π–φ Equilibrium · Unified Seed Geometry

πr² =φ

Unified Seed Geometry · Final Edition · 2026

The π–φ Equilibrium

A single algebraic seed — πr²=φ — generates a complete family of exact geometric structures: equilibrium radius, golden spiral, bridge constant d*, π-star lattice, φ-torus, catenoid throat, hyperbolic inversion, and a 294-layer scale hierarchy from Planck to cosmos.

§2 Seed

§3 Bridge d*

§4 Spiral

§5 π-Star

§6 Torus

§6 Catenoid

§7 Inversion

§8 Ruler

§9 Hierarchy

§10 Intervals

§12 Prime-Time

By Gregory J. DeCarlo · Interactive Edition

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π–φ Equilibrium · Navigation

Modes

1Hierarchy (§9 · Planck→Universe elevator) 2Seed (§2 · πr² = φ) 3Spiral (§4 · golden logarithmic) 4π-Star (§5 · C₄×C₅ lattice) 5Torus (§6 · R/a = φ) 6Catenoid (§6 · equilibrium throat) 7Inversion (§7 · Euclidean↔Hyperbolic) 8Ruler (§8 · universal midpoint) 9Bridge d* (§3 · log_φ π) 0Intervals (§10 · F₁/F₂/F₃) −Prime-Time (§12 · Tier III)

Hierarchy Navigation

↑Up 1 layer ↓Down 1 layer PgUpUp 20 layers (one π-star period) PgDnDown 20 layers PJump to Planck (n=0) EJump to Equilibrium midpoint (n=147) UJump to Universe (n=294) wheelScroll through layers · fine grain

Global

iInfo — details for current mode or floor tTools — open mode-specific panel cChapters — paper section navigator mMute / unmute elevator audio hHome / reset view ?Toggle this help EscClose overlays

URLs are shareable — the address bar encodes the current mode and floor (e.g. #mode=hier&n=147). Copy the link to send any exact view. Based on Gregory J. DeCarlo's π–φ Equilibrium (2026).

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Modes

Hierarchy Navigation

Global

Equilibrium Ruler

Paper Sections

Equilibrium Ruler ✕

Paper Sections ✕

Equilibrium Ruler

Paper Sections