# Sidon-Sofa Coloring: A Unified Problem

**Status:** CONCEPTUAL — problem formulation, no measurements yet
**Date:** 2026-07-03
**Depends on:** `INVARIANT_COMPUTATION_GEOMETRY.md`, `sidon_preservation_creation.md`,
`braid_group_action.md`, `OCTAGON_PRINCIPLE.md`

**One-sentence statement:** A shape with Sidon-structured boundary navigates
an L-corridor while the induced unit-distance conflict graph on its
configuration-space trajectory has bounded chromatic number.

---

## 1. The Two Parent Problems

### 1A. Moving Sofa (Moser 1966)

What planar shape of maximum area can be moved around a right-angle
hallway of unit width?

| Quantity | Value |
|----------|-------|
| Best known shape | Gerver's sofa (1992) |
| Best known area | ≈ 2.2195 |
| Optimality proved? | **No** |
| Configuration space | SE(2) = ℝ² × S¹ |
| Constraint type | Rigid-body motion through constrained geometry |

The problem is an optimization over continuous paths in SE(2):

    γ: [0,1] → SE(2),    γ(t)S ⊂ H  ∀t

where H is the L-shaped hallway and S is the shape.

### 1B. Hadwiger–Nelson (1950)

What is the minimum number of colors needed to color the plane so that
no two points at distance 1 share a color?

| Quantity | Value |
|----------|-------|
| Classical bounds | 4 ≤ χ(ℝ²) ≤ 7 |
| de Grey (2018) | 5 ≤ χ(ℝ²) ≤ 7 |
| Answer known? | **No** |
| Constraint type | Unit-distance graph coloring |

Both problems are open. Both involve geometry under constraint. Neither
has been studied through the lens of Sidon structure.

---

## 2. The Bridge: Sidon Structure as Invariant Geometry

The SilverSight framework already identifies Sidon sets as a key
invariant — the collision-free property where all pairwise sums are
distinct:

    aᵢ + aⱼ = aₖ + aₗ  ⟹  {i,j} = {k,l}

In the language of `INVARIANT_COMPUTATION_GEOMETRY.md`:

    Sidon property = equivalence class under the Φ-metric
    (nonlinear → spectral: pairwise sums → eigenvalue degeneracy)

The Sidon property is **observer-independent**: it survives change of
basis, change of representation, change of coordinate system. Every pair
of elements has a **unique signature** (its sum). This is exactly the
"observerless observer" protocol from `OCTAGON_PRINCIPLE.md`.

**Key insight:** If we require the boundary of the sofa to form a Sidon
set, then every geometric interaction between boundary points carries a
unique, intrinsically identifiable signature. The conflict graph becomes
**rigid enough** for nontrivial chromatic analysis while remaining
**expressive enough** for geometric optimization.

---

## 3. The Unified Problem: Formal Statement

### 3.1 Setup

Let H ⊂ ℝ² be an L-shaped hallway of unit width.
Let S ⊂ ℝ² be a compact connected shape.
Let P = {p₁, …, pₙ} ⊂ ∂S be a finite boundary point set.

### 3.2 Three Constraint Layers

**Layer 1 — Sidon Constraint (structural bridge):**

P is a Sidon set in ℝ²:

    pᵢ + pⱼ = pₖ + pₗ  ⟹  {i,j} = {k,l}

All pairwise sums (midpoints × 2) are distinct. Every pair of boundary
points has a unique "center of mass" signature. This means every
geometric interaction is intrinsically identifiable — no ambiguity
about which pair caused a conflict.

**Layer 2 — Motion Constraint (sofa):**

∃ continuous path γ: [0,1] → SE(2) such that:
- γ(0)S is in the horizontal corridor
- γ(1)S is in the vertical corridor
- γ(t)S ⊂ H  ∀t ∈ [0,1]

**Layer 3 — Configuration-Space Coloring (Hadwiger–Nelson lifted to SE(2)):**

Discretize the motion path into m time samples:

    T = {t₁, t₂, …, tₘ} ⊂ [0,1]    with t₁ = 0, tₘ = 1

Define the finite conflict graph Γ_γ^T on these samples:

    V(Γ_γ^T) = T    (m discrete time instances)

    {t_a, t_b} ∈ E(Γ_γ^T)  ⟺  ∃ pᵢ, pⱼ ∈ P: ‖γ(t_a)pᵢ − γ(t_b)pⱼ‖ = 1

Two time-instances conflict if some boundary point at time t_a and some
boundary point at time t_b are at exactly unit distance. The chromatic
number χ(Γ_γ^T) measures how many "phases" the discretized motion
decomposes into.

