The Loop That Improves the Model

Why This Matters

A model generates candidate solutions. A checker separates useful experience from mistakes. Training turns that experience into a better model, which supplies the next round of candidates. Methods such as STaR and Absolute Zero make versions of this loop concrete. [1, 2]

The stronger possibility is that the system also becomes better at deciding what to learn, which experiments to run, and how to train its successor. Now the improvement process itself is part of what improves.

The frontier is the second loop: an AI research system that improves not only its outputs but also how it discovers, tests, and learns future improvements. This page connects verified synthetic learning to recursive agents, learned learning algorithms, formal self-reference, statistical certification, and conditional models of research acceleration.

The central question: Does the loop produce better independently evaluated capability, or merely more outputs that its own evaluator approves?

Reading paths: For the RSI frontier, start at Section 8. For mathematical theory, start at Section 9. For the conditional 2027+ agenda, start at Section 13. Sections 1–7 establish the synthetic-learning and verification foundations.

Evidence key: Definitions and elementary derivations are distinguished from source-reported experiments, theoretical frameworks, and proposed future research. No independent replication or Lean verification is claimed.

Before you start: Conditional probability, empirical risk minimization, and the distinction between training and evaluation. Helpful companions: Test-Time Compute and Search, Scaling Laws, and The Loop Inside the Model.

First identify what changes

“Self-improvement” can refer to several different mechanisms. The following are working distinctions for this page, not mutually exclusive categories or a universal naming standard.

A better answer is not automatically a better model. A better model is not automatically a better research process. Reusing the same neural block inside an inference call is a different kind of recurrence again.

STaR exemplifies training from generated reasoning. The Darwin Gödel Machine explores inherited changes to agent code. Both are relevant, but they update different objects. Neither result should be silently promoted into a claim of unlimited improvement. [1, 3]

Here, recursive self-improvement (RSI) means a feedback process in which persistent improvements to an AI system help it produce further improvements to itself or its successors. Stronger autonomy, broader transfer, and faster improvement are additional properties to measure, not consequences of the acronym.

The anatomy of a synthetic-data flywheel

A useful loop has more structure than “train on your own answers.”

Choose tasks → generate attempts → obtain feedback → select or weight experience → update the learner → evaluate → repeat.

For a supervised, filtered-data version, let the current model be $p_{\theta_t}(y\mid x)$, the task distribution be $Q_t$, and the acceptance rule be $V_t$. Generate attempts and retain accepted examples:

$$x_i\sim Q_t,\qquad y_i\sim p_{\theta_t}(\cdot\mid x_i),\qquad D_t^+=\{(x_i,y_i):V_t(x_i,y_i)=1\}.$$

One possible training distribution mixes that experience with an anchor corpus:

$$\mu_t=(1-\lambda_t)\mu_{\mathrm{anchor}}+\lambda_t\widehat\mu_{D_t^+},\qquad 0\leq\lambda_t\leq1.$$

The learning algorithm updates the model using $\mu_t$. If no examples are accepted, the synthetic component is undefined: the implementation must skip it or fall back to the anchor corpus, not silently divide by zero.

This is one mechanism, not a definition of all self-improving AI. Reinforcement learning can use graded rewards and unsuccessful trajectories. A system can improve executable procedures without retraining model weights. The chosen representation of experience matters. [2, 3]

The anchor corpus is also not a magic safety device. Its relevance, representation of rare cases, and sampling weight determine what it preserves. Accumulation-based results support retaining original data in the settings studied; they do not supply a universal mixing ratio. [4]

Where does the improvement signal come from?

Start with a deliberately narrow idealization. For one fixed input, let a teacher have distribution $p$, and fit an unrestricted student distribution $q$ by exact maximum likelihood on infinitely many unfiltered samples from that teacher. Assume a finite output space. Then

$$\mathbb E_{Y\sim p}[-\log q(Y)]=H(p)+D_{\mathrm{KL}}(p\Vert q).$$

The minimum occurs at $q=p$. Exact population self-imitation reproduces the teacher distribution. It contains no automatic improvement step. This follows directly from nonnegativity of KL divergence.

The statement is intentionally limited. Finite samples, changed architectures, regularization, optimization, different inference budgets, and external feedback change the problem. It is not an impossibility theorem for learning from generated data.

A productive loop must explain what changes the effective target or learning problem: search that finds rare successes, a checker that distinguishes success from failure, a new curriculum, useful environmental feedback, or a better learning procedure. STaR selects reasoning traces using correct final answers; Absolute Zero obtains task and answer feedback from code execution. [1, 2]

“External feedback” need not mean a fresh human annotation or physical experiment every time. A fixed formal system or executable environment can evaluate new attempts. The novelty may be a newly found solution under already supplied rules, rather than a newly observed fact about the world.

AlphaGeometry illustrates the structured-domain route: large-scale synthetic geometry training supports neural proposals combined with symbolic deduction. That is evidence for useful synthetic learning and search, not autonomous redesign of the full learning pipeline. [8]

Search can expose competence that one attempt misses

If each of $K$ independent attempts solves a fixed problem with probability $p$, then

$$\Pr(\text{at least one correct attempt})=1-(1-p)^K.$$

For $p=0.05$ and $K=64$, this probability is about 96.25%. This is a calculation under an independence assumption, not an empirical model result.

It also does not say the system can identify the correct attempt. A verifier must do that. Correlated attempts reduce the usefulness of the independent-trial calculation; a model that never proposes a valid solution cannot be rescued by filtering alone.

Training on selected successes may make expensive search behavior cheaper to reproduce later. That possibility is different from claiming that an unfiltered student must exceed its teacher.

A verifier is a statistical filter

Consider one fixed proposal distribution. Let $C$ denote a genuinely correct candidate and $A$ acceptance by the filter. Define

$$p=\Pr(C),\qquad \alpha=\Pr(A\mid C),\qquad \beta=\Pr(A\mid C^c).$$

Here $\alpha$ is the true-acceptance rate and $\beta$ the false-acceptance rate. Bayes' rule gives the correctness rate among accepted candidates:

$$p_{\mathrm{accepted}}=\frac{p\alpha}{p\alpha+(1-p)\beta}.$$

Proposition

A Filter Raises Accepted Precision Exactly When It Prefers Correct Proposals

Statement

Write $p_{\mathrm{accepted}} = p\alpha / (p\alpha + (1-p)\beta)$ for the correctness rate among accepted proposals. Then $p_{\mathrm{accepted}} > p$ holds if and only if $\alpha > \beta$.

