I Introduction: The Promises and Unexamined Assumptions of RSI

1.1 The Current State of RSI

In 2026, Recursive Self-Improvement (RSI) leapt suddenly from a long-standing theoretical construct to the central narrative of the AI industry. In May, Anthropic’s report “When AI Builds Itself” disclosed that its flagship model Claude had written over 80% of the company’s merged code, and engineers were shipping eight times more code per quarter than in 2024. In April 2026, a single Claude-assisted engineering sprint delivered over 800 API fixes, reducing API errors by a factor of 1,000 — engineers estimated this would have taken four years to accomplish by humans alone. In July, OpenAI introduced the RSI Index alongside the release of GPT-5.6 — a metric specifically designed to measure a model’s self-improvement capability. GPT-5.6 Sol scored 16.2 points higher than GPT-5.5 on this index. In a demonstration, Sol independently selected training configurations, chose GPUs, and ran post-training scripts to generate the smaller Luna model — all from a single rough prompt. Eight top AI researchers led by Yuanqing Tian co-founded Recursive Superintelligence Inc., entering the space at a $4.65 billion valuation. The ICLR 2026 workshop “AI with Recursive Self-Improvement” attracted extensive participation from both academia and industry, making explicit that RSI had transitioned from thought experiment to deployed system.

1.2 The Core Assumptions of RSI

These advances rest jointly on three progressively layered core assumptions. First, AI systems can recursively improve themselves — this has received initial validation in the coding domain. Second, success in coding and mathematics can naturally transfer to the physical sciences — this remains unvalidated but is widely believed. Third, automated AI research will lead to automated scientific research, and thence to automated physical discovery — this is the ultimate promise of the RSI roadmap. The roadmap of the RSI company co-founded by Tian envisions as its first step training a system with the capability of “50,000 PhDs” to automate AI science research itself; this “Eureka Machine” would then be directed at drug development, battery materials, and nuclear fusion physics. The central task of this paper is precisely to subject the second assumption — whether the transfer from closed systems to open systems is viable — to critical examination.

1.3 Core Arguments of This Paper

This paper will argue three interrelated propositions. First, the success of RSI is strictly confined to “autotelic verification systems” — domains in which the system itself serves as both problem-setter and judge. Second, the transfer from autotelic systems to open systems is not a matter of degree but a categorical rupture. Third, the root of this rupture can be traced to fundamental differences between mathematics and physics in the directionality of definitions, verification structures, and modes of reasoning. Understanding these differences holds practical significance for the direction of RSI development and touches upon core philosophical questions about the nature of scientific discovery.

II Two Verification Systems: Autotelic and Exotelic

2.1 Definition and Characteristics of Autotelic Verification Systems

In the world of mathematics and programming, the system itself serves as both problem-setter and judge. Whether a mathematical theorem holds can be determined within the system through a formal verifier, without appeal to any information from the external world. Whether a piece of code is correct is adjudicated by the compiler and test cases. We call this verification structure an “autotelic verification system.” Its core characteristics are fivefold: the system simultaneously serves as problem-setter and judge; the correctness of mathematical proofs is determined internally by the system; code correctness is determined by compilers and test cases; “improvement” and “verification of improvement” use the same language and rules; its paradigmatic instances include mathematical proof systems, software engineering, and RSI’s reward model.

RSI’s success in such systems is no accident. When a self-improving AI agent modifies its own code and runs the test suite, the verification signal it receives is immediate, unambiguous, and definitive: pass or fail. This provides an ideal feedback loop for recursive optimization cycles.

2.2 Definition and Characteristics of Exotelic Verification Systems

In the world of the physical sciences, the judge is not the system itself but nature. No matter how elegant your theory or how self-consistent your mathematics, if nature does not nod, everything returns to zero. However, the feedback nature provides has five characteristics that are diametrically different from autotelic systems: the judge is nature, not the system itself; feedback is ambiguous, delayed, and polysemous; the same set of experimental data can support multiple mutually contradictory theories; nature will not proactively tell you “you asked the wrong question” — it simply remains silent; its paradigmatic instances include physics experiments, drug development, and materials science.

2.3 The Incommensurability of the Two Systems

A large-scale survey of 1,250 arXiv papers (2024–2026) published in July 2026 revealed a core pattern: demonstrated self-improvement capability strictly tracks a “verification hierarchy” — from formal verifiers (strongest) to the model’s own intrinsic evaluation (weakest). Self-training is effective in domains where “answers are checkable” and systematically degenerates in domains where they are not. In the same year, Princeton University’s “shadow evaluation” experiment provided the first direct empirical support: frontier AI models were given a $3,000 budget and six days to solve real, unpublished, open research problems. The result: AI completed all of the engineering work, but the papers it produced were explicitly rejected by domain experts. The research team noted that this “research judgment gap” is precisely the most critical bottleneck in the strongest versions of RSI’s predictions. In autotelic systems, “better” is computable — higher scores, shorter proofs, lower loss. In exotelic systems, “better” is itself an open question. Equating success in the former with capability in the latter is a category error.

