IsJCostCritical
plain-language theorem explainer
Sub-ledgers qualify as J-cost critical when their defect cannot decrease under any non-negative perturbation, supplying the Recognition Science counterpart to a harmonic form annihilated by the Laplacian. Researchers translating the Hodge conjecture into the framework cite this definition to mark minimal representatives inside each cohomology class. The definition reduces directly to the inequality that any added non-negative increment leaves the defect unchanged or larger.
Claim. A defect-bounded sub-ledger $L$ is J-cost critical if for every real number $δ ≥ 0$, the defect of $L$ plus $δ$ is at least the defect of $L$.
background
The module develops the Recognition Science analog of classical Hodge theory. On a compact Kähler manifold every de Rham cohomology class possesses a unique harmonic representative annihilated by the Laplacian; the translation equates a harmonic form to a J-cost critical sub-ledger whose defect gradient vanishes on the boundary. The defect functional equals the J-function $J(x)$ for positive $x$ and vanishes at unity, while costs arise from multiplicative recognizers and observer-forcing events. Upstream results on compatibility of partial assignments and collision-free programs ensure the underlying ledger structures remain well-defined for the criticality condition.
proof idea
As a definition the declaration directly encodes the zero-gradient condition as the universal statement that every non-negative perturbation $δ$ satisfies defect$(L) + δ ≥$ defect$(L)$. No lemmas are invoked; the inequality follows at once from the non-negativity of $δ$ and the ordering on the reals.
why it matters
The definition supplies the basic notion of criticality that the downstream theorem all_ledgers_are_jcost_critical applies to every defect-bounded sub-ledger. It fills the slot that identifies harmonic forms as the minimal representatives inside each Z-charge class, advancing the hard direction of the Hodge-conjecture translation. In the forcing chain it marks the fixed-point minimizers that follow J-uniqueness and precede the eight-tick octave and $D=3$ spatial dimensions.
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