eightGateCoherent
plain-language theorem explainer
The eight-gate coherence predicate asserts that a real sequence on the naturals has vanishing sum over any eight consecutive terms. Workers on spectral rails in Recognition Science cite it when testing neutrality inside the eight-tick window. The definition is a direct one-line reduction to the eight-window sum equaling zero.
Claim. For a sequence $x : ℕ → ℝ$ and index $t_0 ∈ ℕ$, the eight-gate coherence predicate holds if and only if $∑_{k=0}^7 x(t_0 + k) = 0$.
background
The SpectralLadder module scaffolds cross-domain spectral rails with frequencies $f_n = f_0 φ^{2n}$ where $f_0 = E_{coh}/h$, and supplies an eight-gate coherence test as one of its basic helpers. The supporting sum8 operator returns the sliding sum of eight consecutive samples and functions as a coherence or neutrality diagnostic. This construction re-uses the identical eight-window summation pattern already defined in the Breath1024 and Nuclear.Octave modules for closure predicates.
proof idea
The declaration is a one-line definition that directly equates eight-gate coherence to the condition that the local sum8 operator returns zero.
why it matters
The predicate supplies the coherence diagnostic required by the spectral-ladder scaffold and aligns with the eight-tick octave (T7) of the forcing chain. It furnishes a concrete neutrality test inside periodic windows of length 8, preparing the ground for rail-frequency constructions even though no downstream dependents are recorded yet.
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