We identify the algebraic structure underlying the Bell correlations of Paper VIII. The condensate suppression weight sin4 θ/φ6 from the prior papers is precisely the quaternionic (ℍ-valued) norm on transition amplitudes: the factor sin4 θ = |sin2 θ|2 is the squared modulus of a quaternion-valued amplitude, contrasting with the complex norm |ψ|2 of standard quantum mechanics. The condensate correlation function xcond(θ) = 2 cos θ/(1 + cos2θ) is identified as the push-forward of the standard quantum correlation xQM(θ) = cos θ under the Hopf fibration S3 → S2, the natural map from quaternionic to complex amplitudes, with the angle relation tan(θℍ/2) = tan2(θℂ/2). Three foundational results are proved in the first part of the paper. Theorem (Hopf push-forward): xcond is the push-forward of xQM under the Hopf fibration, with xcond = xQM / (1 − ⅓(1 − xQM2)) from the Jacobian of S3 → S2. Theorem (Born rule from complexification): The Born rule P = |ψ|2 emerges from the condensate suppression weights when amplitudes are complexified via the topological winding phase Φ = Q · ω, where Q = 10 is the Hopf charge and ω is the solid angle subtended at mismatch angle θ; the quantisation of Q ∈ π3(S2) = ℤ is proved in companion Paper X. Theorem (Tsirelson gap decomposition): The raw CHSH of the condensate is CHSHraw ≈ 0.886; after rescaling by the planar geometric factor 8/3 this gives CHSHrescaled ≈ 2.363, leaving a gap of ≈ 0.47 below Tsirelson’s bound 2√2 ≈ 2.828. The gap decomposes exactly into a Fourier harmonic mismatch between xcond and xQM, and a geometric projection error from the ℍ → ℂ averaging; both vanish under the pointwise ℍ → ℂ projection. The exact Fourier coefficients are A2k+1 = (4 − 2√2) · (√2 − 3)k, with Parseval sum ∑An2/2 = 1/√2 (exact). Three further results extend the framework to measurement strength and density feedback. Theorem (CHSH(f) transition): A one-parameter family of condensate probabilities pf(θ) interpolates from the standard Born rule (f = 1, CHSHrescaled = 1.742) through the quaternionic condensate (f = 2, CHSHrescaled ≈ 2.363) to the projective limit (f → ∞, ceiling 8/3 ≈ 2.667 for leptons); the rescaled CHSH is strictly monotone in f and the ℍ → ℂ projection recovers Tsirelson’s bound at each f. Theorem (density feedback): Incorporating the density-feedback coupling β* = 0.452 from the Bogomolny self-consistency condition gives CHSHrescaled ≈ 2.341 at f = 2 and ≈ 2.60 at f = 4, in agreement with the Monte Carlo result of Paper VIII. Four results derive the photon coupling from first principles. Theorem (interaction Hamiltonian): The condensate–photon interaction Hamiltonian Hint = e JμAμ follows from the shift-symmetry Noether current of the k-essence condensate action P(X) = X/[φ6(1 + β*X)], giving matrix element ⟨φn|Hint|α⟩ = geff cos(φn − α), with effective coupling geff = 2eA0|∇ρ0|/[φ6(1 + β*X0)2]. The same factor φ6 suppresses both the Bell correlation law sin4 θ/φ6 and the photon coupling — a microscopic unity of Bell and Malus phenomena arising from the single condensate action. Theorem (Malus’s law): Normalising the interaction amplitude over two polarization outcomes gives the transmission probability T(φn,α) = cos2(φn − α); Malus’s law is a theorem, not an assumption. Theorem (photon winding number): The photon polarization observable in the condensate is uniquely the m = 2 Fourier mode of S1, determined by two constraints: the Malus transmission probability and the π-periodicity of polarization (photon polarization lives on ℝP1 = S1/ℤ2, giving a 2:1 covering of the condensate circle). Theorem (Tsirelson exact): The Stokes/Malus observable cos(2(φn − α)) gives CHSH = 2√2 = Tsirelson exactly; the Parseval factor ⅓ for the m = 2 mode on