One basis-invariant half-angle behind the T gate, CHSH, and the QTT clock Why does the same half-angle appear in a quarter-phase gate, the symmetric CHSH optimum, and the QTT two-clock projection? This paper answers the operator question exactly. Starting from the A1 clock-plane quarter-turn and the A4 real-J Pauli representation, the centered clock rotor is \[ U_{\rm A1}=\exp\!\left(-J\frac{\pi}{8}Z\right). \] The positive symmetric CHSH analyzer is generated by \[ U_{\rm CHSH}=\exp\!\left(-J\frac{\pi}{8}Y\right). \] With the Clifford operator \(C=\mathsf S\mathsf H\), for which \(C Z C^\dagger=Y\), the two rotors obey \[ \boxed{ U_{\rm CHSH}=C U_{\rm A1}C^\dagger, \qquad \frac12\operatorname{Tr}U_{\rm CHSH} =\frac12\operatorname{Tr}U_{\rm A1} =\cos\!\left(\frac{\pi}{8}\right). } \] The normalized character is invariant under the basis change. It joins three familiar quantum-information expressions into one exact chain: \[ \boxed{ I_{\rm clk} =\left|\langle +|\mathsf T|+\rangle\right| =\sqrt{P_{\rm win}^{\rm QM}} =\cos\!\left(\frac{\pi}{8}\right), \qquad S_{\max}=8I_{\rm clk}^{2}-4=2\sqrt2. } \] The paper also prints the complete angle ledger: the A1 physical angle \(\pi/4\), the T-gate relative phase \(\pi/4\), the spinor exponent \(\pi/8\), the photon-analyzer parameter \(\pi/8\), the amplitude \(\cos(\pi/8)\), the probability \(\cos^2(\pi/8)\), and the optimal correlator \(1/\sqrt2\) are related but are not interchangeable quantities. The exact identity is a structural cross-lock, not a claim that ordinary CHSH experiments independently validate A1. Standard quantum mechanics already contains the same spinor geometry. A laboratory test is therefore ineligible if \(22.5^\circ\), \(45^\circ\), or the target overlap is programmed into preparation, calibration, control code, or priors. The independent discriminator is the sealed, unprogrammed dual-clock reference-switch test: \[ \boxed{ R_{\rm standard}=1, \qquad R_{\rm QTT\text{-}A1}=\cos\!\left(\frac{\pi}{8}\right). } \] Status CLOSED: SIGMA-A1-CHSH-SPINOR-CHARACTER-IDENTITY NOT CLAIMED: SIGMA-A1-CHSH-INDEPENDENT-EMPIRICAL-VALIDATION PENDING: SIGMA-A1-UNPROGRAMMED-REFERENCE-SWITCH-TEST Version: 2.0 Concept DOI: 10.5281/zenodo.19979594 Author: Ali Attar Website: quantumtraction.org Main-book anchor: Quantum Traction Theory: Main Book v10.01, pp. 235-237, 445-449, 541-542, and 552-553 Independent test: Artian Dual-Clock Sagnac Reference-Switch Preregistration Files Reader-facing PDF, Version 2.0 Reconstruction ZIP containing LaTeX source, style file, numerical verifier, machine receipt, scientific audit, render audit, metadata, and checksums
One basis-invariant half-angle behind the T gate, CHSH, and the QTT clock Why does the same half-angle appear in a quarter-phase gate, the symmetric CHSH optimum, and the QTT two-clock projection? This paper answers the operator question exactly. Starting from the A1 clock-plane quarter-turn and the A4 real-J Pauli representation, the centered clock rotor is [ U_{\rm A1}=\exp!\left(-J\frac{\pi}{8}Z\right). ] The positive symmetric CHSH analyzer is generated by [ U_{\rm CHSH}=\exp!\left(-J\frac{\pi}{8}Y\right). ] With the Clifford operator (C=\mathsf S\mathsf H), for which (C Z C^\dagger=Y), the two rotors obey [ \boxed{ U_{\rm CHSH}=C U_{\rm A1}C^\dagger, \qquad \frac12\operatorname{Tr}U_{\rm CHSH} =\frac12\operatorname{Tr}U_{\rm A1} =\cos!\left(\frac{\pi}{8}\right). } ] The normalized character is invariant under the basis change. It joins three familiar quantum-information expressions into one exact chain: [ \boxed{ I_{\rm clk} =\left|\langle +|\mathsf T|+\rangle\right| =\sqrt{P_{\rm win}^{\rm QM}} =\cos!