Odlyzko's Sign-Alternation Theorem — Complete Proof

Odlyzko’s Sign-Alternation Theorem — Complete Proof presents a self-contained reconstruction of the Stanton–Zeilberger–O’Hara route to the sign alternation theorem attributed to Odlyzko. The manuscript is written as a complete proof package: it derives the KOH identity from the corrected O’Hara–Goodman product-decomposition corridor, turns that corridor into a q-enumeration recurrence, expands the recurrence into the Stanton–Zeilberger multiplicity form, proves the finite sign theorem by a sign-cone argument, and finally passes to the coefficientwise limit giving Odlyzko’s reciprocal q-factorial product.

The theorem proved is the following sign-alternation statement. For every integer k ≥ 0, the Maclaurin coefficients of

1 / ((1 + q)(1 + q + q²)⋯(1 + q + ⋯ + qᵏ⁻¹))

alternate in sign. Equivalently, with

(q)ₖ = ∏₁≤j≤k (1 − qʲ),

the target is

(1 − q)ᵏ / (q)ₖ ∈ 𝒜,

where 𝒜 is the cone of formal power series

𝒜 = { f(q)=∑ₙ≥0 aₙqⁿ : (−1)ⁿaₙ ≥ 0 for every n }.

The manuscript treats the result as a known theorem in the Stanton–Zeilberger–O’Hara lineage, and its contribution is a complete reconstruction of the proof corridor in one coherent manuscript. The proof does not merely quote the KOH identity as an external black box: it reconstructs the corrected product-decomposition mechanism that produces KOH, then proves the sign theorem and limiting passage explicitly.

The proof spine is:

corrected O’Hara–Goodman corridor
⇒ q-enumeration recurrence
⇒ KOH identity
⇒ finite sign theorem
⇒ coefficientwise limit
⇒ Odlyzko sign alternation.

The first structural layer is the alternating sign cone. The cone 𝒜 is shown to be closed under finite sums, Cauchy products, even shifts q²ʳ, multiplication by powers (1 − q)ˢ for s ≥ 0, and coefficientwise limits. These closure properties are the formal algebraic engine used in the final sign argument.

The second layer introduces Gaussian polynomials

G(n,k) = [n+k choose k]q
= ∏₁≤i≤k (1 − qⁿ⁺ⁱ) / ∏₁≤i≤k (1 − qⁱ),

with the convention G(n,k)=0 for n<0 and k≥0, and G(n,0)=1 for n≥0. Combinatorially, G(b,a) is the rank-generating function of partitions fitting inside an a × b rectangle.

The third and most delicate layer is the corrected O’Hara–Goodman product decomposition. For rectangular partition sets

U(b,a) = { p=(p₁,…,pₐ) : 0 ≤ p₁ ≤ ⋯ ≤ pₐ ≤ b },

with boundary conventions pⱼ=0 for j≤0 and pⱼ=b for j>a, the corrected spread is

spread(p) = max₁≤j≤a+2 (pⱼ − pⱼ₋₂).

The active set is

M(p) = { j : pⱼ − pⱼ₋₂ = spread(p) },

and the degree is computed by decomposing M(p) into maximal consecutive intervals D and setting

deg(p) = ∑D floor((|D|+1)/2).

The corrected endpoint convention is essential. Endpoints 1 and a+2 may attach to active intervals, but the deletion operation must use the internal active index

h = max(M(p) ∩ {2,…,a+1}).

This internal-active deletion avoids the endpoint ambiguity created by the corrected spread. The manuscript proves that the corrected endpoint convention is regression-safe, that active intervals transform correctly under deletion, and that the auxiliary boundary states U(0,−1) and U(0,−2) are exactly the terminal singleton states required by the corrected decomposition.

The central product-decomposition theorem is the rank-shifting bijection

U(b,a,m,d) ≅ U(b−md, a−2d, ≤m−1) × U(am+2m−2b, d),

with rank shift

|p| = |q| + |r| + 2bd − md(d+1).