L2(S1) provides the geometric explanation of the QM normalisation convention, and is now derived from the condensate action. The condensate model predicts two distinct CHSH ceilings depending on the observable and particle species. For leptons measured with the quaternionic pcond observable xcond(θ), the projective ceiling is 8/3 ≈ 2.667 (a structural prediction for loophole-free lepton Bell tests). For photons measured with the Stokes/Malus observable, the ceiling is exactly 2√2 = Tsirelson; all existing loophole-free photon Bell experiments — including Poh et al. (2015) who reached S = 2.82759 ± 0.00051, within 0.03% of Tsirelson — are fully consistent with the condensate prediction. The structural gap 2√2 − 8/3 ≈ 0.161 is a falsifiable prediction for lepton Bell tests where the ℍ-valued observable applies. A post-quantum conjecture predicts CHSH = 8√2/3 ≈ 3.771 if a future experiment couples to the condensate via the pcond (quaternionic) weight rather than Malus transmission for a spin-1 particle; this exceeds Tsirelson’s quantum bound while satisfying no-signaling. The three Hurwitz division algebras ℂ, ℍ, 𝕆 play distinct and irreplaceable roles. ℍ supplies the |·|4 suppression giving the pcond observable and the lepton CHSH ceiling 8/3. ℂ supplies the topological winding phase enabling quantum interference and, crucially, the tensor-product structure of composite entangled systems — standard quantum mechanics occupies the ℂ slot not by postulate but by necessity, since ℍ ⊗ ℍ is not a quaternionic Hilbert space while ℂ ⊗ ℂ ≅ ℂ2. 𝕆 forces the three-generation structure and constrains the Hopf charge to Q ∈ π3(S2) = ℤ via the exceptional Albert algebra J3(𝕆) and its automorphism group G2 ⊂ F4. The Born rule, Tsirelson bound, and two-ceiling structure are all derived from, and consistent with, the parent condensate action of Papers I–IV.
We identify the algebraic structure underlying the Bell correlations of Paper VIII. The condensate suppression weight sin4 θ/φ6 from the prior papers is precisely the quaternionic (ℍ-valued) norm on transition amplitudes: the factor sin4 θ = |sin2 θ|2 is the squared modulus of a quaternion-valued amplitude, contrasting with the complex norm |ψ|2 of standard quantum mechanics. The condensate correlation function xcond(θ) = 2 cos θ/(1 + cos2θ) is identified as the push-forward of the standard quantum correlation xQM(θ) = cos θ under the Hopf fibration S3 → S2, the natural map from quaternionic to complex amplitudes, with the angle relation tan(θℍ/2) = tan2(θℂ/2). Three foundational results are proved in the first part of the paper. Theorem (Hopf push-forward): xcond is the push-forward of xQM under the Hopf fibration, with xcond = xQM / (1 − ⅓(1 − xQM2)) from the Jacobian of S3 → S2. Theorem (Born rule from complexification): The Born rule P = |ψ|2 emerges from the condensate suppression weights when amplitudes are complexified via the topological winding phase Φ = Q · ω, where Q = 10 is the Hopf charge and ω is the solid angle subtended at mismatch angle θ; the quantisation of Q ∈ π3(S2) = ℤ is proved in companion Paper X. Theorem (Tsirelson gap decomposition): The raw CHSH of the condensate is CHSHraw ≈ 0.886; after rescaling by the planar geometric factor 8/3 this gives CHSHrescaled ≈ 2.363, leaving a gap of ≈ 0.47 below Tsirelson’s bound 2√2 ≈ 2.828. The gap decomposes exactly into a Fourier harmonic mismatch between xcond and xQM, and a geometric projection error from the ℍ → ℂ averaging; both vanish under the pointwise ℍ → ℂ projection. The exact Fourier coefficients are A2k+1 = (4 − 2√2) · (√2 − 3)k, with Parseval sum ∑An2/2 = 1/√2 (exact). Three further results extend the framework to measurement strength and density feedback. Theorem (CHSH(f) transition): A one-parameter family of condensate probabilities pf(θ) interpolates from the standard Born rule (f = 1, CHSHrescaled = 1.742) through the quaternionic condensate (f = 2, CHSHrescaled ≈ 2.363) to the projective limit (f → ∞, ceiling 8/3 ≈ 2.667 for leptons); the rescaled CHSH is strictly monotone in f and the ℍ → ℂ projection recovers Tsirelson’s bound at each f. Theorem (density feedback): Incorporating the density-feedback coupling β* = 0.452 from the Bogomolny self-consistency condition gives CHSHrescaled ≈ 2.341 at f = 2 and ≈ 2.60 at f = 4, in agreement with the Monte Carlo result of Paper VIII. Four results derive the photon coupling from first principles. Theorem (interaction Hamiltonian): The condensate–photon interaction Hamiltonian Hint = e JμAμ follows from the shift-symmetry Noether current of the k-essence condensate action P(X) = X/[φ6(1 + βX)], giving matrix element ⟨φn|Hint|α⟩ = geff cos(φn − α), with effective coupling geff = 2eA0|∇ρ0|/[φ6(1 + βX0)2]. The same factor φ6 suppresses both the Bell correlation law sin4 θ/φ6 and the photon coupling — a microscopic unity of Bell and Malus phenomena arising from the single condensate action. Theorem (Malus’s law): Normalising the interaction amplitude over two polarization outcomes gives the transmission probability T(φn,α) = cos2(φn − α); Malus’s law is a theorem, not an assumption. Theorem (photon winding number): The photon polarization observable in the condensate is uniquely the m = 2 Fourier mode of S1, determined by two constraints: the Malus transmission probability and the π-periodicity of polarization (photon polarization lives on ℝP1 = S1/ℤ2, giving a 2:1 covering of the condensate circle). Theorem (Tsirelson exact): The Stokes/Malus observable cos(2(φn − α)) gives CHSH = 2√2 = Tsirelson exactly; the Parseval factor ⅓ for the m = 2 mode on L2(S1) provides the geometric explanation of the QM normalisation convention, and is now derived from the condensate action. The condensate model predicts two distinct CHSH ceilings depending on the observable and particle species. For leptons measured with the quaternionic pcond observable xcond(θ), the projective ceiling is 8/3 ≈ 2.667 (a structural prediction for loophole-free lepton Bell tests). For photons measured with the Stokes/Malus observable, the ceiling is exactly 2√2 = Tsirelson; all existing loophole-free photon Bell experiments — including Poh et al. (2015) who reached S = 2.82759 ± 0.00051, within 0.03% of Tsirelson — are fully consistent with the condensate prediction. The structural gap 2√2 − 8/3 ≈ 0.161 is a falsifiable prediction for lepton Bell tests where the ℍ-valued observable applies. A post-quantum conjecture predicts CHSH = 8√2/3 ≈ 3.771 if a future experiment couples to the condensate via the pcond (quaternionic) weight rather than Malus transmission for a spin-1 particle; this exceeds Tsirelson’s quantum bound while satisfying no-signaling. The three Hurwitz division algebras ℂ, ℍ, 𝕆 play distinct and irreplaceable roles. ℍ supplies the |·|4 suppression giving the pcond observable and the lepton CHSH ceiling 8/3. ℂ supplies the topological winding phase enabling quantum interference and, crucially, the tensor-product structure of composite entangled systems — standard quantum mechanics occupies the ℂ slot not by postulate but by necessity, since ℍ ⊗ ℍ is not a quaternionic Hilbert space while ℂ ⊗ ℂ ≅ ℂ2. 𝕆 forces the three-generation structure and constrains the Hopf charge to Q ∈ π3(S2) = ℤ via the exceptional Albert algebra J3(𝕆) and its automorphism group G2 ⊂ F4. The Born rule, Tsirelson bound, and two-ceiling structure are all derived from, and consistent with, the parent condensate action of Papers I–IV.