Intuition

Acceptance is evidence only when it happens more often to correct proposals than to incorrect ones. A checker that accepts both kinds at the same rate carries no information, so the accepted pool keeps the correctness rate it started with.

Proof Sketch

Subtract $p$ from the displayed ratio and put the difference over one denominator:

$$p_{\mathrm{accepted}}-p=\frac{p(1-p)(\alpha-\beta)}{p\alpha+(1-p)\beta}.$$

The denominator is the acceptance probability, positive by assumption, and $p(1-p)$ is positive because $0<p<1$. The sign of the difference is therefore the sign of $\alpha-\beta$.

Why It Matters

It sets the lowest bar a checker must clear to be worth anything, and it needs no independence between generator and checker: whatever dependence exists is already inside the conditional rates $\alpha$ and $\beta$.

Failure Mode

The conclusion concerns the precision of retained data, not what a model trained on that data learns. Coverage, sample size, optimization and transfer stay separate questions. The two rates also belong to one proposal distribution, and they can move once the generator changes, which is exactly what a flywheel does to it.

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Worked example: a useful filter

Suppose 20% of attempts are correct. The checker accepts 90% of correct attempts and 10% of incorrect attempts. Among 10,000 proposals, the expected accepted counts are 1,800 correct and 800 incorrect. Accepted-data precision is 69.23%, up from 20%.

The base-rate trap

Now suppose only 1% of proposals are correct. A checker accepts 95% of correct proposals and 5% of incorrect proposals. Among 10,000 attempts, the expected accepted counts are 95 correct and 495 incorrect. Precision is only 16.10%.

The filter is useful relative to the starting point, yet most accepted examples are still wrong. A report of “95% checker accuracy” would not answer the operational question: what fraction of the retained corpus is correct under this proposal distribution?

For a target accepted precision $\tau\in(0,1)$, positive acceptance, and $0<p<1$, the required false-acceptance rate satisfies

$$\beta\leq\frac{p\alpha(1-\tau)}{\tau(1-p)}.$$

For the rare-success example, achieving 90% precision requires $\beta\leq0.001066$, approximately 0.107%, far below 5%.

Correct, useful, and representative are different properties

A formally valid proof can establish an irrelevant theorem. A passing program can exploit incomplete tests. A perfectly labeled synthetic dataset can omit the mechanisms that matter at deployment. These are logical gaps between the certificate supplied and the claim someone wants to make from it.

For formal mathematics, distinguish proof validity, statement fidelity, and scientific value. Lean's logic permits explicit axioms; therefore the declaration, assumptions, and axiom dependencies are part of the certificate's meaning. A proof of a weakened or incorrectly formalized statement does not establish the intended informal claim. [5]

For executable tasks, keep the target behavior and evaluator outside the candidate's modification privileges. Generated tests can help development, but tests designed by the same search process should not become the sole basis for final claims.

For simulation, ground truth is exact only relative to the generator. Changing random seeds tests new draws from that generator; changing mechanisms tests a broader form of transfer. Neither substitutes automatically for evidence from the deployment setting.

Design implication: maintain separate records for validity, novelty, task coverage, and transfer. A single “verified” badge hides too much.

Why a flywheel can stall or run backward

Distributional degradation. Shumailov and colleagues show settings in which recursive training on generated data loses information, including distribution tails. Gerstgrasser and colleagues find that retaining original data and accumulating successive synthetic generations avoids collapse in their studied settings, with a supporting analysis for a linear-model framework. The justified conclusion is conditional: replacement and accumulation behave differently; synthetic data is neither automatically destructive nor automatically safe. [4, 6]

Selection-driven narrowing. Under the filter above, a task with acceptance probability $a(x)$ appears in the accepted corpus with frequency proportional to $Q_t(x)a(x)$. Even an accurate checker can therefore overrepresent easy tasks. More accepted rows need not mean better coverage of the original task distribution.

Proxy improvement. Approval by an evaluator and success on the intended task are different random variables. Optimizing the former is only useful to the extent that the relationship survives the search. The solution is not merely to rename the proxy “ground truth.”

Adaptive evaluation. Once checkpoint scores influence subsequent task generation or training choices, that evaluation stream participates in development. Protecting item text does not make unlimited score reuse statistically independent. This is the problem studied in adaptive data analysis. [7]

Resource bottlenecks. Generation, verification, training, and final evaluation all consume budget. A more accurate checker can improve data quality while reducing throughput enough to make the entire loop less effective. The comparison has to count both effects.

What would make the loop recursively self-improving?

Represent the system at round $t$ by its model, task-selection policy, training procedure, and agent machinery. Call this state $M_t$. Let $U_t$ be the procedure that proposes the next change. A schematic iteration is

$$(M_{t+1},U_{t+1})=\mathcal F(M_t,U_t,E_t;B_t),$$

where $E_t$ is the available experience and $B_t$ the round's resource budget. This notation describes a system boundary; it does not prove any improvement.

Ordinary iteration can reuse a fixed $U$ indefinitely. A stronger recursive claim says that the inherited state makes future improvement more effective: better experiment selection, cheaper training, more useful task generation, or better search over candidate procedures.

The improvement machinery need not be a separate file that gets rewritten. An unchanged scaffold can invoke a more capable model whose improved research decisions change the effective improvement policy. Conversely, editing a file named self_improve.py establishes nothing on its own.

A useful causal question: from the same starting checkpoint and with the same available evidence and budget, does the inherited improvement machinery produce a better next candidate than the ancestor's machinery?

That test is stronger than observing that round five scores higher than round one. Extra compute, inherited training data, or a better base checkpoint can explain the latter without establishing a better improvement procedure.

The frontier: improving the research process itself

Literature cutoff: September 14, 2026. The results below are selected primary-source reports, not an exhaustive survey or independent replications. A paper called “towards RSI” may establish a useful component without demonstrating repeated improvement of the entire loop.

The important transition is from an AI that produces a better artifact to an AI that leaves behind a better way of producing the next artifact. The inherited object could be an experiment-selection policy, a learned optimizer, a data curriculum, a compiler transformation, or a program that rewrites the research agent. Treat these as distinct experimental claims.