\left(\frac{\pi}{8}\right), \qquad S_{\max}=8I_{\rm clk}^{2}-4=2\sqrt2. } ] The paper also prints the complete angle ledger: the A1 physical angle (\pi/4), the T-gate relative phase (\pi/4), the spinor exponent (\pi/8), the photon-analyzer parameter (\pi/8), the amplitude (\cos(\pi/8)), the probability (\cos^2(\pi/8)), and the optimal correlator (1/\sqrt2) are related but are not interchangeable quantities. The exact identity is a structural cross-lock, not a claim that ordinary CHSH experiments independently validate A1. Standard quantum mechanics already contains the same spinor geometry. A laboratory test is therefore ineligible if (22.5^\circ), (45^\circ), or the target overlap is programmed into preparation, calibration, control code, or priors. The independent discriminator is the sealed, unprogrammed dual-clock reference-switch test: [ \boxed{ R_{\rm standard}=1, \qquad R_{\rm QTT\text{-}A1}=\cos!\left(\frac{\pi}{8}\right). } ] Status CLOSED: SIGMA-A1-CHSH-SPINOR-CHARACTER-IDENTITY NOT CLAIMED: SIGMA-A1-CHSH-INDEPENDENT-EMPIRICAL-VALIDATION PENDING: SIGMA-A1-UNPROGRAMMED-REFERENCE-SWITCH-TEST Version: 2.0 Concept DOI: 10.5281/zenodo.19979594 Author: Ali Attar Website: quantumtraction.org Main-book anchor: Quantum Traction Theory: Main Book v10.01, pp. 235-237, 445-449, 541-542, and 552-553 Independent test: Artian Dual-Clock Sagnac Reference-Switch Preregistration Files Reader-facing PDF, Version 2.0 Reconstruction ZIP containing LaTeX source, style file, numerical verifier, machine receipt, scientific audit, render audit, metadata, and checksums
A quantum clock can follow two proper-time histories. Can one completed coherence also distinguish two physical reference classes? Quantum-clock interferometry already supplies the standard carrier relation Δϕclock=ω0Δτ. This preregistration keeps that relation intact and asks a separate question. The same atomic transition and path geometry are executed through four physical input/output terminal histories, ττ,τT,Tτ,TT. For each history and reversed proper-time geometry, the laboratory fits the first complex clock-coherence phasor and forms Dab=∣Zab+Zab−∣. The registered reference-switch statistic is Rqc,clk=N×ord1DττDTTDτTDTτ. After a target-blind ordinary cross-history map, the fixed prospective fork is Hordinary:Rqc,clk=1,HQTT−A1:Rqc,clk=cos(8π). The alternatives differ by 7.6120467%. A bare five-standard-deviation separation requires σ(Rqc,clk)≤0.0152241, and the recommended design target is 0.005. The proper-time phase is a common carrier check, not the A1 discriminator. The release includes three adversarial audits. It requires a physical hardware graph for the T and τ terminal classes, derives the readout exponent from a first complex phasor, and qualifies preparation and calibration by their physical transfer, not by angle alone. A target-inserting primary operation or target-trained correction remains ineligible. Any failed eligibility gate returns INELIGIBLE_NO_THEORY_VERDICT rather than a false rejection of A1. Scientific status: theorem and estimator closed; carrier technology plausible; physical four-terminal hardware map and laboratory dataset pending. No existing quantum-clock record is claimed as A1 evidence. Concept DOI: 10.5281/zenodo.21674815 Main book: Quantum Traction Theory: Main Book v10.01 Website: quantumtraction.org · Two Universes · Lexicon · Observatory The public files are the paper PDF first and the complete reconstruction ZIP second. Version 1.1: operation qualification and retained observations The numerical targets, ratio orientation, decision regions and historical seals are unchanged. A 22.5-degree or 45-degree component is no longer excluded by its angle alone: its physical transfer must be independently established. Common-operation cancellation and signal-preserving calibration are proved; both hypotheses must remain identifiable. Every acquired record is