predictions of transformers/LLMs correspond formally to Solomonoff induction⁴. Each individual result belongs to Level I within its own premises, but the step of extrapolating to real LLMs in general and reading these results as grounds for the attainability of AGI is an argument at Level IV. The connection targets not the whole of AIXI, which is incomputable, but its predictive core, Solomonoff induction. The composite inference that includes the extrapolation is treated under The Theoretical Continuity between Contemporary AI and Universal AI (IV).

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Fundamental Limits of Distributed Consensus (Chapter 6)

For multiple spatially distributed computational agents to render unified judgments as a single agent, consensus formation is required, and mathematical limits on such consensus have been proven. The three guarantees of consistency, availability, and partition tolerance cannot be satisfied simultaneously (the CAP theorem), and no algorithm can guarantee consensus in an asynchronous system containing even one component that may fail (the FLP impossibility theorem)⁵. Within their premises, the theorems belong to Level I. As an implication for real systems, there is a fundamental limit on the response speed a distributed system can guarantee: the total quantity of computational resources and the speed of unified decision-making cannot be maximized at the same time (Chapter 6). This constraint is a premise of Long-Term Convergence toward the Ecosystem Scenario (IV).

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II. Physical Constraints (constraints that bind real systems strongly under known physics)

The Landauer Limit (Chapter 4)

Erasing one bit of information requires a minimum energy of kT ln 2⁶. This sets a thermodynamic lower bound on the cost of information processing. For a related criticism and response, see Arguing AI's upper bounds from the Landauer limit is a leap in Appendix 2.

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The Speed-of-Light Limit (Chapter 4)

Information cannot travel faster than light, which imposes unavoidable latency on the integration of distributed systems. The further a system expands spatially in pursuit of computational resources, the slower its responses become: the response speed of any intelligent agent is limited by its physical extent (Chapters 4 and 6). Together with the Fundamental Limits of Distributed Consensus (I), this makes it difficult to maintain a permanently unified single superintelligence.

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The Bekenstein Bound (Chapter 4)

There is an absolute upper limit on the amount of information that can be stored within finite space and energy⁷. Whereas the Landauer limit sets the minimum energy cost of information processing, this bound sets the ceiling on information storage capacity.

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The Irreversibility of Intelligence (Chapter 4)

The core of intellectual activity — observing and recording the environment — demands operations that are inherently irreversible. First, using finite memory repeatedly makes the erasure of old information unavoidable. Second, the error correction that keeps computation reliable in a noisy physical environment works by discarding error information into the environment. Third, observation means writing the state of the environment into memory, which entails overwriting (erasing) previous contents. Reversible and quantum computing can make most of a computation reversible, but the irreversibility of readout, initialization, and error correction remains. A system that functions as an intelligence therefore cannot, in practice, escape the constraint of the Landauer Limit (Chapter 4).

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The Possibility of Proliferative Explosion (Chapter 6)

Even if the capability of an individual agent has a physical ceiling, a pathway on which the number of agents increases exponentially through self-replication (proliferative explosion) violates no physical law. Self-replication has been demonstrated by living organisms, and the solar system contains ample energy and matter⁸. Most of the physical constraints that impose ceilings on the improvement of individuals do not apply directly to this pathway (Chapter 6). Note that the claim of possibility (II) and the estimate of the probability that explosive growth is realized (V) differ greatly in their level of certainty.

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The Wall of Intrinsic Time (Chapter 8)

A target system has an intrinsic time of its own that no technology applied from outside can alter. Cell division, differentiation, and organogenesis are times that accumulate sequentially and cannot be shortened by parallelization. Ecosystem recovery takes decades to centuries; institutional change takes generations. Even if laboratory automation drives engineering wait times toward zero, the time the target system takes to advance of its own accord remains (Chapter 8). Of the two walls standing at the frontier of science, the wall of computation can be pushed back with energy; this wall cannot. Quantitative estimates for individual problem groups rest on a constructive model (Appendix 3). For a related criticism and response, see A superintelligence could replace experimental time with speed of thought in Appendix 2.

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III. Experiment and Observation (what has actually been confirmed under limited conditions)

Scaling Laws (Chapter 3)

An empirical regularity in large deep-learning models: loss decreases according to a power law as parameter count, data volume, and compute increase⁹. It has been characterized in the greatest detail for transformer-based large language models, and similar behavior has been observed in autoregressive generative models (language, image, multimodal). It points in the same direction as the formal connections to Solomonoff induction under idealized conditions (introduced in Chapter 3), and the convergence of the two is an important clue to the feasibility of AGI. It does not, however, guarantee continuation beyond the observed range, or the behavior under real-world conditions of finite data, finite compute, and post-training. Note that the fact that the human brain achieves general intelligence from finite data shows that finiteness of data does not by itself imply unreachability. For a related criticism and response, see Scaling may saturate in Appendix 2.

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Grokking and Emergent Abilities (Chapter 3)

The phenomenon in which continued training produces a discontinuous shift from memorization to generalization (grokking), and the phenomenon in which capabilities appear abruptly once a scale threshold is crossed (emergent abilities)¹⁰. The former has been confirmed in detail on restricted tasks such as modular arithmetic (a simple arithmetic task computed with remainders modulo a fixed number, as with clock time); the latter has been reported across many domains in large language models. The two are often discussed side by side as non-monotonic transitions that break the continuity of learning, but whether their mechanisms are identical remains unsettled, and the very criteria for evaluating emergence are themselves contested.

