---
title: "QX-NEPHRIEL / Seventh-String Closure"
subtitle: "Quantitative rank-three Chollet inequalities"
author: "AI-generated derivation by Eve; prepared for Maciej Nowicki"
date: "Release v1.0.0 | 3 October 2026"
lang: en-GB
---

\clearpage

# Main theorem and proof

**Research status.** This is an AI-generated mathematical derivation with an executed exact-arithmetic certificate. It has not received independent mathematical peer review or proof-assistant formalization. The argument proves the stated restricted theorem, not the unrestricted Chollet conjecture. Priority of the restricted result has not been established. All hypotheses, reductions, and finite checks used for the main theorem are included below.

## 1. The target and the claim

For an \(n\times n\) complex matrix \(A\), its permanent is

\[
\operatorname{per} A=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.
\]

Chollet's 1982 conjecture asks whether

\[
\operatorname{per}(A\circ B)\le\operatorname{per}A\,\operatorname{per}B
\]

for all complex Hermitian positive semidefinite matrices A and B of the same order. The product symbol means the entrywise, or Hadamard, product, not matrix multiplication. Conjugation below is entrywise.

An author-maintained progress report consulted on 3 October 2026 explicitly lists complex rank at most three as covered in every order except seven. A recent preprint supplies general inequalities for estimating the same permanent ratio using spherical integrals and eigenvalues. The result below combines these ingredients with a new-to-this-derivation scalar-envelope closure at order seven. These observations support targeting the case; they do not constitute a comprehensive priority audit. See the source and attribution section at the end of this manuscript.

### Theorem 1

For every pair of complex Hermitian positive semidefinite \(7\times7\) matrices \(A\) and \(B\), each of rank at most three,

\[
\boxed{\operatorname{per}(A\circ B)\le\frac{24}{25}\operatorname{per}A\operatorname{per}B.}
\]

In particular, the ordinary Chollet inequality holds on this entire class. The coefficient 24/25 is a certified convenient coefficient; it is not claimed optimal.

The stronger self-conjugate statement is

\[
\boxed{\operatorname{per}(C\circ\overline C)\le\frac{24}{25}\operatorname{per}(C)^2}
\tag{1}
\]

for every order-seven positive semidefinite C of rank at most three.

## 2. Reduction to correlation matrices and self-conjugate pairs

Positive semidefiniteness implies \(|a_{ij}|^2\le a_{ii}a_{jj}\). Consequently a zero diagonal entry forces its entire row and column to vanish. The permanent and the permanent of any Hadamard product containing that matrix both vanish. Thus such cases are immediate.

For positive diagonal entries, write \(A=DCD\), where \(D=\operatorname{diag}(\sqrt{a_{ii}})\), and \(C\) has diagonal one. Then \(C\) is a correlation matrix, has the same rank, and with \(h=\prod_i a_{ii}\),

\[
\operatorname{per}A=h\operatorname{per}C,\qquad
\operatorname{per}(A\circ\overline A)=h^2\operatorname{per}(C\circ\overline C).
\]

Therefore the self-conjugate ratio is unchanged by diagonal normalization.

For arbitrary A,B, Cauchy-Schwarz applied to their permutation monomials gives

\[
|\operatorname{per}(A\circ B)|^2
\le \operatorname{per}(A\circ\overline A)
     \operatorname{per}(B\circ\overline B).
\tag{2}
\]

Permanents of positive semidefinite matrices are nonnegative, as also follows from the Gaussian identity proved in Section 4. The Hadamard product is positive semidefinite because, for Gram representations \(a_{ij}=\langle v_i,v_j\rangle\) and \(b_{ij}=\langle w_i,w_j\rangle\), it is the Gram matrix of \(v_i\otimes w_i\). Applying (1) to both factors in (2) proves Theorem 1. Notice that both factors are assumed rank at most three; the Hadamard product need not have rank at most three.

It suffices from here onward to consider an order-seven correlation matrix C of rank at most three. Set

\[
P=\operatorname{per}C,\qquad Q=\operatorname{per}(C\circ\overline C).
\]

## 3. A combinatorial upper bound for Q

Define \(b_k=(k!)^{1/k}\) for integers \(k\ge1\), and let b be linear between successive integer arguments.