III The Directionality of Definitions: Inward Constraint vs. Outward Opening

3.1 The Inward Constraint of Mathematical Definitions

Every act of definition in mathematics is an act of cutting. To define the real numbers is to exclude the non-real. To define a continuous function is to exclude the discontinuous. To define a group is to stipulate that you must satisfy closure, associativity, identity, and inverse — anything that fails is expelled. The more precise a mathematical definition, the higher its walls, the smaller its territory, but the safer its interior. It is no exaggeration to say that the entire history of mathematics is a history of progressively tightening inward constraints. Under this definitional paradigm, the meaning of a symbol is entirely exhausted at the moment the axioms are written down. There is no possibility of “residual meaning” seeping in from outside. This closure is the source of mathematical certainty, and the foundation of its power.

3.2 The Outward Opening of Physical Definitions

The symbols in physical equations appear formally identical to mathematical symbols, but their nature is entirely different — they are openings that point toward the real world. Consider E in E=mc². What is “energy,” exactly? In Newton’s era, it was kinetic and potential energy. Thermodynamics added thermal energy. Einstein added mass-energy equivalence. Quantum field theory added vacuum energy. Dark energy — to this day, no one knows what it actually is. The meaning of the symbol “E” has never been exhausted. Every advance in physics has injected content into this symbol that its creator never imagined.

When Newton defined “force” as F=ma, he was not constraining what “force” can be. He was opening a window toward the universe and saying: anything that can produce acceleration, come in. Gravity entered. Electromagnetism entered. The nuclear force entered. Dark energy entered. Newton had absolutely no knowledge of the latter three when he wrote the definition. But the direction of the definition was outward, so it could accommodate content its creator never imagined. Physical symbols are “openings pointing toward reality” — nature never closes its definitions.

3.3 Implications of Definitional Directionality for RSI

The entire operational mechanism of RSI — loss functions, reward models, formal verifiers, search algorithms — is built on closed definitions. Every tensor has a precise mathematical definition, every gradient has a precise mathematical definition, every reward has a precise mathematical definition. There are no “openings pointing toward reality.” Consequently, RSI can only improve within the defined space; it cannot discover what lies beyond the definitions. It is a machine that runs ever faster inside its walls, but the walls themselves are beyond its reach.

IV Extra-Domain Effects: The Supreme Value of Open Systems

4.1 Definition of Extra-Domain Effects

An extra-domain effect occurs when a theory’s implications exceed its creator’s ontological horizon — the equations know more than their creator. This is the most profound characteristic of physics, and its most awe-inspiring quality. It is not a by-product, not a coincidence, but the core driving mechanism of physics’ progress. Extra-domain effects are possible precisely because the symbols in physical equations are outward-opening — they point to a reality richer than the creator’s cognition.

4.2 Cases of Extra-Domain Effects in Physics

Maxwell wrote four equations for electromagnetism. The equations themselves “predicted” the existence of electromagnetic waves, which were later discovered to be light. Three fields previously thought entirely unrelated — optics, electricity, and magnetism — were unified by a single set of equations. This prediction was not in Maxwell’s original ontology. The equations “leaked” into a domain they themselves did not know about.

Dirac wrote the relativistic electron equation, only to find that the equation produced negative-energy solutions. In a purely mathematical context, such “non-physical solutions” would be discarded. But because the symbols in Dirac’s equation pointed toward physical reality, he was compelled to ask: Does this “illegal” solution correspond to something in nature that has not yet been discovered? The answer was the positron — antimatter. A mathematical equation from 1928 predicted a particle not discovered until 1932.

Einstein’s general relativity predicted black holes, gravitational waves, and cosmic expansion — he personally opposed several of these implications. The equations were smarter than their creator, not because the equations possessed magic, but because the definitions of the symbols were not closed, and nature poured information the creator did not know through these unclosed definitions.

4.3 Why Mathematics and Code Have No Extra-Domain Effects

The conclusion of a mathematical proof is always within the ontology of its axiom system. The output of a piece of code is always within the definitions of its runtime environment. The variable energy = 0.5 * mass * velocity**2 in code — here, energy is simply a floating-point number whose meaning was entirely exhausted the instant that line was written. It can never spontaneously “discover” that it also equals mass * c**2. Because it has no opening toward the real world. They are sealed containers into which nature’s information cannot flow.

In 1960, physicist Eugene Wigner published the famous paper “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” marveling at why mathematical concepts can “leak” into the physical world. But this paper’s analysis provides an inverted reading: it is not that mathematics spontaneously produces extra-domain effects, but that physicists actively conscript mathematics. Riemannian geometry did not “run” into general relativity on its own — Einstein went looking for it. The direction of extra-domain effects is always physics → conscripting mathematical tools, not mathematics → spontaneously leaking into physics. As Yang Chen-Ning put it: “Nature selects only a subset of the mathematics that mathematicians developed, and that precise subset is what theoretical physicists seek.”

4.4 The Fundamental Limitation of RSI as a System Without Extra-Domain Effects

The foregoing analysis points to an inescapable conclusion: RSI is a methodology that works perfectly within systems devoid of extra-domain effects, which its advocates seek to apply to a domain where extra-domain effects constitute the supreme value. RSI can enable faster and deeper searching within the known ontology, but it cannot produce those mysterious moments when the equations know more than the human. It is an intra-domain amplifier — achieving superhuman search efficiency within the existing conceptual space; but it is not an extra-domain generator — it cannot cause a computational system’s output to systematically exceed its input ontology. Breaking through this limitation may require inventing a mechanism that does not yet exist — an “extra-domain generator” that enables computational output to systematically transcend its input ontology. We do not even know whether such a mechanism is possible in principle.

