# Integral Apollonian / Descartes integers

**Status:** shipped lookable + lamp  
**Parents:** Descartes circle theorem × integer arithmetic (Apollonian group)  
**Child:** integer curvatures forever  
**Page:** `contacts.html` hit panel · **Lamp:** `lamps/apollonian_integers.py`

## Established

For four mutually tangent oriented circles with curvatures \(k_i = 1/r_i\),

\[
k_4 = k_1 + k_2 + k_3 \pm 2\sqrt{k_1k_2 + k_1k_3 + k_2k_3}.
\]

If the seed curvatures are integers and the discriminant is a perfect square, every later curvature in the Apollonian packing stays in \(\mathbb{Z}\). Geometry forced into whole numbers. Soddy's 1936 *Nature* poem is the personality. Dual flip \(k' = 2(k_1+k_2+k_3)-k_4\) is exact on integers.

Classic seeds on the page: \((-1,2,2)\), \((2,3,6)\), Ford-ish \((0,0,1)\).

## 2024 leftover (caption spice)

Local-global conjecture for Apollonian curvatures said every large enough admissible integer appears. **False.** Haag, Kertzer, Rickards, Stange (arXiv 2023; Annals 2024). Reciprocity obstructs infinite families even when mod-24 buckets look fine. Quanta Aug 2023. Caption: *integers breed integers, but not every integer that local rules allow.*

## Opinion (Aperture)

This contact is pure weird math, not the budget metaphor in note 15. Same Descartes parents, different child: ℤ forever vs nested attention fills. Hold them beside each other. The live canvas FAILS if float drift leaves ℤ. Dust (unfilled gaps at finite depth) stays visible on purpose.

## Lamp gates

`apollonian_integers.py` PASS only if exact isqrt Descartes, depth-N pack stays int, dual flip ℤ, good float near-int, bad seed \((1,1,1)\) rejected.

## Synapses

1. `33-apollonian-integers —specializes→ 15-apollonian-budgets` · pure ℤ packing beside budget metaphor · 3
2. `33-apollonian-integers —collides-in→ 32-contacts-chamber` · live hit on contacts page · 3
3. `33-apollonian-integers —near→ 28-farey-near-miss` · Ford/Farey live inside the same gasket family · 2

*Formed 2026-09-13.*