{"kind":"parent_provers","proof":{"attempts":[{"detail":"the rule is not P_(n+k) = c(x) P_n with c free of n","route":"periodic_product","success":false,"variant":"P_(n+k) = c(x) P_n"},{"detail":"P_n = P_(n-1) + 3 t P_(n-3); P_0 = 1, P_m = 0 for m < 0; generating function 1/(1 - x R(t x^2)) (paper_proof_not_lean_checked): For all n, s >= 0 with 1*s <= 1*2+1: P_n << P_(n+s) (oriented weak interlacing; shared and repeated zeros allowed). Every P_n has nonnegative coefficients and constant term 1 and is real-rooted.; the identifying seeds equal the canonical rows (Canonical), so every parent row is a canonical row","route":"paper_theorem","success":true,"variant":"weighted_renewal_window"},{"detail":"the parent rows are not the canonical boundary rows (Mismatch)","route":"paper_theorem","success":false,"variant":"fixed_delay_predecessor_window"}],"conditional_on":[],"interlacing":{"from_index":1,"justification":{"premises":["homogeneous, index-independent, no derivatives or sign factors; every t^j term has lag 1 + 2*j with ell = 2 >= 1 (exact_structural)","R(z) = 1 + 3 z has R(0) = 1 and nonnegative coefficients (exact_rational)","R(z) has only real zeros (exact_root_count)","window: P_n << P_(n+s) for every n >= 0 and 0 <= s <= 3 (1*s <= 1*2+1 (d*s <= d*ell+1))","the identifying parent seeds equal the canonical rows (Canonical), so every parent row is a canonical row and consecutive parent rows are consecutive canonical rows"],"theorem":"weighted_renewal_window (paper_proof_not_lean_checked; https://github.com/PerAlexandersson/RealRooted/issues/2037): For all n, s >= 0 with 1*s <= 1*2+1: P_n << P_(n+s) (oriented weak interlacing; shared and repeated zeros allowed). Every P_n has nonnegative coefficients and constant term 1 and is real-rooted."},"kind":"weak","orientation":"previous_interlaces_next"},"route":"paper_theorem","rows_checked":10,"statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","status":"proved"},"record":{"first_index":0,"initial_polynomials":[["1"],["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3"]]},"deriv_order":0,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"try_root_domain":true}