We extend the density-feedback Faddeev–Niemi Hopfion framework of Papers I–VII to quantum Bell correlations. The suppression law S(θ) = sin4θ/φ6, derived from the Hopf map Jacobian in Paper I, together with measurement-induced density feedback 1/(1 + β(ρ + δρ)), generates a local deterministic model that reproduces quantum Bell correlations. The model avoids Bell’s theorem because the hidden variable is not a classical probability distribution over independent local properties but a single shared geometric flux line, a deterministic field structure whose angular correlations emerge from the suppression law. Structurally this is the Aharonov–Bohm situation: a single global, source-fixed connection fixes the correlation through its holonomy, with no local field or signal passing between the sites. In planar laboratory geometries, the configuration of every existing loophole-free Bell test, the model achieves CHSH ≈ 2.6 for density feedback β ≈ 0.452 & β ∈ [0.1, 1.0], within 7–9% of Tsirelson’s quantum bound 2√2 ≈ 2.828. The framework-fixed coupling β* ≈ 0.452 (determined from first principles by the Bogomolny structure of Papers I–VII) gives CHSHrescaled = 2.611, sitting smoothly within the scan range and confirming the result is not fine-tuned. The scaling 2.57–2.62 of the geometric rescale factor is a genuine geometric correction, not a fit parameter. Fully isotropic 3D sampling – where the hidden flux-line direction is distributed uniformly over the sphere rather than confined to the measurement plane – yields CHSH ≈ 1.7 even with standard detectors. Standard quantum mechanics predicts CHSH = 2.828 independent of geometry; this is the model’s primary falsifiable distinction. The model makes three testable predictions: (1) reduced Bell violations in multi-satellite quantum networks probing true 3D geometries; (2) continuous CHSH variation from ≈ 1.8 to ≈ 2.6 as detector back-action is varied from weak to projective measurements; and (3) φ-periodic corrections to the exact cos4 correlation form at the few-percent level. Monte Carlo simulations (N = 100,000) confirm robust violations in planar and biased geometries with smooth degradation toward the classical regime. Python simulation code is provided for independent verification.
The complete mathematical structure of non-relativistic quantum mechanics — Hilbert space, canonical commutation relations, the uncertainty principle, the de Broglie relation, the Schrödinger equation, the Born rule, spin-statistics, and quantum entanglement — is shown to be derivable from three geometric axioms: the spatial universe is an S³ hypersurface propagating at velocity c through the quaternionic vacuum H ≅ ℝ⁴, subject to an isoclinic rotation enforced by the extremality condition. Phase space and quantisation of ℏ. The null constraint |v₄| = c forces the phase space to be T*S³. The Hopf fibration S¹ ↪ S³ → S² with first Chern class c₁ = 1 quantises the symplectic flux, determining Planck's constant ℏ as both the minimum symplectic area and the conserved Noether charge of the contact-conformal symmetry. The universality of ℏ — its identical value across all particles and interactions — is proved to be a consequence of the isoclinic condition: only isoclinic rotation preserves the Hopf fibration structure required for symplectic flux quantisation. Five principal results. (i) Born rule. The probability measure |ψ|² arises from marginalisation over the inaccessible Hopf fibre coordinate; it is not postulated. (ii) Quantum entanglement. Entanglement is topological, forced by the universal Hopf linking number lk = 1. (iii) Bell correlations. The correlator E(a,b) = −a·b is a consequence of SO(4) rotational invariance on S³, without hidden-variable supplementation. (iv) Classical–quantum decomposition. Classical physics is the radial projection of the membrane dynamics; quantum mechanics is its rotational projection. The two regimes are not separate postulates but orthogonal projections of a single geometric structure. (v) Quantisation obstruction. The Groenewold–van Hove quantisation obstruction does not apply to S³ because C^∞(S³) = U(su(2)) by the Peter–Weyl theorem; the algebra of observables is globally well-defined. Falsifiable predictions. Two predictions are stated with explicit observational protocols: — No gravitational decoherence: the geometric origin of ℏ implies that gravitational interaction cannot degrade entanglement coherence. Testable with the MAQRO space mission via matter-wave interferometry at mass scales m ~ 10⁻¹⁷ kg over baseline ~ 10³ km. — Perfect Bell correlations at arbitrary separation: the topological origin of entanglement (lk = 1) predicts no geometric suppression of Bell correlations with increasing spatial separation, in contrast to models coupling entanglement to background curvature.
