Mathematical Methodology & Lean Proofs

Mathematical Methodology & Formal Verification

Standard polling and social data analysis traditionally rely on conventional hypothesis testing and basic margins of error. However, when evaluating high-stakes collective decisions, traditional statistical models often fail to capture the true reliability of the outcome. This platform introduces a stricter mathematical framework—including the Accuracy Equation —designed to stress-test data and measure the definitive statistical safety of group decisions.

To guarantee absolute structural validity, the core theorems, semantic models, and accuracy equations presented on this site have been formally verified using the Lean 4 Interactive Theorem Prover.

While traditional academic publishing relies on human peer review, formal verification provides a high standard of proof. Lean 4 utilizes a deterministic computational kernel to machine-check every logical step of a mathematical claim. By encoding our binary vote semantics and accuracy schemas into Lean, we ensure that the underlying mathematics powering our interactive calculators are mathematically sound and free from human or algorithmic error.

In the interest of complete scientific transparency, all mathematical definitions and formal proof scripts are open-source. Independent researchers, statisticians, and developers can directly inspect, compile, and verify these .lean files via the public repository below.

Formally Verified Mathematical Theorems

The underlying mathematics of the Decision Accuracy framework have been separated into three core computational layers. The following primary theorems have been machine-checked for absolute logical soundness:

Layer 1: Structural Foundations & Baseline Mathematics

• mean_signal_closed_form: Proves the exact closed-form algebraic calculation for the binary vote mean difference, ensuring the baseline signal is always mathematically exact.

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• binary_level3_closed_form: Verifies the rigorous deduction of the primary "Vote Accuracy" equation ($u - M$) for binary options, locking in the formula powering our calculators.

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• accuracy_decreases_with_M: Formally guarantees monotonicity—proving definitively that as the statistical margin of error expands, the resulting vote accuracy strictly and predictably decreases.

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Layer 2: Semantic Voting Models

• probs_sum_to_one: Establishes absolute, machine-checked bounds on population voting behavior, ensuring the underlying binary probabilities always adhere to fundamental probability laws.

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• semanticSignal_eq_meanSignal: Formally proves that the semantic signal of a population perfectly matches the baseline mean signal for binary models, guaranteeing that the conceptual model aligns with the raw mathematics.

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Layer 3: Empirical Error & Observational Limits

• meanSignal_estimation_error: Computes the exact linear propagation of sampling error from empirical data, proving that observational deviations are strictly and deterministically bounded.

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• reportedAccuracy_conservative_if_M_covers_error: The reported accuracy acts as an absolute, conservative lower bound on the true population signal.

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• exact_observation_recovers_true_signal: Proves that in the event of an exact observation (where estimation error is zero), the mathematical model perfectly recovers the true population signal.