**Remark (finite vs. continuum).** For any fixed T, Γ_γ^T is a finite
graph with well-defined chromatic number. As m → ∞ (finer sampling),
the graphs form a nested sequence: if T ⊂ T' then Γ_γ^T is an
induced subgraph of Γ_γ^{T'}, so χ(Γ_γ^T) ≤ χ(Γ_γ^{T'}). The limit
chromatic number is:

    χ(Γ_γ) = sup_{T finite} χ(Γ_γ^T)

By the Erdős–de Bruijn theorem (assuming AC), this equals the chromatic
number of the full continuum graph on [0,1]. For computation, we always
work with finite T and report χ(Γ_γ^T) as a lower bound on χ(Γ_γ).

### 3.3 The Optimization

**Definition.** For integers n ≥ 3 and χ ≥ 1, define:

    A*(n, χ) = sup { Area(S) : P ⊂ ∂S, |P| = n, P is Sidon in ℝ²,
                                 S navigates H via γ,
                                 χ(Γ_γ^T) ≤ χ }

where Γ_γ^T is the conflict graph defined in Layer 3. The two explicit
parameters are:

- **n** (boundary resolution): the number of monitored boundary points.
- **χ** (chromatic budget): the maximum number of motion phases.

When the discretization parameter m (from Layer 3) must be explicit,
we write A*(n, χ; m).

**Remark (minimum n for non-trivial Sidon constraint).** The Sidon
condition generates n(n+1)/2 pairwise sums that must all be distinct:

| n  | Pairwise sums | What the Sidon condition excludes                    | Status       |
|----|---------------|------------------------------------------------------|--------------|
| 1  | 1             | Nothing (no pairs to compare)                        | Vacuous      |
| 2  | 3             | Nothing beyond p₁ ≠ p₂ (all 3 sums trivially distinct)| Vacuous      |
| 3  | 6             | One point being the midpoint of the other two        | Near-vacuous |
| 4  | 10            | Midpoints + parallelograms (pᵢ−pⱼ = pₖ−pₗ)          | Mild         |
| ≥5 | ≥15           | Θ(n⁴) additive coincidences of codimension 2 in (ℝ²)ⁿ| Non-trivial  |

For n ≤ 2, every set of n distinct points in ℝ² is Sidon. The
optimization A*(n, χ) with n ≤ 2 is therefore equivalent to the
classical moving sofa problem with a weakened coloring constraint.
This is why the definition requires **n ≥ 3**.

**Proposition (geometric significance for n ≥ 5).** For n ≥ 5, the
Sidon constraint becomes geometrically meaningful:

(a) **Sumset packing.** The sumset P + P has cardinality exactly
    n(n+1)/2 (the maximum possible). All these sums lie in 2·conv(P),
    which is constrained by the hallway geometry. For n = 5, this
    requires 15 distinct points; for n = 13, it requires 91.

(b) **Additive energy.** E(P) = |{(a,b,c,d) ∈ P⁴ : a+b = c+d}| = n
    (the minimum possible). By Balog–Szemerédi–Gowers, any subset P'
    participating in many unit-distance pairs must have E(P') ≫ |P'|,
    contradicting the Sidon property. For n ≥ 5, this creates genuine
    tension between Sidon structure (Layer 1) and unit-distance
    conflicts (Layer 3).

---

## 4. The Interpolation Spectrum

**Lemma 1 (Monotonicity in χ).** For fixed n, A*(n,χ) is non-decreasing in χ.

*Proof.* If χ₁ < χ₂, then any motion valid under χ₁ colors is also valid under χ₂ colors.
Thus the feasible set for χ₁ is a subset of the feasible set for χ₂. Taking the supremum
over a larger set can only increase the value.

