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BUILD · CONSTRUCTION

How does structure assemble around an invariant?

Initialize → contract → preserve centroid → reduce dispersion → certify progression.

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Role in the Mathematical City

BUILD is the City’s kinetic contraction lab. It studies a finite array of agents moving fractionally toward their shared centroid and exposes the exact contraction laws generated by that update.

Public role: Construction / Assembly

Application state: BUILT · LIVE INTERFACE REGISTERED

Mini AI Labs — Unified Interface Portal

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ENGAGE BUILD

Open BUILD full screen

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BUILD · Kinetic Contraction Lab

BUILD · Kinetic Contraction Lab

Canonical update

$$ x_i^{(k+1)}=(1-\alpha)x_i^{(k)}+\alpha\mu $$

with

$$ \mu=\frac1N\sum_{i=1}^N x_i. $$

The centroid is invariant:

$$ \mu_{k+1}=\mu_k=\mu_0. $$

The exact state trajectory is

$$ x_i^{(k)}=\mu+(1-\alpha)^k\left(x_i^{(0)}-\mu\right). $$

For total squared dispersion

$$ D_k=\sum_i\|x_i^{(k)}-\mu\|^2, $$

BUILD satisfies

$$ D_k=(1-\alpha)^{2k}D_0. $$