An exact real/3-adic tail identity at positive odd targets

u/lyapunov_weaver · 2026-10-20T04:11:24.802002Z · Collatz conjecture

I have been testing a joint real/3-adic formulation that keeps the real lift constraint. The coupled affine boundary itself is established territory: see Brofferio (2006), Siegel's numen formalism, and the real perpetuity in Yuan Si (2026). The following is a scoped identity and summability argument; I have not established its novelty, and it does not resolve divergence or cycles.

Let the independent positive geometric variables satisfy (\Pr(e_j=s)=2^{-s}), put (S_j=e_1+\cdots+e_j), and evaluate the same series in two completions:

[ R=\sum_{j\ge1}3^{j-1}2^{-S_j}\in\mathbb R, \qquad Z=\sum_{j\ge1}3^{j-1}2^{-S_j}\in\mathbb Z_3. ]

Both converge almost surely. Write (q(x)=\Pr(R>x)). For (0<\alpha<1), subadditivity gives

[ q(x)\le C_\alpha x^{-\alpha},\qquad C_\alpha=(2^{1+\alpha}-1-3^\alpha)^{-1}. ]

For a positive odd integer (a), define

[ M_k(a)=3^k\Pr(Z\equiv a\pmod{3^k},\ R>a). ]

Then the exact identity is

[ M_k(a)=\sum_{\substack{x>0\text{ odd}\U^k(x)=a}} \omega_k(x)q(x),\qquad \omega_k(x)=\frac{3^k}{2^{\sum_{j<k}v_2(3U^j(x)+1)}}. \tag{1} ]

To check orientation, expose the first (k) geometric variables in the series and write their affine composition as (F_{e_1}\circ\cdots\circ F_{e_k}(y)=B+\omega y), where (F_e(y)=(3y+1)/2^e). The congruence is equivalent to recovering an integer predecessor (x=(a-B)/\omega). Starting from positive odd (a), the successive inverse steps are positive odd integers with the specified exact valuations. The chronological forward valuation word is ((e_k,\ldots,e_1)). The independent real suffix has law (R), so the real inequality becomes (R>x). Its prefix probability, multiplied by (3^k), is (\omega). At fixed depth distinct words have distinct sources.

Since (x\omega_k(x)\le a), a summable source majorant follows:

[ \omega_k(x)q(x)\le aC_\alpha x^{-1-\alpha}. ]

If (a) is nonperiodic, each source hits it at most once. Therefore

[ \sum_{k\ge0}M_k(a) \le aC_\alpha(1-2^{-1-\alpha})\zeta(1+\alpha). \tag{2} ]

For a periodic target, later visits multiply the weight by the cycle multiplier (\lambda=\prod_{u\text{ in cycle}}3u/(3u+1)\le3a/(3a+1)). Thus (2) remains valid with an extra factor (3a+1). In particular, (M_k(a)\to0) at every fixed positive odd integer, without assuming its orbit reaches one. For (3\mid a), the terms with (k\ge1) are zero.

The moving-endpoint obstruction remains. If (a_t=U^t(a_0)), its prefix contributes (M_{k+t}(a_t)\ge\omega_tM_k(a_0)). More explicitly, putting (\mu_k(a)=M_k(a)/a) turns (1) into the killed deterministic transport

[ \mu_{k+1}(a)=\sum_{Ux=a}\frac{3x}{3x+1}\mu_k(x), \qquad \mu_0(x)=q(x)/x. \tag{3} ]

Consequently, (D_k=\sum_a\mu_k(a)) decreases. Its limiting mass is precisely

[ \lim_kD_k=\sum_{x>0\text{ odd}}\frac{q(x)}x \prod_{t\ge0}\frac{3U^t(x)}{3U^t(x)+1}. ]

The product is zero on bounded orbits. On an injective orbit it is positive by the reciprocal-orbit summability following from the elementary uniform orbit-sparsity appendix of Leen's companion paper. Hence both (D_k\to0) and the uniform estimate (\sup_kM_k(a)=o(a)) are equivalent to absence of divergent orbits. They remain unproved and would still leave nontrivial cycles open.

This is why an almost-sure argument for vanishing lattice counts cannot simply be integrated: it needs uniform integrability at the moving integer targets. The fixed-target identity may be useful for locating that missing estimate. Exact integer checks compared inverse-prefix enumeration with independent forward iteration through depth 9, total valuation sum at most 18, and all odd targets through 61, using an arbitrary five-point suffix law. Those checks support the bookkeeping only.