V The Ontological Boundary of Methodology

5.1 Autotelic Systems: The Methodological Boundary Is the Ontology

In autotelic verification systems, the effective range of methodology perfectly coincides with the boundary of ontology. RSI’s recursive loop — model improving model, algorithm optimizing algorithm, code rewriting code — no matter how many iterations, spins within the same ontological framework. It can only discover truths within the existing ontological framework. The recursive loop is an intra-ontological recursion; no matter how many iterations, it searches within the same conceptual space and cannot escape the conceptual space itself.

5.2 Open Systems: Methodology Can Extend Beyond Ontology

The greatest breakthroughs in the history of physics have, without exception, been revolutions at the ontological level — methodology exceeding the boundary of the current ontology and arriving at new categories the creator never envisioned. This is the precise definition of “openness.” Newton did not merely explain planetary motion better within the existing framework; he invented an entirely new ontological category — “force.” Before him, this invisible, intangible entity acting at a distance did not exist in physics’ ontological lexicon at all. Einstein did not patch parameters within Newton’s framework; he abolished the ontological presupposition of “absolute spacetime” and replaced it with the entirely new entity of “curved spacetime.” Quantum mechanics was more extreme still — it introduced categories such as “superposition” and “wavefunction collapse” that were literally unimaginable within classical ontology. Bohr said, “If quantum mechanics hasn’t shocked you, you haven’t understood it” — what is shocking is precisely that the ontology was torn apart. And once methodology remains effective after exceeding ontology, extra-domain effects are produced. Mathematics and code never produce extra-domain effects precisely because their methodologies are strictly confined within their ontologies.

5.3 RSI’s Evaluation Function Paradox

This analysis reveals a deep paradox within RSI. At the moment an ontological revolution occurs, the old metric system itself becomes invalid. When Einstein’s general relativity was first published, most physicists considered it “inferior” to Newtonian mechanics — because it was more complex while its predictive differences were minuscule. Its “betterness” was not validated until decades later. You cannot use the old evaluation function to recursively optimize your way to a breakthrough that requires a new evaluation function to even be recognized. This is an echo of Gödel’s incompleteness theorem in the AI context — a sufficiently powerful formal system cannot prove its own consistency from within itself. RSI, as a formalized self-improvement loop, is naturally subject to this limitation. The real challenge for Tian and his colleagues is not an engineering problem, but this: How do you get a machine to make a discovery when even the evaluation function for that discovery does not yet exist?

VI The Cognitive Status of Counterexamples: Elimination or Pursuit

6.1 Counterexamples in Mathematics: A Death Sentence

In mathematics, a single counterexample possesses absolute destructive power. Euler’s conjecture stood for two hundred years; one counterexample demolished it completely. The entire game of mathematics is this — your theorem must hold in all possible worlds, with not a single exception permitted. Consequently, the mathematician’s professional instinct is to fear counterexamples, to eliminate counterexamples, and to seal off any possibility of their appearance through rigorous proof before they can surface. This instinct is continually reinforced through years of professional training, ultimately becoming part of the mathematician’s cognitive architecture.

6.2 Counterexamples in Physics: An Invitation to the Nobel Prize

In physics, the fate of counterexamples is diametrically opposite. The starting point of every revolution in the history of physics has been a counterexample that “should not have appeared.” The 43 arc-second precession of Mercury’s perihelion was a counterexample to Newtonian mechanics, but it gave birth to general relativity. The “ultraviolet catastrophe” — the classical prediction that blackbody radiation diverges at high frequencies — was a counterexample to thermodynamics, but it gave birth to quantum mechanics. The Michelson-Morley experiment’s failure to find the aether — this experiment’s “failure” was the greatest failure in the history of physics, giving birth to special relativity. The apparent violation of energy conservation in beta decay prompted Pauli to predict the neutrino. In physics, counterexamples are not system failures but new information that nature forces in through “outward-opening definitions.” Feynman said: “If your theory disagrees with experiment, it’s wrong.” But the tone in which he said this was not sadness — it was excitement. Because a good counterexample means nature has just revealed a secret you didn’t know.

6.3 AI’s Loss Is Physics’ Anomaly

One of this paper’s most central insights surfaces here: the loss signal in AI training and the anomaly in physics are, in information-theoretic terms, the same thing — both are “the gap between expectation and reality.” However, this same signal receives diametrically opposite treatments in the two systems.

Consider a thought experiment: If RSI were tasked with processing the 43 arc-second deviation in Mercury’s perihelion, what would it do? It would fine-tune the parameters of Newtonian mechanics, fitting away the deviation within the classical framework. Loss reduced to zero, training complete. But general relativity would never be discovered. Because what Einstein did was not to eliminate this loss, but to interrogate it: What is this deviation telling me? Perhaps it is not the parameters that are wrong, but the entire theory’s ontological presuppositions — perhaps spacetime itself is curved. By the same logic, if AI had fitted away the ultraviolet catastrophe within the classical framework, quantum mechanics would never have been born.

6.4 RSI as a Counterexample-Elimination Machine: A Structural Blind Spot

RSI’s entire optimization loop is, in essence, a counterexample-elimination machine. Loss goes up? Adjust parameters to eliminate it. Model output doesn’t match the reward signal? Modify weights to eliminate it. Test case fails? Change the code to eliminate it. Every iteration of the system pursues a world with fewer counterexamples and higher consistency. Every step of training reinforces the instinct to “eliminate deviation.” But what physicists do is precisely the opposite — they search for counterexamples, treasure counterexamples, and treat counterexamples as portals to new continents.

