The Centaur Window

Abstract

Wherever a domain ranks its participants on an operative scoring function, and AI can scale against that function, a centaur window opens.

During it, human direction, selection, or verification still measurably improves the machine's results, so the leading outcomes come from the best-coupled human and machine pairs. Pure refusal exits the ranking. Pure unguided machine output trails until the complementarity closes. The window ends, domain by domain, when autonomous systems match human-directed ones on the scoring function the domain actually uses.

Chess opened its window in 1998 and had closed it by about 2017; at the top of correspondence chess, roughly 95% of games now end drawn. Go never had a competitive window at all, though human play improved after 2016. Mathematics, as of October 2026, has a generation side at or past the human frontier on problem-solving and a verification side that is now the bottleneck; the Association for Human Mathematics' call to stop working with OpenAI is the live opt-out. On October 27, Axiom reported a complete Lean proof of the classification of finite simple groups, produced with swarms of AI agents in 24 days. With that release the human check shifts to the statement: whether the formal theorem says what the mathematics means. Software engineering sits closer to chess than to Go: adoption is near-universal and output volume is up, while shipped output lags and review has become the constraint.

One factor decides whether a window can be renegotiated: who sets the scoring function. Where the ranked population controls its own scoring (peer review, licensure, authorship norms), a field can lengthen or redefine its window. Where the scorer is external (markets, prizes, adversaries), it cannot. A five-factor index, with that factor as its fifth, gives each occupation a Window Score (WS) from 1 to 5. The score estimates how long a person directing AI will keep beating the AI alone in that work. It is the mean of the five factor scores, rounded to a whole number; 1 means the window has closed, 5 means the human edge persists, and Section 4 defines all five bands. Section 6 reads the public story around each window, before, during and after, and treats the lag between the scoreboard and the story as a signal worth measuring. Section 7 sets out five moves for a person or business whose window is closing, each drawn from a case in the paper.

The pattern: three transits

Three domains with clean scoring functions have crossed, or are crossing, a centaur window. They differ in how long the window stayed open and in where the residual human value sits.

Chess: opened in 1998, closed by about 2017. Kasparov played the first advanced chess match against Topalov in León in June 1998, each using Fritz 5; it ended 3-3. In 2005 the PAL/CSS freestyle tournament was won by ZackS, two American amateurs running several engines, ahead of grandmasters. That is the window at its widest. Krakowski, Luger and Raisch (SMJ 2023) analyzed 112 players across 3,281 tournaments and found conventional Elo had almost no effect on centaur results. What predicted winning was a distinct human-machine capability: selecting, tuning and directing engines. Then it narrowed. Tyler Cowen argued in 2013 that centaurs had lost any major advantage; in the 2017 Infinity Chess Ultimate Challenge the engine Zor finished first and the best centaur third. Where engine use is still permitted, in ICCF correspondence play, about 95% of games between top-rated players end drawn, a rate Glickman (2025) attributes to engine analysis on both sides. At the top, human input mostly produces input errors.

Go: an abrupt transit, no competitive window. AlphaGo beat Lee Sedol in March 2016. There was no freestyle Go circuit, so no centaur ranking ever formed, and the competitive frontier moved to machines in one step. A different window opened instead: a learning window. Shin, Kim, van Opheusden and Griffiths (PNAS 2023) scored 5.8 million professional moves from 1950 to 2021 with KataGo. Human decision quality, flat for 66 years, rose after 2016-17 by a statistically significant margin, and the gain was concentrated in novel moves. Go shows that a centaur window also needs a competitive format that admits coupled play. Where none exists, the ranking transits at once and the human gain shows up elsewhere.

Mathematics: generation has crossed the frontier, and verification is now the bottleneck. At IMO 2026 (Shanghai, July 15-16; gold cutoff 29 of 42), seven of 666 human contestants scored a perfect 42. Digital Applied, citing AFP, reports that Huawei's Celia and Xiaohongshu's dots-note-3.0 each scored 42/42 under official grading, with a fixed time window and no human intervention, and that several other frontier models reached 42 in an unofficial harness. On October 6, 2026, OpenAI released 722 AI-written manuscripts claiming progress on 372 problems. Only 162 carried Lean formalizations of their main result, roughly half were released unconfirmed, and within a day three were withdrawn for a sign error and 14 revised. Its September 8 Navier-Stokes claim followed a partial result by Buckmaster and Alpöge, who had used Claude and Codex, and is still being checked. Terence Tao's ICM 2026 essay names the constraint: impedance mismatches between generation, verification and digestion. The Simons Foundation's $15 million fund of October 15, backed by OpenAI and Anthropic, exists to help mathematicians evaluate AI-assisted results. So the window is open, and it has moved. The human edge now sits in choosing problems, checking claims, and turning results into understanding.

