Measured, not inferred: the clause-generated Gauss–Newton basin boundary was sampled at six increasing raster resolutions. Its high-resolution box-counting estimate remains noninteger under the tested boundary definitions and fit windows.
| Quantity | Measured value |
|---|---|
| High-resolution mean $D$ | 1.1498975519 |
| 192²–512² range | 1.1356637850–1.1588068412 |
| 4/8-neighbor robustness band | 1.1252727299–1.1588068412 |
| Nearest-integer separation of the high-resolution mean | 0.1498975519 |
| Unresolved pixels | 0 at every tested resolution |
| Minimum regression $R^2$ | 0.9948629036 |
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E1 RESULT
The measured 2-D basin boundary is numerically consistent with a noninteger box-counting dimension over the tested finite-resolution scaling window. The conclusion persists under 4-neighbor and 8-neighbor boundary definitions and fit-window variation.
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The starting lineage is the Babylonian / Newton–Heron square-root map recorded in Babylonian Mathematics — Sexagesimal Computation, Square Roots, and the Newton–Heron Lineage.
For the SAT lift, each Boolean variable is constrained by
$$ b_i(x)=x_i(1-x_i)=0. $$
For a clause $C=(\ell_1\lor\ell_2\lor\ell_3)$, define its polynomial residual as the product of literal-falsity factors:
$$ c_C(x)=\operatorname{false}(\ell_1)\operatorname{false}(\ell_2)\operatorname{false}(\ell_3). $$
For example,
$$ (x_1\lor\neg x_2\lor x_3) \quad\longmapsto\quad (1-x_1)x_2(1-x_3). $$
The full residual vector contains the Boolean residuals and the clause residuals. The executable flow is damped Gauss–Newton:
$$ (J^T J+\lambda I)\Delta=-J^TR, \qquad x_{k+1}=x_k+\eta\Delta, $$
with backtracking that accepts residual-decreasing steps.
$$ (x_1\lor x_2\lor x_3) \land (\neg x_1\lor\neg x_2\lor x_3) \land (x_1\lor\neg x_3\lor x_2). $$
The satisfying Boolean roots used only for post-convergence labeling and independent checking are
$$ 010,\;011,\;100,\;101,\;111. $$
They are not used to construct or drive the clause-generated polynomial dynamics.