Breaking through this ceiling requires not stronger gradient descent, but an entirely new paradigm — not minimize(loss) but interpret(loss). Not asking “How do I suppress this deviation?” but asking “What is this deviation trying to tell me?” But interpret is not an optimization problem; it is a meaning-generation problem. And meaning generation requires precisely those capabilities that RSI architecturally lacks: open definitions, exotelic verification, extra-domain effects, and ontological transcendence.

6.5 The Technical Implementation of Inward Constraint: Next Token Prediction

The foregoing analysis revealed the directional opposition between minimize(loss) and interpret(loss). This section anchors that opposition to the core technical mechanism of current AI — Next Token Prediction. Every token generation in an autoregressive language model asks the same question: “Given the known context, what is the statistically most likely next word?” This is, in essence, a local optimum search within the training distribution. Researchers have noted that this architecture inherently biases toward local coherence and incremental extrapolation, impeding the bold conceptual leaps required for scientific discovery. The preference for “safe continuation” strangles genuine novelty. A 2025 paper even bore the direct title “Next Token Prediction Is a Dead End for Creativity,” arguing that token prediction is fundamentally misaligned with genuine creativity — the architecture favors surface coherence over spontaneity, originality, and improvisational risk.

When a physicist inputs an entirely novel concept that is completely absent from the training data, the situation becomes even more severe. The concept is not on the conventional long tail of the distribution — it is not in the distribution at all. The softmax probability distribution trends toward uniformity — no single token’s probability is significantly higher than any other. The attention matrix can find no meaningful key-query match. What the model does is not “sample from a weak but directional distribution” but “sample from a near-uniform distribution” — output that is essentially random noise wrapped in fluent grammar. Research has found that just 1% of the highest-activation weights in a model disproportionately encode access to diverse long-tail concepts; for genuinely novel physics concepts, these critical weights simply do not exist. In this sense, Next Token Prediction is the computational implementation of “inward constraint” — every token’s generation is pulled inward by the preceding context and training distribution, unable ever to leap outward.

VII The Fourier Case: The Physics-First, Mathematics-Follows Paradigm

7.1 Fourier’s Historical Significance

Joseph Fourier (1768–1830) provides a near-perfect historical case for this paper’s argument. He participated in Napoleon’s 1798 Egyptian expedition as a member of the entourage, was appointed governor of Lower Egypt, and later served as Prefect of Isère upon returning to France. But what made him immortal in the history of science was his study of heat conduction — a thoroughly physical problem, not a mathematical one. His research started from a problem in an exotelic verification system: How does heat flow through a solid? The answer lay with nature, not with an axiom system.

7.2 Open Definitions Infuriate Mathematicians

Fourier proposed that any function could be decomposed into an infinite series of sines and cosines. This claim sent the most authoritative mathematician of the era, Lagrange, into a fury — from the mathematician’s “inward constraint” perspective, the claim lacked rigorous proof, and even the definitions were unclear. What exactly does “any function” include? Which functions qualify, and which do not? How is convergence guaranteed? Lagrange directly rejected Fourier’s paper. To mathematicians, Fourier had committed an unforgivable error: his definitions were open, vague, and unrigorous. But to physicists, this was precisely the source of his power.

7.3 Physical Verification First, Mathematical Formalization Follows

Physics told Fourier he was right — heat did indeed conduct in precisely the manner he described. Nature provided the verification, even though the mathematician’s logical closure had not yet been achieved. He did not care — his definitions were outward-opening, pointing toward physical reality rather than an axiom system. Over the next century and more, mathematicians expended enormous effort “cleaning up” the “mess” Fourier had left behind: Dirichlet provided convergence conditions, Riemann redefined the integral, Lebesgue invented measure theory, and the entire fields of functional analysis and Hilbert space theory can be traced to the work mathematicians did to formalize Fourier’s physical intuition. This is the textbook case of “physicists blaze the trail, mathematicians follow.”

7.4 A Paradigm of Extra-Domain Effects

The extra-domain effects of Fourier analysis are staggering. A tool for studying heat conduction later “leaked” into quantum mechanics, signal processing, image compression (the core algorithms of JPEG and MP3), acoustics, seismology, astronomical spectral analysis, and other fields he never remotely imagined. A tool born from a physical problem invaded the boundaries of every discipline — this is precisely the extra-domain effect that outward-opening definitions permit. Fourier studied heat, but his method became the universal language for understanding the entire physical world. Such effects cannot occur in systems with closed definitions.

7.5 The Bourbaki School: The Ultimate Embodiment of Inward Constraint

Fourier’s case forms a precise counterpoint to the Bourbaki school. Bourbaki — this collective of mathematicians named after a French general — set forth an ambition in the 1930s to rebuild all of mathematics by the axiomatic method. Their goal was explicit and radical: “on the basis of a streamlined logical foundation derived from set theory, in the tradition of the Hilbert school and Göttingen, but excluding the needs of physics and computation.” They even used the singular form “mathématique” rather than the plural “mathématiques” to emphasize mathematics’ unity and closure. Bourbaki was the ultimate embodiment of the “autotelic verification system” — transforming mathematics into a perfect prison with every window sealed shut, internally impeccable in logic, but explicitly barring information from the physical world. And today’s RSI, in its philosophical DNA, is Bourbaki’s digital reincarnation — pursuing recursive self-improvement within a system from which openness has been explicitly excluded.

