We characterize, within quantum theory, the two-party correlation E(θ_A, θ_B) = −cos(θ_A − θ_B) that saturates the Tsirelson bound |S| = 2√2. Let E be the correlation kernel of a U(1)-covariant, perfectly anticorrelated bipartite quantum realization (a jointly invariant state with covariant local dichotomic observables). We prove that −E is then automatically a positive-definite function on U(1), because it equals a diagonal matrix coefficient of an associated unitary representation. We prove a converse: every normalized nonnegative harmonic spectrum is realized by an explicit bipartite quantum model with genuinely local observables. In the real-even category we prove that spectral extremality of the kernel is equivalent to irreducibility of its real cyclic (GNS) representation, and we separate this from minimality of the cyclic realization: a reducible multi-harmonic kernel still has a minimal cyclic realization. Adding one axiom — Primitive Modal Simplicity, that the response occupies a single irreducible real sector — together with faithfulness of that active carrier, singles out −cos Δ. We prove by an explicit, reproducible counterexample (a spin-3/2 singlet with the dichotomic observable sign(S_x), giving the kernel −(7/8)cos Δ − (1/8)cos 3Δ) that modal simplicity is independent of purity, covariance, perfect anticorrelation, one reference dichotomic observable per party, absence of classical randomness, and minimality of the cyclic realization. We state carefully what is and is not established: this is a characterization of a scalar correlation-function family, not a reconstruction of the quantum state/observable framework, the Born rule, or a new empirical prediction. Code: the accompanying verification suite (routeB_verification_suite.tar.gz) is licensed MIT; this manuscript is licensed CC BY 4.0.
We characterize, within quantum theory, the two-party correlation E(θ_A, θ_B) = −cos(θ_A − θ_B) that saturates the Tsirelson bound |S| = 2√2. Let E be the correlation kernel of a U(1)-covariant, perfectly anticorrelated bipartite quantum realization (a jointly invariant state with covariant local dichotomic observables). We prove that −E is then automatically a positive-definite function on U(1), because it equals a diagonal matrix coefficient of an associated unitary representation. We prove a converse: every normalized nonnegative harmonic spectrum is realized by an explicit bipartite quantum model with genuinely local observables. In the real-even category we prove that spectral extremality of the kernel is equivalent to irreducibility of its real cyclic (GNS) representation, and we separate this from minimality of the cyclic realization: a reducible multi-harmonic kernel still has a minimal cyclic realization. Adding one axiom — Primitive Modal Simplicity, that the response occupies a single irreducible real sector — together with faithfulness of that active carrier, singles out −cos Δ. We prove by an explicit, reproducible counterexample (a spin-3/2 singlet with the dichotomic observable sign(S_x), giving the kernel −(7/8)cos Δ − (1/8)cos 3Δ) that modal simplicity is independent of purity, covariance, perfect anticorrelation, one reference dichotomic observable per party, absence of classical randomness, and minimality of the cyclic realization. We state carefully what is and is not established: this is a characterization of a scalar correlation-function family, not a reconstruction of the quantum state/observable framework, the Born rule, or a new empirical prediction. Code: the accompanying verification suite (routeB_verification_suite.tar.gz) is licensed MIT; this manuscript is licensed CC BY 4.0.
Route B characterized covariant perfect anticorrelation as the singlet cosine E(θ_A, θ_B) = −cos(θ_A − θ_B), whose optimized CHSH value is 2√2, attaining the Tsirelson bound, and reduced its foundational content to a single adopted axiom: Primitive Modal Simplicity (real-cyclic irreducibility — one active nontrivial harmonic sector). This paper maps the mathematical structure surrounding that primitive and reports the outcome of a dedicated attempt to derive it. Primitive Modal Simplicity is a local, selected-response condition; it is surrounded by two distinct global conditions — observable uniformity (End₀(V) irreducible), a genuine strengthening once the invariant mode is removed (w₀ = 0), and state homogeneity (G_cov transitive on pure states), a state-side condition the primitive does not imply, both holding for the physical qubit. We prove the observable side is sharply classified: for a nontrivial irreducible complex representation of a connected compact group with simple Lie algebra g, End₀(V) is irreducible if and only if (g, V) ≅ (su(d), d or d*); within this class observable uniformity implies state homogeneity, while a computed Sp(2) = USp(4) witness shows the converse false. Typing the primitive to the covariance group as PMS_G (real-cyclic irreducibility of the response's carrier), the same USp(4) carries, at the response level, an admissible perfectly anticorrelated dichotomic response whose cyclic carrier is reducible (5 ⊕ 10): state homogeneity does not force PMS_G for all admissible seeds, closing that implication in the negative while leaving a canonical-selector form open. State homogeneity is characterized as pure-state controllability by the covariance generators, and the residual Z_n ambiguity is resolved by automorphism transitivity (intrinsic non-selection). Finally, an external investigation asked whether a deeper informational principle grounds either condition. Its result is negative but precise: forgetting the designated covariance group provably loses the carrier (a non-identifiability theorem), and the one surviving canonical-enrichment escape — covariance completeness, identifying the covariance group with the connected reversible group — yields observable uniformity for a complex matrix factor, but only as a maximality choice with no independent informational ground. Primitive Modal Simplicity therefore remains adopted rather than derived, now with a precisely delimited conditional bridge and an explicit open problem. Code: the accompanying reproducibility suite (in modal_simplicity_note_package.zip) is licensed MIT; this manuscript is licensed CC BY 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.