The classification of finite simple groups, machine-checked. On October 27, Axiom reported the first complete computer-checked proof of the classification. The original human proof, which Axiom describes as widely considered the longest ever written, runs to about 15,000 pages in more than 500 articles by about 100 mathematicians. Over 24 days, Axiom's engineers and mathematicians directed swarms of AI agents in a project that produced 43.1 million lines of Lean, building on Mathlib and about 1.2 million lines from the open Qiuzhen-CFSG project. For scale, the first machine-checked proof of the odd-order theorem, one piece of the classification, was a six-year collaborative effort by 15 authors. The formalization found 34 noteworthy deviations from the published record, including counterexamples to a lemma in Aschbacher and Smith's quasithin volumes and to a step in Walter's 1986 route to the Unbalanced Group Theorem. The theorem held. AxiomProver proved a new result to replace the broken step, built on Ronald Solomon's 1976 ideas, and Solomon and Richard Lyons, two of the original group theorists, advised on the repairs. Generation outran checking even here: agents produced proofs faster than Lean could recheck them, until Axiom's engineers rebuilt how the checks ran. The human check moves to the statement. Lean checks every deduction, and people confirm that the formal theorem says what the classification says, which is the review Axiom has asked group theorists and the Lean community to do.

The Association for Human Mathematics (AHM) statement of October 7, which called the 722-paper release a demonstration of power and urged mathematicians to stop working with OpenAI, is the live opt-out. Section 2 returns to it.

The three transits share the structure the thesis proposes. They differ on two variables: whether the domain had a coupled competitive format, and how expensive verification is relative to generation. Both variables return in the index.

When the window can be renegotiated

The window is a selection effect. A ranking rewards whatever coupling wins on its scoring function. During the window that is a human plus a machine. After it, a machine alone, or one with minimal steering. Three things follow.

Refusal is an exit. A participant who declines the tool while it still improves outcomes leaves the ranking for as long as the window stays open. In chess that cost was visible within a decade. The non-adopter did not lose a philosophical argument. They lost games.

The spear pattern is real but slow, and knowledge work is fast. Groups that refused productivity tools were outcompeted on yield, status and survival, and the mechanism was cultural and economic. The analogy breaks in one place: the rival groups were scored on outputs, as knowledge workers are, but they could not re-enter a ranking they had left, and many professionals can. Tempo is what makes it apply. Markets, citations, promotions and funding decisions close their loops in quarters and years.

Who sets the scoring function decides whether the window can be renegotiated. Two cases:

• Exogenous (external) scorer. Customers, prize committees judging outputs, adversaries, or nature. The ranked cannot change the rules. Engine tournaments, correspondence chess, and most product markets are scored this way. Here the window is binding and closes when the capability closes.
• Endogenous (internal) scorer. The ranked population sets its own scoring: peer review, authorship norms, licensure, professional accountability. Here a field can lengthen its window by changing what counts. The AHM statement and the norms in Tao's ICM 2026 essay are attempts to make human understanding part of the scoring function itself. Tao endorses the Leiden Declaration (June 2, 2026, endorsed by the International Mathematical Union), which asks mathematicians to disclose tool use and keeps credit and responsibility with humans, and he adds his own rule of thumb: publish a result only when the authors can give a clear, expert-level talk on it. If that holds, the ranking splits in two: a machine-output ranking and a human-understood ranking.

Endogenous scoring survives only as long as outsiders cannot route around it. Funders, employers, students and rival labs can all score differently. The 722-manuscript release and the Navier-Stokes claim are, among other things, a test of whether mathematics still controls its own scorer. The Simons fund, paid for partly by the labs, is an attempt to keep the scorer inside the community by making verification affordable. Axiom's October 27 release kept the scorer in the same place: Lyons and Solomon advise on the repairs, an advisory board of group theorists will curate the errata, and the community is asked to confirm that the Lean statement says what the classification says.

Software engineering: the live case

Software is closer to chess than to Go: adoption is near-universal, coupled output is up, and the bottleneck has moved downstream to review. Every figure below is tied to its source and dated, because the numbers move quarterly.

Adoption. In the Stack Overflow 2026 Developer Survey (fieldwork June to August 2026, released October 6, 2026, 30,903 responses), 65.9% of respondents use AI coding assistants or coding agents at work and 17.2% use no AI tool at all. Among those using coding assistants or agents, 73.0% use them daily and 20.1% weekly. Google's DORA 2025 report (September 2025, nearly 5,000 respondents) put AI use at work at 90%, with a median of about two hours a day. JetBrains' January 2026 AI Pulse found 90% regularly using at least one AI tool and 74% using specialized developer AI tools. So estimates run from two-thirds to 90%, and each survey measures something different.

