A public series of mathematical letters to the giants I refuse to leave behind.
This repository is not a journal submission.
It is not a paper archive.
It is a conversation — with the brilliant minds I admire who built the world I now extend.
✔ New mathematical structure and/or discovery
✔ A formal proof as a human-readable letter written to its historical originator continuing and honoring them.
✔ Executable code that reproduces mathematical claims in seconds (packaged for wheel imports).
✔ Cryptographic signatures and hashes for proof of authorship and integrity.
✔ Each letter is mostly independent and does not require any knowledge of other letters. Though they all fit together and provide explanatory power to interleaved concepts.
You can master them individually if you like, however, they do cohesively fit together in retrospect.
This project is structured as a long-form series of mathematical letters. Each letter stands alone, but together they build the full architecture of Runtime Mathematics — a mathematics of construction and deterministic execution.
- Letter 1: Georg Cantor — Golden Continuum (Completed)
- Letter 2: Augustin-Louis Cauchy — (Completed, coming soon)
- Letter 3: Leonhard Euler — (Completed, coming soon)
- Letter 4: ??? — (In progress)
Future letters will be addressed but not limited to:
- Zermelo/Fraenkel
- Riemann
- Fourier
(Please note the order is subject to change)
The overarching mathematics behind the series is largely complete;
what remains is deeper exploration, packaging, and polish.
LettersToMyHeroes/
├── Cantor_GoldenContinuum/ → A constructive continuum built from φ and factorial recursion
├── Cauchy_???/ → (coming soon)
├── CITATION.cff → How to cite me and my work
├── CONTRIBUTING.md → How to contribute to Runtime Mathematics
├── crypto.md → PGP signatures and SHA256 verification steps and quick scripts
├── REPO_MAP.md → Full repo directory graphs and links for quick reference
├── LICENSE → all code licensing information (GPLv3-or-later)
├── LICENSE-docs → all docs licensing information (CC-BY-SA 4.0)
└── README.md → you are here
See Repo Map for directions!
Introduces:
-
$\mathbb{G}$ the φ-scheduled factorial continuum: a single lawful generator whose recursion can reach any real target through pure analysis without changing its rules. - A multiplicity powered calculus that converges to classical analytic operations
- A constructive, computable, φ-powered calculus that transcends the prior.
- Amortized cost for achieving D digits is
$O(log D − loglog D)$ — effectively constant. - Precision scales superfactorially
- Total function evaluations scale only with layer count
- Functions are evaluated exactly in rational form, precision is only lost when converting for display.
- For differentiation, 12 evaluations routinely yield thousands of exact digits.
- Integration is also exact in rational form (locally) and expands globally by dyadic paneling.
- Digit guarantees do not come from numeric approximation; they come from exact rational contraction of analytic moment laws, always eliminating 2N−2 Taylor terms deterministically.
These demonstrate the factorial-layer engine exactly as described in my Letter to Cantor.
This continuum executes and exploits factorial structure itself to produce exact rationals from a generator.
📂 Directory: Cantor_GoldenContinuum/
📖 README: Golden Continuum README, PyPI φ-Engine README
📄 Letter (proof): LetterToCantor.pdf
💻 Code:
• phi_engine/
• Multiplicity Calculus Demo (this is NOT the φ-Engine)
❓ Common Questions: FAQ
⏳ TLDRs for the gist of it: Mathematicians, Engineers, [Cryptographers](Cantor_GoldenContinuum/Capability Brief — Cryptographers.md), Physicists
Every Letter is:
- Signed with my PGP key
- Hash-verified (SHA256)
- Reproducible by quick commands in markdown
Reproducible builds and deterministic signatures ensure that any clone of this repo can verify every artifact bit-for-bit.
See Cantor crypto.md for verification instructions.
Linux / MacOS
./Cantor_GoldenContinuum/crypto/verify_all.shWindows
./Cantor_GoldenContinuum/crypto/verify_all.ps1This script validates PGP signatures and SHA256 hashes for every file in the Golden Continuum project.
Mathematical truth should be provable — not trusted.
This project is dedicated to public, reproducible review.
- Discovery is not licensed — it is revealed. I publish proofs, code, data, signatures, and reproducible harnesses so anyone can verify claims.
- Peer review = public review: fork the repo, run the tests, and open issues / PRs. All reviews are visible, timestamped, and welcome.
- No gatekeepers. I will not submit this work for exclusive review or negotiate with journals that insist on paywalls, opaque formatting, or editorial gatekeeping.
- If any journal or corporate entity requests exclusivity: I will decline — public verification is the authority.
- Verification is trustless: cryptographic signatures, checksums, CI, and reproducible containers are provided so the claims speak for themselves.
If you want to criticize or extend this work — do it openly. I'll cite, link, and credit public contributions.
- All code: GPLv3-or-later
- All documents, letters, and images: CC BY-SA 4.0
This project was written — start to finish — by Alex B (Purrplexia) alone.
No collaborators, no editors, no institutional support.
Every idea, proof, and line of code is original work.
Some earlier mathematical documents authored by me exist outside this repository. They represent early explorations, not components of Runtime Mathematics.
Those works may contain ideas that later matured into the formal structures presented here, but they do not reflect the standards of rigor, verification, or methodology of this project. They should therefore be read as historical artifacts, not as formal or authoritative references.
Readers are welcome to explore them for context, intuition, or insight into the evolution of these ideas, but all definitive statements, proofs, and definitions live in this repository alone. Future letters may revisit earlier concepts with full rigor when appropriate.
Curiosity welcomed.
Gatekeeping ignored.
Issues can be opened directly in this repository.
Public peer commentary is greatly encouraged!
Pull requests improving clarity, code efficiency, or validation coverage are also welcome.
Formal inquiries may be directed to mathsisbeautiful@proton.me
This project will always remain free, open-source, and publicly signed.
If you'd like to support ongoing research and future Letters:
- GitHub Sponsors: https://github.com/sponsors/Purrplexia
- ETH (MetaMask):
0x663D8288b4Aa6F3A72FF4FE67d1a7B080cD5097d - BTC (Electrum):
bc1q0ntv4zvpdprlxexvz20mf7ajdydmp9ke9ulgep
Never required — always appreciated. Every dollar goes toward building, teaching,
and releasing more of this work openly.
I will never gate mathematical truth behind paywalls or NDAs.
This work belongs to everyone.
Forever.
The following supporters helped launch Runtime Mathematics. Thank you for believing in open, reproducible, executable mathematics.
- <Your name could be here> (See GitHub Sponsors for details on this limited offer)