V07·Act II · Space centrepiece

The damping race, and why three survives

Twelve points drift in a scattered cloud. Two mode-clouds dim and die, the third brightens, and the points lock into an exact icosahedron.

The relation on screen

δ_{k} (n) \propto (1 - λ_{k})^{n}, dim = dim ker (L - λ_{m i n})

Every mode decays geometrically at its own eigenvalue, so the surviving subspace is the slowest one, and its multiplicity is three.

Symbols

$L$: repair Laplacian on the port space
$λ_{k}$: eigenvalue of mode k, its damping per step
$δ_{k}$: amplitude remaining in mode k

preparing the damping race

what to look for · Three dimensions arrive as the multiplicity of the slowest mode. Watch the 3 curve stay above the others by a widening margin.

Consensus to physics

01repair is the smoothing operator T = I - L/60
02each A5 block is damped at its own rate: 1, 0.953934, 0.900000, 0.879399
03iterate and every non-constant block dies except the slowest
04the surviving block is three-dimensional, so distinguishable directions number exactly three

In standard physics

The derivation of three spatial dimensions

Repair is a low-pass filter on symmetry blocks and the survivor is the standard 3-dimensional representation, the one under which the icosahedron sits in ordinary Euclidean space. Nothing here was tuned to make three come out on top; it follows from the symmetry of the carrier.

boundary · Not a spectral-dimension measurement and not a fit. At finite n the kernel rank is 11; three appears only in the normalised limit.

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