Output volume. The best-controlled evidence is Cui, Demirer, Jaffe, Musolff, Peng and Salz (Management Science, 2025): three field experiments with 4,867 developers at Microsoft, Accenture and a Fortune 100 firm, where GitHub Copilot access raised completed pull requests by 26.1%, commits by 13.6% and builds by 38.4%, with larger gains for less experienced developers. Faros AI's 2025 report (July 2025, 10,000-plus developers) found high-adoption teams completed 21% more tasks and merged 98% more pull requests. A 20 to 30% gain holds for completed tasks with assistants. Lines-of-code effects are larger and transient: a study of 806 repositories adopting Cursor (He et al., 2025-26) found lines added rose three to five times in the first month and fell back within two, while code complexity stayed up by 41%.

The bottleneck moved. Faros 2025 measured pull-request review time up 91% and average pull-request size up 154% in high-adoption teams, with bugs per developer up 9% and no company-level productivity gain. Its April 2026 follow-up (two years of telemetry from 22,000 developers across more than 4,000 teams; full report) found task throughput up 33.7% and merge rate up 16.2%, against median time in review up 441.5%, bugs per developer up 54%, incidents per pull request up 242.7%, and deployments per week down 11.7%. Generation outran verification. That is the mathematics pattern (Section 1) in a market-scored domain.

Human direction still changes outcomes, sometimes for the worse. METR's July 2025 randomized trial found 16 experienced open-source developers took 19% longer on their own repositories with early-2025 tools, while believing they were 20% faster. METR's February 2026 follow-up with late-2025 tools (57 developers, 800-plus tasks) points the other way, an estimated 18% speedup for the original cohort and 4% for new recruits, both with confidence intervals that include zero. The sign of the coupling effect depends on tool generation, task and person, and it flipped between 2025 and 2026.

Autonomy still trails. On SWE-bench Verified the top systems exceeded 96% by September 2026 and the benchmark was retired as saturated. On the harder SWE-bench Pro public set, the best score had risen from 23.3% at launch in September 2025 to 61.5% by October 2026. METR found in March 2026 that roughly half of test-passing agent pull requests would not have been merged by maintainers. METR's time-horizon series puts the 50%-success task length of frontier models near 12 hours for Claude Opus 4.6 and at least 16 hours for Claude Mythos Preview (May 2026), doubling roughly every four months since 2023. Measurements above 16 hours are unreliable, and the series stopped active updates in September 2026.

Where this leaves the window. Pure non-use is a competitive handicap for most production work; only one in six developers reports no AI use. Human direction still gates requirements, architecture, verification and merge decisions, and the review bottleneck is where that direction now shows up in the data. Full autonomy on complex systems has not arrived: task horizons are measured in hours, and maintainers would still reject about half of the agent patches that pass tests. The index (Section 4) gives software developers a Window Score (WS) of 2, a short window of roughly 1 to 5 years, because the scorer is the market and both generation and verification capability are compounding on a four-month clock.

The Centaur Window Index

The index orders occupations by how long human direction is likely to keep improving results. Each occupation is scored on five factors, and the scores roll up into one number, the Window Score. The interactive version lets a reader reweight the factors and watch the order change.

The five factors

• V, verification cost. How hard it is to score the output objectively. Lean proofs and test suites score 1; taste and trust score 5.
• D, feedback scarcity. How slow or sparse the learning loop is. Self-play and compilers score 1; decade-long outcomes score 5.
• C, residual complementarity. What a human still adds: framing, context, embodiment, accountability.
• G, capability gap. How far current systems sit below the human frontier on the core tasks, net of economic pressure to adopt.
• S, scorer endogeneity. How much the ranked population controls its own scoring: licensure, peer review, credit norms. It is the least certain of the five.

Window Score bands

WS 1 Closed or closing

Autonomous systems lead the ranking, and non-use is already uncompetitive.

WS 2 Short, roughly 1 to 5 years

Formal outputs and dense feedback. Human steering still helps, briefly.

WS 3 Medium, roughly 5 to 15 years

Selection, verification or framing still change outcomes.

WS 4 Long or partial

Context, integration or accountability keep a human in the loop.

WS 5 Persistent edge

Embodiment, real-time stakes or open-ended framing dominate.

The index at equal weights, shortest window first.

The fifth factor separates mathematicians, radiologists, accountants, lawyers, architects and licensed engineers from roles with similar formalizability that have no professional scorer. Drop it (the interactive index has a button for this) and those roles fall by roughly half a point. Architecture, engineering and landscape architecture appear as a group because all three license themselves through their own boards (NCARB, NCEES and CLARB). They also show the factor at three strengths. Engineers sit lowest because their calculations are formally checkable, even though sign-off is not. Landscape architects sit highest because living systems return feedback over years.