From self-editing code to an improved improver

STOP, Self-Taught Optimizer (2023; revised 2024). A language-model-based program improves candidate programs, then applies that improvement procedure to itself. The resulting improver performs better on a small set of downstream tasks. The foundation model stays fixed. This is a concrete early example of recursive scaffold improvement, rather than model-weight recursion. Its importance is the changed search procedure, not simply an agent editing a file. [11]

Darwin Gödel Machine (2025; revised 2026). Candidate coding agents modify their code and accumulate in an archive from which subsequent candidates are generated. The archive allows branching and reuse of intermediate solutions rather than requiring a single monotone lineage. The original system retains externally designed parts of its search machinery; its coding gains do not independently identify every component of an improved improver. [3]

HyperAgents / DGM-H (March 2026). Task-agent code and meta-agent code share an editable program. The meta-agent can change how it proposes future modifications, including memory and performance-tracking procedures. The authors report transfer of evolved improvement mechanisms to a new domain. This is directly relevant evidence for bounded meta-improvement, not merely answer refinement. The underlying foundation models remain frozen. Cross-domain transfer and indefinite acceleration are different claims: the longer-run experiment does not establish a statistically significant final advantage for transferred initialization. [12]

That evidence should neither be inflated into “unlimited RSI has arrived” nor erased by defining RSI so strongly that every working subsystem becomes irrelevant. The useful questions are what was inherited, where it transferred, and whether it improved the next search under comparable resources.

Learning how learning should work

Learned optimizers. Andrychowicz and colleagues formulate optimizer design as learning a parameterized update rule, demonstrating an LSTM-based optimizer on classes of optimization problems. This places part of the learning algorithm inside the learned system. A learned optimizer is not automatically a recursively improving optimizer: the next question is whether it improves the procedure that will learn its successor. [14]

DiscoRL (Nature, October 2025). The system discovers reinforcement-learning update rules through a meta-learning process, with reported transfer beyond its discovery settings. This moves beyond choosing a scalar learning rate: the output is a learned learning rule. The discovery process is still a designed outer system. A stronger recursive experiment would reuse the discovered advance in that outer process and measure its effect on subsequent algorithm discovery. [13]

AlphaEvolve (2025). Program search with automated evaluation can find changes relevant to AI-training infrastructure, according to DeepMind's developer report. This supplies a different feedback channel: an AI-designed implementation can make later training or search cheaper even without changing the statistical learning objective. Whether those savings cause additional research progress is a separate end-to-end measurement. [9]

These mechanisms can compose. A more capable model may write a faster kernel; the faster kernel may allow more candidate training runs; those runs may produce a better research model. No individual component needs to contain all of “intelligence” for the combined system to exhibit feedback.

AI systems conducting AI research

Frontis-MA1 / OpenMLE (July 2026). This work couples executable ML-engineering tasks, post-training on program-evolution operators, and long-horizon search. The authors train a 35B model around Draft, Improve, Debug, and Crossover operations. Held-out comparisons separately replace the model and the search framework, with gains attributed to both. It is a useful AI4AI research substrate; component transfer is not yet evidence that repeatedly rebuilding the entire stack produces accelerating returns. [15]

A-Evolve-Training (June 2026; revised September 8). The report describes autonomous post-training of a 30B model over four rounds. Its inherited research policy changes after internal development improvements stop predicting external gains. The fixed worker substrate is not overwritten by each winner. This makes research-policy inheritance, rather than unrestricted self-rewriting, the relevant recursive mechanism. The competitive result is specific to one challenge and base model. Larger-model runs are presented as infrastructure evidence, not an established competitive outcome. [16]

NeoHorse-1 (September 8, 2026). Routing and interaction records inform agent post-training and the next training mixture. Its authors explicitly frame the system as an initial feedback-driven prototype. It illustrates how deployment evidence can guide subsequent learning; a long chain of recursively improving research policies is not established by the reported first prototype. [18]

Measurement substrates. RE-Bench evaluates ML research-engineering work, while MLE-bench evaluates ML engineering through competitions. These test whether agents can perform parts of AI R&D. RSIBench-Data focuses more narrowly on budgeted data-strategy research for a specified training stack and reports that iterative revisions are not consistently better. None of these benchmark labels alone certifies recursive inheritance. [10, 19, 20]

The evaluator can improve too

Red Queen Gödel Machine (June 2026). This preliminary preprint makes evaluation part of the evolving system. Its mechanism divides search into epochs with a fixed within-epoch criterion and allows changes between epochs. The authors report results in coding, scientific writing/reviewing, and mathematical proof writing/grading. Its guarantees are epoch-relative, not a theorem that arbitrary changes of utility preserve progress toward an unchanged external objective. [17]

The design problem is substantial. Freezing every evaluator indefinitely can preserve blind spots; allowing unrestricted replacement can redefine success. An appropriate architecture can permit new judges, tests, and rubrics while retaining an independent reference for deciding whether those changes are better. Evaluator improvement and evaluator authority should be separate permissions.

From laboratory prototypes to industrial feedback

OpenAI's September 6, 2026 research-acceleration report describes increasing delegation of research execution while humans retain priority-setting, interpretation, and consequential decisions. Its March 2028 automated-researcher target is a stated organizational goal, not a demonstrated capability or a calibrated probability. Reported workflow changes also coincide with changes in tools and compute; they are not a clean causal estimate of whole-laboratory acceleration. [23]

Anthropic's 2026 account likewise discusses substantial automation within AI development while distinguishing execution from autonomous successor development. Such reports matter because research feedback can operate within a laboratory before a standalone system controls every stage. They remain developer accounts, with limited public observability of the full counterfactual. [24]

The strongest practical unit of analysis may therefore be the research organization plus its AI systems, not one model checkpoint. This is an analytical choice, not an assertion that current organizations have achieved unrestricted RSI. Broader system boundaries require broader accounting for human input, compute, infrastructure, and inherited knowledge.

A compact map of the literature

Two recent survey entry points are Duan and colleagues' September 2026 autonomy-centered framework, and Chen, Wang, and Qu's survey revised September 6, 2026. The first separates control over execution, strategy, experience acquisition, adaptation, and recursive inheritance; the second distinguishes the object improved from the degree of loop closure. These are proposed organizing frameworks, not a settled universal scale or substitutes for the original experiments. [21, 22]

A mathematical definition of improvement skill

A model's score on a task and its ability to improve another system are different functionals. Make the latter explicit.

