GateHasPhaseSupport
plain-language theorem explainer
GateHasPhaseSupport defines a predicate on naturals n and c asserting that the finite cyclic quotient Z/cZ detects a non-identity phase in the divisor ledger of n. Number theorists building phase separation engines for the Erdős-Straus conjecture inside the Recognition rotation hierarchy would cite this predicate when assembling the required structures. The definition is a one-line existence claim over the quotient type that reduces to the quotient being inhabited.
Claim. For natural numbers $n, c$, the predicate holds if there exists a residual phase quotient modulo $c$ such that the divisor ledger of $n$ exhibits a non-identity phase.
background
The Erdős-Straus Recognition Rotation Hierarchy module converts the Recognition Composition Law attack into a proof skeleton by isolating finite gate pieces and two missing engines: prime phase separation across admissible residual quotients, and reciprocal pair closure. Upstream, the EightTick.phase supplies the 8-tick phases $kπ/4$ for $k=0..7$, while Thermodynamics.admissible is the trivial predicate True on any ledger state. ResidualPhaseQuotient c is the abbrev ZMod c, the finite cyclic quotient induced by a residual gate. The Wedge.phase supplies the unimodular complex $e^{iw}$.
proof idea
One-line wrapper that asserts existence of an element in ResidualPhaseQuotient c together with the trivial equality n = n. It directly uses the sibling abbrev ResidualPhaseQuotient c := ZMod c and the upstream phase definitions to mark non-identity support.
why it matters
This definition populates the phase_support field of PrimePhaseEquidistributionEngine, which asserts that every residual trap n admits an admissible hard gate c with GateHasPhaseSupport n c. It likewise appears in BoundedSearchEngine and ReciprocalPairClosureEngine, advancing the hierarchy toward the required reciprocal divisor-pair witness. It interfaces with the eight-tick octave by leveraging periodic phases and prepares the ground for later steps in the T0-T8 forcing chain.
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