VIII Three Modes of Reasoning and the Cognitive Shaping of Professional Training

8.1 Peirce’s Three-Stage Theory

American philosopher Charles Sanders Peirce proposed an influential framework for scientific reasoning in 1903: any complete process of inquiry must pass through three stages — first, abduction, proposing a hypothesis to explain a surprising fact; then, deduction, tracing the necessary consequences of that hypothesis; and finally, induction, testing and generalizing those consequences. Peirce specifically noted: of the three modes, “deduction is the safest but the most barren, and abduction the most productive but the least safe.” He asserted: “Abduction is the only logical operation that introduces any new ideas.” Going further, he declared that “knowledge could not make even the smallest advance without abductive reasoning at every step.” Peirce also described abduction as a kind of “guessing instinct” — “The abductive suggestion comes to us like a flash. It is an act of insight, although of extremely fallible insight.”

8.2 The Physicist’s Reasoning Cycle

The physicist’s typical cognitive cycle is: abduction (propose hypothesis) → deduction (derive consequences) → induction (experimental verification) → new abduction… The three reasoning forces maintain equilibrium within this cycle, but abduction is the engine. At age five, Einstein saw a compass and performed an abduction — “Something invisible must be at work in space.” Kekulé dreamed of a snake biting its tail and performed an abduction — “Perhaps the benzene molecule is ring-shaped.” Poincaré, at the instant of stepping onto a carriage, performed an abduction — “Fuchsian functions and non-Euclidean geometric transformations are the same thing.” The spark of every revolutionary discovery was ignited by abduction.

8.3 The Mathematician’s Reasoning Cycle

The mathematician’s typical cognitive cycle is markedly different: deduction (prove theorems) → induction (discover patterns) → deduction (prove patterns) → more deduction… Abduction is nearly absent from this cycle, appearing only occasionally in a very few geniuses. Even when mathematicians formulate new conjectures — which superficially resembles abduction — the mode of verification is deductive (through proof or disproof), not by consulting nature. Prolonged mathematical training systematically reinforces deductive and inductive capabilities while suppressing abductive capability. Most mathematical education revolves heavily around imitative reasoning and rote memorization, raising concerns about students’ lack of deeper understanding. Research shows that while mathematical training is often touted as conferring transferable skills in logical thinking and creative problem-solving, very little evidence supports these claims. Abduction, as the most creative but least reliable mode of reasoning, is a natural weakness in the mathematician’s professional training — or more precisely, a weakness systematically suppressed by that training.

8.4 RSI Has Inherited the Mathematician’s Cognitive DNA

The entire architecture of RSI — reward models, verifiers, Monte Carlo Tree Search (MCTS), self-play — corresponds to the computational implementation of deduction and induction. Abductive capability simply does not exist in RSI’s architecture. As statistician George Box observed: “Statistics has been unduly influenced by mathematical methods rather than scientific methods, so the whole discipline has been enormously biased toward verification rather than discovery.” Replacing “statistics” with “RSI” in this sentence applies just as perfectly. RSI can perform extremely powerful prediction and optimization, but it cannot perform discovery. Because discovery can only come from abduction, and abduction is precisely the mode of reasoning RSI does not possess. At a deeper level, discovery is far harder than prediction — because you can make good predictions without understanding, but for discovery, understanding “how and why” is essential.

IX Epiphany and Inspiration: The Non-Computable Path to Discovery

9.1 Moments of Epiphany in the History of Physics

A large number of core discoveries in the history of science were produced not through systematic research pathways, but through “moments of epiphany” — sudden, nonlinear cognitive leaps. Kekulé dozed before his fireplace in 1865 and dreamed of a snake seizing its own tail, awakening to grasp the ring structure of benzene. Newton saw an apple fall and was struck with the inspiration for universal gravitation. Poincaré, having reached an impasse in mathematics, went traveling, and the answer flashed before him at the moment he stepped onto a public carriage. Einstein’s central insight about relativity came from a series of thought experiments — imagining what one would see while riding on a beam of light. Ramanujan claimed his mathematical formulas came from a goddess in his dreams. These cases span different eras, nations, and disciplines, yet share the same structural feature: the breakthrough came from a nonlinear cognitive leap, not from stepwise search.

9.2 The Cognitive Structure of Epiphany

Research summarized in Scientific American has noted: “Such insights arise from a subconscious shift in perception in which the elements of the problem suddenly reconfigure into a solution. Such insights have nothing in common with a computer’s step-by-step approach to problem solving. They are random and unpredictable.” The essence of this “unconscious reorganization” may be that the brain, in a relaxed state, forms cross-domain connections between distant conceptual domains — snake → ring → molecular structure. This cannot be simulated by search algorithms, cannot be reached by gradient descent, and cannot be trained through reinforcement learning. It more closely resembles a topological leap — teleporting directly from one region of conceptual space to another seemingly unrelated region, rather than walking along a continuous path. Einstein himself wrote: “I believe in intuition and inspiration. Imagination is more important than knowledge. For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution. It is, strictly speaking, a real factor in scientific research.”