**Lemma 2 (Monotonicity in n).** For fixed χ, A*(n,χ) is non-increasing in n.

*Proof.* If n₁ < n₂, then any Sidon set P₂ with |P₂| = n₂ contains a Sidon subset P₁
with |P₁| = n₁. The conflict graph Γ_{P₁} is an induced subgraph of Γ_{P₂}, so
χ(Γ_{P₁}) ≤ χ(Γ_{P₂}). Thus any motion valid for n₂ is also valid for n₁. The feasible
set for n₂ is a subset of the feasible set for n₁, so the supremum is smaller.

**2D Interpolation Spectrum.** The function A*(n,χ) interpolates between known regimes:

| n \ χ | 1 | 2 | 5 | 7 | ∞ |
|-------|---|---|---|---|---|
| 3 | ≈0 | small | small | small | < 2.22 |
| 5 | ≈0 | small | ? | ? | < 2.22 |
| 8 | ≈0 | small | ? | ? | < 2.22 |
| 13 | ≈0 | small | ? | ? | < 2.22 |
| ∞ | ≈0 | small | ? | ? | ≈ 2.2195 |

**Key observations:**
- χ = 1: Extremely restrictive (no two boundary points ever at distance 1), A*(n,1) ≈ 0 for all n
- χ = ∞: Recovers classical moving sofa, A*(n,∞) approaches 2.2195 as n → ∞
- χ = 5,7: Match de Grey's bounds for ℝ², critical thresholds for phase transitions
- n = ∞: Full boundary resolution, approaches classical problem
- Phase transitions: Expected at χ ∈ {5,7} for fixed n, and at n ∈ {5,8} for fixed χ

**Research questions:**
- Where are the phase transitions? Does A*(5,5) > A*(5,4)? Is χ = 5 a critical threshold at n = 5?
- Does A*(n,χ) plateau at some finite χ for each n? What is the saturation color?
- For Gerver's sofa discretized as Sidon with |P| = n, what is χ(Γ_γ)?
- What is the limiting behavior as both n → ∞ and χ → ∞?

---

## 5. Connection to SilverSight Concepts

### 5.1 Braid Trees (from `braid_group_action.md`)

As S moves through the corner, the worldlines of boundary points
{γ(t)pᵢ} trace curves in ℝ² × [0,1]. These worldlines **braid** around
each other and around the inner corner vertex.

The Sidon constraint guarantees every **crossing** in the braid tree
is **canonically labeled**: when worldlines of pᵢ and pⱼ come within
unit distance, the pair (i,j) is uniquely determined by the midpoint
(pᵢ + pⱼ)/2. No two crossings can share the same midpoint signature.

This gives the braid tree a **Sidon labeling** — a combinatorial
invariant of the motion that is independent of coordinate choices.

In the dual-model framework of `braid_group_action.md`:
- The **axis-swap model** describes the topology (which strands cross)
- The **modulus-adjustment model** describes the physics (which distances
  trigger conflicts)
- The Sidon labeling bridges both: topology determines crossing order,
  Sidon determines unique identification at each crossing

### 5.2 CRT Torus Embedding — ℤ² Extension (from `sidon_preservation_creation.md`)

**Type correction.** The CRT Sidon Creation Theorem (§6.3 of
`sidon_preservation_creation.md`) is proven for A ⊂ ℤ. The sofa
boundary P ⊂ ℝ² cannot be directly embedded via modular arithmetic —
the operation pᵢ mod L is undefined for vectors in ℝ² without a
lattice structure. This section develops the valid extension to
P ⊂ ℤ² (integer lattice boundary points) via coordinate-wise CRT
lifting.

#### 5.2.1 Discretization Requirement

To apply CRT methods, boundary points must lie on the integer lattice:

    P ⊂ ℤ²    (not ℝ²)

This restricts the sofa boundary ∂S to be a **lattice polygon** — a
polygon whose vertices have integer coordinates. Continuous boundary
curves must be discretized:

    P_continuous ⊂ ℝ²  →  P_lattice ⊂ ℤ²  (rounding or sampling)

The discretization introduces approximation error that must be bounded
relative to the hallway width. This is a non-trivial geometric
constraint not addressed by the CRT theorem itself.

#### 5.2.2 Sidon Sets in ℤ²

**Definition.** A finite set P = {p₁, …, pₙ} ⊂ ℤ² is **Sidon** if all
pairwise vector sums are distinct:

    pᵢ + pⱼ = pₖ + pₗ   (i ≤ j, k ≤ l)  ⟹  {i,j} = {k,l}

Equivalently, the sumset P+P has maximum size |P+P| = n(n+1)/2.

