Probability is Lack of Information: Part IV - The Second Derivative of Binary Entropy as the Curvature of Determinism
Diğer, ss.1-8, 2026
- Yayın Türü: Diğer Yayınlar / Diğer
- Basım Tarihi: 2026
- Sayfa Sayıları: ss.1-8
- Ankara Üniversitesi Adresli: Evet
Özet
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Abstract: Following the axiomatic foundation established in Part I, the formal proof via the Binary Entropy Function in Part II, and the introduction of the first derivative as the Gradient of Determinism in Part III, this fourth installment examines the second derivative of the binary entropy function. We demonstrate that H''(p) = −1/(p(1−p) ln 2) quantifies the curvature of determinism, revealing the rate at which the gradient of determinism itself changes.
This second derivative is always negative, confirming the inherent concavity of the entropy function, with its most negative value at p = 0.5 — the point of maximum epistemic uncertainty. This region represents the highest “acceleration” toward deterministic resolution. We interpret this curvature as a dynamic measure of how rapidly informational deficit collapses as observational resolution increases within the Hendeca-Tier Cosmic Signal Architecture (HCSA).
By extending the analysis from gradient to curvature, we provide a more complete mathematical framework for understanding the transition from epistemic probability to ontological determinism. This installment strengthens the central thesis of the series: probability is not a fundamental property of nature, but a measurable indicator of missing information whose rate of resolution can be rigorously quantified.
The philosophical motto of the series continues to guide our inquiry: Dubito ergo cogito ergo sum → Dubito ergo determinismus est. I doubt, therefore I think, therefore I am — I doubt, therefore determinism exists.