diff --git a/docs/research/SIDON_SOFA_COLORING.md b/docs/research/SIDON_SOFA_COLORING.md index ccda0993..a4cb7d4d 100644 --- a/docs/research/SIDON_SOFA_COLORING.md +++ b/docs/research/SIDON_SOFA_COLORING.md @@ -665,46 +665,6 @@ 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