This paper proves a finite-dimensional structural recovery-and-obstruction theorem for Bell-local factorization inside projective phase structure. The primitive object is a completion-locked projective probability datum: a bipartite state, fixed local projective contexts, the Born–Haar reduction rule, and fixed normalization. A projective measurement context carries a compact relative phase torus, and Haar averaging over that torus gives the nonselective Born/Lüders projection channel. The Born probability table is therefore determined before any hidden-variable or Bell-local representation is tested. Bell-local factorization is determined as a derived classical recovery representation of the locked Born table. The paper proves canonical recovery regimes: pure product rays are exactly Segre-factorized projective phase classes, mixed separability yields Bell-local recovery, and one-sided commutative local readout pairs yield Bell-local recovery through a joint-outcome hidden variable. Conversely, a Clauser–Horne–Shimony–Holt (CHSH) value larger than 2 proves obstruction: no Bell-local representation exists for the chosen datum. No-signaling follows from the trace identities of the projective Born law, while Tsirelson’s bound 2√2 follows from the operator geometry of admissible binary projective observables. The theorem chain therefore determines the hierarchy: projective phase carrier → Born–Haar reduction → Segre/separable recovery or Bell-factorization obstruction → no-signaling and Tsirelson admissibility. This hierarchy also determines the intermediate cases: Bell–CHSH violation is a property of the complete projective measurement datum, not of entanglement alone. Product states, separable mixtures, commutative readouts, and nonoptimal settings can remain below the classical threshold even inside quantum theory. The constructive spin instantiation is derived explicitly. For the diagonal SU(2) action on ℂ² ⊗ ℂ², rotational invariance selects the unique spin-zero singlet line. The singlet is entangled by failure of Segre factorization, its Pauli spin correlations satisfy E(a,b) = −a·b, and optimal admissible spin readouts attain the exact Tsirelson value 2√2. In this derivation, 2√2 is not introduced by experiment-fitting; it is the Hilbert-space admissibility ceiling of the projective phase datum, realized exactly by the spin-zero SU(2) sector. License note: Distributed under CC BY-NC-ND 4.0.
This paper proves a finite-dimensional structural recovery-and-obstruction theorem for Bell-local factorization inside projective phase structure. The primitive object is a completion-locked projective probability datum: a bipartite state, fixed local projective contexts, the Born–Haar reduction rule, and fixed normalization. A projective measurement context carries a compact relative phase torus, and Haar averaging over that torus gives the nonselective Born/Lüders projection channel. The Born probability table is therefore determined before any hidden-variable or Bell-local representation is tested. Bell-local factorization is determined as a derived classical recovery representation of the locked Born table. The paper proves canonical recovery regimes: pure product rays are exactly Segre-factorized projective phase classes, mixed separability yields Bell-local recovery, and one-sided commutative local readout pairs yield Bell-local recovery through a joint-outcome hidden variable. Conversely, a Clauser–Horne–Shimony–Holt (CHSH) value larger than 2 proves obstruction: no Bell-local representation exists for the chosen datum. No-signaling follows from the trace identities of the projective Born law, while Tsirelson’s bound 2√2 follows from the operator geometry of admissible binary projective observables. The theorem chain therefore determines the hierarchy: projective phase carrier → Born–Haar reduction → Segre/separable recovery or Bell-factorization obstruction → no-signaling and Tsirelson admissibility. This hierarchy also determines the intermediate cases: Bell–CHSH violation is a property of the complete projective measurement datum, not of entanglement alone. Product states, separable mixtures, commutative readouts, and nonoptimal settings can remain below the classical threshold even inside quantum theory. The constructive spin instantiation is derived explicitly. For the diagonal SU(2) action on ℂ² ⊗ ℂ², rotational invariance selects the unique spin-zero singlet line. The singlet is entangled by failure of Segre factorization, its Pauli spin correlations satisfy E(a,b) = −a·b, and optimal admissible spin readouts attain the exact Tsirelson value 2√2. In this derivation, 2√2 is not introduced by experiment-fitting; it is the Hilbert-space admissibility ceiling of the projective phase datum, realized exactly by the spin-zero SU(2) sector. License note: Distributed under CC BY-NC-ND 4.0.
