# CRT Torus Embedding: Iteration Regime

Open Direction #1 — defining and analyzing the re-embedding cascade.

---

## 1. Problem

F is defined from A ⊂ ℤ into R = ℤ/Mℤ. For the k-torus, F(A) lives in a
different space than A. To iterate, we need:

1. An **extension** of F to the integer lift of any finite set
2. A **regeneration rule** for parameters (L₁,…,Lₖ, S) at each step
3. A **stability condition** that determines when the cascade terminates

---

## 2. Domain Extension

Define a family of maps indexed by moduli:

$$
F_{L_1,\dots,L_k,S}(a) = \text{CRT-1}(a \bmod L_1,\; S-a \bmod L_2,\; \dots,\; S-a \bmod L_k)
$$

for any integer a (or any residue a ∈ ℤ/Mℤ lifted to ℤ). This extends F from
A ⊂ ℤ to all of ℤ/Mℤ via the same congruence rule.

**Iteration step n:**

$$
A_{n+1} = \{\, F_{L_1^{(n)},\dots,L_k^{(n)},S^{(n)}}(a) \mid a \in \text{lift}(A_n) \,\}
$$

where $\text{lift}(A_n)$ maps the current set to ℤ (the CRT integer lift).

---

## 3. Regeneration Rule

The simplest deterministic rule: a **geometric modulus cascade**.

Fix initial moduli (L₁⁽⁰⁾, L₂⁽⁰⁾) and growth factors (α, β) ≥ 1:

$$
L_1^{(n)} = \lfloor \alpha^n \cdot L_1^{(0)} \rfloor,
\qquad
L_2^{(n)} = \lfloor \beta^n \cdot L_2^{(0)} \rfloor
$$

and S fixed or adapted:

- **Fixed S**: the involution center remains constant across steps. The
  reflection constraint S−a may not hold in Aₙ for n ≥ 1 — this is fine,
  the constraint only needs to hold in A₀.
- **Adaptive S**: at step n, choose Sₙ = max(Aₙ) + min(Aₙ) to keep Aₙ
  reflection-closed.

### Regime types

| Growth | Behavior | Use case |
|--------|----------|----------|
| α > 1, β > 1 | **Expanding cascade** — torus grows, finer resolution | Multi-scale embedding |
| α = β = 1 | **Fixed torus** — F² = id on ℤ/Mℤ, sequence stabilizes at A₁ | Single-step transformation |
| α, β alternating | **Oscillating cascade** — cycles between resolutions | Searching for Sidon creation |

---

## 4. Stability Condition

A cascade stabilizes at step n if:

$$
F_{L_1^{(n)},\dots,L_k^{(n)},S^{(n)}}(A_n) = A_n \quad\text{(as sets of integers)}
$$

Sufficient condition for stability:

If the moduli at step n+1 are the same as step n and Aₙ is F-invariant
(i.e., Aₙ is a union of F-orbits), then F² = id on the torus forces
A_{n+2} = A_n — a 2-cycle.

**Terminal state:** A cascade converges to a fixed point when:

1. Aₙ is closed under S-reflection (the original constraint), AND
2. F(Aₙ) = Aₙ (set invariance under F)

This is equivalent to: every element of Aₙ is either a fixed point of F
or paired with its F-image within Aₙ.

---

## 5. Example: Expanding Cascade with k = 1

For a single-modulus system (k = 1), F reduces to the identity. The
cascade does nothing — trivial. The interesting case starts at k = 2.

---

## 6. Open Questions

1. **Convergence rate** — for α > 1, does the cascade reach a terminal
   state in finite steps, or does the expanding torus prevent stabilization?

2. **Optimal growth** — what α, β minimize the number of steps needed
   to achieve a target property P in Aₙ?

3. **S-adaptation** — does adaptive S always outperform fixed S for
   reaching Sidon/B_h/Golomb properties?

4. **Braid connection** — does the expanding cascade correspond to
   iterating braid crossings (adding one crossing per step)?