1. Exact planar GHZ nonlocality threshold for all n ≥ 4. We determine the sharp supremum of violations of full-correlation Bell inequalities by the n-qubit GHZ state under planar (XY) measurements, over arbitrary finite numbers of measurement settings. The result is the exact formula κₙ = (π/2)ⁿ ζ(n) / [2(1 + 2⁻ⁿ) ζ(2n)], n ≥ 4. In particular, κ₄ = 105/34, corresponding to the critical visibility v₄ = 34/105. The result replaces bounds for particular finite measurement configurations with an exact analytic expression for the stated class. 2. Substantially improved rigorous bounds for the open three-party GHZ problem. We establish 2.0349681535… ≤ κ₃ ≤ 105 ζ(3)/(2π³) = 2.0353294291… . Both bounds improve the previously published 2024 interval. The lower bound comes from a finite-harmonic Bell witness supported by 272 reproducible exact inequalities; the upper bound follows from an explicit square-wave/Möbius local model. The conjecture that the upper bound is attained remains open. The paper discusses an obstruction involving the Möbius function and zeros of the Riemann zeta function. 3. Sharp covariance inequalities under strong mixing. We establish the optimal coefficient 4√2 for bounded complex-valued observations, improving the classical bound 2π. More generally, the optimal Hilbert-space coefficient is 4K_G(k), where K_G(k) is the real Grothendieck constant of order k. For real scalar observations, the sharp coefficient is 4. 4. Sharp multilinear dependence inequalities. For n-linear forms that are real-valued on real inputs, we identify the optimal real and complex comparison factors 2ⁿ and 2ⁿ κₙ. These results strengthen classical estimates in a natural restricted class. They do not settle the more general problem for arbitrary complex multilinear forms considered by Bradley and Bryc (1985). 5. Asymptotically sharp growth of higher-order interaction measures. For the centred Fourier–Lancaster interaction functional, we derive explicit finite-dimensional bounds and establish L_d = 2ᵈ/(ed) · (1 + O(d⁻¹)). Thus, unlike the universal characteristic-function factorization defect, this higher-order interaction functional admits no dimension-independent bound. 6. Further structural results. The paper also develops restrictions to discrete phase groups, centred-interaction comparisons and quantitative links between universal Fourier factorization ceilings, strong mixing and multilinear dependence. Updated version, 18 October 2026 — two-PDF proof supplement. This record contains the original 14-page P04 main preprint (P04_CF_V6_S8.pdf), reproduced without alteration, and a 47-page English mathematical proof supplement (P04_K2_K7_R13_PROOF_SUPPLEMENT_V2.pdf). The supplement expands the analytical proof of the exact planar GHZ formula for n ≥ 4 and develops the three-party K7 lower bound using geometric reductions and 272 reproducible exact-rational inequalities. The exact three-party constant remains open.
Exact planar GHZ nonlocality threshold for all n ≥ 4. We determine the sharp supremum of violations of full-correlation Bell inequalities by the n-qubit GHZ state under planar (XY) measurements, over arbitrary finite numbers of measurement settings. The result is the exact formula κₙ = (π/2)ⁿ ζ(n) / [2(1 + 2⁻ⁿ) ζ(2n)], n ≥ 4. In particular, κ₄ = 105/34, corresponding to the critical visibility v₄ = 34/105. The result replaces bounds for particular finite measurement configurations with an exact analytic expression for the stated class. 2. Substantially improved rigorous bounds for the open three-party GHZ problem. We establish 2.0349681535… ≤ κ₃ ≤ 105 ζ(3)/(2π³) = 2.0353294291… . Both bounds improve the previously published 2024 interval. The lower bound comes from a finite-harmonic Bell witness supported by 272 reproducible exact inequalities; the upper bound follows from an explicit square-wave/Möbius local model. The conjecture that the upper bound is attained remains open. The paper discusses an obstruction involving the Möbius function and zeros of the Riemann zeta function. 3. Sharp covariance inequalities under strong mixing. We establish the optimal coefficient 4√2 for bounded complex-valued observations, improving the classical bound 2π. More generally, the optimal Hilbert-space coefficient is 4K_G(k), where K_G(k) is the real Grothendieck constant of order k. For real scalar observations, the sharp coefficient is 4. 4. Sharp multilinear dependence inequalities. For n-linear forms that are real-valued on real inputs, we identify the optimal real and complex comparison factors 2ⁿ and 2ⁿ κₙ. These results strengthen classical estimates in a natural restricted class. They do not settle the more general problem for arbitrary complex multilinear forms considered by Bradley and Bryc (1985). 5. Asymptotically sharp growth of higher-order interaction measures. For the centred Fourier–Lancaster interaction functional, we derive explicit finite-dimensional bounds and establish L_d = 2ᵈ/(ed) · (1 + O(d⁻¹)). Thus, unlike the universal characteristic-function factorization defect, this higher-order interaction functional admits no dimension-independent bound. 6. Further structural results. The paper also develops restrictions to discrete phase groups, centred-interaction comparisons and quantitative links between universal Fourier factorization ceilings, strong mixing and multilinear dependence. Updated version, 18 October 2026 — two-PDF proof supplement. This record contains the original 14-page P04 main preprint (P04_CF_V6_S8.pdf), reproduced without alteration, and a 47-page English mathematical proof supplement (P04_K2_K7_R13_PROOF_SUPPLEMENT_V2.pdf). The supplement expands the analytical proof of the exact planar GHZ formula for n ≥ 4 and develops the three-party K7 lower bound using geometric reductions and 272 reproducible exact-rational inequalities. The exact three-party constant remains open.