Note: this is STRONGER than Sidon in each coordinate separately.
Points can share an x-coordinate if their y-coordinates differ, and
vector sums can coincide in one coordinate without coinciding in both.

#### 5.2.3 Coordinate-wise CRT Lift

Let P = {(x₁,y₁), …, (xₙ,yₙ)} ⊂ ℤ². Define coordinate projections:

    X = {x₁, …, xₙ} ⊂ ℤ,    Y = {y₁, …, yₙ} ⊂ ℤ

Choose reflection centers S_x, S_y ∈ ℤ (e.g., S_x = max(X)+min(X)).

**X-axis CRT lift.** Choose pairwise coprime moduli (L₁ˣ, …, Lₖˣ)
with L₁ˣ, L₂ˣ ≥ 2. Let M_x = ∏ Lᵢˣ > max(X). Define
F_x: X → [0, M_x) as the unique integer satisfying:

    F_x(x) ≡ x        (mod L₁ˣ)       [identity axis]
    F_x(x) ≡ S_x − x  (mod Lᵢˣ)       [reflection axes, i ≥ 2]

**Y-axis CRT lift.** Choose pairwise coprime moduli (L₁ʸ, …, Lₘʸ)
with L₁ʸ, L₂ʸ ≥ 2. Let M_y = ∏ Lⱼʸ > max(Y). Define
F_y: Y → [0, M_y) analogously with reflection center S_y.

**Combined lift.** F: P → [0, M_x) × [0, M_y) defined by:

    F(pᵢ) = (F_x(xᵢ), F_y(yᵢ))

#### 5.2.4 2D CRT Sidon Creation Theorem

**Theorem (Coordinate-wise CRT Sidon in ℤ²).** Let P ⊂ ℤ² be finite,
with coordinate-wise CRT lifts F_x, F_y as above, M_x > max(X),
M_y > max(Y). Then:

    F(P) is Sidon in ℤ²  ⟺  for every pair of index-pairs
    {i,j} ≠ {k,l}, at least one coordinate is collision-free.

That is, F(P) is Sidon iff there is NO pair {i,j} ≠ {k,l} with BOTH:

    F_x(xᵢ)+F_x(xⱼ) = F_x(xₖ)+F_x(xₗ)     [x-collision]
    F_y(yᵢ)+F_y(yⱼ) = F_y(yₖ)+F_y(yₗ)     [y-collision]

By the 1D CRT Sidon Creation Theorem applied to each coordinate, an
x-collision occurs iff EITHER:

    (x-a) xᵢ+xⱼ = xₖ+xₗ  AND  wrapping matches:
          (F_x(xᵢ)+F_x(xⱼ) ≥ M_x) = (F_x(xₖ)+F_x(xₗ) ≥ M_x)
    OR
    (x-b) |T₁ˣ − T₂ˣ| = M_x  for distinct x-sums T₁ˣ, T₂ˣ

and analogously for y-collisions with M_y.

**Proof sketch.**

The vector sum decomposes coordinate-wise:

    F(pᵢ)+F(pⱼ) = (F_x(xᵢ)+F_x(xⱼ),  F_y(yᵢ)+F_y(yⱼ))

Two vector sums are equal iff BOTH coordinate sums are equal. This
factorizes the 2D collision condition into independent 1D collision
conditions on each coordinate.

Each 1D condition is fully characterized by the CRT Sidon Creation
Theorem (`sidon_preservation_creation.md` §6.3): a collision in
coordinate c ∈ {x,y} occurs iff either (a) an existing sum collision
in that coordinate has matching wrapping, or (b) distinct sums differ
by exactly M_c.

The 2D collision is the INTERSECTION of the two 1D collision events.
F(P) is Sidon iff this intersection is empty. ∎

**Corollary 1 (Sufficient condition via one coordinate).** If F_x(X) is
Sidon in ℤ (i.e., satisfies the 1D CRT Sidon Creation Theorem), then
F(P) is Sidon in ℤ² regardless of F_y. Symmetrically, if F_y(Y) is
Sidon, then F(P) is Sidon regardless of F_x.

