The Unfair 0-1 Polynomial Problem and High-Degree Trinomials

Abstract: The unfair $0$--$1$ polynomial conjecture asks whether a factorization [C(x)=A(x)B(x),] with $A$ and $B$ monic and having nonnegative real coefficients, must already be a factorization into $0$--$1$ polynomials. Let $k$ be odd and $0<a<1$. We study the possibility that [1+a x^2+x^k] divides a $0$--$1$ polynomial with a nonzero cofactor having nonnegative real coefficients. Ghidelli settled the first nontrivial case $k=5$, and the cases $k=7,9,11$ were treated subsequently by finite recurrence and spectral arguments. We prove that no such factorization exists for any odd $k\ge 341$.

• The paper proves that for every odd k≥341 and 0<a<1, no nonzero polynomial with nonnegative coefficients can make (1+ax²+xᵏ)Q(x) a 0-1 polynomial.
• It replaces exponent-by-exponent Boolean continuation searches with a uniform analytic method combining first-zero reduction, negative-binomial packets, exact strict log-concavity, and theta-kernel limits.
• The proof establishes a rigid pre-zero coefficient profile and then uses deterministic continuation to derive a negative terminal coefficient, while leaving odd exponents 13≤k≤339 and broader factor families open.

The problem and its background

The unfair $0$--$1$ polynomial conjecture (Problem 28 in Ben Green's list) asks whether a factorization of a polynomial $C(x)$ with coefficients in $\{0,1\}$ into monic polynomials with nonnegative real coefficients must already be a factorization into $0$--$1$ polynomials. Equivalently, a uniform finitely supported distribution on $\mathbb Z$ cannot be a convolution of two nonuniform finitely supported probability distributions.

The paper studies the trinomial test family

$$P_{k,a}(x)=1+ax^2+x^k,\qquad 0<a<1,\ k\ \text{odd},$$

and proves that for every odd $k\ge341$ there is no nonzero polynomial $Q$ with nonnegative coefficients such that $1$0 has all coefficients in $1$1. This is the first uniform-in-$1$2 result for this family. Prior work covered only individual exponents: Ghidelli settled $1$3 via forced recurrences, resultants, and numerical analysis; Hare's computational framework eliminated all but $1$4 of $1$5 candidate factor patterns up to degree $1$6; and the author's earlier work handled $1$7 by finite spectral boxes and exact Farkas certificates. Those methods scale poorly: characteristic roots become crowded, retained-mode boxes grow with $1$8, and continuation trees deepen rather than shrink. The present proof abandons both strategies in favor of an analytic, global argument that does not enumerate Boolean continuations at all.

The first-zero reduction

Writing $1$9 for the product digits and assuming $C(x)$0, the initial forcing lemma shows all odd coefficients vanish below $C(x)$1, even coefficients equal alternating geometric sums, and $C(x)$2. Letting $C(x)$3 be the first index $C(x)$4 with $C(x)$5, the cofactor coincides before $C(x)$6 with a forced sequence $C(x)$7 defined by a linear recurrence, and nonnegativity forces three summands to vanish simultaneously:

$C(x)$8

These two "anchor" equations convert a Boolean continuation problem into two exact constraints on one analytic sequence. A short argument also gives the uniform bound $C(x)$9. In canonical block coordinates $\{0,1\}$0, the key parameter is $\{0,1\}$1, together with the packet variance $\{0,1\}$2 and image spacing $\{0,1\}$3.

The negative-binomial packet and exact log-concavity

The generating function of the forced sequence,

$\{0,1\}$4

admits an exact decomposition into truncated sums $\{0,1\}$5 over packet images indexed by $\{0,1\}$6. Subtracting the full sum $\{0,1\}$7 isolates remainders $\{0,1\}$8 with a one-sign integral representation, so their absolute values $\{0,1\}$9 satisfy the adjunction $0$0 where $0$1 is an unnormalized negative-binomial mass with mode near $0$2. A binomial-expectation identity for the correction factor $0$3 yields the paper's principal rigidity tool:

Exact strict log-concavity. For every $0$4, $0$5, and interior $0$6, the ratios $0$7 strictly decrease with $0$8.

This holds for all parameters, not asymptotically, and drives the entire isolated-regime argument: pinning the anchor-adjacent ratio near $0$9 via the two anchor equations and the exact seam identity forces quantitative loss at every neighboring sample.

Regime decomposition of the proof

The proof splits according to $1$0 and $1$1, and each regime is closed differently.

Compact range ($1$2). Here the stationary term matters. A limiting block profile $1$3 approximates the actual coefficients to within $1$4, boundary coordinates are excluded exactly, and a Pascal-type relation reduces the two anchors to the algebraic system $1$5 plus one additional equation. A finite certificate — $1$6 candidate pairs filtered by exact integer arithmetic, directed interval evaluation giving residual separation above $1$7, and an independent modular gcd check modulo $1$8 — excludes this range entirely. This is the only finite search in the paper.

Isolated packets ($1$9, and later $\mathbb Z$0). Dominant-image reindexing locates the packet image nearest its mode; remote images are suppressed because they begin more than six standard deviations away ($\mathbb Z$1 in the finite strip). A coarse bootstrap using nine explicit envelope factors proves $\mathbb Z$2 without any search over packet indices. The seam identity then pins the dominant ratio $\mathbb Z$3 to within $\mathbb Z$4 of $\mathbb Z$5, and strict log-concavity converts this into curvature losses exceeding $\mathbb Z$6. Every non-anchor coefficient in the last pre-zero cycle is thereby confined to $\mathbb Z$7, with the adjacent sample satisfying $\mathbb Z$8.

