[Richard777 (OP)]

A stationary, massive object is assumed to impose a “force” upon surrounding space. The force may be called the “Hawking force” as it is associated with Hawking temperature. This force may be represented as a vector.

If one condition applies, and if the components are suitably defined, then the components will give the definition of Hawking temperature and length contraction.
See reference attachment.

[Bored chemist]

"Can Hawking temperature be associated with a force?"
Not easily.
One is a vector and the other is scalar.

[evan_au]

“Hawking force” as it is associated with Hawking temperature.

Without having read the attachment...

Hawking radiation does produce a force, but:
- The temperature is in the nanoKelvin range, so the force is exceedingly small
- The event horizon has a uniform temperature, so the force is equal in all directions, so there is no net force on the black hole
- A nearby object will feel an outward radiation pressure from the black hole (at a temperature of nanoKelvins), and a radiation pressure from every other direction due to Cosmic Background radiation at 2.7 Kelvin. The latter overwhelms any radiation pressure from the black hole (by a factor of billions)

The inwards gravitational force of the black hole overwhelms the outward radiation pressure from the black hole by a factor much larger than billions.

[alancalverd]

The quantization rule states that only an integer number of waves may be imposed upon a
circumference; nλ = 2πr

seems irrelevant to Hawking radiation but an integral part of the derivation.
You also have a problem in beginning with a vector and ending with a scalar, as BC pointed out. Your argument loses its direction early on, where the force vector mysteriously loses its direction. 

But full marks for neatness, as usual.

[Bored chemist]

It says
"A condition is required for emission; A1 = A2 "
Why?