Let $U_\phi$ be an improvement policy with persistent state $\phi$. On research task $\mathcal T$, it starts from checkpoint $\theta$, receives a fixed evidence package $E$, and spends budget $B$ to produce a candidate. The candidate may be a model, agent, or learning procedure. Define

$$J(\phi;\theta,E,B)=\mathbb E_{\mathcal T,\omega}\left[V_{\mathcal T}\!\left(U_\phi(\theta,E,\mathcal T;B,\omega)\right)-V_{\mathcal T}(\theta)\right].$$

Here $V_{\mathcal T}$ is an independently specified assessment and $\omega$ includes search, training, and evaluation randomness. Scores across task families must be put on a predeclared comparable scale. This is a proposed operational definition, not a universal measure of intelligence.

A controlled meta-improvement contrast is

$$\Delta_{\mathrm{meta}}=J(\phi_t;\theta^*,E^*,B)-J(\phi_0;\theta^*,E^*,B).$$

The same checkpoint, evidence package, and budget appear on both sides. A positive contrast means that inherited improvement machinery helps under this research-task distribution. It does not establish that every future generation improves, that the effect transfers everywhere, or that discovering $\phi_t$ was economical.

Acquisition and reuse are different estimands. The comparison above measures the marginal value of reusing an existing improver. A campaign-level comparison must also charge the experiments that discovered it. A procedure that costs a thousand runs to develop and saves one run per reuse needs enough reuse to justify its cost.

For a model-based improver, $\phi$ includes its model checkpoint as well as executable scaffolding. A fixed Python scaffold can implement a changing effective research policy when its underlying model changes. Conversely, changed code with unchanged behavior is not meta-improvement.

When checkpoint and scaffold co-adapt, add a factorial transfer test: ancestral and evolved improvers crossed with ancestral and evolved starting systems. This identifies specialization and interaction. A strong result need not win in every cell, but a claim of general improvement skill needs more than one favorable pairing.

The theoretical lineage of recursive self-improvement

Gödel machines: prove that switching is worth doing

Schmidhuber's Gödel-machine framework allows self-modification when the system proves an appropriate utility-improvement statement within its formal model. Schematically, the relevant comparison is

$$\mathbb E_{\mathcal A}[U\mid\text{switch now}]>\mathbb E_{\mathcal A}[U\mid\text{continue the current search}].$$

The formal setup $\mathcal A$ encodes the machine, objective, resource costs, and assumptions about its environment. The comparison accounts for continuing proof search rather than merely comparing two static programs. Its optimality claim is conditional on that setup and the proof found. It is not a guarantee that a practical proof search will discover the best program or finish on a useful timescale. [25]

The appealing idea is proof-carrying improvement: a modification arrives with a certificate of a specified benefit. The difficult part is supplying a faithful specification and tractable certificate. Formalizing a compiler transformation or bounded scheduler is a different challenge from proving that a new training recipe will improve a broadly deployed model.

Incompleteness and undecidability constrain universal certification procedures. They do not imply that no useful self-modification can be certified, or that empirical RSI is impossible. The engineering target is a meaningful class of changes with explicit assumptions, not a universal oracle for all beneficial changes.

Vingean reflection: trusting a successor you cannot fully simulate

A weaker system cannot generally verify a stronger successor by predicting every action it will take. Vingean-reflection research studies how resource-bounded agents might reason about such successors without exhaustive simulation. The central tension is between requiring adequate justification and demanding a form of justification that defeats the purpose of delegation. [26]

For AI research, the analogous question is concrete: what evidence should permit a research agent to replace its own experiment-selection policy with one it cannot fully explain? Possible components include restricted interfaces, invariants, performance guarantees on defined tasks, and independent monitoring. None alone establishes universal successor reliability; this is a research problem, not an assurance supplied by the word “reflection.”

Reflective oracles and logical uncertainty

Reflective-oracle models use randomized answers to handle self-referential probabilistic queries without the usual diagonalization contradiction. Their existence results illuminate which forms of self-reference are mathematically coherent. They do not hand an engineer a practical, efficiently implementable all-purpose oracle. [27]

Logical induction studies computable probabilistic beliefs about mathematical statements, including patterns involving the reasoner's own beliefs. It offers a framework for uncertainty before proof. Computability and useful runtime are separate: this is not a plug-in solution for frontier-model training or a general guarantee of safe self-trust. [28]

Together these lines distinguish three issues that are often conflated: whether self-reference is logically coherent, whether beneficial change can be established in a formal model, and whether the necessary reasoning is computationally feasible. Progress on one does not automatically settle the others.

Open-endedness: who supplies the next worthwhile problem?

A closed loop can exhaust a fixed benchmark. POET co-evolves environments and their solvers, transferring solutions between environments so that one search trajectory can supply useful intermediate solutions for another. This is a concrete mechanism for expanding a curriculum rather than repeatedly sampling one fixed task distribution. Its environment space and evolutionary machinery remain designed. [29]

Hughes and colleagues argue that open-endedness is central to artificial superhuman intelligence, proposing novelty and learnability as useful concepts. This is a position paper and research framework, not a theorem that any particular open-ended algorithm must yield general superintelligence. [30]

Proposed research criterion: a new task is useful for an RSI curriculum when learning it changes future capability or improvement efficiency outside the task's own acceptance test. Novel syntax, increasing problem length, and endless generation alone do not establish that.

When does recursive improvement accelerate?

“Improves every round,” “grows exponentially,” and “reaches a finite-time singularity in a model” are different statements. None should be smuggled into another by calling the process recursive.

A simple counterexample: improvement with shrinking gains

For a bounded score, suppose $0<c_0<1$, $0<\eta<1$, and

$$c_{t+1}=c_t+\eta(1-c_t),\qquad c_t=1-(1-c_0)(1-\eta)^t.$$

Then every round improves, but

$$\Delta_t=\eta(1-c_0)(1-\eta)^t$$

decreases. This is a mathematical counterexample to “iteration implies acceleration.” It is not an AI forecast; bounded scores also compress improvements near their ceiling.

