The Unified Theorem of Correlation Determination: Mathematical and Physical Anatomy and Systematic Reconstruction of Bell's Theorem and CHSH Experiments

Abstract

Abstract: "God does not play dice—he decrees the rules" “上帝不掷骰子——他制定规则” Peering through the veil of a century of history, this study provides a profound anatomy and comprehensive reconstruction of Bell’s theorem and the CHSH inequality from both mathematical and physical perspectives. Building upon this foundation, we establish the unified theorem of correlation determination (UTCD) for photon polarization. This framework comprehensively illuminates the underlying mechanism driving polarization correlation effects—including the long-mythologized phenomenon of quantum entanglement—thoroughly explains the physical phenomena and deep-seated paradoxes of CHSH experiments, and delivers four falsifiable experimental predictions (4.3 Testable experimental predictions) with highly distinct boundaries, capable of immediate, direct, and unambiguous validation across contemporary optics laboratories while keeping existing standard configurations unchanged. Crucially, while mainstream literature takes the non-local interpretation for granted, the historical record for an unmanipulated hardware baseline using classical localized states remains completely non-existent. This critical absence of a direct control group provides the primary empirical justification for the testable predictions presented herein—human science now stands before Galileo’s Leaning Tower of Pisa, watching in breathless anticipation of this ultimate verdict for the physics of our time: 1 – CHSH experiment with conventional photon pairStrictly adhering to the standard measurement configuration defined in the experimental protocol, the classical paired two-photon states predefined in CH74 are adopted to substitute the conventional entangled photon pairs in general CHSH experiments, to implement joint measurements for parallel or orthogonal correlations.Experimental predictions:- As long as the requirement of Condition #1 of the UTCD theorem (the intrinsic root cause behind the generation of the “correlation effect” between paired photons is that they possess parallel, antiparallel, or orthogonal polarization azimuth) can be truly satisfied, the final parallel correlation S-value will definitely exceed 2, and will approach the following maximum theoretical value just like the “entangled photon pairs” commonly used in CHSH experiments: S=2\sqrt{2}\approx 2.828- This experiment can also use any other paired photons as long as they satisfy Condition 1. For example, the method of single-photon-level synchronous pulse + splitting the same pulse can naturally realize event-by-event co-polarization binding; or use the idle port of the entangled photon source as a trigger marker to “decohere” the original entangled photon pairs into co-polarized paired photons, and directly mark the pairing events with the trigger signal. This can also achieve the same effect, and all can obtain the predicted result of S>2. 2 – “Failure Test” of entangled photon pairsWith the standard CHSH experimental setup unchanged, the commonly used “entangled photon pairs” in CHSH experiments are adopted to carry out the “failure test” of quantum entanglement effect: Alice and Bob select exactly the same measurement angles (for example, both are 35° or any other arbitrary angle).Experimental predictions:- Whether the photon at end A transmits first or the photon at end B transmits first — assuming the photon at end A transmits first, the transmission probability of the photon at end B will by no means achieve the claimed 100% efficiency/accuracy. Instead, it will fluctuate drastically around the baseline of 0.5 according to \cos^{2}\left(\theta -35^{\circ }\right), unless someone intentionally or unintentionally modulates its polarization angle to 35^{\circ } before the photon at end B enters the PBS. - The aforementioned determination will explicitly emerge within the macroscopic statistical results of coincidence counts via the law of large numbers.- Analogously, this reasoning holds for an orthogonal polarization correlation scenario. 3 – “Incapacitation test” of entangled photon pairsStrictly follow the standard measurement configuration and procedure specified in the CHSH experiment protocol, and use the “entangled photon pairs” commonly adopted in CHSH experiments: before they reach their respective PBS, change the polarization angle of one of the photons, for example, by not applying polarization compensation at a single end or applying unbalanced polarization compensation, so that the photons at ends A and B cannot maintain their original parallel or orthogonal polarization angles.Experimental predictions: The S parameter in this experiment is impossible to exceed 2. 