No problem here is you proof - although a bit long:

1. THEOREM: Let a semantic frame be defined as Ω = (Σ, R), where

Σ is a finite symbol set and R is a finite set of inference rules.

Let Ω′ = (Σ′, R′) be a candidate successor frame.

Define a frame jump as: Frame Jump Condition: Ω′ extends Ω if Σ′\Σ ≠ ∅ or R′\R ≠ ∅

Let P be a deterministic Turing machine (TM) operating entirely within Ω.

Then: Lemma 1 (Symbol Containment): For any output L(P) ⊆ Σ, P cannot emit any σ ∉ Σ.

(Whereas Σ = the set of all finite symbol strings in the frame; derivable outputs are formed from Σ under the inference rules R.)

Proof Sketch: P’s tape alphabet is fixed to Σ and symbols derived from Σ. By induction, no computation step can introduce a symbol not already in Σ. ∎

2. APPLICATION: Newton → Special Relativity

Let Σᴺ = { t, x, y, z, v, F, m, +, · } (Newtonian Frame) Let Σᴿ = Σᴺ ∪ { c, γ, η(·,·) } (SR Frame)

Let φ = “The speed of light is invariant in all inertial frames.” Let Tᴿ be the theory of special relativity. Let Pᴺ be a TM constrained to Σᴺ.

By Lemma 1, Pᴺ cannot emit any σ ∉ Σᴺ.

But φ ∈ Tᴿ requires σ ∈ Σᴿ \ Σᴺ

→ Therefore Pᴺ ⊬ φ → Tᴿ ⊈ L(Pᴺ)

Thus:

Special Relativity cannot be derived from Newtonian physics within its original formal frame.

3. EMPIRICAL CONFLICT Let: Axiom N₁: Galilean transformation (x′ = x − vt, t′ = t) Axiom N₂: Ether model for light speed Data D: Michelson–Morley ⇒ c = const

In Ωᴺ, combining N₁ and N₂ with D leads to contradiction. Resolving D requires introducing {c, γ, η(·,·)}, i.e., Σᴿ \ Σᴺ But by Lemma 1: impossible within Pᴺ. -> Frame must be exited to resolve data.

4. FRAME JUMP OBSERVATION

Einstein introduced Σᴿ — a new frame with new symbols and transformation rules. He did so without derivation from within Ωᴺ. That constitutes a frame jump.

5. FINALLY

A: Einstein created Tᴿ with Σᴿ, where Σᴿ \ Σᴺ ≠ ∅

B: Einstein was human

C: Therefore, humans can initiate frame jumps (i.e., generate formal systems containing symbols/rules not computable within the original system).

Algorithmic systems (defined by fixed Σ and R) cannot perform frame jumps. But human cognition demonstrably can.

QED.

BUT: Can Humans COMPUTE those functions? (As you asked)

-> Answer: a) No - because frame-jumping is not a computation.

It’s a generative act that lies outside the scope of computational derivation. Any attempt to perform frame-jumping by computation would either a) enter a Goedelian paradox (truth unprovable in frame),b) trigger the halting problem , or c) collapse into semantic overload , where symbols become unstable, and inference breaks down.

In each case, the cognitive system fails not from error, but from structural constraint. AND: The same constraint exists for human rationality.