COMPUTATIONAL OBSERVATORY · E1
SAT → Newton → Basin Boundary
A clause-generated polynomial system, solved by damped multivariate Gauss–Newton iteration, then measured rather than merely admired.
Babylonian→Newton→3-SAT polynomials→basins→box dimension
MEASURED RESULT · 2026-09-22
The measured 2D basin boundary is numerically consistent with a noninteger box-counting dimension over the tested finite-resolution scaling window. The result persists under 4-neighbor and 8-neighbor boundary definitions and fit-window variation.
Evidence boundary: finite-resolution E1 numerical result for one explicit 3-SAT polynomial system and one 2-D initial-condition slice. It is not a proof of a universal fractal dimension and does not establish a polynomial-time algorithm for general SAT.
BOOLEAN GEOMETRY
Clause system
xᵢ(1−xᵢ)=0 C=(ℓ₁∨ℓ₂∨ℓ₃) c_C(x)=false(ℓ₁)false(ℓ₂)false(ℓ₃)
ITERATION
Damped Gauss–Newton
(JᵀJ+λI)Δ = −JᵀR x ← x + ηΔ
Backtracking accepts only residual-decreasing steps.
INCREASING RESOLUTION
Boundary dimension estimates
| Raster | D | 95% fit interval | R² | Unresolved |
|---|
PROVENANCE