Here's the actual structure of the theorem. A frame function is defined as f: (unit vectors) → [0,1] such that for every orthonormal basis {eᵢ}, Σᵢ f(eᵢ) = 1. That condition — non-negative, additive over any resolution of the identity, normalized to 1 — is literally the definition of a non-contextual probability measure on the lattice of subspaces. So yes, in that sense he's right that probability-like structure is baked into the premise. Gleason isn't deriving "there should be a probability measure" out of pure logic with no measure-theoretic assumptions at all. That's a fair and fairly sophisticated observation.
But that's not the same as saying the theorem "is just about the modulus of the projections" or that it presupposes the modulus. It's the opposite. The premise is completely agnostic about what mathematical form f takes — it could a priori be anything satisfying that additivity condition. The theorem's entire content, the hard part Gleason actually proved, is that any function satisfying that weak structural axiom is forced to equal Tr(ρP) for some density operator ρ — and for pure states, that collapses to |⟨ψ|φ⟩|², the modulus squared. The modulus-squared form is the conclusion, not an input assumption.
So the accurate way to put it: Gleason assumes you have something behaving like a probability measure (additive, non-contextual), and derives that it must take the specific trace/modulus-squared form. Your friend has collapsed those two into each other — treating the output (the modulus) as if it were smuggled into the input. It isn't. That's precisely why the theorem is considered a nontrivial derivation of the Born rule rather than a restatement of it: you get the specific quantitative rule out of a much weaker qualitative constraint.
If he wants to press the strongest honest version of his objection, it's this: Gleason's theorem shows the Born rule is the unique probability measure compatible with quantum logic — but it doesn't explain why nature uses a probability measure at all, or why measurement outcomes are stochastic in the first place. That's a legitimate limitation. But "nothing to do with probability" and "assumes the modulus" are not that objection — they're just inaccurate."
John K Clark