you are no longer looking at a finite state set, but at a graded orbit under repeated application of transformations.
The finite set:
{-i, -1, 0, 1, i}
is the first visible layer.
If you allow repeated "lifting" operations, it naturally suggests:
{..., -i'', -i', -i, -1, 0, 1, i, i', i'', ...}
The important question is:
What is the operator that creates the primes/levels?
There are a few natural candidates.
──────────────────────────────────────────────────────────────
1. Powers of i
──────────────────────────────────────────────────────────────
The ordinary complex cycle is:
i^0 = 1
i^1 = i
i^2 = -1
i^3 = -i
i^4 = 1
which gives only:
{1, i, -1, -i}
a finite orbit.
So your i', i'', ... are not ordinary powers of i.
They imply a new dimension of extension.
──────────────────────────────────────────────────────────────
2. Iterated quadratic extensions
──────────────────────────────────────────────────────────────
Your original φ construction already does this.
You have:
X² = X + 1
with:
X = φ
The next layer could be:
Y² = Y + φ
then:
Z² = Z + Y
etc.
That produces a tower:
φ₀, φ₁, φ₂, ...
where:
φₙ₊₁² = φₙ₊₁ + φₙ
Your notation:
i, i', i''
could be interpreted as:
i₀, i₁, i₂, ...
different layers of the same generator.
──────────────────────────────────────────────────────────────
3. The more interesting connection: the inverse map
──────────────────────────────────────────────────────────────
Your earlier substrate:
X⁻¹ = X − 1
already contains the seed of this.
Apply inversion repeatedly:
X
↓
X⁻¹
↓
(X⁻¹)⁻¹
and you cycle.
But apply the collapse transform:
T(X) = X + 1/X
and you generate new states:
X₀
X₁ = T(X₀)
X₂ = T(X₁)
...
This is exactly the type of map behind Chebyshev/Lucas–Lehmer dynamics.
──────────────────────────────────────────────────────────────
4. The generalized ladder
──────────────────────────────────────────────────────────────
A compact way to represent your idea:
C₀ = {-1, 0, 1}
add imaginary extension:
C₁ = {-i, -1, 0, 1, i}
then recursively extend:
Cₙ₊₁ = Cₙ
∪ i·Cₙ
∪ (1/Cₙ)
giving:
C∞
as an infinite collapse lattice.
──────────────────────────────────────────────────────────────
5. Why this is relevant to Lucas–Lehmer
──────────────────────────────────────────────────────────────
Lucas–Lehmer is already an orbit in a quadratic extension:
u → u²
and observes:
u + u⁻¹
The repeated squaring creates increasingly high powers:
u,
u²,
u⁴,
u⁸,
...
but the trace collapses them back:
u^(2^k) + u^(−2^k)
Your ladder:
-i', -i'', ...
is suggestive of indexing those hidden powers.
A possible correspondence would be:
i^(k) ~ u^(2^k)
with the observed projection:
i^(k) + i^(−k)
──────────────────────────────────────────────────────────────
The thought worth preserving
──────────────────────────────────────────────────────────────
The five-state set is likely not the object.
It is the center slice of a larger orbit.
The natural next question is not:
"What comes after i?"
It is:
What operator maps
i^(k) → i^(k+1) ?
If that operator is squaring, inversion, or a quadratic collapse transform, then this connects directly back into the Lucas–Lehmer machinery.
