sync: update from blackroad-operator 2026-03-14

Synced from BlackRoad-OS-Inc/blackroad-operator/orgs/personal/lucidia
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RoadChain-Full: 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

codex/mirror/number_mirror_mu.py

@@ -0,0 +1,65 @@
+"""
+number_mirror_mu.py
+
+This module implements a simple Möbius mirror demonstration.
+It defines functions to compute the Möbius function µ(n), split
+positive and negative values, compute the Mertens function, and
+verify the Dirichlet generating identity.
+
+Functions:
+    mobius(n) -> int
+    mirror_split_mu(N) -> (pos_indices, neg_indices)
+    mertens(N) -> list[int]
+    dirichlet_sum(s, N) -> complex
+"""
+
+import cmath
+
+def mobius(n: int) -> int:
+    """Compute the Möbius function µ(n)."""
+    if n == 1:
+        return 1
+    primes = {}
+    i = 2
+    m = n
+    while i * i <= m:
+        while m % i == 0:
+            primes[i] = primes.get(i, 0) + 1
+            m //= i
+        i += 1
+    if m > 1:
+        primes[m] = primes.get(m, 0) + 1
+    for exp in primes.values():
+        if exp > 1:
+            return 0
+    return -1 if len(primes) % 2 else 1
+
+def mirror_split_mu(N: int):
+    """Return indices where µ(n) = +1 and µ(n) = -1 up to N."""
+    pos = []
+    neg = []
+    for n in range(1, N + 1):
+        mu = mobius(n)
+        if mu == 1:
+            pos.append(n)
+        elif mu == -1:
+            neg.append(n)
+    return pos, neg
+
+def mertens(N: int):
+    """Compute the Mertens function M(x) for x = 1..N."""
+    total = 0
+    M = []
+    for n in range(1, N + 1):
+        total += mobius(n)
+        M.append(total)
+    return M
+
+def dirichlet_sum(s: complex, N: int):
+    """Compute the partial Dirichlet sum \u2211_{n=1..N} µ(n)/n^s."""
+    total = 0+0j
+    for n in range(1, N + 1):
+        mu = mobius(n)
+        if mu != 0:
+            total += mu / (n ** s)
+    return total