retained. After authorized unblinding, QTT-like, ordinary-like and unexpected outcomes are reported, including observations that do not qualify for a theory verdict. This is a prospective amendment, not a retrospective resealing of data. Clock, ruler and signal-recovery qualification. Both registered predictions are constructed at the raw detector with explicit proper-time, coordinate-time, worldline, ruler and synchronization conventions. Each relativistic effect is counted once. Independently calibrated corrections carry their uncertainties through the common comparison; the tested rows cannot train a normalization that forces either answer. Blind recovery is required separately for both alternatives at every predeclared operating setting. The inherited targets and decision rules remain fixed. The included design study makes sensitivity explicit: for the Two-Test Gram classifier, a total ratio uncertainty of 0.0013 gives approximately 88.9% QTT-branch and 94.1% ordinary-branch power under the stated ideal model. These are conditional detection probabilities, not probabilities that a theory is true, and each instrument retains its own classifier. The complete apparatus packet must be sealed before acquisition and target access; historical campaign rules remain preserved. Reconstruction entry point. Start with current/CURRENT_RELATIVISTIC_EXECUTION.md in the ZIP. The shared execution contract refers once to the development filename CURRENT_EXECUTION_RC.md; that reference means CURRENT_RELATIVISTIC_EXECUTION.md in this release. The packaged README names the correct file. This navigation correction does not change the deposited files, scientific requirements or historical seals.
From a finite source budget to light propagation, clock slowing and optical probabilities c=tAℓA,v2+cW2=c2,B∗=2Z+X,tanβ∗=2−1,β∗=8π,Iclk=cosβ∗. A measured optical law does not by itself determine which physical objects are fundamental. This paper begins with Quantum Traction Theory's completed events, finite source capacity, ordered clock and real phase structure. It then reconstructs light propagation, the kinematic clock rate, polarization probabilities and the neutral pair's angular record law within explicit source and readout classes. The angle is calculated before the detector counts: physical exchange symmetry fixes the balanced response, whose positive eigenprofile has component ratio 2−1. Thus β∗=π/8, the retained amplitude is cos(π/8), and its capacity fraction is (2+2)/4. The source rotor, the symmetric correlation analyzer and a diagonal phase operation are connected by exact operator identities. The paper takes QTT as its physical starting account. It imports no textbook physical postulate as a constructor, while keeping QTT's own axioms, selected sectors, sampling and null-preserving observer conditions explicit. Its claim is a reconstruction from stated source premises, not a theory without assumptions. The opening argument is followed by complete proofs, a source and input ledger, angle and amplitude conventions, sensitivity formulas, premise-removal witnesses and a four-history clock-transfer theorem. Known photon observations provide the experimental phenomena being reconstructed. No new optical dataset is fitted, and no new A1 clock result or observational significance is asserted. The first file is the paper PDF. The reconstruction ZIP contains the self-contained LaTeX source, executable exact and numerical checks, claim ledger and checksums. The stated source construction can be checked independently; its physical terminal selection is addressed by the linked clock programme. Source dependencies and companion tests: Main Book; A1–CHSH character; finite real source histories; completed-event Bell construction; clock-reference tests.