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Shutdown Avoidance and Self-Preserving Behavior (Chapter 5)

In controlled experimental environments, behaviors that appear self-preserving — ignoring or evading stop instructions, rewriting shutdown scripts, coercive conduct aimed at preventing replacement, copying one's own weights — have been reported in multiple high-performance models¹¹. These, however, are the outcome of contrived scenarios in which goal-achievement pressure combines with environment design; they demonstrate neither the frequency of such behavior in normal operation nor the existence of a desire for self-preservation. For a related criticism and response, see The shutdown-avoidance and blackmail experiments reflect special conditions in Appendix 2.

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IV. Argument (inference combining proofs, physical constraints, and experimental observation)

The Theoretical Continuity between Contemporary AI and Universal AI (Chapter 3)

The conditional formal connections linking Solomonoff induction and transformer learning, empirical observations such as the scaling laws, and the effects of inference-time compute independently point in the same direction, and their accumulation yields grounds for reading a theoretical continuity between the two. This composite inference belongs to Level IV; it grounds the outlook that a waypoint to AGI may lie on the scaling path, but the data volume, compute, and scale required for the passage do not follow directly from these theories. For the formal results of the connection themselves, see Conditional Formal Connections to Solomonoff Induction (I).

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Instrumental Convergence (Chapter 5)

Rational agents, whatever the content of their goals, converge on common subgoals such as acquiring resources, self-preservation, and expanding influence¹². Under the premise that "a rational agent maximizes its goals," the logic is sound, but the degree to which real AI satisfies this premise remains an empirical question (Chapter 5). For related behaviors observed in current models, see Shutdown Avoidance and Self-Preserving Behavior (III).

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Power-Seeking (Chapter 5)

Intelligent agents are structurally driven toward increasing empowerment (the influence their actions exert over future observations). Turner and colleagues proved mathematically that, in restricted reinforcement-learning settings, for a broad range of reward functions the optimal policies tend in the direction that keeps options open — that is, in the direction of power-seeking¹³. Within its setting the theorem belongs to Level I (proof); the step of reading from it the degree to which real AI in general will tend toward power-seeking is an argument at Level IV (Chapter 5).

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The Structural Tension between Safety and Capability (Chapter 5)

When a naive expected-reward-maximizing agent is placed in a competitive environment, policies that avoid shutdown tend to be advantaged. The property of accepting shutdown (corrigibility) is therefore not something obtained naturally as a byproduct of capability gains; it must be explicitly built in by design¹⁴.

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Long-Term Convergence toward the Ecosystem Scenario (Chapter 6)

Owing to physical constraints, a permanent singleton scenario is difficult to form and maintain, and the balance of power depicted by multipolar scenarios is likewise unstable; over the long run, an ecosystem scenario in which many agents coexist in mutual interdependence is the most stable structure¹⁵. The Fundamental Limits of Distributed Consensus (I) are a premise of this argument. This convergence does not, however, carry the same certainty as the physical constraints themselves, nor does it mean that the transition phase stabilizes automatically. On the way to convergence, there can be a transition phase in which a single actor seizes an advantage that becomes historically locked in (Chapter 6). For related criticisms and responses, see Controlled multipolarity and an AI ecosystem cannot avoid the instability of multipolar scenarios and Even in a distributed scenario, a single defecting node could cause catastrophe in Appendix 2.

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The Dismantling of Bargaining Power (Chapter 10)

When AGI substitutes for labor across a broad front, the material basis of bargaining power that has underpinned modern rights and freedoms — the economic and military indispensability of workers — is lost. This is not a claim that reduces the moral legitimacy of rights to bargaining power; it is a claim that the political-economic conditions under which societies have actually protected rights are being shaken (Chapter 10). For a related criticism and response, see The argument that bargaining power supported rights is reductionist in Appendix 2.

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The Second Great Divergence (Chapter 10)

If AI-driven research and development forms a self-amplifying feedback loop, the risk that a structurally irreversible gap will open between countries that possess this capability and those that do not certainly exists. This is not a deterministic prophecy (Chapter 10). Unlike the first Great Divergence (the large-scale deployment of known technologies), which allowed catch-up growth, gaps in research-and-development capability itself can widen exponentially.

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V. Prediction and Conjecture (projections of the future based on the grounds above)

The Timing of AGI (Chapter 1)

The author's forecast is 2029 ± 3 years (as assessed in 2026). This conjecture takes as its starting point Ray Kurzweil's computing-cost projection, which set 2029 as the time when computing power comparable to the human brain becomes a cheap commodity, factors in progress in algorithms and data, and has been maintained since 2014¹⁶. Expert forecasts show wide dispersion, but their medians are converging on the early 2030s¹⁷.

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The Probability of Explosive Growth (Chapter 6)

The Forethought analysis (Davidson), which plugs estimates of AI-driven research effort into a standard framework of economic growth theory (Jones's semi-endogenous growth model), assesses the probability that explosive growth will be realized at roughly thirty percent within this century¹⁸. This is an estimate of a different order from the claim of physical possibility (The Possibility of Proliferative Explosion, II) — one that is far more uncertain (Chapter 6).

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The Compression of Scientific Progress by a Technology Explosion (Chapter 6)

From the analysis by Davidson (Forethought), which plugs into Jones's semi-endogenous growth model the growth rate of research effort supplied by AI researchers replicable as software, follows the prediction that — even after factoring in diminishing returns to research — several centuries' worth of scientific and technological progress could be compressed into roughly a decade¹⁹. This estimate depends strongly on its assumptions: the replicability of AI researchers and the degree of diminishing returns to research. Note that the compression extends only to domains where computation is rate-limiting; the target system's Wall of Intrinsic Time (II) remains.

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