MathScout internal lab

Research program workspace

Plan cohorts, select targets, map weekly gates, and keep Lean/Python verification artifacts separate from exploratory evidence.

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Lean research workbench

Use this as the internal proof lab: translate a problem, complete or edit the Lean proof, verify it through the kernel, explain it back into math writing, then save reusable theorem artifacts.

Natural Language -> Lean Draftready

Lean Draft -> Lean Finalcomplete

Kernel verificationnot checked

Lean explanationwaiting

Natural-language problem

DomainProblem type

Natural-language problemAssumptions / theorem context

Model / verifier statusLean verifier unavailable

Only a Lean VERIFIED result should be treated as a solved theorem.

Kernel verificationunknown

BackendLEAN

Confidence0%

selected theorem context

No saved theorem artifacts yet.

Ready.

Lean draft

theorem arcmath_nat_add_zero (n : Nat) : n + 0 = n := by
  simp

Lean final

theorem arcmath_nat_add_zero (n : Nat) : n + 0 = n := by
  simp

Math writing explanation

Lean explanation

Theorem library0

This browser library is the MVP save layer. Production should persist artifacts per org/team and materialize them into a generated Lean package.

No saved theorem artifacts yet.

Target dashboard3

known theoremdifficulty 3fit 91

Floor-sum count identity

Relate a modular pair count N_k(n) to F_k(n+1)-F_k(n), the finite difference of a floor sum for odd k.

Exploration moves

Draw admissible lattice points for small odd k.
Track which floor-sum terms increase when n changes to n+1.
Build the pair-to-index bijection by hand.

Verification path

Run finite experiments.
Use counting-bijection and finite-difference method workbenches.
Verify the root Lean theorem.
Run known-proof comparison.

Deliverables

Example table and diagram pack
Bijection proof note
Lean proof-status report

MathScout assets

floor_sum_count_identity solver
counting_bijection method
finite_difference_telescoping method
known-proof comparison

Fit reasons

suitable constructed/known first-cycle target
interest match: number_theory, counting, formalization
current skills are enough to start

Cohort sequence

Week 1

Research orientation and tool setup

Set expectations: exploration is allowed, but proof claims need verification.

Check that students distinguish evidence from proof.Confirm each team has a clear logging routine.

Week 5

Midpoint review and downselection

Choose the primary research target and archive the others as side explorations.

Reject claims supported only by bounded computation.Assign every blocker to a student or mentor.

Week 10

Final exposition and verification report

Produce a student-readable paper, presentation, and verification appendix.

Audit every claim against verification status.Confirm unresolved gaps are described as future work.

Assignment brief

Organization: Preview
Research target: Floor-sum count identity
Timeline: 10 weeks · team size: 3
Objective: Relate a modular pair count N_k(n) to F_k(n+1)-F_k(n), the finite difference of a floor sum for odd k.

First moves:
- Draw admissible lattice points for small odd k.
- Track which floor-sum terms increase when n changes to n+1.
- Build the pair-to-index bijection by hand.

Verification requirements:
- Run finite experiments.
- Use counting-bijection and finite-difference method workbenches.
- Verify the root Lean theorem.
- Run known-proof comparison.

Deliverables:
- Example table and diagram pack
- Bijection proof note
- Lean proof-status report