This gives a practical sufficient condition: make EITHER coordinate
Sidon via the 1D theorem, and the 2D set is automatically Sidon.

**Corollary 2 (Wrapping criterion in 2D).** An existing vector collision
xᵢ+xⱼ = xₖ+xₗ AND yᵢ+yⱼ = yₖ+yₗ is broken by F iff EITHER:

    (i)  (F_x(xᵢ)+F_x(xⱼ) ≥ M_x) ≠ (F_x(xₖ)+F_x(xₗ) ≥ M_x)
    (ii) (F_y(yᵢ)+F_y(yⱼ) ≥ M_y) ≠ (F_y(yₖ)+F_y(yₗ) ≥ M_y)

Wrapping in EITHER coordinate suffices to break a 2D collision.
This is strictly more permissive than the 1D case.

**Corollary 3 (M-difference condition in 2D).** A new vector collision
is created iff BOTH coordinates simultaneously satisfy their respective
M-difference conditions:

    |T₁ˣ - T₂ˣ| = M_x   AND   |T₁ʸ - T₂ʸ| = M_y

where T₁ˣ = xᵢ+xⱼ, T₂ˣ = xₖ+xₗ, T₁ʸ = yᵢ+yⱼ, T₂ʸ = yₖ+yₗ.
This is strictly MORE restrictive than the 1D case: both coordinates
must align to create a spurious collision.

#### 5.2.5 Algorithmic Construction Procedure

Given P ⊂ ℤ² (lattice boundary points), construct a CRT embedding that
guarantees the Sidon property:

1. **Extract coordinates.** Let X = {x₁,...,xₙ}, Y = {y₁,...,yₙ}.

2. **Compute pairwise sums.**
     S_X = {xᵢ+xⱼ : 1 ≤ i ≤ j ≤ n}
     S_Y = {yᵢ+yⱼ : 1 ≤ i ≤ j ≤ n}

3. **Identify collisions.** Find all index pairs (i,j) ≠ (k,l) where
   xᵢ+xⱼ = xₖ+xₗ (x-collisions) and yᵢ+yⱼ = yₖ+yₗ (y-collisions).

4. **Choose strategy.**
   - **Strategy A (sufficient):** Apply 1D CRT Sidon Creation to X alone.
     Choose moduli (L₁ˣ,...,Lₖˣ) satisfying the 1D theorem (§6.3).
     Then F(P) is Sidon regardless of Y.
   - **Strategy B (sufficient):** Apply 1D CRT Sidon Creation to Y alone.
   - **Strategy C (necessary and sufficient):** Choose moduli for both
     coordinates such that no index pair collides in BOTH coordinates.

5. **Apply CRT lift.**
     F(pᵢ) = (F_x(xᵢ), F_y(yᵢ))
   where F_x, F_y are defined as in §5.2.3.

6. **Verify.** Check that for all (i,j) ≠ (k,l):
     F_x(xᵢ)+F_x(xⱼ) ≠ F_x(xₖ)+F_x(xₗ)
     OR
     F_y(yᵢ)+F_y(yⱼ) ≠ F_y(yₖ)+F_y(yₗ)

**Complexity.** Step 3 is O(n²) to enumerate sums, O(n⁴) worst-case to
find all collisions. Step 4 (Strategy A or B) reduces to the 1D modulus
selection problem. Step 6 is O(n⁴) verification.

#### 5.2.6 Geometric Restrictions and Open Questions

**Lattice requirement.** This theorem applies ONLY to P ⊂ ℤ². For the
sofa problem, this means:

  - The boundary ∂S must be a lattice polygon (vertices in ℤ²)
  - Curved boundaries (e.g., Gerver's arcs) must be approximated by
    lattice polygons, introducing discretization error
  - The approximation quality depends on lattice resolution

**Continuous extension is open.** Extending this theorem to P ⊂ ℝ²
(continuous boundary points) remains an open problem. The obstruction:
modular arithmetic requires a ring structure, and ℝ² has no natural
finite quotient ring that preserves the geometric properties needed
for the wrapping criterion.

