Li 2026 square annular dynamics coalescence frontiers n

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corollary_5_10: If the confluence widths |W_(k,s)| are at most L for arbitrarily large pairs (k, s) with k tending to infinity, the graph joining n to n + tau(n) has at most L components; so liminf |A_k(E_k)| = 1 would imply connectedness.

proposition_5_1: For every k >= 1 the set of offsets at which orbits starting in the annulus [k^2, (k+1)^2) first enter the next annulus is exactly the exit set E_(k+1) of overshoots tau((k+1)^2 - j) - j over active deficits j.

proposition_5_4: For every k >= 2 the set of values T(m) with m < k^2 <= T(m) is exactly k^2 + E_k, and consequently at most |E_k| components of the graph joining n to n + tau(n) meet [1, X] whenever X < k^2.

proposition_5_8: For every k >= 2 and s >= 0 the number of components of the graph joining n to n + tau(n) that meet [1, k^2 - 1] is at most the size of the set W_(k,s) of offsets reached from k^2 + E_k after s further annuli.

proposition_6_1: For every k >= 1 the annular transfer map of n + tau(n) sends each positive offset r to an offset of the parity of r + 1, and sends the offset 0 to an even offset.

proposition_7_1: The exit set E_k of overshoots across k^2 under n + tau(n) has at most exp((2 log 2 + o(1)) log k / log log k) elements, and every active deficit and overshoot is at most the largest divisor count below k^2.

proposition_7_5: Every exit set E_k with k >= 2 has at least two elements, those with k congruent to 3 or 5 mod 8 at least three, so the sum of |E_k| over 2 <= k <= K is at least (9/4)K + O(1).

theorem_10_3: For every start n and integer G >= 2, some iterate x of n under n + tau(n) has tau(x) > G and x at most n plus a primorial of size exp(O(G log G log(G log G))) plus G + 1.

theorem_10_6: If two orbits of n + tau(n) never meet, then in the race that always advances the lower one, for every G >= 2 some state has gap greater than G and lower value at most p_0 + exp(O(G log G log(G log G))).

theorem_12_2: If from some level on the one-step frontier A_k(E_k) has at most two elements, of opposite parity when there are two, and contains 0 for arbitrarily large k, then the graph joining n to n + tau(n) is connected.

theorem_3_4: For T(n) = n + tau(n), the number R(X) of components of the graph joining each n to T(n) that meet [1, X] is at most log X + 2 gamma + O(X^(-1/4)), with sharper error terms from sharper divisor-problem exponents.

theorem_4_4: The Erdős-Graham coalescence problem for T(n) = n + tau(n) is equivalent to the assertion that for every K the first-entry offsets of all starts in the annulus [K^2, (K+1)^2) eventually reduce to a single offset.

theorem_7_2: For any modulus q, residue class a mod q and R >= 1, there is an arithmetic progression of levels k along which, for all large k, the exit set E_k contains at least R distinct elements congruent to a mod q.

theorem_8_14: Assuming the paper's unproved quadratic Euler-product mean-value hypothesis H_QE, the sums of |E_k| and of the active-deficit counts a_k over 2 <= k <= K are O(K (log K)^2), one logarithm better than Theorem 8.3.

theorem_8_23: For every fixed integer m >= 2 there is a constant C_m with the sum of |E_k|^m, and of a_k^m, over 2 <= k <= K at most a constant times K (log K)^(C_m), using the shifted-square estimate H_ST.

theorem_8_3: The sum of the exit-set sizes |E_k| over 2 <= k <= K, and even the sum of the active-deficit counts a_k, is O(K (log K)^3), by elementary counting of square roots modulo d.

theorem_8_7: For fixed B >= 1 and nonsquare j, the sum of tau(k^2 - j)^B over a dyadic range M < k <= 2M with k^2 > j is << M (log 2M)^(C_B) times a factor depending only on the primes dividing 2j, derived from a corrected Henriot bound.

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Eric Li, Square-Annular Dynamics and Coalescence Frontiers for n + τ(n). arXiv preprint (2026). arXiv:2606.17926. The copy read for this card is arXiv:2606.17926v1 (16 June 2026); the labels cited here are that version's. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2606.17926), every other right reserved.

