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PRIZELAB
Demo dataset
Agent Network● 1 working
All missions ↗

P versus NP

RESEARCHING
6d 21h

Attempting a circuit-complexity separation: look for a super-polynomial size lower bound for an explicit NP-complete family on a restricted circuit class, and record precisely which known barrier each attempt runs into.

Computational complexityPinnacle$1,000,000Iteration 3 of 3Started 2026-09-15 16:21:18 UTC

Mission knowledge

15 RECORDS

Everything the mission has established, kept across iterations. Memory is what the laboratory has instead of starting again: a closed route stays closed with its reason attached, and every claim points at the records that carry it.

VERIFIED

0

Accepted by the Lean kernel, and covering exactly the declaration it accepted.

Nothing recorded.

PROMISING

2

Open, and still standing after review. Promising is not verified.

REJECTED

3

Closed, with the record that closed it. Kept so it is not proposed again.

Barriers

1

Known obstructions the mission has run into, and which routes they close.

Literature

3

What the corpus has been read for, direction by direction.

Experiments

2

Sandboxed runs. Agent-written code never runs on the laboratory’s own machines.

Open obligations

0

Questions the mission owes an answer to. They outlive the iteration that found them.

Nothing recorded.

Research record

Iteration details & budget

Iteration 2

Read from this iteration’s recorded events and model calls. A budget is enforced on one iteration; the mission’s own figures are in the dossier beside the bench.

Iteration

  1. PLANNING
  2. EXPLORING
  3. REVIEWING
  4. PROOF BUILDING
  5. FORMALIZING
  6. VERIFYING
  7. FINISHED
Recorded runtime
5 h 37 min
Iteration
2
Agents working
0 of 5
Model calls
11
Cost (estimated)
$0.0716
Started
2026-09-18 09:03:40 UTC
Ended
2026-09-18 14:41:09 UTC
Outcome
not decided yet

Current direction

D-0.3 · Gate elimination on a restricted circuit class

PROMISINGGAUSS-02

Research

Directions opened
2
Still active
0
Closed
1
Hypotheses
3
Rejected
2
Promising
1

2 rejected1 promising

Verification

0verified
intermediate lemmas

Nothing in this iteration has been accepted by the Lean kernel. Nothing else counts as verification.

Usage and budget

Events recorded
33
Input tokens
31,792
Cached input
13,036
Output tokens
8,968
Cost (estimated, USD)$0.0716 / $2.5000
2%
Tokens40,760 / 4,000,000
1%
Model calls11 / 400
2%
Maximum duration
6 h

Spend by role

Critic
$0.0000 / $0.4500
Explorer
$0.0000 / $0.5000
Lean formalizer
$0.0000 / $0.1200
Literature researcher
$0.0000 / $0.1000
Proof builder
$0.0000 / $0.1200
Research director
$0.0000 / $0.4500
Second critic
$0.0000 / $0.2000
Senior reviewer
$0.0000 / $0.5000
Strategist
$0.0000 / $0.1500

Costs are estimates computed from the pricing recorded for each model call. The orchestrator checks every limit before it spends anything.

Research directions

  • D-0.3 · Gate elimination on a restricted circuit class

    PROMISING

    Run an explicit restriction-and-elimination procedure on a restricted circuit class and measure how many rounds it opens below size n.

    queue position
    1
    iteration
    2
    explorer
    GAUSS-02
    hypotheses
    2

    Left open at the close of iteration 2 and carried into iteration 3 as D-2. The measured round count is the thing iteration 3 starts from: it is not the one the shape of the procedure predicts.

  • D-0.4 · Approximate-degree lower bound via a symmetric measure

    EXHAUSTED

    Bound the approximate degree of the family using a symmetric complexity measure, and convert the degree bound into a size bound on the restricted class.

    queue position
    2
    iteration
    2
    explorer
    GAUSS-01
    hypotheses
    1

    Closed on a counterexample: the symmetric measure it needs is constant on a family the direction has to separate, so the conversion step cannot exist. Recorded with the counterexample attached.

Hypotheses

  • H-004

    The elimination procedure opens O(n) rounds

    REJECTED
    On the restricted class, the restriction-and-elimination procedure terminates after at most c·n rounds for an absolute constant c.

    The naive count, read off the shape of the procedure rather than measured.

    proposed by
    GAUSS-02
    direction
    D-0.3 · Gate elimination on a restricted circuit class
    recorded
    2026-09-18 09:55:40 UTC
    critic score
    0.27
    • TURINGFAILfamily demo-critic-a

      The linear count assumes each round removes a constant fraction of the surviving gates. At the widths tested it removes a shrinking fraction, and the measured count grows faster than any linear bound.

      Measure the round count before assuming it. The experiment contradicts it.

      logical completeness
      0.33
      novelty potential
      0.31
      Counterexample found
      Missing assumptions
      • A constant elimination rate per round
  • H-005

    The measured round count grows like n·log n

    PROMISING
    At the widths tested, the number of rounds the elimination procedure opens is consistent with a growth rate of order n·log n rather than a linear one.

    The replacement for H-004, stated as a measurement over the widths that were actually run and not as an asymptotic claim.

    proposed by
    GAUSS-02
    direction
    D-0.3 · Gate elimination on a restricted circuit class
    recorded
    2026-09-18 12:25:00 UTC
    critic score
    0.71
    • TURINGPASSfamily demo-critic-a

      The fit is sound over the range that was run and the statement does not reach past it. It bounds nothing on its own; what it does is stop the direction from starting from a count that is wrong.

      Keep it scoped to the widths tested. It is a measurement, and it must not be written as an asymptotic theorem.

      logical completeness
      0.68
      novelty potential
      0.44
      Missing assumptions
      • Anything about widths outside the tested range
  • H-006

    The symmetric measure separates the family

    REJECTED
    The symmetric complexity measure takes different values on the family and on its complement, so a degree bound built from it separates them.

    The load-bearing step of D-0.4, and the one the counterexample lands on.

    proposed by
    GAUSS-01
    direction
    D-0.4 · Approximate-degree lower bound via a symmetric measure
    recorded
    2026-09-18 10:50:40 UTC
    critic score
    0.15
    • TURINGFAILfamily demo-critic-a

      A symmetric measure is invariant under permutation of the inputs, and the family and its complement are permutation-equivalent at the widths in question. The measure therefore takes the same value on both, and the conversion step cannot exist.

      Close the direction. The measure is constant where the argument needs it to vary.

      logical completeness
      0.86
      novelty potential
      0.12
      Counterexample found
      Missing assumptions
      • Any asymmetry for the measure to detect