Finite Binary Selector Classification, Identifiability, and Exhaustive 256-Model Comparison

Research Overview This research studies a mathematical system called a Finite Binary Selector Model. The system contains four labelled sectors, with two binary indicators for each sector. Together, these indicators produce exactly 256 possible models. The main purpose of this study is to examine all 256 models, understand their mathematical properties, and determine how well they match selected numerical data. 1. What Problem Does This Research Study? When a mathematical model matches some observations, it does not automatically mean that the model is unique or physically correct. This research asks three simple questions: How many different binary selector models are possible? Under what conditions can we identify a model uniquely? Which model gives the closest mathematical match to the available numerical values? Instead of selecting only one preferred model, the study examines every model in the defined 256-model family. 2. Mathematical Method Each model uses eight binary indicators and one common mathematical weight, represented by (u). The study develops exact mathematical formulas to calculate the predictions and errors of every model. It also establishes conditions for identifying a unique model. When (u) is known, lies strictly between 0 and 1, and is not equal to 0.5, the indicator values can be uniquely identified from exact model outputs. Special cases where different models give identical predictions are also examined. 3. Data Used in This Study The numerical example comes from two archived sets of binary measurement results. Each measurement basis contains 5,000 observations: X basis: 2,739 HIGH and 2,261 LOW observations. Z basis: 2,763 HIGH and 2,237 LOW observations. These counts produce four numerical values: 0.5478, 0.5526, 0.4522, and 0.4474 These values represent two separate binary marginal distributions. They are not one joint four-outcome probability distribution. The data are used for mathematical model comparison, not as direct evidence of a new quantum mechanism. 4. Main Mathematical Results The study produces several exact results. Result 1 — Complete Model Classification Exactly 256 labelled binary selector models exist within the defined mathematical family. Result 2 — Unique Model Identification In a synthetic example with a fixed weight of (u=0.49757556), exactly one model matches the target values: Mask 105. Result 3 — Multiple Models at Balance When the weight is (u=0.5) and all four target values are 0.5, exactly 16 different models produce the same result. Result 4 — Comparison With Archived Data None of the 256 models matches the four archived numerical values exactly. However, two mathematically equivalent models, Mask 90 and Mask 165, give the smallest error. Their minimum squared error is: SSE = 0.00002304 The corresponding root mean squared error is: RMSE = 0.0024 No third model reaches the same minimum error. 5. Why Are These Results Important? This study shows why it is necessary to examine all possible models before accepting one mathematical explanation. It demonstrates that: Some conditions allow a model to be identified uniquely. Different models can sometimes produce identical predictions. A model may fit numerical data closely without matching them exactly. A mathematical mismatch must be interpreted within the assumptions of the model being tested. The work provides a clear and reproducible method for studying a restricted family of binary mathematical models. 6. Scientific Interpretation and Limitations The results establish mathematical properties of the defined 256-model family. They do not demonstrate a new quantum force, backward-time influence, or a verified physical selection mechanism. The absence of an exact match means that the archived numerical values do not fit this particular restricted mathematical model family exactly. It does not mean that quantum mechanics or another physical theory has been experimentally disproved. The available observations are also insufficient to establish a new causal or time-reversed physical process. Further independent experiments would be necessary to investigate possible physical applications. 7. Data, Code, and Reproducibility The research is designed so that other researchers can independently reproduce its mathematical calculations. The supporting materials are: Manuscript PDF: Complete explanation of the mathematical framework, proofs, results, and limitations. All256_Exact_Results.csv: Numerical results for all 256 models. Input_Basis_Marginals.csv: The four input values and their measurement-count information. reproduce_paper10_selectors.py: Python program for reproducing the model comparisons and checking the results. LaTeX source: Editable source of the research manuscript. The complete underlying photon-level laboratory data are not distributed with these materials. 8. Connection With the Quantum Diverter Loop Project This study forms part of the broader Quantum Diverter Loop (QDL) research project. QDL explores mathematical structures, model comparisons, and questions relating to physical interpretation. The present paper focuses specifically on exact mathematical classification, model identifiability, and reproducible numerical comparison. Its mathematical conclusions are separate from any unverified physical hypotheses associated with the wider project. 9. Future Research Future work may investigate larger families of binary models, additional identifiability conditions, and comparisons with independently collected experimental data. Any proposed physical interpretation would require clearly defined experimental settings, independently verified observations, and appropriate statistical testing. 10. Author Information Author: Roshankumar Chandaliya Affiliation: Independent Researcher, India Research Project: Quantum Diverter Loop (QDL) Research Area: Mathematical Modelling, Binary Systems, Model Identifiability, and Computational Verification Funding: This research received no external funding. Competing Interests: The author declares no competing interests.