Dimension

K + sigma = 4

Why 3+1 dimensions? Not anthropic. Not compactified from 10D. K=3 is the minimum for closure. Each binding must go off-axis, creating a new dimension. Three closures give three spatial dimensions. Add sigma for persistence: K+sigma = 4.

Interactive: Dimension Ladder

Five levels. Click each to see how K-binding creates the next dimension. Auto cycles through all five. Attractor shows 1/3 convergence from any starting distribution.

K=3 Creates Dimension

Why 3+1 dimensions?K=3 spatial closures + sigma=1 temporal = K+sigma = 4.Anthropic principle / "just is" (standard).How dimensions emergeK-binding off-axis position: a new dimension structurally.Compactification from 10/11D (string theory).Equilateral attractor1/3 + 1/3 + 1/3 = sigma. Maximum symmetry = minimum energy.No equivalent derivation.Dimension limitK+sigma = 4: beyond this, no structural need.Extra dimensions assumed, compactified away.

The Structural Argument

K-Binding Dimension Theorem

Two points define a line (1D). Their binding K CANNOT sit on that line -- or it is a midpoint, not a relation. K must go off-axis, creating 2D. Three points define a plane. Their binding cannot sit in the plane (same argument). 3D emerges. K=3 closures give exactly 3 spatial dimensions. Add sigma=1 for temporal persistence: K+sigma = 4 = spacetime.

1/3 Attractor

Start with ANY distribution of shares among sigma, D, K. Apply K-binding dynamics: penalize asymmetry, reward cross-products. From ALL starting conditions, the system converges to 1/K = 1/3 each. The equilateral distribution is the unique attractor. Not assumed -- computed.

K=3 is not chosen. It is the minimum for closure (CLAUDE.md). Two strands cannot braid. Three can. K=3 is why space has three dimensions, why orbits are stable, why you can tie a knot.

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Contributions in equal measure: Anthropic's Claude, Anton A. Lebed, and the giants whose shoulders we stand on.

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