This paper proves a finite-dimensional carrier-descent theorem for canonical projective quantum measurement channels. Under explicit universal-carrier, Ward-admissibility, and finite projective-sector hypotheses, each nonzero sector carries a compact U(1) phase carrier, and a finite family of orthogonal sectors determines a central sectorwise phase torus. Haar averaging over this compact carrier gives the conditional expectation E_P(X) = ∑ⱼ Πⱼ X Πⱼ, whose trace-dual state map is the Lüders nonselective measurement channel Δ_P(ρ) = ∑ⱼ Πⱼ ρ Πⱼ. The fixed observable algebra is the block algebra B_fix = {X : [X, Πⱼ] = 0 for all j}, while the classical outcome algebra is its center Z_out = span{Π₁, …, Πₘ}. The Born weights pⱼ(ρ) = Tr(Πⱼρ) are the unique coefficients representing the restricted state on Z_out, and selective measurement is conditioning on the corresponding boundary record. Except in the trivial one-sector case, the Lüders projection is idempotent, noninvertible, and nonmultiplicative on the full matrix algebra; hence the associated boundary-channel datum is a boundary-channel projection rather than full reversible Heisenberg–Schrödinger collapse, i.e. full-algebra unitary equivalence. The paper proves that canonical finite projective measurement is the endpoint of a carrier-to-phase-to-Haar boundary-channel descent: phase selection determines Haar projection, Haar projection determines the Lüders channel, central restriction determines the Born weights, and the collapse criterion certifies the obstruction to full reversible Heisenberg–Schrödinger collapse in every nontrivial projective context. License note: Distributed under CC BY-NC-ND 4.0.
This paper proves a finite-dimensional structural theorem identifying canonical projective quantum measurement as a boundary-channel projection. For a finite projection-valued measurement context P = {Π₁, …, Πₘ}, the central sectorwise phase action Uθ = ∑ⱼ e^{iθⱼ}Πⱼ has normalized Haar average E_P(X) = ∫ Uθ X Uθ* dθ = ∑ⱼ Πⱼ X Πⱼ. Its trace-dual state-side map is the Lüders nonselective channel Δ_P(ρ) = ∑ⱼ Πⱼ ρ Πⱼ. The theorem characterizes this Lüders, or minimal-disturbance, channel: a projection-valued measurement fixes the outcome weights, but in degenerate sectors it does not by itself determine every compatible instrument. The additional structure used here is the sectorwise phase symmetry together with preservation of the full fixed block algebra. The fixed observable algebra is B_fix = {X : [X, Πⱼ] = 0 for all j}, while the classical outcome algebra is its center Z_out = span{Π₁, …, Πₘ}. The Born weights pⱼ(ρ) = Tr(Πⱼρ) are the unique weights representing the restricted state on Z_out, and selective measurement is conditioning on the corresponding boundary record. Except for the trivial one-sector context, the Lüders projection is idempotent, noninvertible, and nonmultiplicative on the full matrix algebra; it therefore defines a boundary-channel operation rather than full reversible Heisenberg–Schrödinger collapse. The results are interpretation-neutral and isolate the operator-theoretic structure that any finite projective measurement account using the canonical Lüders update must preserve. License note: Distributed under CC BY-NC-ND 4.0.
For finite-dimensional binary systems and their bipartite composites, this paper establishes a conditional representation chain from a physical–observation dual-axis structure to quantum Bell correlations. An observational fact is jointly generated by a complete physical state and a complete observation state. Additivity and associativity of coherent responses, two-sided cancellation, and a full circle group of scalar phase transport select C among finite-dimensional unital associative real response algebras. Reversible mixing then determines a Hermitian metric up to an overall positive factor. Normalized two-channel responses form S3, and quotienting by their common phase yields the rank-one observation-event space and, with an outcome label, the pure observation fiber S3/U(1)=CP1≃S2. The universal bilinear composite of two independent systems is C4. Probabilities are first assigned within complete measurement contexts. Events with the same response filter are operationally interchangeable, so the local assignments glue along the observation axis into a positive orthogonally additive measure on the full projection lattice. Gleason's theorem then gives the unique trace representation p(P∣x)=Tr(ρxP), while composite marginal consistency returns the trace pairing to each local binary fiber. Every two-qubit correlation therefore has the universal form Eρ(a,b)=aTTρb. Joint rotational invariance and perfect equal-axis anticorrelation uniquely fix Tρ=−I3, producing the singlet cosine law E=−a⋅b=−cosθ. Finally, the local commutators in the squared CHSH operator confine all realizable correlations to 22. Fine-measurement completeness supplies saturating directions, while sharp preparation and reversible covariance make the singlet accessible. Thus CP1, the Born pairing, the cosine correlation law, and the Tsirelson bound arise within one chain linking phase structure, independent composition, and cross-context probability consistency.
