AI’s Contribution to Cyber, Math, and Algorithms: Which Theory Fits?

Note

This is a draft. It asks a single question: across three domains where AI systems have produced candidate autonomous contributions — cybersecurity, mathematics, and algorithms — which theory of AI’s contribution best fits the evidence? The evidence itself lives in a companion source log; this document cites it by anchor and stays focused on the argument. Load-bearing claims are dated and sourced there so a later reader can check and update them.

Scope

We want to know how AI is going to bend these efficiency curves.

We plot declining cost (increasing efficiency) for a bunch of different technologies, and most of them fall in fairly consistent pattern [→ OWID cost curves]. We’re expecting AI to bend those curves down, and it’s starting to happen, but hard to say how much.

Unit cost over time for a selection of the 66 technologies in the Santa Fe Performance Curve Database. Log vertical axis, so a straight line is a constant exponential rate. Each series is priced in its own unit, so only slopes are comparable, never levels.

Four theories of AI’s contribution

Before looking at the evidence it helps to be explicit about the competing accounts of what AI is doing to R&D. Four are worth putting head-to-head. Some are formal models and some are verbal mechanisms; they overlap, but they make different enough predictions to earn separate columns in the evidence below. Three further accounts — a verification bottleneck, recombinant search, and recursive feedback — are set out in Other theories at the end, either because they refine task replacement rather than rivalling it or because they speak to aggregate dynamics rather than to the shape of an individual result.

Human acceleration: AI multiplies human output by a constant.

On this view AI is a better tool, like a faster compiler or a search engine. It makes every researcher some factor more productive but contributes nothing a human didn’t drive, and it doesn’t change the shape of what gets done — the mix of easy and hard results is the same, just arriving faster. Signature prediction: no result appears that a human didn’t direct step-by-step.

Human replacement: AI is a researcher, and the only limit is cost.

The “late-RSI” view (Davidson and Houlden 2025; Eth and Davidson 2025): AI has already reached human-level research ability, so the binding constraint is the compute cost of running more of it. Signature prediction: contributions are additive and scale with spend — N agents are worth roughly N humans — and there is no reachable task a sufficiently-funded agent cannot do.

Task replacement: AI substitutes on component tasks while bottlenecks determine the result.

The standard R&D version is Benjamin Jones’s task model [→ Jones]. Research progress combines a continuum of complementary task outputs. Humans can perform every task; AI performs a share \(s_t\), with productivity \(M_t\) on those tasks and bottleneck strength \(\theta\). AI therefore produces inputs into a result, not generally a standalone result: while \(s_t<1\), humans still perform the remaining tasks, which can severely constrain progress. This is not an intrinsic human requirement — at \(s_t=1\), the model permits fully automated R&D — nor is the task boundary fixed, since Jones explicitly studies it expanding. Signature prediction: gains track task coverage, relative task productivity, and complementarity; extreme performance on a narrow subset has limited effect when human-only tasks remain bottlenecks.

Apple-picking: AI picks real but low fruit, with a moving ceiling.

The apple-picking model: an AI agent is like a robot that picks apples cheaply but can only reach up to a height \(\lambda\). It finds genuine, previously-unpicked fruit, but disproportionately low fruit; its returns diminish sharply and are non-additive (running it twice is not worth two humans); it contributes most where humans have picked least; and each model generation raises \(\lambda\) by a band. Formally, apples remaining after human and agent expenditure \(x_H, x_A\) is \[\text{apples remaining}= \underbrace{\lambda\, e^{-r_Hx_H-r_Ax_A}}_{\text{apples on bottom}}+\underbrace{(1-\lambda)\,e^{-r_Hx_H}}_{\text{apples on top}}.\] Signature prediction: genuine new finds, but shallow, bounded, non-additive, front-loaded, and staircased by model generation.

The theories are separated by eight questions.

Everything below is an attempt to answer these, because they are where the four accounts disagree. The table is deliberately outside this definition-list cell so it can use the full page width.

Question	Human acceleration	Human replacement	Task replacement	Apple-picking
Rate of efficiency growth	One-off level jump, slope unchanged	Slope rises with AI spend, no ceiling	Slope rises only as far as the bottleneck allows	Burst then decay, re-steepening each generation
Autonomous contribution	None; every result is human-directed	Complete results at any reachable difficulty	Only where AI covers the whole task bundle	Yes, but concentrated in easy results
Demand for humans	Rises; AI complements every researcher	Falls; researchers substitutable at the AI price	Falls on automated tasks, rises on bottleneck tasks	Falls for low fruit, holds for high fruit
Expertise of discoverers	Unchanged; the same experts, working faster	Irrelevant; a budget substitutes for expertise	Shifts to whoever holds the non-automated skills	Lower on average; experts keep the hard remainder
Diminishing returns to AI spend	None to trade off; AI spend scales with headcount	Constant returns; N agents ≈ N humans	Sharp, once AI’s own tasks are saturated	Sharp; repeat runs re-find picked fruit
Intertemporal substitution of AI spend	None; the multiplier applies as work arrives	High; spend earlier to pull results forward	Deferred as coverage widens, but early spend costs nothing later	Deferred, and early spend cannibalizes later yield
Where the impact lands	No differential beyond routine workload	Where research labor is the largest cost	Where the automatable task share is highest	Where humans have searched least
Benchmarks most predictive of value	Human time saved on a real task	Cost per solved task, at any difficulty	End-to-end workflow coverage, not per-task scores	Realistic starting points and repeat-run yield

The last row asks about measurement rather than about the world, but the theories disagree about it sharply enough to be worth asking, and it is the row a reader can act on soonest. Rows two, three, five, seven, and eight are the ones the evidence below can actually speak to; the expertise and intertemporal rows are thinner, sketched more from each theory’s logic than from anything measured. The clean reading is unchanged at the extremes: autonomous end-to-end results kill human acceleration, and non-additivity plus a ceiling kills human replacement. The live comparison is between the last two columns, which agree about autonomy and about diminishing returns and differ on why — task coverage versus a depleting stock of reachable results.

The next section scores the eight questions by domain, as a map of what follows. Then each domain section takes the same eight questions in the same order, one or two slugs to a question, so a question can be read across cyber, math, and algorithms. Each opens by saying what “efficiency” means in that domain, because the OWID curves have a unit and these domains need one too.

Evidence summary

The same eight questions, now scored by domain, as a map of the three sections that follow. Each cell asks one thing only: is the evidence in that domain consistent with apple-picking? ✅ consistent, ❌ inconsistent, ❓ unclear or untested. The scores are judgments about the sources set out below, not measurements, and a reader who weighs the counterexamples differently should expect to move two or three cells.

Consistency is a weaker test than discrimination.

Most ✅ cells are equally consistent with task replacement, because the two theories agree about far more than they disagree about — as the formal section shows, they differ in whether the ceiling on AI’s contribution scales with headcount, not in whether returns diminish. So the table below is a scorecard for one theory, not a comparison between theories. The three tests at the end of the formal section are the comparison, and none of them has been run. The table sits outside this cell so it can use the page width.

Question	Cyber	Math	Algorithms
Q1 growth rate	✅ known bug classes dominate	❓ low walls, but one deep disproof	✅ 1.72× and surface-level edits
Q2 autonomy	✅ autonomous, mechanical classes	❓ autonomous, but hard-fought	✅ only where verifiers are cheap
Q3 demand	✅ triage revalued, enumeration devalued	✅ each result pulled in more experts	✅ the deep gains are still human
Q4 expertise	❌ expertise shifted, not lowered	❌ verification is expert-only	❌ wins came from expert harnesses
Q5 returns	✅ duplicates, once disaggregated	❓ no measured return curve	✅ humans overtake by 8 hours
Q6 intertemporal	❓ smooth path is the wrong test	❓ generation clusters confounded	❓ harness effects, also wrong test
Q7 incidence	❓ under-audited fits, hardened code doesn’t	✅ obscurity rather than difficulty	✅ collapse on mature repositories
Q8 benchmarks	✅ denominators travel, scores don’t	❓ no benchmark carries a denominator	✅ real baselines predict, multipliers don’t

Thirteen cells consistent, three inconsistent, eight unclear. The distribution matters more than the count, because the marks are not scattered: they cluster by row.

Apple-picking fails the same question in all three domains.

Q4 is the only row that is uniformly ❌, and it fails for the same reason each time. The model predicts that the average expertise of people making discoveries with AI falls, because low fruit becomes reachable by non-experts. What all three domains show instead is the expertise requirement moving: detection of Mythos-class bugs democratized to public models while weaponization did not [→ Vidoc]; sorting a real solve from a plausible one stayed expert-only, with roughly nine incorrect among forty-seven AI-standalone cases [→ Erdős wiki]; and the kernel records came from a test-time-search harness built by researchers rather than from anyone prompting a model [→ TTT-Discover]. Shifting rather than falling is task replacement’s prediction, not apple-picking’s. This is the clearest place where the two separate, and apple-picking loses.

The intertemporal row is untested everywhere, and the obvious test is the wrong one.

Q6 is uniformly ❓, and the formalization explains why the reader should not read that as a failure. The evidence people reach for is the staircase — do gains arrive in discrete bands at model releases? — and all three domains say no: cyber shows a smooth exponential [→ AISI cyber], the algorithms domain shows harness effects swamping generation effects [→ PERFOPT]. But the staircase is a claim about the time path of the reach ceiling, not a prediction of the production function. The actual intertemporal prediction is that past spend lowers the current return to spend on the same targets, and no source in this log measures it.

Where apple-picking wins, it wins on shallowness and on incidence.

The ✅ marks concentrate in Q1, Q2, Q7, and Q8 — how deep the finds are, where they land, and what predicts value. That is the part of the model that is really a claim about the distribution of available results: disproportionately easy, disproportionately in places humans have not looked, and therefore best measured by benchmarks with realistic starting points. Those cells are also where the model earns its keep over Jones, since task coverage alone does not predict that finds should cluster in under-audited web surfaces, neglected Erdős problems, and unoptimized baselines.

Math is where the evidence is thinnest, not where the theory does worst.

Five of math’s eight cells are ❓ or ❌, but for measurement reasons rather than adverse findings. There is no return curve, because nobody has run repeated equal-budget attempts on a fixed problem population [→ Tao]. There is no benchmark with a human-time or cost denominator. And the single deepest counterexample in the whole post — an 80-year-old conjecture disproved by a model that was not aimed at it — lives here [→ unit-distance]. Adopting bounds rather than problem counts as the outcome variable is what would change this, since bounds give the dated continuous series the domain currently lacks [→ ANTEDB].