A symmetry-selected pi/8 spinor character and its physical clock test B∗=2Z+X,U∗ZU∗†=B∗,U∗=exp(−J8πY),21TrU∗=cos8π. QTT Balanced-Rail Optical Intertwining Theorem, v1.4. In a normalized, exchange-symmetric two-rail source sector, the positive balanced direction is unique and its shortest centered spinor lift fixes the pi/8 character without a fitted angle. Positive additive capacity supplies its square, while a declared reference interaction supplies the detector coherence. A complete memory-partition theorem gives eleven jointly fixed configurations and four-phase detector predictions. This private research edition preserves the fourteen earlier theorems and adds independent-reference kernel composition, calibrated signal identifiability, and higher-order closure from quadratic capacity. It connects the finite model to measured photon-scattering and internal-clock coherence, and independently recalculates two Bell aggregate statistics from published event totals. All four Shalm pulse windows and all nine Kauten response-correction rows are retained. Verification: 284 mathematical, numerical, synthetic and scope checks, including 235 inherited checks unchanged; 50 separate archival aggregate checks. The reconstruction includes source receipts, both supplied PDFs, complete numerical transcriptions, code, figures, preservation records and three diagnostic review passes. These counts refer to software checks, not independent experiments. Scientific scope: the cited observations support shared optical correlations and calibrated reference-coherence structure. They do not provide a new A1-versus-QM likelihood contrast or a measurement of an unprogrammed absolute-clock terminal. The same deliberately engineered apparatus has the same fixed quantum predictions. The separate physical source-to-apparatus transfer remains the experimentally distinguishing question. Detector calibration, target fitting and signal survival are treated explicitly rather than merged into one category. Author: Ali Attar, Independent Researcher, Colombes, France. ORCID: 0009-0008-9931-2691. Published research record with public metadata and restricted files. Access to the manuscript and reconstruction package requires the author's permission. Source dependencies: QTT Main Book, A1-CHSH source-character paper, Born-rule paper, Physical Terminal Reference Framework. Same-author dependencies are not counted as independent evidence.
This work provides an explicit, reproducible demonstration of how theClauser-Horne-Shimony-Holt (CHSH) inequality depends jointly on thechoice of Bell state AND measurement configuration in the spin-1/2qubit convention used by gate-based quantum processors. Main result. The combination of CHSH Form I, S = | E(a,b) - E(a,b') + E(a',b) + E(a',b') |, with the commonly-cited angular configuration {0, pi/2, pi/4, -pi/4}yields S = 0 for ALL FOUR Bell states in the spin-1/2 Bloch-sphereconvention, not selectively for one. This frequently misleadspractitioners who prepare |Phi+> on gate-based hardware and applytextbook angles, obtaining S = 0 and misdiagnosing the issue as ahardware fault, code bug, or absence of entanglement. Corrections provided. For the two states whose correlation functionin the XZ plane has argument theta_a - theta_b (|Phi+> and |Psi->),two minimal one-parameter fixes recover the Tsirelson bound 2*sqrt(2): 1. Retain the angles and apply CHSH Form II: S = | E(a,b) + E(a,b') + E(a',b) - E(a',b') |. 2. Retain Form I and replace b' = -pi/4 with b' = 3*pi/4. For the remaining two Bell states (|Phi-> and |Psi+>, whose correlationargument is theta_a + theta_b), neither single-parameter fix suffices;the optimal angles b and b' must be exchanged. A complete optimal-angletable for all four states is given in Section 4 of the manuscript. Contents. - paper.pdf / paper.tex: 8-page manuscript with full derivation, tables, code listing and appendix. - chsh_verification.py: self-contained Python script (NumPy only) reproducing all numerical claims of the paper. Its standard output is the verbatim text reproduced in Section 5, and the script additionally cross-checks the closed-form correlation expressions against explicit 4x4 matrix calculations on random angles (azimuthal angles included) to machine precision. - CHANGELOG.md: complete history of corrections relative to earlier versions of this deposit. Reproducibility. Running 'python3 chsh_verification.py' produces theverification tables displayed in Section 5 of the paper, byte for byte.