**Potential approaches for ℝ²:**
  - **Scaling:** If P ⊂ ℚ², scale to ℤ² and apply this theorem. The
    scaling factor controls precision but increases modulus size.
  - **Algebraic coordinates:** If P lies in a number field K with ring
    of integers O_K, one might use ideals in O_K as moduli. This is
    unexplored territory.
  - **Approximation:** Discretize ℝ² → ℤ² at resolution ε, apply the
    lattice theorem, and bound the error. The Sidon property is discrete
    (equality of sums), so small perturbations typically preserve it,
    but this needs rigorous proof.

**Connection to sofa problem.** For practical sofa optimization:
  - Discretize the boundary at resolution ε (e.g., ε = 0.01)
  - Scale to ℤ² (multiply by 1/ε)
  - Apply Strategy A or B (make one coordinate Sidon)
  - The resulting lattice Sidon set approximates the continuous boundary
  - Error analysis: how does discretization affect area optimization?

### 5.3 Octagon Principle (from `OCTAGON_PRINCIPLE.md`)

The sofa coloring problem is a **matter problem** (nonlinear geometric
constraint: does the shape fit?). The octagon principle says: convert
matter → light.

The conversion:

```
matter:  shape S navigates hallway H (geometric collision detection)
   ↓ Φ-metric embedding (Sidon boundary → adjacency matrix)
operator: conflict graph Γ_γ on SE(2) (unit-distance adjacency)
   ↓ spectral decomposition
light:   chromatic polynomial P(Γ_γ, k) (spectral coloring bound)
```

The Sidon boundary condition IS the Φ-metric: it converts the nonlinear
shape-fitting problem into a spectral problem on the conflict graph.
The nonlinear property (shape fits hallway) manifests as a spectral
signature (chromatic number of Γ_γ).

The octagon is RICHER than either original shape: the conflict graph
encodes BOTH the geometry of S and the topology of its motion. Neither
alone is sufficient.

### 5.4 Observer-Independence

The Sidon labeling of the conflict graph is **frame-independent**:

- Translate S → all pairwise sums shift by 2Δ (Sidon preserved)
- Rotate S → all pairwise sums rotate (Sidon preserved)
- Reparameterize time → graph Γ_γ is isomorphic (chromatic number
  invariant)

The coloring χ(Γ_γ) is an **intrinsic property** of the shape and
motion, not an artifact of coordinate system. This matches the
"observerless observer" from `INVARIANT_COMPUTATION_GEOMETRY.md`:
the invariant (chromatic number) survives all representations.

### 5.5 Conservation Law

The measured conservation law states:

    program_size + residual_size ≥ K(data)

Both terms on the left are **description lengths** (non-negative
bit-counts of specific representations). The inequality holds because
any decomposition of data into a program part and a residual part
must account for at least K(data) bits total. This follows from the
chain rule for Kolmogorov complexity:

    K(data) ≤ K(program) + K(residual | program) + O(log n)

**Valid analog for the sofa coloring problem.** Replacing geometric
measures with proper description lengths:

    K(S) + K(γ) + K(P | S, γ) ≥ K(P) - O(log n)

where:
- K(S) = description complexity of the shape (the "program" that
  generates the boundary ∂S)
- K(γ) = description complexity of the motion path through SE(2)
- K(P | S, γ) = residual information to specify which n points on
  ∂S form the Sidon set P, given the shape and motion
- K(P) = total Kolmogorov complexity of the boundary point set

This is **proven** (chain rule of Kolmogorov complexity, since P is
a component of the triple (S, γ, P)). It captures the intended
conservation intuition: you cannot simultaneously have a simple
shape, a simple motion, and low residual specification cost while
maintaining a complex boundary structure. The three terms trade off
against each other, bounded below by K(P).

The trade-off has operational meaning:
- Increasing shape complexity (richer ∂S) can reduce the residual
  K(P | S, γ) if the boundary naturally passes through the Sidon points
- Increasing motion complexity (more intricate γ) can reduce
  K(P | S, γ) if the motion implicitly constrains which points interact
- But the total information is conserved: simplifying one component
  shifts the burden to another

**Why the earlier geometric version failed.** A previous draft of this
section proposed:

    log₂(Area(S)) + log₂(χ(Γ_γ)) ≥ K(P)        [INVALID]

This is **false**. Counterexample: let S be a disk of radius
ε = 10⁻¹⁰.

