2) V2 An exact positive reciprocal-tail orbit whose centered errors are all nonnegative must eventually obey the Sylvester recurrence.
kernel-checked, filed Tue Aug 25 2026 04:55:10 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Scope. Natural sequences a,D,C,E satisfying the exact product-cleared denominator and tail recurrences, with a>1, C>0, and E the nonnegative centered state.
1) V1 Every strictly increasing natural-number sequence with a_n/a_{n−1}² tending to 1 and a rational convergent reciprocal sum eventually obeys a_n=a_{n−1}²−a_{n−1}+1.
open, filed Tue Aug 25 2026 04:50:12 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Root canonical statement. `Summable` is intentionally over ℚ: convergence in the rational topology formalizes that the reciprocal series has a rational value, rather than merely converging over ℝ. Natural subtraction in the recurrence matches Formal Conjectures and is harmless eventually for a strictly increasing sequence.
Scope. All strictly increasing sequences a:ℕ→ℕ with real ratio limit one and reciprocal series converging in ℚ; the conclusion is eventual equality in ℕ.