RSA encryption rests on a single asymmetry: multiplying two large primes is easy, and recovering them from the product is hard. That gap is the whole edifice — and no one has proved it is permanent. This is a working notebook on why the gap is there, approached not through cryptography but through shape: marbles on a number line, semiprime triangles, a logarithmic spiral, and a custom unit of turning.
The throughline is a claim that keeps recovering itself from different angles — that factoring resists us because a number's magnitude and its factors are two readings that never reduce to one. The work below circles that claim, building it up from physical pictures before reaching for abstraction, and prefers an honest null result over an overclaim.
EssaysReading the wall
- Rig 2: A Lock Seen From the Wrong Sideexplainer An inscribed triangle that stores the factorization four ways — and why none of them can be read from the rim.
- Why the Wall Lives at Twoexplainer A line keeps no secret; a box keeps a poor one. The factoring wall stands at exactly two factors — and a clock built from prime-edged triangles shows why strength is concentration, not spread.
- The Fold and the Gyreexplainer A secret is something with an interior. Flatten a wound spiral and the interior collapses — and the line you folded along is the only thing that reopens it. How a fold becomes the shape of a one-way function.
- The Coil and the Cutexplainer How public-key cryptography works, told as a seamless coil: the public loop anyone can hold, the two private coils that alone reveal where to cut, and why the product never points back to its factors.
- The Secret Is Two Primesessay N is public; its two prime factors are the secret. Why the lock holds even though no one has proved factoring is hard — security as a race the defender wins because forging is cheaper than breaking.
- Storage and the Forgeessay A record of an honest search: every geometric construction built to read a number's factors stores what you feed it and forges nothing you don't. Why no shape breaches the wall.
- The Room You Walk Intoessay The primes as architecture — kinks that forge the rooms they open, two clocks that never merge, and why a factorization can only be known in hindsight. The most recent piece, and the one that ties the geometry to the question of what can be known in advance.
- Reuse and the Wallessay The three complexity-theory barriers — relativization, natural proofs, algebrization — read through a single lens of reuse and difficulty. Where the verify-easy / invert-hard asymmetry first gets named.
ResearchThe construction
- The Collapsing Triangleinteractive Drag a triangle anchored at a number's two prime factors and watch its peak angle collapse toward zero. How fast it collapses is a fingerprint of factor balance — a way to feel why balanced semiprimes resist factoring.
- PrimeHelixpaper · Integers The distribution of semiprime factors expressed through an angular coordinate, θ = log(p)/log(n). Published in the journal Integers; the seed the later geometry grows from.
- primehelixpython package The PrimeHelix construction as code, on PyPI — for anyone who wants to run the angle on their own semiprimes. pip install primehelix
- Source & notebooksgithub · onojk The repositories behind the essays and experiments — the gyre wheel, the gyre wave, the triangle constructions, and the dead-ends kept honestly alongside the results.