Here U(b,a,m,d) denotes the set of partitions in U(b,a) with corrected spread m and degree d. The proof gives the deletion map, the insertion inverse, the monotonicity of the deleted charge tuple

0 ≤ r₁ ≤ r₂ ≤ ⋯ ≤ r_d ≤ am + 2m − 2b,

the active-interval transport table for the degree drop, the tail-maximality lemma ensuring that insertion creates the maximal internal active index, and the full rank-shift calculation.

Taking ranks in this product decomposition gives the q-enumeration recurrence. If

v(b,a;m) = ∑_{p∈U(b,a,≤m)} q^|p|,

then for a,b ≥ 0 and m ≥ 1,

v(b,a;m) = v(b,a;m−1)

• ∑_{d≥1} q²ᵇᵈ⁻ᵐᵈ⁽ᵈ⁺¹⁾ G(am+2m−2b,d) v(b−md,a−2d;m−1).

The d=0 branch is treated separately as the lower-layer term v(b,a;m−1), avoiding any ambiguity in the Gaussian convention.

Iterating this recurrence from

G(n,k) = G(k,n) = v(k,n;k)

gives the KOH identity in Stanton–Zeilberger multiplicity form. If

λ = (1^{d₁} 2^{d₂} ⋯ k^{dₖ}) ⊢ k,

and

Dᵣ = ∑_{s≥r} d_s,

then

2n(λ) = ∑₁≤r≤k (Dᵣ² − Dᵣ),

and the recurrence expands to

G(n,k) = ∑_{λ⊢k} q²ⁿ⁽λ⁾ ∏₁≤s≤k G(A_s,d_s),

where

A_s = sn − 2(k−s) + 2∑_{t>s}(t−s)d_t.

This is the precise KOH identity needed for the sign-alternation proof.

The finite sign theorem is then proved from KOH. If Nk is even, then

(1 − q)^m G(N,k) ∈ 𝒜,

where

m = min(⌈k/2⌉, ⌊(N+1)/2⌋).

For N=2r, each KOH summand is analyzed through the alpha deficit

α(λ) = m − ∑ₛ⌈d_s/2⌉.

For r ≥ ⌈k/2⌉, a packet argument gives α(λ) ≥ 0 for every λ⊢k. For r < ⌈k/2⌉, nonzero KOH summands satisfy a small-r bound on the number of parts. The only apparent exceptional case occurs when λ has exactly r+1 distinct parts; then one KOH factor is exactly G(0,1)=1, and this factor refunds one missing power of (1−q). Thus every nonzero summand of (1−q)^mG(2r,k) lies in the sign cone 𝒜. Symmetry G(N,k)=G(k,N) covers the remaining even-product case Nk even.

The final passage to Odlyzko’s reciprocal product is coefficientwise. For fixed k,

G(N,k) → 1/(q)ₖ

coefficientwise as N→∞, because for any fixed degree d, the numerator

∏₁≤j≤k (1 − qᴺ⁺ʲ)

has no nonconstant terms of degree ≤ d once N>d. Taking N through even integers and using cone closure under coefficientwise limits gives

(1 − q)^⌈k/2⌉ / (q)ₖ ∈ 𝒜.

Multiplication by the remaining cone element

(1 − q)^{k−⌈k/2⌉}

yields

(1 − q)ᵏ / (q)ₖ ∈ 𝒜,

which is exactly Odlyzko’s sign-alternation statement.

The appendix records the sharpness of the exponent threshold. For k≥1 and e≥0,

(1 − q)^e / (q)ₖ ∈ 𝒜 ⇔ e ≥ ⌈k/2⌉.

Sufficiency follows from the finite sign theorem and the coefficientwise limit. Necessity follows from the dominant pole at q=1: if e<⌈k/2⌉, then q=1 is the unique dominant pole of order k−e, while every other root of unity contributes pole order at most ⌊k/2⌋. Therefore the coefficients are eventually positive, including along odd degrees, contradicting membership in 𝒜.

This manuscript is intended as a polished proof reconstruction and publication-candidate proof package. Its mathematical core is the corrected O’Hara–Goodman corridor, internal-active product decomposition, q-enumeration recurrence, explicit KOH iteration, finite sign-cone theorem, and coefficientwise limiting exit.