Does this research claim to prove God exists?

No. The programme contains a cosmological speculation, the Hyperspace Recursive Intelligence Hypothesis, and it is graded as speculation on every surface that carries it. The shape of the idea: if intelligence can recursively generate intelligence, then what religious traditions describe as a creator could be conceived as a more advanced intelligence rather than a conventional supernatural deity. The site's registers hold that at concept level, flag it as theoretical and untested, and credit the deep prior work: John Wheeler's participatory universe, Teilhard de Chardin's Omega Point, Lee Smolin's cosmological natural selection.

The logical gap is stated rather than blurred. Even if every empirical claim in the programme were confirmed, that would establish at most that recursive intelligence can create intelligence, which makes a creator-like intelligence conceivable. Conceivable is not demonstrated: showing that creation by intelligence is possible would not show that our universe was actually created that way, and neither result would identify any such intelligence with the God of any tradition. Those are three separate steps, and the programme claims none of them as science.

Structurally, the cosmological step is deliberately severable: the synthesis is built so that no empirical claim, no measurement and no alignment result rests on HRIH being true. The measurable programme, scaling exponents, correction races and their kill conditions, stands or falls on its own evidence. The speculation is kept because the book is a synthesis and the question is old and serious; it is kept severable because the difference between a research programme and a belief system is that the first survives the removal of its most interesting story.