Union Dipole Theory postulates two distinct channels of transfer: the particle moves through the permeable medium at a bounded speed, while a disturbance within that medium is stated to propagate at an unbounded one (Al-Mayahi, 2014). This paper asks what that structure implies for correlated pairs, and obtains the singlet joint distribution and the Tsirelson value from a single principle. The principle is single-valuedness of the internal phase — the same condition that produced the phase-closure result of the companion paper. Applied to a pair, it does two things at once. It indexes admissible transitions by integers, so that a shared pair quantum resolves into one channel and not partly into both. And, applied to the two pair channels as degrees of freedom of the medium, it fixes their action integrals, which for a linear medium are proportional to the stored energy. The channel weights are therefore not an added rule about rates: they are what closure gives. One condition supplies both the counting and the quadratic weighting. The consequences follow without further assumption. Writing the pair on its shared internal phase, the two channel amplitudes are 2cos(θ/2)cos φ and −2sin(θ/2)sin φ. Their closed action integrals stand in the ratio cos²(θ/2) to sin²(θ/2); a single shared quantum normalises them; and the correlation is E = −a·b with a CHSH maximum of exactly 2√2. An inversion symmetry then fixes the four joint outcome probabilities uniquely and makes the single-station marginals exactly one half at every angle, so no-signalling holds for the construction and not only for the architecture. Three negative controls are computed rather than asserted, and each is derived in closed form. Extracting outcomes by any local rule gives −(1 − 2θ/π) and |S| = 2 exactly. Sending the indivisible transition to the channel of larger instantaneous drive gives the same linear correlation and the same value 2, and no threshold weight repairs it. Weighting the channels by amplitude rather than by action gives S = 1.657 at the standard settings and, on optimisation over the four settings, exactly 2. All three controls therefore terminate at the Bell bound, and the distance between 2 and 2√2 is attributable to the closure measure and to nothing else in the construction. Two structural results complete the account. The correlation is independent of distance because it is a ratio: any common factor multiplying both channel weights — the inverse-square attenuation of the medium included — cancels identically between numerator and denominator, so the disturbance law is obeyed in full while the correlation does not register it. And a residual anisotropy of the medium leaves every prediction unchanged: perfect anticorrelation at zero angle survives at every order, and the CHSH maximum remains exactly 2√2 on the circular sections of the response ellipsoid. Finally, generalising the channel weighting to an arbitrary exponent shows that the correlation is an ordinary cosine for exactly one value, n = 2, and that the maximal CHSH value is strictly increasing in the exponent and passes through 2√2 at exactly that value. The position of the quantum ceiling is thereby tied to a constitutive property — the quadratic, that is linearly elastic, response of the medium — which is itself what the closure condition selects. The framework is not a local hidden-variable theory, does not evade Bell’s theorem, and lies outside the class the theorem constrains; this is stated as a classification and not as an achievement. What is read rather than derived is named, and what remains open is named: in particular, no dynamical account is given of how the shared medium delivers the single transition to one channel across two separated stations. Two appendices follow. The first compares the predicted angular form with Bell correlations reported on three physically unrelated platforms, spanning CHSH values from 2.07 to 2.71, all of which report the same angular form: strength varies, shape does not. The second sets out a two-coordinate hardware diagnostic which follows from the result and which is the subject of United Kingdom patent application GB 2619873.9, filed 26 August 2026. Companion papers: Al-Mayahi, A. (2026a). Internal Time as a Counted Quantity. https://doi.org/10.5281/zenodo.22110704 — Al-Mayahi, A. (2026b). Union-Dipole Continuum Electrodynamics. https://doi.org/10.5281/zenodo.22110528
Part 6 of this record supersedes Parts 1-5. Earlier parts conjectured that Information Causality (IC) alone enforces the positive-semidefinite (PSD) structure of translation-invariant correlation kernels; the Sole-Origin and Eigenvector versions (Parts 1, 4, 5) claimed constructive proofs. This paper withdraws those proofs and replaces them with what can be proven. Two structure theorems show the withdrawal is structural, not incidental. The Barrier Lemma: any protocol whose measurement settings are independent of the encoded data satisfies S ≤ E(0)²‖u‖₂² ≤ 1, for every Bob strategy and every resource model (verified by exhaustive maximization over all Bob functions at N=4). The Pyramid Factorization Theorem: the Pawłowski concatenation satisfies S_{2ⁿ} = Π levels ≤ (sup S₂)ⁿ, so nesting can never rescue a kernel that a single level cannot detect. Two