IndisputableMonolith.NumberTheory.CompositionDivergence
The module formalizes the Composition Closure Hypothesis (CCH) asserting that iterated RCL self-compositions on off-critical-line zeta zeros generate defects absorbable only by a finite carrier budget. Researchers deriving the Riemann Hypothesis from cost symmetries cite this setup. The module organizes the hypothesis together with budget-violation and cascade-growth auxiliaries as a collection of declarations without a terminating proof.
claimThe Composition Closure Hypothesis states that for each nontrivial zero $\rho$ with $\operatorname{Re} \rho \neq 1/2$, the defect produced by the $n$-th iterate of the Recognition Composition Law self-composition is bounded above by the carrier budget scale $B$ of the AnnularCost framework.
background
The module operates in the NumberTheory domain and imports the $\xi(s)$-$J(x)$ bridge (where $\xi(s)=\xi(1-s)$ maps to $J(x)=J(1/x)$ via the coordinate change $x=\exp(2(\operatorname{Re}s-1/2))$), the Recognition Composition Law inducing a composition law on zero defects, and the zero-location dictionary that converts deviation to defect. ZeroDeviation $\rho$ is defined as $2(\operatorname{Re}\rho-1/2)$ and zeroDefect $\rho$ as defect of the corresponding exponential. The local theoretical setting treats iterated RCL applications as defect accumulation that must remain inside a finite carrier budget.
proof idea
This is a definition module, no proofs. It declares the central hypothesis together with auxiliary predicates for budget violation and exponential cascade growth.
why it matters in Recognition Science
The module supplies the Composition Closure Hypothesis that sibling results such as rh_from_composition_closure employ to obtain the Riemann Hypothesis. It connects iterated RCL defect growth to the carrier-budget constraint, linking directly to J-uniqueness (T5) and the eight-tick octave (T7) in the forcing chain.
scope and limits
- Does not prove that every nontrivial zero lies on the critical line.
- Does not supply explicit numerical bounds for any concrete zero.
- Does not treat trivial zeros.
- Does not quantify the precise growth rate of iterated defects beyond the budget statement.