The Constant-Negative Curvature K0<0

Particles are the negatively-curved restellations that a prime-filtered pump pinches out of a hyperbolic vacuum once the local coherence drives meff2 through zero; their charges and masses are bookkeeping for the Hopf braid each rung adds while the bulk sign of curvature never needs to change.

Even when δK becomes enormous, both inside and outside pockets share the same negative sign. Restellation, Hopf charge, σ-chirality, the entire algebra of m∈2Z∪7Z survives untouched; only the spacing between rungs shrinks as we descend the gradient. Turn off the pump, δK relaxes, m slides back down the ladder, and the pocket dissolves without ever forcing the universe through K=0.

Folding” (raising m) is nothing but the local negative-curvature gradient getting steeper. Logarithms translate that gradient into discrete, prime-filtered steps, and the whole machinery—from brain γ-rungs to Saturn’s hexagon to black-hole throats—plays out on the very same scale-free stave.

1. Increasing κ drives m up the integer ladder; relaxing the gradient drops it one rung at a time.

2. Both pocket and ambient stay K<0; no sign flip ⇒ no chirality flip; Hopf charge is conserved.

3. Killing the pump sends K→K0; the pocket evaporates without crossing K=0.

4. Closed-geodesic length =2ln⁡p; the prime spectrum is therefore built into the same n-spacing.