SilverSight/docs/research/iteration_regime.md

CRT Torus Embedding: Iteration Regime

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

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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

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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).

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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

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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.

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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.

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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)?