W.27 3D Planes, De Gua’s Quadrea Law & Platonic Dihedral Spreads — Code

In 3D space over F, quadrance is Q(A_1, A_2) = dx^2 + dy^2 + dz^2, a right-corner tetrahedron satisfies De Gua’s Quadrea Theorem A_base = A_xy + A_yz + A_zx, and the dihedral (face) spreads of ALL FIVE regular Platonic solids—Tetrahedron (8/9), Cube (1), Octahedron (8/9), Icosahedron (4/9), and Dodecahedron (4/5)—are exact rational numbers in Q!

Exact geometor.model SVG output of model.py for W.27 (3D Planes, De Gua’s Quadrea Law & Platonic Dihedral Spreads).

Symbolic Construction & Verification Script (model.py)

"""GEOMETOR Wildberger symbolic model and proof script for W.27: 3D Planes, De Gua's Quadrea Law & Platonic Dihedral Spreads.

        Enunciation: In 3D space over F, quadrance is Q(A_1, A_2) = dx^2 + dy^2 + dz^2, a right-corner tetrahedron satisfies De Gua's Quadrea Theorem A_base = A_xy + A_yz + A_zx, and the dihedral (face) spreads of ALL FIVE regular Platonic solids—Tetrahedron (8/9), Cube (1), Octahedron (8/9), Icosahedron (4/9), and Dodecahedron (4/5)—are exact rational numbers in Q!
        """

        from __future__ import annotations

        import sys
        from pathlib import Path

        _ROOT_SRC = Path(__file__).resolve().parents[3] / "src"
        if str(_ROOT_SRC) not in sys.path:
            sys.path.insert(0, str(_ROOT_SRC))

        from geometor.wildberger.models import (  # noqa: E402
            TopicBuildResult,
            assert_zero_floats,
            build_principle_model,
        )
        from geometor.wildberger.render import render_svg  # noqa: E402


        def construct_w_27() -> TopicBuildResult:
            """Construct the exact symbolic model and verifications for W.27.

            Returns:
                Populated `TopicBuildResult` for W.27 (3D Planes, De Gua's Quadrea Law & Platonic Dihedral Spreads).
            """
            result = build_principle_model("W.27")
            assert_zero_floats(result.model)
            return result


        def main() -> None:
            """Execute W.27 symbolic construction, render `w-27.svg`, and print invariants."""
            here = Path(__file__).resolve().parent
            bundle = construct_w_27()
            svg_out = render_svg(
                bundle.model,
                here / "w-27.svg",
                extra_curves=bundle.extra_curves,
                annotations=bundle.annotations,
            )
            print("W.27: 3D Planes, De Gua's Quadrea Law & Platonic Dihedral Spreads")
            print(f"  Elements in model: {len(bundle.model)}")
            print(f"  Rendered SVG:      {svg_out.name}")
            for label, val, status in bundle.verifications:
                print(f"  [OK] {label}: {val} ({status})")


        if __name__ == "__main__":
            main()