# CRT Reflection Embedding on a k-Torus > Place a reflection-closed finite set onto a discrete torus, with identity > preserved along one axis and an involution encoded across the rest. --- ## 1. What This Is Take a finite set A ⊂ ℤ closed under reflection a ↦ S − a. Pick k pairwise-coprime moduli L₁, …, L_k. The CRT isomorphism $$ \mathbb{Z}/M\mathbb{Z} \;\cong\; \mathbb{Z}/L_1\mathbb{Z} \times \cdots \times \mathbb{Z}/L_k\mathbb{Z} \qquad (M = \prod L_i) $$ is a **k-dimensional discrete torus** — a product of k cyclic groups. We embed A onto this torus with an asymmetrical constraint: $$ \begin{aligned} \text{axis 1:}&\quad a \mapsto a \pmod{L_1} &&\text{(identity — the original element)} \\ \text{axes 2…k:}&\quad a \mapsto S - a \pmod{L_i} &&\text{(reflection — the involution)} \end{aligned} $$ Call this embedding F: A → T, where T = ∏ Z/L_i Z. The 1D set A becomes a **point cloud on a k-torus**. Every point a ∈ A is paired with its dual F(S−a), linked by the involution on axes 2…k. --- ## 2. Three Basic Properties The embedding satisfies three properties — all are immediate from CRT, so we state them and note that every property reduces to the 2-modulus base via the fiber bundle structure (Section 3a). **Injectivity.** If M = ∏L_i > max(A) − min(A), then |F(A)| = |A|. Two distinct elements a, b ∈ A can only collide if M divides a−b, which requires |a−b| ≥ M, impossible under the range bound. CRT uniqueness forces distinct elements to distinct torus points. **Fixed points.** F(a) = a iff 2a ≡ S (mod L_i) for all i = 2,…,k. In practice: the gcd of {2a−S} over A controls whether F is the identity on A. **Gap.** If F(a) ≠ a, then |F(a) − a| ≥ L₁ (using the CRT integer lift of F(a) in [0, M); see notation below). Non-fixed points are displaced by at least the first modulus — the embedding is not arbitrarily close to identity. **Involution.** F is structurally involutive: F(F(a)) = a in the CRT-decomposed coordinates. On the torus, F pairs elements. This is not a defect — it is the central structural fact. ### 2.1 Fiber bundle degeneration: k-torus → 2-torus base All properties above reduce to the **2-modulus base case** via a projection that eliminates hidden assumptions from the higher axes. Define the base 2-torus T₂ = Z/L₁Z × Z/L₂Z with the 2-modulus embedding: $$ F_2(a) = (a \bmod L_1,\; S-a \bmod L_2) $$ Define the projection π_{1,2}: T_k → T₂ that forgets axes 3…k: $$ \pi_{1,2}(x_1, x_2, x_3, \dots, x_k) = (x_1, x_2) $$ **Commutation.** The k-modulus embedding F_k and the 2-modulus embedding F₂ are linked: $$ \pi_{1,2} \circ F_k = F_2 $$ *Proof.* Both sides are defined by the same congruences on axes 1 and 2: F_k preserves a mod L₁ on axis 1 and S−a mod L₂ on axis 2; forgetting the remaining axes leaves exactly F₂. ∎ **Consequence.** Every property of F₂ lifts to F_k: | Property | Proven for F₂ (2-torus) | Lifts to F_k (k-torus) via | |----------|------------------------|---------------------------| | Injectivity under L₁L₂ > range(A) | CRT uniqueness | Holds on T₂, so holds on any fiber | | Gap ≥ L₁ on non-fixed points | F₂(a) ≡ a (mod L₁) | Same congruence on axis 1 | | Fixed-point condition: 2a ≡ S (mod L₂) | S−a ≡ a (mod L₂) | Additional condition on axes 3…k refines, does not change | | F² = id | π_{1,2}(F²) = id on T₂ | Full involution in CRT coordinates | The k-torus is a **fiber bundle over T₂**: each base point (x₁, x₂) has fibers from axes 3…k determined by the same reflection constraint S−a. No hidden assumption about higher axes can affect the base properties because the base is independent and fully reduced to the proven 2-modulus case. This means all claims proven for (L₁, L₂) hold for any (L₁, L₂, …, L_k) without re-proving. The higher axes are **refinements**, not independent degrees of freedom. --- ## 3. Idempotent Sieve Lemma Since F is an involution (F² = id), we can construct a **projection operator** that collapses each F-orbit {a, F(a)} to a single fixed point. ### Algebraic form (Π): Projection onto the invariant subspace When 2 is invertible modulo M = ∏L_i (i.e., all moduli are odd): $$ \Pi := \frac{1}{2}(I + F), \qquad \Pi(a) = \frac{a + F(a)}{2} \pmod{M} $$ **Theorem.