state Ω
# ════════════════════════════════════════════════════════════════════════
# HDGL — REVERSIBLE SUBSTRATE
# One ring. Two inverse operators. One invariant.
# Expansion and return are directions of the same machine.
# ════════════════════════════════════════════════════════════════════════
glyph AXIOM
origin : X = 0
translate : x + 1
gather : x - 1
invert : 1 / x
T : translate ∘ invert # T(x)=1+1/x
S : invert ∘ gather # S(x)=1/(x-1)
invariant :
S = T⁻¹
T∘S = Identity
S∘T = Identity
x = T(x)
⇒ x² = x + 1
x = S(x)
⇒ x² − x − 1 = 0
end
glyph RING
element : Ω=(a,b) ≡ aφ+b
add : (a,b)+(c,d)=(a+c,b+d)
mul : (a,b)(c,d)
= (ac+ad+bc, ac+bd)
conj : (a,b)↦(-a,a+b)
norm : N(a,b)=−a²+ab+b²
φ : (1,0)
ψ : (-1,1)
1 : (0,1)
0 : (0,0)
invariant :
ψ = 1−φ
Ω and Ψ are coordinates
of the same substrate.
end
glyph FLOW
forward :
Ω ← T(Ω)
reverse :
Ω ← S(Ω)
invariant :
every forward step
possesses one exact reverse step.
no information is created.
no information is destroyed.
end
glyph LADDER
forward :
Ω·φ=(a+b,a)
reverse :
Ω/φ=(b,a−b)
invariant :
φ is carried.
forward and reverse
require only integer addition.
end
glyph NORM
N(Ω)=−a²+ab+b²
N(xy)=N(x)N(y)
Trinity :
+1 unit (even parity)
0 origin
−1 unit (odd parity)
invariant :
dynamics change.
norm does not.
end
glyph ORBIT
state :
...
← Ω₋₂
← Ω₋₁
Ω₀
→ Ω₁
→ Ω₂
...
primitive :
orbit admits no
shorter decomposition.
composite :
orbit decomposes
into primitive segments.
end
glyph EULER
e^(iπ)=(0,-1)
√5=(2,-1)
invariant :
phase is a projection
of the exact lattice.
end
glyph PRIME
conjecture :
prime
=
primitive reversible orbit
composite
=
decomposable orbit
split
=
orbit possesses
local fixed point
inert
=
orbit never closes
locally
end
glyph HDGL
substrate : Z[φ]
operators :
T
S=T⁻¹
invariant :
N
ladder :
φ
dynamics :
reversible
conjecture :
arithmetic is orbit topology.
primes are not numbers.
primes are irreducible reversible orbits.
end
