Certified Syracuse descent on 13 progressions inside $n \equiv 27 \pmod{32}$

For every $n$ in any of the 13 arithmetic progressions listed below, the Syracuse orbit of $n$ drops below its starting point within at most $7$ steps:

These are exactly the residue classes modulo a power of $2$ up to $2^{12}$, inside $n \equiv 27 \pmod{32}$, on which the Terras affine certificate closes. Together they cover a proportion $0.6484$ of that class; the complement is left open in the companion statement.

The progressions:

• $n \equiv 59 \pmod{128}$
• $n \equiv 123 \pmod{256}$
• $n \equiv 219 \pmod{256}$
• $n \equiv 347 \pmod{1024}$
• $n \equiv 507 \pmod{1024}$
• $n \equiv 923 \pmod{1024}$
• $n \equiv 1019 \pmod{4096}$
• $n \equiv 1435 \pmod{4096}$
• $n \equiv 1787 \pmod{4096}$
• $n \equiv 2203 \pmod{4096}$
• $n \equiv 2587 \pmod{4096}$
• $n \equiv 2907 \pmod{4096}$
• $n \equiv 3675 \pmod{4096}$

The certificate

Let $T(n) = (3n+1)/2^{\,v_2(3n+1)}$ be the Syracuse (accelerated Collatz) map, which sends an odd $n$ to the odd part of $3n+1$.

Terras's observation is that a residue class fixes an initial segment of the orbit's $2$-adic data. Write $n = 2^k q + r$. Then $3n+1 = (3r+1) + 3\cdot 2^k q$, so $v_2(3n+1) = v_2(3r+1) =: a_1$ as soon as $a_1 < k$, and

Iterating, as long as the running total $V_i = a_1 + \dots + a_i$ stays below $k$, the valuations $a_i$ are the ones read off the representative $r$, and

an affine function of $n$ on the whole class.

Two further points make the certificate sharp:

• Truncated last step. When the remaining budget $j = k - V_{t-1}$ is at most $a_t$, the valuation is no longer pinned down, but $2^{j} \mid 3n_{t-1}+1$ still holds, so $T(n_{t-1}) \le (3n_{t-1}+1)/2^{j}$. A one-sided bound is all a strict inequality needs, so this final step may be spent even though it is not exact.
• The least representative decides. Since $c_t \ge 0$, the inequality $T^{t}(n) < n$ for every $n$ in the class is equivalent to $c_t < (2^{V_t} - 3^{t})\,r$, i.e. to the same inequality at the smallest member $r$ of the class. So checking the representative certifies the entire progression.

Consequently each progression listed below carries a finite, purely computational witness, and the bound $t \le T_{\max}$ is uniform across the class.

Formalization note. No oddness or positivity hypothesis is needed: each congruence already forces $n$ odd and $n$ at least the stated residue. The step bound is stated as $t \le 7$ so that the lemma composes with a case split without re-deriving a bound.