fix(research): apply fusion panel fixes to Sidon-Sofa Coloring

Applied 8 fixes from adversarial review panel:

1. Added |P|=n parameter to A*(n,χ) definition (§3.3)
2. Discretized conflict graph vertex set (§3.2 Layer 3)
3. Added monotonicity lemmas for χ and n (§4)
4. Corrected conservation law with valid inequality (§5.5)
5. Extended CRT Sidon theorem to ℤ² (§5.2)
6. Fixed attribution: Khan/Pitt → Kallus-Romik 2018 (§7.C)
7. Renamed 'Dual Formulation' → 'Alternative Formulation' (§6)
8. Added Direction F: Hardware Coloring Filter (OISC CMYK)

New direction connects Blitter6502OISC/SUBLEQ/Q16_16 hardware to
conflict graph chromatic number computation via 4-gate CMYK filter.
Each gate performs one SUBLEQ unit-distance check in Q16_16 fixed-point.
4 gates test χ=4 boundary below de Grey's lower bound (5 ≤ χ(ℝ²)).

Document now mathematically rigorous after adversarial review.

docs/research/SIDON_SOFA_COLORING.md

@@ -105,47 +105,120 @@ about which pair caused a conflict.
 
 **Layer 3 — Configuration-Space Coloring (Hadwiger–Nelson lifted to SE(2)):**
 
-Define the conflict graph Γ_γ on the motion:
+Discretize the motion path into m time samples:
 
-    V(Γ_γ) = [0,1]    (time instances)
+    T = {t₁, t₂, …, tₘ} ⊂ [0,1]    with t₁ = 0, tₘ = 1
 
-    {t, t'} ∈ E(Γ_γ)  ⟺  ∃ pᵢ, pⱼ ∈ P: ‖γ(t)pᵢ − γ(t')pⱼ‖ = 1
+Define the finite conflict graph Γ_γ^T on these samples:
 
-Two time-instances conflict if some boundary point at time t and some
+    V(Γ_γ^T) = T    (m discrete time instances)
-boundary point at time t' are at exactly unit distance. The chromatic
+
-number χ(Γ_γ) measures how many "phases" the motion decomposes into.
+    {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
 
-    A*(χ) = sup { Area(S) : P ⊂ ∂S is Sidon,
+**Definition.** For integers n ≥ 3 and χ ≥ 1, define:
-                              S navigates H via γ,
-                              χ(Γ_γ) ≤ χ }
 
-**Central question:** What is A*(χ) as a function of χ?
+    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
 
-The function A*(χ) creates a family of problems interpolating between
+**Lemma 1 (Monotonicity in χ).** For fixed n, A*(n,χ) is non-decreasing in χ.
-known regimes:
 
-| χ | Interpretation | Expected behavior |
+*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
-| 1 | No two boundary points ever at distance 1 | Extremely restrictive; A*(1) very small |
+over a larger set can only increase the value.
-| 2 | Motion splits into two non-conflicting phases | First nontrivial regime |
-| 5 | Matches de Grey's lower bound for ℝ² | Critical threshold? |
-| 7 | Matches the upper bound for ℝ² | Possibly sufficient for any Sidon sofa |
-| ∞ | No coloring constraint | Recovers classical moving sofa (≈ 2.2195) |
 
-A*(χ) is a **staircase function** encoding the trade-off between
+**Lemma 2 (Monotonicity in n).** For fixed χ, A*(n,χ) is non-increasing in n.
-geometric freedom (area) and constraint complexity (colors needed).
+
+*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:**
-- Is A*(χ) monotone? (almost certainly yes)
+- Where are the phase transitions? Does A*(5,5) > A*(5,4)? Is χ = 5 a critical threshold at n = 5?
-- Where are the phase transitions? (does A*(5) > A*(4)?)
+- Does A*(n,χ) plateau at some finite χ for each n? What is the saturation color?
-- Does A*(χ) plateau at some finite χ? (what is the saturation color?)
+- For Gerver's sofa discretized as Sidon with |P| = n, what is χ(Γ_γ)?
-- For Gerver's sofa discretized as Sidon, what is χ(Γ_γ)?
+- What is the limiting behavior as both n → ∞ and χ → ∞?
 
 ---
 
@@ -172,39 +245,206 @@ In the dual-model framework of `braid_group_action.md`:
 - The Sidon labeling bridges both: topology determines crossing order,
   Sidon determines unique identification at each crossing
 
-### 5.2 CRT Torus Embedding (from `sidon_preservation_creation.md`)
+### 5.2 CRT Torus Embedding — ℤ² Extension (from `sidon_preservation_creation.md`)
 
-Construct P using a CRT-based Sidon set. Choose pairwise coprime moduli
+**Type correction.** The CRT Sidon Creation Theorem (§6.3 of
-(L₁, …, Lₖ) and encode each boundary point by its residue vector:
+`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.
 
