Complete factorization and mu(30)

Goal and scope

mu(30)=-1, mu(12)=0, mu(1)=1; the sum over every divisor of 30 is zero.

Complete positive-integer factorization and divisor cross-checks; incomplete factors give no classification.

Givens and symbols

• calc

• G1 — Main n=30, with 12 and 1; mu is zero for a repeated prime factor and (-1)^k for a squarefree integer with k distinct primes.

• \mu — Mobius function
• D(n) — all positive divisors

Rules and sources

Some rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged.

• calc · prerequisite — Assume exact integer arithmetic, division with remainder, and definitions of divisibility and gcd; every congruence has an explicit modulus.
• mko_definition · cited — A complete unique-prime-factorization certificate supplies multiplicities and the distinct-prime count, independently of factor order. The exact divisor-sum cross-check is sum over d dividing n of mu(d)=1 for n=1 and 0 otherwise. Zero and negative integers are outside the domain. Incomplete trial division does not justify classification. · mko-mobius-function

Main derivation

30=2\cdot15=2\cdot3\cdot5

30=2^1 3^1 5^1

(e_2,e_3,e_5)=(1,1,1)

30\text{ squarefree},\quad k=3

k=3

\mu(30)=(-1)^3=-1

30=2^1 3^1 5^1

D(30)=\{1,2,3,5,6,10,15,30\}

d=1,2,3,5,6,10,15,30

\mu(d)=1,-1,-1,-1,1,1,1,-1\text{ respectively}

\sum_{d\mid30}\mu(d)

1-1-1-1+1+1+1-1=0

12=2^2\cdot3

2^2\mid12\Rightarrow\mu(12)=0

D(1)=\{1\},\quad\mu(1)=1

\sum_{d\mid1}\mu(d)=1;\quad n\le0\text{ outside domain}

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S05, S07 — mu(0) is not assigned zero; it is outside the classification domain.
• S07 · premise_audit · G1, S02 — Complete multiplicities are 2,1; distinct count 2 alone is insufficient.
• S06 · premise_audit · G1, S05 — This finite check certifies 30, not the general divisor-sum theorem.
• S05 · premise_audit · G1, S03, S04 — 6 has two primes giving +1 and 30 three giving -1; numerical parity is irrelevant.
• S04 · premise_audit · G1, S01 — The list includes both 1 and 30, not just prime or proper divisors.
• S03 · premise_audit · G1, S02 — Reordering factors changes neither the count nor multiplicities.
• S02 · premise_audit · G1, S01 — Check multiplicities before counting types; three distinct primes alone would not ensure squarefreeness.
• S01 · premise_audit · G1 — No remaining factor is unclassified, so multiplicities can be read.

Error controls and negative controls

Multiplicity first

Any prime multiplicity at least 2 forces zero; distinct count then does not determine a sign.

Complete factorization

Do not label a remainder prime before trial division reaches its square-root bound.

• mu(12)=1 because there are two prime types. → reject — 2^2 divides 12, forcing zero.
• mu(1)=-1. → reject — The empty prime product has k=0, so (-1)^0=1.
• mu(0)=0 under this definition. → reject — The domain is positive integers; zero is unclassified.

Result and limitations

mu(30)=-1, mu(12)=0, mu(1)=1; the sum over every divisor of 30 is zero.

Verification and source binding

2 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 3505226a6366d7f0fe4ff0cdea84d0829765aebda7ef89862b6364d7922604cd
• original source SHA-256: 59850a6096a974b7810ab0b6adb6274aa72c4c80f42e39bd8ed038c2ccad9727
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

Same-source full JSON (formulas, ASTs, controls and revision log)