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Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
112 lines
3.3 KiB
Markdown
112 lines
3.3 KiB
Markdown
# CRT Torus Embedding: Iteration Regime
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Open Direction #1 — defining and analyzing the re-embedding cascade.
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---
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## 1. Problem
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F is defined from A ⊂ ℤ into R = ℤ/Mℤ. For the k-torus, F(A) lives in a
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different space than A. To iterate, we need:
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1. An **extension** of F to the integer lift of any finite set
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2. A **regeneration rule** for parameters (L₁,…,Lₖ, S) at each step
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3. A **stability condition** that determines when the cascade terminates
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---
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## 2. Domain Extension
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Define a family of maps indexed by moduli:
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$$
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F_{L_1,\dots,L_k,S}(a) = \text{CRT-1}(a \bmod L_1,\; S-a \bmod L_2,\; \dots,\; S-a \bmod L_k)
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$$
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for any integer a (or any residue a ∈ ℤ/Mℤ lifted to ℤ). This extends F from
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A ⊂ ℤ to all of ℤ/Mℤ via the same congruence rule.
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**Iteration step n:**
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$$
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A_{n+1} = \{\, F_{L_1^{(n)},\dots,L_k^{(n)},S^{(n)}}(a) \mid a \in \text{lift}(A_n) \,\}
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$$
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where $\text{lift}(A_n)$ maps the current set to ℤ (the CRT integer lift).
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---
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## 3. Regeneration Rule
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The simplest deterministic rule: a **geometric modulus cascade**.
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Fix initial moduli (L₁⁽⁰⁾, L₂⁽⁰⁾) and growth factors (α, β) ≥ 1:
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$$
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L_1^{(n)} = \lfloor \alpha^n \cdot L_1^{(0)} \rfloor,
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\qquad
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L_2^{(n)} = \lfloor \beta^n \cdot L_2^{(0)} \rfloor
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$$
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and S fixed or adapted:
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- **Fixed S**: the involution center remains constant across steps. The
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reflection constraint S−a may not hold in Aₙ for n ≥ 1 — this is fine,
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the constraint only needs to hold in A₀.
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- **Adaptive S**: at step n, choose Sₙ = max(Aₙ) + min(Aₙ) to keep Aₙ
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reflection-closed.
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### Regime types
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| Growth | Behavior | Use case |
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|--------|----------|----------|
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| α > 1, β > 1 | **Expanding cascade** — torus grows, finer resolution | Multi-scale embedding |
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| α = β = 1 | **Fixed torus** — F² = id on ℤ/Mℤ, sequence stabilizes at A₁ | Single-step transformation |
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| α, β alternating | **Oscillating cascade** — cycles between resolutions | Searching for Sidon creation |
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---
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## 4. Stability Condition
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A cascade stabilizes at step n if:
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$$
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F_{L_1^{(n)},\dots,L_k^{(n)},S^{(n)}}(A_n) = A_n \quad\text{(as sets of integers)}
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$$
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Sufficient condition for stability:
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If the moduli at step n+1 are the same as step n and Aₙ is F-invariant
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(i.e., Aₙ is a union of F-orbits), then F² = id on the torus forces
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A_{n+2} = A_n — a 2-cycle.
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**Terminal state:** A cascade converges to a fixed point when:
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1. Aₙ is closed under S-reflection (the original constraint), AND
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2. F(Aₙ) = Aₙ (set invariance under F)
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This is equivalent to: every element of Aₙ is either a fixed point of F
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or paired with its F-image within Aₙ.
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---
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## 5. Example: Expanding Cascade with k = 1
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For a single-modulus system (k = 1), F reduces to the identity. The
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cascade does nothing — trivial. The interesting case starts at k = 2.
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---
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## 6. Open Questions
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1. **Convergence rate** — for α > 1, does the cascade reach a terminal
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state in finite steps, or does the expanding torus prevent stabilization?
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2. **Optimal growth** — what α, β minimize the number of steps needed
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to achieve a target property P in Aₙ?
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3. **S-adaptation** — does adaptive S always outperform fixed S for
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reaching Sidon/B_h/Golomb properties?
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4. **Braid connection** — does the expanding cascade correspond to
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iterating braid crossings (adding one crossing per step)?
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