Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== Assistant: In the context of spherical geometry — and the way we’ve been talking about QAT — a geometric half-radius usually means: ===
In the context of spherical geometry — and the way we’ve been talking about QAT — a geometric half-radius usually means:

It’s not literally half the size of the physical radius, but rather a point or length in the geometry that is defined by symmetry or proportional relationships inside the sphere, rather than direct measurement.

Here are some examples of what that could mean:
# Chord / Arc Geometry Inside a Sphere
If you have a circle (cross-section of the sphere), a “half-radius” could mean the radius of a circle inside that circle such that certain geometric proportions hold. For example:
* The distance from the center to a point where the arc subtends a special angle (like the Golden Angle or 90°).
* A point halfway along the surface arc, not halfway in straight-line distance.
# Surface Area or Volume Halving
Instead of cutting the linear radius in half, you can halve the surface area or volume and then find the radius that encloses that fraction.
* For a sphere, the radius that encloses half the volume is ≈ 0.7937 × R.
* For half the surface area, it’s exactly R / √2 ≈ 0.7071 × R.
# Standing Wave / Mode Geometry
In QAT terms, the “half-radius” might mean the location where a standing wave mode changes phase (node or antinode), so it’s geometrically halfway in the wave pattern, not the solid ball. This is common in atomic orbitals, where nodes are not physical shells but geometric loci defined by the wave equation.
# Golden-Ratio Constructions
You can also have a “half-radius” defined by Golden-ratio splits of the diameter or circumference. For example, a square inside a circle that touches at special points will create lengths that are “fractional radii” but come directly from geometric constraints rather than measuring halfway.

If I translate this back into QAT language:
A geometric half-radius is the radius implied by a symmetry, conservation, or proportional rule inside the spherical photon–electron interaction surface, not simply R/2. It’s one of the distances that appears naturally when the surface is divided according to wave-phase, probability (Ψ²), or special constants like the Golden Ratio.

If you want, I can draw the main QAT version of this half-radius as it appears in your electron-sphere + golden-rectangle model so it’s visually clear how it emerges from the geometry, rather than just from number-crunching. That would make the link to the Fine Structure Constant and Golden Angle much more direct. Would you like me to do that?