larger_sync_flatter
plain-language theorem explainer
Larger synchronization periods produce strictly smaller values of the landscape flatness parameter, hence flatter J-cost landscapes in the Gap-45 region. Researchers modeling degenerate attractors and free-will basins cite the monotonicity to bound energy differences below phi to the minus 45. The proof is a direct term reduction that unfolds the reciprocal definition and applies the standard inequality for positive reciprocals.
Claim. For positive integers $s_1 < s_2$, the landscape flatness parameter satisfies $1/s_2 < 1/s_1$, where flatness is defined as the reciprocal of the synchronization period.
background
The Gap-45 module studies near-degenerate attractors that arise when the 8-tick cadence meets the 45-rung phi-ladder, producing the combined period lcm(8,45)=360. At this 360-tick boundary the J-cost landscape becomes nearly flat, with energy differences smaller than phi^{-45}, creating the degenerate basin required for free will.
proof idea
The term proof first applies simp to unfold the definition landscapeFlatness(sync) := 1/sync, reducing the goal to 1/s2 < 1/s1. It then invokes div_lt_div_of_pos_left together with positivity witnesses and the cast of the hypothesis s1 < s2.
why it matters
This monotonicity result directly implements the module statement that the 360-tick period creates a flat cost landscape and that flat-region width scales with gap tolerance. It reinforces the Recognition Science landmarks T7 (eight-tick octave) and the phi-ladder that together generate the Gap-45 degeneracy. No downstream theorems are recorded yet.
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