Observer First Physics

We derive finite-dimensional quantum mechanics from five constraints on observers and one invariance principle. The constraints: observers are finite physical systems inside the world they observe, with finite causally-ordered memory, bounded information storage, interaction-only epistemology, and physical states defined by indistinguishability under all admissible protocols. The invariance principle: any transformation that changes no outcome of any physical process has no physical content — redescription cannot do physics.

The central result is the redecomposition argument. Once the state space is established as ℝⁿ, any orthogonal rotation of its description basis is pure redescription by the invariance principle, so the distinguishability measure must be O(n)-invariant. The unique O(n)-invariant norm on ℝⁿ is Euclidean. This forces the inner product, the complex structure, and the Hilbert space. Group structure on the state space is derived via a cancellativity lemma and the Grothendieck construction. The phase group U(1) is derived from the requirement that free dynamics of isolated systems preserve probability assignments — a consequence of the observer constraints, not an assumption. The qubit Born rule gap is closed internally via observer embedding: the observer's presence guarantees every system appears in a joint space of dimension ≥ 4, making Gleason's theorem universally applicable.

No alternative survives all constraints: classical probability fails at U(1), real and quaternionic Hilbert space fail at the scalar field selection, super-quantum theories fail at the norm, the exceptional Jordan algebra fails at local tomography.

The derivation is finite-dimensional throughout. The infinite-dimensional limit is open.

The conclusion: in the finite-dimensional setting, quantum mechanics is not a feature of the universe that requires explanation. It is the unique structure observation necessarily has when the observer is inside the world.