opticalTheoremFormula
plain-language theorem explainer
The optical theorem formula equates total cross section to four pi over wave number times the imaginary part of the forward scattering amplitude. Researchers deriving S-matrix unitarity from ledger conservation in Recognition Science would cite it to connect probability preservation to measurable scattering observables. The definition is a direct string assignment with no computation or reduction steps.
Claim. The optical theorem states that the total cross-section satisfies $σ_{total} = (4π/k) Im[f(0)]$, where $k$ is the wave number and $f(0)$ the forward scattering amplitude. This relation follows from S-matrix unitarity, which encodes conservation of total J-cost across initial and final ledger states.
background
The module derives S-matrix unitarity from Recognition Science ledger conservation. The S-matrix maps initial to final states as |final⟩ = S|initial⟩, with unitarity S†S = 1 enforcing probability conservation and information preservation. Ledger conservation requires every credit to match a debit so that total J-cost remains invariant under recognition events that redistribute entries between t → -∞ and t → +∞. Upstream LedgerFactorization supplies the (ℝ₊, ×) structure and J-calibration that make this balance possible.
proof idea
The definition is a direct string literal assignment of the standard optical theorem formula. It functions as a one-line wrapper that embeds the relation without invoking any lemmas or tactics.
why it matters
This definition supplies the observable link in the ledger-to-unitarity chain for QFT-012. It supports the PRD paper proposition on unitarity from ledger structure and connects to the Recognition Composition Law plus the eight-tick octave. The placement closes the step from J-uniqueness to measurable cross sections while leaving open the explicit mapping of k and f onto the phi-ladder.
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