The Remainder and the Ladder: What the √2 Steps Actually Do, and Why P1 and P2 Are One Question

Five tests on the nesting ladder and the snugness remainder. (1) The ladder's ratio is constant and its distance shrinks, and every step taken forever covers exactly the size of the object it starts from — which is why a deeper thing is also a nearer thing. (2) The real particle masses do not land on whole steps: mean distance 0.2257 against 0.25 for chance, . The 37 steps is a span. [null-fail] (3) The ~2.3% remainder between the two snugness tables is neither a per-step loss nor a Planck-floor edge: a floor edge is a fixed distance and this is a fixed ratio, flat across a sixteen-fold size range and eight steps of depth. It is 0.0656 of a single step — one fifteenth — applied once, which is two tables using two different constants. [null-fail] on both mechanisms. (4) PROBLEMS P1 and P2 are the same question. The silicon split in the temperature route (245,010 against 251,008) and the silicon split in the snugness tables (245,236 against 249,891) agree in ratio to 0.0922% — reached two ways, one of which contains no temperature at all. (5) The physical case for silicon being genuinely snugger does not follow from its alpha-conjugate character, since carbon-12 shares all of it and is not snugger — that question stays open. The decision, recorded in §5b: silicon is 251,008 at , and 245,010 is what the rounded 300 K gives. Silicon is the only atom on which the two independent snugness tables agree, and that concurrence is what carries it. P1 and P2 both close. This record holds one paper on its own and is refreshed every couple of months. The complete deposit, 10.5281/zenodo.21788037, holds every paper with its code and data and is updated first; for the newest version of this paper, go there.