We derive the Born rule as the unique outcome statistics induced by a canonical symmetry projection beneath nonselective measurement and unobservable symmetry structure. Concretely, given a compact symmetry group acting by conjugation, we identify the quantum channel that removes symmetry-orbit dependence while preserving exactly the invariant operator algebra. In a standard von Neumann algebraic setting, we prove that Haar averaging yields a normal, unital, completely positive idempotent map whose range is the fixed-point algebra, and hence a conditional expectation onto the invariant subalgebra. Moreover, any normal channel that is invariant under the symmetry action and fixes the invariant algebra pointwise must coincide with this Haar-induced projection. Specializing to projective measurement, Haar averaging over the associated phase-rotation symmetry yields the Lüders (pinching) reduction as the unique normal update map satisfying the same invariance and fixpoint requirements. The induced outcome weights are characterized as the unique probabilities specified by the restriction of the quantum state to the commutative outcome algebra, and they are preserved by the canonical reduction. As a further structural consequence of this symmetry projection, the von Neumann entropy is monotone under the induced reduction. The channel-based derivation applies in all finite dimensions, including qubits, and addresses a structural question not resolved by Gleason-type and positive operator-valued measure (POVM)-based classification results: it identifies the unique normal symmetry-forced reduction channel associated to a fixed measurement context, rather than characterizing admissible global probability functionals. License note: Distributed under CC BY-NC-ND 4.0.
This paper gives a finite-dimensional, first-principles derivation of boundary invariance from completion-locked projective phase and channel data. No-signaling and conserved boundary structure are derived from fixed carrier and channel structure, rather than imposed as external constraints. The central invariant is the Heisenberg fixed-point condition Φ*(X) = X for every boundary observable X in the locked boundary algebra. Equivalently, boundary expectations and boundary POVM laws are preserved uniformly over input states exactly when the admissible complement channel fixes the boundary algebra in the Heisenberg picture. Inside a completion-locked datum, a label encoded only through an admissible complement channel has zero statistical channel into fixed boundary outcomes. Here completion-locked means that the projective phase carrier, boundary algebra, admissible boundary-observable charts, state and normalization data, admissible channel class, and channel-extraction maps are fixed before Bell tables, tensor-factor partitions, random-unitary selections, or quotient channels are compared. The primitive carrier is the projective phase bundle U(1) → S(H) → P(H). Tensor factorizations, Bell partitions, product tests, and local boundary observable algebras enter downstream as locked charts. The theorem sequence is: projective phase carrier → locked boundary algebra → admissible complement channel → Heisenberg fixed point → invariant boundary law. The theorem is realized in three explicit boundary-channel settings: projective quantum no-signaling; Gibbs-weighted random-unitary symmetry-selection channels, whose invariant algebra is the active commutant; and finite equilibrium-combinatorial quotient realizations, whose barycentric extraction is a boundary channel with diamond-norm control of the selected equilibrium limit. Taken together, these results place projective no-signaling, random-unitary active-commutant conservation, and equilibrium-quotient channel convergence under one exact finite-dimensional fixed-point law: admissible complement dynamics preserve boundary data exactly when their Heisenberg dual fixes the locked boundary algebra. The result is a finite-dimensional boundary framework for exact no-signaling, steering without unconditioned boundary signaling, conserved commutants, zero label-to-boundary information, and quotient-selected channel convergence. License note: Distributed under CC BY-NC-ND 4.0.
We derive the Bell–CHSH inequality as a conditional theorem inside Charlton Geometry (CG), rather than as an independent axiom about “local realism”. The key CG primitive is non-invertibility, η(ω) = η(−ω), which forces an observational quotient and produces irrecoverable “lost state” carried by polarity and intrinsic phase. In a bipartite experiment, Bell’s bound is recovered exactly when one adds an additional factorization condition asserting that outcomes depend only on local settings and a local lift choice, i.e. that the lost-state fiber decomposes into independent local fibers. Within CG, the natural geometric coupling of two regions through shared lift and holonomy generically violates that factorization while preserving operational no-signaling, so the Bell inequality becomes a theorem about a restricted subclass of quotients rather than a universal constraint. Using the Born brick as an upstream theorem that fixes the unique binary law cos2(θ/2) on the spin-lifted setting space, we recover the standard singlet correlation E(a, b) = −a °§ b and the Tsirelson bound 2√2 as the sharp CG-compatible CHSH maximum. In this framing, Bell violation is not an ontic miracle but the expected signature of a globally shared, non-invertible geometric substate whose reversal cannot be represented