The 420 Lattice

108 = D² × K³

The lattice has exactly 108 rings with λ=420. Partition: 32 all-5-axiom + 76 partial = D&sup5; + D²·f(E).

Diameter = 10 = Decality

Any ring reachable from any other in at most 10 steps. The vocabulary of 10 terms MATCHES the geometry. Unique among all lattice levels.

Gap = 0.016 (universal)

ALL 108 rings share spectral gap 4·sin²(π/49) = 0.016. The depth channel b²=49 controls the gap universally.

Center = 420420

Highest betweenness: N = 420,420 = heartbeat × b×L×13. The heartbeat written twice. Pattern: center = λ × boundary primes.

Triangle-Free

Zero triangles. Every short cycle is a 4-cycle (girth = D²). 538 squares. The lattice has square geometry, not triangular.

Width = 21 = K×b = DNA

Widest rank has 21 rings at the waist (rank 4). Shape: diamond, peak at middle. 21 = codon ring Z/21.

λ	rings	edges	diameter	height	max width	center factor
6	16	24	6	5	4	b=7
12	84	210	9	8	16	—
60	184	530	11	9	35	b·L=77
420	108	282	10	8	21=K×b	b·L·13=1001
5460	60	133	9	8	12	b·L·13=1001

λ=420 is the unique level where diameter = Decality count (D×E=10). Triangle-free at ALL levels. Diameter not monotone — peaks at λ=60.

Traditional vs Axiom

Before: Lattice theory studies abstract posets with generic properties.

After: 108 concrete rings share λ=420 = heartbeat. Every metric is axiom-named.

Before: Divisibility lattices have no preferred center or vocabulary size.

After: Center = 420420 (heartbeat²). Diameter = 10 = vocabulary size. Unique.

Before: Spectral gaps depend on the specific ring.

After: Gap = 0.016 in ALL 108 rings. b²=49 controls universally.

Number Theory Threads

The lattice connects to deep number theory. Class numbers along the D-chain follow the axiom: h(−11)=1, h(−23)=3, h(−47)=5, h(−191)=13. Partition smooth zone breaks at p(13). Ramanujan’s tau classifies by mod 23 — the first excluded Cunningham prime.

D-chain class numbers → Partitions & the gate → Modular forms →

The Lattice at a Glance

Ring Inspector