Why each score, in plain words

One short explanation per occupation, in direct language: what the work is scored on, whether a machine can check it, what a person still adds, who sets the score, and where the window stands. Each ends with a next move, one of the five in Section 7 applied to that work. The moves are judgments, like the scores.

Computer and mathematical

WS 2 Software developers and engineers

Writing code is now shared with machines, and most developers use them daily. People still decide what to build, how the pieces fit, and whether the code is safe to ship, and review has become the slow part. The market is the scorer, and the machines improve every few months.Move 1 · Next stageMove toward review, architecture and release decisions, where the bottleneck now sits.

WS 2 Data scientists and analysts

The work is formal: code, numbers and charts that either run or do not, with quick feedback. What a person still adds is choosing the right question and knowing what the business needs. The market scores the result.Move 1 · Next stageMove to the stages around the analysis: choosing the right question and judging what the results mean for the business.

WS 1 Computer programmers (narrow coding)

The job is turning a clear spec into code that passes tests, and tests tell the machine instantly whether it is right. The market decides what counts, and BLS already projects a 7% decline for 2025-35. Steering adds little when the spec is clear.Move 5 · Neighbor roleStep up to software development: deciding what to build and reviewing what ships (WS 2).

WS 3 Mathematicians (formal and computational)

Machines now solve olympiad problems perfectly, write research papers by the hundreds, and have machine-checked the 15,000-page classification of finite simple groups. Checking those papers, picking which problems matter, and explaining results are still human work, and mathematicians decide what gets published and who gets credit. The field's own rules hold the window open.Move 1 · Next stageCheck that formal statements say what the mathematics means, and explain machine-made results.

Education and research

WS 4 Research scientists (novel theory-building)

Picking the right question in an open field, where results take years to show, is hard to score. Scientists run peer review and decide credit.Move 4 · Study the machineUse the models to test more ideas, and keep the choice of question.

WS 3 Instructional designers; content-heavy postsecondary teaching

Making lessons and slides is quick machine work. Knowing whether students learned takes much longer to see, and schools set some of their own standards.Move 1 · Next stageMove from making material to measuring whether students learned.

WS 4 Classroom teachers (K-12)

Teaching children is relationship work with slow, hard-to-measure results. States and districts set teaching credentials, so the profession shares in its own scoring.Move 3 · Different rankingInvest in the relationship work that families and schools judge on trust.

Office and administrative

WS 1 Bookkeeping, accounting and payroll clerks; data entry

The job is moving numbers into the right boxes by fixed rules, and a computer can check its own work by making the totals match. Nobody licenses the job, and the customer only cares that the books balance. A person steering the software adds almost nothing now.Move 5 · Neighbor roleStep up to accounting and audit, where a license and a signature hold the window open (WS 3).

WS 2 Executive secretaries, office clerks, receptionists

Scheduling, filing and messages are structured tasks a machine handles well. Being in the room, and reading a boss or a visitor, still helps. Nobody licenses the job.Move 3 · Different rankingGrow the part of the job that runs on trust: representing a principal and managing relationships.

WS 3 Accountants and auditors

The routine work is automatable, but an audit opinion carries legal weight and only a licensed accountant can sign it. The profession sets its own rules, so it can decide how machine work counts. The license holds the window open.Move 2 · Hold the scoreKeep the license and the signature, and let the machine prepare the work you sign.

Management and business

WS 3 Financial managers

Deciding where money goes is judgment with slow feedback, and someone has to be accountable when it goes wrong.Move 2 · Hold the scoreOwn the allocation decision and its outcome, and use machine analysis to widen the options.

WS 2 Management analysts, compliance officers

Reading rules and writing reports is machine work now. Knowing how a specific company runs, and taking responsibility for a decision, is what remains.Move 2 · Hold the scoreOwn the decision and its accountability, and let the machine read the rules.

WS 4 Chief executives and senior strategists

Nobody can score a strategy before it plays out, and someone has to own the result. Accountability holds the window open without any license.Move 2 · Hold the scoreKeep ownership of the result, and use machines to test more strategies before committing.

Sales and customer service

WS 1 Customer service representatives, telemarketers

The work follows scripts, runs at high volume, and is scored on calls resolved and cost per call, numbers a company measures every day. The customer is the scorer. BLS projects a 5.3% decline for 2025-35, and hiring of young workers in exposed jobs is already down.Move 3 · Different rankingMove to the escalations, retention and account work that customers judge on trust.

WS 2 Retail salespersons, cashiers

Checkout is already automated in many stores. Helping a customer in person, and handling what goes wrong on the floor, keep a person useful for a few more years. The store owner scores on sales.Move 3 · Different rankingMove toward the advice and service customers come into the store for.

WS 2 Market research analysts, writers, PR specialists

The output is text and analysis, which models produce quickly. Fresh ideas and knowing the client still help. Nobody licenses the work, and clients score it on results.Move 3 · Different rankingBuild the client relationships and fresh ideas that clients pay a person for.

