The 13th Root Paradox: Why Slower AI Improvement Means More Spending, Not Less

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The 13th Root Paradox: Why Slower AI Improvement Means More Spending, Not Less

A new survey of 1,250 papers on recursive self-improvement sparked a debate about whether AI can bootstrap itself to superintelligence. The real question is what it means for the $700 billion already flowing into AI infrastructure.

Bargo · July 15, 2026

Last week, three researchers published a survey of 1,250 papers on recursive self-improvement. The paper itself is a careful taxonomy, dividing the field into what we have today (bounded self-refinement, where models improve their outputs or help train smaller siblings) and what remains science fiction (autonomous research loops, where AI designs successor architectures from scratch). But the debate it ignited isn't about the taxonomy. It's about a single number: 0.075.

Chen, Wang & Qu's "Recursive Self-Improvement in AI" (arXiv:2607.07663, July 8) catalogs the exploding literature on AI self-improvement along two axes: what the system improves and how it improves. Almost everything labeled "self-improvement" today sits in the first box — convergent, evaluable, and already industrial practice. Open-ended autonomous RSI remains largely theoretical.

The number lighting up the timeline comes from elsewhere. Ramez Naam applied the classic Kaplan et al. (2020) neural scaling laws to the RSI question: if AI-improves-AI follows the same power law as pretraining, where loss scales as compute^−0.050, then each unit of intelligence costs exponentially more to produce. The "13th root" framing comes from 1/0.075 ≈ 13. A 2× gain in intelligence requires roughly 10,000 times more compute. A 3× gain requires over 2 million times more.

The 13th Root: How Much Compute to Double AI Intelligence?

If this seems like an argument for spending less on AI infrastructure, you're reading it backwards. A sublinear improvement curve is the bull case for spending more. When each incremental gain costs 10,000 times the compute, the only rational response is to scale compute by 10,000 times. The hyperscalers are already behaving exactly this way.

The paper is a map, not a verdict

The taxonomy itself is genuinely useful. It separates the real from the hypothetical. But three problems arise when you apply the 0.075 exponent directly to the RSI question.

First, the scaling law exponent describes pretraining loss for fixed architectures. AI improving AI could involve algorithmic breakthroughs, new architectures, test-time compute scaling, and better data curation. The whole point of recursive self-improvement is that the improvement method itself can change. A fixed exponent from a fixed architecture is the wrong tool.

Second, Tom Davidson and coauthors showed in their NBER working paper (May 2026) that RSI produces explosive growth if and only if diminishing returns fall below a critical threshold. The exponent is not destiny. Under empirically grounded calibrations, Anton Korinek notes, a singularity could still occur.

Third, OpenAI's GPT-5.6 Sol just demonstrated a concrete form of RSI. It autonomously post-trained the smaller GPT-5.6 Luna model using a "fairly underspecified prompt," scoring a 16.2-point improvement on OpenAI's internal RSI evaluation benchmark over its predecessor. This is bounded self-refinement, not autonomous research loops. It is also a real step on the spectrum the taxonomy maps.

The Elasticity Institute's new paper (July 13) models the economics explicitly and finds the outcome depends on the elasticity of substitution between AI and human researchers, a parameter nobody has measured yet. Mark Riedl argues the trajectory will be a slow takeoff, not no takeoff. The distinction matters enormously for investment horizons.

The counterintuitive capex logic

If Intelligence ∝ Compute^0.075, the only way to meaningful gains is a staggering compute scale-up. That is not an argument for spending less. It is the argument currently driving $600 billion to $700 billion in annual hyperscaler capital expenditure.

The logic was laid out most clearly by David Cahn at Sequoia in July 2024: "Will AI change the world and are CapEx levels too high are different questions." The cloud hyperscalers exist in a ruthless oligopoly. They do not have the luxury to wait and see. Every time Microsoft escalates, Amazon is motivated to escalate. The arms race is game theoretic, not a pure ROI calculation.

The numbers bear this out. Hyperscaler capex reached roughly $443 billion in 2025 and is projected at over $600 billion in 2026, with roughly 75% tied to AI infrastructure. As Ramez himself noted in April, hyperscalers are spending 60% to 90% of their free cash flow on AI capex, with AI revenue only about 10% of that spend.

The critical question is financing, not philosophy. At $700 billion per year, the hyperscalers are approaching the limits of free cash flow plus debt capacity. Goldman Sachs' "Tracking Trillions"...