### Lemma 2: the interpolated row bound

If N is an \(n\times n\) matrix with entries in \([0,1]\) and row sums \(s_i\ge1\), then

\[
\operatorname{per}N\le\prod_i b(s_i).
\tag{3}
\]

**Proof.** First suppose N is binary, with \(d_i\) ones in row \(i\). Bregman's theorem gives \(\operatorname{per}N\le\prod_i(d_i!)^{1/d_i}\), with a zero row treated separately. Here is the standard entropy argument for that binary result. If N has no perfect matching, the conclusion is immediate. Otherwise choose a perfect matching uniformly, and expose its chosen columns in a uniformly random order of rows, independent of the matching. The chain rule bounds the matching's entropy by the sum of the logarithms of the numbers of still-available neighbors. For a fixed full matching and row i, the rows matched to its \(d_i\) neighboring columns have uniformly random relative order. The number remaining when i is exposed is therefore uniform on \(\{1,\ldots,d_i\}\). The expected logarithm is \(\log(d_i!)/d_i\). Taking expectations and exponentiating proves the binary bound. This is the established entropy proof of Bregman's theorem, not a new theorem claimed here.

Now fix each row sum of a matrix in \([0,1]^{n\times n}\). The row polytope is compact; the permanent is affine in each row separately. Starting with a maximizer, replace its rows one by one by vertices of their row polytopes without decreasing the permanent. A row vertex has at most one fractional entry: two fractional entries would admit a nontrivial opposite perturbation preserving the row sum. A row of sum \(k+t\), \(0\le t<1\), is therefore a convex combination of binary rows of sums \(k\) and \(k+1\). Multilinearly expanding these combinations and applying the binary bound gives \(\prod_i[(1-t_i)b_{k_i}+t_i b_{k_i+1}]\), which is (3). QED.

The exact root brackets in Section 7 certify that b is increasing and concave on \([1,7]\). For \(N=C\circ\overline C\), its row sums are

\[
s_i=\sum_j|c_{ij}|^2=(C^2)_{ii}\ge1.
\]

Each \(s_i\le7\) because all entries of C have modulus at most one. Consequently (3), the arithmetic-geometric mean inequality, and concavity imply

\[
Q\le\left(\frac{1}{7}\sum_i b(s_i)\right)^7
\le b\left(\frac{\operatorname{tr}C^2}{7}\right)^7.
\tag{4}
\]

Pad the nonzero eigenvalues to three, writing them as \(t\ge\lambda_2\ge\lambda_3\ge0\). Since \(\operatorname{tr}C=7\),

\[
7/3\le t\le7,\qquad \lambda_2+\lambda_3=7-t.
\]

For fixed t, the sum \(\lambda_2^2+\lambda_3^2\) is maximized by making these two numbers as unequal as their bounds permit. Thus

\[
\frac{\operatorname{tr}C^2}{7}\le M(t),
\qquad
M(t)=\begin{cases}
\dfrac{2t^2+(7-2t)^2}{7},&7/3\le t\le7/2,\\[6pt]
\dfrac{t^2+(7-t)^2}{7},&7/2\le t\le7.
\end{cases}
\tag{5}
\]

For clarity, in the first range the extremal remaining pair is \((t,7-2t)\), and in the second it is \((7-t,0)\). Both are admissible in their stated ranges. Because b is increasing, (4) becomes

\[
\boxed{Q\le b(M(t))^7.}
\tag{6}
\]

This is the first certificate: a global upper bound requiring only one eigenvalue.

## 4. Two Gaussian-sphere lower bounds for P

The following spherical estimates are known ingredients; the spectral-tilt version appears in the recent work recorded in the source and attribution section. Their derivation is supplied to expose every normalization.

Write \(c_{ij}=\langle v_i,v_j\rangle\) with unit vectors \(v_i\) in \(\mathbb C^3\), padding zero coordinates when the rank is smaller. Let \(g\) be a standard complex Gaussian vector in \(\mathbb C^3\), so \(\mathbb E g_j\overline{g_k}=\delta_{jk}\). Expansion into complex Gaussian monomials yields

\[
P=\mathbb E\prod_{i=1}^7|\langle v_i,g\rangle|^2.
\tag{7}
\]

One can see (7) directly by pairing each of the seven unconjugated linear factors with a conjugated factor; the possible pairings are precisely the permutations in the permanent.