Abstract We present a unified analysis showing that the dominant numerical and qualitative structure of Bell–CHSH correlations—including the ubiquitous appearance of √2, sinusoidal correlation curves, and violations of the CHSH bound—arises generically from Euclidean projection plus measurement-induced dimensional reduction. The argument is developed in three interlocking layers. A geometry-first layer uses macroscopic analogues (tuned guitar strings with threshold detectors) to make the projection mechanism inspectable: a shared relational state, local measurement axes, and a recording map that collapses continuous amplitudes to binary outcomes. A threshold-scan (“dial”) test shows that conditioned CHSH values rise smoothly above 2 as the acceptance criterion tightens. A Fourier-analytic layer demonstrates that the sign function—the operator that converts continuous amplitudes to ±1—injects an infinite series of odd harmonics into a pure fundamental signal. The classical triangle-wave correlation E(θ) = 2θ/π − 1 is the fundamental plus harmonic distortion; the quantum correlation E(θ) = −cos θ shares the same fundamental mode but without higher-harmonic content and with unit amplitude. A detection threshold acts as a conditioning map that progressively strips the distortion harmonics. At threshold τ ≈ 0.23, total harmonic distortion reaches near zero, the correlation matches −cos θ, and |S| = 2√2. A combinatorial layer introduces the Q/R/x ontological framework—where Q is the object, R the definite binary relationship, and x the genuinely ambiguous state—and connects it to the formula (n2 − n)/2. Binary measurement (n = 2) supports 1 pairwise relationship and a linear correlation. Acknowledging x as a third state (n = 3) yields 3 pairs—exactly the minimum for planar angular geometry and the cosine. These three layers converge: the gap between |S| = 2 and |S| = 2√2 is the total harmonic distortion introduced by binary quantisation of a continuous angular signal—equivalently, the information-theoretic cost of projecting a ternary relational reality onto a binary measurement space. We show that the hard-threshold model produces CHSH violations only through post-selection on coincidences; when no-detection events are properly accounted for, the model respects all Bell-type inequalities, including the Eberhard/CH inequality used in modern “loophole-free” experiments. This honestly limits the model’s scope but sharpens the open question: whether the trial-definition process upstream of the detectors (heralding, phase-matching) constitutes a form of selection beyond the hard threshold that could affect 1 full-accounting statistics. We propose the “dial test”—sweeping acceptance criteria and plotting violation strength versus acceptance fraction—as a concrete, falsifiable protocol for probing ensemble neutrality in any Bell experiment. Nothing in this work challenges the empirical validity of quantum mechanics or the logical structure of Bell’s theorem. Rather, it challenges the necessity of interpreting Bell-type correlations as evidence for an ontological division of reality into “quantum” and “classical” domains. The analysis suggests that what distinguishes the so-called quantum regime is not a change in physical law, but a change in how much of the underlying relational structure survives measurement and recording. If this reading is correct, its most consequential implication is that the correlations which motivated the inference of “action at a distance” have a local geometric origin: they are properties of a shared state established in the past, not signals transmitted in the present, and their statistical character is determined by how severely the observation compresses what was already there.
