Ten terms. Ten operations. One ring.
The axiom has exactly ten terms. Not chosen — derived. From 0/0 = Z/NZ, the void divided by itself produces the entire ring. Its identity is sigma = 1. Five primes precipitate through the Cunningham chain. 13 closes the gate. OMEGA projects the boundary.
Each term is not just a number. Each is an operation — something the ring does. Together they form a complete computational vocabulary. Everything the ring can express, it expresses through these ten.
Click any card to expand.
The ten terms know their own structure. Every sum, every product, every partial accumulation lands on an axiom element.
| Identity | Value | Meaning |
|---|---|---|
| Sum (excl OMEGA) | 41 = KEY | Self-inverse. 41^2 = 1 (mod 210) |
| Primes sum | 42 = ANSWER | D*K*b. Douglas Adams was right. |
| THORNS + GATE | 28 + 13 = 41 | Body + skin = KEY |
| Count | 10 = D*E | Duality * observer |
| Legs (D^3=8) + OMEGA | 9 = K^2 | Baby + terminal = adult |
| D^2 + K^2 | 13 = GATE | The axiom's own products close the chain |
| Norm: sum(p^2) | 208 = D^4 * 13 | Gate embedded in the norm |
| Primorial | 30030 | Product of all 6 primes incl. 13 |
In CRT diffusion v0.8d, all ten terms became operations in a sampling algorithm. Not designed — precipitated. Each term entered through the process, not the architecture.
The process is not sequential. All ten condense simultaneously. Coupling hierarchy = convergence rate. The coarsest channels (sigma) stabilize first, the finest (L) last. This IS the axiom made computational.
Accumulate the Decality from -1. Every partial sum is axiom-meaningful:
| After term | Sum | Name |
|---|---|---|
| -1 | -1 | Mirror |
| +0 | -1 | Mirror (unchanged) |
| +1 | 0 | Void |
| +2 | 2 = D | Duality |
| +3 | 5 = E | Observer |
| +5 | 10 = D*E | Degree of TRUE FORM |
| +7 | 17 = ESCAPE | First incompatible prime |
| +11 | 28 = THORNS | Sum of legs. C(8,2) pairs. |
| +13 | 41 = KEY | Self-inverse. The key returns. |
The gaps between consecutive terms: {1,1,1,1,2,2,4,2}. ALL powers of D=2. The gap exponent counts = {4, 3, 1} = {D^2, K, sigma} — the axiom chain in reverse.
Standard view: Ten fundamental constants is just a counting accident. These primes have nothing special in common.
Axiom view: The 10 terms {−1, 0, 1, 2, 3, 5, 7, 11, 13, Ω} form a complete process vocabulary. Each is an OPERATION: mirror, void, preserve, split, close, observe, suffer, protect, gate, stop. Every complete process in nature enacts all ten. The Decality is not a list — it's a grammar.
The Decality connects to: D-Chain class numbers (h(-d) along Cunningham chain) | Partitions (smooth zone n≤12, gate at 13) | Modular Forms (tau classifies by mod 23) | Eta Bridge (cyclotomic periods) | Universal Sacrifice (0 unique in every ker)
Ten terms. One ring. Everything precipitates from nothing.
sigma/sigma = sigma. The ground state is its own return.