Degrees of the order-eleven quasi-Fibonacci polynomials

Wituła and Słota define the five polynomial sequences by system (3.12) on printed page 5:

A_{n+1}(δ) = A_n(δ) + 2δB_n(δ) - δE_n(δ),

B_{n+1}(δ) = δA_n(δ) + B_n(δ) + δC_n(δ) - δE_n(δ),

C_{n+1}(δ) = δB_n(δ) + C_n(δ) + δD_n(δ) - δE_n(δ),

D_{n+1}(δ) = δC_n(δ) + D_n(δ),

E_{n+1}(δ) = δD_n(δ) + (1 - δ)E_n(δ),

with the initial values printed immediately below it:

A_0(δ) = 1, B_0(δ) = C_0(δ) = D_0(δ) = E_0(δ) = 0.

The sentence on printed page 19 is:

Problem. Is it true that deg A_n(∆) = deg B_n(∆) = deg C_n(∆) = deg D_n(∆) = deg E_n(∆) = n for every n = 5, 6, . . .?

Section 6 introduces ∆ := 1/δ and writes the same polynomial family as A_n(∆),...,E_n(∆). The formal module names the indeterminate X. It reads degree in WithBot ℕ, rather than natDegree, takes the printed threshold literally as 5 ≤ n, and defines the recurrence over the coefficient ring ℤ.

The source proves only that the maximum of the five degrees is n. Its final Problem asks whether every coordinate has full degree from index five onward. The frozen declaration D5/S1/Recurrence/WitulaSlotaQuasiFibonacciOrderElevenDegree.result answers that universal question affirmatively for all natural n ≥ 5, including the nonvanishing assertion implicit in equality with a finite polynomial degree.

Issue #9270 preregisters this first-tier published Problem and its literature check. OpenAlex record W80948396 lists nine citing works, and no available abstract mentions the degree question. Four accessible citing full texts were read: the 2009 AADM paper on δ-Fibonacci numbers, the 2017 paper on two-parametric quasi-Fibonacci numbers, the 2019 paper on quaternion equivalents, and the 2021 paper on split-quaternions. None answers the printed Problem.

The 2017 Mathematica Slovaca paper on δ-Fibonacci and δ-Lucas polynomials and the 2013 Part II paper were unreachable and remain ASSUMED-UNVERIFIED. OEIS A062883 and A189235 contain no statement about these polynomial degrees. Three MathDB queries returned no entry for the Problem. These checked surfaces do not establish exhaustive worldwide literature coverage, publication priority, or the absence of an independent proof.

Let l_n be the vector of coefficients of X^n in (A_n,B_n,C_n,D_n,E_n). The live escape witness (W) is the sign-normalized leading-coefficient invariant

w_n = ((-1)^(n+j) [X^n] X_n)_j.

At index five, w_5 = (1,9,1,4,1). If its coordinates are (a,b,c,d,e), the recurrence gives

w' = (2b+e, a+c-e, b+d+e, c, d+e).

Positivity is preserved together with e < a+c, because a'+c'-e' = 3b+e > 0. A separate induction proves that every one of the five polynomials has degree at most n. The invariant makes every coefficient of X^n nonzero for n ≥ 5, and Polynomial.degree_eq_of_le_of_coeff_ne_zero upgrades each upper bound to an equality.

Under CLAUDE.md section 3.2, result has proof_shape: content: (W) is proved inside the live have chain, is not an instance, projection, or normalization of an existing frozen or pinned-upstream statement, is not definitionally equivalent to the five degree equalities, and is consumed by the five nonvanishing deductions. The module basis is open-problem-resolution under the preregistration in issue #9270.

Any n ≥ 5 and any one of the five coordinates whose degree differs from n would refute the result. Exact-integer recomputation of (3.12) through n = 200 found all five degrees equal to n for every 5 ≤ n ≤ 200.

Below the asserted range, the degree vectors (deg A_n, deg B_n, deg C_n, deg D_n, deg E_n) are (0,-∞,-∞,-∞,-∞), (0,1,-∞,-∞,-∞), (2,1,2,-∞,-∞), (2,3,2,3,-∞), and (4,3,4,3,4) for n = 0,1,2,3,4, respectively. Thus the printed threshold n = 5 is sharp.

The exact-integer recurrence sweep gives l_5 = (-1,9,-1,4,-1), l_6 = (19,-1,14,-1,5), and l_7 = (-7,28,-7,14,-6), matching the alternating-sign normalization used in (W). The source statement, printed-page locators, system (3.12), initial values, and Section 6 notation are recorded in Library/Recurrence/witulaslota2007order11.md.

The frozen theorem has statement ID sha256:83f74a96350070824c0f10bdbd200bb2ea415d554083885ac853128ba9800f31. Its axiom closure is exactly propext, Classical.choice, and Quot.sound; the Freeze event has no prerequisite frozen project nodes.

First tier: a Problem printed by Wituła and Słota in Journal of Integer Sequences 10 (2007), Article 07.8.5, preregistered in issue #9270. Resolution: proved.

| Declaration | proof_shape | escape_witness | direct frozen dependencies | admission_basis | | --- | --- | --- | --- | --- | | result | content | (W) | none | open-problem-resolution |

The public surface is exactly quasi, A, B, C, D, E, claim, and result. This is a symbolic theorem for every n ≥ 5; it is not bounded enumeration, checker infrastructure, numeric reduction, or a certified finite instance, so utility: none applies.

The 2017 Mathematica Slovaca δ-Fibonacci/δ-Lucas paper and the 2013 Part II paper are ASSUMED-UNVERIFIED because their full texts were unreachable. The bounded literature check does not establish exhaustive worldwide novelty, priority, or the absence of an independent proof. The Lean kernel does not authenticate the external PDF, its printed pagination, the literature-search coverage, or publication history.