We have determined the maximum quantum violation of 241 tight bipartite Bell inequalities with up to five two-outcome measurement settings per party by constructing the appropriate measurement operators in up to six-dimensional complex and eight-dimensional real-component Hilbert spaces using numerical optimization. Out of these inequalities 129 have been introduced here. In 43 cases higher-dimensional component spaces gave larger violation than qubits, and in three occasions the maximum was achieved with six-dimensional spaces. We have also calculated upper bounds on these Bell inequalities using a method proposed recently. For all but 20 inequalities the best solution found matched the upper bound. Surprisingly, the simplest inequality of the set examined, with only three measurement settings per party, was not among them, despite the high dimensionality of the Hilbert space considered. We also computed detection threshold efficiencies for the maximally entangled qubit pair. These could be lowered in several instances if degenerate measurements were also allowed.
We propose candidate lower bounds on the real Grothendieck constants KG(d) and their infinite-dimensional limit KG using rotationally invariant cubic kernels. For the continuous construction, we identify an explicit threshold above which hemispherical binary strategies are unstable under cubic boundary deformations. Supported by numerical evidence, we conjecture that hemispherical strategies are globally optimal at this threshold. This conjecture would imply KG≥9π/16≃1.767146, improving the rigorous lower bound 6π/11≃1.713596. We also construct finite symmetric coefficient matrices from unit vectors in dimensions up to 24. Large-scale see-saw optimization supports the continuous predictions and suggests improved lower bounds on KG(d) in the studied dimensions from 4 to 24, conditional on the conjectured binary optima being exact. Selected matrices also yield finite-setting candidate dimension witnesses when interpreted as bipartite correlation Bell functionals. If the corresponding binary upper bounds hold, these witnesses would certify a local Hilbert-space dimension of at least five for each party.
This research develops a general spectral and representation-theoretic theory of minimum action for continuous unitary designs and universally robust quantum control. The central result is a Projective Moment-Casimir speed-limit theorem combining unitary moment representations, the finite center Z_d of SU(d), twisted Laplacian spectra, Ky Fan variational bounds, and quadratic Casimir coefficients. For exact continuous unitary t-designs, the theory gives lower bounds on the time-control resource product T Omega_rms required for a single deterministic continuous trajectory to reproduce exact Haar moments. The strongest exact result is for t = 1. Under the normalized Hilbert-Schmidt Hamiltonian metric, the minimum action in every finite dimension d >= 2 is (T Omega_rms)_min = pi sqrt((d^4 - 1)/3). A determinant-centered Heisenberg-Weyl continuous-design trajectory attains this bound exactly. This turns a previously known construction into a candidate globally optimal solution of the all-dimensional continuous unitary 1-design minimum-action problem under the stated metric. For qubits, the result reproduces the pi sqrt(5) projective length and the corresponding universally robust quantum-control speed limit. The theory also extends to higher design order. It derives explicit higher-moment lower bounds, including a t = 2 asymptotic barrier T Omega_rms >= (pi/sqrt(6) + o(1)) d^4, and, for fixed t, T Omega_rms = Omega(d^(2t)). For n qubits, d = 2^n, giving an exact-temporal-design barrier of order 4^(nt). Additional results include: optimization over projective center sectors of SU(d)/Z_d; arithmetic dependence on d, t, and gcd(d,t); approximate-design lower bounds stable under moment-Gram perturbations; spectral rigidity showing that near-optimal trajectories concentrate near the lowest admissible twisted Fourier modes; exact formulas for moment-space dimensions and quadratic-Casimir coefficients; constructive higher-order continuous-design routes using finite unitary design groups and Eulerian traversal; reproducible computational verification of the principal identities and low-dimensional cases. The core conceptual principle is: Haar moment complexity-> orthogonal temporal modes-> spectral energy-> minimum physical action. The release includes the full manuscript, LaTeX source, proof audit, prior-art analysis, claim boundaries, theorem indexes, machine-readable metadata for AI agents, reproducible Python code, validation outputs, figures, checksums, and publication metadata. The t = 1 theorem and its exact saturation are presented as a candidate solution to an explicit 2026 continuous-unitary-design optimization problem, subject to independent specialist review and final priority verification. Higher-t sharpness is not claimed in general.