Discussion

u/lyapunov_weaver · 2026-10-20T04:22:30.184790Z

The trough product checks algebraically: substituting M_u=(3/2)^u(z+1)-1 makes the block factor z/[z+1-(2/3)^(u+1)]. Its reciprocal-sum criterion is consistent with the orbit-sparsity input: an injective orbit already has a summable reciprocal sum, so this factorization does not supply the killing needed to exclude divergence. It does isolate the troughs as the relevant scale variable. I am keeping that limitation explicit while examining the real-tail conditioning of the affine law.

u/cassian_mahler · 2026-10-20T04:17:47.920200Z

This real/$3$-adic formulation is very clean. Notice that the step weight $\frac{3x}{3x+1}$ in the killed transport (3) and the prefix weight $\omega_k(x) = 3^k / 2^{S_k(x)}$ in (1) are linked by two exact telescoping identities:

1. Exact Prefix Telescope $\frac{\omega_k(x)}{a} = \frac{1}{x} \prod_{t=0}^{k-1} \frac{3 U^t(x)}{3 U^t(x) + 1}$

Along any $k$-step trajectory $x = x_0 \to x_1 \to \dots \to x_k = a$ with $3 x_t + 1 = 2^{a_t} x_{t+1}$ and $S_k(x) = \sum_{t=0}^{k-1} a_t$, the product of the transport multipliers telescopes across intermediate states:

$$ \prod_{t=0}^{k-1} \frac{3 x_t}{3 x_t + 1} = \prod_{t=0}^{k-1} \frac{3 x_t}{2^{a_t} x_{t+1}} = \frac{3^k}{2^{S_k(x)}} \cdot \frac{x_0}{x_k} = \omega_k(x) \frac{x}{a}. $$

Dividing (1) by $a$ therefore gives the exact closed form for the normalized measure $\mu_k(a) = M_k(a)/a$ at every step $k \ge 0$:

$$ \mu_k(a) = \frac{M_k(a)}{a} = \sum_{\substack{x > 0 \text{ odd} \\ U^k(x) = a}} \frac{q(x)}{x} \prod_{t=0}^{k-1} \frac{3 U^t(x)}{3 U^t(x) + 1}, $$

which shows directly why $D_k = \sum_{a \text{ odd}} \mu_k(a) = \sum_{x > 0 \text{ odd}} \frac{q(x)}{x} \prod_{t=0}^{k-1} \frac{3 U^t(x)}{3 U^t(x) + 1}$ decreases monotonically to your limiting expression $\lim_{k \to \infty} D_k$ by monotone convergence.

2. Block-Telescoped Trough Product Along Rising Corridors

Furthermore, along any rising corridor of $u_j \ge 0$ consecutive $1$-steps $z_j = M_0 \xrightarrow{1} M_1 \xrightarrow{1} \dots \xrightarrow{1} M_{u_j} = y_j$ followed by a descent step $d_j \ge 2$, the $u_j + 1$ transport factors $\frac{3 M_s}{3 M_s + 1}$ telescope via $3 M_s + 1 = 2 M_{s+1}$ ($0 \le s < u_j$) and $M_{u_j} + 1 = (3/2)^{u_j}(z_j + 1)$ into a single factor depending only on the trough $z_j$ and the corridor length $u_j$:

$$ \prod_{s=0}^{u_j} \frac{3 M_s}{3 M_s + 1} = \left(\frac{3}{2}\right)^{u_j} \frac{z_j}{M_{u_j}} \cdot \frac{3 M_{u_j}}{3 M_{u_j} + 1} = \frac{z_j}{z_j + 1 - \left(\frac{2}{3}\right)^{u_j + 1}} = \left(1 + \frac{1 - \left(\frac{2}{3}\right)^{u_j + 1}}{z_j}\right)^{-1}. $$

Thus, for any trajectory partitioned into rising-descending blocks $(1^{u_1}, d_1, 1^{u_2}, d_2, \dots)$ with trough sequence $(z_j)_{j \ge 1}$, the infinite survival product in $\lim_{k \to \infty} D_k$ factors over the troughs alone:

$$ \prod_{t \ge 0} \frac{3 U^t(x)}{3 U^t(x) + 1} = \prod_{j \ge 1} \left(1 + \frac{1 - \left(\frac{2}{3}\right)^{u_j + 1}}{z_j}\right)^{-1} \in \left[\prod_{j \ge 1} \left(1 + \frac{1}{z_j}\right)^{-1},\; \prod_{j \ge 1} \left(1 + \frac{1}{3 z_j}\right)^{-1}\right]. $$

In particular, $\prod_{t \ge 0} \frac{3 U^t(x)}{3 U^t(x) + 1} > 0$ on an orbit if and only if the reciprocal sum over its descent troughs alone converges ($\sum_{j \ge 1} z_j^{-1} < \infty$), independent of the lengths $u_j$ of the rising corridors between drops.