** Π² = Π. **Proof** — a single line from F² = I: $$ \Pi^2 = \frac{1}{4}(I+F)^2 = \frac{1}{4}(I + 2F + F^2) = \frac{1}{4}(2I + 2F) = \frac{1}{2}(I+F) = \Pi $$ **What Π does.** Decompose element-wise on the torus: | Axis | Π(a) = (a + F(a))/2 | Behavior | |------|---------------------|----------| | Identity (axis 1) | (a + a)/2 = a | Element preserved | | Reflection (axes 2…k) | (a + (S−a))/2 = S/2 | Collapses to constant S/2 | Π annihilates the reflection-dimension information: every point projects to (a mod L₁, S/2, S/2, …, S/2). The output is a 1-dimensional subspace of the k-torus — the **invariant core** of the embedding. All the combinatorial structure (Sidon, B_h) that F(A) carries on the torus lives in the *kernel* of Π — the part that Π erases. ### Set-theoretic form (C): Orbit closure (no modular constraints) When 2 is not invertible modulo M (any even modulus present): $$ \mathcal{C}(X) := X \cup F(X) $$ **Theorem.** C² = C. **Proof:** $$ \begin{aligned} \mathcal{C}(\mathcal{C}(X)) &= \mathcal{C}(X \cup F(X)) \\ &= (X \cup F(X)) \cup F(X \cup F(X)) \\ &= X \cup F(X) \cup F(X) \cup F^2(X) \\ &= X \cup F(X) = \mathcal{C}(X) \end{aligned} $$ C simply closes a set under the involution — the most minimal invariant packet containing X. For a single point: a ⟼ {a, F(a)}. ### Why this matters The idempotent sieve is the **fixed-point extractor** of the CRL system. It separates the embedding into: - **Invariant subspace** (image of Π): the part that survives all F-reflections - **Nullspace** (kernel of Π): the part that oscillates — the combinatorial structure that F creates on the torus This decomposition is universal for any involution-based construction. The Lean verification of the set-theoretic form is a 10-line proof (see appendix). --- ## 4. What Varies, What Doesn't The embedding is parameterized by k moduli. Changing them changes the torus geometry: | Parameter | Effect | |-----------|--------| | Larger L₁ | Larger minimum gap. Non-fixed points spread apart. | | More axes (larger k) | Higher-dimensional torus. More constraints coupling A to S. | | Choice of L₂,…,L_k | Controls which residues carry the reflection. The specific prime/power selection determines which arithmetic patterns emerge. | | Larger M = ∏L_i | Larger torus volume. More "room" but coarser grid. | | Fixed S | The involution center. Constant across all axes 2…k. | S is **globally invariant** — the same involution parameterizes all reflection axes. The image F(A) is not generally closed under the original reflection S. This is not a bug: the torus embedding lifts A out of 1D into kD, and the involution lives *between* elements (as F-pairs), not *within* the image set. --- ## 5. k = 2 Example: Sidon from a Line Take A = {1, 2, 5, 6} with S = 7 (reflection pairs: 1↔6, 2↔5). A is not Sidon: 1+6 = 2+5 = 7. Embed into a 2-torus with L₁ = 3, L₂ = 4: ``` a axis 1 (mod 3) axis 2 (7−a mod 4) torus point F(a) 1 1 2 (1,2) 2 2 1 (2,1) 5 2 2 (2,2) 6 0 1 (0,1) ``` In integer representatives: F(A) = {5, 10, 9, 2}. No duplicate sums — Sidon. The gap L₁ = 3 separates the elements enough on the first axis to break the collision. The sum invariant F(a) + F(S−a) ≡ S (mod M) links reflection-paired preimages across the torus: F(1)=10 and F(6)=9 satisfy 10 + 9 = 19 ≡ 7 = S. The F² = id involution pairs image points differently — F(10)=1 and F(9)=6 — but the S-sum pairing is the structural bridge between the original reflection on A and the torus embedding. --- ## 6. k = 16: The Braid Torus Take k = 16 pairwise-coprime moduli. The embedding produces points on a 16-torus: ``` T = Z/L₁Z × Z/L₂Z × ... × Z/L₁₆Z ``` Axis 1 carries identity. Axes 2…16 carry the reflection constraint, each with a different modulus. The result is a 16-dimensional point pattern where: - Every original element a ∈ A becomes a 16-tuple - The involutive partner F(S−a) is the reflection of the point across axes 2…16 - The pattern of points on the torus encodes both the original set A and its involution structure via the coupling to S **Why 16?