---
id: C-0065
type: claim
title: "Degree seven is the sharp all-rank Karlin polynomial threshold"
status: proved-unconditional
family: adjacent-unconditional
interestingness: 3
statement: >
  Let f be a positive-leading real polynomial of degree at most six
  whose zeros all lie in (-infinity,0]. For every integer m>=1, define
  K_m(f;x)=(-1)^(m(m-1)/2) det[f^(i+j)(x)]_(i,j=0)^(m-1). Then
  K_m(f;x)>=0 for every m>=1 and every x>=0. Degree seven is minimal
  for failure: a counterexample exists at rank six, even among
  positive-leading polynomials with simple strictly negative zeros.
verifier: canon/witnesses/C-0065/verify.py
depends: ["C-0043", "C-0056", "C-0059"]
supersedes: []
sources:
  - literature/2026-07-19-karlin-all-order-sharp-degree-search.md
  - scratch/adjacent-unconditional--karlin-all-order-sharp-degree-six/RESULT.md
tags: [karlin, wronskian, hankel-determinant, all-rank, sharp-degree, real-rooted]
caveats: >
  The positive-leading restriction is essential because odd-rank
  determinants scale by an odd power. The theorem concerns the complete
  family of signed centered consecutive-derivative Wronskian/Hankel
  determinants, not arbitrary Toeplitz minors, full PF_m, or
  nonconsecutive derivative Wronskians. Sharpness means that some rank
  fails in degree seven, not that every rank does. The result has no
  implication for RH.
---

## Statement

For every integer \(m\geq1\), define
\[
K_m(f;x)
=(-1)^{m(m-1)/2}
\det[f^{(i+j)}(x)]_{i,j=0}^{m-1}.
\tag{1}
\]
Let \(f\) be a positive-leading real polynomial such that
\[
\deg f\leq6,
\qquad
Z(f)\subseteq(-\infty,0].
\]
Then
\[
\boxed{
K_m(f;x)\geq0
\quad\text{for every }m\geq1\text{ and }x\geq0.
}
\tag{2}
\]

The degree bound is sharp. There are positive-leading degree-seven
polynomials with simple strictly negative zeros for which
\(K_6(f;0)<0\).

## Ranks one through three

Write
\[
f(t)=a\prod_{\nu=1}^d(t+r_\nu),
\qquad
a>0,\quad r_\nu\geq0,\quad d\leq6.
\]
For \(x>0\), put
\[
\alpha_\nu=(x+r_\nu)^{-1},
\qquad
p_k=\sum_\nu\alpha_\nu^k.
\]
Then \(f(x)>0\) and
\[
\frac{f(x+z)}{f(x)}
=\prod_\nu(1+\alpha_\nu z).
\tag{3}
\]

Direct determinant expansion gives
\[
\begin{aligned}
K_1(f;x)&=f(x),\\
\frac{K_2(f;x)}{f(x)^2}&=p_2,\\
\frac{K_3(f;x)}{f(x)^3}
&=2(p_2^3-3p_2p_4+2p_3^2)\\
&=8\sum_{\mu<\nu}\alpha_\mu^3\alpha_\nu^3
+12\sum_{\lambda<\mu<\nu}
\alpha_\lambda^2\alpha_\mu^2\alpha_\nu^2.
\end{aligned}
\tag{4}
\]
Every expression in (4) is nonnegative. Continuity gives \(x=0\).

## Ranks four through six

Three exact canonical theorems apply on the present domain:

- C-0043 proves \(K_4=H_4\geq0\) through degree twelve.
- C-0056 proves \(K_5=H_5\geq0\) for positive-leading polynomials
  through degree seven.
- C-0059 proves \(K_6=-H_6\geq0\) through degree six.

Their degree bounds contain the present degree-at-most-six domain, and
their sign conventions agree with (1).

## Rank seven and higher

Suppose first that \(\deg f=6\), and write its leading coefficient as
\(a>0\). In the \(7\times7\) Hankel matrix
\[
H=[f^{(i+j)}(x)]_{i,j=0}^6,
\]
an entry vanishes when \(i+j>6\). Any contributing determinant
permutation \(\sigma\) must satisfy \(i+\sigma(i)\leq6\) for every
\(i\). But
\[
\sum_{i=0}^6(i+\sigma(i))=42=7\cdot6,
\]
so every inequality is an equality and
\(\sigma(i)=6-i\). Therefore
\[
\det H=(-1)^{21}(6!a)^7=-(720a)^7
\]
and
\[
\boxed{K_7(f;x)=(720a)^7>0.}
\tag{5}
\]

If \(\deg f\leq5\), the last column of \(H\) is zero and
\(K_7(f;x)=0\). For every \(m\geq8\), the last column of
\([f^{(i+j)}(x)]_{i,j=0}^{m-1}\) starts at derivative order
\(m-1\geq7>\deg f\). Hence
\[
\boxed{K_m(f;x)=0\quad(m\geq8).}
\tag{6}
\]
Equations (4)--(6), together with C-0043, C-0056, and C-0059, prove
(2).

