Paper V — Angular Information Locality and Spin

DOI: 10.5281/zenodo.17925682 Angular Information Locality and Spin Paper V of the Ordered-Dynamics Reconstruction Program Paul Cooneya aIndependent Researcher, Innisfil, Ontario, Canada E-mail: paul.co[email protected]to.ca Contents 1 Introduction and Scope 1 2 Internal Orientation and Operational Distinguishability 2 3 Angular Information Locality (AIL) 2 4 Compactness and Discrete Spin Sectors 2 5 Spin-1 2as Minimal Angular Capacity 3 6 Causal Factorization and Dirac Dynamics 3 7 Statistics from Bounded Angular Information 3 8 Conclusion 3 DOI: 10.5281/zenodo.17925682 apers I–IV reconstructed nonrelativistic and relativistic scalar quantum mechanics from bounded information growth and operational locality. The present paper extends the reconstruction to intrinsic angular degrees of freedom. We introduce Angular Information Locality (AIL): a bound relating intrinsic angular distinguishability to orbital angular resolution for localized systems. AIL forces compact internal rotation structure and discrete spin sectors. We show that the minimal nontrivial solution is spin-1 2, identified as the minimal carrier of directional information. Relativistic dynamics is then constrained by causal factorization: a quadratic invariant transport bound cannot propagate locally unless it factorizes into a first-order law. This necessity uniquely yields Dirac dynamics. Finally, bounded angular information forces indistinguishability, and the topology of rotation space yields the spin–statistics dichotomy. Spin is thus reconstructed as an informational and causal necessity rather than a group-theoretic postulate. Contents 1 Introduction and Scope Papers I–IV established that quantum dynamics arises as the saturation of sharp informational constraints. In particular: (i) bounded distinguishability growth forces complex amplitudes and Schr¨odinger dynamics (Paper I–II); (ii) relativistic compatibility refines transport to the invariant mass shell (Paper III); (iii) operational locality restricts interactions to local potentials (Paper IV). The present paper addresses the remaining structural ingredient at the single-particle level: intrinsic angular degrees of freedom. What is not assumed. We do not assume spinors, Clifford algebras, Pauli matrices, or Wigner classification. No representation theory is postulated. – 1 – What is derived. We show that bounded angular distinguishability forces compact rotation structure, discrete spin sectors, and selects spin-1 2as the minimal nontrivial carrier of orientation. Relativistic causality then forces first-order dynamics, yielding the Dirac equation uniquely. Statistics follow from bounded angular labeling. Logical position. This paper closes the single-particle reconstruction tier. Many-body entanglement and field theory follow in Papers VI–VII. 2 Internal Orientation and Operational Distinguishability Definition 1 (Internal orientation sector).Let denote the operational Hilbert space of a localized system. An internal orientation sector is a factor int such that ∼ =spatial ⊗int, and laboratory rotations act as U(R) = Uorb(R)⊗Uint(R) up to a global phase. Remark 1.This expresses operational independence: spatial motion and intrinsic orientation are independently controllable for localized systems. We quantify distinguishability using any operationally monotone metric (e.g. Fubini–Study). For small rotations generated by A, d(ψ, e−iθAψ) = |θ|qψ(A)+O(θ2). Thus angular distinguishability is controlled by generator variance. 3 Angular Information Locality (AIL) [Angular Information Locality] There exist constants κ>0 and σ2 0≥0 such that for any normalized state Ψ with finite variances and any axis ˆn, Ψ(ˆn· S)≤κΨ(ˆn· L)+σ2 0. Remark 2 (Operational meaning).Intrinsic angular resolution cannot exceed orbital angular resolution arbitrarily. Orientation information is bounded relative to spatial localization. AIL is the angular analogue of OIL (Paper II). 4 Compactness and Discrete Spin Sectors [Compact internal rotation structure] AIL forces the intrinsic rotation action to be a continuous projective unitary representation of (3) with compact image. Proof. Unbounded distinguishability under finite rotations would violate AIL. Compactness of the operational orbit follows. Continuity lifts the action to (3) or (2). [Discrete spin decomposition] The internal space decomposes as int =M s∈{0,1 2,1,... } Vs⊗ Ms, with finite-dimensional irreducible spin-ssectors. – 2 – 5 Spin-1 2as Minimal Angular Capacity [Minimal nontrivial angular carrier] Spin-1 2is the unique minimal nontrivial intrinsic sector compatible with AIL. Proof. Spin-0 carries no directional information. Spin-1 2provides exactly two distinguishable orientations — the minimal binary alternative required to encode a directional degree of freedom. Higher spins encode redundant angular information exceeding the minimal bound allowed by AIL for localized systems. Remark 3 (Qubit of orientation).Spin-1 2is the minimal quantum carrier of spatial orientation. It is selected by informational efficiency, not group theory. 6 Causal Factorization and Dirac Dynamics Paper III showed that relativistic transport satisfies the invariant quadratic bound E2=p2+m2. [Causal factorization] A quadratic transport constraint cannot generate local causal evolution unless it factorizes into a first-order law. Proof. Local propagation requires generators linear in derivatives. A quadratic generator propagates norms but not amplitudes locally. To enforce order-by-order causal evolution, the invariant constraint must factorize. [Dirac equation] The unique first-order Lorentz-covariant factorization is (iγµ∂µ−m)ψ= 0, with {γµ, γν}= 2ηµν. Remark 4.Spinors arise as the bookkeeping structure required to linearize invariant transport. 7 Statistics from Bounded Angular Information [Spin–statistics] Bounded angular labeling forces indistinguishability. Exchange corresponds to a 2πrotation. Hence: integer spin →bosons,half-integer spin →fermions. Proof. Indistinguishable angular labels imply exchange equivalence to rotation. The topology of (3) yields the dichotomy. 8 Conclusion Spin is reconstructed as an informational necessity. AIL forces compact rotation structure and discrete spin sectors. Minimality selects spin-1 2. Relativistic causality forces factorization of invariant transport, yielding Dirac dynamics. Statistics follow from bounded angular distinguishability. Spin, fermions, and the Dirac equation are thus not postulates, but the unique solutions to bounded angular information in a causal universe. – 3 –