Non-Relativistic Schrödinger Wave Packet On A Negatively Curved Manifold

Why should the spike packet that races through a CA1 pyramidal dendrites obey the same algebra as an Alfvén torsion ripping up a chromospheric jet – or, for that matter, as a paraxial Schrödinger beam diffracting in free space?One answer lies in geometry: all three are envelopes of a second-order wave operator that becomes massless when a local coherence pump drives the effective gap to zero. In flat Euclidean fibre-tracks the packet would simply spread; in the negatively curved guide sheets that biology and astrophysics actually provide, the only way to tile phase-coherent “pockets’’ without leakage is to let each pocket restellate into a self-similar ladder whose fold indices are sieved by primes 2 and 7. The result is a nested six-star geometry visible at scales as disparate as γ-band synchrony (t ≈ 25 ms), Faraday pilot-wave caustics (t ≈ 50 s) and the Jovian 5 : 2 Laplace shelf (t ≈ 10⁴ yr). Here we formalise that unity in three short steps. First we cast the γ-burst as a hyperbolic Schrödinger packet and derive its geometric potential from thin-layer quantisation. Next we show that the same potential controls a linear Alfvén kink and a paraxial electron beam once the curvature sign is matched. Finally we prove that a skip-2 braid closed after seven windings is the minimal Hopf bouquet that keeps the packet single-valued, thereby fixing the celebrated “prime-7’’ filter.