primePhaseBoxDistribution_of_effectivePrimePhaseInput
plain-language theorem explainer
This definition constructs the prime phase box distribution required for the residual Erdős-Straus argument directly from an effective prime phase input structure. Researchers closing the number-theoretic part of the Recognition Science chain cite it to bridge the input hypothesis to the distribution predicate. The construction copies the bound function and lifts each generator supply to a balanced phase hit via the conversion theorem on subset-product phase hits.
Claim. Let $I$ be an effective prime phase input. The associated prime phase box distribution is the structure whose bound function equals that of $I$, and whose hits predicate asserts that for every residual trap $n$ there exists a bounded admissible gate $c$ such that the subset-product phase hit supplied by $I$ yields a balanced phase hit.
background
The module states the exact prime-distribution input needed for the residual Erdős-Straus proof and shows it implies the prime phase box distribution. An effective prime phase input consists of a bound function together with a supplies_generators predicate: for every residual trap $n$ there exists $c$ bounded by the function, admissible, and carrying a nonempty subset-product phase hit. The target prime phase box distribution reuses the same bound and requires instead that each such $c$ produces a hits-balanced-phase witness.
proof idea
The definition builds the prime phase box distribution by setting its bound field to the input bound. For the hits field it introduces $n$ and the residual trap hypothesis, destructures the supplies_generators call to extract $c$, the bound proof, admissibility, and the phase hit, then applies the conversion theorem generated_phase_hit_gives_HitsBalancedPhase to obtain the required balanced-phase witness.
why it matters
This definition supplies the bridge from the effective prime phase input hypothesis to the distribution predicate used by the residual Erdős-Straus solver. It is invoked to obtain both the bounded balanced search engine and the theorem that the input solves the residual trapped class. In the Recognition framework it advances the eight-tick phase structure into the divisor-box distribution required for the T7 octave step of the forcing chain.
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