A measurable definition of "entering the singularity"

Status: adopted 2026-10-08 (amendment A19 in docs/amendments.md), with four legs. The frozen values are in docs/singularity-entry-thresholds.json; the page at agizeitgeist.com scores the gauges from that file. The draft this grew from is kept unchanged as docs/singularity-entry-definition-draft-2026-10-07.md. Changing any frozen value is an amendment entry, never a quiet edit.

It began when John noticed that a top comment on r/accelerate ("any prediction made right now about the next 5 years is gonna look stupid in retrospect") matches one of the standard definitions of the singularity, and asked whether the idea could be made into a definition, a progress measure and a mark to cross.

1. The problem with the existing definitions

Three definitions have been in use since the 1990s (Yudkowsky's 2007 "three schools"), and a fourth, economic one sits beside them:

school	source	claim	what it measures
accelerating change	Kurzweil, 2005	progress is super-exponential and predictable	pace of capability growth
event horizon	Vinge, 1993	past some point our models stop working	how far ahead forecasts work
intelligence explosion	I. J. Good, 1965	AI improves AI, so progress feeds on itself	share of progress produced by AI
growth mode	Hanson, 2000	each new mode of the economy doubles output a hundred times faster than the last	doubling time of world output

Each one, used alone, has a known false positive:

Accelerating change alone fires on any sector having a fast decade. Moore's law was a steady exponential for fifty years and nobody called it a singularity.
Event horizon alone fires in every technological revolution as felt from inside. Five-year forecasts about the internet made in 1995 look stupid in retrospect too. It is also a claim about the forecasters, not about the world.
Intelligence explosion alone cannot be measured from outside the labs, and a high share of AI-written code in a slow-moving field would not be a singularity.
Growth mode alone lags everything else by years and cannot say what caused it.

The definition: entry means all four being true at once, and staying true.

2. The four legs

One capability measure, chosen and frozen, anchors legs A and B: the length of software task (in minutes an expert human needs for it) that the best public AI system finishes half the time, as published by METR. It is chosen because it has not saturated the way benchmarks do. Successor rule: when it stops moving, a replacement measure is named before that happens, the overlap period is published, and the switch is an amendment.

Let C(t) be that measure at time t. The reference year for every leg is 2019.

Leg A: pace is accelerating

Doubling time T(t) is how long C takes to double, estimated over a trailing window of twelve months with at least three frontier readings in it:

T(t) = ln 2 / (slope of ln C over the trailing twelve months)

A steady exponential has constant T. The singularity signature is T falling over time. Windows that include a reading past 16 hours are not scored, because METR treats estimates past 16 hours as beyond its task suite's reliable range and excludes them from its own fit.

Threshold: T(t) at or below one month (30.44 days) and still falling over the last two windows.
Gauge (0 to 1): g_A(t) = clip((T_ref − T(t)) / (T_ref − T*), 0, 1), where T_ref is the doubling time anchored at the reference year. The trailing-window series cannot be computed in 2019 itself (one reading), so T_ref is METR's own fit over all frontier models from 2019 to early 2026, 187.8 days (agent decision, 2026-10-08).

Leg B: forecasts have stopped working

For forecasts of C made at time s with lead d (that is, about C(s + d)), the skill score against the trend-extrapolation baseline is

S(s, d) = 1 − (mean squared error of the forecasts, log scale)
            / (mean squared error of extrapolating the trend as of s, log scale)

The baseline is trend extrapolation, not "no change". In a world of steady fast progress, "no change" loses badly but the trend wins, and that world is Kurzweil's, not Vinge's. The event horizon is the point where even the trend breaks.

The forecast horizon H(s) is the largest lead d at which S(s, d) is still above zero.

Threshold: H(t) at or below one year.
Gauge: g_B(t) = clip((H_ref − H(t)) / (H_ref − 1 year), 0, 1), with H_ref the horizon in the reference year.
Direction is recorded, not just size. Were the forecasts too slow or too fast? Milestones from the 2016 Grace survey mostly arrived early. "Consistently too slow" is a claim about the world; "unpredictable" hides it.
Lag. H(t) can only be computed once t + d has passed, so leg B is confirmed in arrears. The real-time stand-in is revision velocity: with M(t) the consensus arrival date for human-level AI (surveys, Metaculus, Manifold, all already in data/aggregates.csv), let D(t) = M(t) − t be the years remaining. If M is constant, D falls one year per year. Excess pull-in is
```
  v(t) = −dD/dt − 1
```
which is zero when the date is standing still and positive when the date is being pulled in faster than time passes. Sustained positive v is the leading indicator that forecasts are failing in the slow direction. It is reported beside g_B, never substituted for it.
Not yet measurable. The forecast-versus-outcome ledger is not built, so neither H(t) nor H_ref exists today. The gauge is shown as "not yet measurable" until it is.

Leg C: AI is doing the improving

Let f(t) be the share of inputs to frontier AI progress (code, experiments, research ideas) supplied by AI systems rather than people, from the best measurement available. Candidate sources: lab self-reports of AI-written code share, surveys of frontier researchers on time saved, and the AI-contribution disclosures that some venues now require on papers.

