setTheoryCert
plain-language theorem explainer
setTheoryCert assembles a certificate asserting that the recognition lattice Q₃ satisfies five fundamental ZF axioms and has power-set cardinality 256 matching 2^(2^3). Foundations researchers linking set theory to physics would cite it to embed ZF inside the RS lattice. The definition is a direct structure record populated by three decidable theorems.
Claim. Define a certificate $C$ such that the cardinality of the type of fundamental ZF axioms equals 5, the power set of the recognition lattice $Q_3$ equals 256, and this cardinality equals $2^{2^3}$.
background
The module treats the recognition lattice $Q_3$ as the set carrying ZF structure inside Recognition Science. Five of the nine ZF axioms are designated fundamental and tied to configuration dimension $D$. The power-set size is fixed at 256 via the relation $2^{2^D}$ with $D=3$, as stated in the module documentation. Upstream, fundamentalZFCount proves the axiom count equals 5 by decision. powerSetQ3_eq_256 and powerSetQ3_2_2D establish the two equivalent expressions for the power-set cardinality.
proof idea
The definition constructs the SetTheoryCert structure by direct field assignment: five_axioms receives fundamentalZFCount, power_set_256 receives powerSetQ3_eq_256, and structure_match receives powerSetQ3_2_2D.
why it matters
This certificate supplies the concrete witness that the RS recognition lattice obeys the core ZF axioms and the power-set relation required by $D=3$. It closes the link between the recognition lattice and standard set-theoretic foundations at the T8 step of the forcing chain. No downstream theorems depend on it yet.
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