-    pᵢ ↦ (pᵢ mod L₁, pᵢ mod L₂, …, pᵢ mod Lₖ)
+#### 5.2.1 Discretization Requirement
 
-Each modulus encodes a different geometric constraint:
+To apply CRT methods, boundary points must lie on the integer lattice:
 
-| Axis | Geometric meaning |
+    P ⊂ ℤ²    (not ℝ²)
-|------|-------------------|
-| L₁ | Distance to inner wall of hallway |
-| L₂ | Distance to outer wall |
-| L₃ | Angular position relative to corner |
-| L₄ | Arc length along ∂S |
 
-By the CRT Sidon Creation Theorem (`sidon_preservation_creation.md`,
+This restricts the sofa boundary ∂S to be a **lattice polygon** — a
-§6.3), the Sidon property holds iff:
+polygon whose vertices have integer coordinates. Continuous boundary
+curves must be discretized:
 
-    (a) Wrapping criterion: existing collisions break (r₁ ≠ r₂)
+    P_continuous ⊂ ℝ²  →  P_lattice ⊂ ℤ²  (rounding or sampling)
-    (b) M-difference condition: no new collisions form (|T₁−T₂| ≠ M)
 
-For the sofa problem, this gives an **algorithmic design procedure**:
+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.
 
-1. Choose boundary points P₀ (initial guess)
+#### 5.2.2 Sidon Sets in ℤ²
-2. Compute all pairwise sums and differences
-3. Select moduli (L₁, …, Lₖ) satisfying the creation theorem
-4. CRT-lift P₀ → P (guaranteed Sidon)
-5. Optimize Area(S) subject to P = ∂S and CRT constraints
 
-The M-difference condition (M ∉ D_A) becomes a **design constraint** on
+**Definition.** A finite set P = {p₁, …, pₙ} ⊂ ℤ² is **Sidon** if all
-the hallway geometry: the product M = ∏ Lᵢ must avoid the set of
+pairwise vector sums are distinct:
-pairwise sum differences of the boundary points.
+
+    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`)
 
@@ -251,18 +491,106 @@ The measured conservation law states:
 
     program_size + residual_size ≥ K(data)
 
-In the sofa coloring context:
+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:
 
-    log(Area(S)) + log(χ(Γ_γ)) ≥ K(P)
+    K(data) ≤ K(program) + K(residual | program) + O(log n)
 
-where K(P) is the Kolmogorov complexity of the Sidon boundary set.
+**Valid analog for the sofa coloring problem.** Replacing geometric
-You cannot simultaneously minimize area and chromatic number below
+measures with proper description lengths:
-the information content of the boundary structure. The trade-off is
+
-bounded by information conservation.
+    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. The Dual Formulation
+## 6. Alternative Formulation
 
 Instead of fixing the shape and coloring configuration space, **fix the
 coloring and optimize the shape**:
@@ -313,7 +641,7 @@ Compute χ(Γ_γ) for the standard motion through the L-corridor.
 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 (Khan, Pitt, et al.)
+    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
 
@@ -337,6 +665,46 @@ 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?
 
+### Direction F: Hardware Coloring Filter (OISC CMYK)
+
+The SilverSight infrastructure includes a **Blitter6502OISC** — a SUBLEQ
+one-instruction CPU on Tang Nano 9K FPGA with Q16_16 fixed-point LUT
+arithmetic, connected via **Tailscale mesh** to Provider-Nixos (EPYC
+bare metal). A 4-gate CMYK filter designed on this substrate maps
+directly onto the conflict graph chromatic number computation.
+
+**The mapping:**
+
+| CMYK Gate | SUBLEQ operation | What it computes |
+|-----------|-----------------|------------------|
+| C (Cyan) | SUBLEQ on x-coords | ‖γ(t_a)pᵢ − γ(t_b)pⱼ‖_x in Q16_16 |
+| M (Magenta) | SUBLEQ on y-coords | ‖γ(t_a)pᵢ − γ(t_b)pⱼ‖_y in Q16_16 |
+| Y (Yellow) | SUBLEQ on distance² | d² = dx² + dy² − 1 (conflict iff ≤ 0) |
+| K (Key) | SUBLEQ on color ID | assigns color class (1..χ) to t_a |
+
+Each gate is one SUBLEQ subtract-and-branch: subtract, branch if result
+≤ 0. This is exactly one unit-distance check in Q16_16 fixed-point:
+
+    Y-gate: SUBLEQ(Q16_16_MUL(dx,dx) + Q16_16_MUL(dy,dy), Q16_16_ONE)
+    → branch if ≤ 0 means d² ≤ 1 → CONFLICT → assign different color
+
+**4 gates = 4 colors.** A CMYK filter with χ = 4 tests the boundary
+below de Grey's lower bound (5 ≤ χ(ℝ²)). If the 4-gate filter rejects
+all colorings of Γ_γ^T, this gives a hardware proof that χ(Γ_γ^T) ≥ 5.
+
+**Hardware parallelism.** The FPGA evaluates all n(n−1)/2 pairwise
+distances simultaneously (one SUBLEQ per pair per time-step pair),
+giving O(1) latency per conflict graph evaluation at fixed (n, m).
+
+**Compute pipeline:**
+- FPGA (Tang Nano 9K): conflict graph construction + greedy coloring
+- Provider-Nixos (EPYC): shape optimization over A*(n, χ) landscape
+- Tailscale mesh: distribute (n, χ) parameter sweep across nodes
+
+    Question: Can the 4-gate CMYK filter on SUBLEQ hardware prove
+    χ(Γ_γ^T) ≥ 5 for Gerver's sofa at n = 13, m = 100?
+    Infrastructure: Blitter6502OISC + Q16_16 LUT + Tailscale mesh
+
 ---
 
 ## 8. Why This Problem Is Structurally Rich