failureMode
plain-language theorem explainer
This definition assigns a concrete empirical failure mode to each of the nine Option A combinations. Researchers building falsifiable cross-domain protocols cite it to attach testable criteria to the C1-C9 theorems. The implementation consists of exhaustive pattern matching on the CombinationID constructors, each returning its paired FailureMode value.
Claim. Define the function $f$ from the set of Option A combinations to failure modes by cases: $f$(cognitive tensor) = single-axis decoder suffices, $f$(planet strata) = no small phi-power ratio, $f$(oncology tensor) = additive therapy response, $f$(quantum molecular depth) = depth exceeds five bits, $f$(attention tensor) = non-forty plateau spectrum, $f$(Erikson reverse) = non-reverse dementia order, $f$(universal response) = non-unit shared coefficient, $f$(Miller span) = non-Q-space span collapse, and $f$(regulatory ceiling) = module exceeds ceiling 70.
background
The Option A Falsifier Registry maintains a finite pairing of each C1-C9 cross-domain theorem with its empirical test class. This registry does not prove the empirical claims. It keeps the falsifiers attached to the Lean theorem bundle so the cross-domain work cannot drift into unfalsifiable numerology. CombinationID enumerates the nine implemented combinations as an inductive type; FailureMode enumerates the corresponding empirical failure patterns as a second inductive type.
proof idea
The definition is realized by direct pattern matching on each of the nine constructors of CombinationID. Each branch returns the matching constructor of FailureMode with no lemmas invoked and no tactics applied.
why it matters
This definition supplies the failure component required by ProtocolFalsifiable, protocolFalsifiable_all, ProtocolMatches, and protocolSpec in the empirical protocol module. It completes the registry step described in the module documentation, ensuring each combination carries an attached falsifier. The construction supports the Recognition Science requirement that cross-domain claims remain empirically grounded rather than drifting into numerology.
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