flatness_pos
plain-language theorem explainer
For any positive integer sync period the landscape flatness parameter is strictly positive. Modelers of near-degenerate attractors in the Gap-45 setting cite this to keep the J-cost measure well-defined and positive. The result follows from the reciprocal definition of flatness and the basic ordering of positive reals. The proof is a one-line wrapper that unfolds the definition and invokes positivity.
Claim. Let $n$ be a positive integer. Then $0 < 1/n$, where the right-hand side is the landscape flatness parameter given by the reciprocal of the sync period.
background
The Gap-45 module studies near-degenerate attractors produced by lcm(8,45)=360. The 8-tick cadence from the forcing chain combines with the 45-rung phi-ladder to create a 360-tick sync boundary at which J-cost differences become smaller than phi to the minus 45, forming a flat basin. landscapeFlatness is defined as the reciprocal of the sync period and therefore quantifies the degree of flatness directly from that period.
proof idea
The proof is a one-line wrapper. It applies simp to unfold landscapeFlatness into the explicit reciprocal 1/sync, then uses the positivity tactic on the hypothesis 0 < sync to obtain the strict inequality.
why it matters
The declaration guarantees that the flatness parameter remains positive, a prerequisite for the degenerate basin that the module documentation links to free-will mechanisms. It rests on the eight-tick octave (T7) and the phi-ladder structure of the Recognition Science chain. No downstream theorems are recorded, so the result functions as a local positivity anchor inside the Gap-45 construction.
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