Structural Postulates

The state space Σ is a finite-dimensional compact smooth manifold. Now derived as Theorem 0.2: the base case S¹ is a Lie group manifold (from U(1) loop closure), the inductive step uses level sets of relational invariants (regular value theorem preserves manifold structure), and finite dimensionality follows from the Delone finite local complexity of the observer network.

Promoted to Theorem 0.2. The manifold structure follows from the constructive mechanism by which observer state spaces are built: U(1) Lie group orbits at the base, bootstrap level-set construction at each step, finite local complexity from aperiodic order bounding the dimension.

A Riemannian metric g on Σ that is G_O-invariant: φ_t* g = g for all t.

Now a theorem (Theorem 0.1): any smooth compact manifold admits a Riemannian metric (partition of unity), and Weyl averaging over the compact U(1) action produces a G_O-invariant metric. Replaces the former structural postulate.

The joint state space Σ₁ × Σ₂ carries a symplectic form ω compatible with the U(1) × U(1) action of the constituent observers.

Now a theorem (Theorem 0.1): Axiom 3 gives each observer a Noether pair (Loop Closure, Theorem 3.1), a Noether pair defines a symplectic form, and the product of symplectic manifolds is canonically symplectic (Abraham & Marsden, Proposition 3.2.10).

The coherence measure C is at least C² (twice continuously differentiable), with a positive-definite Hessian metric on state space.

Now a theorem (Theorem 0.1): the Born Rule (derived via coherence-operational) forces statistical regularity for finite-dimensional systems, and the Fisher metric chain (Čencov uniqueness + Proposition 4.1 of Fisher Information Metric) identifies the coherence Hessian as ℏ × Fisher metric, which is C∞ and positive definite.

The coherence measure C and the metric g are invariant under spacetime translations xμ → xμ + aμ.

Follows from the axioms' location-independence: coherence conservation and observer definitions do not reference absolute position, so the derived geometry inherits translation symmetry.

The metric g_μν is locally determined by the relational invariant density ρ_I, with curvature proportional to ρ_I.

Derived from coherence subadditivity (Axiom 1, C4): the action duality links the spacetime metric to the coherence Hessian; subadditivity ensures the Hessian varies with observer content; action equality transfers this to the spacetime metric. Locality from finite c; coupling to ρ_I from Noether identification.

The self-consistency relation between curvature and coherence content involves at most second derivatives of the metric g_μν.

Now a theorem (Coherence Lagrangian, Theorem 6.0): Ostrogradsky's instability theorem shows higher-derivative Lagrangians have unbounded Hamiltonians, violating the Lyapunov stability required by Axiom 3 (loop closure). Second-order locality is therefore forced by loop closure stability.

The coherence measure restricted to transition amplitudes satisfies C(|ψ⟩) = ⟨ψ|ψ⟩ = Σ_k |ψ_k|².

Now a theorem (Theorem 0.1): the unique continuous, U(1)-invariant, composition-compatible coherence functional on quantum states is the squared norm, via the Cauchy multiplicative equation. Derived in Coherence as Physical Primitive (Theorem 4.1).

Transition amplitudes are single-valued on the universal cover of configuration space. The wave function is a section of a flat line bundle with holonomy Hom(π₁(Q), U(1)).

Now a theorem (Theorem 0.1): Axiom 3 (loop closure) applied to configuration space forces single-valued phases on the universal cover; the holonomy classification follows from standard covering-space theory. The Laidlaw–DeWitt condition is derived, not postulated.

Each fermion generation corresponds to a class of half-integer winding loops classified by dominant alignment with one of three independent generators of so(3), partitioned by Voronoi decomposition of S².

Since d = 3 spatial dimensions provide exactly three independent rotation axes, the half-integer winding loops of fermions naturally partition into three sectors, one per generation.

Phase comparison between observers at distinct spacetime points requires smooth parallel transport, described by a connection 1-form A on a principal U(1) bundle P → ℳ over the spacetime manifold.

Now a theorem (Theorem 0.1): local gauge freedom is inevitable from relational physics + finite signal speed (Theorem 2.1), and the classification theorem for connections on principal bundles (Kobayashi & Nomizu) shows the principal U(1) bundle with connection is the unique smooth implementation.

The self-consistency condition for the U(1) gauge connection involves at most second derivatives of A_μ — equivalently, at most first derivatives of F_μν.

Now a theorem (Coherence Lagrangian, Theorem 6.0): Ostrogradsky instability excludes higher-derivative gauge Lagrangians, as the resulting unbounded Hamiltonian violates loop closure stability (Axiom 3).

The observer’s phase algebra 𝒶 forms a normed division algebra over ℝ: for all a, b ∈ 𝒶, |ab| = |a| · |b|. Combined with three independent imaginary units (one per spatial axis), this selects 𝒶 = ℍ (quaternions).

Packages three physical requirements: (i) bilinearity of phase composition, (ii) norm-preservation from coherence conservation, (iii) invertibility of phase transformations. With exactly three imaginary units, Hurwitz’s theorem uniquely selects the quaternions.

The self-consistency condition for the SU(2) gauge connection involves at most second derivatives of Wᵃ_μ — equivalently, at most first derivatives of the field strength Wᵃ_μν.

Now a theorem (Coherence Lagrangian, Theorem 6.0): Ostrogradsky instability excludes higher-derivative gauge Lagrangians, as the resulting unbounded Hamiltonian violates loop closure stability (Axiom 3).

Each level of the bootstrap hierarchy saturates the phase algebra to the next normed division algebra via the Cayley-Dickson construction: ℝ → ℂ → ℍ → 𝕆. Hurwitz’s theorem terminates the sequence at 𝕆.

Extends the algebraic completeness postulate (Weak Interaction S1) to the full bootstrap hierarchy. Now derived as a theorem: Bootstrap → Division Algebras proves that coherence conservation forces Cayley-Dickson doubling at each level.