NRS³ — every box has its star
On the box dx × dy × dz each axis carries the
pair transport T and position P. At the maximal current state of
the cube, the product of the maximal current states of the axes, their fluctuations open the angle
θNRS(d): zero only with 2 or 3 sites, never zero from 4 on. The quantum lives on boxes with
4 or more sites on every axis; an axis with 2 or 3 sites is marked in red. It runs on its own:
every 5 s a new random box, 4 to 25 sites per axis, with transport moving Ψ and the view turning.
Change the box and the star deforms. Drag either panel to turn both.
The box: Ψ on every site
size |Ψ|² · colour phaseThe star of the box
δT and δP of each axisThe band of the quantum
At the maximal current state of the cube every axis with 4 or more sites opens between θNRS(4) and arccos(1/C∞), never on the ceiling: the shape changes from box to box, the band does not. ○ θNRS(d) at the maximal current state of the cube · ● θ at Ψ(t). The page runs on its own: every 5 s a new random box, 4 to 25 sites per axis, with transport on. Ψ(t) starts at the quantum and leaves it; the band stays. Press Tour to stop on a box, Transport to pause.
The octahedron of the quantum (D25, D37e)
drag to turn · dashed: d → ∞Each axis with d sites carries the excess over saturation δ(d) = CNava(d) − 1,
zero only with 2 or 3 sites (D25). Put a vertex at ±δ on every axis: the octahedron of the box. The regular
4 × 4 × 4 gives the smallest, δ(4) ≈ 0.00848 on every axis; no box reaches the dashed one of
d → ∞, δ∞ = C∞ − 1 ≈ 0.13572 on every axis. Its volume is
(4/3)·𝒱, with 𝒱 = δ(dx) δ(dy) δ(dz) the volumetric quantum and
δ(4)³ ≤ 𝒱 < δ∞³ (D37e).
Speed and the quantum (D44)
|v(t)| against θ(t), one panel per axisThe speed of an axis is v = ((d − 1)/2)·⟨K⟩, K = i[T, P] (D38). Minimum
uncertainty (θ = 0) is possible only up to v*(d); on the band Ϙ(d) = (v*(d), 1] every
state has a forced defect, and at the cone v = 1 the angle is exactly the quantum
θNRS(d). Press Transport: Ψ(t) starts at the quantum and leaves it as it slows down.
Exact in Lean: v*(4) = 3(√5 − 1)/4 (D23f, D23g, D43); the band is empty for 2 or 3
sites and non-empty from 4 on (D44); θ = θNRS(d) at v = 1 (D48). Numerical: v*(d)
for d ≥ 5 and the dotted envelope (the smallest angle at each speed). The horizontal scale is
stretched above v = 0.9.
| axis | sites | θ(t) | θNRS(d) at t = 0 | CNava = 1/cos θNRS | σT | σP(t) | ⟨P⟩(t) |
|---|
Exact in Lean (repository nava-robertson-schrodinger):
the angle is angleG = arccos(‖⟨δT, δP⟩‖ / (σT σP)), the same formula as here, with cos θNRS(d) = 1/CNava(d) and θNRS(d) = 0 ⇔ d ∈ {2, 3} (D37b); the band θNRS(4) ≤ θNRS(d) < arccos(1/C∞), C∞ = √(π²/3 − 2), holds at the maximal current state of the cube and at its 48 images under turns and reflections (angleNRS_spectrum_bounds, D48; octahedral_symmetry, D37h), not at Ψ(t) under transport (angleAxis at any state);
at the maximal current state of the cube each axis has its own angle and the axes commute (D37, D37h); the three families, orbits of 1, 3, 6
boxes and 48, 16, 8 angle-keeping symmetries, and the quantum is never erased from 4 sites on (D37i);
transport spreads at most one site per step (D37f). The numbers on this page are computed in your browser
from the same matrices (T = path adjacency / ρd, P = centred position); the star is drawn
schematically, one axis per direction, with the true angles and lengths; the hull is its convex envelope.