An endogenous research-productivity model

For a more useful discussion of acceleration, define $A(t)>0$ as algorithmic efficiency on a specified workload family, with a fixed quality target and resource accounting. It is not a scalar measure of all intelligence. Consider the illustrative local model

$$\dot A=\kappa R^\lambda A^\phi,\qquad R=R_0A^\psi.$$

Here $R$ is effective research input, $\phi$ describes how existing efficiency affects the difficulty of producing further advances, and $\psi$ describes feedback from better AI into effective research input. Substitution gives

$$\dot A=kA^\gamma,\qquad k=\kappa R_0^\lambda>0,\qquad\gamma=\phi+\lambda\psi.$$

This is a pedagogical power-law reduction, not an estimated law of AI progress. Cunningham and colleagues' Economics of Recursive Self-Improvement develops a richer economic treatment involving research inputs and multiple feedback channels; its conclusions depend on the specified elasticities and calibration. The simple model here is not their full model or an empirical fit to it. [34]

For initial efficiency $A_0>0$, separation of variables gives

$$A(t)=\begin{cases} \left[A_0^{1-\gamma}+(1-\gamma)kt\right]^{1/(1-\gamma)},&\gamma\ne1,\\ A_0e^{kt},&\gamma=1, \end{cases}$$

on the interval where the bracket is positive. The proportional growth rate is

$$g(t)=\frac{\dot A(t)}{A(t)}=kA(t)^{\gamma-1}.$$

For $\gamma>1$, that formal singularity occurs at

$$t^*=\frac{A_0^{1-\gamma}}{k(\gamma-1)}.$$

It means the assumed equation stops giving a finite solution. It is not a prediction of physically infinite compute, intelligence, or output. Resource constraints, delays, shifting workloads, and changing elasticities are precisely the factors suppressed by the model.

A cleaner observable: the next doubling time

Starting from efficiency $A$, the model's time to double is

$$T_2(A)=\begin{cases} \dfrac{A^{1-\gamma}(2^{1-\gamma}-1)}{k(1-\gamma)},&\gamma\ne1,\\ \dfrac{\log 2}{k},&\gamma=1. \end{cases}$$

Consequently,

$$\frac{T_2(2A)}{T_2(A)}=2^{1-\gamma}.$$

This makes the forecasting disagreement explicit. The issue is not whether better AI can contribute to research at all. It is how strong that contribution is relative to diminishing returns and non-automated bottlenecks, over what range, and at what cost.

Increasing external compute can also produce apparent acceleration. Adding an exogenous factor such as $C(t)^\nu$ to the right-hand side changes the trajectory without establishing that the endogenous feedback exponent exceeds one. Identifying the mechanism requires measurements or interventions that separate those channels.

Several feedback channels can amplify one another

A separate local model tracks small deviations in research capability, implementation efficiency, and available resources:

$$z_{t+1}=Gz_t+u_t.$$

For a fixed matrix $G$, if its spectral radius $\rho(G)<1$, then $G^t\to0$; with constant forcing $u$, the response converges to $(I-G)^{-1}u$. If $\rho(G)>1$, some perturbations grow. For the two-channel example

$$G=\begin{pmatrix}0&a\\b&0\end{pmatrix},\qquad a,b\ge0,\qquad G^2=abI,$$

one complete two-step loop multiplies a perturbation by $ab$. The relevant question concerns the product around the loop, not just the existence of a positive arrow between two components.

This is elementary linear-systems reasoning. Its gain condition is not interchangeable with the exponent condition in the nonlinear growth model: one describes local perturbation propagation, the other describes growth of a level. Neither identifies actual parameter values without evidence. Per-cycle amplification also says nothing about wall-clock speed until cycle durations are specified.

Fast subroutines are not a fast research organization

If fraction $f$ of a serial workflow is sped up by factor $s$, leaving the remainder unchanged, total speedup is

$$S=\frac{1}{(1-f)+f/s}.$$

Speeding up 30% of that workflow tenfold gives only about 1.37 times the end-to-end speed. This follows by adding remaining execution times under the stated serial model. Real research also has parallel work, queues, failed experiments, and critical paths; the equation is a warning against equating one component's throughput with overall research acceleration.

Statistically certified improvement under adaptive search

A search process can generate many apparently beneficial modifications simply by trying enough noisy alternatives. Repeatedly reading a development score makes it part of the search, even when the test items themselves are hidden. Adaptive data analysis formalizes why unqualified reuse is problematic. [7]

The Statistical Gödel Machine preprint proposes statistical acceptance tests and a cumulative error budget for recursive modifications. It is relevant as an attempt to connect self-modification to sequential inference. Its statistical certificates concern defined performance claims; they should not be interpreted as universal safety certificates. [32]

The following is a separate elementary sufficient construction, not a reproduction or validation of that paper's complete algorithm.

Proposition

A Lifetime Error Budget for a Stream of Candidate Approvals

Statement

Give candidate $j$ the error allowance $\alpha_j=\alpha/[j(j+1)]$ and approve it only when its test supermartingale crosses $1/\alpha_j$. Then the probability that the campaign ever approves any non-improving candidate is at most $\alpha$, however long the campaign runs and whenever it stops.

Intuition

Each candidate gets its own slice of one fixed error budget, and the slices sum to the whole budget. Betting on a claim that is false in expectation cannot make money on average, so a large payout is unlikely, and Ville's inequality bounds the chance of ever reaching the payout that triggers approval.

Proof Sketch

For candidate $j$, let $X_{j,i}\in[-1,1]$ be its score minus the comparator's on confirmation case $i$, and choose nonnegative predictable $\lambda_{j,i}$. Define

$$E_{j,n}=\exp\left\{\sum_{i=1}^{n}\lambda_{j,i}X_{j,i}-\frac12\sum_{i=1}^{n}\lambda_{j,i}^{2}\right\}.$$

Conditional Hoeffding's lemma on an interval of width two makes $E_{j,n}$ a nonnegative supermartingale starting at one under the null, so Ville's inequality bounds the chance it ever crosses $1/\alpha_j$ by $\alpha_j$, with data-dependent stopping allowed. Since $1/[j(j+1)]=1/j-1/(j+1)$, the allowances telescope to $\alpha$. A union bound over candidates finishes it, and no independence between candidates is required.

Why It Matters

A search that proposes enough noisy modifications will produce apparent winners by chance. This construction keeps the campaign's lifetime false-approval rate fixed no matter how many candidates the search invents.

Failure Mode

It controls false approvals for one declared mean-performance comparison. It does not promise that good changes get accepted, that individual capabilities survive, or that anything transfers beyond the confirmation distribution. Recycling development observations breaks the conditional null, resetting the budget after a disappointing result voids the bound, and a changed evaluator or target population changes the claim being made.

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What this certificate does, and does not, buy

It controls false positive approvals for the specified mean-performance comparisons. It does not guarantee eventual acceptance of every good change, preserve every individual capability, or establish transfer beyond the confirmation distribution. Poorly chosen bets may have little power. Repeatedly recycling development observations can invalidate the conditional null argument.

The procedure must not reset its error budget after a disappointing result or reuse an index for a newly chosen candidate. A changed evaluator, target population, or definition of success changes the statistical claim. Improvements to the verifier itself need their own evidence; the verifier cannot manufacture independence by labelling its next query “fresh.”

