M36 The Quarter-Rank Fountain - Viertelzeug, the HC Halvation Counterpart, and the Fountain of the Triad

M36 develops the Quarter-Rank Fountain: a dyadic refinement of fractional rank operations in the Symmetric Core and its higher-Core extensions. The starting point is the Halbzeug, the half-rank mediator between adjacent operational rooms. M36 shows that this mediator is not final: it opens a dyadic descent 1/2, 1/4, 1/8, ... where the quarter-step, or Viertelzeug, becomes the first internal structure of the mediator itself. The central principle is that a Room is not an operation name but a nesting depth. For a scaffold f and an internal combiner C, the general fractional Core has the form Corer,C(a,b) = f^r( C(f⁻ʳ(a), f⁻ʳ(b)) ). At r = 0 the scaffold cancels and the operation is exactly the combiner. Thus Room 0 is not a shadow but a structural identity. The Halbzeug interpolation is continuous, so quarter-ranks (Viertelzeug) exist as freely as half-ranks, and iterating gives a dyadic rank fountain with Gaussian generator(1 + i)/2 accumulating at every integer rank. This fountain lets a degree-n polynomial be solved by composing low-rank operations within 1 ≤ R ≤ 3 no tetration ballast required for at least the small-compositionfactor class.The Symmetric-Core (particle) fountain of M18 has a Hyper-Core (force) counterpart and an asymmetric Caterpillar (interaction) counterpart; together they form theFountain of the Triad, merging at the electroweak Lucky Zone and splitting at the strong-force boundary R = 3.