Cyber

Cyber is the domain with the most public, dated, dollar-denominated evidence, because 2025–2026 saw both vendor demonstrations and independent government and academic measurement. It is also where the apple-picking model is most sharply tested.

Efficiency here is vulnerabilities discovered per unit of search effort, on a fixed codebase.

Fixing the codebase is what makes the metric conceptually clean. The tempting alternative — bugs found per year across all software — confounds discovery capability with how much new code got written, and code volume is the faster-moving term. Holding the target fixed, the unit is cost per confirmed vulnerability, and the record already supplies both numerator and denominator in places: AISI budgets about $12,500 per attempt, XBOW’s ~1,060 submissions yielded ~132 confirmed-and-resolved, and AIxCC’s organizers report about $152 per competition task [→ AISI runs, XBOW, AIxCC]. The three differ by two orders of magnitude, which is itself informative: cost per bug is set by the difficulty of the target far more than by the price of tokens.

The one clean year-over-year measurement on a fixed task set shows a large gain.

Almost every capability series in this domain confounds model generation with harness, prompt, and task mix. DARPA’s AIxCC is the exception, because the organizers ran the same competition structure twice: between the August 2024 semifinal and the August 2025 final, the share of planted vulnerabilities the autonomous systems identified rose from 37% to 86%, and the share of those they patched from 25% to 68% [→ AIxCC]. Two caveats hold the inference down. The teams changed their systems as well as their models, so this is progress in the whole stack rather than in the models alone; and the denominator itself was revised after publication, from 70 planted bugs to 63, which moved the headline rate from 77% to 86%. Taken with the per-generation AISI series, the reading is that capability on cyber tasks is rising fast and smoothly, which every theory here predicts and none is distinguished by.

A fixed codebase holds a finite stock, which the OWID curves do not.

This is the one structural disanalogy worth keeping in view. A cost curve for photovoltaics has no exhaustion built in; a fixed codebase holds a finite, depleting stock of bugs, so yield falls over time even for a searcher of constant capability [→ OWID cost curves]. Any efficiency series in this domain therefore mixes two effects — capability rising and stock depleting — and separating them is exactly what distinguishes apple-picking from the alternatives. Depletion is a prediction of the model, not a nuisance to be differenced away, but it does mean a falling bug-per-dollar series is not by itself evidence that agents got worse.

The finds are mostly novel instances of known bug classes — apple-picking over human replacement.

The shape of the finds favors “shallow, low fruit,” which caps how far the curve bends. The headline bugs are mechanical memory-safety classes — stack buffer underflow (SQLite), out-of-bounds write (FFmpeg), DoS (OpenBSD) — and Vidoc reproduced the detection of Mythos-class bugs with public models “but they didn’t build the weapon” [→ Vidoc]. There is no clear public case of an agent discovering a genuinely novel vulnerability class or a deep architectural flaw. That absence is what human replacement would not predict.

Depth is not literally absent, though. — against a too-strong apple-picking

The counter-pressure keeps “shallow” from becoming unfalsifiable. Anthropic presents the OpenBSD SACK bug as requiring reasoning about signed-integer overflow, and AISI scores a 32-step attack chain rather than isolated one-step finds [→ Mythos, AISI cyber]. “Shallow only” is too strong; “disproportionately shallow” is what the record supports.

Agents cut off a live zero-day and found flaws humans hadn’t. — against human acceleration

The autonomy question has the clearest answer here, and it rules out pure acceleration. Google’s Big Sleep found CVE-2025-6965, a live SQLite memory-corruption zero-day “known only to threat actors,” and cut it off before exploitation — reportedly the first time an AI agent foiled an in-the-wild exploit — and separately filed 20 previously-unknown OSS bugs with a human only in final review [→ Big Sleep]. The government anchor is stronger still: DARPA’s AIxCC finalists processed 54M lines of code and uncovered 18 previously unknown real-world flaws [→ AIxCC], and CyberGym’s agents incidentally exposed 15 zero-days in current software [→ CyberGym]. These are apples humans had not picked, found without step-by-step human guidance. Every remaining theory clears this bar; human acceleration does not.

Discovery got cheap and triage became the scarce input — task replacement’s reallocation, not human replacement’s substitution.

Demand moved rather than fell. curl ended its bug-bounty program after AI “slop” reached ~20% of submissions while genuine yield fell: the flood of candidates consumed more human validation time than the finds were worth [→ curl]. Pricing shows the substitution pressure at the low end, with ARTEMIS reporting AI pentesters at $18/h against $60/h for humans and ranking one 2nd of 11 [→ ARTEMIS]. Both patterns are the same reallocation: routine enumeration is worth less in human hands, while judging what is real is worth more. Nothing here measures net demand, so treat the direction as suggestive.

The finds still route through expert hands at both ends.

Someone expert chose the target and someone expert judged the output. Big Sleep’s twenty OSS filings had a human in final review [→ Big Sleep], and the AISI protocol budgets about $12,500 per attempt on an expert-built harness rather than a prompt [→ AISI runs]. The sharpest split is Vidoc’s: public models reproduced the detection of Mythos-class bugs, “but they didn’t build the weapon” [→ Vidoc]. Detection has democratized; exploitation has not. That is task replacement’s prediction — the expertise requirement shifts rather than disappearing — and it is squarely against human replacement’s claim that a budget substitutes for expertise.

More compute keeps paying off — against a naive apple-picking ceiling, for now.

This is the single most important tension in the post. The naive apple-picking gloss — “agents hit a compute wall fast” — is contradicted by the best independent evidence. AISI finds cyber performance scales log-linearly with inference-time compute with no observed plateau, up to 100M tokens (a 10M→100M increase improved performance by up to 59%) [→ AISI cyber]; summarizing the UK AISI runs, none of the 100M-budget models “showed signs of diminishing returns,” the basis for the “cybersecurity is proof-of-work now” framing [→ AISI runs]. Taken literally this looks most like human replacement (spend more, get more). The apple-picking model survives only if “diminishing returns” is restated carefully — see the next slug.

But value per run collapses, even where raw compute keeps paying — against human-replacement additivity.

The additivity question separates the two survivors, and the answer is no. XBOW’s autonomous pentester produced ~1,060 submissions but only ~132 confirmed-and-resolved, with ~208 duplicates: running the picker again mostly re-finds picked apples [→ XBOW]. The reconciliation: three things get conflated under “diminishing returns” — (a) returns to compute on one attempt (not falling, to 100M tokens), (b) marginal value of repeated runs on a fixed target (falling hard, via duplicates), and (c) a capability ceiling on the deepest tasks. Non-additivity lives at (b) and (c), and it is exactly what human replacement denies. The apples-remaining formulation (per tree) already encodes (b)/(c); the loose “hit a wall” gloss wrongly claimed (a).

Per-generation gains are real but look smooth, not staircased.

Apple-picking’s distinctive dynamic prediction — clear a band, then plateau until the next model, so that waiting buys a taller picker — is the weakest-supported. Gains per generation are real (AISI: average steps 1.7 → 9.8 from GPT-4o to Opus 4.6; Lyptus: offensive-cyber horizon ~30s → ~3h, accelerating) [→ AISI cyber, Lyptus]. But these look like a steady exponential, not discrete cliffs, and AISLE finds rankings “reshuffle completely across tasks,” complicating any clean monotonic per-generation story [→ AISLE]. A smooth exponential still rewards waiting, but it gives no band to wait for, which is the specific claim the staircase makes.

Starting-point dependence holds on average but has striking exceptions.

Apple-picking predicts agents contribute most where humans have picked least. XBOW’s web-app haul fits the under-audited surface exactly [→ XBOW]. But the marquee finds cut the other way: a 27-year-old OpenBSD bug and a 16-year-old FFmpeg bug that survived millions of fuzz executions sit in heavily-hardened code [→ Mythos]. Once the picker got tall enough, apples that survived decades of expert review turned out to be reachable — the sharpest tension with a naive starting-point story. (Vendor “N× more bugs” framing, e.g. Palo Alto, is a separate and contested claim [→ Palo Alto].)

The frontier is jagged, and verifiability predicts the jags — task replacement and apple-picking both fit.

“There is no stable best model across cybersecurity tasks… capability rankings reshuffle completely” [→ AISLE]. Low apples: systematic enumeration, known-CWE web classes, reproducing known memory bugs (a 3.6B-active-parameter model detected the showcased overflow). High apples: exploit weaponization, deep data-flow logic, full real-world reproduction (CyberGym’s 11.9% ceiling), and — increasingly — triage of a flood of candidates. Jones’s model reads this as heterogeneous task coverage with human-only bottlenecks; apple-picking adds that reachable work should be front-loaded and depleted by repeated runs. Jaggedness alone does not distinguish them.

Human-time and dollar denominators travel; capability scores don’t.

Cyber has the best measurement discipline of the three domains, and the split is clear. What predicts value is a benchmark with a human-time or dollar denominator: Lyptus gives independent human-time horizons (P50 ≈ 3 h in 2026, saturating on GPT-5.5), ARTEMIS prices agents against human pentesters, and AISI’s $12,500-per-attempt budgeting fixes what a “success” cost [→ Lyptus, ARTEMIS, AISI runs]. What does not travel is a score on any single task set, because AISLE shows rankings reshuffle completely across tasks [→ AISLE]. The detailed cost-per-bug figures remain vendor-reported or unverified [→ Mythos, Palo Alto], which is the reason to prefer the denominators someone outside the vendor can reproduce.

Math

Mathematics has the most vivid single data point — a model-generated disproof of an 80-year-old conjecture — and the most articulate skeptic in Terence Tao, whose “long tail” and “jumping machines” framings closely track apple-picking. The discipline here is separating three senses of “solved”: a genuinely new AI result, AI retrieval of a forgotten human solution, and human–AI collaboration with substantial human framing.

Efficiency here has two candidate units, and tightened bounds are the better one.

The obvious unit is open problems solved per unit of effort, which is what the Erdős-problem counts measure. It is a bad outcome variable: solved-or-not is binary, the problem population is not fixed, and selection dominates, as the Erdős wiki says in its own disclaimers [→ Erdős wiki]. The better unit is a tightened bound. Tao, Trudgian, and Yang’s exponent database records the best known value of each exponent in analytic number theory with a date and a proof, so progress on any one of them is a monotone dated series — the same object as a cost curve, with a number on the vertical axis [→ ANTEDB]. The kissing-number sequence, 592 → 593 and then a further human improvement, is one such series with an AI step in the middle of it [→ kissing number].

Collating the literature was itself worth new bounds, before any model was involved.