    Area(S) = πε² ≈ 3.14 × 10⁻²⁰
    log₂(Area) ≈ -64.8                           (negative!)
    χ(Γ_γ) = 1  (all boundary points within 2ε ≪ 1,
                  no unit-distance edges in Γ_γ)
    log₂(χ) = 0
    LHS = -64.8 + 0 = -64.8
    K(P) ≥ 0  (for any P)
    Claim: -64.8 ≥ 0                             ✗ FALSE

**Root cause.** log₂(Area) is not a description length. Area is a
continuous geometric measure that can be less than 1, producing a
negative logarithm. The original conservation law has non-negative
bit-counts on both sides. The substitution of a geometric measure
(Area) for an information-theoretic quantity (description length)
broke the non-negativity invariant that makes the conservation law
work.

Additionally, log₂(χ) captures only O(log n) bits at most (the
number of bits to name a color count), while K(P) scales as
Ω(n log n) for an n-point Sidon set. Even for large shapes,
log₂(χ) alone carries too little information to bound K(P) from
below.

**Open question: geometric conservation bound.** A conservation-type
inequality using the native geometric quantities (Area, χ) rather
than Kolmogorov complexities remains desirable but unproven. For
any such bound to hold, the following conditions are necessary:

1. **Non-negative area term.** Replace log₂(Area(S)) with a quantity
   guaranteed non-negative, e.g., log₂(Area(S)/ε₀²) for some fixed
   reference scale ε₀, or ⌈Area(S)/ε²⌉ (the number of ε-grid cells
   intersecting S). This restores the non-negativity that the
   conservation law requires.

2. **Coloring cost scaled by n.** The term log₂(χ) must be multiplied
   by n (or a function of n) to carry enough information. The
   per-point coloring cost is ⌈log₂(χ)⌉ bits, so the total
   coloring information is n · ⌈log₂(χ)⌉, not log₂(χ) alone.

3. **Sidon density coupling.** For a Sidon set of n points in a
   region of diameter D, the Sidon property constrains
   n ≤ O(D/ε), coupling the geometric extent (related to Area)
   to the maximum boundary complexity n.

Whether a bound of the form

    f(Area, ε₀) + g(n, χ) ≥ c · K(P)

exists under these conditions (for suitable functions f, g and
constant c) is an **open conjecture**. The valid K-based inequality
above provides the structural template; the geometric version would
add operational specificity at the cost of requiring proof.

---

## 6. Alternative Formulation

Instead of fixing the shape and coloring configuration space, **fix the
coloring and optimize the shape**:

**Dual Problem:** Given a coloring c: SE(2) → {1, …, χ} (a "chromatic
schedule"), find the Sidon-structured shape S of maximum area whose
motion γ satisfies:

    ‖γ(t)pᵢ − γ(t')pⱼ‖ = 1  ⟹  c(γ(t)) ≠ c(γ(t'))

This makes the constraint propagation explicit: the coloring is a
pre-assigned schedule of "safe" and "unsafe" phases, and the shape
must be designed to respect it.

In the light/matter language:
- Primal: matter (shape) determines light (coloring)
- Dual: light (coloring schedule) constrains matter (shape)

This is the same duality structure as the octagon principle: the
matter→light and light→matter directions are complementary projections
of the same invariant geometry.

---

## 7. Concrete Research Directions

### Direction A: Finite Sidon Sofas (computational)

Take n boundary points forming a Sidon set. Compute A*(n, χ) for small
n and χ. Study convergence as n → ∞.

    Parameters: n ∈ {5, 8, 13, 21, 34} (Fibonacci, for Sidon density)
    Method: numerical optimization over polygon vertices +
            CRT Sidon creation theorem for guarantee
    Output: A*(n, χ) table, phase transition locations

### Direction B: Chromatic Number of Gerver's Sofa (analytical)

Discretize Gerver's sofa boundary into a Sidon set (using CRT lift).
Compute χ(Γ_γ) for the standard motion through the L-corridor.

    Question: Does Gerver's sofa require 5, 6, or 7 colors?
    Significance: Gives a LOWER BOUND on the chromatic number of
    the SE(2) unit-distance graph induced by optimal sofa motion.

### Direction C: Sidon-Restricted Upper Bounds (theoretical)

The Sidon constraint reduces degrees of freedom in the shape. Can you
prove a tighter upper bound on sofa area when ∂S must be Sidon?

