Observer First Physics

We derive the mathematical structure of finite-dimensional quantum mechanics—complex Hilbert space, tensor-product composition, the Born rule, and unitary dynamics—from a single invariance principle (Axiom A0) together with five operational constraints on a minimal embedded physical observer: a bounded, finite-energy record-keeping system that updates its causally ordered internal memory solely through local physical interactions along its continuous worldline. Axiom A0 asserts that physical reality consists solely of what is invariant under physically inert transformations—pure redescription.

The central technical result is the redecomposition argument: any orthogonal rotation of the description basis is pure redescription, so the distinguishability measure must be O(n)-invariant, and the unique O(n)-invariant norm on ℝⁿ is Euclidean. Continuous proper time is constituted by the observer's memory and worldline, not assumed as background. Group structure is derived via a cancellativity lemma and the Grothendieck construction. The phase group U(1) is shown to be the physically inert freedom of free dynamics: an observable-algebra dichotomy establishes that whether or not off-diagonal observables couple distinct invariant planes, the global U(1) rotation is the operationally inert residual freedom. The qubit Born-rule gap is closed by the observer's necessary presence in any joint measurement.

Three consequences extend the core derivation. First, the Principle of Least Action is not merely recovered formally: recent work of Lohmiller and Slotine establishes the exact converse, that the Schrödinger equation can be solved exactly from classical multi-valued action and density with no semi-classical approximation; that construction is integrated here. Second, the Born rule is identified as precisely the structure the action-based construction cannot supply, so the two frameworks are exactly complementary. Third, local U(1) phase freedom—a consequence of the observer's locality—forces a gauge-invariant interaction structure with a massless mediating field; the identification of this structure with electromagnetism is confirmed empirically by the Standard Model, but that identification is not derived here.

Classical probability, real and quaternionic Hilbert spaces, super-quantum theories, and the exceptional Jordan algebra J₃⁸ are each excluded at a specific identified step. The framework is finite-dimensional throughout; extensions to quantum field theory and general relativity are identified as open problems.