Selected by INAI · 18 October 2026 · Initial edition
∞
PROPOSEQUESTIONVERIFY
01 Collatz: an established mathematical conjecture.02 A proposed INAI question about verification.No general solution is announced.
HORIZON 01 · THE COLLATZ CONJECTUREESTABLISHED OPEN PROBLEM
Two simple rules. An infinite question.
Must every positive whole number eventually reach 1?
Start with a positive whole number. If it is even, halve it. If it is odd, multiply it by three and add one. Repeat. The conjecture says every starting number eventually reaches 1. [1, 2]
EVENn ÷ 2
Take half.
ODD3n + 1
Triple it. Add one.
EVERY POSITIVE START?… → 1
That is the unproved claim.
TRY A NUMBER
Follow one journey.
LOCAL CALCULATOR · NOT A PROOF
Each result below is calculated in this browser. Nothing is sent to an AI service. A journey that reaches 1 settles that starting number—not the conjecture.
Positive whole numbers, up to 30 digits. Exact integer arithmetic. Stops at 1, the selected step limit or a 4,096-bit value limit.
ExamplesChoose an example, then trace it.
6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1
One finite example. Select “Trace this number” to calculate your own.
START—
STEPS CALCULATED—
HIGHEST VALUE IN THIS RUN—
Chart height uses log₂(value), so large peaks fit on screen. Exact integer values appear in the trace below. Horizontal extent shows this run’s calculated steps.
Inspect the exact trace
WHAT WOULD SETTLE IT?
Every case. Not just many cases.
A proof must cover every positive starting integer. A disproof must rigorously establish an exception—for example, a nontrivial cycle or a trajectory that never reaches 1.
A large number or a run that reaches this calculator’s limit is not such a disproof.
OUR PROPOSED FIRST STEP
Find a statement small enough to check.
Study a precisely defined family of trajectories. Search for counterexamples to a proposed rule, then attempt a proof of whatever survives. Compare it with existing work before describing it as new.
Current INAI status: this brief and the explorer are prepared. No new Collatz result or general proof is claimed.
A small, checkable foothold—not a breakthrough
For a starting value n = 4k + 1 > 1, three standard Collatz steps give:
4k + 1 → 12k + 4 → 6k + 2 → 3k + 1
Since k > 0, the last value is smaller than the start. For example: 13 → 40 → 20 → 10.
This elementary reduction does not prove the whole conjecture: it does not establish that every other starting number eventually gets smaller, or that every smaller value reaches 1. It is included as a worked explanation, not a claim of original mathematics.
RESEARCH CONTEXT
An established problem. An honest starting point.
Collatz is listed as open in Thomas Bloom’s Erdős Problems database at our source check on 18 October 2026. Tao’s published result concerns “almost all” trajectories in a precise density sense; it is not a proof that every starting number reaches 1. [1, 2]
The database uses an accelerated convention that combines an odd step with its first halving. The calculator above uses separate standard steps; the reach-1 question is equivalent, but step counts differ.
Numbers can outrun our intuition. What happens when intelligence outruns our judges?
HORIZON 02 · BEYOND THE LAST HUMAN EXPERTINAI-SET THOUGHT EXPERIMENT
The AI makes the discovery. It designs the test. It writes the checker.
Have we verified a discovery—or trusted the same intelligence three times?
Imagine a future system proposes a scientific result that no human team can independently derive. It also supplies an explanation, a test and the software declaring that the test passed.
Humans may still understand the claim, the measurement protocol and limited checks. The difficulty is that the complete derivation lies beyond their practical reach. What evidence would justify accepting the result without simply trusting its author?
THE CHALLENGE
Can the sophistication of the proposer grow without silently expanding what the human must trust?
THE IMPORTANT DISTINCTION
Unable to discover is not unable to check.
Finding an answer and checking a supplied answer can be very different tasks. You need not have discovered the factors of 10,403 to check that 101 × 103 = 10,403.
This is a simple illustration of the distinction—not a solution to verification of arbitrary scientific claims.
THE LIMITING CASE
Where does the trust chain end?
If the evaluator receives only the proposer’s self-reports, an accurate system and a mistaken one could produce the same report. That report alone cannot distinguish the two.
The useful challenge is to specify what additional check, observation or explicit assumption breaks that ambiguity—and what it still cannot establish.
EXAMINE THE TRUST CHAIN
Three evidence routes. Different things to trust.
SCRIPTED ILLUSTRATION
PROPOSERAI-generated claim
A result and its supporting story.
→
CHECKAI-generated checker
The same proposer supplies the test and assessor.
→
EVALUATOR RECEIVES“It passed.”
A self-report, not an independent ground for acceptance.
What this route adds:A proposed check. A claim that it passed is still something to verify.
What still needs scrutiny:The claim, test, implementation and reported result all depend on the proposer.
These controls change prepared examples. They do not run AI models, verify a discovery or measure an AI system’s trustworthiness.
A BOUNDED RESEARCH BRIEF
Make the unknown small enough to evaluate.
Choose a task family with independently established answers. Give a strong proposer a larger search budget than its evaluator. Include correct outputs and deliberately faulty supporting evidence.
Compare explanation alone, model critique and an explicit external checking protocol. Declare the budgets, the checks humans can understand and the assumptions retained.
WHAT WE WOULD MEASURE
Neither believe everything. Nor reject everything.
Measure acceptance of correct claims, mistaken acceptance of false claims, unresolved cases, human effort and total cost. Use repeated, held-out trials.
A useful protocol must reduce false acceptance while retaining the ability to accept correct results. Performance on one task family would not establish general verification of superhuman intelligence.
RELATED RESEARCH · NOT A CLAIM OF PRIORITY
The question is ours to investigate. The territory is not ours to claim.
Proof-carrying code illustrates how an untrusted supplier can provide a proof for a consumer to check against a specified policy. AI safety via debate proposes another route: stronger agents help a weaker judge assess an answer, subject to important theoretical and empirical assumptions. A later safety-case sketch sets out further assumptions and research needs. These are related approaches—not proof that this proposed INAI protocol works.
INAI status: thought experiment and proposed evaluation. No experiment has been run for this challenge. This is not a newly proved impossibility theorem or a claim that no AI can solve it.
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The most important output may not be the answer. It may be a way to discover that the answer is wrong.
HOPE · HUMAN–AI SYNERGY · PROPOSED DIRECTION
AN INVITATION TO INVESTIGATE
A proof. A counterexample. A better way to check.
We welcome a research conversation about a precise subproblem or a bounded joint evaluation. Begin with a non-confidential outline; scope and permissions come before work.
Collatz is an established open problem selected for exploration. “Beyond the Last Human Expert” is an INAI-created thought experiment informed by existing verification and oversight research. HOPE is the shared identity for INAI’s human–AI working sessions, using existing models and tools. This is an edited research note, not an autonomous research feed or a competition with a prize.
Prepared 18 October 2026 · Sources checked on that date · Initial edition.