Detection Threshold for Q5 Phase-Cycling Signal
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| Natura: | Recurso digital |
| Lingua: | inglese |
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2026
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| _version_ | 1866902068299563008 |
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| author | Holdway, Craig Edwin |
| author_facet | Holdway, Craig Edwin |
| contents | <p>T74 establishes the statistical detection threshold for the Q5 phase-cycling signal developed in T72 and T73. Starting from the single-scan fringe residual<br>\[<br>\epsilon(\phi)<br>=<br>2\sqrt{P_1P_2}\bigl[(A-1)\cos\phi + B\sin\phi\bigr],<br>\]<br>The theorem defines the intrinsic bias magnitude<br>\[<br>\eta<br>=<br>\sqrt{(A-1)^2+B^2},<br>\]<br>and proves that the maximum balanced-path residual amplitude satisfies<br>\[<br>\epsilon_{\max}=\eta.<br>\]<br>A key refinement is introduced: measurement of a first-harmonic bias alone is not operationally sufficient for Q5 detection, because such a bias can be absorbed into standard interferometric fitting parameters. The correct discriminator is therefore the phase-cycling invariant<br>\[<br>\mathcal{I}_{\mathrm{Q5}}<br>=<br>\sum_{k=0}^{3}|v_{k+1}-Jv_k|^2,<br>\]<br>which tests whether the measured quadrature evolution obeys the expected \(\mathbb Z_4\) cycling structure established in T73.</p> <p>T74 proves that Q5 detectability requires two simultaneous conditions. First, the intrinsic signal scale must exceed the statistical shot-noise floor,<br>\[<br>\eta<br>\gtrsim<br>\frac{N_\sigma}{\sqrt N},<br>\]<br>where \(N\) is the photon count and \(N_\sigma\) is the required detection significance. Second, the observed phase evolution must remain consistent with the Q5 phase-cycling law through sufficiently small \(\mathcal{I}_{\mathrm{Q5}}\), distinguishing genuine Q5-compatible structure from ordinary calibration artifacts. Under shot-noise scaling, the minimum detectable bias obeys<br>\[<br>\eta_{\min}<br>\sim<br>\frac{N_\sigma}{\sqrt N}.<br>\]</p> <p>T74 is structurally important because it completes the minimal experimental prediction arc initiated in T72 and T73 by separating signal shape, structural discriminator, and statistical detectability. The theorem converts the earlier transport-cycling framework into a falsifiable interferometric prediction program with explicitly identified observables, structural consistency conditions, and detection scaling laws.</p> <p>Status: solid for the fringe-residual amplitude relations and shot-noise detection-threshold scaling under the stated assumptions; solid for the necessity of combining signal magnitude with structural phase-cycling discrimination; conditional on the T73 \(\mathbb Z_4\) cycling framework and practical extraction of quadrature vectors; speculative for the existence or magnitude of any physical Q5 signal prior to empirical observation.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_20369348 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Detection Threshold for Q5 Phase-Cycling Signal Holdway, Craig Edwin Q5 hypercube penteract Gray code Hamiltonian cycle hypercube graph discrete geometry combinatorial topology defect bond holonomy transport operator phase defect linked-pair reduction ordered residue kernel structure fibre reduction Lie algebra representation theory spinor commutator projection operator orientation invariant grading phase transport barrier crossing paired transport continuous limit discrete-to-continuous unitary evolution phase accumulation Born rule interference entanglement decoherence quantum measurement reduced sector observable interferometry quadrature cycling first-harmonic bias falsifiable prediction fringe visibility ZPH independent research preprint <p>T74 establishes the statistical detection threshold for the Q5 phase-cycling signal developed in T72 and T73. Starting from the single-scan fringe residual<br>\[<br>\epsilon(\phi)<br>=<br>2\sqrt{P_1P_2}\bigl[(A-1)\cos\phi + B\sin\phi\bigr],<br>\]<br>The theorem defines the intrinsic bias magnitude<br>\[<br>\eta<br>=<br>\sqrt{(A-1)^2+B^2},<br>\]<br>and proves that the maximum balanced-path residual amplitude satisfies<br>\[<br>\epsilon_{\max}=\eta.<br>\]<br>A key refinement is introduced: measurement of a first-harmonic bias alone is not operationally sufficient for Q5 detection, because such a bias can be absorbed into standard interferometric fitting parameters. The correct discriminator is therefore the phase-cycling invariant<br>\[<br>\mathcal{I}_{\mathrm{Q5}}<br>=<br>\sum_{k=0}^{3}|v_{k+1}-Jv_k|^2,<br>\]<br>which tests whether the measured quadrature evolution obeys the expected \(\mathbb Z_4\) cycling structure established in T73.</p> <p>T74 proves that Q5 detectability requires two simultaneous conditions. First, the intrinsic signal scale must exceed the statistical shot-noise floor,<br>\[<br>\eta<br>\gtrsim<br>\frac{N_\sigma}{\sqrt N},<br>\]<br>where \(N\) is the photon count and \(N_\sigma\) is the required detection significance. Second, the observed phase evolution must remain consistent with the Q5 phase-cycling law through sufficiently small \(\mathcal{I}_{\mathrm{Q5}}\), distinguishing genuine Q5-compatible structure from ordinary calibration artifacts. Under shot-noise scaling, the minimum detectable bias obeys<br>\[<br>\eta_{\min}<br>\sim<br>\frac{N_\sigma}{\sqrt N}.<br>\]</p> <p>T74 is structurally important because it completes the minimal experimental prediction arc initiated in T72 and T73 by separating signal shape, structural discriminator, and statistical detectability. The theorem converts the earlier transport-cycling framework into a falsifiable interferometric prediction program with explicitly identified observables, structural consistency conditions, and detection scaling laws.</p> <p>Status: solid for the fringe-residual amplitude relations and shot-noise detection-threshold scaling under the stated assumptions; solid for the necessity of combining signal magnitude with structural phase-cycling discrimination; conditional on the T73 \(\mathbb Z_4\) cycling framework and practical extraction of quadrature vectors; speculative for the existence or magnitude of any physical Q5 signal prior to empirical observation.</p> |
| title | Detection Threshold for Q5 Phase-Cycling Signal |
| topic | Q5 hypercube penteract Gray code Hamiltonian cycle hypercube graph discrete geometry combinatorial topology defect bond holonomy transport operator phase defect linked-pair reduction ordered residue kernel structure fibre reduction Lie algebra representation theory spinor commutator projection operator orientation invariant grading phase transport barrier crossing paired transport continuous limit discrete-to-continuous unitary evolution phase accumulation Born rule interference entanglement decoherence quantum measurement reduced sector observable interferometry quadrature cycling first-harmonic bias falsifiable prediction fringe visibility ZPH independent research preprint |
| url | https://doi.org/10.5281/zenodo.20369348 |