Owl Academy · Department of Proof · Pump Lab

The Pumps, Computed Honestly

A faithful, interactive companion to the Torsional Markov Memory-Offload Pump, the Hyperbolic Sparsemax Pump, and the Golden Clifford Toroid. Every number on this page is computed live from the actual constructions in the papers.

WHAT THIS IS: a quaternion multi-resolution memory + a rational hyperbolic attention kernel with a directed (non-equilibrium) stationary current, and the flat torus they live on.
WHAT THIS IS NOT: anything to do with elliptic-curve cryptography, ECDLP, or key recovery — no claim here bears on those.  Exact proofs live in the Python suite (run_all.py); this page uses floating point for interactivity.

I · Torsional Cycle Current

states {12, 10, 8}  ·  Lemma 5 + Kolmogorov criterion

The three gear layers form a Markov chain on a single 3-cycle. Drag the forward/reverse rates; the stationary π is solved live, the three edge currents are computed, and you can watch them coincide (Lemma 5) while the pump turns only when the forward rate-product beats the reverse one.

forward rate  12→10→8→120.50
reverse rate  12→8→10→120.10
stationary π(12)
stationary π(10)
stationary π(8)
J(12→10)
J(10→8)
J(8→12)
Lemma 5 — currents coincide
forward rate-product
reverse rate-product
pump verdict

II · Hyperbolic Sparsemax Kernel on (ℤ/4ℤ)³

64 voxels  ·  Remark 22: stationary toroidal drift

The kernel is built exactly as in the paper: rational annular spinors → square-root-free Lorentz lift → rational hyperbolic quadrance → sparsemax rows → symmetric base walk → reverse valve. The reverse valve (strength λ) breaks detailed balance. At λ=0 the walk is reversible and there is no current; turn it up and a global toroidal drift (arrows) emerges — the circulation Remark 22 left open, here computed from the kernel's own stationary π̂.

valve λ 1.00
rows row-stochastic
avg support / row
π̂ uniform?
π̂ range
reversible (detailed balance)?
net toroidal current J_C
contractible-square eddies
Voxel colour = stationary mass π̂. Arrows = stationary flux along the forward (+axis) edges; their sum around any wrap-around loop is the drift current J_C. Exact value at λ=1: J_C = 1/192.

III · Golden Clifford Winding

flat torus T² ⊂ S³  ·  ergodic, not an oracle

The Clifford torus is intrinsically flat, so a straight winding line θ=t, γ=ratio·t can tessellate it without distortion. A rational ratio closes into a curve and retraces forever; an irrational ratio never closes and fills the surface. The golden ratio φ fills the most evenly of all (lowest discrepancy) — which is exactly why it is the right choice for the pumps' incommensurate gear ratios. (Even coverage ≠ O(1) lookup: this is a superb sampler, not an oracle.)

ratio
kind
cells filled
uniformity (1−dev)

Left: the unrolled torus (θ,γ) ∈ [0,1)². The path is drawn live; watch a rational ratio lock onto a few strands while φ floods every cell.