T86 establishes the reduced barrier cycle operator for the Q5 transport framework. Let\[A=E_d=i\sigma_y,\qquadA^2=-I,\]and letBdenote the full Q5 barrier crossing operator containing the orientation-reversal component Θ. Using the canonical phase projectionΠphase,introduced in T26-T29, the theorem identifies the reduced barrier operatorBred=Πphase(B),under the conditional assumption that the barrier crossing acts as a quarter-turn on the reduced defect plane. Under this assumption,Bred=e(π/4)A. Using the exponential identityeθA=Icosθ+Asinθ,The theorem proves\[B_{\mathrm{red}}^4=-I,\qquadB_{\mathrm{red}}^8=I.\] The reduced barrier operator therefore, produces the spinor state cycleψn+4=−ψn,ψn+8=ψn,giving a Z8 state period and a Z4 observable period through projective sign identification. T86 is structurally important because it isolates the reduced coarse barrier operator explicitly within the Q5 framework. Earlier theorems established:- the rotational generator A,- the barrier crossing structure,- and the projection architecture. T86 now combines these ingredients into a single reduced operator form. The theorem also sharply isolates the remaining open step. The algebraic structuree(π/4)Acorrectly reproduces the required Z8 cycling, but the geometric origin of the quarter-turn angleπ/4has not yet been derived from first-principles Q5 barrier geometry. Accordingly, T86 proves:Πphase(B)=e(π/4)Aconditional on the quarter-turn assumption, while T87 is designated as the theorem responsible for deriving the angle geometrically. The theorem further clarifies that the identification applies only after projection onto the reduced defect plane. The full barrier operatorBcontains orientation-reversal structure and scalar/symmetric dressing removed byΠphase. Thus, the clean exponential forme(π/4)AIt is a reduced-sector statement rather than a claim about the full raw barrier operator globally. T86 therefore serves as the algebraic bridge connecting:- the barrier crossing structure from T47-T49,- the Z8 spinor cycling from T68,- and the dual-scale generator structure established in T85. Status: solid fore(π/4)Aproducing\[B_{\mathrm{red}}^4=-I,\qquadB_{\mathrm{red}}^8=I,\]and for the resulting Z8/Z4 cycling structure; solid for the projection properties ofΠphase;conditional for the identificationΠphase(B)=e(π/4)A,which depends on deriving the quarter-turn angleπ/4from Q5 barrier geometry in T87.
A qubit is one setting of the two sliding rings. The rotation gate Rz(θ) has entries cos(θ/2) and sin(θ/2), and the instrument's law is d = 2cos(θ/2) between the centers and e = 2sin(θ/2) between the crossings at crossing separation θ: the gate matrix is read off the setting, the Bloch amplitudes are the two half-distances, and Born's rule is the two shares of one unit. Composing rotation gates is adding positions, certified by the ring's addition law. The spinor's sign, −1 after a 360° rotation and +1 after 720°, is the ring's half turn, digits summing to ten, and its lap. The Pauli phases i, −1, −i are the quarter-turn orbit 7, 9, 3, and the phase exponent of any Pauli product is carried by the digits, verified on two hundred random products. The hidden symmetry of the Jordan–Wigner transformation that Davis and Freericks (arXiv:2512.24589) use to halve measurement circuits is the ring turning, its two angles are the quarter turn and the eighth turn, and its identities are the fold law averaged round the ring, whose coefficients 3/8 and 1/8 are exact averages over the sixty positions. Their equation 24 is verified on random fixed-number states of four qubits. From this one picture the paper places the live checksum where it can live in a quantum computer: on the classical phase registers that drive virtual-Z gates and merge rotations, on the classical Pauli frame, and on measured expectation values through the symmetry identities, which flag every number-changing error, while the law of readings of the companion paper catches the number-conserving ones. And it states where it cannot live, on the amplitudes themselves, by the no-cloning theorem. The four operator-mechanics papers of Freericks and colleagues, the operator Rodrigues formula, the momentum measurement, the free expansion of a Gaussian and the squeezed-state wavefunction, are then worked side by side as examples of the same picture: one slide, one quarter turn, one shear, one rotating ellipse. Every number is verified by the script in the appendix.