A three-scale quaternionic delta chain — 720 → 72 → 10 gears (the
Torsional Markov Memory-Offload Pump). The coarse layers are ordered products of deltas
(telescoping, exact); fine detail is recovered by SLERP fractional-power lifts. The logged
delta chain is the replayable archive that re-creates the structure; the mean is the
lossy centroid summary. Geometry summarizes; the deltas reconstruct.
Stream source
Rate–distortion dial — store one scale, reconstruct 720
telescoping err (Prop 2)—
72→10 partition (Prop 3)7/8 ✓
stored / compression—
reconstruction RMSE—
holonomy ‖H‖ → closed—
centroid (mean) w—
Delta chain log — the replayable archive
Telescoping is shown here in floating point (~1e-15);
the Sparsemax source is exact-rational in the reference, so its offload telescopes to
exactly 0 (parity-safe). Storing the outer chain is lossless; coarser scales are compact
but SLERP-approximate the interior — low RMSE for smooth/structured, high for noise. No factoring
speedup is claimed (TQF jump-ahead / harmonic bias measured ≤ 0 elsewhere).