Spectral bootstrap ($\mathbb Z$9). Before any theta approximation is used — deliberately, to avoid circularity — a sparse spectral projection onto a characteristic root of $$P_{k,a}(x)=1+ax^2+x^k,\qquad 0<a<1,\ k\ \text{odd},$$0 near $$P_{k,a}(x)=1+ax^2+x^k,\qquad 0<a<1,\ k\ \text{odd},$$1 proves the striking estimate

$$P_{k,a}(x)=1+ax^2+x^k,\qquad 0<a<1,\ k\ \text{odd},$$2

This exploits sparsity of $$P_{k,a}(x)=1+ax^2+x^k,\qquad 0<a<1,\ k\ \text{odd},$$3: the Lagrange projector has $$P_{k,a}(x)=1+ax^2+x^k,\qquad 0<a<1,\ k\ \text{odd},$$4 norm below $$P_{k,a}(x)=1+ax^2+x^k,\qquad 0<a<1,\ k\ \text{odd},$$5 uniformly, and the residue coefficient is bounded below by $$P_{k,a}(x)=1+ax^2+x^k,\qquad 0<a<1,\ k\ \text{odd},$$6. Combined with modal-amplitude bounds when $$P_{k,a}(x)=1+ax^2+x^k,\qquad 0<a<1,\ k\ \text{odd},$$7, the exponential smallness holds throughout the large-parameter range. Its logical role is essential: after it, changing the packet index from $$P_{k,a}(x)=1+ax^2+x^k,\qquad 0<a<1,\ k\ \text{odd},$$8 to $$P_{k,a}(x)=1+ax^2+x^k,\qquad 0<a<1,\ k\ \text{odd},$$9 alters the normalized profile by less than $k\ge341$0, legitimizing "freezing" the slowly varying packet parameter.

Overlapping regime ($k\ge341$1): the theta kernel. With $k\ge341$2 exponentially small, the antiperiodic cyclic packet admits a parameter-uniform $k\ge341$3 local limit:

$k\ge341$4

Up to normalization this is the time-one Dirichlet heat kernel on $k\ge341$5; the Jacobi theta product formula gives the sharp uniform curvature bound $k\ge341$6 on the positive hump. Anchor saturation (which requires no log-concavity assumption) places both zeros high on the hump ($k\ge341$7), adjacent-anchor matching makes their log-heights agree to $k\ge341$8, and a discrete extraction lemma converts strong concavity into a lattice loss $k\ge341$9 dominating the exponentially small error budget. Again every non-anchor coefficient lands in $Q$0 and $Q$1.

First-harmonic regime ($Q$2). In the heat-kernel picture this is ground-state dominance: an exact multiplier analysis shows the third harmonic is relatively smaller than $Q$3 and higher harmonics contribute below $Q$4 in relative $Q$5. The frozen profile becomes a cosine with curvature $Q$6 up to negligible perturbation, and the same corollary applies with enormous margin.

All four regimes converge on the common pre-zero interior profile: $Q$7, $Q$8, $Q$9 for $1$00, and $1$01.

Post-zero determinism and the terminal contradiction

A notable structural point is that once the static pre-zero profile holds, no branching remains: a short positivity induction forces all product digits through degree $1$02 to equal $1$03 ("no-explosion"). Under this continuation, two exact identities give

$1$04

and since $1$05 implies $1$06, the coefficient $1$07 is negative — contradicting nonnegativity of $1$08. Thus the hard part is genuinely the global pre-zero shape, not the post-zero Boolean tree.

Computer assistance and verification

The numerical content is deliberately minimal and structured: one finite algebraic certificate for the compact range (interval arithmetic plus an independent modular check), and outward-rounded scalar bounds elsewhere — seven subinterval bounds for the modal-product inequality, moment and tail constants for the $1$09 transfer, cross-image envelope majorants. Accompanying verifier archives record software versions, hashes, and margins. No step enumerates continuation trees whose depth grows with $1$10; the cutoff $1$11 reflects nonoptimized constants in regime passage, not intrinsic significance.

Limitations and open questions

Several boundaries of the result deserve emphasis. First, the theorem covers only $1$12; the cases $1$13 remain open, and the small-case machinery does not obviously extend to them. Second, the restriction to the exponent $1$14 is partly technical: the author remarks that the argument appears to use separation of delays rather than the special value $1$15, and conjectures analogous results for $1$16 with $1$17 fixed and $1$18 sufficiently large, but notes that interactions among $1$19 residue classes pose real difficulties that are not pursued. Third, the compact-range exclusion depends on computer-assisted certificates whose reproducibility rests on the archived verifiers. Finally, the author explicitly cautions against reading the result as evidence that the full conjecture is computationally accessible: for general factors with several interacting delay directions, neither exact log-concavity nor a scalar theta kernel should be expected automatically. Whether any analogue of the one-dimensional packet structure exists beyond three-term factors is left open.

Conclusion

The paper proves that the trinomial $1$20 admits no unfair factorization for all odd $1$21 and all $1$22, replacing per-exponent combinatorial computations with a uniform analytic argument built on an exact negative-binomial decomposition, strict log-concavity, a sparse spectral bootstrap driving $1$23 exponentially small, and a theta-kernel local limit identifiable with a midpoint Dirichlet heat kernel. The reduction of the entire difficulty to a static pre-zero profile — after which a deterministic continuation yields an exact terminal contradiction — suggests that for this family the crowded spectrum is best understood as a single physical-space object rather than a collection of nearly dominant modes.