p_preserves_cost
plain-language theorem explainer
Parity transformation on a ledger entry leaves its recognition cost unchanged, establishing the spatial reflection symmetry required for CPT invariance in the Recognition Science ledger model. Physicists deriving discrete QFT symmetries from first principles would cite this when assembling the full CPT theorem. The equality follows by direct reflexivity because applyP permutes only the position coordinates while copying the cost field verbatim.
Claim. For any ledger entry $e$, the cost of the parity-transformed entry satisfies $cost(P(e)) = cost(e)$.
background
A LedgerEntry records a recognition event via its 3D position (Fin 3 → ℝ), phase in the 8-tick cycle, charge indicator, and non-negative cost field. The cost is the J-cost of the underlying state, where J satisfies the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y) and is induced by the multiplicative recognizer comparator. Parity symmetry follows from the isotropy of the 3D voxel lattice forced by D = 3 in the unified forcing chain (T8).
proof idea
The proof is a one-line reflexivity reduction. It applies because the definition of applyP leaves the cost field of the LedgerEntry unchanged while only reflecting the position coordinates.
why it matters
This supplies the P-invariance component of the CPT theorem derived from ledger double-entry structure in the QFT-005 module. It rests on the eight-tick octave (T7) and D = 3 (T8) from the forcing chain, together with the J-cost definition from ObserverForcing and MultiplicativeRecognizerL4. The module targets a PRL paper on CPT from discrete ledger symmetry; the result closes one symmetry leg without additional hypotheses.
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