The Principia Orthogona series spans nine volumes, three books, 121+ chapters, three languages, and formal proofs. Each area has open problems—gaps waiting for someone to fill them. Below are the current opportunities, arranged by difficulty and type.
Quick Entry Points (No Special Skills)
Easy · 5 min each
Grammar & Spelling
English text has typos, comma splices, inconsistent hyphens. Read a chapter. Fix what you find.
Examples: "teh" → "the", missing comma before "however", inconsistent em-dash usage
Files: Any .html chapter in Books 3, 4, 6, 8, 9
Impact: High — typos undermine credibility
Find Issues ↗
Easy · 10 min each
Broken Links
Cross-chapter links point to the wrong files. Chapters reference sources that don't exist.
Examples: href="ch5.html" should be href="ch5-immune.html", Zenodo DOI returns 404
Tools: Read the file, check the link, report broken URLs
Impact: Medium — breaks navigation
Report Link ↗
Easy · 20 min each
Missing References
Chapters cite papers, theorems, or datasets that should have Zenodo DOIs or arXiv links but don't.
Examples: "Baker (1966)" cited but no link to Mathematika paper, Pacioli mentioned but no DOI
Process: Find the source, get the DOI, add it
Impact: High — citations are currency
Add Reference ↗
Medium Difficulty (Language & Translation)
Medium · Hours
Portuguese Translation
Book 4 (GTCT) exists in Portuguese. Other books need translation. Start small: one chapter or one section.
Status: Book 4 (GTCT) 100% · Books 3, 6, 8, 9 need 0-30%
PTEN Required
Format: Duplicate chapter, translate, send PR
Impact: Very high — unlocks the work for 250M Portuguese speakers
Translate Section ↗
Medium · Hours
Other Language Translation
Spanish, French, German, Mandarin, Japanese—any language. Start with Chapter Zero or the Prelude.
Status: English (100%) · Portuguese (30%) · Others (0%)
ESFRZHJA
Start with: Chapter Zero (500 lines, foundational)
Impact: Very high — globalizes the work
Start Translation ↗
Medium · 30 min–1 hour
Clarity Improvements
A chapter is technically correct but confusing. Rewrite a section for clarity without changing the meaning.
Examples: A theorem proof has 5 steps but could be 3. An explanation uses jargon that could be simpler.
Process: Open issue, explain what's confusing, propose rewording, send PR
Impact: Medium — improves pedagogy
Improve Clarity ↗
Advanced (Mathematics & Code)
Hard · Days
Verify Lean 4 Proofs
Book 4 (GTCT) claims zero `sorry`s. Verify the chain by running the Lean kernel yourself. Find gaps or errors.
Status: 1,165 formalized · 1,004 sorry-free in source · 61 kernel-audited — 31 re-run by CI (geometry), 30 registry-only (AXLE has no CI)
LEAN4MATHLIB4
Setup: `git clone https://github.com/TOTOGT/GTCT` + install Lean 4
Impact: Very high — proof rigor is everything
GTCT Repo ↗
Hard · Days–Weeks
Find & Fix Math Errors
A theorem is stated wrong, a derivation has a gap, a citation is incorrect. Identify the error and propose the fix.
Process: Run the calculation, verify against sources, document the error, send PR
Impact: Very high — correctness is everything
Report Math Error ↗
Hard · Weeks
Add Computational Examples
A theorem exists in abstract form. Build a Python/Julia/Lean example that instantiates it concretely.
Examples: Plot the Reeb orbit, simulate a tribonacci sequence, visualize the Leech lattice
Skills: Python (matplotlib, numpy), visualization, or Lean
Impact: High — examples transform abstract into intuitive
Add Example ↗
Hard · Weeks–Months
Complete Open Proofs
A chapter has a theorem marked `[OPEN]`. Prove it. Or prove it's impossible and document why.
Examples: Finish the proof of the g₃₃ = 33 conjecture, complete the large-cardinal hierarchy bridge
Skills: Advanced mathematics (topology, algebra, category theory)
Impact: Highest — closes research gaps
View Open Issues ↗
How to Contribute
1
Find something
Read a chapter. You'll find a typo, a broken link, a confusing section, or a mistake. (We left them on purpose.)
2
Open an issue (or go straight to PR)
For small fixes (grammar, spelling): Open a GitHub Issue or fork and send a PR directly.
For big changes (translation, new proofs): Open an issue first to discuss approach.
3
Make your change
Fork the repo, make the fix, commit with a clear message. For translations, please note the language and percentage complete.
4
Send a Pull Request
We'll review it, discuss if needed, and merge. You'll be credited in the acknowledgments.
5
You're a co-author
The moment your PR merges, you're part of the series. Grammar fix, proof completion, or full translation—all count.
Questions?
Reach out via GitHub issues. Or if you want to discuss a big contribution before starting, open an issue with the label `discussion` and we'll talk it through.
Remember: This is a living work. The fact that you found a mistake means you understand the material well enough to know it's wrong. That's the exact person we want fixing it.