SilverSight/docs/crt-torus-embedding.md

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.

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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.

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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.

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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).

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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.

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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.

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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.)

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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.

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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.

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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.