From 98af17f50aa7fdb20656f7e73fe0a95aeee652e8 Mon Sep 17 00:00:00 2001
From: Eric Coissac <eric@coissac.eu>
Date: Sat, 12 Sep 2026 11:12:28 +0200
Subject: [PATCH] Update mixing function to avoid fixed point 0

This modification ensures that the mixing function no longer allows the canonical value 0 to always win window comparisons.
---
 theory-minimizer_selection.md | 10 +---------
 1 file changed, 1 insertion(+), 9 deletions(-)

diff --git a/theory-minimizer_selection.md b/theory-minimizer_selection.md
index f30afc2..469e412 100644
--- a/theory-minimizer_selection.md
+++ b/theory-minimizer_selection.md
@@ -20,15 +20,7 @@ The hash function is a 64-bit mixing function (splitmix64-style finalizer) appli
 
 $$H(x) = \text{mix64}(x \oplus s), \quad s = \lfloor 2^{64}/\varphi \rfloor = \texttt{0x9e3779b97f4a7c15}$$
 
-```
-H(x):
-    x ← x  ⊕  0x9e3779b97f4a7c15
-    x ← x  ⊕  (x >> 30)
-    x ← x  ×  0xbf58476d1ce4e5b9
-    x ← x  ⊕  (x >> 27)
-    x ← x  ×  0x94d049bb133111eb
-    return x ⊕ (x >> 31)
-```
+$$\begin{aligned} x &\gets x \oplus \texttt{0x9e3779b97f4a7c15} \\ x &\gets x \oplus (x \gg 30) \\ x &\gets x \times \texttt{0xbf58476d1ce4e5b9} \\ x &\gets x \oplus (x \gg 27) \\ x &\gets x \times \texttt{0x94d049bb133111eb} \\ x &\gets x \oplus (x \gg 31) \end{aligned}$$
 
 The XOR seed avoids the finalizer's fixed point at 0 ($\text{mix64}(0) = 0$), which would otherwise make an all-A m-mer (canonical value 0) win every window comparison.