isClosure
isClosure defines the predicate that a real sequence x on naturals is neutral at starting index i0 precisely when its eight consecutive terms sum to zero. Nuclear modelers working inside Recognition Science would cite it when testing octave closure on valence-cost sequences assembled from levelEnergy. The definition is a direct one-line equality that delegates the summation to the period-8 window operator supplied by the Breath1024 foundation.
claimLet $x : ℕ → ℝ$ and $i_0 ∈ ℕ$. The sequence $x$ is closed at $i_0$ when $∑_{k=0}^7 x(i_0 + k) = 0$.
background
The Nuclear.Octave module realizes the eight-tick octave (period 2^3) of the Recognition Science forcing chain by means of window neutrality predicates. The central auxiliary is the sum8 operator, which returns the sum of any real-valued function over the eight consecutive indices starting at a given point and is explicitly labeled the eight-window neutrality predicate. Sequences x are understood to be valence-cost proxies built from levelEnergy and the J-cost functions imported from MultiplicativeRecognizerL4 and ObserverForcing.
proof idea
The declaration is a direct definition. It simply sets the predicate equal to the statement that the eight-window sum supplied by sum8 vanishes at the given index.
why it matters in Recognition Science
isClosure supplies the atomic closure test used by the sibling isMagic definition, which declares an index Z magic precisely when some aligned 8-window ending at or before Z is neutral. It therefore implements the T7 octave step of the UnifiedForcingChain inside the nuclear domain, furnishing the discrete neutrality condition that later feeds mass-ladder and magic-number constructions. The predicate remains open to empirical calibration against observed nuclear closures once concrete cost sequences are substituted.
scope and limits
- Does not assign any physical meaning to the input sequence x.
- Does not prove invariance, monotonicity, or other properties of the predicate.
- Does not reference the phi-ladder, J-function, or Recognition Composition Law directly.
- Does not extend to non-integer indices or continuous limits.
formal statement (Lean)
30def isClosure (x : ℕ → ℝ) (i0 : ℕ) : Prop :=
proof body
Definition body.
31 sum8 x i0 = 0
32
33/‑ Magic‑number predicate (display‑level): index `Z` is magic if some aligned
34 8‑window around it is neutral. In practice, this will be evaluated on a
35 fit‑free valence‑cost proxy assembled from `levelEnergy`. -/