Evidence Paper VI of the Existence Equation series. From Born rule to Bell inequality: one observer to two. In the parent work (The Existence Equation), the Born rule P = |Ψ|² is not an axiom but a derived consequence: the condensation term α|Ψ|²Ψ projects the continuous phase of the deviation field onto discrete binary outcomes (±1), and time-averaging the projected phase recovers cos θ correlations — verified by simulation on a 256³ lattice (hydrogen s-orbital, 30,000 steps). The Born rule emerges from the measurement geometry of one observer. If the Born rule for one observer is a consequence of ED's condensation mechanism, then for two observers sharing a single conserved field (Ψ_A + Ψ_B = 0), the Tsirelson bound S = 2√2 follows as pure measurement geometry. The Bell inequality violation is not a separate mystery — it is the natural two-observer extension of the same mechanism that produces the Born rule. Input: 2 items. Nothing else. #InputDescription 1Continuous conserved fieldΨ_A + Ψ_B = 0 (single deviation field shared between two sites) 2Binary measurementDiscrete outcomes ±1 from nonlinear condensation α|Ψ|²Ψ (same mechanism that produces the Born rule) No Hilbert space. No C*-algebra. No Born rule postulate. No tensor product. The measurement geometry decides. The logical chain: StepWhat happensWhere demonstrated 1α|Ψ|²Ψ projects continuous phase onto ±1Main paper (axiom + simulation) 2Time-averaging → P = |Ψ|² (Born rule)Main paper (256³ lattice, 30k steps) 3One observer: cos θ projectionMain paper (consequence of steps 1–2) 4Two observers + conserved field → cos θ correlationThis paper 5cos θ + binary outcomes + unit vectors → S = 2√2This paper Output: S_max = 2 × √2 — decomposed, not just calculated. FactorValueOrigin 2Bob's constraintTwo unit measurement vectors: |b|² + |b'|² = 2 √2Alice's diagonalDiagonal of Alice's 2D choice space (2 settings) 2√2ProductGeometric ceiling of binary measurement Why this matters: The geometric ceiling 2√2 exists for all theories, including classical ones. The "Classical" configuration in the table below has |u| + |v| = 2√2 — the ceiling is fully available — yet achieves only S = 2. What differs is the correlation function. Separable hidden variables produce piecewise-linear correlations. Only cos θ correlations — which ED derives from field conservation, not from Hilbert space — can saturate the ceiling. Ceiling vs. Saturation — the key separation: Configuration|S||u||v||u|+|v| Optimal (quantum)2.82841.41421.41422.8284 Uniform2.59811.00001.73212.7321 Tight2.38900.76541.84782.6131 Classical2.00001.41421.41422.8284 The constraint |u|² + |v|² = 4 verified to 10⁻¹⁶ precision across 2D, 3D, and 10D measurement spaces — the bound is dimension-independent and reflects the structure of binary measurement, not of physical space. The structural reading of Bell violation: Standard quantum mechanics treats the Born rule as an axiom and calculates 2√2 through Hilbert space operator norms. The Existence Equation derives the Born rule from condensation, then explains 2√2 as the geometric consequence of applying that same mechanism to two observers sharing a conserved field. The Bell inequality violation is not evidence for non-locality — it is evidence for non-separability of a single conserved deviation field. The Bell inequality violation is not spooky. It is geometric. It measures the diagonal of the binary measurement lattice. Evidence Program — Complete Summary: EPED mechanismWhat emergesKey result I α|Ψ|²Ψ strong-coupling → binary occupation; ∇²Ψ → nearest-neighbor exclusion PXP Hamiltonian (not assumed, derived) ‖T − H_PXP‖ = 0 II Same exclusion on 2D torus; spatial phase closure → winding number sectors Topological degeneracy (without Coulomb interaction); anyon braiding q-fold at ν = p/q; 2π/q phase (0.0% error) III ∇²Ψ + (1/c²)Ψ̈: simultaneous spatial and temporal phase closure Helical scars with spontaneous chirality |ΔF| = 1.78 × 10⁻¹⁵ IV α ≠ 0 → amplitude–phase coupling; commutative operators projected onto constrained subspace Non-commutative algebra with integer eigenvalues j_max = ⌊L/2⌋ (exact) V ½A²|∇Φ|² (phase energy) persists where |∇A|² (amplitude) has decayed Topological phase persistence as structural origin of dark matter ⟨v_θ·r⟩ = 2.0000 ± 0.0003 VI Axiom 1.1 (discreteness) → ±1 projection of continuous phase; 2 axes × √2 diagonal Tsirelson bound S = 2√2 without Hilbert space Exact (analytic + numerical) VII (1/c²)Ψ̈ → 0 (NR limit): ED reduces to GP; ε = k²/(4λ) controls validity boundary Josephson effect (DC + AC); GP as low-energy ED; f_Φ̇ force channel (no GP counterpart) DC: R² = 0.998; 4 falsifiable predictions at ε ~ 0.1 All code and figures: https://github.com/Galileo-leo/existence-equation Part of the Existence Equation Evidence Paper series (EP I–VII). Parent work: The Existence Equation: The Grammar of Persistence (doi:10.5281/zenodo.19364254)
