# Extensions to all orders n >= 7 for rank at most three

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 by n rank-at-most-three correlation matrix C, set P=per C, Q=per(C circle conjugate(C)), t=lambda_max(C), and D_n=(n+2)!/2. Embedding in C^3 gives n/3<=t<=n. The derivation of the two lower bounds in PROOF.md 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, with denominator 10000, are b_8 in [37643,37644]/10000 and b_9 in [41471,41472]/10000. Integer-power checks and the neighboring-slope inequalities establish the required concavity.

For rational 0<=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 >= 10, without an interval grid

A coarser bound suffices for the infinite tail. Since C^2 is at most t C in positive semidefinite order, s_i=(C^2)_ii<=t. The interpolated row bound and increasingness of b give

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

The arithmetic–geometric mean inequality applied to 1,...,k gives b_k<=(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 <= 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

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

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:

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

In particular the single coefficient 24/25 works uniformly for all n>=7. 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<=6, 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<=6 analytic proof is an external dependency and has not been independently audited in this package. No conclusion for all ranks is drawn.