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

Eric Li (Trinity College; University of Cambridge)

arxiv: 2606.17926 · v1 · pith:M7EUEZSPnew · submitted 2026-06-16 · 🧮 math.NT · math.DS

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

Eric Li (Trinity College , University of Cambridge) This is my paper

Pith reviewed 2026-06-26 22:44 UTC · model grok-4.3

classification 🧮 math.NT math.DS

keywords Erdős–Graham coalescencedivisor functiondivisor-successor graphsquare-annular systemsexit sets E_ktransfer mapsovershoot bounds

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The pith

Coalescence for T(n)=n+τ(n) equals connectedness of the divisor-successor graph and synchronization of annular systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper recasts the Erdős-Graham coalescence question for the map T(n)=n+τ(n) as a question about connectivity in a graph whose edges are divisor successors. It proves that coalescence occurs precisely when this graph is connected and when a sequence of square-annular transfer maps synchronizes. The work supplies unconditional upper bounds on the largest gap R(X) in the image of T and on the sizes of certain overshoot sets E_k that measure how far T jumps over squares. It also isolates a simple criterion: if the image of the transfer map A_k applied to E_k has size tending to 1 infinitely often, then the graph must be connected.

Core claim

Coalescence is equivalent both to connectedness of the divisor-successor graph and to synchronization along an infinite non-autonomous sequence of finite annular systems. The basic identities are im(A_k)=E_{k+1} and F_{k^2}=k^2+E_k, where E_k is the set of square-crossing overshoots from below k^2. The paper proves a transfer parity law, dynamic frontier bounds for the widths W_{k,s}, and the criterion that liminf_k |A_k(E_k)|=1 would imply connectedness. Unconditionally R(X)≤log X+2γ+O(X^{-1/4}), |E_k|≤k^{o(1)}, and the stated sum bounds on |E_k| hold, along with fixed-moment bounds under the shifted-square estimate HST.

What carries the argument

The divisor-successor graph whose edges connect each n to n+τ(n), together with the square-annular transfer maps A_k that encode finite-level obstructions.

If this is right

If liminf_k |A_k(E_k)|=1 then the divisor-successor graph is connected and coalescence holds.
The maximal gap satisfies R(X)≤log X+2γ+O(X^{-1/4}).
The exit sets obey |E_k|≤k^{o(1)} and (9/4)K+O(1)≤sum_{k≤K}|E_k|≪K(log K)^3.
Under the HST estimate the moment sums satisfy sum_{k≤K}|E_k|^m≪_m K(log K)^{C_m} for every fixed m≥2.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The residue-universality of E_k suggests that the image of T eventually hits every arithmetic progression.
Numerical enumeration of the graph for moderate k could locate any early disconnections.
Establishing the unproved HQE hypothesis would improve the first-moment bound on sum |E_k| to K(log K)^2.
The square-gated two-branch criterion supplies a possible alternative route to proving connectedness by splitting into two cases.

Load-bearing premise

The modeling choice that coalescence is exactly equivalent to connectedness of the divisor-successor graph and synchronization of the annular systems.

What would settle it

An explicit k where |A_k(E_k)| stays at least 2 for all larger k, or a disconnected component visible in the divisor-successor graph that prevents reachability from 1 to all positive integers.

read the original abstract

Let $T(n)=n+\tau(n)$, where $\tau$ 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, \[ R(X)\le \log X+2\gamma+O(X^{-1/4}), \] 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives new unconditional bounds on exit sets and a liminf criterion via annular maps, but the load-bearing equivalence of coalescence to graph connectedness is presented as encoding rather than derived.

read the letter

The two things to know are that this paper produces concrete new bounds R(X) ≤ log X + 2γ + O(X^{-1/4}), |E_k| ≤ k^{o(1)}, and the stated sum and moment estimates on |E_k|, plus a liminf criterion that liminf |A_k(E_k)| = 1 would imply connectedness of the divisor-successor graph. These come from the square-annular transfer maps, the identities im(A_k) = E_{k+1} and F_{k^2} = k^2 + E_k, a transfer parity law, and dynamic frontier widths W_{k,s}.

It does a clean job extracting these from the corrected Henriot–Nair–Tenenbaum theorem in specialized form and from separate square-shift estimates. The large-jump and lower-runner race theorems, interval filling, and square-gated two-branch criterion add quantitative content without overclaiming a full solution to the Erdős-Graham problem. The conditional HQE hypothesis is flagged explicitly.

The soft spot is the opening modeling step. Coalescence is stated as equivalent to graph connectedness and annular synchronization right at the start, and this equivalence carries the criterion and all later claims. If the graph misses some finite-level obstructions or if synchronization needs extra conditions not implied by the components, the translation back to the original problem weakens. The abstract treats it as definitional rather than a theorem to be checked, which matches the stress-test note. Derivations and error terms cannot be inspected from the abstract alone, so the bounds' sharpness remains to be verified in the full text.

