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THE RESEARCH STUDIO

Gao Math Lab

Connecting
RESEARCH, IN MOTION v2.1

Follow the idea.
Check every step.

A living workspace for mathematical exploration. Search for counterexamples, test conjectures, and trace every result back to its evidence.

3 research enginesExact integer arithmeticSaved checkpoints

01 / The big picture

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Collatz starting values checked—Finite observations, not a global proof
Goldbach even numbers checked—Exact primality checks
Residue classes examined—Symbolic derivation
Local descent lemmas—Conditional, independently checkable results

02 / Inside the engines

Research pipeline
01

Collatz conjecture

3n + 1 · Counterexample search

EXPERIMENT
Next starting integer—
Longest trajectory —
Checks per second —
An inconclusive trajectory pauses the scan. Step limits are never reported as counterexamples.
02

Goldbach conjecture

Even number = prime + prime

EXPERIMENT
Next even integer—
Verified range —
Checks per second —
Any candidate counterexample is checked by independently enumerating every possible prime pair.
03

Automated lemma discovery

Collatz · Symbolic modular proofs

PROOF CHECK
Analyzing—
Progress for the current modulus—
Proves descent conditions for specific residue classes after a fixed number of steps. These local lemmas do not prove the global conjecture.

03 / Research workspace

v2.1 · Bounded research loop
01 Conjecture→02 Challenge→03 Verify→04 Refine

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Export research records ↗
No runs yet

Finding no counterexample is not a proof; finding no papers does not establish novelty. Revisions adjust conjecture parameters and reviewed strategies. Generated content is never executed as code.

04 / The evidence library

Independent certificates
Browse local proof certificates Independent, reproducible records ↗
Analyzing residue classes…

Verification applies only to the local algebraic statement in each certificate. Novelty requires a literature review and peer assessment.

Research controls

Admin token required

Research log

Latest 20 entries
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RESEARCH HANDOFF

Continue research with Codex

Export current progress, local proofs and next steps for Codex to analyze and propose new hypotheses.

Export research brief ↗
Mathematical proof ≠ numerical experiment

This system does not guarantee breakthroughs on open problems. Finite checks do not prove statements about all integers. Local lemmas do not constitute a complete Collatz proof, and independent software checks do not establish novelty. Independently recheck any candidate counterexample and seek expert review.