The exponent database’s launch paper obtained new exponent pairs, zero-density estimates, and additive energy estimates by computer-assisted search over the relations it had collected — in Tao’s words, “in a largely automated fashion (without introducing any substantial new inputs from analytic number theory)” [→ ANTEDB]. This is not an LLM result and should not be reported as one; it is optimization over a relation database. But it is a clean measurement of something apple-picking asserts and the other three theories do not: that an uncollated literature holds reachable fruit which nobody had picked because nobody had systematically combined what was already known. The reach ceiling did not move, and results appeared anyway.

AI clears low walls but cannot build on its own progress — task replacement, not human replacement.

The shape-of-finds question gets its sharpest answer from Tao. AI tools are “jumping machines that can jump two meters in the air… reaching the tops of the lowest walls,” but “what they can’t do is jump a little bit, reach some handhold, stay there, pull other people up, and then try to jump from there. There isn’t this cumulative process.” And: “these tools either succeed or they fail. They’ve been really bad at creating partial progress” [→ Tao]. In Jones’s terms, AI performs some research tasks while humans still supply framing, verification, and cumulative direction; the machine task output is not yet an independent research process. A genuine human replacement would do cumulative, partial-progress work, so the absence of it favors both task replacement and apple-picking.

An AI disproved an 80-year-old conjecture without being aimed at it. — against human acceleration, and against a too-strong apple-picking

The autonomy bar is cleared, and by an unusually deep result. A general-purpose OpenAI model disproved Erdős’s 1946 unit-distance conjecture, producing a construction with growth \(n^{1+\delta}\); nine mathematicians produced a human-verified account and Will Sawin made the exponent explicit (>1.014) [→ unit-distance]. OpenAI says the model was not math-specialized, not scaffolded to search proof strategies, and not aimed at this problem. This simultaneously kills human acceleration and stress-tests apple-picking’s “shallow, neglected” story: the problem was hard-fought for 80 years and the result is deep. Erdős #728 is a second, weaker autonomous solve, hedged by an operator-in-the-loop convention [→ Erdős #728].

Each headline result pulled in more expert human time, not less.

The autonomous solve was the beginning of the human work rather than the end of it: nine mathematicians produced the human-verified account of the unit-distance construction, and Will Sawin was the one who made the exponent explicit [→ unit-distance]. On the kissing-number problem AlphaEvolve pushed the 11-D bound 592 → 593 and a human then improved it further [→ kissing number]. Both are complementarity, and both point the same way as cyber’s triage story: the machine output raises the value of the human step that follows it. What none of this settles is whether the number of mathematicians rises or falls, only that each result still consumes expert attention.

Sorting a real solve from a plausible one is still specialist work.

Apple-picking predicts the average expertise of AI-assisted discoverers falls, and math gives the strongest reason to doubt that in the near term: the verification step is expert-only and unavoidable. The Erdős wiki’s tally of AI-standalone cases includes about nine incorrect among roughly forty-seven, so someone qualified has to separate them [→ Erdős wiki], and Erdős #728 is hedged by an operator-in-the-loop convention rather than counted as a clean solve [→ Erdős #728]. The October-2025 GPT-5 episode is the cautionary case: retrieval of forgotten human solutions was briefly reported as solving, and it took domain experts to notice [→ GPT-5 retrieval].

There is a harvest, but no measured return curve.

The non-additivity question can’t be answered cleanly in math. Tao’s arithmetic — “fifty-odd problems solved with AI assistance… but there’s like six hundred to go” — is consistent with an initial harvest leaving a harder remainder, but it is not a success rate: problem selection, effort, and failed attempts are unobserved [→ Tao]. The kissing-number sequence shows continued human headroom after an AI improvement, but one sequence is not a crossover curve [→ kissing number]. No source shows repeated equal-budget runs on a fixed problem population, so non-additivity is neither confirmed nor refuted here.

Generation-linked clusters are visible but confounded.

The staircase prediction is again only suggestive. GPT-5.2-era Erdős contributions and the later unit-distance disproof cluster in the timeline, but model release, prompting, scaffolding, researcher attention, and reporting all move together, so attribution to the base-model generation is not identified. FrontierMath gives a capability series but its v2 revision (errors in 42% of problems) blocks a clean time series [→ FrontierMath]. No primary source asserts a “clear a batch, then plateau” dynamic for research math, so the case for deferring effort to the next generation rests on the theory rather than on the record.

Most solved problems were neglected, not hard-fought — strong apple-picking, with selection bias.

Starting-point dependence is unusually well-attested here, from primary sources. Tao frames unsolved problems as a “long-tail distribution” with harvesting “at the very end of the tail” — the easy, neglected problems clear first [→ Tao]. The Erdős-problems wiki says so in its own disclaimers, verbatim: it “is not a benchmark,” and “Absence of past progress may reflect obscurity rather than difficulty” [→ Erdős wiki]. A problem “open” for 50 years often means nobody seriously tried. The unit-distance disproof is the standing exception — genuinely hard-fought — so the mechanism is a tendency, not a law.

Well-posed and verifiable is easy; deep and cumulative is hard.

Jaggedness, strongly. Low apples: construction/search problems (AlphaEvolve matched or beat SOTA on many open problems), literature retrieval, competition-style problems. High apples: deep conceptual long proofs, mathematical taste, and partial-progress hill-climbing — Tao’s cumulative-process failure [→ Tao, AlphaEvolve]. The ordering is by feedback-loop length and verifiability, exactly as in cyber and algorithms.

Counts and contest scores proxy capability but not value.

Math is the domain where the available benchmarks predict value least well, and where the sources say so themselves. The Erdős wiki documents ~47 AI-standalone cases (≈13 full, ~25 partial, ~9 incorrect) while explicitly disclaiming that it “is not a benchmark” [→ Erdős wiki]; IMO 2025 saw an officially-graded gold on problems chosen to be solvable in hours [→ IMO]; FrontierMath rose from <2% to the 40–90% range on hard tiers but revised 42% of its own problems [→ FrontierMath]. All are capability markers; none carries a human-time or cost denominator, and the GPT-5 retrieval episode shows a count can move without any new mathematics behind it [→ GPT-5 retrieval].

Algorithms

Algorithmic and code efficiency is the domain the original apple-picking post was built around, and it has the best-instrumented benchmarks — spanning unoptimized numerical code, ML research tasks, GPU kernels, and mature repositories. The breadth is useful but it is not a controlled experiment: task distribution, metric, scaffold, and budget change together.

Efficiency here is already a series of well-known curves, each with a fixed task and a resource denominator.

Unlike the other two domains, this one does not need a metric invented for it. Compression gives bits per byte on a fixed corpus, the Hutter Prize being the standing version. Pretraining gives compute to reach a fixed loss. Reinforcement learning gives environment steps to a fixed score, as in the Atari-100k convention. Inference gives latency or tokens per dollar for a fixed model, which is what the kernel and speedrun records above actually move [→ TTT-Discover, nanogpt]. Classical complexity supplies bounds in the same form as math’s: the matrix-multiplication exponent, max-flow running time, SAT solving. Each is a fixed task with a resource denominator, which is exactly the structure the cyber and math metrics had to be constructed to have.

The pretraining curve is the one that has actually been measured.

Erdil and Besiroglu estimate that the compute needed to reach a fixed language-model performance level has halved roughly every 8 months, far faster than Moore’s law, while attributing only 5–40% of observed gains to algorithms and 60–95% to compute and data scaling [→ algorithmic progress]. Two things make this the right reference point. It is a real efficiency curve of the OWID kind, with a slope that a new technology could bend. And it ends in 2023, covering pretraining only, so it is the pre-agent baseline rather than any measurement of AI’s contribution to itself.

The most decision-relevant curve is the one nobody outside the labs can see.

What matters for every question in the table is the labs’ internal algorithmic efficiency: the compute-equivalent gain per year on their own training and inference stack, which is where AI contributions to AI research would show up first and compound fastest. That series is essentially unobserved. The closest public glimpse is AlphaEvolve’s self-reported 23% kernel speedup and recovery of about 0.7% of fleet compute [→ AlphaEvolve], which is a real number attached to a real deployment but is one firm’s self-report on two components of a stack. Every public benchmark below is a proxy for this curve, and the gap between the proxies and the thing itself is the largest measurement problem in the post.

Agents tune and substitute libraries; they don’t invent algorithms — strong apple-picking.

Shallowness is the most consistent finding in this domain, and it directly caps the achievable bend in the curve. AlgoTune averages only 1.72× over expert solvers and reports that models “fail to discover algorithmic innovations, instead preferring surface-level optimizations” — its 142× outlier is a BLAS-for-Python swap, not a better algorithm [→ AlgoTune]. MLGym: models “improve on the given baselines, usually by finding better hyperparameters, but do not generate novel hypotheses, algorithms, architectures” [→ MLGym]. Karpathy’s autoresearch stacked ~20 additive edits — real, transferable, but “cagey and scared” on open-ended problems [→ autoresearch].

Randomized against real work on mature repositories, the measured effect was negative.

This is the only randomized estimate in the post and the only one whose sign is negative. METR gave 16 experienced maintainers 246 real issues from their own large repositories and randomized whether AI was allowed: allowing it increased completion time by 19% [→ METR RCT]. Three features make it load-bearing rather than a curiosity. The setting is a high starting point, so it is the human-side counterpart to the collapse of agent gains on mature repos. The tooling was early 2025 and non-agentic, so it bounds a moment rather than the present. And the perception gap is enormous: the same developers forecast a 24% speedup and, having been slowed down, still believed they had been sped up by 20%, while economics and ML experts forecast 39% and 38% speedups. Any efficiency estimate in this post that rests on self-report or expert forecast rather than measurement should be discounted accordingly.

Agents set genuine records where the verifier is cheap. — against human acceleration

Autonomy is cleanest where a fast automatic verifier exists. AlphaEvolve found a 4×4 complex matmul in 48 scalar multiplications, the first improvement on Strassen in that setting in 56 years [→ AlphaEvolve]. TTT-Discover’s kernels beat the best human TriMul submissions by >15% on every GPU type [→ TTT-Discover]. AI-agent companies hold official verified records on the modded-nanogpt leaderboard [→ nanogpt]. Every win here has a cheap, automatable verifier — kernel latency, validation loss, matmul rank.

On the same leaderboard, the deep gains are still human.

modded-nanogpt is the closest thing to a like-for-like comparison in the record, because humans and agents compete on one metric under one set of rules. The AI-held records are shallow-mechanical, around 1% from a fused triton kernel, while the large gains are human: Muon at ~21% and U-Net skips at ~8% [→ nanogpt]. Demand for humans is highest exactly where the improvement requires a new idea rather than a faster implementation of an existing one, which is what both task replacement and apple-picking predict and what human replacement does not.

The wins came from experts building harnesses, not from users prompting.