positive results take their place. A visibility dichotomy: at sub-unit visibility, explicitly non-PSD kernels are quantum-realizable via a Tsirelson factorization with sign-flipped harmonics, so no correlation principle can enforce PSD there; at unit visibility (E(0) = −1), Cauchy-Schwarz rigidity forces single-family Gram structure. The main theorem: Macroscopic Locality implies kernel positivity at unit visibility (via the Q¹ characterization of ML plus the rigidity argument). The axiomatic gap open since Pawłowski et al. (2009) is thereby closed, by Macroscopic Locality. Corollary: every unit-visibility non-PSD kernel lies outside the Almost-Quantum set, confirmed numerically by robust NPA-1+AB semidefinite programming (N = 7 settings, Clarabel, PSD control at 10⁻⁹). Whether the strictly weaker principle IC suffices remains open, and is sharpened by seven benchmark kernels, unit visibility, non-PSD, provably outside Almost Quantum, that resist every IC protocol tested: optimized N=2/3 functionals, general random-access codes up to four bits, the full concatenation pyramid with free angles, linear parity encodings, entropic and mutual-information functionals, and continuous-angle protocols. Either a fundamentally new protocol detects them (IC ⇒ PSD), or one of them satisfies IC (IC strictly weaker than ML here). Both outcomes are discoveries; the benchmark coefficients are given in full and machine-readable. An open verification suite (6 self-contained Python scripts, MIT license, ~90 s runtime) is included and reproduces every numerical claim: kernel validity, the Barrier Lemma (three exact checks), the Pyramid Factorization, the Tsirelson factorization of the dichotomy, the NPA-1+AB distances for all seven benchmarks, and the H² bound. Following the documentation practice of this series, all withdrawn claims and their precise failure modes are recorded, including one hypothesis raised and refuted during review of this work.
The quantum structure sub-programme comprises three companion papers on the binary icosahedral group 2I, plus one independently published result. Q1 proves an exact, unconditional Fourier-support rigidity theorem in the finite Weil representation of the Heisenberg group, with a precise no-go corollary for support-, rank-, and Gram-based diagnostics; it establishes no phase coherence, singlet correlator, Tsirelson bound, or Born rule. Q2 proves an exact finite-group fact: the spin-21 and spin-23 representations of 2I induce the same Cayley-graph Laplacian eigenvalue; it establishes no Casimir/isotropy normalisation, Born rule, Tsirelson-type bound, or SU(2) fixed-point/sector-selection argument. Q3 proves a theorem conditional on an explicit, undemonstrated hypothesis: given that a bipartite state on a supplied composition is invariant under the diagonal action of 2I — not derived from admissibility or from Born–Infeld indiscernibility — the state is uniquely the SU(2) singlet, with an explicit Casimir correlator, for all five admissible sectors of 2I; it does not derive that invariance hypothesis, phase coherence, or the Born rule. No chain currently connects these three results to each other or to the structures of quantum mechanics. The Bell paper, published independently in Quantum Reports, establishes the structural non-applicability of Bell-type factorizability in non-injective effective descriptions; the proof of that central theorem uses only non-injectivity and informational completeness. This note records the current, narrow scope of each constituent and the open problems that remain.
Abstract. In a recent work, we established a phase-coherent directional amplitude framework that natively reproduces the full spectrum of bipartite quantum correlations, exactly saturating the Tsirelson bound . While that work demonstrated the predictive power of retaining continuous geometric phase information, the structural origin and uniqueness of the underlying signed amplitude distributions remained implicit. In the present paper we show that these distributions are uniquely determined by symmetry and measurement geometry. We propose a geometric interpretation of quantum measurement in which state reduction arises from marginalization over operationally inaccessible degrees of freedom. An arriving particle and a measurement apparatus uniquely determine a plane of relevance, while orthogonal degrees of freedom are physically filtered and marginalized from the effective description. Using harmonic analysis on the sphere, we prove that this geometric marginalization—together with rotational covariance and normalization—uniquely fixes the post-selection amplitude distributions. For spin-½ systems, rotational symmetry restricts the Legendre expansion to its lowest harmonics, forcing the exact signed distribution previously employed. Extended to bipartite product spaces , global rotational invariance uniquely determines the joint amplitude structure that reproduces the standard singlet correlation law. These results provide a kinematic explanation of the amplitude distributions underlying quantum spin correlations and offer a geometric account of quantum state reduction as marginalization over degrees of freedom that are operationally inaccessible to the measurement interaction.