This is for specialists already working on the Erdős-Graham coalescence or divisor dynamics. A reader in that subfield gets value from the new constructions and explicit bounds. The work shows clear thinking and honest engagement with the literature, so it deserves a serious referee even if the equivalence needs more justification.

Recommendation: send it to peer review.

Referee Report

2 major / 1 minor

Summary. The paper studies the Erdős-Graham coalescence problem for T(n)=n+τ(n) by encoding finite-level obstructions in the divisor-successor graph and square-annular transfer maps. It asserts that coalescence is equivalent to connectedness of this graph and to synchronization along annular systems, establishes the identities im(A_k)=E_{k+1} and F_{k^2}=k^2+E_k, proves a transfer parity law and dynamic frontier bounds for widths W_{k,s}, and gives the criterion that liminf_k |A_k(E_k)|=1 implies connectedness. Unconditionally it proves R(X)≤log X+2γ+O(X^{-1/4}), |E_k|≤k^{o(1)}, and the sum bounds (9/4)K+O(1)≤∑_{k≤K}|E_k|≪K(log K)^3; using the shifted-square estimate HST from the corrected Henriot–Nair–Tenenbaum theorem (specialized form of Proposition 8.4) together with separate square-shift estimates, it obtains the moment bounds ∑_{k≤K}|E_k|^m≪_m K(log K)^{C_m} for m≥2, with a conditional refinement to K(log K)^2 under the uniform quadratic Euler-product hypothesis HQE. Additional results include quantitative large-jump and lower-runner race theorems, isolation of interval filling, and a square-gated two-branch criterion. No proof of the full problem is claimed.

Significance. If the modeling equivalence is rigorously justified and the external estimates are applied correctly, the work supplies a new dynamical framework for analyzing coalescence via graph connectedness and annular synchronization, together with explicit unconditional bounds on exit-set sizes and moments and a concrete criterion that could be tested numerically. The explicit flagging of the conditional HQE hypothesis and the reliance on a corrected external theorem are strengths in transparency. The results isolate transferable features (transfer parity law, frontier bounds) that may guide further study even if the full coalescence question remains open.

major comments (2)

[Abstract] Abstract, opening paragraph: the statement that coalescence is equivalent both to connectedness of the divisor-successor graph and to synchronization along the annular systems is presented as an encoding rather than a derived theorem. Because this equivalence is load-bearing for the criterion that liminf_k |A_k(E_k)|=1 would imply connectedness (and for all subsequent claims about the original problem), the manuscript must supply a self-contained derivation verifying that every finite-level obstruction is captured by the graph components and that annular synchronization follows from the stated identities without additional conditions.
[Section discussing application of corrected Henriot–Nair–Tenenbaum theorem] The paragraph on HST and Proposition 8.4: the fixed-moment bounds ∑_{k≤K}|E_k|^m ≪_m K(log K)^{C_m} (m≥2) rest on the shifted-square estimate HST obtained from the corrected Henriot–Nair–Tenenbaum theorem in specialized form. The manuscript should state Proposition 8.4 in full, including the precise error terms and data-exclusion steps, and confirm that these align exactly with the square-shift estimates used here without introducing hidden dependencies.

minor comments (1)

[Notation and basic identities] The basic objects A_k, E_k, F_{k^2}, W_{k,s} and the transfer maps are introduced inline; a short preliminary section or table collecting their definitions and the identities im(A_k)=E_{k+1}, F_{k^2}=k^2+E_k would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, the positive assessment of the dynamical framework, and the recommendation for major revision. We address each major comment below, agreeing that explicit derivations and full statements of external results strengthen the manuscript. Revisions will be incorporated accordingly.

read point-by-point responses

Referee: [Abstract] Abstract, opening paragraph: the statement that coalescence is equivalent both to connectedness of the divisor-successor graph and to synchronization along the annular systems is presented as an encoding rather than a derived theorem. Because this equivalence is load-bearing for the criterion that liminf_k |A_k(E_k)|=1 would imply connectedness (and for all subsequent claims about the original problem), the manuscript must supply a self-contained derivation verifying that every finite-level obstruction is captured by the graph components and that annular synchronization follows from the stated identities without additional conditions.

Authors: We agree that the equivalence requires an explicit, self-contained derivation rather than being presented solely as an encoding. In the revised version we will insert a new subsection (placed after the basic identities in Section 2) that derives the equivalence from first principles: it verifies that every finite-level obstruction appears as a disconnected component in the divisor-successor graph, and shows that annular synchronization is a direct consequence of the two identities im(A_k)=E_{k+1} and F_{k^2}=k^2+E_k together with the transfer parity law, without extra hypotheses. The connectedness criterion will then be restated as a theorem following this derivation. revision: yes
Referee: [Section discussing application of corrected Henriot–Nair–Tenenbaum theorem] The paragraph on HST and Proposition 8.4: the fixed-moment bounds ∑_{k≤K}|E_k|^m ≪_m K(log K)^{C_m} (m≥2) rest on the shifted-square estimate HST obtained from the corrected Henriot–Nair–Tenenbaum theorem in specialized form. The manuscript should state Proposition 8.4 in full, including the precise error terms and data-exclusion steps, and confirm that these align exactly with the square-shift estimates used here without introducing hidden dependencies.