Write \(g=RU\), with U uniform on the complex unit sphere and independent of R. Then \(R^2\) has the \(\operatorname{Gamma}(3,1)\) distribution and

\[
D:=\mathbb E R^{14}=(3)_7=\frac{9!}{2}=181440.
\]

Set \(T_i=|\langle v_i,U\rangle|^2\) and \(X=\prod_i T_i\). Equation (7) becomes

\[
P=D\mathbb E X.
\tag{8}
\]

For every unit vector \(v_i\), \(T_i\) has density \(2(1-x)\), \(0<x<1\), hence \(\mathbb E\log T_i=-3/2\). Jensen's inequality gives

\[
\boxed{P\ge D e^{-21/2}.}
\tag{9}
\]

This is valid without independence of the seven \(T_i\).

For the second bound, let \(w\) be a unit top eigenvector of \(S=\sum_i v_iv_i^*\), whose nonzero eigenvalues equal those of C. Define

\[
T_w=|\langle w,U\rangle|^2,\qquad d\nu=3T_w\,d\mu.
\]

This is a probability measure since \(\mathbb E_\mu T_w=1/3\). Write \(\gamma_i=|\langle w,v_i\rangle|^2\). Then \(\sum_i\gamma_i=t\). Conditional rotational symmetry in the two-dimensional orthogonal complement gives

\[
\mathbb E(T_w\mid T_i=x)=\gamma_i x+(1-\gamma_i)\frac{1-x}{2}.
\]

The elementary integrals

\[
\mathbb E(T_i\log T_i)=-5/18,\qquad
\mathbb E((1-T_i)\log T_i)=-11/9
\]

therefore imply

\[
\mathbb E_\nu\log T_i=\gamma_i-11/6,
\qquad
\mathbb E_\nu\log T_w=-5/6.
\]

The logarithms are integrable; the density is bounded and the relevant zero sets have spherical measure zero. Changing measure in (8) and applying Jensen once more yields

\[
\begin{aligned}
P&=\frac D3\mathbb E_\nu(X/T_w)\\
&\ge\frac D3\exp\left(\sum_i\mathbb E_\nu\log T_i-\mathbb E_\nu\log T_w\right)\\
&=\frac D3e^{t-12}.
\end{aligned}
\tag{10}
\]

Consequently

\[
\boxed{P\ge D\max\{e^{-21/2},\tfrac13e^{t-12}\}.}
\tag{11}
\]

In particular, P is strictly positive for correlation matrices in the stated class. Equation (11) is the second certificate, a lower bound using the same eigenvalue as (6).

## 5. The scalar problem

Combining (6) and (11),

\[
\frac Q{P^2}\le R(t)
:=\frac{b(M(t))^7}{D^2\max\{e^{-21},\tfrac19e^{2t-24}\}},
\qquad7/3\le t\le7.
\tag{12}
\]

The crossing point of the two denominator branches is

\[
t_0=\frac32+\log3.
\]

Thus

\[
R(t)=\begin{cases}
e^{21}b(M(t))^7/D^2,&t\le t_0,\\
9e^{24-2t}b(M(t))^7/D^2,&t\ge t_0.
\end{cases}
\tag{13}
\]

The only interior knot of b(M(t)) between \(t_0\) and 7/2 is M(t)=3, occurring at

\[
t_1=\frac{7+\sqrt7}{3}.
\]

We now prove the exact global reduction

\[
\boxed{\sup_{7/3\le t\le7}R(t)
\le\max\{R(t_0),R(t_1),R(7/2)\}.}
\tag{14}
\]

This is a continuous-domain argument, not a numerical sample.

## 6. Why three points suffice

The finite root certificate below establishes the following elementary properties of b:

* b is increasing and concave on \([1,7]\);
* its slopes are less than 5/12 throughout \([1,7]\);
* its slopes are less than 2/5 on \([3,7]\);
* \(b(x)\ge(x+2)/3\) on \([3,7]\).

### 6.1 Before \(t_0\)

M(t) is increasing on \([7/3,7]\). On [7/3,\(t_0\)], the denominator branch in (13) is constant, so R increases. Its maximum there is R(\(t_0\)).

