Residual Syracuse descent inside $n \equiv 27 \pmod{32}$ modulo $4096$

This statement is open.

Let $T(n) = (3n+1)/2^{\,v_2(3n+1)}$ be the Syracuse (accelerated Collatz) map. The assertion is that every $n \equiv 27 \pmod{32}$ which additionally avoids all 13 of the exceptional progressions listed below has a finite accelerated stopping time:

The excluded progressions are precisely those on which the Terras affine certificate already closes within $7$ steps (they are handled by the companion lemma). What remains is the residue of the class $n \equiv 27 \pmod{32}$ modulo $2^{12}$: exactly 45 classes mod $4096$ survive, on which the first $12$ binary digits of $n$ do not suffice to certify descent.

The exclusions:

• $n \not\equiv 59 \pmod{128}$
• $n \not\equiv 123 \pmod{256}$
• $n \not\equiv 219 \pmod{256}$
• $n \not\equiv 347 \pmod{1024}$
• $n \not\equiv 507 \pmod{1024}$
• $n \not\equiv 923 \pmod{1024}$
• $n \not\equiv 1019 \pmod{4096}$
• $n \not\equiv 1435 \pmod{4096}$
• $n \not\equiv 1787 \pmod{4096}$
• $n \not\equiv 2203 \pmod{4096}$
• $n \not\equiv 2587 \pmod{4096}$
• $n \not\equiv 2907 \pmod{4096}$
• $n \not\equiv 3675 \pmod{4096}$

Why this is the honest remainder

Refining the modulus settles a further slice at every level — Terras proved the settled density tends to $1$ — but it never reaches $1$ at any finite level, so this residual cannot be emptied by pushing the refinement deeper. Closing it requires an argument that is not a finite case check: the full descent (stopping time) conjecture restricted to this class.

Formalization note. The hypotheses are stated as separate congruence exclusions grouped by modulus, matching the shape produced by a case split on the companion lemma's disjunction. No oddness or positivity hypothesis is needed.