Authors: We will expand the relevant paragraph (currently referencing the specialized form of Proposition 8.4) to reproduce the full statement of Proposition 8.4, including all error terms and data-exclusion conditions from the corrected Henriot–Nair–Tenenbaum theorem. A subsequent sentence will explicitly verify that the square-shift estimates employed in the paper match these terms exactly and introduce no additional dependencies beyond those already listed in the proposition. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper encodes the Erdős-Graham problem via the divisor-successor graph and annular systems, stating their equivalence in the abstract as the modeling framework. However, the claimed results (R(X) bound, |E_k| size and moment bounds) are derived using external results such as the corrected Henriot–Nair–Tenenbaum theorem (HST) in Proposition 8.4 and separate square-shift estimates, without any reduction of predictions or first-principles claims to internally fitted quantities or self-defined inputs by construction. No self-citation load-bearing steps, ansatz smuggling, or renaming of known results appear in the derivation chain. The estimates are independent of the internal encoding choice.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard analytic number theory estimates and the modeling assumption that the graph and annular maps correctly capture the coalescence obstructions; no free parameters or new physical entities are introduced.

axioms (2)

domain assumption The divisor function τ(n) and the map T(n)=n+τ(n) generate a well-defined directed graph whose connectedness encodes the Erdős-Graham coalescence property.
Invoked in the first sentence of the abstract as the encoding of the problem.
standard math The shifted-square estimate HST holds in the form given by the corrected Henriot–Nair–Tenenbaum theorem (Proposition 8.4).
Used to obtain the fixed-moment bounds on |E_k|.

pith-pipeline@v0.9.1-grok · 5914 in / 1540 out tokens · 35731 ms · 2026-06-26T22:44:21.234546+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 5 canonical work pages

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Erdős and R

P. Erdős and R. L. Graham,Old and New Problems and Results in Combinatorial Number Theory, Monographie No. 28 de L’Enseignement Mathématique, Université de Genève, 1980

1980

[2] [2]

G. H. Hardy and E. M. Wright,An Introduction to the Theory of Numbers, 5th ed., Oxford University Press, 1979

1979

[3] [3]

and Soundararajan, K

A. Granville and K. Soundararajan,The distribution of values ofL(1,χd), Geom. Funct. Anal.13(2003), no. 5, 992–1028,https://doi.org/10.1007/s00039-003-0438-3

work page doi:10.1007/s00039-003-0438-3 2003

[4] [4]

Henriot, Nair–Tenenbaum bounds uniform with respect to the discriminant,Math

K. Henriot, Nair–Tenenbaum bounds uniform with respect to the discriminant,Math. Proc. Cambridge Philos. Soc.152(2012), no. 3, 405–424,https://doi.org/10.1017/S0305004111000752

work page doi:10.1017/s0305004111000752 2012

[5] [5]

Henriot, Nair–Tenenbaum uniform with respect to the discriminant–erratum,Math

K. Henriot, Nair–Tenenbaum uniform with respect to the discriminant–erratum,Math. Proc. Cambridge Philos. Soc.157(2014), no. 2, 375–377,https://doi.org/10.1017/S0305004114000280

work page doi:10.1017/s0305004114000280 2014

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M. N. Huxley, Exponential sums and lattice points III,Proc. London Math. Soc.(3)87(2003), no. 3, 591–609, https://doi.org/10.1112/S0024611503014485

work page doi:10.1112/s0024611503014485 2003

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Tenenbaum,Introduction to Analytic and Probabilistic Number Theory, 3rd ed., Graduate Studies in Mathe- matics 163, American Mathematical Society, 2015

G. Tenenbaum,Introduction to Analytic and Probabilistic Number Theory, 3rd ed., Graduate Studies in Mathe- matics 163, American Mathematical Society, 2015

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Nair and G

M. Nair and G. Tenenbaum, Short sums of certain arithmetic functions,Acta Math.180(1998), no. 1, 119–144, https://doi.org/10.1007/BF02392880

work page doi:10.1007/bf02392880 1998

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Wigert, Sur l’ordre de grandeur du nombre des diviseurs d’un entier,Ark

S. Wigert, Sur l’ordre de grandeur du nombre des diviseurs d’un entier,Ark. Mat. Astr. Fys.3(1907), no. 18, 1–9

1907