Where a record fell, the credit sits with whoever engineered the search around the model. TTT-Discover’s kernel wins came from a test-time-search harness applied to an older open model rather than from a new frontier model [→ TTT-Discover], and PERFOPT-Bench shows the same model’s per-task speedup changes materially with the agent framework around it [→ PERFOPT]. Karpathy’s autoresearch is the same story at smaller scale [→ autoresearch]. The expertise required has moved from the domain to the scaffold, which is a reallocation rather than a removal.

Humans overtake agents as the budget grows — the cleanest non-additivity evidence, against human replacement.

RE-Bench measures the human-vs-agent crossover directly and shows the tortoise–hare shape almost exactly: at 2 h the best agents score ~4× human experts; humans exceed the best agent by 8 h; at 32 h humans reach ~2× the top agent [→ RE-Bench]. This is the strongest single refutation of human replacement across all three domains: more agent budget does not keep matching more human hours. The one caveat runs the other way — PERFOPT-Bench’s relay pilot recovered headroom by restarting an agent from an externalized summary, suggesting some apparent within-run plateau is a context-management failure rather than true exhaustion [→ PERFOPT].

The record clustering is a harness effect, not a model generation.

The staircase prediction is if anything cut against here, which matters because it is the prediction that would justify deferring spend. TTT-Discover’s win came from the test-time-search harness on an older open model, not a new base generation [→ TTT-Discover]; PERFOPT-Bench shows the same model’s per-task speedup changes materially with the agent framework [→ PERFOPT]. If observed jumps come from scaffolding and task mix rather than model generations, then waiting for the next generation is the wrong way to buy a taller picker; building a better harness is the cheaper one.

Gains collapse from unoptimized baselines to mature repos — starting-point dependence, uncontrolled.

The direction matches apple-picking, though the endpoints come from different benchmarks. Unoptimized baselines yield large reported gains (AlgoTune’s 142× outlier; Karpathy’s ~20 stacked edits) [→ AlgoTune]. Expert-optimized real repos collapse: SWE-fficiency agents reach “less than 0.15× the expert speedup on average” across 498 tasks, and GSO agents achieve less than 5% success [→ SWE-fficiency, GSO]. This is consistent with value shrinking as the starting point gets more optimized, but it does not isolate the effect: a clean test would run the same agent, scaffold, and budget on staged pre- and post-optimization versions of one codebase.

Fast, verifiable tasks fall first; long-feedback engineering resists.

Jaggedness along the axis of feedback-loop speed. Low apples: GPU/kernel optimization, hyperparameter and recipe tuning, naive-code library substitution. High apples: repo-level performance engineering in mature codebases (SWE-fficiency <0.15×, GSO <5%) and novel architecture, for which this record has no comparable autonomous success case [→ SWE-fficiency, GSO].

Benchmarks with a real starting point and a real denominator are the ones that predict value.

This domain has both the best and the most misleading benchmarks, and the difference between them is the answer to this question. The predictive ones fix a realistic baseline and a cost: RE-Bench’s 2/8/32-hour human budgets, SWE-fficiency’s expert-speedup denominator across 498 real tasks, GSO’s real repositories, and AlgoTune’s hard $1-per-task ceiling, under which only surface gains emerge [→ RE-Bench, SWE-fficiency, GSO, AlgoTune]. The misleading ones report a raw multiplier over an arbitrary baseline, which is why AlgoTune’s 142× says more about the baseline than the agent. Two caveats bound even the good ones: a reliability audit found absolute multipliers and rankings are machine- and scoring-rule-dependent, though “agents struggle on repo-level optimization” survives [→ reliability audit], and self-reported deployment figures such as TTT-Discover’s “few hundred dollars” and AlphaEvolve’s 23% kernel speedup and ~0.7% fleet compute are the least reproducible numbers here [→ TTT-Discover, AlphaEvolve].

Which theory fits?

Reading the three domains against the four theories:

Human acceleration is dead.

In every domain, agents produced results humans did not drive — live zero-days, an 80-year-old disproof, records beating the best human kernels. Whatever AI is doing, it is not merely a faster tool for humans.

Human replacement is refuted by non-additivity and a ceiling.

RE-Bench’s human overtake by 8 hours, XBOW’s duplicate-swamped yield, and the absence of any deep novel bug class or cumulative math progress all say the same thing: N agents are not worth N humans, and there are reachable-looking tasks agents still can’t do. The one apparent exception — cyber compute scaling to 100M tokens with no plateau — is about returns to compute on one attempt, not about matching human hours, and dissolves once the three senses of “diminishing returns” are separated.

Task replacement carries most of the explanatory weight, once its bottleneck is named.

Jones cleanly explains why spectacular performance on some tasks can coexist with modest overall progress: AI covers only part of the workflow, and remaining tasks bottleneck the composite result. The evidence repeatedly identifies the same remaining task, which is why the verification refinement in Other theories is worth reading alongside it. Cheap automatic scoring accompanies the strongest results — exploit tests, formal or inspectable proofs, kernel latency, validation loss — while human review, triage, implementation, and cumulative direction bind elsewhere.

Apple-picking adds depletion and starting-point dependence, and fits better than task replacement alone — barely.

Jones and the validation account do not by themselves predict that machine finds should be concentrated where humans have searched least or depleted by repeated runs. Apple-picking earns its edge from those additional patterns — the Erdős neglected tail, unoptimized baselines, under-audited web surfaces, RE-Bench’s crossover, and duplicate-heavy vulnerability discovery. But it pays for that edge with two live counterexamples — the hardened-code cyber finds (27-year OpenBSD, 16-year FFmpeg) and the unit-distance disproof show that “low and neglected only” is too strong — and with one uniform failure. On expertise, all three domains show the requirement shifting rather than falling, which is what task replacement predicts and apple-picking does not; see the evidence summary. The honest reading is layered: task coverage is the general production structure; verification is the most visible current bottleneck; apple-picking captures additional front-loading and depletion that fit much, but not all, of the record.

Apple-picking’s own weakest prediction is the staircase.

Nowhere is the “clear a band, then plateau until the next model” dynamic cleanly demonstrated. Per-generation gains are real but look like a smooth exponential; and in algorithms the biggest recent jumps came from harnesses on old models, not new generations. This is the prediction most worth targeting with future data.

What would move the verdict.

The cleanest experiments are still unrun. A matched pre/post-optimization test — same model, scaffold, budget, and metric on staged versions of one codebase — would isolate starting-point dependence and separate task replacement from apple-picking. A standardized repeated-run protocol on a fixed target would measure non-additivity directly instead of inferring it from duplicates. An experiment varying automation breadth separately from model productivity would test recombinant search’s inverted-U. And a dated, per-generation capability series at fixed scaffold would test the staircase and provide the raw material for measuring recursive feedback. Until then, the fair summary is: human acceleration and human replacement are out; task coverage plus a verification bottleneck explains the broad pattern; and a moving, front-loaded, non-additive boundary adds explanatory power, with its ceiling less sharp and its dynamics less punctuated than the simplest apple-picking version claims.

Cross-cutting limitations. Public examples are selected for success; vendors choose what to disclose; failed-attempt denominators are usually missing. The expertise question is worse off than the others, because its most-cited empirical result — that AI’s gains to scientific discovery concentrate among the ablest researchers — came from a paper MIT has disavowed for fabricated data [→ Toner-Rodgers]; everything said about expertise above is assembled from domain-specific observation instead. Benchmarks may be contaminated, revised, or saturated; model, scaffold, budget, and evaluator often change together; and “autonomous” almost always means autonomous execution after humans chose the target and verifier. This evidence can establish existence and suggest mechanisms far more readily than it can estimate representative rates or causal effects — which is why the verdict above is a ranking of theories, not a measurement.

Formalizing the theories

Start from the Jones research production function and ask where AI goes.

Let \(A\) be the stock of results in a domain and \(\dot A\) the flow of new ones — bugs found, bounds tightened, speedups shipped — so \(\dot A\) is the slope of the efficiency curve in. The workhorse is

\[\dot A = L^{\gamma}A^{1-\beta},\]

with \(L\) research labor, \(\gamma\le1\) the duplication or stepping-on-toes exponent, and \(\beta>0\) the fishing-out exponent. The growth rate is \(g_A = L^{\gamma}A^{-\beta}\), so at constant \(L\) growth dies away, and with population growing at \(g_L\) the balanced rate is \(\gamma g_L/\beta\) — long-run progress is pinned by the number of researchers.

Two arguments are missing: AI expenditure \(K\) and model capability \(C\). Each of the four theories is an answer to where they belong. Throughout, \(\gamma\) and \(\beta\) are reserved for the two Jones exponents, the automated task share is \(s\), and the discount factor is \(\delta\).

Three of the four theories put AI in the labor slot and leave the stock alone.

They share the form

\[\dot A = X(L,K;C)^{\gamma}\,A^{\,1-\beta},\]

where \(X\) is effective research input, and they differ only in \(X\). They do not change the interaction between effort and the stock of ideas, in the production of new ideas.

Human acceleration: \(X = L\cdot G(K,C)\).

Spend and capability combine into a per-researcher multiplier, with \(G_K>0\) and \(G_C>0\). Buying more AI does raise output, since \(X_K = L\,G_K>0\), so this is not the claim that AI spend is useless. It is the stronger claim that spend acts only through \(L\): \(X=0\) whenever \(L=0\), at every level of capability.

The identifying restriction is testable. The ratio of the two output elasticities, \(\big(\partial\ln\dot A/\partial\ln K\big)\big/\big(\partial\ln\dot A/\partial\ln L\big)\), equals \(\varepsilon_{G,K}(K,C)\) and so is independent of \(L\). No other theory here implies that.

Human replacement: \(X = L + m(C)K\).

Humans and agents are one effective input, with \(m(C)\) human-equivalents bought per unit of spend and \(m'>0\). Two static signatures: the marginal rate of substitution \(X_K/X_L=m(C)\) is constant, which is the formal content of “N agents are worth N humans,” and \(X\) is unbounded in \(K\) at \(L=0\).

This implies that once \(C\) rises above some threshold then all human labor will be instantly replaced with computer labor (a slightly weaker interpretation: if the supply of \(L\) and \(K\) isn’t perfectly elastic, then the replacement will be a bit slower).

Task replacement: \(X\) is a CES aggregate of human and machine tasks.