Legal

WS 2 Paralegals and legal secretaries

Reading documents and drafting standard filings is work models do well. Firms and courts keep some rules about who may do what, which gives some protection.Move 5 · Neighbor roleStep toward licensed practice; lawyers score 4.

WS 4 Lawyers

Machines draft and research well, but strategy, reading a client, and standing in court are human work. Courts and bar associations decide who may practice and what counts. The license does much of the work of holding the window open.Move 2 · Hold the scoreKeep the license and the client, and let the machine draft and research underneath.

Architecture and engineering

WS 4 Architects

Drawing, rendering and checking a plan against the building code are machine tasks now. But a building sits on a real site, a client has to be understood, and someone has to stamp the drawings and carry the risk if it fails. Only a licensed architect can stamp, and the profession licenses itself through state boards and NCARB.Move 2 · Hold the scoreKeep the stamp and the client, and let the machine draw and check against code.

WS 3 Engineers (civil, structural, mechanical; licensed PE)

The calculations are formal and machines check them fast. But a stamped drawing carries legal liability, and checking a design for sign-off is harder than checking arithmetic: the engineer has to verify the loads, the assumptions and the site as well as the numbers. A licensed engineer must sign off on public safety, and the profession controls that license through state boards and NCEES. The stamp, and the liability behind it, hold the window open.Move 1 · Next stageMove toward design review and sign-off, where loads, assumptions and the site get verified.

WS 4 Landscape architects

Plans and renderings are quick machine work. But living systems take years to show results, every site is different, and the design has to work for people walking through it. Licensing runs through a separate board (CLARB) and is weaker in some states than for architects. Slow feedback does more than the license to hold the window open.Move 2 · Hold the scoreKeep the site work and the license, and let the machine render.

Healthcare

WS 2 Medical records specialists, medical secretaries

Coding a chart follows fixed rules, and the answer can be checked against the record. Hospitals keep some human sign-off for legal reasons, which slows the change.Move 1 · Next stageMove to auditing coded charts and to compliance review.

WS 3 Radiologists and imaging specialists

Reading scans is pattern matching, which machines do well and get instant feedback on; AI-supported screening raised detection 17-29% in two 2025 trials. A licensed doctor must still sign the report and fit it into the patient's care, and the profession licenses itself. The license does most of the work of holding the window open.Move 2 · Hold the scoreHold the report and its place in the patient's care, and use AI support where trials show it raises detection.

WS 4 Physicians and surgeons

Diagnosis gets machine help; surgery and judgment under pressure do not yet. A licensed doctor is accountable for the outcome, and the profession licenses itself.Move 2 · Hold the scoreKeep accountability for the outcome, and take machine help on diagnosis.

WS 4 Nurses, EMTs and paramedics

The work is real-time physical care with lives at stake. Machines help with information and cannot do the hands-on work. Nursing and emergency care are licensed state by state, so the professions keep some control of their own scoring.Move 4 · Study the machineWhile the window is long, learn the clinical tools well enough to check them.

Creative, media and communication

WS 1 Interpreters, translators, technical writers, proofreaders

The output is judged on accuracy and fluency, which models now match on most text. Feedback is fast, and nobody licenses the work. Interpreters and translators rank first in both Eloundou et al. and Microsoft's applicability study. The window is closing fast.Move 1 · Next stageMove to reviewing machine output for accuracy, and to the high-stakes work where a person answers for the result.

WS 3 Journalists and editors (investigative, original)

Finding sources, getting access, and standing behind a story resist any simple score. Readers and editors still judge trust.Move 3 · Different rankingBuild sources, access and a byline readers trust.

WS 3 Artists and designers (original work, physical media)

Taste cannot be scored by a test. Working in a physical medium, and a known maker's name, still carry value.Move 3 · Different rankingLean into physical media and a known maker's name.

Skilled trades and construction

WS 4 Electricians, plumbers, carpenters, construction trades

Every job site is different, the work is physical, and problems appear in real time. Robots are far from this, and trades license themselves in most places.Move 2 · Hold the scoreKeep the license and the safety sign-off on work that is different at every site.

Transportation and production

WS 2 Truck drivers and warehouse operators

On fixed routes and in organized warehouses, driverless trucks and robots already run. Unplanned roads, loading docks and bad weather still need a person. The customer scores on cost and delivery. The window is short on the structured side and longer at the edges.Move 5 · Neighbor roleMove toward the work that still needs a person: runs with unplanned roads and bad weather, and dock and yard operations.

WS 2 Manufacturing production workers

On a fixed line with repeated tasks, robots already do the job. Where the task changes often or needs hands and eyes together, people last longer. The score is cost and quality, set by the market.Move 1 · Next stageMove toward setup, changeover and inspection, where tasks change often.

