Square-Annular Dynamics and Coalescence Frontiers for $n+τ(n)$

Abstract

Let $T(n)=n+τ(n)$, where $τ$ is the divisor function. We study the Erdos-Graham coalescence problem by encoding finite-level obstructions in the divisor-successor graph and in square-annular transfer maps. Coalescence is equivalent both to connectedness of this graph and to synchronization along an infinite non-autonomous sequence of finite annular systems. The basic identities are \[ \operatorname{im}(\mathcal A_k)=E_{k+1}, \qquad \mathcal F_{k^2}=k^2+E_k, \] where $E_k$ is the set of square-crossing overshoots from below $k^2$. We prove a transfer parity law, dynamic frontier bounds for the widths $W_{k,s}$, and the criterion that $\liminf_k|\mathcal A_k(E_k)|=1$ would imply connectedness. Unconditionally, \[ R(X)\le \log X+2γ+O(X^{-1/4}), \] and the exit sets are residue-universal, satisfy $|E_k|\le k^{o(1)}$, and obey \[ \frac94K+O(1)\le \sum_{k\le K}|E_k|\ll K(\log K)^3. \] Using the shifted-square estimate HST, obtained from the corrected Henriot--Nair--Tenenbaum theorem in the specialized form of Proposition 8.4 and from separate square-shift estimates, we obtain fixed-moment bounds \[ \sum_{k\le K}|E_k|^m\ll_m K(\log K)^{C_m}\quad(m\ge2). \] A further first-moment refinement to $K(\log K)^2$ is conditional on the additional, currently unproved, uniform quadratic Euler-product mean-value hypothesis HQE. We also prove quantitative large-jump and lower-runner race theorems, isolate interval filling, and formulate a square-gated two-branch criterion. No proof of the full Erdos-Graham problem is claimed.