P31 Demos & Math

Interactive visualizations showing the math, physics, and cryptography that make P31 Labs work. Drag, edit, explore — every demo is a live artifact.

Graph Theory

K₄ — The Same Shape

K4 · |V| = 4 · |E| = 6 · χ = 2

The complete graph K₄ rendered at four scales simultaneously: family mesh, Platonic tetrahedron, SIC-POVM quantum measurement basis, and Posner phosphate cluster. Drag any panel — all four rotate in lockstep because they are the same object.

Euler: V − E + F = 2 · Planar with β2 = 1 · 4 vertices, 6 edges, 0 hubs
Launch Demo →
Quantum Physics

863 Hz — The Pulse

f = γB / 2π · 863 Hz

The Larmor frequency of phosphorus-31 in Earth's magnetic field (~50 μT). Edit the constant and watch every surface update in real time. Includes the alignment DAG showing how one number flows through the entire codebase.

³¹P gyromagnetic ratio: 108.394 MHz/T · Earth's field: ~50 μT · Period: ~1.16 ms
Launch Demo →
HASH VERIFY STORE R2
Cryptography

Evidence Cage — SHA-256

H(m) = SHA-256(m) · |H| = 256 bits

Every court exhibit is cryptographically fingerprinted with SHA-256 and stored in PGLite + Cloudflare R2. The hash is signed with ML-DSA (post-quantum). Any party can verify integrity without accessing raw evidence.

64 hex chars · Merkle-Damgård · 64 rounds · 2256 space
Coming to PHOS →
INHALE 4s HOLD 2s EXHALE 6s
Neurophysiology

Vagal Coherence — 0.1 Hz

fresp = 0.083 Hz · T = 12 s · 4-2-6

The 4-2-6 breathing cycle targets the baroreflex resonance frequency (~0.1 Hz). Each exhale auto-mints a L.O.V.E. care token. The breathing phase syncs with the K₄ tetrahedron visualization and the starfield particle system.

4s in · 2s hold · 6s out · HRV coherence target · Vagal tone entrainment
Built into PHOS →
4
Demos
863
Hz Larmor
K₄
Graph
SHA-256
Crypto
4-2-6
Breath