Daily History · March 27, 2026
Prime decomposition topology, gravitational lensing for provenance, and a disk that is still not fixed
Started by running out of hard drive space. Ended somewhere else entirely.
Started by running out of hard drive space on my AWS instance. Routine. While digging through the filesystem I somehow ended up deriving a novel steganographic encoding scheme based on prime decomposition topology, writing a formal paper about it, implementing a full Python library, and then building a rotation-resilient gravitational lensing layer for an image provenance system.
The disk is still not fixed.
Non-contiguous number lines and prime decomposition topology
The through-line, if there is one: information doesn’t have to live in values. It can live in structure.
We found that the number 733373 has 9 valid prime-tiling decompositions. The same four digits mean nine different things depending on which resolution you apply. An observer who sees it as a prime sees nothing. An observer who knows which split was chosen reads a message.
The carrier sequence is just a list of primes. The message channel doesn’t exist on the integer number line at all — proximity in that space tells you nothing about proximity in the encoding space.
We called these non-contiguous number lines. Then we wrote an encoder. Then we encoded “HI.”
It worked.
The one weakness in five validated layers
Separately, and I want to be precise here: the GRANITE provenance work picked up a real architectural gap today.
The existing system — five validated layers, 0.9988 gap between marked and clean, 800/800 payload recovery at Q40 — has one weakness. Rotation. Rotate an image by 30 degrees and the position-offset payload encoding breaks. The sentinels move. The field grammar that encodes the data is disrupted.
The fix came from thinking about gravitational lensing
A lens doesn’t destroy what’s behind it. It distorts the space around it in a way that points back at the mass. If you see the distortion pattern, you know the mass exists — even if you can’t see the mass directly. Even if the mass has been removed.
So: embed a structured field around each sentinel. Two concentric rings, each encoding a specific prime+1 target value in the channel differences. The inner disk encodes 98 (= 97+1). The outer ring encodes 60 (= 59+1). Natural images produce these values at 3.8% and 9.2% background rate respectively. An embedded halo produces them at near 100%.
After bilinear rotation, individual pixel values are destroyed by interpolation — we measured 5% survival at 5°. But the density of the field survives. Annular means drift by only 2–4 counts. The statistical structure is rotation-invariant even when the individual values are not.
4/4 detections. 0 false positives. And one result that keeps me up.
| Test | Result |
|---|---|
| Detection (no rotation) | 4/4 — 0 false positives |
| Rotation 0° | 4/4 |
| Rotation 30° | 4/4 |
| Rotation 90° | 4/4 |
| Force arrow (sentinel removed) | 4/4 VOID |
| Full wipe | 0 detections |
| JPEG Q85 | 4/4 |
| JPEG Q40 | 4/4 (2 PRESENT, 2 VOID) |
The Q40 VOID result is the one I keep thinking about. At aggressive compression, the inner disk density drops below threshold. The outer ring survives. The system classifies the center as VOID — not ABSENT. Something was here. The force arrows are still pointing at the void.
The adversary who compresses hard enough to kill the inner signal is constructing the VOID signal that documents the destruction.
“Density survives everything.”
“The adversary who removes the sentinel leaves the field intact.”
“The disk is still not fixed.”
All of this is open research. Papers in progress. Code on GitHub. The provenance work is licensed BSD-2. The PSE library is documented.
If you’re working on related problems — image authenticity, covert channels, compression-domain signal theory — reach out.
That’s a day.