Benjamin Jones’s continuum of tasks, with AI covering a share \(s(C)\) at productivity \(m(C)\) and a CES aggregator of elasticity \(\theta\), reduces to [→ Jones]

\[X = \Big[s\,y_1^{\rho} + (1-s)\,y_2^{\rho}\Big]^{1/\rho}, \qquad y_1 = \frac{L_1+m(C)K}{s}, \quad y_2 = \frac{L_2}{1-s}, \qquad \rho = \frac{\theta-1}{\theta} < 0,\]

where \(y_1\) and \(y_2\) are output per task in the automatable and non-automatable groups and labor divides between them, \(L_1+L_2=L\). Humans can work an automatable task as well, AI being merely cheaper on it, and that carries more weight than it looks: aggregating over \(mK\) and \(L\) directly would put \(X=0\) at \(K=0\), describing a world in which no research was possible before AI. With \(\rho<0\) the groups are complements. This is Jones’s own technology rather than a relative of it — his budget-constrained solution is a CES unit cost function, and the expression above is what that cost function is dual to. Read the two side by side with care, because he writes the CES exponent as \(\theta\), which is this post’s \(\rho\), and uses \(\rho\) for something else again. Implications:

• Before AI, \(X = L\): the theory nests the Jones baseline rather than replacing it.
• While any labor is still working automatable tasks the optimal split equalizes \(y_1\) and \(y_2\), and then \(X = L + m(C)K\) exactly. Over that range task replacement is human replacement; the two separate only at \(K = sX/m\), where the last human leaves the automatable tasks.
• Labor is a necessary input (while \(s<1\))
• There’s a ceiling on AI-produced progress (\(\lim_{K\to\infty}X = (1-s)^{1/\rho-1}L\)).

Apple-picking: it splits \(A\).

Partition results by difficulty into those AI can reach and those it cannot, \(A = A_1 + A_2\), and let researchers divide between the two, \(L_1+L_2=L\):

\[\begin{aligned} \dot A_1 &= \big(L_1 + m(C)K\big)^{\gamma}\big(\bar A_1(C) - A_1\big)^{\eta},\\ \dot A_2 &= L_2^{\,\gamma}A_2^{\,1-\beta}. \end{aligned}\]

Two departures, logically independent, and neither forced — other formalizations of the same intuitions are set out at the end of this section. The first is the decomposition itself: \(A\) is not one stock, and AI’s productivity depends on one component rather than on the whole. The second is that in the reachable component the fishing-out term is replaced by a remaining-stock term, \(\bar A_1(C)-A_1\) rather than \(A_1^{1-\beta}\), where \(\bar A_1(C)\) is the mass of results below the reach height \(\lambda\) and rises with capability. Note that the input index \(L_1+mK\) is the human-replacement index: within the reachable zone the two are the same theory.

The four theories are four shapes in \((K,L)\) space, and the axis intercepts carry the difference.

Hold the stock \(A\) and capability \(C\) fixed, fix a quantity of results \(\bar V\) to be produced, and ask which pairs of AI spend and research labor deliver it — putting spend only where it can do work, and allocating labor across tasks or components to maximize output. For the three single-stock theories \(A^{1-\beta}\) is then a constant, so the contours are just isoquants of the input index \(X\). For apple-picking they are not, because the reachable component is depleted as it is picked, and that is what makes its panel a different shape rather than a different parameterization of the same shape.

All four are normalized to the same pre-AI world, \(X=L\) at \(K=0\), so a contour can be labelled by its own height on the labor axis and every difference between the panels is something \(K\) did. They are normalized again at the margin: the multiplier is scaled so that \(G'(0)=m\), and apple-picking’s picking hazard is set so that the first unit of picking effort yields one result — which is also what lets labor sit in \(A_2\) and leaves AI to do the picking in \(A_1\). At the single point \((K,L)=(0,1)\), then, all four theories agree that the marginal dollar of AI is worth \(m\) and the marginal researcher is worth one, and everything that follows is disagreement about what happens away from that point. The figure sits outside this cell so it can use the page width.

The red path in each panel is the expansion path at \(w=r=1\): the best split of a budget \(B\) between the two inputs, traced as \(B\) rises. It requires \(m>1\) to be defined at all. At \(m=1\) a dollar of AI and a unit of labor are worth exactly the same at the margin, and three of the four theories make them perfect substitutes over some region, so half the budget line is optimal and there is no path to draw; the figure therefore uses \(m=2\), an AI dollar worth two human ones, which is Jones’s adoption condition holding strictly rather than with equality.

One column cannot hold \(m\) fixed, and the exception is worth stating rather than hiding. Under human replacement \(m(C)\) is not only the exchange rate but the single parameter through which capability enters, so raising \(C\) in that panel is raising \(m\). It is drawn crossing \(m=1\), at \(0.5\) above and \(2\) below, because that crossing is the only event the theory contains. Its lower row is therefore the one at the figure’s shared \(m\), and its upper row is a world in which AI is not yet worth buying: the lines there are flatter than their neighbours because \(m\) is smaller, not because the theory differs in kind.

Isovalue curves in AI spend \(K\) and research labor \(L\), holding the stock \(A\) and capability \(C\) fixed. Contours run upward from \(\bar V=0.5\) in steps of \(0.5\), far enough past the corner to fill each box. Since all four theories are normalized to \(\bar V=L\) at \(K=0\), a contour is labelled by the height at which it meets the labor axis, and the dots on that axis stand at the same heights in all eight panels. Two contours are picked out in dark blue, \(\bar V=2\) and \(\bar V=4\), and the dotted rule marks the floor the second is heading for — one that most panels have not reached by \(K=4\). Dots on the spend axis mark contours reached with no labor at all. The bottom row raises capability through the parameter named in each panel, repeating the two marked contours’ old positions in grey. The red arrow is the expansion path at \(w=r=1\), the optimal split of a budget as the budget rises. Drawn with \(m=2\) and \(\theta=0.5\), an elasticity that is Jones’s own worked example, except in the human-replacement column, where \(m\) is itself the capability parameter and the two rows straddle the switching rate at \(m:0.5\to2\); the shapes rather than the numbers are the content.

Only two of the four ever touch the spend axis, and one of those only below its ceiling.

Reading the top row left to right. Human acceleration is the only one whose contours leave the labor axis at different angles: \(K\) enters as a multiplier on \(L\), so the first dollar is worth \(\bar V\) times as much on the \(\bar V\) contour as on the unit contour, and the contours are vertical rescalings of one curve rather than a family with a common slope. It never meets the spend axis, flattening instead onto \(\bar V/G_\infty\). Human replacement is a straight line all the way to the spend axis at \(\bar V/m\), and is the only theory in which money alone reaches any target whatever. Task replacement leaves the labor axis along exactly that line, drawn at the same \(m\), because AI and labor are perfect substitutes within an automatable task, and departs from it only at \(K = s\bar V/m\), where the last human leaves those tasks; from there the CES bends it onto a floor of \((1-s)^{1-1/\rho}\bar V\). Apple-picking also leaves along that line — inside the reachable zone the two are the same theory — and then bends as the reachable stock runs down. Its contours below \(\bar A_1\) reach the spend axis, and every contour above \(\bar A_1\) stops short.

The two floors then say what the ceiling test says. Task replacement’s floor is proportional to the target, so its contours fan out from the origin and doubling the target doubles the labor that unlimited spend still requires. Apple-picking’s floor is \(\bar V-\bar A_1\), so its contours are vertical translates spaced by a constant rather than by a factor, and every target below \(\bar A_1\) has no floor at all. Apple-picking hands over \(\bar A_1\) results with no labor attached to them; task replacement never hands over anything with no labor attached. That is the whole content of a ceiling that contains no \(L\), drawn.

The bottom row is a new model generation, and the four theories move in four ways. Every panel’s labor-axis intercepts are pinned, since capability does nothing without spend. Below that, acceleration’s and task replacement’s floors scale down, multiplying what a given headcount can reach; human replacement’s line rotates about its intercept, a better model being simply a cheaper human, and here the rotation is drawn carrying it past the rate at which the human stops being worth hiring; and apple-picking’s contours translate downward by \(\Delta\bar A_1\), the newly reachable band, by the same absolute amount at every target, so that targets which previously required labor now fall below the ceiling and reach the spend axis. A staircase, if there is one, is a sequence of those translations.

Optimal spending traces four different paths, and only under apple-picking does the AI budget stop growing.

Give a lab a budget \(B\) to split at \(w=r=1\) and raise it. Under human acceleration the marginal dollar of AI is worth \(LG'(K)\) while the marginal researcher is worth \(G(K)\), so AI is not worth buying at all until headcount passes \(1/m\): the first dollars go entirely to labor, and only then does the path bend right, along \(L = G/G'\). It stays labor-heavy throughout — at a headcount of four the optimal AI budget is one — and AI’s share of spend falls toward zero as the budget grows, because the value of a dollar of AI is proportional to the people it is multiplying. Human replacement has no interior to find. The two inputs are perfect substitutes at a fixed rate, so the whole budget goes to whichever is cheaper per effective unit, at every budget, and the panel is drawn crossing the rate at which the answer changes: at \(m=0.5\) every dollar hires people and the path runs up the labor axis, at \(m=2\) every dollar buys AI and it runs along the spend axis. The path does not tilt as capability rises. It rotates through ninety degrees at \(m=1\), which is what the theory’s claim that labor is replaced all at once looks like when it is drawn.

Task replacement sits between them, and its path is a straight ray from the origin. The technology is constant-returns and homothetic, so the optimal split is a set of expenditure shares rather than a sequence of decisions: the optimum equates per-task outputs at the ratio \(y_1/y_2 = m^{1/(1-\rho)}\), which is a property of \(m\), \(s\) and \(\rho\) and not of \(B\). Doubling the budget doubles both inputs. Raising the automatable share tilts the ray toward spend — AI’s share goes from \(0.41\) to \(0.68\) as \(s\) goes from \(0.5\) to \(0.75\) — but never turns it into either axis, because the bottleneck tasks keep buying labor and the automatable ones keep buying AI.

Apple-picking is the only one where AI spend is bounded. Its marginal dollar is worth \(me^{-mK/\bar A_1}\), which falls to the price of labor at \(K^{*}=(\bar A_1/m)\ln m\) — a threshold that depends on the reachable stock and the exchange rate and not at all on the budget. Below it every dollar goes to AI; above it every further dollar goes to labor, and the path turns vertical and stays there. This is the result the original post drew: buy the cheap agents first, then hire people, with the agent budget saturating at a level that a larger budget cannot move. It is also the sharpest observable difference between the theories, since a lab’s split of its own budget is visible in a way that its isoquants are not. The staircase reappears here as a prediction about that split: each new model raises \(\bar A_1\) and so moves \(K^{*}\) right, so AI spend should step up on release and then plateau until the next one, rather than growing with the budget.

Fishing out and running out are not the same assumption.