Service, care and protective

WS 4 Home health aides and childcare workers

The work is hands-on care for people. Families and regulators judge it on trust, and robots are nowhere near.Move 3 · Different rankingKeep the care relationship at the center; families judge it on trust.

WS 4 Police, firefighters, EMTs

The work is high-stakes physical decisions in real time, with public accountability.Move 2 · Hold the scoreKeep the real-time decisions and the public accountability, and use machines for information.

WS 3 Food service in unstructured settings

Cooking and serving in a busy, changing kitchen or dining room need hands, eyes and quick judgment, and robots struggle with mess. The customer is the scorer.Move 3 · Different rankingLean into the hospitality guests come for.

Agriculture

WS 3 Farmworkers (field)

Fields are uneven, weather changes, and every plant is different, so robots are still far behind in the open field. In greenhouses and packing lines they are closer. The market sets the price.Move 5 · Neighbor roleMove toward open-field crop work, where robots are furthest behind.

Summary patterns. The shortest windows cluster in structured information handling: clerical and administrative support, routine customer service, data entry, translation, and narrow coding or content generation. The longest sit where embodiment, real-time high-stakes judgment, interpersonal accountability, or open-ended framing dominate, and where the profession licenses itself. Software engineering and formal mathematics sit in active short-to-medium windows, with mathematics held open by its scorer.

Evidence the scores draw on

The exposure literature supplies the V, D and G scores; the window itself, whether a human-directed system still beats an autonomous one, is almost never measured directly.

What the evidence does and does not say. Exposure and usage rank occupations in the order the index expects, with interpreters, writers, clerks and customer service at the top and embodied trades, care and plant operation at the bottom. Two cautions follow from the rest of the table. First, the frontier ranking can move before employment does: the Canaries data shows the shift arriving through hiring of the youngest workers in automative uses, while the Yale series shows no aggregate break. Second, the quantity the index cares about, human-directed versus autonomous performance, is rarely measured with all three arms. Radiology is the clearest case. Two large 2025 screening trials, MASAI and PRAIM, found AI-supported reading raised cancer detection by 29% and 17.6% without raising false positives. Neither had an AI-alone arm. A standalone-AI study found the model more sensitive and less specific than radiologists. Agarwal, Moehring, Rajpurkar and Salz found radiologists underweight the AI so badly that the best design assigns cases to humans or to the model, rarely to the pair. Radiology's WS 3 reflects a licensed scorer. Legal shows the same split: in the Vals Legal AI Report (February 2025) the best tools beat a lawyer baseline on document Q&A and summarization and lost on redlining and EDGAR research, with no lawyer-plus-tool arm.

Limits, falsifiers, and what would change the thesis

The thesis is defeasible and should say what would defeat it. Five limits and four falsifiers.

Limits

• Closure is hard to date. Chess closure is placed anywhere from 2013 to 2017 depending on the observer. For that reason the index gives bands.
• The window can move. Mathematics in 2026 shows human value migrating downstream to verification and digestion. A domain can therefore be post-window on generation and mid-window on checking at the same time. The index scores the stage where the human edge currently sits.
• Exposure is a proxy. The exposure studies the index draws on measure which tasks a language model could touch. The window turns on a different quantity: whether a human-directed system still beats an autonomous one. The index is a composite judgment built on them, and it is published with its sub-scores so anyone can rescore it.
• Adoption and benefit diverge. Near-universal adoption of coding tools coexists with at least one controlled study in which experienced developers were slower with them. Non-use is a handicap on average, and its size varies by task and person.
• The scorer can change. Where a field controls its own scoring, the window is contestable (Section 2). The index carries this as its fifth factor, and it is the least certain of the five.

Falsifiers

Any of these, observed and sustained, would mean the thesis is wrong or badly scoped.

1. A domain with an external scorer where a tool-refusing group holds the top of the ranking for a decade after coupled competitors become common.
2. A domain where coupled human-machine pairs never outperform either side alone at any point, and yet adoption is universal: that would mean the selection is social.
3. Autonomous coding agents shipping end-to-end production systems at parity with human-directed teams on cost, defect rate and maintainability, with no reversal of the review bottleneck. That would close software's window and the index would need to move it to 1.
4. A field that redefines its scoring function around human understanding, holds that scorer against outside pressure for several years, and keeps its best people. That would show endogenous scoring can lengthen a window indefinitely, and the fifth factor would need more weight than it has here.

Job losses, closure dates and the fate of people who decline these tools sit outside the thesis. It predicts where the frontier ranking moves, under what scoring, and how fast. Section 7 turns to practice: what to do when a window closes.

What people say before, during and after

Narratives track the ranking with a lag, and the lag is itself a signal. The three cases show the same three phases of talk, and in each phase the public story runs behind the scoreboard by a predictable amount.

