sum8
plain-language theorem explainer
The sum8 definition computes the total of eight consecutive terms in a real-valued sequence indexed by naturals, starting at a given index. Nuclear and spectral modelers in Recognition Science cite it to build neutrality diagnostics over the eight-tick window. It is realized as a direct finite summation over the range from zero to seven.
Claim. Let $x : ℕ → ℝ$ be a sequence and $i_0 ∈ ℕ$ an index. The eight-window sum is $∑_{k=0}^{7} x(i_0 + k)$.
background
In the Nuclear.Octave module this supplies the basic sliding sum used for coherence checks. It replicates the eight-window sum from Breath1024, defined identically as the sum over period-8 terms, and the version in SpectralLadder described as a sliding sum over 8 samples for coherence/neutrality diagnostics. The upstream for structure records the meta-realization axioms that justify applying such periodic windows inside the forcing chain.
proof idea
The definition is a one-line wrapper that applies the finite sum over the range of eight indices starting from i0.
why it matters
This supplies the core summation for closure and coherence predicates in the nuclear octave. It feeds isClosure, which sets the magic-number predicate to the eight-window sum equaling zero, and neutral8 together with eightGateCoherent. In the framework it realizes the eight-tick octave (T7) that supports the phi-ladder and alpha-band constraints.
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