Three-Block Mechanical Families and p-Adic Levers in the Collatz Carry Equation

We extend the finite-block obstruction mechanism of the archived two-block notes [7, 9] to
three-block mechanical families D(ϕ,ψ,ω)
a,b,c = U a
ϕ V b
ψ W c
ω in the accelerated odd-only Collatz cycle
equation. The two-block master interpolation identity generalizes to an exact telescoping
identity among the three pure-block constants, and wall contact becomes a four-monomial
exponential Diophantine equation in which only the two extreme coefficients involve the
cycle seed m. Extreme 2-adic and 3-adic levers therefore survive, with the same single
quoted two-logarithm irrationality measure (Rhin, as used by Simons–de Weger) controlling
the endpoint defect ∆ = L log2 3 − S independently of block count. A new phenomenon
appears at three blocks: wall contacts of known cycles can form infinite resyllabification
lines rather than isolated points, the key example being (1, 2, 1)(2, 1)b(2) = (1, 2)b+2, a line
of −5 contacts. Two exact lemmas tame this: a power-reduction principle (Q(P r ) = Q(P ),
so imprimitive contacts reduce to shorter, already-classified words) and a resyllabification-
line criterion. A theorem-grade sweep then exhausted a finite-depth dataset of 8 families and
74 phase triples to exact thresholds with no budget caps: 4,871,267 critical-core members
were checked exactly, with zero unresolved triples, zero primitive contacts, and all contacts
lying on the known −5 line. The theorem applies to the tested finite-depth dataset only; it
does not prove the Collatz conjecture, does not classify all three-block families, and does not
constrain positive-entropy words.