### 6.2 Between \(t_0\) and 7/2

Put u=t-7/3. In this interval,

\[
M(t)=7/3+6u^2/7,\qquad 0\le u\le7/6.
\]

Within any linear segment of b, write \(b(M)=aM+c\), with \(0<a<5/12\). Since \(\log R\) differs from \(7\log b(M)-2t\) only by a constant, its second derivative has the sign of

\[
b(M)-2a(M-7/3).
\]

That expression satisfies

\[
b(M)-2a(M-7/3)
\ge1-2\frac5{12}\frac76=\frac1{36}>0.
\tag{15}
\]

To check the derivative explicitly: if \(B(u)=d+(6a/7)u^2\), then \[B''B-(B')^2=\frac{12a}{7}\left[d-\frac{6a}{7}u^2\right],\] which is the stated sign condition. Therefore \(\log R\) is convex separately on [\(t_0\),\(t_1\)] and [\(t_1\),7/2]. A convex function is bounded above by the larger of its endpoint values on a closed interval. This leaves only \(t_0\),\(t_1\),7/2 as possible maxima in the middle range. No convexity across the knot itself is assumed.

### 6.3 After 7/2

Put v=t-7/2, so \(0\le v\le7/2\) and

\[
M(t)=7/2+2v^2/7.
\]

Away from the finitely many knots, the derivative is

\[
\frac d{dt}\log R=\frac{4a v}{b(M(t))}-2.
\]

Using \(a<2/5\) and \(b(M)\ge(M+2)/3\),

\[
\frac d{dt}\log R
\le \frac{24v}{5(11/2+2v^2/7)}-2
\le\frac{28}{15}-2=-\frac2{15}<0.
\tag{16}
\]

The second inequality follows because \(v/(11/2+2v^2/7)\) increases on \([0,7/2]\): its derivative has numerator \(11/2-2v^2/7\ge2>0\). Its largest value occurs at \(v=7/2\). R is continuous at all knots; hence it decreases on the entire tail. This completes the proof of (14).

## 7. The exact finite certificate

No floating-point estimate is needed in the proof. Define lower and upper rational brackets by dividing the following vectors by 10000:

\[
(\ell_1,\ldots,\ell_7)=
\frac1{10000}(10000,14142,18171,22133,26051,29937,33800),
\]

\[
(h_1,\ldots,h_7)=
\frac1{10000}(10000,14143,18172,22134,26052,29938,33801).
\]

Exact integer arithmetic verifies \(\ell_k^k\le k!\le h_k^k\). It also verifies:

\[
\ell_{k+1}>h_k\quad(1\le k\le6),
\]
\[
2\ell_k>h_{k-1}+h_{k+1}\quad(2\le k\le6),
\]
\[
h_{k+1}-\ell_k<5/12\quad(1\le k\le6),
\]
\[
h_{k+1}-\ell_k<2/5\quad(3\le k\le6),
\]
\[
\ell_k\ge(k+2)/3\quad(3\le k\le7).
\]

These prove all properties of b used in Sections 3 and 6; the last property extends by interpolation.

Let \(E=27183/10000\) and \(q=2599/1000\). A rational Taylor tail yields

\[
e\le\sum_{j=0}^9\frac1{j!}+\frac{1/10!}{1-1/11}<E<3.
\]

Also

\[
\sum_{j=0}^8\frac{(1099/1000)^j}{j!}>3
\]

implies \(\log3<1099/1000\), so \(t_0<q\). The inequality \(e<3\) gives \(\log3>1\), so \(t_0>5/2>7/3\). Finally \((13/5)^2<7\) gives \(t_1>16/5\); these observations verify the asserted ordering of the intervals.

Write

\[
m_q=M(q)=\frac{2q^2+(7-2q)^2}{7},\qquad
\beta=(3-m_q)h_2+(m_q-2)h_3.
\]

Here \(2<m_q<3\). Increasingness of M and b gives

\[
R(t_0)\le\frac{E^{21}\beta^7}{D^2}<\frac{24}{25}.
\tag{17}
\]

For the knot \(t_1\), since \(t_1>16/5\),

\[
e^{24-2t_1}<e^{88/5}=\frac{e^{18}}{e^{2/5}}
<\frac{25}{37}E^{18},
\]

where \(e^{2/5}>1+2/5+(2/5)^2/2=37/25\). Thus

\[
R(t_1)<\frac{9h_3^7 E^{18}}{D^2}\frac{25}{37}<\frac{24}{25}.
\tag{18}
\]

At the spectral midpoint,

\[
R(7/2)\le
\frac{9[(h_3+h_4)/2]^7E^{17}}{D^2}<\frac{24}{25}.
\tag{19}
\]

Every quantity in (17)-(19) is rational. Cross-multiplication proves the inequalities using integers. For readable reporting only, the three upper bounds are approximately 0.954617923041444, 0.793755502346409, and 0.891681828686875. These decimals are not premises.

