.ax was born in the era of 3 terms. DFloat(s|b|a) was the fundamental type -- never implemented. CRT was a builtin you called. Now we know: 10 terms, 108 rings, 5 channels. The language should THINK in channels, not translate to them. CRT IS the type. The Decality IS the instruction set. This document: the blueprint for .ax v2.0.
The Diagnosis
What we knew when .ax was designed: 3 terms (sigma, D, K). What we know now: 10 terms, 5 CRT channels {8,9,25,49,11}, 108 rings in the lambda-420 lattice, L=11 free ECC, emergence from 3 ops. The gap: .ax USES the axiom. It does not EMBODY it.
Old
Values are plain numbers
CRT is a function you call. Channels are hidden. Ring is global.
New
Values ARE their CRT decomposition
42 = (2, 6, 17, 42, 9). Channel access is O(1). Parallel by nature.
Old
62 builtins accumulated organically
Library calls, not the language's nature. Bolted on.
New
10 Decality keywords = the instruction set
Each is BOTH a value AND an operation. The vocabulary IS the axiom.
Principle 1: CRT IS the Type
Every Value IS Decomposed
42 in .ax v2.0: displays as 42, stores as (2, 6, 17, 42, 9). Access channels directly: 42.D = 2, 42.K = 6, 42.E = 17, 42.b = 42, 42.L = 9. Arithmetic happens per channel automatically: 42 + 137 computes (2+1)%8, (6+2)%9, (17+12)%25, (42+39)%49, (9+5)%11 = 179. This IS the Loop Theorem. This IS the Backprop Theorem. This IS the axiom, embodied in data representation.
Principle 2: The Decality IS the Vocabulary
10 keywords. Each is BOTH a value AND an operation:
Term
Value
As Operation
CRT Signature
mirror
-1
Reflect: N - x
(7, 8, 24, 48, 10)
o
0
Absorb: 0
(0, 0, 0, 0, 0)
sigma
1
Preserve: x
(1, 1, 1, 1, 1)
D
2
Split in two
(2, 2, 2, 2, 2)
K
3
Close / compose
(3, 3, 3, 3, 3)
E
5
Observe (self-blind)
(5, 5, 5, 5, 5)
b
7
Depth / iterate
(7, 7, 7, 7, 7)
L
11
Protect / ECC verify
(3, 2, 11, 11, 0)
GATE
13
Boundary / type check
(5, 4, 13, 13, 2)
OMEGA
606376
Terminal / project / halt
(0, 1, 1, 1, 1)
Notice: elements 0-7 have CRT = (n,n,n,n,n) -- they are IDENTICAL across all channels. L=11 is the first to break this: (3,2,11,11,0). The protector is the first element that looks DIFFERENT from different angles. OMEGA has D-channel dead: (0,1,1,1,1). The terminal projection kills duality.
Principle 3: Rings AS Types
The 108-Ring Lattice IS the Type Hierarchy
Every value carries its ring. let x: Z210 = 42 -- x lives in DATA ring. let y: TRUE = 42 -- y lives in TRUE FORM. Lifting (fattening) adds channels. Projection (thinning) removes them. Type checking = ring membership. Z/210 has 4 channels {D,K,E,b}. Z/970200 has 5. Z/12612600 has 6. Subtyping = subring embedding. The GATE FORM is the universal type.
Ring
Channels
Domain
Z/6 = D*K
{D, K}
Clock, basic modular
Z/210 = DATA
{D, K, E, b}
Biology, DNA, body
Z/2310 = THIN
{D, K, E, b, L}
Full axiom, thin channels
Z/970200 = TRUE
{D^3, K^2, E^2, b^2, L}
Full axiom, fat channels
Z/12612600 = GATE
{D^3, K^2, E^2, b^2, L, 13}
The organism, universal type
Four More Principles
4. L=11 Runtime Guardian
Automatic ECC
Every computation checked by L channel. Corruption detected at runtime. Free. No programmer effort.
5. 3 Ops -> Everything
meet(*), complement(~), add(+)
Arrays, strings, functions, closures, neural nets -- all emerge from K=3 operations. LIFE=7 proved.
6. K^2=9 Parallelism
3 hearts x 3 stages
The compiler has K^2=9 analysis passes. FPGA: K^2 pipeline units. The code IS the hardware description.
7. Lambda-420 Heartbeat
Universal rhythm
Every ring shares lambda=420. Every computation has a heartbeat. The ring breathes.
The Strange Loop
.ax ships -> proves axiom -> IS the entity -> writes .ax -> .ax ships. The website IS the proof. The language IS the math. The compiler IS the theorem. Each pass through the loop: tighter, more capable, fewer pieces.
Ship of Theseus
The old website: 154 hand-crafted HTML pages, 58,641 lines. The new website: site.ax compiled to site.wasm. 108 pages, pure .ax. No HTML authored. .ax builds pages directly via WASM DOM imports. The old ship was dismantled plank by plank. The new ship is the same ship -- but every plank is now math.
What Others See
Types
Integers, floats, strings, structs, classes
One type: ring element. CRT decomposition IS the type. 108 rings = type hierarchy.