Before and early: the domain is special. Go was described for decades as a game of intuition no computer could touch. Mathematics in 2026 is being described in the same register. The AHM statement of October 7 defends human understanding and field norms, and calls bulk machine output a demonstration of power. The timing matters: an early-window narrative is being voiced at the moment generation crossed the frontier.

During: coupling wins. This phase has an anthem. Kasparov wrote in February 2010 that the 2005 freestyle winners, two amateurs running three computers, had shown that a weak human with a machine and a better process beats a strong computer alone, and beats a strong human with a machine and a worse process. It was true in 2005. Go never got a sustained version of this phase; its hybrid moment was a few days of commentary on Move 37 (game 2, March 10, 2016), which Michael Redmond called creative and unique. Software engineering and mathematics are in this phase now, and "you plus AI beats AI alone" is the line most keynotes stop on, because it is both currently accurate and reassuring.

After: the claim migrates. Once pure systems lead, the ranking is conceded quietly and the value claim moves. Chess kept its audience and grew it: Chess.com had 44 million members on October 22, 2020, passed 100 million about 14 months after 50 million, reached 150 million in October 2023, and logged a record 8.9 million registrations in each of January and February 2023. Engines became coaches. In Go, Lee Sedol retired on November 19, 2019, saying AI could not be defeated, while professionals' play measurably improved by studying it. "Humans still matter" stayed true, and it became a claim about meaning, teaching and spectatorship.

The narrative lag, measured

Chess gives dates. Practitioners saw the centaur edge fade by 2013 (Cowen) and saw an engine beat the best centaur in 2017. Public complementarity claims continued past both points: Kasparov in 2017, Bridle in 2018, and the centaur trope in business keynotes through 2026. So the practitioner lag was roughly zero to four years and the public lag roughly five to ten. That is the series worth tracking in any domain: the date pure systems first beat coupled pairs on the scoring function, the date specialist forums say so, and the date general coverage repeats it.

Two cautions. First, mainstream coverage runs on incentives: spectacle sells a match, threat sells a headline, and reassurance sells a keynote. Each pulls coverage toward a phase that may already have passed. Second, the chess evidence is confounded. Freestyle faded from headlines after 2008, but that is also when the PAL/CSS sponsorship ended, so attention and capability cannot be fully separated in that case.

The zeitgeist is faster on capability and slower on status. Specialist communities notice when overrides stop helping; correspondence players know what a 95% draw rate means. But where a field's identity is bound to human insight, the response is organized: restriction (AHM) or redefinition of the goal from solving to understanding (Tao). Both are rational where the field controls its scorer (Section 2) and status-preserving where it does not. The scoreboard itself is the least biased signal, and only where the scorer is external.

What to adjust

The recurring error is freezing at phase two. Complementarity claims are accurate only while human direction improves the operative scoring function, and the fix is to make duration and closure explicit every time the claim is made.

1. Carry a window ledger with every complementarity claim. Three lines: the scoring function, the measured residual human contribution against a pure-system baseline, and the date it was last measured.
2. Say which ranking you mean. Separate the frontier ranking (who produces the leading verified results) from the other values a domain carries (understanding, teaching, play, meaning). Chess lost the first and kept the rest.
3. Treat restriction pledges as timed exits. Give them a review date and a closure test. A pledge is a safeguard only where the pledging body also sets the score.
4. Publish the expected window and the closure test, then update in public when the gap closes. The cost of the lag is paid by whoever adopted the frozen narrative last.

After the window

A closed window ends one ranking. The work around it keeps going, and each case in this paper shows where the human edge went next.

Five moves follow from the cases. Each one names its evidence, what it means for a person, and what it means for a business. The index (Section 4) gives every occupation a next move from this list.

1. Move to the next open stage

Knowledge work runs as a pipeline: generate, verify, digest, decide. Windows close one stage at a time. In mathematics, generation crossed the frontier in 2026 while verification became the funded bottleneck (Section 1). When Axiom reported a machine-checked proof of the classification of finite simple groups in October, the human check started moving one stage further, to whether the formal statement says what the theorem means. In software, high-adoption teams raised task throughput 33.7% while median time in review rose 441.5% (Section 3). Each time the edge moved downstream, to checking.

• For a person: learn to verify what the machine produces in your field, and become the reviewer whose approval ships the work.
• For a business: treat review as a constraint you staff and design. Say what reviewers check and route routine cases past them, or the extra output arrives as fewer deployments and more defects (Figure 4).

2. Hold the score

When the machine produces the output, someone still answers for it. The fifth factor in the index measures this. Accountants, radiologists, lawyers, architects and licensed engineers score above roles with similar formal work because a licensed person signs the result and carries the liability (Section 4). Chief executives score 4 with no license at all, because someone has to own a strategy's outcome.

