jcostEntangled
plain-language theorem explainer
jcostEntangled supplies the explicit J-cost formula for an N-particle fully entangled state as N times the single-particle cost plus a quadratic cross-term. Researchers deriving classical emergence in Recognition Science would cite this expression when comparing entangled versus product-state costs. The definition is introduced as a direct algebraic sum that encodes the N² scaling of entanglement penalties.
Claim. For positive integer $N$ and real numbers $j_0$, $a$, the J-cost of a fully entangled state of $N$ particles is $N j_0 + a N (N-1)/2$.
background
In Recognition Science the J-cost of any state is the value of the cost function induced by a multiplicative recognizer, equivalently the J-cost assigned to a recognition event. The module develops classical emergence from many-body J-cost minimization: single-particle superpositions remain low-cost while correlated N-particle states incur quadratic penalties, so that product states minimize total cost for large N. The local setting is QF-011, whose core claim is that the environment acts as a J-cost regulator selecting classical pointer states.
proof idea
This is a direct definition that adds the linear single-particle contribution to the pairwise entanglement cross-term. No lemmas or tactics are invoked; the expression is the primitive that downstream theorems unfold.
why it matters
The definition supplies the concrete cost expression required by entangled_higher_cost and cost_difference_scales_quadratically, which establish the quadratic advantage of product states, and by the QM-interpretation structure qm_interpretation_from_ledger asserting that classical descriptions emerge as J-cost minima. It instantiates the many-body mechanism of QF-011 and connects to the Recognition Composition Law through the quadratic scaling of cross-terms.
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