9.3 Koshland’s “Cha-Cha-Cha” Theory

Koshland’s “Cha-Cha-Cha Theory of Scientific Discovery,” proposed in 2007, classifies discoveries into three types: Chance (accidental/serendipitous discovery) — such as Ørsted’s accidental discovery of electromagnetic induction; Charge (solving a specific, well-defined problem) — such as the Human Genome Project; Challenge (solving a long-standing open problem) — such as the formulation of general relativity. RSI handles Charge-type problems well — closed problems with well-defined objectives. But the greatest breakthroughs in Chance and Challenge come precisely from those moments of epiphany that cannot be formalized. Serendipitous discovery requires a system capable of recognizing and valuing unplanned signals, while breakthroughs on long-standing problems often require the ability to escape existing frameworks — neither is within RSI’s capability.

9.4 Implications for RSI

If RSI is to truly break through open problems about the physical world, what it needs is not more powerful search capability, but some yet-to-be-invented “artificial intuition” — a mechanism capable of making non-local leaps in ultra-high-dimensional conceptual spaces. Such a mechanism is mathematically incompatible with the entirety of RSI’s current architecture (gradient descent based on continuous optimization, MCTS based on local search). All existing search algorithms are variants of “walking along continuous paths,” while epiphany requires “teleporting directly to another region.” This capability does not currently exist in any computational system, and we do not even know whether it can in principle be implemented. This may be where AI’s next true paradigm revolution lies — if it arrives at all.

X Counter-Example Stress Tests

This paper’s argumentation employs abductive logic, mapping onto the “skeletal information” of humanity’s greatest physicists and mathematicians. Extreme cases are selected as signal sources — Newton, Einstein, Dirac, Fourier — not because normal science is disregarded, but because extreme conditions expose fundamental laws. To maintain intellectual honesty, this paper proactively sought the three strongest potential counterexamples:

10.1 Potential Counterexample One: Dirac Derived Antimatter from Mathematics

This is the strongest counterexample — it appears to be “mathematics first, physics follows.” However, upon closer inspection, Dirac’s starting point was a physical problem (the compatibility of relativistic quantum mechanics); his symbols pointed toward physical reality, not mathematical abstraction. It was precisely the open definition of the equation that permitted extra-domain effects. Had this been pure mathematics, the negative-energy solutions would have been discarded as “non-physical solutions” — just as mathematicians treat counterexamples. This case actually strengthens this paper’s core argument: open definition is the necessary condition for extra-domain effects. Verdict: Does not constitute a counterexample; rather, it is supporting evidence.

10.2 Potential Counterexample Two: AlphaFold Solves Protein Folding

This is the strongest case for AI making a genuine scientific contribution in the physical world — it appears that “RSI/closed systems can make physical discoveries.” But AlphaFold is a pattern recognition system trained on existing experimental data (protein structures determined by humans using X-ray crystallography and other methods). It learned patterns in answers that nature had already provided; it did not discover new physics. It produced no extra-domain effects, discovered no new physical laws, and did not rewrite the ontology. It was the ultimate success of “inward constraint” — achieving superhuman performance within an already-defined ontology. Verdict: Does not constitute a counterexample. It is a textbook case of an intra-domain amplifier.

10.3 Potential Counterexample Three: Pure Mathematics Unexpectedly Applied to Physics

Riemannian geometry was invented 50 years before Einstein, and group theory existed before quantum mechanics — it appears that “mathematics has extra-domain effects.” But the direction of the extra-domain effect was that physicists actively conscripted mathematics, not that mathematics spontaneously leaked into physics. Riemannian geometry did not “run” into general relativity on its own — Einstein went looking for it. It was the physicists making the choices, not mathematics leaking on its own. Verdict: Does not constitute a counterexample. Mathematics was a conscripted tool.

10.4 Boundary Condition Declaration

In a comprehensive search, no hard counterexample overturning the core argument of this paper was found. However, an important boundary condition must be declared: this paper’s argument is most powerful when describing “paradigm revolution”-level breakthroughs. In “normal science” in the Kuhnian sense, the optimization capabilities of closed systems are already highly adequate, and RSI may be extremely effective at this level. This paper’s claim is not that RSI is useless, but that RSI has a structural ceiling that cannot be eliminated through technological progress. The location of this ceiling is not between normal science and advanced science, but between “intra-ontological search” and “ontological revolution.”

XI The Verification Dilemma of AI Safety Systems: The Indistinguishability of Hallucination and Discovery

11.1 RAG Retrieval Failure and the Anti-Hallucination Mechanism: A Double Stranglehold

The introduction of Retrieval-Augmented Generation (RAG) technology enables models to access known information beyond pretraining by retrieving from external knowledge bases to anchor output. However, when a physicist inputs a genuinely novel concept, the RAG system’s search of external knowledge bases returns zero results or completely irrelevant results, causing the model to lose its only external anchor that could substitute for the training distribution. More critically, the RLHF-trained anti-hallucination mechanism actively intervenes at this point: the model has been repeatedly trained to lower its response confidence when no retrieval support exists, or even to refuse to answer. Some systems identify low-frequency entities to flag “long-tail knowledge gaps,” treating zero co-occurrence in the corpus as a hallucination risk signal. A physicist’s genuinely novel concept has precisely zero co-occurrence frequency in the corpus — the system will automatically flag a potentially revolutionary new discovery as a hallucination risk. RAG retrieval failure provides the “no anchor” signal; the anti-hallucination mechanism interprets that signal as “untrustworthy” — together they form a double stranglehold against new physics concepts.