4 – Single-sided measurement of entangled photon pairsThis single-end characterization is performed strictly adhering to the canonical measurement configuration defined in the standard CHSH experimental protocol, utilizing polarization-entangled photon pairs — the standard entangled resource universally adopted in CHSH testbeds — as the input quantum state. Specifically, the optical path of side B is deliberately extended to introduce a well-controlled temporal delay, such that the photon propagating to side A impinges first on PBS pre-calibrated to operate at a 0° measurement basis, before being registered by the single-photon counting module.Experimental predictions:- The transmission and reflection probabilities of the photon at end A will absolutely never be exactly (or approximately) 0.5. Instead, they will fluctuate drastically (unlike the close-around) around the baseline of 0.5 in accordance with \cos^{2}\left(\theta -0^{\circ }\right) , rather than closely around it— unless someone intentionally or unintentionally modulates the polarization angle of the photon at end A to 45^{\circ } before it enters the PBS.- The above determination will clearly emerge in the macroscopic statistical results of the two coincidence counts for transmission and reflection through the law of large numbers, which can be obtained by comparing the distribution of the coincidence counts at both ends within a unit time window.- When the single-end PBS is changed to other measurement angles, similar predicted results will emerge by the same logic. A “present-day” prediction/assertionIt is time for quantum computing and quantum computers to stage an exit, since the quantum entanglement effect simply does not exist—human scientific productivity and precious scientific resources, which have been imprisoned for a century, should be completely liberated and released into genuinely meaningful scientific endeavors. The classical photon pairs defined by CH74 for local realism can perfectly yield the maximum violation of CHSH inequality under both parallel and orthogonal polarization correlations. This verification profoundly reveals the fundamental mechanism underlying the “correlation effect” of twin photons, while simultaneously elucidating why any peer-paired photons, regardless of whether they originate from homologous or non-homologous sources, can perfectly demonstrate this correlation. Leveraging rigorous mathematical and physical reasoning, the UTCD theorem provides an a priori constraint for determining the correlation between a pair of photons generated through an underlying stochastic mechanism. Its tautological expression is given below (in tribute to Bell): #1 The intrinsic root cause behind the generation of the “correlation effect” between paired photons is that they possess parallel, antiparallel, or orthogonal polarization azimuth.#2 As long as they satisfy Condition #1 before arriving at the PBS, their S value will be greater than 2.#3 As long as they fail to satisfy Condition #1 before arriving at the PBS, their S value will not be greater than 2.#4 Any instance with S>2 necessarily satisfies Condition #1, while any instance with S≤2 invariably stems from the non-fulfillment of Condition #1. Condition #1 satisfied⟺S>2Condition #1 not satisfied⟺S≤2 The objective completeness of this theorem lies in its capability to comprehensively and exhaustively encompass all the probability dimensions of photon transmission event, thereby allowing for a complete and precise determination of the polarization correlation between the two paired photons. This “tautological” mathematical logic rule serves as the intrinsic explanation for “classical” photons breaking through Bell inequality, and moreover, provides the realistic explanation for all the entangled photons breaking through Bell inequality during CHSH experiments. Simultaneously, the extension of this theorem can likewise elucidate the experimental paradoxes in multi-setting Bell test, high-dimensional Bell tests and three-dimensional spin geometric correlation of high-energy particles (such as the quantum entanglement of top-quark pairs asserted by CERN in 2024): once the N-dimensional probability events encompassed by the UTCD theorem are fully incorporated, the so-called quantum entanglement effect will be demystified, entirely degenerating into the correlation described herein and becoming mere subsets within the UTCD theorem. This work stands in tribute to Bell, and the pioneering minds of CHSH and CH74: it is precisely their relentless exploration of the quantum realm and their monumental achievements that laid the foundation for the successful formulation of the UTCD theorem. Enshrouded beneath the triple iron curtains of the Planck limit, the photon indivisibility principle, and the no-cloning theorem, this represents humanity’s sole triumph to date—relying strictly on profound conceptual insights and razor-sharp experimentation—in catching a glimpse of the world hidden within the quantum scale. For the first time in history, light quanta have pierced through these triple iron curtains to present humanity with a macroscopic manifestation of their intrinsic, microscopic properties. There can be no higher tribute paid to Dr. Bell, whose profound insights were first br