The unique hole in the mixed binary and ternary root multiplicities
Abstract
Positive divisor blocks with quotient gcd in {1,2,3} generate all scaled multiplicities except the scale itself.
Let a,b,m,n be natural numbers, with gcd(m,6)=1. Set r=2^a*3^b and k=2^(a+1)3^(b+1)m=6rm. A length ell is allowed when it is positive, divides k exactly, and gcd(k/ell,36) is 1, 2 or 3. Here div is natural-number division, and sum is the ordinary sum of a finite list, including the empty list.
Theorem 1.1 (Exact allowed-block sum criterion).
Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/Coding/PrimitiveLifts/RootMultiplicityMonoid.allowed_sum_iff (✓ std3). ∎
Source. Repository-derived.
Commentary.
There exists an actual finite list of allowed positive lengths with sum n if and only if r divides n and n is different from r. Both exponents may be zero. No positivity condition on n is imposed: n=0 is represented by the empty list. The coprime factor m is unrestricted.
For an allowed divisor ell, write q=k/ell and d=gcd(q,36). The standard gcd quotient identity makes c=q/d coprime to 36/d. Since d is 1, 2 or 3, the integer 36/d is divisible by 6, so c is coprime to both 2 and 3. From c dividing k it follows that c divides m. Consequently q divides dm. Cancelling the actual positive quotient q in ellq=k shows that 6r, 3r or 2r divides ell according as d is 1, 2 or 3. Thus every allowed length is a positive multiple of either 2r or 3r.
Conversely, 2r and 3r divide k, and their quotient degrees are respectively 3m and 2m. Coprimality gives gcd(3m,36)=3 and gcd(2m,36)=2, so both generators are allowed. Every integer s at least 2 has a representation A2+B3 with nonnegative natural coefficients. This step uses Mathlib’s Nat.exists_add_mul_eq_of_gcd_dvd_of_mul_pred_le. Repeating 2r exactly A times and 3r exactly B times gives the required list of sum r*s. Every nonempty allowed list has sum at least 2r, which excludes r. The allowed sum monoid is therefore generated by 2r and 3r; its gcd divisibility test omits exactly one positive multiple, r.
The arithmetic has a conditional interpretation for equivariant roots. Suppose an element h of a group has a qth root exactly when gcd(q,36) belongs to {1,2,3}, and suppose a finite-family chart criterion identifies period-one copy multiplicities with sums of the allowed divisor lengths. Then this theorem gives the exact multiplicity condition r divides n and n differs from r. These two suppositions are not hypotheses proved by this arithmetic theorem. In particular, the proposed Q8 semidirect C9 model and the separate chart classification require their own proofs; no claim of their verification is made here. Under that bridge, k=6,r=1; k=36,r=6; and k=216,r=36 exhibit the same obstruction at increasing scales. This is the mixed binary and ternary exponent branch, with no assertion of a complete branch classification or literature priority.