informationPreservation
plain-language theorem explainer
Information preservation is encoded as the norm equality ||Sv|| = ||v|| for any state vector v. QFT researchers deriving scattering from ledger balance would cite this when linking Recognition Science conservation to S-matrix properties. The definition is a direct string assignment of the algebraic identity that follows from S†S = I.
Claim. The information preservation property states that for any initial state vector $v$, the S-matrix satisfies $||(Sv)|| = ||v||$, which follows from the unitarity condition $S^†S = I$.
background
The module derives S-matrix unitarity from Recognition Science ledger conservation, where the ledger is a balanced double-entry system preserving total J-cost across recognition events. The S-matrix maps initial states at $t → -∞$ to final states at $t → +∞$, with unitarity encoding probability conservation and information preservation. Upstream, LedgerFactorization.of supplies the structure of $(ℝ₊, ×)$ and J calibration, while PhiForcingDerived.of gives the derived J-cost properties used in the ledger balance.
proof idea
This is a direct string assignment of the algebraic identity $(Sv)^†(Sv) = v^†(S^†S)v = v^†v$, which holds once $S^†S = I$ is assumed. No additional lemmas or tactics are invoked beyond the definition of the adjoint and norm.
why it matters
The definition supports the module target of deriving S-matrix unitarity from ledger conservation and connects to sibling results such as unitarity_means_probability_conserved and ledgerUnitarityConnection. It fills the explicit statement of information preservation as a consequence of unitarity, consistent with the forcing chain landmarks T5 (J-uniqueness) and T7 (eight-tick octave) that underpin the ledger structure.
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