A Proposed Solution to Erdős Problem 486

The argument is internally coherent throughout. Definitions (B, q_S, E_j, U_j, F_{t-1}, V_{t-1}, X_q, A) are stable and used consistently. The two competing bounds are cleanly separated: the recovery cutoff at x_t=2^{a_t-1} ensures all epoch-s>=t moduli exceed x_t and hence are inactive, so B agrees with B_{F_{t-1}} below x_t, yielding L_B(x_t)>=49/50; the deletion cutoff at y_t=2^{2a_t+1} exploits the a_t+1 installed scales' harmonic deletion mass to give L_B(y_t)<177/200. Scale separation (21/20)2^j<(19/20)2^{j+1} is invoked correctly to guarantee moduli do not collide across scales, and endpoint interval disjointness (19/10)2^j<(11/10)2^{j+1} is used correctly for counting distinct deletions. Remark on non-summability is consistent with the introduction's exclusion of the summable regime. The strict/inclusive convention remark is logically sound. Since 177/200=0.885<49/50=0.98, the liminf<limsup conclusion (density fails to exist) is coherent.

The main derivation chain is mathematically sound and largely reproducible: (i) Lemma 3.2 constructs q_S near Q with prescribed prime-divisor pattern using primes in separated ranges and a CRT-like adjustment R_S≡1 (mod P); the bounds P^2/Q→0 given k≈(1/4)√j ensure |q_S−Q|≪Q. (ii) Lemma 3.3 applies McDiarmid correctly to Z=|E|/|J| using bounded differences over independent ε_i(b); constants are plausible though somewhat compressed. (iii) Lemma 3.4’s small-footprint estimate is the technical heart and is proved with a careful conditioning argument: collision sets C_{S,i} control dependence, then candidate probabilities are bounded by 2^{-(k−|S|)}, and the number of candidate S⊂T is bounded via an entropy estimate; Tonelli + Markov turn pointwise probability into an expectation bound for μ(U). (iv) Lemma 3.5 then selects a deterministic labeling (probability-of-success >0) and establishes summability of η_j. (v) The global section uses Lemma 2.1 to choose epochs where the finite-past survivor has logarithmic average ≥49/50 at x_t, and uses explicit deletion of E_j points (each excluded by its assigned active modulus) to force L_B(y_t) ≤ 177/200.

Main mathematical weaknesses are not outright errors but compression around numerical/exponential constants: the entropy-to-constant step leading to (3.12) and the numeric margins (0.29, 0.33, 0.04) could warrant independent checking, since they are load-bearing for μ(U_j) being summably small. However, the argument structure is robust: even if constants shift, one can likely re-tune parameters (k scaling, thresholds like 3k/5, the ε choice) to recover an exponentially small footprint, suggesting the proof is not balanced on a knife-edge. No fundamental derivation error was found.

Scored using the VERIFIABILITY rubric (pure mathematics). The central claims are concrete numerical inequalities with explicit constants (3Q/8 deletions, footprint e^{-Ω(√j)}, 177/200, 49/50) obtained through recomputable steps: Bertrand-postulate prime selection, CRT modulus construction, McDiarmid and Hoeffding concentration, entropy bounds, and Lemma (periodic recovery). An independent reader can re-derive each constant (e.g., 15/76, 5/14, the H(1/6) entropy margin) and check the arithmetic. Failure/boundary conditions are stated precisely (scale separation eq., recovery cutoffs x_t, deletion cutoffs y_t, and the non-summability remark distinguishing it from the Araújo summable regime). It falls short of 5 because several numerical margins rely on stated but tightly-hand-computed inequalities (e.g., 0.49·log2>0.33, 3/5·H(1/6)<0.28) and no independent numerical/experimental table is provided to corroborate the block construction; some intermediate reconstruction is required.

For a specialist mathematics audience, the paper is clearly structured and communicative. The introduction states the problem, the claimed theorem, the role of strict versus inclusive activation, and the two-stage proof strategy. Sections are logically ordered, notation is mostly introduced before use, and the paper repeatedly explains why each lemma matters for the global construction. The narrative around 'periodic recovery,' 'finite block,' 'epochs,' and 'recovery/deletion cutoffs' is especially helpful. The reason this is not a 5 is that some arguments are dense and optimization-heavy, with many constants and inequalities that require careful rereading; a scientifically literate but non-specialist graduate reader would likely struggle without more intuition or examples.

The paper proposes a genuinely new construction technique — separating local harmonic deletion mass from the global Haar measure of the completed periodic footprint — to answer an open Erdős problem in the negative. The 'gliding hump' assembly with recovery cutoffs, combined with the arithmetic-skeleton primes and the multi-residue grouping that keeps the profinite footprint stretched-exponentially small while deleting a fixed proportion, is a non-trivial and specific mechanism. It situates itself correctly against Davenport–Erdős, Besicovitch, Behrend, and Araújo, and clarifies why it does not settle the singleton Problem 25. Not a 5 because the individual tools (concentration inequalities, CRT, entropy counting, periodic-density arguments) are standard; novelty lies in their synthesis for this specific problem.

As a completeness matter, the paper is largely self-contained and purpose-driven. It states the target theorem clearly, separates the proof into a local block construction and a global assembly, and tracks the needed parameters through to the final liminf/limsup contradiction. Core variables are introduced before use, assumptions are explicit, and the paper addresses a boundary convention issue (strict versus inclusive activation) rather than leaving that ambiguity unresolved. It also states where its construction sits relative to known positive results, especially the nonsummable regime.

The main reasons not to award a 5 are secondary but real. There is at least one apparent typographical/sign error in Lemma 2's final interval comparison ('q_S \le 21Q/20 < 11Q/10'), which is numerically false as written even though the intended conclusion is clear. There is also a dangling reference to the definition of B ('defined in \eqref{eq:defB}') without the displayed definition visible in the provided text. More importantly for scholarly completeness, the reference list has two automated red flags: a likely fabricated DOI for the 1951 Davenport-Erdős citation and a malformed/fabricated item associated with Hoeffding ('1459.1963'). These do not appear central to the new construction itself, since the corresponding tools are standard and the argument is presented in-text, but they do weaken the submission's completeness as a formal paper. Overall, the core argument is fully developed, but the presentation has enough citation/integrity and proofreading issues to keep it below the top score.