The program `verify_q7.py` executes these checks and writes `exact_certificate.json` containing each full numerator, denominator, and exact positive margin. It reported 49 successful exact checks. Four of those checks concern the optional analytic-tail extension; the rest cover this theorem's finite obligations.

## 8. Completion

Equations (14) and (17)-(19) imply \(R(t)<24/25\) for every admissible top eigenvalue t. Equations (6) and (11) therefore give \(Q<(24/25)P^2\) for every correlation matrix in the theorem. Diagonal normalization extends the non-strict form to all positive semidefinite matrices of the stated order and rank, including zero-diagonal cases. Cauchy-Schwarz as in (2) then proves Theorem 1 for arbitrary pairs A,B. QED.

If both matrices have positive diagonal, their permanents are positive and the proof in fact gives strict inequality with coefficient 24/25. Equality with that coefficient can occur only through the trivial zero-permanent cases. The coefficient is deliberately not presented as extremal.

## 9. What has and has not been established

The argument above is a complete proof of a precisely stated rank-restricted theorem, with explicit finite verification. It is not a redefinition of the conjecture or an algorithm assuming a global feasibility oracle. The essential new step in this derivation is the order-seven scalar-envelope closure; the permanent estimates and the entropy argument have precedents.

The proof does not cover arbitrary ranks at order seven. It does not settle unrestricted Chollet, the permanent-on-top conjecture, or other stronger permanent proposals. It does not establish a new optimal constant. A search found an explicit report listing the targeted case as missing, but could have missed other sources. Publication-level novelty remains unconfirmed.

The appendices supply an independent derivation for all orders \(n\ge10\) and exact interval certificates for \(n=8,9\). Coupled with the cited preprint theorem for all matrices of orders at most six, this would give the complete rank-at-most-three case in every order. That last small-order input is attributed, not independently re-proved by this package.

## 10. Relation to the Qingxiao-inspired design

The operational structure that survives is the dual-certificate gate: the numerator is controlled from above and the denominator from below, using the same spectral coordinate. A switch between two lower bounds replaces a heuristic change of search mode. The final certificate replaces an optimization guess with a globally valid statement.

The actual domains used are combinatorial matching entropy, complex Gaussian moments, spherical change of measure, eigenvalue geometry, one-variable convexity, and exact rational arithmetic. Persistent homology, braid groups, and symplectic integration are not used in this proof and are not claimed as hidden ingredients. The inspiration helped organize a search; it is not itself mathematical evidence.


\clearpage

# Appendix: extensions to all orders \(n\ge7\)

This appendix records additional bounds, proved with the same two-certificate method. The new-to-this-derivation targeted gap is n=7; the other orders are included to make the high-order rank-three statement independent of the progress report's assertions.

For an \(n\times n\) rank-at-most-three correlation matrix \(C\), set \(P=\operatorname{per}C\), \(Q=\operatorname{per}(C\circ\overline C)\), \(t=\lambda_{\max}(C)\), and \(D_n=(n+2)!/2\). Embedding in \(\mathbb C^3\) gives \(n/3\le t\le n\). The derivation of the two lower bounds in the main proof is unchanged except for the number of factors:

\[
P\ge D_n\max\{e^{-3n/2},\tfrac13e^{t-1-11(n-1)/6}\}=:F_n(t).
\tag{A1}
\]

The expectation under the tilted spherical measure is \(\sum_i(\gamma_i-11/6)+5/6=t-1-11(n-1)/6\), which verifies the exponent in (A1).

## A. Orders eight and nine

For \(n=8,9\), the exact factorial-root brackets extended through nine certify that b is concave and increasing on \([1,n]\). Therefore

\[
Q\le b(M_n(t))^n,
\quad
M_n(t)=\begin{cases}
[2t^2+(n-2t)^2]/n,&n/3\le t\le n/2,\\
[t^2+(n-t)^2]/n,&n/2\le t\le n.
\end{cases}
\tag{A2}
\]

Both \(M_n\) and \(F_n\) are nondecreasing. A closed interval \([l,u]\) is thus certified by

\[
Q/P^2\le b(M_n(u))^n/F_n(l)^2.
\tag{A3}
\]

This covers every admissible spectrum with top eigenvalue in the interval. It is not a pointwise test.

