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.

---

## 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.