Hypergendered Logic Source
Hypergendered Logic — Page 165
Allowed outside: • ordinary Boolean equivalences on state nodes; • IH projection as an implication rule, not a destructive rewrite. Forbidden destructive rewrites: • PA distribution; • IH distribution; • PH distribution; • IH flattening; • PH flattening into component profiles; • NOT migration through PA; • NOT migration through IH; • NOT migration through PH; • replacement of PH(NOT S) with NOT PH(S) or the converse. K.4 Two proof graphs Maintain: 1. a truth proof graph for ordinary consequences; 2. a provenance graph for IH source inheritance; 3. a phenotype-profile graph for PH source identity, profile membership, expression/maturation status, and any explicit causal mappings. A truth proof can prove T from IH(S) and S=>T without creating a provenance edge IH(T). Likewise, a visible trait resemblance does not create a PH(S) node unless the whole-profile criterion is actually satisfied. K.5 Memoization For a finite or ordinal-general successor SF(alpha+1), the canonical six-conjunct rule contains six recursive operand occurrences: four references to the same-side predecessor SF(alpha) through IH, PH, UD, and NOT-FD, and two references to the opposite-side predecessor SM(alpha) through IH(NOT SM(alpha)) and NOT PH(SM(alpha)). A naive copied syntax tree therefore satisfies approximately T(n+1)=6T(n)+O(1), giving exponential textual/tree growth Theta(6^n). This is not the correct storage model. A provenance-preserving DAG stores only two unique predecessor-state objects per successor and lets multiple typed operator edges point to them, making stage-object growth approximately linear before limit/SIH symbolic-domain effects are considered. K.5a Literal-expansion complexity: if every recursive occurrence copies its complete source subtree, a current Stage-3+ successor formula produces approximately six predecessor-subtree copies: four own-side occurrences through IH, PH, UD, and NOT-FD, plus two opposite-side occurrences through IH(NOT opposite) and NOT PH(opposite). Thus T(n+1)=6T(n)+O(1) and naive textual/tree size is Theta(6^n). This is only a warning model and should never be used for high-order computation. K.5b DAG complexity: finite Stage n stores one SF node and one SM node per order. Each successor side carries six typed recursive operator edges, but those edges target only two unique predecessor-state objects: the own-side predecessor and the opposite-side predecessor. Repeated references do not duplicate the referenced stage object. Provenance remains exact because edge labels preserve IH/PH/NOT-PH/UD/NOT-FD role and source identity. K.5c Limit-closure complexity: a limit stage is one LimF/LimM object whose predecessor domain is represented intensionally by an ordinal bound, not extensionally by an impossible-to-enumerate list when the bound is transfinite. A query such as beta<lambda? generates the relevant IH/PH/UD/FD closure facts on demand.
Source Canon: Verbatim mathematical and modal logic. Interactive navigation and operator lookups available at /#hgl-part:165.