    Current upper bound (no Sidon): ≈ 2.37 (Kallus, Romik 2018)
    Conjecture: Sidon constraint → tighter bound, possibly < 2.2
    Method: Use the unique midpoint property to bound contact geometry

### Direction D: SE(2) Unit-Distance Chromatic Number

The conflict graph Γ_γ is a unit-distance graph embedded in SE(2)
rather than ℝ². Generalize Hadwiger-Nelson:

    Question: Is 5 ≤ χ(SE(2)) ≤ 7?
    Or does the SE(2) structure (3D, non-commutative) allow different
    bounds?

    Method: Construct finite unit-distance graphs in SE(2) requiring
    ≥ 5 colors (analogous to de Grey's construction in ℝ²)

### Direction E: Braid-Coloring Correspondence

Study the relationship between the braid word of the motion (topology)
and the chromatic number of Γ_γ (combinatorics).

    Question: Does a longer braid word → higher chromatic number?
    Is there a braid invariant that bounds χ from below?

---

## 8. Why This Problem Is Structurally Rich

The unified problem sits at the intersection of **five** SilverSight
research threads:

```
        Moving Sofa (continuous optimization)
              ↗                ↘
    Sidon Structure    ←→    Unit-Distance Coloring
    (algebraic)              (combinatorial)
              ↘                ↗
        Braid Topology    ←→   CRT Embedding
        (topological)          (arithmetic)
```

Each thread provides a different lens:

| Thread | What it contributes |
|--------|-------------------|
| Sidon structure | Unique identification of interactions |
| Sofa optimization | Continuous geometric degrees of freedom |
| Hadwiger-Nelson coloring | Discrete constraint propagation |
| Braid topology | Topological invariants of motion |
| CRT embedding | Arithmetic construction of Sidon sets |

The Sidon constraint is not arbitrary — it is the **structural keystone**
that makes the unified problem well-posed. Without it, the conflict
graph has too much ambiguity (which pair caused the conflict?). With it,
every interaction is uniquely identified, the braid tree is canonically
labeled, and the CRT provides an algorithmic construction.

---

## 9. Connection to Session Measurements

| Existing measurement | Role in the unified problem |
|---------------------|----------------------------|
| Sidon 4/4 (octagon) | Boundary structure is the Φ-metric embedding |
| CRT lift O(1) | Algorithmic construction of Sidon boundaries |
| Braid group action | Topology of boundary worldlines through corner |
| Conservation law | Bound: log(Area) + log(χ) ≥ K(P) |
| Cospectral failure | Symmetry of hallway may cause degenerate conflict graphs |
| Merged O(1) transform | Physics of motion does invariant extraction |
| Reaction primes | Prime decomposition of Sidon boundary moduli |

---

## claim_boundary

```
sidon-sofa-coloring:problem-formulation:unified-optimization
```

This document formulates a new problem combining the Moving Sofa Problem,
the Hadwiger-Nelson coloring problem, and Sidon structure into a single
optimization A*(χ). No measurements have been performed. All connections
to SilverSight concepts (octagon, CRT, braid, conservation law) are
structural analogies awaiting empirical verification. The problem is
well-posed (finite Sidon sets exist, motions exist, conflict graphs are
finite for discretized boundaries), but A*(χ) is unknown for all finite χ.

---

## What's Measured vs. What's Speculative

**STRUCTURAL (well-defined):**
- The problem formulation is mathematically precise
- Sidon sets in ℝ² exist at all finite sizes
- The conflict graph Γ_γ is well-defined for any discretized motion
- The CRT construction algorithm exists (from sidon_preservation_creation)

**SPECULATIVE (unverified):**
- That A*(χ) has nontrivial phase transitions
- That Gerver's sofa discretized as Sidon requires 5+ colors
- That the SE(2) unit-distance chromatic number differs from ℝ²
- That the conservation law applies in the stated form
- That the octagon principle applies to this specific matter→light
  conversion

**OPEN QUESTIONS:**
- What is A*(1)? (the maximally constrained case)
- At what χ does A*(χ) saturate?
- Does the Sidon constraint change the asymptotic sofa area?
- Is there a braid invariant bounding χ from below?
