sub_ledger_exists
plain-language theorem explainer
The theorem establishes existence of a non-empty DefectBoundedSubLedger, ensuring the Recognition Science translation of the Hodge conjecture is not vacuously true. Researchers formalizing algebraic geometry inside the Recognition Science framework cite it to confirm the setup admits concrete instances. The proof constructs an explicit minimal ledger with one recognition event of zero defect and verifies the bounds via norm_num, linarith, and le_refl.
Claim. There exists a DefectBoundedSubLedger $L$ such that the length of its list of recognition events is positive.
background
DefectBoundedSubLedger is a finite list of recognition events equipped with a real defect satisfying defect < phi^44 and 0 ≤ defect; it serves as the RS analog of a smooth projective algebraic variety. The module translates the classical Hodge conjecture by identifying Hodge classes with CoarseGrainingStableClass and algebraic cycles with JCostMinimalCycle, where the defect budget bridge requires surviving features to anchor at zero-cost minima. Upstream results include the defect functional from LawOfExistence (equal to J for positive arguments) and le_refl from ArithmeticFromLogic.
proof idea
The proof is a direct construction that instantiates a DefectBoundedSubLedger with a single RecognitionEvent, sets defect to zero, proves the upper bound via pow_pos phi_pos 44 followed by linarith, invokes le_refl for nonnegativity, and finishes with simp to discharge the length condition.
why it matters
This result confirms the RS Hodge conjecture applies to at least one concrete sub-ledger, closing the non-vacuity gap in the formalization. It supports the module's program of showing every CoarseGrainingStableClass arises from JCostMinimalCycles via the defect budget bridge. The construction aligns with the eight-tick octave and phi-ladder landmarks but does not resolve the open direction of the conjecture.
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