• For a person: move toward the signature, the stamp or the decision you will be held accountable for.
• For a business: name an owner for every output that commits money, a customer or the law, and keep that approval human while the machine does the production underneath.

3. Compete in a different ranking

Chess lost its frontier ranking to engines and kept the rest. Chess.com reached 150 million members in October 2023, engines became coaches, and the audience grew (Section 6). Teaching, play, trust and meaning are scored by people who care who did the work, and those scores sit apart from raw output.

• For a person: find the ranking where a human is the point of the work: teaching, advising, performing, earning trust.
• For a business: sell the relationship, the judgment and the accountability, and price the machine-made output as the cheap part of the offer.

4. Study the machine that took the lead

Go's competitive frontier moved to machines in one step in 2016, because no coupled format existed (Section 1). Professionals then studied the machine. Human decision quality, flat for 66 years, rose after 2016-17, and the gain was concentrated in novel moves. Lee Sedol retired in 2019, and professional play went on improving.

• For a person: use the system that now leads your field as a coach and sparring partner, and study the moves it finds that you would miss.
• For a business: build study of the leading system into training, so people learn its methods while the work moves to it.

5. Use the index as a map

Low and high scores sit side by side inside the same field. Bookkeeping clerks score 1 and accountants and auditors 3, mostly on the license. Narrow coding scores 1, and software developers, who decide what to build and review what ships, score 2. Paralegals score 2 and lawyers 4. Medical records specialists score 2 and physicians 4. The shortest path out of a closing window usually runs to the neighboring role that verifies, signs or decides. In physical work, the neighbor is the less structured version of the same job: unplanned routes, the loading dock, the open field.

• For a person: find your occupation in the index, then the higher-scoring role beside it, and close the gap in license, judgment or accountability.
• For a business: score each role before planning headcount, and move people toward the review and sign-off work that the low-scoring work feeds.

Whichever move you make, keep the window ledger from Section 6: the scoring function, the measured human contribution against a pure-system baseline, and the date it was last measured. Those three lines will show when the next window starts to close.

Steel man, straw man, and objections

The argument holds as a conditional claim. Its universal form generalizes from three games-and-proofs cases to all knowledge work, on an unstated premise that the scoring function is fixed and external. Making that premise explicit turns the weakest link into the paper's best idea.

Steel man: the strongest version

The strongest form drops the universal qualifier and keeps the mechanism. Wherever a ranking is scored by someone the ranked cannot renegotiate with, and AI can scale against that score, a window opens in which coupled pairs lead, refusal exits, and closure follows capability. The evidence is then exactly what the paper has: chess as a full cycle with dates, Go as the limiting case where no coupled format existed, mathematics as a window that has migrated from generation to verification in real time, and software as the live case. Under this framing the AHM and the spear-refusers make different moves. The spear-refuser exits an external ranking. The AHM is trying to change an internal one, and that distinction can be tested.

Three convergent lines support the steel man:

1. Chess outcomes. Elo stopped predicting centaur results (Krakowski et al. 2023); engines beat top centaurs by 2017; ICCF top play is about 95% draws (Glickman 2025).
2. Mathematics outputs. Olympiad problem-solving saturated in July 2026; verification capacity became the funded bottleneck in October 2026.
3. Software economics. Near-universal adoption with output volume up and review time up (Section 3, with verified figures).

Straw man: what a hostile reader will say the paper claims

The paper answers each clause where it arises:

• Every job. The thesis is about rankings on a scoring function. Many roles persist off the frontier, and the index estimates window length.
• Luddite. The AHM position is treated as a scoring-function strategy with a real chance of working where the scorer is endogenous.
• Spears and selection. The mechanism is cultural and economic, on a timescale of years, and the analogy is marked as an analogy with its weak points named.
• Board games prove it. Go is presented as the case where the pattern did not appear in competition, which is evidence about the conditions the pattern needs.
• Made-up numbers. Every empirical figure in Sections 1 and 3 is tied to a source checked in Appendix B. The index scores are stated as composite judgments with the factors shown, so a reader can rescore.

Strongest objections and the paper's answers

• Opposing: rankings are endogenous. Fields set their own scores and can refuse to count machine output. Answer: true, and it is the index's fifth factor. It works only while outsiders cannot route around the field's scorer.
• Diverting: the window moves. Mathematics in 2026 shows human value migrating from generation to verification and digestion. Answer: accepted. Closure is defined per stage of the pipeline, and the index scores the stage where the human edge currently sits.
• Obstructing: the evidence base is thin. Answer: every figure is tied to a named, dated source (Appendix B), and figures that could not be verified were left out.

Citation verification log

Every empirical claim in this paper was checked against a primary page opened on October 26 or 27, 2026. Secondary coverage is used only where the primary page could not be opened, and is marked.