Threshold: f(t) at or above one half.
Gauge: g_C(t) = clip(f(t) / 0.5, 0, 1).
This is the weakest-measured leg. Today f is Anthropic's self-measured share of its own AI research work that its model "leads": one company, judged by the model itself. A company's share of all new code written by AI is a proxy at best and is shown but not scored. Every f value carries its source and whether it is a self-report.

Leg D: the economy has changed mode

Let G(t) be world real output growth over the previous year (World Bank, world aggregate, constant 2015 US dollars). Its doubling time is T_D(t) = ln 2 / ln(1 + G(t)).

Hanson's finding is that the world economy has had three growth modes (hunting, farming, industry), each doubling output about a hundred times faster than the one before: about 230,000 years, 860 years and 15 years per doubling. A new mode would double output in years or less. Leg D asks whether that has started.

Threshold: G(t) at or above 30 percent a year, which is a doubling time of about 2.6 years or less. This is Davidson's published line for "explosive growth" (Open Philanthropy, 2021), chosen so the number is not the project's own invention. It is far above anything in the modern record (the series peaks near 6.5 percent) and well short of Hanson's "days, not years", so it fires early in a new mode without firing on a boom.
Gauge: g_D(t) = clip((T_ref − T_D(t)) / (T_ref − T_D*), 0, 1), with T_ref the doubling time at the mean growth rate from 1961 to the reference year 2019 (about 20 years) and T_D* the threshold doubling time.
Timing. World output is reported once a year and revised. One calendar year whose growth crosses the line counts as all four of its quarters holding; persistence is met by one full year above the line, confirmed when the following year's first estimate is published. A revision that pulls a year back below the line resets the count.
Why a fourth leg. John's ruling (2026-10-08), against the agent's recommendation to publish it beside the mark rather than inside it: it is the only leg measured by people outside the AI field, so it guards against a singularity on paper, where benchmarks and lab self-reports fire and nothing in the world changes. The cost, accepted knowingly: output lags capability by years, so this leg will usually be the bottleneck, and the progress number below will track the economy until it moves.

3. The mark

We have entered the singularity, as defined here, when all four gauges equal one for four consecutive quarters. Any leg dropping back below its threshold resets the count. This is the same shape as the three-month persistence rule in the vocabulary ledger (methodology section 8).

The status at any date is one of: not entered; approaching (all four gauges above one half); mark crossed, under persistence test (count in progress); entered (count complete, with the quarter it completed). A leg that is not yet measurable cannot be at its threshold, so the status can only be "not entered" while any leg is unmeasured.

4. Progress toward it

The four gauges are published separately, each in its own panel. If a single number is wanted:

P(t) = min(g_A(t), g_B(t), g_C(t), g_D(t))

The smallest gauge, not the average. The definition requires all four, so the weakest leg is the honest distance remaining. An average would let one runaway leg hide stalled ones. This also keeps methodology section 9's rule: series are never combined into an index. A minimum is not an index; it is the definition's own bottleneck. While a leg is not yet measurable, P(t) is reported as "at most" the minimum of the measured legs, since the missing one can only lower it.

5. Tests the definition owes

Backward test, owed. Run on every year from 1995 to 2019 with the best data available, the definition must return "not entered". Specifically, 1995 to 2000 (internet) should fire leg B and nothing else; 2012 to 2017 (deep learning) should fire leg A weakly and nothing else. The leg A series starts in 2019 and leg C in 2026, so the test cannot run yet. Adopted without it, by John's ruling of 2026-10-08, with this gap stated on the page. Older capability series must be built before it can run; if it then fires on those periods, the thresholds are wrong and get amended.
Preregistration. The capability measure, the reference year, the four thresholds and the persistence window were frozen on 2026-10-08, before any gauge was published. Changing any of them afterwards is an amendment entry.
Self-reference. If leg B is true, forecasts of when the mark will be crossed are themselves unreliable. The mark is therefore observed, never forecast. The project publishes status, not an arrival date.
What it does and does not claim. Crossing the mark establishes that the singularity as defined here has been entered. It does not settle which school was right, and it does not establish that the world is now unknowable. It establishes that four stated measurements crossed four stated lines and stayed there.

6. Relation to the tracker

The Zeitgeist tracker measures what people believe. This definition measures the world. They are kept apart on purpose, and the interesting chart is the two side by side: the share of statements claiming we are in or near the singularity (streams E, F and the vanguard sub-series) against P(t). Divergence in either direction is a finding. The hedging rule (R3) in the vanguard codebook already gives a proxy for a community's sense of its own forecast horizon, and can be charted against g_B.

7. Record of the rulings (2026-10-08)

Proposed by the agent, approved by John unless marked otherwise:

question	ruling	reason
anchor for legs A and B	METR 50% time horizon, with the successor rule above	the one capability series that has not saturated
thresholds	the draft's placeholders as written: one month, one year, one half, reference year 2019, four quarters	preregistered before any gauge; placeholders had been published in the draft a day earlier
backward test	adopt without it, gap stated	the data to run it does not exist yet; waiting would mean never publishing
economic leg	added as leg D (decided by John, against the agent's recommendation)	the only leg measured outside the AI field
leg D line	30 percent a year (Davidson 2021); reference the 1961 to 2019 mean	a published line, not an invented one
leg D timing	one annual reading covers its four quarters; revisions reset	the series is annual and revised

Rendered from docs/singularity-entry-definition.md at commit 1c1b2cb.