How does a purely geometric projection — without Hilbert space, tensor products, or the Born rule — recover the exact quantum correlation E(a,b) = −cos θ and saturate Tsirelson's bound 2√2?We derive both results from the involution φ(x) = 9 − x on the digit alphabet {1, …, 9}, which generates four inversion pairs summing to 9 with barycenter μ = 4.5 and projection constant k_arc = 1/2. A hidden variable λ emitted by the source is projected sequentially onto detector axes a and b, each projection costing k_arc = 1/2. The product of two such projections yields E(a,b) = −cos θ, matching the quantum prediction exactly. The CHSH sum of four correlations at optimal angles gives S = 2√2, saturating Tsirelson's bound.The projection constant k_arc = 1/2 is established by seven independent paths: (I) global — barycenter over closure, 4.5/9; (II) local — half-span of the maximal pair, 3.5/7; (III) per-pair — every inversion pair individually encodes half-span/span = 1/2; (IV) positional — exact center of the sequence 1–8, with four elements on each side; (V) cadastral — the barycenters of the three row-pairs {2,4}, {5,7}, {8,1} average to 4.5; (VI) geometric — the interface mirror 11 divided by π = 22/7 gives 3.5, then 3.5/7 = 1/2; (VII) linker — 5.5/11 = 1/2, the barycenter of the phase cycle over the interface operator. Seven paths, sharing neither data nor method, converging to the same constant. This excludes numerical coincidence.No Hilbert space is assumed. No tensor product is constructed. No Born rule is invoked. The correlation is geometric, not probabilistic.New in v4.0: We add the complete arithmetic foundation — the Universal Fold Theorem proving that every multiplication table folds into inversion pairs summing to 9, the visual multiplication table mirrors, and the proof that base 10 is not arbitrary. We introduce entanglement as bipolar inversion: if x + φ(x) = 9, knowing x determines φ(x) instantaneously at any distance — not by causal signal but by structural constraint. This is quantum entanglement without mystery. Einstein's objection to "spooky action at a distance" is resolved: the correlation is not action, it is structure. The minimum of relation is 2 = DR(11), the interface operator — the smallest cost of knowing the other.Keywords: Tsirelson's bound, quantum correlations, Bell inequality, CHSH, geometric projection, hidden variables, digital roots, inversion symmetry, entanglement, EPR, sevenfold convergence.
What quantum mechanics calls "observation" has lumped together three interactions of different character: one that switches the discrete state of the partner, one that leaves the partner's state unswitched and shifts only the internal phases, and one that severs the mutual phase relation. This paper places the substance of observation in the second, the readout of the interference of a phase difference, and adopts only three premises. (P1) The distance between two parties is set by the ratio of a shared viewing angle to the partner's size; only ratios are readable and the absolute scale is not. (P2) An interaction is a single relation in which emission and absorption hold simultaneously; the speed of light appears on the readout side as a conversion factor between local clocks. (P3) The phase of a relation carries a harmonic order n that depends on the kind of wave. From these, the fact that a probability is the product of an amplitude and its conjugate (the form of the Born rule), the two-party correlation E = ∓cos nθ, the upper bound 2√2 of the CHSH quantity (the Tsirelson bound), and no-signaling follow in a few lines. The normalization that turns this into a probability is the division that discards the size ratio, i.e. the choice of the η = 0 section. As a consequence, the bound is 2√2 independently of the kind of wave, and the detector angle giving the maximal violation shrinks in proportion to 1/n (45° for spin-½, 22.5° for photons, 11.25° for gravitational waves). For a localized relation that is a sum of harmonics the correlation is the autocorrelation of the pulse shape; the bound holds for any weights, equality is attained only for a single harmonic, and a rectangular pulse reduces to a local hidden-variable model with S ≤ 2. Japanese original (authoritative) and English translation, with TeX sources, PDFs, five figures (Japanese and English labels) and the figure-generating program.
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.