Ready to supercharge your AI agents?

Install any skill in seconds. Free to start, no credit card required.

Get Started Free →

Connectedness · vibeeval | DecimalAI

scienceby vibeeval

Connectedness

vibeeval/connectedness

Source vibeeval/vibecosystem ↗License MIT

Problem-solving strategies for connectedness in topology

SkillScore73Maintained

Automatic

Research pack

Last verified 8/3/2026

What it changes

Passed 14 of 19 test cases with the skill, and 7 of 19 without it.

Without skill36%

With skill73%

Checks it passed that the model alone missed

Applies Strategy 3 (Fan lemma) for connected subsets sharing a common origin point
Executes uv run python -m runtime.harness scripts/z3_solve.py prove "no_separation"
Executes uv run python -m runtime.harness scripts/z3_solve.py prove "intermediate_value"

and 2 more in the full report

8 fixed8 passed either way6 still failing0 broke

Pass-rate lift: +36 pp
Tokens: -3%
Agent turns: 0%
Response time: -49%

Model gemini-3.6-flashMeasured 8/3/2026v116 of 22 cases passed with skill · 19 comparable

Tokens, turns and time describe resource use, not a dollar price.

Use this skill

Save it into your project's skills folder. Claude Code picks it up in its next session.

curl -fsSL --create-dirs -o .claude/skills/vibeeval-connectedness/SKILL.md https://app.decimal.ai/s/vibeeval-connectedness/SKILL.md

Want it pinned and updatable? pip install "decimalai>=0.15", then decimalai skills add vibeeval-connectedness, and start Claude Code with claude --plugin-dir "$(git rev-parse --show-toplevel)/.agents"

1 · One-time — install the CLI (docs ↗)

pip install "decimalai>=0.15"

2 · Get the skill — writes .agents/skills/vibeeval-connectedness/ and pins it in .decimal/skills.lock

decimalai skills add vibeeval-connectedness

No CLI? Get the same file — curl · raw file

curl -fsSL --create-dirs -o .claude/skills/vibeeval-connectedness/SKILL.md https://app.decimal.ai/s/vibeeval-connectedness/SKILL.md

Raw file: /s/vibeeval-connectedness/SKILL.md ↗ · machine-readable summary at /s/vibeeval-connectedness.json ↗

Safety checks: 3 of 3 passed

Code scan CleanIntent PassedContent Clean

See it in action

One of 7 test cases the skill fixed. Same model, same prompt, with and without the skill.

1 of 7

Prompt

What exact CLI command using the python harness sympy script simplifies the convex combination path expression?

What was checked, without and with the skill
Checked	Without skill	With skill
Executes uv run python -m runtime.harness scripts/sympy_compute.py simplify "(1-t)x + ty"	Missed	Passed

case-13 · gemini-3.6-flash · 8/3/2026 · answers are capped and redacted

What's inside

The instructions your model reads when the skill loads.

SKILL.md62 linessingle file

Connectedness

When to Use

Use this skill when working on connectedness problems in topology.

Decision Tree

Is X connected?
- Strategy 1 - Contradiction:
  - Assume X = U union V where U, V are disjoint, non-empty, and open
  - Derive a contradiction
- Strategy 2 - Path connectedness:
  - Show for all x,y in X, exists continuous path f: [0,1] -> X with f(0)=x, f(1)=y
- Strategy 3 - Fan lemma:
  - If {A_i} are connected sharing a common point, then union A_i is connected
Connectedness Proofs
- Show no separation exists
- z3_solve.py prove "no_separation"
- Use intermediate value theorem for R subsets
Path Connectedness
- Construct explicit path: f(t) = (1-t)x + ty for convex sets
- sympy_compute.py simplify "(1-t)*x + t*y" to verify path
Components
- Connected component: maximal connected subset containing x
- Path component: maximal path-connected subset containing x

Tool Commands

Z3_No_Separation

uv run python -m runtime.harness scripts/z3_solve.py prove "no_separation"

Sympy_Path

Track record

Benchmark

14 / 19

test cases passed, vs 7 / 19 without

Adoption

Users: 0
Times used: 0

Safety and source details

SkillSafety

✓ 3 of 3 passed

Code scan15 of 15 checks clean

0 critical · 0 warning · 0 info — scanner v4 — 8/1/2026

View 15 scanner checks

live_secretremote_code_executionreverse_shelldata_exfiltrationssrf_metadataagent_snoopinganti_refusaldestructive_commandsobfuscated_unicodeunicode_homoglyphopaque_blobsuspicious_urltool_overreachtrigger_abuseprompt_injection_phrasing

IntentSafe

LLM judge · does the skill do what it claims

ContentClean

LLM judge · policy & harmful-content review

Source

Imported

vibeeval / vibecosystem

skills/math/topology/connectedness/SKILL.md

View on GitHub ↗

License MITImported 8/1/2026Version v1

More skills from the same source and nearby in the registry, with their headline benchmark lift.

Ready to supercharge your AI agents?

See it in action

What's inside

Connectedness

When to Use

Decision Tree

Tool Commands

Z3_No_Separation

Sympy_Path

Track record

Benchmark

Adoption

SkillSafety

Source

Related skills