For meaningful margins, test whether the mean gain exceeds a predeclared threshold rather than zero, adjusting the centered observations and betting construction accordingly. Track critical regressions and resource costs separately. A narrow valid statistical claim is more useful than a broad unsupported “safe self-improvement” label.

2027 and beyond: a conditional research agenda

Everything in this section is a proposed research agenda or conditional scenario written on September 14, 2026. The dates describe planning horizons, not predictions that particular capabilities will exist then.

Near-horizon agenda: transferable improvement skill

The most informative next target is not another isolated benchmark record. It is a reusable research policy that earns its development cost across fresh problems.

A decisive experiment would evolve an improver on several ML tasks, freeze it, and transfer it to new task families, model sizes, or training regimes. Compare against the ancestral improver at matched starting checkpoints and budgets. Then allow both arms to continue improving. This separates an initial transfer advantage from an advantage in subsequent improvement rate.

The resulting artifact could be a research-model checkpoint, a program for allocating experiments, a library of verified transformations, or a combination. Success means more externally confirmed progress per unit budget over repeated use, not merely a richer experiment log.

A second target is joint model-and-scaffold inheritance. Models can learn operators suited to their own search system; the search system can allocate work according to the models' strengths. The difficult experiment measures whether jointly inherited components keep helping when one component, the task family, or the resource regime changes. That exposes brittle co-adaptation rather than hiding it behind aggregate gains.

Near-horizon agenda: evolving curricula and evaluators

A promising curriculum should discover the learner's next productive difficulty, not maximize task novelty alone. A proposed evaluation would compare fresh tasks from familiar generators, shifted mechanisms, and an independently curated external set. Improvements on the first without the others would identify a closed synthetic curriculum rather than broadly useful research progress.

Evaluator evolution should be tested as an intervention. Does a new judge reduce false approvals or evaluation cost on independently audited cases? Does that improvement survive proposals optimized against the new judge? Keep an invariant external criterion for such comparisons, even as development judges change.

This suggests an evaluation ecology rather than one omnipotent verifier: proof checkers for formal claims, executable tests for behavior, statistical confirmation for noisy effects, and external measurements for deployment relevance. This is a proposed division of responsibilities. No ordering of checkers makes a formally true but irrelevant claim scientifically important.

Beyond the first few generations: successor co-design

A stronger longer-horizon target would let an AI research system jointly propose changes to architecture, optimizer, curriculum, inference procedure, and implementation. The resulting successor would then become a demonstrably better participant in the next research cycle.

The key observable is causal closure across generations: removing an inherited research advance should measurably reduce subsequent research performance under otherwise comparable conditions. Showing that a later model is stronger after more training is insufficient.

The loop need not be a single agent rewriting its own weights. A population of specialist researchers could contribute training rules, kernels, test suites, and experiment-selection policies. An archive could preserve useful alternatives that are not individually best on the current score. The challenge is to evaluate the collective feedback without attributing extra models, compute, or human curation to autonomous improvement skill.

Beyond software-only loops: external grounding and the economy

A software loop can rapidly evaluate changes whose relevant consequences are executable. A claim about material performance, biological mechanisms, or deployment behavior requires evidence tied to that external target. Simulated success and internally judged scientific writing are not interchangeable with those measurements.

A longer-horizon research program could connect AI experiment design, physical measurements, model revision, and improved experiment design. Its recursive unit is the scientific workflow. The evidence would need to show that inherited methodological advances improve later experiments, not only that the system discovers one useful result.

Economic feedback adds another layer: useful AI can change available research resources, while research changes AI. Distinguish algorithmic progress, additional compute purchases, improved hardware utilization, and laboratory organization. Their interaction may amplify progress, but measuring one does not identify the others.

Oversight that can follow the successor

As improvement systems become stronger, verifying their work can itself become a limiting research task. Weak-to-strong generalization experiments investigate settings where a stronger learner can exceed the capability of weak supervision. That is relevant evidence for a possible oversight route, not a guarantee that weak supervisors can reliably certify every stronger system. [31]

A proposed successor-evaluation program would measure whether supervision techniques retain calibrated error rates as the candidate grows stronger, changes strategy, and becomes better at exploiting the evaluator. Proof-carrying components, fresh adversarial tasks, and statistical acceptance gates address different parts of this question.

Human authorization over objectives, spending, access, and deployment is not a failure to study RSI. It is part of the experimental system boundary. Autonomy in generating an improvement does not require permission to alter every safeguard or deploy every candidate.

Three conditional trajectories worth distinguishing

These are hypotheses to distinguish, not assigned probabilities. The power-law model gives a vocabulary for the disagreement, not evidence selecting a row.

Forecasting work such as the AI Futures Model, including its August 2026 update, makes assumptions about milestones and feedback explicit. The AI 2027 scenario is another example of conditional scenario construction. A calendar in a scenario title is not a publication from that future year or proof that its sequence will occur. Use dated model versions, sensitivity analysis, and stated update criteria rather than treating one storyline as the literature's conclusion. [35, 36]

The experiment worth running

Use a verifiable domain with enough headroom, fix the initial checkpoint, and predeclare the capability and cost measures. Compare four budget-matched arms:

These comparisons estimate effects of the declared packages, not every ingredient. For an RSI claim, add the same-checkpoint improver comparison in Section 9, a transfer test, and at least one further generation in which the inherited policy actually controls research. Predeclare the number of rounds, stopping rules, and selection procedure.

Freeze the reference, not all experimentation. Development agents may propose tests and evaluators, but candidate selection must not rewrite the final scoring criteria or access hidden evaluation artifacts. Log evaluator versions so that scores under changed development criteria are not joined into a misleading progress curve.

Charge generation, failed attempts, verification, training, evaluation, and orchestration to the budget. Record wall-clock time and human intervention separately. For total-return claims, charge the acquisition of the evolved improver; for reuse claims, explicitly condition on its availability. Neither comparison should masquerade as the other.

Select the candidate before final evaluation. Replicate complete search campaigns, not only the last selected checkpoint. Retain and report negative results, failed modifications, and rejected branches. A best-of-many candidate's uncertainty is not described by a confidence interval that ignores how it was selected. Sequential confirmation offers one route when its assumptions hold. [7, 33]

A useful result sheet would report final capability at fixed cost, cost to a target, same-checkpoint meta-improvement, transfer, regression rates, and the share of research decisions made by humans. The headline should match the strongest causal comparison that was actually performed.