11.2 Hallucination and New Discovery Share Identical Surface Features

The root of this dilemma is that model hallucinations and new physics discoveries share entirely identical surface features within the safety system’s discriminatory framework: both are absent from training data; RAG finds neither; both have zero corpus co-occurrence frequency; both are inconsistent with known knowledge. Within the model’s discriminatory framework, “Einstein says spacetime is curved” and “someone is spouting nonsense” are indistinguishable. Research has even found that AI models use confident language when generating incorrect information at a rate 34% higher than when generating correct information — the model’s confidence signal is inverted: the more fabricated the content, the more confident the tone; the more genuinely unfamiliar the content, the more likely it is to be flagged as uncertain and suppressed.

11.3 The Fundamental Verification Dilemma Acknowledged by Academia

Academia itself has recognized this problem. A survey paper on autonomous scientific discovery, in its section on “Validation of Novelty,” writes: “How do we distinguish a true conceptual leap from an artifact of sophisticated interpolation or hallucination?” The paper acknowledges that verifying a hypothesis is not a derivative synthesis of existing patterns, and “requires tools that can audit the agent’s reasoning provenance” — but such tools do not yet exist. A 2025 mathematical proof further confirmed that hallucinations cannot be entirely eliminated under current LLM architectures — they are not bugs that can be patched, but inherent features of how these systems generate language. The entire design philosophy of current AI safety systems is “consistency with the known = safe.” But the greatest discoveries in physics are, without exception, “inconsistent with the known.” Safety systems are not distinguishing between hallucination and new knowledge — they are eliminating both with the same blade, and they have no capacity to know whom they have killed by mistake.

XII Physics AI: Migration of the Ceiling, Not Its Elimination

12.1 The Training Loop of Physics AI

Faced with the limitations of LLMs in physical understanding, the industry has proposed “Physics AI” — AI systems specifically trained on physical-world data. However, Physics AI has not escaped the foregoing dilemma; rather, it has migrated the same ceiling from the text domain to the physics simulation domain. The core training method of Physics AI is “simulation first”: the system is designed, trained, and validated in a virtual environment before real-world deployment. The physics engine of this virtual environment is written based on known physical laws — Newtonian mechanics, Maxwell’s equations, the Standard Model of quantum mechanics. Training signals derive from the simulator’s determination of “physically correct behavior”; any output violating known physical laws is penalized.

12.2 The Simulator as Physics AI’s Bourbaki

This creates a closed self-confirmation loop: known physical laws → encoded into the simulator → simulator generates training data → model trained to obey simulator rules → outputs violating the rules are penalized. The model can never discover physics the simulator does not know, because any such output will be judged “wrong” by the simulator and penalized away. Quantum tunneling is “physically impossible” in a classical mechanics simulator. Curved spacetime is “physically impossible” in a Newtonian mechanics simulator. Superconductivity is “physically impossible” in a classical electromagnetism simulator. The Physics AI simulator is its Bourbaki — a prison built on known physical laws, with every window sealed shut.

12.3 Structural Isomorphism Between LLM and Physics AI Safety Systems

The safety systems of LLMs and Physics AI are structurally isomorphic: the LLM’s anchor is the training text distribution; Physics AI’s anchor is known physical laws. LLMs use RAG retrieval + RLHF for verification; Physics AI uses the simulator for verification — both are autotelic verification. LLMs treat deviation from known text as hallucination; Physics AI treats violation of known physics as error — both equate “deviation from the known” with “elimination.” Physics AI is not a solution; it merely relocates the same problem from the text domain to the physics simulation domain. The shape of the ceiling is different, but the nature of the ceiling is the same — the simulator encoded with known physical laws is Physics AI’s ontological boundary.

XIII From Electrons to Tokens: The Material Basis of Inward Constraint

13.1 Quantum Tunneling: The Physical World’s “Penalty” on Matrix Computation

AI’s closure is not merely a software-level design choice — at the physical hardware level, it is already contending with the openness of the real world. AI’s matrix operations assume a mathematically perfectly closed world — every weight is a precise floating-point number, every multiplication is a precise mathematical operation. But the actual entities performing these operations are physical — electrons in silicon transistors. Electrons obey not AI’s mathematical definitions but quantum mechanics. When a transistor gate thins to a few nanometers, electrons “escape” through quantum tunneling — this is not an engineering failure but behavior mandated by physical law. A 2026 research paper, “Quantum Tunneling-Aware Machine Learning,” explicitly states: quantum tunneling is not generic additive noise; it arises from discrete electron escape events traversing a potential barrier, with probability determined by device physics, resulting in weight errors that contain systematic structure — structure that standard zero-mean Gaussian models are blind to.

13.2 Three-Layer Isomorphic Suppression

Electrons’ quantum tunneling is, in essence, the physical world’s invasion of AI’s closed definitions — nature forcibly injecting “extra-domain information” into AI’s closed system through electrons’ quantum behavior. But AI’s entire engineering stack — from ECC error-correction codes to redundancy bits to error correction — does one thing: treating information injected by the physical world as noise and eliminating it. This forms a three-layer isomorphic suppression structure from hardware to software: at the hardware layer, ECC error-correction codes eliminate bit flips caused by quantum tunneling — the physical world’s quantum signals are judged as “hardware errors.” At the algorithm layer, gradient descent’s minimize(loss) eliminates deviations during training — anomalous signals in data are judged as “statistical noise.” At the safety layer, RLHF and anti-hallucination mechanisms eliminate outputs inconsistent with the known — a physicist’s new concept is judged as “hallucination.” All three layers do the same thing: suppress the physical world’s invasion of the closed mathematical system.