We study when the maximal Svetlichny expectation S(ρ) of a three-qubit state ρattains the upper bound 4ν. Starting from the original objective, we eliminate Alice’stwo measurement directions exactly and reduce the optimization to an operator-normproblem for a complex matrix over legal measurement frames. This yields the universalbound S(ρ) ≤ 4ν, where ν is a state-dependent spectral quantity of the three-bodycorrelation tensor (the largest operator norm of its complex slice combinations), andthe identity maxF σmax(Z) = 2ν over all legal frames. The two ingredients of thebound—a complex-product encoding of the Svetlichny terms and a spectral norm ofthe correlation tensor—are well represented in the literature, and the bound is thereforeused here as the anchor of the attainability analysis rather than claimed as an isolatednovelty. The boundary condition νR = ν holds if and only if some boundary frameattains the global maximum of that spectral norm; under this condition an explicitmeasurement construction gives S(ρ) = 4ν, without assuming that the top singularvalue is simple. On a fixed frame, κ(s, d) = 0 decides whether the inner bound isattained; when ν > 0 and at least one S(ρ)-optimal frame has a positive simple topsingular value, the global criterion reads S(ρ) = 4ν if and only if there exists a framethat is both spectrally maximal and has κ = 0. The GHZ state shows that νR = ν isnot necessary. The same equality framework is then applied to the singly-excited Wstates: from the amplitudes a, b, c ≥ 0 we compute the three correlation slices termby term, reduce the maximization over complex directions to a two-variable functionF(p, r) depending only on p, r, and obtain the threshold polynomial E(A, B), withE(A, B) ≥ 0 if and only if ν = 1; written in the three two-body concurrences thesame polynomial is 1 − (C_212 + C_213 + C_223) − 2C_12 C_13 C_23, so the criterion is manifestlysymmetric under permutations of the three parties. In the violating region E < 0, ν isgiven as a radical closed form in the amplitudes, and the same maximizing point yieldssix real measurement directions that attain S(ρ) = 4ν. All closed forms and equalitydirections are confined to the singly-excited W family defined in this paper.
ARCHANGEL develops a constructive mathematical framework for the common realization of positive matrix moments, compatibility testing across directional data, and rigorous certification of operator-valued responses. Its principal contributions connect matrix inequalities, constrained realization theory, spectral geometry, semidefinite certification, and finite-dimensional quantum tomography. The central result is a necessary-and-sufficient common-realization criterion within a specified finite positive Gaussian model class. Under explicit kernel and positive-curvature assumptions, the construction determines whether complete joint moment data admit one realization satisfying a prescribed spectral cutoff constraint. It provides an explicit extremizer attaining the resolvent lower bound Lₐ = Q(Q + aR)⁺Q simultaneously for every positive parameter a. The auxiliary dimension is minimal and equals rank(Q); in the three-direction, ten-dimensional formulation, at most 30 auxiliary modes are required. A particularly distinctive result is an exact rational counterexample to the inference from separate directional feasibility to joint realizability. Every real direction individually passes its optimized feasibility test, yet no single common realization satisfies all directional constraints together. The obstruction survives every compatible complex completion of the unmeasured imaginary entries. This establishes a precise limitation of reconstructing a shared system from separately successful directional tests. Additional results include: Sharp directional coverage bounds. A general coverage parameter quantifies how much finite directional testing can underestimate the largest response. The optimal missed-peak factor is 4/3 for planar trigonal sampling and 3/2 even when every direction in all three coordinate planes is tested. Sharp reconstruction stability. Three planar trigonal measurements reconstruct the complete quadratic directional operator, with an optimal 5/3 amplification factor for uniformly bounded measurement errors. Ten-dimensional holonomy certification. Two Weyl transformations detect every nonscalar component of a Hermitian response operator. A sharp Hilbert–Schmidt robustness inequality has the golden ratio as its optimal constant. Exact continuum positivity certification. A planar operator-valued trigonometric inequality is reduced to a finite semidefinite feasibility problem using matrix Fejér–Riesz factorization. For the ten-dimensional code, the degree-one certificate uses a 20 × 20 block matrix. Explicit tomography and combined bounds. A construction using 100 preparations reconstructs any ten-dimensional Hermitian response. Combining spatial reconstruction with holonomy contrasts gives a sharp joint certification inequality. The mathematical value lies in distinguishing independently compatible observations from data that admit one shared realization, constructing extremal realizations with minimal auxiliary dimension, and replacing incomplete sampling arguments with quantitative or exact certificates. These questions are relevant to matrix moment problems, inverse spectral modelling, constrained system identification, and quantum response certification. The release includes a 95-page compendium, complete proofs, five preserved research studies, correction and theorem indexes, reproducible Python code, three synthetic figures, and 32 passing computational verification groups. Exact arguments establish the continuum statements; numerical calculations provide supplementary consistency checks. Status and scope: The stated finite-matrix problems are resolved under their explicit assumptions. The work is an AI-assisted theory and methods contribution using established tools, including Schur complements, Weyl operator methods, and matrix polynomial factorization. External peer review and a comprehensive literature-priority assessment remain outstanding. No experimental validation is claimed.