Hypergendered Logic Source
Hypergendered Logic — Page 268
HGL now distinguishes at least four independent kinds of “later”: (1) movement from alpha to alpha+1 on one
active coordinate axis; (2) movement from one dimensional stratum n to a higher dimensional stratum n+1;
(3) movement from one completed minor category to the seed of the next minor category; and (4) movement
from one completed major stage to the next major-stage envelope. A proof that uses a constructor for one
kind of movement to justify another kind is ill-typed even when the notation looks superficially similar.
CoordinateStep: C<n>[alpha,Omega^n] -> C<n>[alpha+1,Omega^n].
DimensionalLift: C_n -> C_(n+1), where the whole prior active axis/domain is superinherited.
CategoryLift: CompletedDimDomain(C) -> NextCategorySeed.
MajorLift: CompletedMajor(M) -> Major(M+1).
Layman analogy: walking one metre along one floor, constructing a new floor, constructing a new building
from the completed old building, and constructing a new city-district containing building systems are all
“going higher” only in a loose English sense. HGL refuses to let those four operations collapse into one
counter.
393. Tail-SIH is not historical prefix closure
Ordinary SIH was introduced as a tail-oriented superinheritance operation. SIH_axis(X[a]) means that every
canonical coordinate of the selected axis at a and after a through the Hyperomega saturation bound is
inherited with provenance. That is exactly what is needed when a higher dimensional stratum superinherits a
completed lower axis. It is not safe for constructing a limit lying inside that same axis, because a tail
beginning below lambda contains lambda and coordinates beyond lambda.
SIH_axis(X[a]) := source-preserving closure over {X[beta] : a <= beta <= Omega}.
Therefore the following attempted limit definition is forbidden: “for every beta<lambda, put SIH_axis(X[beta])
into X[lambda].” Each such SIH tail would reach through lambda into the future. The definition could make
the limit inherit itself and states not yet developmentally reached. HGL names this the Tail-at-Limit
Self/Future-Contamination Error.
394. Bounded Superinheritance and Prefix Superinheritance
HGL therefore generalizes superinheritance with an interval-bounded form. BSIH_i(X;[a,b)) means: inherit
every state of typed source axis i whose coordinate beta satisfies a<=beta<b, preserving each coordinate,
polarity, category, dimensional order, and source key separately. The right boundary is exclusive. Prefix
Superinheritance is the special historical case from the beginning of the active axis up to but not including
the target limit.
BSIH_i(X;[a,b)) := SuperInheritanceClosure_i({X[beta] : a <= beta < b}).
PSIH_i(X;<lambda) := BSIH_i(X;[1,lambda)).
Ordinary SIH_i(X[a]) := BSIH_i(X;[a,Omega+]) where Omega+ denotes inclusion of the canonical
Omega endpoint.
PSIH is not the logical negation of SIH and it is not an ordinary IH. It is a bounded source-domain closure.
PSIH_i(NOT X;<lambda) preserves structured negation on each inherited source; it does not become NOT
PSIH_i(X;<lambda).
PSIH_i(NOT X;<lambda) != NOT PSIH_i(X;<lambda).
PSIH_i(X;<lambda) != SIH_i(X[1]) whenever lambda is below the axis endpoint.
395. Prefix non-selfinheritance and non-futureinheritance theorems
The most important safety property of PSIH is what it does not contain. Because the bound is strict, the limit
coordinate is excluded from its own defining predecessor domain, and every later coordinate is excluded as
well.
PSIH_i(X;<lambda) => IH(X[beta]) for every 1<=beta<lambda.
Source Canon: Verbatim mathematical and modal logic. Interactive navigation and operator lookups available at /#hgl-part:268.