Open problems that would materially change the field

Transfer of improvement policies. Under what assumptions does an improver learned on one task family improve research on another? Is useful transfer governed by shared optimization geometry, evaluator structure, implementation primitives, or properties not captured by task labels? A theorem would need a specified task class; an experiment needs genuine held-out mechanisms.

Meta-overfitting and sample complexity. How much independent research-task diversity is needed to distinguish a general improvement policy from one that has memorized the peculiarities of a small suite? More trials on the same tasks do not necessarily answer that question.

Verifier–generator co-adaptation. How can a system gain the efficiency of an evolving evaluator while preserving a meaningful reference for progress? Can one derive useful error-control guarantees when proposals and development judges adapt, but only sparse external confirmation is available?

Identifiability of recursive acceleration. Which observational measurements can distinguish improved research productivity from compute expansion, shifted targets, and easier opportunities? Where are randomized interventions or staged rollbacks necessary? The feedback exponent is not read directly off a benchmark curve.

Open-ended task value. Can a curriculum optimize expected downstream learning value rather than novelty or immediate solvability? What prevents the task generator and solver from creating an increasingly elaborate but externally irrelevant game?

Composable certificates. When do local guarantees about a compiler change, optimizer, evaluator, and allocation policy compose into a useful statement about the research loop? Which interactions defeat composition? The answer must preserve the distinction between formal correctness, statistical performance, and objective fidelity.

These are proposed research questions. They are not claims that the page has solved them or that a single existing paper supplies a general answer.

Check your understanding

The agent edits its own improvement function. Has meta-improvement been established?

No. It establishes access and possibly structural inheritance. Compare the resulting improver against the original on independently evaluated research tasks under controlled starting conditions.

A foundation model is frozen. Can the larger system still exhibit recursive improvement?

Yes, under the system-level definition used here. Persistent changes to an agent's improvement machinery can help generate further changes. That is not evidence that the model's weights improved.

Every generation has a higher score. Does that establish shorter research doubling times?

No. The bounded recurrence improves with shrinking gains. Even exponential efficiency growth has a constant doubling time. State the measured quantity and time scale.

A finite-time singularity appears in a growth equation. Has a physical intelligence explosion been predicted?

No. It is a conditional mathematical property of the equation. Establishing relevance requires evidence for the assumptions over the extrapolated range and treatment of resource constraints and delays.

A statistical gate bounds false approvals by 5%. Is the resulting agent 95% safe?

No. The bound concerns a defined family of null hypotheses under stated assumptions. It does not cover every hazard, capability regression, distribution shift, or misspecified objective.

A 2027 scenario describes autonomous successor development. Is it frontier experimental evidence?

No. It is a forecast or scenario. Record its publication date, assumptions, and update history separately from empirical results.

ExerciseCore

Problem

A generator proposes solutions that are correct 4% of the time. A checker accepts 92% of correct proposals and 8% of incorrect ones. Out of 50,000 proposals, how many accepted examples are correct, and what fraction of the accepted pool is that? Would you train on this pool?

Hint

Reveal Solution

ExerciseAdvanced

Problem

Keep $p = 0.04$ and $\alpha = 0.92$. You want 90% of the accepted pool to be correct. What false-acceptance rate does that require, and what does the requirement do as $p$ falls?

Hint

Reveal Solution

Takeaway

Synthetic flywheels make generated experience reusable. Recursive self-improvement asks whether inherited advances also make the next round of improvement more effective.

The strongest frontier is not simply more synthetic text. It is the co-development of research policies, learning rules, evaluation, infrastructure, and task generation. There are bounded empirical examples worth taking seriously, formal models worth studying, and stronger multi-generation claims that require different evidence.

Generate better experience. Improve the learner. Improve the research process. Then measure what each inherited advance makes possible next.

References

Registry ids in monospace resolve in data/content/sources.json. The literature cutoff for this page is September 14, 2026, and every record was re-checked by identifier or URL on 2026-09-15. Surveys are evidence for their own taxonomies, developer reports for what their authors state, and forecasts for their stated assumptions. None of them substitutes for the original experiments.