Jones’s \(A^{1-\beta}\) says the next idea is harder than the last but there is always a next one. At constant \(L\) it gives \(A_t = (\beta L^{\gamma}t)^{1/\beta}\) — growth that slows forever without ever stopping. The apple-picking term says the reachable results are finite: \(A_1\to\bar A_1(C)\) and halts there. With \(\eta=1\) and constant input \(X\) it integrates to \(A_1(t) = \bar A_1\big(1-e^{-X^{\gamma}t}\big)\), which is the exponential in the original post; a constant hazard of finding is all the exponential ever was. Everything distinctive follows from boundedness rather than from that functional form.

The CES question has a clean answer once the framework is set up this way.

Three of the four input indices are built out of a CES and differ only in \(\sigma\): human replacement has \(\sigma=\infty\); task replacement is CES across task groups with \(\sigma=\theta<1\), which in \((L,K)\) means perfect substitution until labor has left the automatable tasks and \(\theta\) only after that; and human acceleration is labor-augmenting rather than a substitution story at all. Apple-picking’s input index, \(L_1+mK\), is also CES with \(\sigma=\infty\) — so the substitution structure is the least distinctive thing about it, and writing that part in CES form does no violence whatever.

What cannot be written as a CES is the other factor. A CES is homogeneous of degree one and therefore cannot be bounded, so \(\bar A_1 - A_1\) has no CES representation; forcing the whole right-hand side into one deletes the finite stock, which is the claim the model exists to make. It is worth saying what this does not buy, correcting an earlier version of this section: the task index satisfies no Inada condition in \(K\) once humans can work the automatable tasks, so the first dollar of AI is worth \(m\) under task replacement exactly as it is under apple-picking, and the two indices leave the labor axis along the same line. So apple-picking is CES on the input side, indistinguishable from task replacement near \(K=0\), and cannot be CES on the stock side — and everything that distinguishes it lives on the side that cannot be.

The ceiling test is sharper than the cross-partial, which duplication contaminates.

The natural test is the sign of \(\partial^2\dot A/\partial L\,\partial K\): positive under task replacement from CES complementarity, negative under apple-picking from depletion. Inside the Jones function that test does not survive. Differentiating the shared form,

\[\frac{\partial^2 \dot A}{\partial L\,\partial K} = A^{1-\beta}\Big[\underbrace{\gamma X^{\gamma-1}X_{LK}}_{\text{substitution}} + \underbrace{\gamma(\gamma-1)X^{\gamma-2}X_LX_K}_{\text{duplication}}\Big],\]

and the second term is negative for every theory whenever \(\gamma<1\). Since duplication is exactly what \(\gamma<1\) encodes, a measured negative cross-partial is consistent with task replacement plus crowded researchers, and the test only bites if \(\gamma\) is known.

The ceiling comparison has no such problem. Under task replacement, unlimited spend yields \(\big((1-s)^{1/\rho-1}L\big)^{\gamma}A^{1-\beta}\), which is proportional to \(L\): double the researchers and you double what infinite AI can achieve. Under apple-picking, unlimited spend yields \(A_1\to\bar A_1(C)\), which does not contain \(L\) at all: double the researchers and the ceiling does not move, the same finite stock is merely exhausted sooner. That is the same economics as the cross-partial and it is estimable without knowing \(\gamma\). It is also the difference between the third and fourth panels of the figure above: floors that fan out from the origin against floors that are parallel.

The four signatures, side by side.

Each theory is a distinct answer to where \(K\) and \(C\) enter, and the answer propagates all the way to the long-run growth rate. The table sits outside this cell so it can use the page width.

Theory	Where \(K\) and \(C\) enter	AI alone, \(L=0\)	Ceiling as \(K\to\infty\)	Long-run effect of spend
Human acceleration	\(X = L\,G(K,C)\)	\(\dot A = 0\) at every \(C\)	\(\big(L\,G(\infty,C)\big)^{\gamma}A^{1-\beta}\)	none except through \(g_G\)
Human replacement	\(X = L + m(C)K\)	unbounded in \(K\)	none	explosive iff \(\gamma>\beta\)
Task replacement	\(X\) CES, share \(s(C)\)	\(0\) until \(s\to1\)	\(\big((1-s)^{1/\rho-1}L\big)^{\gamma}A^{1-\beta}\)	none unless \(s\to1\)
Apple-picking	splits \(A = A_1+A_2\)	positive, capped	\(A_1\to\bar A_1(C)\)	none except through \(\dot{\bar A}_1\)

Capability enters all four the same way, which is why they are easy to confuse.

In every case \(C\) expands the domain over which AI is useful: the multiplier \(G\), the exchange rate \(m\), the automated share \(s\), or the reachable stock \(\bar A_1\). The pair the argument cares about is the pair whose capability parameters are structurally analogous — \(s\) and \(\bar A_1\) are both “the part of the job AI can touch,” and both rise with each model generation. The theories are not distinguished by what capability does. They are distinguished by whether the untouched remainder is a complementary task that AI’s output must pass through, or a separate stock that AI simply never reaches.

Dynamics

Every theory here has a state variable; what is distinctive is that apple-picking’s is asymmetric.

It is tempting to say that only apple-picking carries a state. In this framework that is false: \(A\) is a state in all four, since past effort of any kind raises it and thereby changes today’s productivity through \(A^{1-\beta}\). The cross-period cross-partial has \(\text{sign}(1-\beta)\), negative for any theory with \(\beta>1\). Depletion across periods is not by itself a fingerprint.

The fingerprint is whose return is depleted. Under the single-stock theories, past effort moves \(A\), which scales the productivity of labor and spend identically; the effect is symmetric across inputs. Under apple-picking, past AI spend depletes \(A_1\) specifically, and \(A_1\) is what AI’s productivity depends on while labor also has \(A_2\) to work in. So the test is a difference: does past AI spend depress the return to AI spend by more than it depresses the return to human labor on the same targets? Only a split stock predicts yes.

A two-period version isolates the difference as a cannibalization wedge.

Take two periods, set \(L=0\) and \(\eta=\gamma=1\) for clarity, and let the reachable stock rise from \(\bar A_1\) to \(\bar A_1 + \Delta\). With discount factor \(\delta\) and picking rate \(m\), discounted results are

\[V = \bar A_1\big(1-e^{-mK_1}\big) + \delta\Big[\bar A_1 e^{-mK_1} + \Delta\Big]\big(1-e^{-mK_2}\big),\]

where \(\bar A_1 e^{-mK_1}\) is what period-1 picking left behind. The exponential is doing real work here, unlike in the static results: a constant hazard makes the remaining stock a sufficient statistic, so the state is one number. With general \(\eta\) you would have to carry the whole distribution of remaining difficulty forward. Differentiating,

\[\frac{\partial V}{\partial K_1} = \bar A_1\, m\, e^{-mK_1}\Big[\,\underbrace{1 - \delta\big(1-e^{-mK_2}\big)}_{\text{cannibalization wedge}}\Big].\]

Spending early is worth less than its immediate yield, because part of what it picks would have been picked next period anyway. Differentiating again,

\[\frac{\partial^2 V}{\partial K_1 \partial K_2} = -\,\delta\,\bar A_1 m^2 e^{-m(K_1+K_2)} \;<\; 0,\]

What identifies the theory is not that this is negative, but that the corresponding derivative with respect to human effort on the same targets is not equally negative, because labor can retreat into \(A_2\).

Deferral alone does not identify apple-picking; the wedge does.

It is tempting to read “wait for the next model” as an apple-picking signature. It is not. Task replacement with a rising automated share defers too, because \(s_2>s_1\) raises tomorrow’s marginal product just as a larger \(\bar A_1\) does — deferral follows from any expanding domain, and both theories have one. What is unique to apple-picking is the wedge, not the direction. Observing that firms should wait tells you almost nothing; observing that last year’s spend lowered this year’s yield on the same targets tells you a lot.

In the long run, spend cannot raise growth under apple-picking, and the reason is the split.

Since \(A_1\le\bar A_1(C_t)\), the reachable component contributes at most \(\dot{\bar A}_1\) once spend is high enough to clear each newly-reachable band as it arrives, and total output settles at

\[\dot A_t \;\approx\; \dot{\bar A}_1(C_t) \;+\; L_2^{\,\gamma}A_2^{\,1-\beta}.\]

The marginal product of \(K\) goes to zero: extra spend clears each band sooner but cannot produce results that are not yet reachable. This is the conclusion most sensitive to the specification — it assumes the tree stands still, and a regenerating stock removes it. Long-run progress is therefore governed by \(\dot{\bar A}_1\) — the rate at which capability rises — and by the human-only sector, never by the AI budget. Set this against human replacement, where \(K\propto A\) gives \(g_A\propto A^{\gamma-\beta}\) and spend growth can carry the whole economy: the two theories disagree about whether the quantity worth forecasting is the capability path or the compute budget.

The staircase is a claim about \(\bar A_1(C_t)\), not a claim of the model.

Whether \(\dot A_t\) looks like a sawtooth or a smooth exponential depends entirely on whether the reachable stock jumps at releases or drifts continuously. Both are consistent with the same production function. This dissolves the tension the evidence sections kept running into: AISI’s and Lyptus’s smooth exponentials are evidence about the time path of capability, not evidence against apple-picking [→ AISI cyber, Lyptus]. The staircase should be dropped as a test of the model and kept only as a question about release dynamics.

Recursive feedback is a law of motion for \(C\), which any of the four can be given.

The closing loop is one line here: let capability depend on accumulated results, \(\dot C_t = g(A_t)\), and the picking process starts raising its own ceiling [→ recursive R&D]. This is why that account sits in Other theories rather than competing here: it does not say where \(K\) enters \(\dot A\), it closes the system in the other direction. Given the result above — that long-run output under apple-picking is governed by \(\dot{\bar A}_1\), which capability drives — pairing the two is where the interesting behavior would come from.

What the formalization buys: three tests, and one it takes away.

Rebasing on the Jones function turns four verbal theories into three tests, none of them yet run. Whether \(\dot A\) at \(L=0\) is zero, positive-and-bounded, or positive-and-unbounded separates the human-only theories, apple-picking, and human replacement. Whether the \(K\to\infty\) ceiling scales with headcount separates task replacement, whose ceiling is proportional to \(L\), from apple-picking, whose ceiling is \(\bar A_1(C)\) and contains no \(L\). And whether past AI spend depresses the return to AI spend by more than it depresses the return to human labor on the same targets isolates the split stock; that is the repeated-run experiment on a fixed target population the evidence sections kept asking for and never found [→ XBOW, RE-Bench].

The test it takes away is the raw within-period cross-partial, which the duplication exponent contaminates for every theory at once. Both surviving comparisons are differences — a ceiling compared across headcounts, a depletion compared across inputs — which is harder to run than a single sign but is what the framework actually licenses. A fourth and much cheaper test falls out of the alternatives below, from data that may already exist.