## Sharpness

C-0059 supplies a degree-seven rank-six counterexample. The boundary
form is
\[
f(x)=x^4(x+1)^3,
\qquad
K_6(f;0)=-1246946918400.
\tag{7}
\]

For a simple strictly-negative-root witness, set \(q=10^7\) and use
the seven positive reciprocal roots
\[
\left(
\frac{53q-3}{q},\frac{53q-1}{q},
\frac{53q+1}{q},\frac{53q+3}{q},
\frac{q-2}{q},1,\frac{q+2}{q}
\right).
\tag{8}
\]
They are pairwise distinct. The pinned verifier evaluates their signed
rank-six determinant as an exact negative rational number and pins the
canonical `numerator/denominator` SHA-256
`3cee62a7790b6341ed2b586e6cf3dd0add9e3e9499bca2116bc1846f2569c5ca`.
The corresponding degree-seven product polynomial has simple strictly
negative zeros. Thus degree seven is the least degree at which any
determinant in (1) can fail.

Route-admission contract:

- `exact_map: PASS` -- the rank partition \(1\)--\(3\), \(4\)--\(6\),
  \(7\), and \(m\geq8\) is exhaustive and uses the common sign
  convention (1).
- `counterfeit: PASS` -- (7)--(8) are degree-seven counterfeits to
  every degree-uniform all-rank extension.
- `counterfeit_failure: PASS` -- failure occurs at rank six immediately
  beyond the proved degree range.
- `success_delta: PASS` -- the complete all-rank degree-at-most-six
  theorem and sharp degree-seven minimum are supplied.
- `failure_delta: PASS` -- any degree-at-most-six counterexample,
  dependency mismatch, or prior all-rank theorem would have stopped
  promotion.
- `stop_condition: PASS` -- every dependency verifier, the synthesis
  verifier, a primary-source audit, and an independent hostile-reader
  reconstruction all passed.
- `backward_heat_openness: N/A` -- this finite-polynomial theorem is
  not an RH criterion.

## Witness

The pinned verifier independently checks:

1. the exact claim and verifier hashes for C-0043, C-0056, and C-0059;
2. successful reruns of all three dependency verifiers;
3. the complete rank-one through rank-three identities (4);
4. the positive monomial expansion of the rank-three expression;
5. the symbolic rank-seven anti-triangular identity (5);
6. zero determinants at ranks eight through ten, with the proof's
   zero-column argument covering every higher rank;
7. the exact boundary value (7); and
8. the simple-root rational witness (8), including its exact-value
   digest; and
9. the primary-source sign convention and the special-class restriction
   on the published Craven--Csordas all-rank theorem.

## Prior-art boundary

The classical Laguerre inequality and Craven--Csordas (1989) provide
general ranks two and three. Their Open Problem 3.4 leaves higher ranks
conjectural. Craven--Csordas (2002) prove a restricted order-four
identity, and their 2005 Theorem 4.5 gives all-rank positivity only for
the special class \(C_0\). Dimitrov's all-rank positive theorem applies
to a proper factorially transformed subclass, while his degree-eight
counterexample is rank eight.

No checked source states (2), gives a degree-seven counterexample, or
identifies degree seven as the sharp all-rank threshold. The claim is
limited to those exact contributions.

## What survives

The theorem concerns only the centered consecutive-derivative family
(1). It does not imply positivity of arbitrary Toeplitz minors, full
\(PF_m\), or arbitrary nonconsecutive Wronskians.

The proof uses only finite negative-real-root geometry. It supplies no
new zero-free region and no proof or disproof of RH.

## Provenance

The exact all-rank degree-at-most-six theorem and sharp degree-seven
minimum were assembled on 2026-07-19 from C-0043, C-0056, and C-0059
plus the explicit low- and high-rank reductions above. A post-promotion
hostile audit replaced a verifier broken by stale dependency hashes
with a dependency-free synthesis checker. It reconstructs the low
ranks in independent root variables, exhausts the rank-seven
permutations, proves the generic higher-rank zero-column condition,
reruns all strengthened dependency verifiers, reproduces both
degree-seven witnesses, and pins the primary-source special-class
boundary. A separate forward-citation audit found no prior all-rank
degree-six/seven threshold.
