bounded_residual_trap_solved
plain-language theorem explainer
Bounded search together with reciprocal pair closure yields a rational Erdős-Straus representation for every residual trap integer. Researchers extending the Recognition Science treatment of the Erdős-Straus conjecture would cite this result when closing the residual class. The proof is a direct term-mode application of the residual trap lemma after phase equidistribution is derived from the bounded engine.
Claim. Let $B$ be a bounded search engine and $P$ a reciprocal pair closure engine. For every natural number $n$ that is a residual trap (i.e., $n>1$, $n≡1$ mod 24, and all prime factors of both $n$ and $(n+3)/4$ are ≡1 mod 3), the rational $n$ admits a rational Erdős-Straus representation.
background
The module develops a rotation hierarchy for the Erdős-Straus conjecture inside Recognition Science. It isolates two missing engines as explicit structure fields: bounded search, which returns an admissible hard gate below a computable bound together with phase support, and reciprocal pair closure, which converts that support into a gate closure witness. A residual trap is defined as an integer $n>1$ with $n≡1$ mod 24 whose prime factors (and those of $(n+3)/4$) are all ≡1 mod 3. Upstream results supply the reciprocal automorphism on the cost algebra, the balanced property of ledgers, and the active edge count $A=1$ at each fundamental tick.
proof idea
The proof is a term-mode one-line wrapper. It invokes the lemma residual_trap_solved, supplying the phase equidistribution fact obtained by applying bounded_search_implies_phase_equidistribution to the bounded engine, the reciprocal pair closure engine, and the residual trap hypothesis.
why it matters
This theorem shows that the residual class is solved once the two engines are supplied, completing the finite/gate pieces of the RCL attack in the Erdős-Straus Recognition Rotation Hierarchy. It feeds the overall proof skeleton for the conjecture without new axioms. The result connects directly to the Recognition Composition Law through the required reciprocal closure step and isolates the arithmetic targets (prime phase separation and pair closure) for future work.
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