all_ledgers_are_jcost_critical
plain-language theorem explainer
Every defect-bounded sub-ledger satisfies the J-cost criticality condition, so its defect cannot decrease under any non-negative perturbation. Researchers translating Hodge theory into Recognition Science cite this as the trivial half of the harmonic-form statement. The proof is a one-line term that feeds the perturbation into linear arithmetic on the defect inequality.
Claim. For any defect-bounded sub-ledger $L$, the defect functional obeys $L.defect + δ ≥ L.defect$ whenever $δ ≥ 0$.
background
The module translates classical Hodge theory to Recognition Science by identifying harmonic forms with J-cost critical sub-ledgers. A sub-ledger is J-cost critical precisely when its defect is already minimal: no compatible change lowers it further. The defect functional itself is taken from the Law of Existence, where defect(x) equals J(x) for positive x and vanishes at unity.
proof idea
The proof is a one-line term that applies linear arithmetic directly to the definition of IsJCostCritical. It constructs the function taking δ and a proof that δ ≥ 0, then invokes linarith to obtain the required inequality.
why it matters
This result supplies the easy direction for the module's harmonic-form theorems and the Hodge decomposition. It sits inside the Recognition Science translation of the Hodge conjecture, where J-cost critical points serve as the unique minimal representatives of Z-charge classes. The declaration closes the trivial half before the hard direction via defect budget is addressed in sibling lemmas.
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