RecognitionCost
plain-language theorem explainer
RecognitionCost sums the J-cost of each bond multiplier over the active bonds of a ledger state. Information theorists deriving Landauer bounds or 8-tick dissipation limits in Recognition Science cite this definition when quantifying total recognition expenditure. It is a direct one-line summation that applies the J-cost operator to every multiplier in the active set.
Claim. Let $s$ be a ledger state whose active bonds $B$ carry positive multipliers $m_b$. The recognition cost is defined by $C(s) = sum_{b in B} J(m_b)$, where $J(x) = (x + x^{-1})/2 - 1$ is the cost function on positive reals.
background
The module anchors Recognition Science cost in thermodynamic entropy via the Landauer limit. LedgerState is the minimal structure consisting of a finite set of active bonds together with a positive real multiplier for each bond; it records the local collection of recognition events whose total cost is to be computed. J-cost is the function supplied by the phi-forcing chain that assigns to each ratio its recognition expenditure; upstream ledger-state structures from variational dynamics and information-is-ledger supply the same interface, while the J-cost definition itself comes from the phi-forcing-derived factorization of the multiplicative group.
proof idea
The definition is a one-line wrapper that applies the summation of the J-cost function over the active bonds in the ledger state.
why it matters
RecognitionCost supplies the cost functional used by the total-dissipation-bound theorem, which proves $C(s) >= (1/2) sum (log m_b)^2$, and by the eight-tick-dissipation-limit theorem, which shows that one complete cycle of the recognition operator saturates the Landauer erasure cost. It therefore occupies the central place in Phase 7.5.2, linking the eight-tick octave (T7) to thermodynamic dissipation bounds. No open scaffolding questions are closed by this definition.
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