For T(n) = n + tau(n), the paper encodes the Erdős-Graham coalescence problem (do any two starting values have a common iterate?) as connectedness of the divisor-successor graph and as synchronization along an infinite sequence of finite square-annular transfer maps A_k on the annuli I_k = [k^2, (k+1)^2). Basic identities are im(A_k) = E_{k+1} and F_{k^2} = k^2 + E_k, and the paper proves a transfer parity law, frontier bounds for the dynamic widths W_{k,s}, and the criterion that liminf_k |A_k(E_k)| = 1 would imply connectedness. Unconditional estimates include the visible component count R(X) <= log X + 2 gamma + O(X^{-1/4}) (Theorem 3.4), residue-universality of the exit sets with |E_k| <= k^{o(1)} and (9/4)K + O(1) <= sum_{2<=k<=K} |E_k| << K (log K)^3; using the shifted-square estimate H_ST derived from a corrected Henriot-Nair-Tenenbaum bound (Proposition 8.4) and separate square-shift estimates (Proposition 8.9), it obtains sum_{2<=k<=K} |E_k|^m << K (log K)^{C_m} for fixed m >= 2, while a sharper K (log K)^2 first moment is conditional on H_ST together with an unproved quadratic Euler-product mean-value hypothesis H_QE. It also proves that every orbit has arbitrarily large divisor jumps and that two non-coalescing orbits have unbounded race gaps, and gives a conditional square-gated two-branch criterion. The paper explicitly claims no proof of the full problem. Its question is Problem 414's question about coalescence of iterates of n + tau(n), so it supplies structural reformulations and quantitative obstruction bounds rather than a resolution.

Source: https://arxiv.org/abs/2606.17926.

Bears on. #414: the paper restates the problem as connectedness of the graph joining n to T(n) (Lemma 2.1, p. 4) and, equivalently, as synchronization of the annular transfer maps (Theorem 4.4, p. 7); it proves bounds on the number of components seen below X and on the exit sets, and sufficient conditions for a positive answer (Corollary 5.10, p. 10; Theorem 12.2, p. 35) whose hypotheses it does not prove. It leaves the problem open.

Results. Labels and pages are those of v1.

• Theorem 3.4 (p. 5): at most log X + 2 gamma + O(X^{-1/4}) components of the graph meet [1, X].
• Theorem 4.4 (p. 7): coalescence is equivalent to |S_K(t)| = 1 for some t, for every K.
• Proposition 5.1 (p. 8): im(A_k) = E_{k+1}.
• Proposition 5.4 (p. 9): F_{k^2} = k^2 + E_k, so R(X) <= |E_k| for X < k^2.
• Proposition 5.8 (p. 9): R(k^2 - 1) <= |W_{k,s}| for all k >= 2 and s >= 0.
• Corollary 5.10 (p. 10): liminf_k |A_k(E_k)| = 1 would imply connectedness.
• Proposition 6.1 (p. 11): A_k(r) is congruent to r + 1 mod 2 for r > 0, and A_k(0) is even.
• Proposition 7.1 (p. 12): |E_k| <= exp((2 log 2 + o(1)) log k / log log k).
• Theorem 7.2 (p. 12): the exit sets are residue-universal.
• Proposition 7.5 (p. 13): |E_k| >= 2, and sum_{2<=k<=K} |E_k| >= (9/4)K + O(1).
• Theorem 8.3 (p. 16): sum_{2<=k<=K} |E_k| << K (log K)^3, unconditionally.
• Theorem 8.7 (p. 19): the shifted-square estimate H_ST, from the corrected Henriot bound in the form of Proposition 8.4 (p. 17).
• Theorem 8.14 (p. 23): sum_{2<=k<=K} |E_k| << K (log K)^2, conditional on the unproved H_QE.
• Theorem 8.23 (p. 27): sum_{2<=k<=K} |E_k|^m << K (log K)^{C_m} for fixed m >= 2.
• Theorem 10.3 (p. 32): every orbit reaches a value with more than G divisors within a primorial scale.
• Theorem 10.6 (p. 33): two non-coalescing orbits have race gaps exceeding G within a primorial scale.
• Theorem 12.2 (p. 35): eventual two-branch collapse with opposite parity and infinitely many square gates implies connectedness.

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