Alternative formalizations of apple-picking

The two-stock split is one way to write the intuitions down. It is worth separating the intuitions from the implementation, because the implementation is doing more work than it looks like, and several of the choices could go the other way without touching the picture the evidence sections are testing. Three of the entries below are restatements — the same theory in different notation, sometimes with something gained. Three are rivals that reproduce much of the same evidence while dropping one of the theory’s claims, and those are the ones worth worrying about.

A continuum of difficulty, instead of two stocks.

Index results by difficulty \(x\) with density \(f(x)\), and let AI reach everything below a height \(\lambda(C)\), so the reachable stock is \(\bar A_1(C)=\int_0^{\lambda(C)}f(x)\,dx\). The two-stock version is this with \(f\) collapsed onto two atoms. The gain is that \(\bar A_1\) stops being a primitive and becomes a property of the difficulty distribution, which then does predictive work: a thin tail just above the current height means each capability band delivers little, a thick one means bands deliver a lot, and “burst then decay” becomes a statement about \(f\) rather than an assumption bolted on. It is also the version that connects directly to Jones’s march down a quality tail (Jones 2023), which is the closest formal cousin to the whole picture. The cost is an unknown function where there was a scalar, and nothing in this evidence identifies \(f\).

Sampling from a finite pool, which microfounds the exponential rather than assuming it.

The \(e^{-rx}\) in the original post reads like a functional-form choice. It need not be. If each unit of AI effort draws at random from the reachable pool and re-finds things already found, then after \(n\) draws the expected number of distinct results is \(\bar A_1\big(1-e^{-n/\bar A_1}\big)\) — the coupon-collector curve, with no assumption beyond sampling with replacement from a finite set. This is worth more than tidiness, because it fuses two claims the argument currently makes separately: diminishing returns and the duplicate rate become the same phenomenon rather than two pieces of corroborating evidence. It also creates a test that does not exist in the current version. Sampling with replacement implies a specific relationship between the observed duplicate share and the marginal yield of another run, so a bug-bounty submission log with duplicates labelled is enough to check it [→ XBOW]. That is a far cheaper experiment than the ones proposed above, and the data may already exist.

A large fishing-out exponent, with no split at all — the alternative most likely to be right.

Ask whether the second stock is necessary. Ordinary Jones with a large \(\beta\) in the AI-relevant sector, \(\dot A = X^{\gamma}A^{1-\beta}\), also delivers sharply diminishing returns, a front-loaded path, and an apparent wall, with no partition of \(A\) anywhere. What it cannot deliver is a genuine stop: for any finite \(\beta\), \(A_t=(\beta X^{\gamma}t)^{1/\beta}\) grows forever, just slowly. Apple-picking halts at \(\bar A_1\). Over the horizons anything in this post observes, those are close to indistinguishable — \(\beta=5\) looks like a wall across any window shorter than the whole record. This should be treated as the leading rival rather than a footnote, because most of what the evidence sections read as depletion is equally consistent with it, and it is more parsimonious. What separates them is not the decay rate but the ceiling: under any single-stock theory the reachable total is unbounded and the \(K\to\infty\) flow scales with \(L\), whereas apple-picking’s ceiling is \(\bar A_1(C)\) and contains no \(L\). So the ceiling test earns its keep twice — it separates apple-picking from task replacement, and it separates the split-stock story from a steep single-stock one.

Cobb-Douglas inside the split, which is the fourth cell of the 2×2.

The entry above and the model differ along two axes at once, and only three of the four combinations have a name. Crossing “one stock or two” against “fishing out or a finite stock” gives plain Jones, the model above, and a third that nobody writes down but that is the obvious first thing to try: keep the split, and give the reachable component an ordinary Cobb-Douglas term, \(\dot A_1 = \big(L_1+m(C)K\big)^{\gamma}A_1^{\,1-\beta}\). (The fourth, a single bounded stock, says all research halts forever and can be set aside.) It is the cheapest way to get depletion, and it is worth being explicit about what the bounded term buys over it.

Two things go wrong, one cosmetic and one fatal. The sign is the cosmetic one: \(A_1^{1-\beta}\) is increasing in \(A_1\) whenever \(\beta<1\), because Jones’s ideas-getting-harder is a claim about the growth rate \(g_A = X^{\gamma}A^{-\beta}\) and not about the flow — in his model the absolute number of results per year rises with the stock, since standing on shoulders dominates. Apple-picking asserts the reverse in levels, which is what a rising duplicate rate measures. Depletion therefore requires \(\beta>1\), outside the usual calibrated range, and that region implies \(\dot A_1\to\infty\) as \(A_1\to0\), making the first result in a fresh domain infinitely cheap.

The fatal one is the ceiling. Since \(\big(L_1+mK\big)^{\gamma}A_1^{1-\beta}\to\infty\) as \(K\to\infty\), apple-picking’s entry in the \(K\to\infty\) column would become “unbounded” — human replacement’s entry. The theory collapses into a two-sector human-replacement model, and the comparison the whole section is built around, a ceiling containing no \(L\) against one proportional to \(L\), has nothing left to compare. Capability also degrades: \(C\) could only enter through \(m(C)\), so a new generation would pick faster rather than higher, while the moving band and the staircase are claims about \(\bar A_1(C)\) specifically. Letting the partition itself move with \(C\) is a legitimate repair, but working out how much mass crosses from \(A_2\) into \(A_1\) requires the mass below \(\lambda(C)\), which is \(\bar A_1(C)\) under another name.

What is not lost is most of the evidence, which is the uncomfortable part. Cobb-Douglas at \(\beta>1\) still delivers depletion, front-loading, and starting-point dependence, so the three domain sections would read much the same under either. The estimable difference is the shape of the decay on a fixed target set: Cobb-Douglas decays as a power law, \(\dot A_1\propto t^{(1-\beta)/\beta}\), and the bounded term decays exponentially, \(\dot A_1\propto e^{-X^{\gamma}t}\). Those separate cleanly over a long enough run, so this adds a quantity to measure in the repeated-run experiment rather than demanding a new one.

A regenerating stock, which the cyber evidence probably requires.

The tree does not stand still. New code ships and new vulnerabilities come with it; new theorems are proved and new bounds become available to tighten. Let the reachable stock be replenished at rate \(\phi\), so that picking no longer exhausts the domain but converges to a steady state in which the picking rate matches regeneration. Long-run output is then \(\phi\) rather than zero, and the marginal product of spend is positive rather than vanishing — it sets how close to \(\phi\) the field runs, not whether it gets there. This is not a technicality in cyber, where the target population grows with the software stock and the evidence of depletion is a duplicate rate rather than an observed exhaustion; it matters least in math, where nothing about AI activity obviously replenishes the stock of open problems. Neither the section above nor any source in the log measures \(\phi\), and the difference between \(\phi=0\) and \(\phi>0\) is the difference between a domain that gets used up and one that reaches an equilibrium.

Assignment rather than depletion, which produces the same cross-section.

Every static observation the argument leans on — machine finds concentrated in easy and neglected places, experts keeping the hard remainder — can be generated with no depleting stock whatsoever. Order problems by difficulty and let each be worked by whichever of human or AI has comparative advantage: AI takes the easy end because that is where its advantage lies, not because easy results are being consumed. That is the genius model (A. K. Agrawal, Gans, and Goldfarb 2025), and it is a relabelling of task replacement with ordered tasks rather than a stock story. The two accounts agree about the cross-section and disagree about the time series. Assignment says a fixed capability keeps yielding easy results indefinitely, as new problems arrive; depletion says the yield falls even with the problem set held fixed. This is the same repeated-run experiment on a fixed target population that the evidence sections kept asking for, arriving from a third direction, which is the best argument that it is the measurement worth buying.

What survives every rewriting.

Three claims are common to every version that is still recognisably this theory — the continuum, the sampling story, and the regenerating stock — and they are what “apple-picking” should be taken to assert: that AI’s reach is bounded at any given capability, that the bound moves with capability, and that repeat effort against the same targets is non-additive. Everything else in the formalization above is an implementation choice that can be traded for something else without disturbing them: the exponential, the two-way split of the stock, the hard ceiling, the fixed allocation of labor between components.

The three rivals are exactly the accounts that drop one of the three. A large fishing-out exponent and Cobb-Douglas inside the split both drop boundedness, keeping the decay while removing the stop; assignment drops depletion, keeping the cross-section while removing the stock. That is a useful way to read the evidence sections, because it says which findings would bite. A result contradicting the exponential or the staircase contradicts a convenience. A result showing unbounded reach, a static bound, or additive repeat runs would contradict the theory.

Other theories

The four theories above are the ones the evidence is scored against. This section collects the other specific theoretical models of AI’s contribution to scientific progress — not general growth models, not models of automation and jobs, not empirical estimates or surveys, all of which the four theories already draw on where relevant. The test for inclusion is that the paper puts AI into a model of how knowledge gets made. Entries run in chronological order, each carrying its citation in the slug, and each says what the paper contributes and which of the eight questions it speaks to.

These are conceptual references rather than evidence, so they are cited by bibliography key rather than by source-log anchor, and their claims are described qualitatively — no figures are quoted from them. The three that also have source-log entries carry both.

The baseline in Formalizing the theories gives a way to place them that is more informative than a list. Every paper below either fits inside \(\dot A = X(L,K;C)^{\gamma}A^{1-\beta}\) or names a part of it the four theories hold fixed, and there turn out to be exactly four places to touch: the input index \(X\), the split of the stock \(A\), the fishing-out exponent \(\beta\), and the treatment of capability \(C\) itself. Each entry says which. Two of the four theories’ shared assumptions get challenged this way — that \(\beta\) is independent of AI, and that the allocation of labor across kinds of research is fixed — and both challenges look right.

A. Agrawal, McHale, and Oettl (2019): AI is a better search tool over the combinatorial haystack.

New ideas come from combining existing ones, so the space of candidates explodes far faster than the stock of ideas, and the binding constraint becomes the capacity to search it rather than the supply of raw material. They model AI as improving that search, which can offset declining research productivity. This is the one entry that does not fit the framework, and the misfit is the interesting part. All four theories put AI in the input index and hold the fishing-out exponent fixed; better search over a combinatorial space is instead a claim that AI changes how fast ideas get harder, \(\dot A = X^{\gamma}A^{1-\beta(C)}\) with \(\beta'(C)<0\). That is a third place AI could enter, and the four theories all assume it away. It is the optimistic counterpart to the theories here: the same combinatorial explosion that makes results hard to find is what makes a better search tool valuable. It also predicts something apple-picking asserts and Jones does not — that an uncollated literature holds reachable results nobody has taken, which is what the automated exponent-database search demonstrated [→ ANTEDB]. Bears on Q1, Q7.