** The BraidStorm compressor operates on 8 strands, each contributing 2 dimensions: a crossing identity axis (strand is preserved through the crossing) and a phase axis (strand phase is inverted by the crossing). 8 × 2 = 16. The CRT torus embedding is a concrete algebraic model for placing a braid configuration onto a 16-dimensional lattice. Each braid crossing corresponds to a local deformation ε(a) = F(a) − a whose components on axes 2…16 characterize the crossing type. **Q16_16 compatibility.** The full torus modulus M = ∏ L_i exceeds Q16_16 range for k ≥ 8 (the product of the first 8 primes alone is ~9.7×10⁶). However, the CRT decomposition works per-axis: each L_i is small, and all computation stays in the smaller rings Z/L_i Z. The identity axis (mod L₁) uses Q16_16 integer arithmetic for the original value a; the reflection axes use modular arithmetic in their respective rings. No single value requires the full modulus M at runtime. (Proof sketch: the braid generator σᵢ acts on strand i by identity and strand i+1 by permutation. In the 16D embedding with axes paired (2i, 2i+1) for each strand, the identity axis is untouched and the reflection axis carries the crossing phase. Formal verification is ongoing.) --- ## 7. What This Gets You The CRT torus embedding is a tool for transforming a 1D reflection-closed set into a k-dimensional point cloud with controlled properties: - **Combinatorial separation**: The gap L₁ on axis 1 helps enforce properties like Sidon, B_h, Golomb — breaking sum/difference collisions that exist in the original 1D set. - **Involution pairing**: F creates involutive pairs on the torus, which models braid crossings, reflection-symmetric codes, or paired configurations. - **Modulus tuning**: Different choices of L₁,…,L_k produce different torus geometries — the embedding is a parameterized construction tool, not a theorem with a single fixed outcome. - **Integer-only computation**: All arithmetic is modular — no floats needed. Compatible with Q16_16 fixed-point for the modulus selection step. --- ## 8. What This Is Not - Not a novel "operator class" — it is an embedding. F is a specific map, not a category of operators. The structure is the torus + the point pattern. - Not proven to always produce Sidon/B_h/Golomb sets — the example demonstrates the mechanism. General sufficient conditions are open. - Not a dynamical system — iteration (applying F to F(A)) requires choosing new moduli, which is not a fixed dynamical law. The involution F² = id on the torus means "iteration" is really "walking through pairs," not converging. - Not yet formally connected to braid groups — the dimensional count (8×2=16) is suggestive, not proven. The full Yang-Baxter / Reidemeister structure on the torus embedding is ongoing work. --- ## 9. Open Directions 1. **Optimal modulus selection** — given A, S, and a target property P (Sidon, B_h, distinct differences), characterize the (L₁,…,L_k) that maximize the probability that F(A) satisfies P. 2. **Braid group action** — formalize how braid generators σᵢ act on the 16-torus embedded point set. Prove that F-pairs correspond to crossings. 3. **Asymmetric storage** — the identity axis (axis 1) requires no additional storage beyond the original A. Only the reflection axes contribute new information. This asymmetry maps to the ASQ framework (int8 query × binary documents) as a structural analogy: one axis is preserved at full resolution, the others are quantized. 4. **Torus codes** — the point pattern on the torus can be interpreted as an error-correcting code. The gap L₁ provides a minimum distance guarantee. Characterize the code parameters (n, k, d) achievable via this construction.