effectivePrimePhaseInput_of_rsPrimePhaseEquidistribution
plain-language theorem explainer
This definition extracts the effective prime phase input from an RSPrimePhaseEquidistribution structure. Researchers closing the residual Erdős-Straus chain in Recognition Science cite it to connect RCL prime-ledger phase distribution to the bounded generator supply required for subset-product hits. The implementation is a direct one-line field projection.
Claim. Let $RS$ be a structure consisting of an effective prime phase input together with a marker that the input originates from the recognition composition law prime-ledger machinery. The map $RS$ to effective input returns the effective prime phase input component of $RS$.
background
The module states the exact prime-distribution input needed for the residual Erdős-Straus proof and proves that it implies PrimePhaseBoxDistribution. EffectivePrimePhaseInput is the structure whose bound function and supplies_generators clause together assert that every residual trap $n$ admits a $c$ at most the bound, with $c$ an admissible hard gate and the subset-product phase hit nonempty. RSPrimePhaseEquidistribution is the intended source structure whose effective_input field is marked as coming from RCL/J-cost prime-ledger phase distribution rather than finite search.
proof idea
The definition is a one-line wrapper that projects the effective_input field from the supplied RSPrimePhaseEquidistribution structure.
why it matters
It supplies the argument to erdos_straus_residual_from_rsPrimePhaseEquidistribution, which in turn yields the rational Erdős-Straus representation for residual traps via the effective-input version of the same theorem. The definition therefore closes the final interface between the Recognition Science RCL machinery and the number-theoretic residual proof, completing the prime phase distribution step required by the framework.
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