Add hash function for robust minimizer selection

Introduces a specific 64-bit mixing hash function for minimizer selection. This function is designed to remove bias favoring AT-rich composition and accounts for low-complexity m-mers. Exhaustive checks confirm that the argmin is never a homopolymer or periodic repeat.

theory-minimizer_selection.md

@@ -16,15 +16,25 @@ The canonical form used as input to $H$ is still the lexicographic minimum of fo
 
 ### Hash function
 
-The hash function is a 64-bit mixing function (splitmix64-style finalizer) applied to the m-mer XORed with a fixed non-zero seed:
+$s$ is a fixed non-zero seed:
 
-$$H(x) = \text{mix64}(x \oplus s), \quad s = \lfloor 2^{64}/\varphi \rfloor = \texttt{0x9e3779b97f4a7c15}$$
+$$s = \lfloor 2^{64}/\varphi \rfloor = \texttt{0x9e3779b97f4a7c15}$$
 
-$$\begin{aligned} H(x) &: \\ x &\gets& x \oplus \texttt{0x9e3779b97f4a7c15} \\ x &\gets& x \oplus (x \gg 30) \\ x &\gets& x \times \texttt{0xbf58476d1ce4e5b9} \\ x &\gets& x \oplus (x \gg 27) \\ x &\gets& x \times \texttt{0x94d049bb133111eb} \\ &\text{return}& x \end{aligned}$$
+The hash function is a 64-bit mixing function (splitmix64-style finalizer) applied to the m-mer XORed with $s$:
 
-Hashing serves two distinct purposes. First, since a kmer's canonical form is a lexicographic minimum, ranking kmers by the raw value of their canonical form introduces bias into minimizer selection: it favors AT-rich composition, which in turn skews the distribution of kmers across partitions. $H$ removes this bias by making the winner depend on avalanche-mixed bits rather than on nucleotide composition, restoring a uniform distribution.
+$$H(x) = \text{mix64}(x \oplus s)$$
 
-Second, independently of that general debiasing, the specific seed $s$ matters in its own right: low-complexity m-mers (homopolymers, short tandem repeats) are disproportionately abundant in real genomes. If any of them were the perpetual argmin of $H$ — as the all-A m-mer is when $s = 0$ ($\text{mix64}(0) = 0$ is a fixed point) — it would win far more windows than a composition-uniform analysis would predict, not because $H$ favors it, but because that pathological input keeps recurring in real sequence data. The exhaustive checks below confirm that, with this seed, the argmin is never a homopolymer or any periodic repeat, for every tested $m$. Consequently, a kmer or super-kmer containing a low-complexity substring has no increased chance of having that substring selected as its minimizer solely because of its low complexity.
+```
+H(x):
+    x ← x  ⊕  s
+    x ← x  ⊕  (x >> 30)
+    x ← x  ×  0xbf58476d1ce4e5b9
+    x ← x  ⊕  (x >> 27)
+    x ← x  ×  0x94d049bb133111eb
+    return x ⊕ (x >> 31)
+```
+
+The choice of $s$ is not arbitrary. Low-complexity m-mers (homopolymers, short tandem repeats) are disproportionately abundant in real genomes; if one of them happened to be the perpetual argmin of $H$ — as the all-A m-mer is when $s = 0$, since $\text{mix64}(0) = 0$ is a fixed point — it would win far more windows than the composition-uniform behavior established above predicts, not because $H$ favors it, but because that pathological input keeps recurring in real sequence data. Exhaustive checks confirm that, with this seed, the argmin is never a homopolymer or any periodic repeat, for every tested $m$:
 
 | $m$ | argmin (canonical) | decoded sequence | minimal period |
 |---|---|---|---|