Gans (2025a): AI interpolates within what is already known, so its reach is the convex hull of the literature.

A model in which AI’s competence is interpolation between existing knowledge points, reducing uncertainty inside the boundary of the known but not outside it. This is the closest thing in the literature to a microfoundation for apple-picking’s split: \(A_1\) is what lies inside the convex hull of known points and \(\bar A_1(C)\) is the interpolation range, so “low apples” are results between things already established. Two of its predictions are not apple-picking’s. First, the growth effect is non-monotonic in capability — a modest interpolation range raises growth by densifying knowledge clusters, while an extensive range pushes researchers toward exploration that thins density and can slow growth despite direct productivity gains (Gans 2025b). Second, and sharper as a critique, scientists “work to the AI”: in an intermediate regime they choose less novel questions to stay inside the tool’s operational range. Both follow from an ingredient apple-picking leaves out. Apple-picking fixes the allocation \(L_1,L_2\) and takes \(\bar A_1\) as given; Gans makes the allocation endogenous and makes \(\bar A_1\) depend on the density of what is already known, so raising capability pulls labor out of \(A_2\) — which is the sector that sustains long-run growth. Apple-picking treats the stock as exogenous, and this says the pickers reshape the tree. Bears on Q1, Q6, Q7.

A. K. Agrawal, Gans, and Goldfarb (2025): scarce genius is allocated to the boundary, and AI genius moves humans further out.

They separate routine knowledge workers, who apply existing knowledge, from genius workers, who create new knowledge at a cost rising with distance from what is known. Scarce genius should be spent at domain boundaries rather than at midpoints between known answers, and introducing AI genius pushes human specialists further out still, to where their comparative advantage is greatest, while routine workers face displacement as AI efficiency approaches human genius efficiency. This is the most direct model of the question the evidence summary found apple-picking failing uniformly, and it predicts what all three domains show: detection democratizing while weaponization does not, verification staying expert-only, harness engineering becoming the scarce skill. In the framework it is task replacement with the tasks ordered by distance from the known, so that \(s(C)\) expands from the near end rather than covering an unlabelled share — the same ordering apple-picking uses for difficulty, but with AI substituting for labor on near tasks instead of depleting a stock of near results. Expertise relocates outward rather than falling. On Q4 it currently outperforms apple-picking. Bears on Q3, Q4.

Gans (2026): a benchmark scores one task distribution, and no one deploys against that distribution.

Because AI ability is jagged across tasks, deployment loss exceeds benchmark loss precisely when the tasks a user leans on most are the ones the system handles worst — an exposure effect rather than noise, with the implication that providers should report performance by task category so users can reweight to their own mix. The model is written for organizations rather than for research, and the transfer is the point: a domain is a task mix, and Q8 is asking which benchmark reweights onto it. That is why the benchmarks this post found predictive are the ones carrying a denominator from the deployment setting — expert speedup, human hours, dollars per task — and why raw multipliers over arbitrary baselines travel worst. In the framework it touches \(C\): capability is not the scalar the four theories assume but a profile over tasks, and what enters \(s\) or \(\bar A_1\) is the overlap between that profile and the domain’s own task mix. That also gives the jaggedness recurring in all three domains a name and a decision-theoretic consequence rather than leaving it an observation. Bears on Q2, Q8.

Bazzichi, Riccaboni, and Castellacci (2026): automation breadth has an inverted-U effect on radicalness.

Research combines ideas at different distances in a knowledge space [→ recombinant innovation]. Greater AI productivity can make distant, radical combinations feasible, but expanding the share of tasks assigned to AI eventually erodes human–AI complementarity and steers researchers toward the same data-rich regions, producing incremental work and duplication. The streetlight and stepping-on-toes mechanisms fit the observed concentration on known vulnerability classes, neglected problems, hyperparameter tuning, and duplicate bug reports, and the model accommodates the counterexamples better than a literal low-fruit account, since higher productivity can enable a distant combination like the unit-distance construction. In the framework this is the same expansion Gans requires, reached independently: raising \(s\) lifts \(\dot A_1\) while crowding labor and attention out of \(\dot A_2\), so breadth has opposing effects on the two components and the net is non-monotonic. Two unrelated papers converging on the need for an endogenous allocation between the components is the strongest argument that the four theories are missing it. Its distinctive comparative static is that inverted-U in automation breadth, and no dataset here varies breadth while holding model productivity, field, and incentives fixed. Bears on Q1, Q5, Q7.

Davidson et al. (2026): automating AI research can close a loop.

Better software and hardware accelerate the research that produces still better software and hardware, while higher output finances more machine researchers [→ recursive R&D]. This can overcome ideas-getting-harder and produce superexponential growth. It is not one of the four theories because it predicts the time path of aggregate progress rather than the shape of an individual bug, proof, or optimization — in the formal section’s terms it is a law of motion for capability, \(\dot C_t = g(A_t)\), that any of the four production functions can be paired with, and it is the fourth and last place to touch the baseline. Nothing in these snapshots establishes or rules it out; the relevant evidence would show an AI-generated improvement measurably lowering the cost of the next round of AI research. Bears on Q1, Q6.

Google DeepMind (2026): agents make candidates cheap, so validation becomes the scarce input.

The “conjecture machines” account is a specific theory of which task becomes limiting [→ validation bottleneck]. Agents can flood a field with hypotheses, proofs, patches, or designs; accepted knowledge grows only as fast as experiments, formal checks, expert review, and tacit implementation can validate them. Signature prediction: AI produces complete contributions first where verification is cheap, fast, and automatable, and elsewhere more agents mainly create a candidate glut that shifts labor into triage. This is the clearest cross-domain mechanism in the record — curl’s submission flood, XBOW’s duplicates, math’s proof indigestion, and the fact that every strongest optimization result came with cheap automatic scoring attached are one pattern, not four. In the framework it touches the input index and nothing else: it is task replacement with two tasks and \(\theta\to0\), so that \(X=\min\{\text{generation}(L_g,K),\ \text{validation}(L_v)\}\) and AI covers only the first. That is a refinement of task replacement rather than a fifth rival, but a substantially more predictive one than bare task coverage, because it names which task is the bottleneck instead of leaving \(s\) unlabelled. Bears on Q3, Q7.

Excluded on scope, and what they are good for instead.

Four bodies of work bear on the argument without being models of AI in knowledge production, and are cited elsewhere in the post rather than here. Weitzman’s recombinant growth (Weitzman 1998) and Jones’s march down a quality tail (Jones 2023) are the growth-theoretic ancestors of the search picture, but neither contains AI; the second is the closest formal cousin to apple-picking and is where anyone formalizing it further should start. Bloom, Jones, Van Reenen, and Webb (Bloom et al. 2020) establish the declining-research-productivity fact every theory here is a story about, but it is a fact and not a model. Autor and Thompson (Autor and Thompson 2025) give the general treatment of which automated tasks de-skill a job, which Q4 relies on, but the object modeled is a job rather than a field of inquiry. And the surveys — Trammell and Korinek (Trammell and Korinek 2023), Erdil and Besiroglu (Erdil and Besiroglu 2023) — map the space these models sit in without adding one.

References

Agrawal, Ajay K., Joshua S. Gans, and Avi Goldfarb. 2025. “Genius on Demand: The Value of Transformative Artificial Intelligence.” Working Paper 34316. National Bureau of Economic Research. https://doi.org/10.3386/w34316.

Agrawal, Ajay, John McHale, and Alexander Oettl. 2019. “Finding Needles in Haystacks: Artificial Intelligence and Recombinant Growth.” In The Economics of Artificial Intelligence: An Agenda, edited by Ajay Agrawal, Joshua Gans, and Avi Goldfarb, 149–74. Chicago, IL: University of Chicago Press. https://www.degruyterbrill.com/document/doi/10.7208/9780226613475-007/html?lang=en.

Autor, David, and Neil Thompson. 2025. “Expertise.” Journal of the European Economic Association, jvaf023. https://www.semanticscholar.org/search?q=Expertise.

Bazzichi, Emanuele, Massimo Riccaboni, and Fulvio Castellacci. 2026. “Bridging Distant Ideas: The Impact of AI on R&D and Recombinant Innovation.” https://arxiv.org/pdf/2604.02189.pdf.

Bloom, Nicholas, Charles I Jones, John Van Reenen, and Michael Webb. 2020. “Are Ideas Getting Harder to Find?” American Economic Review 110 (4): 1104–44. https://doi.org/10.1257/aer.20180338.

Davidson, Tom, Basil Halperin, Thomas Houlden, and Anton Korinek. 2026. “When Does Automating AI Research Produce Explosive Growth? Feedback Loops in Innovation Networks.” https://basilhalperin.com/papers/singularities.pdf.

Davidson, Tom, and Tom Houlden. 2025. “How Quick and Big Would a Software Intelligence Explosion Be?” https://www.forethought.org/research/how-quick-and-big-would-a-software-intelligence-explosion-be.

Erdil, Ege, and Tamay Besiroglu. 2023. “Explosive Growth from AI Automation: A Review of the Arguments.” https://arxiv.org/abs/2309.11690.

Eth, Daniel, and Tom Davidson. 2025. “Will AI r&d Automation Cause a Software Intelligence Explosion?” https://www.forethought.org/research/will-ai-r-and-d-automation-cause-a-software-intelligence-explosion.

Gans, Joshua S. 2025a. “A Quest for AI Knowledge.” Working Paper 33566. National Bureau of Economic Research. https://doi.org/10.3386/w33566.

———. 2025b. “Growth in AI Knowledge.” Working Paper 33907. National Bureau of Economic Research. https://doi.org/10.3386/w33907.

———. 2026. “Artificial Jagged Intelligence: When AI Benchmarks Misstate Deployment Value.” Working Paper 34712. National Bureau of Economic Research. https://doi.org/10.3386/w34712.

Google DeepMind. 2026. “Conjecture Machines: AI Agents and the New Validation Bottleneck in Science.” July 2026. https://deepmind.google/public-policy/conjecture-machines-ai-agents-and-the-new-validation-bottleneck-in-science/.

Jones, Charles I. 2023. “Recipes and Economic Growth: A Combinatorial March down an Exponential Tail.” Journal of Political Economy 131 (8): 1994–2031. https://doi.org/10.1086/723631.

Trammell, Philip, and Anton Korinek. 2023. “Economic Growth Under Transformative AI.” National Bureau of Economic Research. https://doi.org/10.3386/w31815.

Weitzman, Martin L. 1998. “Recombinant Growth.” The Quarterly Journal of Economics 113 (2): 331–60. https://doi.org/10.1162/003355398555595.