Information-Energy Equivalence

Information-Energy Equivalence is the relation E = ℏ × I × ω: I bits, each one binary transition at angular frequency ω, carry I quanta of energy ℏω. It is derived in Part 3.1 from the Data Energy Transition. A data bit is one binary transition, 0 → 1 or 1 → 0 (Part 1.2), and each transition carries one quantum, E⊶ = ℏ × ω₁ × n○, with n○ = 1 for a transition that occurs. Information of I bits is I such transitions, and energy adds, so for transitions at a common angular frequency ω, E = Σₖ ℏ × ω × n○,ₖ = ℏ × ω × Σₖ n○,ₖ = ℏ × I × ω. Its difference form is the Energy Change from Information Change, ΔE = ℏ × ΔI × ω. E⊶ is the energy of one transition at the Tempo; the Data Energy of a mass m is a different quantity, ↁ⚕ = m(2𝒞→)² (Data Energy Mass Equivalence, Part 1.11), and the bits a mass is worth at a temperature is the Data–Matter Equivalence (Part 2.9). The absolute scale of energy enters the theory through ℏ here. The book writes E = ℏ × I × ω [𝕄·𝕃²·𝕋⁻²], where E is the energy carried by the information [𝕄·𝕃²·𝕋⁻²], ℏ the reduced Planck constant [𝕄·𝕃²·𝕋⁻¹], I the information in bits, the count of binary transitions [∅], and ω the angular frequency of the transitions [𝕋⁻¹]: [𝕄·𝕃²·𝕋⁻¹] × [∅] × [𝕋⁻¹] = [𝕄·𝕃²·𝕋⁻²]. It connects to Data Energy Transition, Energy Change from Information Change, Data Energy Mass Equivalence, Data–Matter Equivalence and Information Conservation.