• [1] Zelikman et al. "STaR: Bootstrapping Reasoning With Reasoning." 2022. Source. supports generated-rationale selection and iterative fine-tuning, not unrestricted RSI. zelikman-2022-star
• [2] Zhao et al. "Absolute Zero: Reinforced Self-play Reasoning with Zero Data." 2025. Source. task proposal, executable feedback, and learning. Algorithm 1 explicitly begins with a pretrained base LLM: https://arxiv.org/html/2505.03335v3 zhao-2025-absolute-zero
• [3] Zhang et al. "Darwin Gödel Machine: Open-Ended Evolution of Self-Improving Agents." 2025; version 3, March 12, 2026. Source. agent-code evolution, archive, empirical coding evaluation, and stated safeguards. zhang-2025-darwin-godel-machine
• [4] Gerstgrasser et al. "Is Model Collapse Inevitable? Breaking the Curse of Recursion by Accumulating Real and Synthetic Data." 2024. Source. conditional accumulation results, not a universal anti-collapse guarantee. gerstgrasser-2024-accumulating-data
• [5] Lean documentation. "Theorem Proving in Lean 4: Axioms and Computation.." Living documentation; accessed September 14, 2026. Source. axiom and foundational scope; statement-fidelity discussion on this page is a logical distinction. lean-4-axioms-and-computation
• [6] Shumailov et al. "The Curse of Recursion: Training on Generated Data Makes Models Forget." 2023; revised 2024. Source. recursive-generation degradation and loss of distribution tails in studied settings. shumailov-2023-curse-of-recursion
• [7] Dwork et al. "Preserving Statistical Validity in Adaptive Data Analysis." 2014; STOC 2015. Source. adaptivity and statistical validity under data reuse. dwork-2015-adaptive-data-analysis
• [8] Trinh and Luong, Google DeepMind. "AlphaGeometry: An Olympiad-level AI system for geometry." January 17, 2024. Source. primary developer account of synthetic data and neuro-symbolic solving. deepmind-2024-alphageometry
• [9] AlphaEvolve team, Google DeepMind. "AlphaEvolve: A Gemini-powered coding agent for designing advanced algorithms." May 14, 2025. Source. primary developer account of program search and infrastructure applications. deepmind-2025-alphaevolve
• [10] Meng et al. "RSIBench-Data: Benchmarking Data-Centric Research for Recursive Self-Improvement." July 28, 2026. Source. budgeted data-strategy research and inconsistent iterative gains; preprint, not a universal statement about all agents. meng-2026-rsibench-data
• [11] Zelikman et al.. "Self-Taught Optimizer (STOP): Recursively Self-Improving Code Generation." 2023; revised 2024. Source. Recursive scaffold improvement with a fixed underlying language model. zelikman-2023-stop
• [12] Zhang et al.. "HyperAgents." 2026; submitted March 19. Source. Editable task and meta agents; frozen foundation models; transfer experiments and a nonsignificant longer-run final contrast. zhang-2026-hyperagents
• [13] Oh et al.. "Discovering state-of-the-art reinforcement learning algorithms." Nature, October 22, 2025. Source. DiscoRL discovers learning rules; held-out generalization does not establish recursive closure of the discovery process. oh-2025-discorl
• [14] Andrychowicz et al.. "Learning to learn by gradient descent by gradient descent." 2016. Source. Parameterized learned optimizers; a predecessor to learning-rule improvement, not unrestricted RSI. andrychowicz-2016-learning-to-learn
• [15] Yang et al.. "Frontis-MA1: Training an AI4AI Model towards Recursive Self-Improvement in Machine Learning Engineering." July 30, 2026. Source. OpenMLE environments, operator learning, search, and component-transfer evidence. yang-2026-frontis-ma1
• [16] Shi et al.. "A-Evolve-Training: Autonomous Post-Training of a 30B Model." June 9, 2026; version 3, September 8. Source. Autonomous multi-round post-training with inherited research policy; competitive evidence is benchmark-specific. shi-2026-a-evolve-training
• [17] Iacob et al.. "The Red Queen Gödel Machine: Co-Evolving Agents and Their Evaluators." June 24, 2026; version 2, June 29. Source. Epoch-wise evaluator evolution; epoch-relative guarantees, not unrestricted utility-change guarantees. iacob-2026-red-queen-godel-machine
• [18] NeoHorse Team et al.. "NeoHorse-1: Towards Recursive Self-Improvement via Agentic Post-Training with Routing Harness." September 8, 2026. Source. Initial routing-informed post-training prototype, not an established long-run recursive trajectory. neohorse-2026-recursive-ai-research
• [19] Wijk et al.. "RE-Bench: Evaluating frontier AI R&D capabilities of language model agents against human experts." 2024; revised 2025. Source. ML research-engineering evaluation; task performance is distinct from recursive inheritance. wijk-2024-re-bench
• [20] Chan et al.. "MLE-bench: Evaluating Machine Learning Agents on Machine Learning Engineering." 2024. Source. ML engineering task environment and evaluation, not a stand-alone RSI certificate. chan-2024-mle-bench
• [21] Duan et al.. "The Last AI Built by Humans: Toward Genuine Recursive Self-Improvement." September 10, 2026. Source. Primary source for its proposed autonomy-centered taxonomy; original studies support experimental claims. duan-2026-rsi-survey
• [22] Chen, Wang, and Qu. "Recursive Self-Improvement in AI: From Bounded Self-Refinement to Autonomous Research Loops." July 8, 2026; version 2, September 6. Source. Primary source for its object-of-improvement and loop-closure taxonomy. chen-2026-bounded-to-open-rsi-survey
• [23] OpenAI. "Research acceleration: The view inside OpenAI." September 6, 2026. Source. Self-reported research workflow; the March 2028 target is a goal, not an achieved capability or calibrated forecast. openai-2026-research-acceleration
• [24] Anthropic. "When AI builds itself." 2026. Source. Industrial research automation and its limits; not a controlled whole-laboratory causal experiment. anthropic-2026-when-ai-builds-itself
• [25] Schmidhuber. "Gödel Machines: Fully Self-Referential Optimal Universal Self-Improvers." 2003; revised 2006. Source. Proof-triggered self-modification, with optimality conditional on the formal setup and proof found. schmidhuber-2003-godel-machines
• [26] Fallenstein and Soares. "Vingean Reflection: Reliable Reasoning for Self-Improving Agents." 2015. Source. Reasoning about more capable successors without exhaustive simulation. fallenstein-2015-vingean-reflection
• [27] Fallenstein, Taylor, and Christiano. "Reflective Oracles: A Foundation for Classical Game Theory." 2015. Source. Randomized self-reference and existence results; not an efficient general-purpose oracle implementation. fallenstein-2015-reflective-oracles
• [28] Garrabrant et al.. "Logical Induction." 2016; revised 2020. Source. Computable uncertainty about logical statements, distinct from practical runtime guarantees. garrabrant-2016-logical-induction
• [29] Wang, Lehman, Clune, and Stanley. "Paired Open-Ended Trailblazer (POET): Endlessly Generating Increasingly Complex and Diverse Learning Environments and Their Solutions." 2019. Source. Environment–solver co-evolution and transfer; designed outer framework. wang-2019-poet
• [30] Hughes et al.. "Open-Endedness is Essential for Artificial Superhuman Intelligence." 2024. Source. Novelty and learnability framework; the necessity claim is a position, not a general proven theorem. hughes-2024-open-endedness
• [31] Burns et al.. "Weak-to-Strong Generalization: Eliciting Strong Capabilities With Weak Supervision." 2023. Source. Selected weak-supervision experiments; no universal successor-oversight guarantee. burns-2023-weak-to-strong
• [32] Wu et al.. "SGM: A Statistical Godel Machine for Risk-Controlled Recursive Self-Modification." October 11, 2025. Source. Statistical gates and cumulative error budgets; the elementary theorem on this page is a separate construction. wu-2025-statistical-godel-machine
• [33] Howard, Ramdas, McAuliffe, and Sekhon. "Time-uniform Chernoff bounds via nonnegative supermartingales." 2018; revised 2020. Source. Nonnegative-supermartingale foundation for time-uniform inference. howard-2020-time-uniform-chernoff
• [34] Cunningham et al.. "The Economics of Recursive Self-Improvement." July 13, 2026. Source. Elasticities, research inputs, and feedback channels; the page uses a separate uncalibrated pedagogical reduction. cunningham-2026-economics-of-rsi
• [35] AI Futures Project. "AI Futures Model." August 2026 update; accessed September 14, 2026. Source. Versioned conditional milestone and feedback assumptions; not experimental evidence. ai-futures-2026-timelines-model
• [36] AI Futures Project. "AI 2027." 2025; accessed September 14, 2026. Source. A future scenario, not a future-dated observation or a consensus arrival date. ai-futures-2025-ai-2027