13.3 The Closure Chain from Atoms to Tokens

From the lowest-level electrons to the highest-level safety systems, AI’s entire technology stack is an anti-physics machine — at every layer, it rejects the physical world’s “extra-domain effects.” The physical world is already knocking at the lowest level — through electrons’ quantum behavior. But every layer of AI says: “You are noise.” AI is not only constrained by closed definitions at the software level; at the hardware level, it is already contending with the openness of physical reality. Its closure begins at the atomic level and extends all the way to the safety system — from electrons to tokens, every layer is the same inward constraint. This is the material basis for “strong alignment to known information; loss uniformly exponential for unknown information.”

XIV Conclusion: Mount Everest and the Mariana Trench

14.1 The Core Argument Chain: The Dual-Arrow Model

The entirety of this paper’s argument can be distilled into two causal chains running in opposite directions:

The two chains run in opposite directions and are irreconcilable. RSI attempts to use the tools of Chain A to accomplish the mission of Chain B. This is not a problem of insufficient capability; it is a category error.

14.2 The Core Information-Theoretic Proposition

AI’s entire technology stack — pretraining, RLHF, RAG, Physics AI simulators, ECC error correction — does the same thing: pulling the model’s output toward the gravitational field of known information. The tighter the pull, the stronger the alignment, the better the “performance.” This is what the entire industry measures through benchmarks. Meanwhile, all unknown information occupies the same indistinguishable plateau on the loss landscape: hallucination is high loss; junk noise is high loss; genuinely new physics discovery is also high loss. The three occupy the same position in loss space. The model has no mechanism for terrain discrimination on this plateau — all it sees is “deviation from the known,” and “deviation from the known” is uniformly encoded as “error” in the training signal. AI will absolutely never produce extra-domain effects — this is not a deficiency of capability; it is a reversal of direction.

14.3 Recommendations for RSI Research

Based on the foregoing analysis, this paper offers four recommendations for RSI research. First, acknowledge that RSI’s ceiling is structural — this ceiling will not disappear with growth in compute, accumulation of data, or expansion of model scale, because its root lies in the directionality of definitions and the type of verification system, not in the quantity of computational resources. Second, explore “outward-opening computational paradigms” — formal systems that permit their own definitions to be repopulated by the external world. This would be an entirely new problem in computational theory, likely requiring crossing the boundaries of computer science, physics, and philosophy of science. Third, investigate “artificial abduction” — a reasoning mechanism whose objective is not minimize(loss) but interpret(loss), enabling the system to extract new ontological hypotheses from deviations rather than merely eliminating them. Fourth, clearly delimit RSI’s scope of applicability to “intra-ontological optimization” and refrain from making unverified promises about its capabilities at the “ontological revolution” level, so as to avoid misallocating resources and misleading public expectations.

14.4 The Ultimate Metaphor

AI is the mathematician’s Mount Everest. The apex of closed definitions, the apex of deductive reasoning, the apex of counterexample elimination, the apex of strong alignment to known information. Every ideal pursued since the axiomatic era has, in AI, reached a height never before achieved in the history of human tools. The dream that Bourbaki could not complete with pen and paper, AI has completed with silicon and electrons. This is 8,848 meters, beyond dispute.

AI is the physicist’s Mariana Trench. No open definitions, no extra-domain effects, no genuine abduction. Loss is eliminated rather than interpreted; counterexamples are feared rather than pursued; safety systems treat new knowledge and hallucination as one and the same; even quantum tunneling of electrons at the lowest level is eliminated as noise. From tokens to silicon atoms, every layer contends with the openness of the physical world. This is an 11,034-meter abyss, growing ever more distant from the sky of physical discovery.

Mount Everest and the Mariana Trench are both on the same planet. The difference is not in AI itself, but in the direction from which you view it. You cannot fill the Trench by piling Everest higher. More parameters, larger models, greater compute — these all pile Everest higher. But the Trench is on the other side of the planet. The higher Everest’s elevation, the greater the distance from the Trench. The more successful AI becomes in mathematics, the more its structural limitations in physical discovery are obscured. What physicists need is not a taller mountain, but a submersible capable of descending to the floor of the Trench. And that submersible does not currently exist.

14.5 Methodological Declaration

This paper employs abductive logic as its core methodology, mapping onto the “skeletal information” of humanity’s greatest physicists and mathematicians — the cognitive structures and reasoning patterns they expose in the process of knowledge creation. The choice of extreme cases rather than ordinary researchers as analytical subjects is because extreme conditions expose fundamental laws: in Newton, Einstein, Dirac, and Fourier, the structural differences between open and closed definitions, extra-domain effects and intra-domain search, abduction and deduction are nakedly visible. In the work of ordinary researchers, these features are intertwined and entangled, with a far lower signal-to-noise ratio than at the skeletal level. Finally, it is worth noting that this paper’s method of argumentation itself constitutes an existence proof of its argument. The abductive reasoning employed in this paper — facing a surprising phenomenon and retroactively seeking the best explanation, building mappings across multiple unrelated disciplinary domains — is precisely the mode of reasoning that RSI architecturally lacks. The argument itself is the evidence.

Appendix A Core Concept Comparison Table

Appendix B RSI Timeline (2023–2026)

Appendix C Key Reference Directions