The additional factorial-root brackets are

\[
b_8\in[37643,37644]/10000,\qquad
b_9\in[41471,41472]/10000.
\]

 Integer-power checks and the neighboring-slope inequalities establish the required concavity.

For rational \(0\le a<66\), the following exact rational expression bounds the exponential from above:

\[
e^a\le\sum_{j=0}^{64}\frac{a^j}{j!}+
\frac{a^{65}/65!}{1-a/66}.
\tag{A4}
\]

The geometric remainder is valid because each subsequent term ratio is at most \(a/66\). Reciprocating (A4) supplies rigorous lower bounds on \(e^{-a}\) in (A1).

The standard-library-only script `verify_q8_q9.py` constructs rational interval certificates, then checks all intervals, their adjacency, and both outer endpoints. Its executed results were:

* \(n=8\): 21 closed intervals cover \([8/3,8]\), with certified ratio strictly below 3/5. The upward-rounded largest interval bound is 18698089279/31250000000.
* \(n=9\): 12 closed intervals cover \([3,9]\), with certified ratio strictly below 2/5. The upward-rounded largest interval bound is 385362547599/1000000000000.

The intervals are in `q8_q9_certificate.json`. Re-executing the script recomputes every inequality with exact fractions; it does not trust recorded floating-point values or the certificate's stated conclusion.

## B. Every order \(n\ge10\), without an interval grid

A coarser bound suffices for the infinite tail. Since \(C^2\) is at most \(tC\) in positive semidefinite order, \(s_i=(C^2)_{ii}\le t\). The interpolated row bound and increasingness of b give

\[
Q\le b(t)^n.
\]

The arithmetic-geometric mean inequality applied to \(1,\ldots,k\) gives \(b_k\le(k+1)/2\). Linear interpolation preserves this bound, so

\[
Q\le((t+1)/2)^n.
\tag{B1}
\]

No concavity assertion for b at arbitrarily large orders is required here. Increasingness holds at every order: since \(b_k\le k<k+1\), the identity \(b_{k+1}^{k+1}=b_k^k(k+1)\) implies \(b_{k+1}>b_k\); linear interpolation preserves increasingness.

Use only the second term in (A1). Maximizing \((t+1)^n e^{-2t}\) over the larger domain \(t>-1\) gives its global maximum at \(t=n/2-1\). Consequently

\[
\frac Q{P^2}\le T_n:=
\frac{36e^{(8n+1)/3}(n/4)^n}{((n+2)!)^2}.
\tag{B2}
\]

The ratio of successive upper bounds is

\[
\begin{aligned}
\frac{T_{n+1}}{T_n}
&=e^{8/3}\frac{n+1}{4(n+3)^2}(1+1/n)^n\\
&<e^{11/3}\frac{n+1}{4(n+3)^2}\\
&<\frac{10(n+1)}{(n+3)^2}<1\quad(n\ge10).
\end{aligned}
\tag{B3}
\]

The elementary estimate \((1+1/n)^n<e\) follows by taking logarithms and using \(\log(1+x)<x\) for \(x>0\). The bound \(e^{11/3}<40\) follows from \(e<87/32\) and the exact integer inequality \((87/32)^{11}<40^3\). The last rational expression in (B3) is less than one at \(n=10\) and decreases thereafter.

Finally,

\[
T_{10}=\frac{36e^{27}(5/2)^{10}}{(12!)^2}
<\frac{36(87/32)^{27}(5/2)^{10}}{(12!)^2}<\frac45.
\tag{B4}
\]

The last comparison is exact rational arithmetic and is executed by `verify_q7.py`. Thus

\[
\boxed{Q\le\frac45P^2\qquad(n\ge10,\ \operatorname{rank}C\le3).}
\]

Diagonal normalization and permutation-monomial Cauchy-Schwarz transfer this to arbitrary pairs.

## C. Unified high-order statement

The main proof and the two appendices establish the following convenient, non-optimal constants for pairs A,B of positive semidefinite matrices of order n and individual ranks at most three:

\[
\begin{gathered}
\operatorname{per}(A\circ B)\le\rho_n\operatorname{per}A\operatorname{per}B,\\[4pt]
\rho_7=24/25,\quad\rho_8=3/5,\quad\rho_9=2/5,\\
\rho_n=4/5\quad(n\ge10).
\end{gathered}
\]

In particular the single coefficient 24/25 works uniformly for all \(n\ge7\). This is not an assertion that the true extremal ratio follows these coefficients or is nonmonotone; the displayed coefficients come from different convenient bounds.

For \(n\le6\), the cited Li-Zhao-Luo preprint states the unrestricted inequality. Accepting that prior theorem, the main proof and appendices yield Chollet for all orders when both matrices have rank at most three. The \(n\le6\) analytic proof is an external dependency and has not been independently audited in this package. No conclusion for all ranks is drawn.


\clearpage

\begingroup\footnotesize\raggedright

# Sources, attribution, and scope

Accessed 3 October 2026. Publication and posting dates are distinguished from access dates.

1. J. Chollet, "Is there a permanental analogue to Oppenheim's inequality?", American Mathematical Monthly 89 (1982), 57-58. DOI: 10.1080/00029890.1982.11995380. Original conjecture; historical metadata checked against the author-maintained survey below.

2. Yair Lavi, "Conjecture 30: Chollet's conjecture", author-maintained Minc's-list page. https://yair-lavi.com/mincs-list/conjecture-30/index.html . The retrieved page explicitly lists the complex rank-at-most-three result as covering every order except n=7. This is evidence of a documented gap, not proof that no other source has closed it. The page's claimed other results are not used to prove the order-seven theorem in this package.

3. Yicen Ma, "Quantitative Chollet Inequalities for Matrices of Rank at Most Four: Spectral bounds and an order-nine tight-frame construction", arXiv:2609.37461v1, posted 28 September 2026. https://arxiv.org/html/2609.37461v1 . In particular, the interpolated row bound and the complex-sphere / spectral-tilt estimates appear in its Sections 2 and 3. These are attributed ingredients, not discoveries claimed by this note. The present main proof specializes those bounds to rank three, eliminates the remaining two eigenvalues, and closes a scalar envelope by convexity and three rational endpoint bounds. The quoted paper's computer-assisted rank-four claims are not required by this main proof.

4. Jaikumar Radhakrishnan, "An Entropy Proof of Bregman's Theorem", Journal of Combinatorial Theory, Series A 77 (1997), 161-164. DOI: 10.1006/jcta.1996.2727. Author repository record: https://repository.ias.ac.in/57644/ . The binary-matrix entropy proof in the main proof is standard background and is attributed accordingly.

5. Qingfeng Li, Qirui Zhao, Wenhao Luo, "Chollet's Permanent Conjecture Through Order Six via Border Induction", Preprints.org 202608.2196, posted 31 August 2026. https://www.preprints.org/manuscript/202608.2196 . The retrieved page labels it a preprint that is not peer-reviewed. It supplies the small-order external theorem when forming the all-order rank-three corollary. The order-seven proof and the \(n\ge7\) appendix are independent of it.

## Novelty boundaries

The order-seven scalar certificate and the quantitative bounds were derived during this conversation. A targeted search did not locate the same stated 24/25 result. That search is not a complete priority audit. Several mathematical ingredients are explicitly existing results, and the result must not be marketed as the unrestricted Chollet conjecture being solved. No claim of independent peer review or Lean formalization is made.

## Computational trust boundaries

`verify_q7.py` checks the finite rational inequalities in the analytic order-seven proof and the \(n\ge10\) tail. `verify_q8_q9.py` supplies exact closed-interval coverage for two auxiliary orders. These programs do not mechanically prove the entropy bound, the spherical change of measure, or the analytic reductions. Those arguments are written out for mathematical review. Floating-point exploration suggested the closure but is not part of the proof certificate.

## Release source check (3 October 2026)

The current arXiv abstract/HTML and Preprints.org landing page were checked again when preparing this release. Their titles, authors, identifiers and posting dates agree with the records above. The Lavi progress page still explicitly reports the complex rank-three exception at order seven. Historical bibliographic metadata were corroborated by that bibliography and the author repository record; access to some publisher pages was restricted. This source check is a metadata and attribution check, not a mathematical audit of the cited preprints or a comprehensive priority search.


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