The papers

The record proves liminf (log L_N)/(N^{3/2} log N) ≥ 2√2/3 for L_N=lcm(2!−1,…,N!−1), by comparing a terminal block of denominators with pairwise gcds rather than an external multiplicity theorem. Specialising a Louwsma–Martino valuation, it exhibits primes 139 and 2593 absent from all later reduced denominators. Coefficient constructions give integer forms MS+k; at fixed moment the remainder changes only by an integer. Rationality criteria use factorial digits and the next integer above a scaled partial sum; the digit argument also recovers the irrationality of e. Two exact calculations give q∤299999! and q≥2^39990>10^12038. These exclusions do not prove irrationality.

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the complete problem-specific reasoning surface for Erdős #68, including all registered result families and their boundaries.

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the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #68, which remains open.

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long68:sec:carry, long68:sec:finite, long68:sec:open, erdos 68 complete family map (our selection, not the paper's own advice)

An increasing sequence of positive integers satisfying a_n^2/a_{n+1}=1+3/n+o(n^{-3}) has an irrational reciprocal sum: rationality would force integer tail numerators to agree eventually with a cubic polynomial, which divisibility, a cubic-field calculation, and congruences modulo seven exclude. For rational reciprocal sums under a_{n+1}/a_n^2→1, the record also develops sufficient conditions for the eventual recurrence a_{n+1}=a_n^2−a_n+1, controlling either increases of an integer tail numerator or a convergent sum at steps setting new maxima. Examples separate numerical growth bounds from the exact recurrences. The cubic proof is distinguished from the theorems formalised in Lean. None of the additional bounds is derived for every rational reciprocal sum in Erdős #243.

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the complete problem-specific reasoning surface for Erdős #243, including all registered result families and their boundaries.

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the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #243, which remains open.

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long243:sec:cubicrate, long243:sec:bounded, long243:sec:open, erdos 243 complete family map (our selection, not the paper's own advice)

The dyadic totient sections through level e≥1 have an explicit rational basis of size 2^e+1. Truncating a scaled tail difference produces an integer whose residue certifies nonintegrality when its distances from both endpoints of the residue interval exceed an explicit error bound. The record develops the resulting finite denominator exclusions, equivalent residue criteria, and sufficient exponential-sum estimates. Rational comparison sequences identify the limitations of particular size, parity, rank, and finite-prefix arguments. The short paper supplies the all-base basis theorem; here each conditional result carries the arithmetic hypothesis still needed to apply it. Those hypotheses are not established in the form required to prove irrationality of S=∑ φ(n)2^{-n}.

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a claim-bounded problem-specific reasoning surface for Erdős #249: selected checked premises, finite certificates, closed routes, surviving obligations, and their exact evidence bands.

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the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #249, which remains open.

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wall, survivors, series, mahler defect (our selection, not the paper's own advice)

A convergent dyadic series with nonnegative integer coefficients can be made rational by sparse nonnegative corrections of upper Banach density zero, eventually bounded by any f(n)→∞, while preserving every fixed modulus and, for a logarithmic allowance, short-block empirical distributions. For prime gaps the cumulative positions remain asymptotic to n log n; no primality conclusion is asserted. Summation by parts reduces the prime series to the gap series, and the tails T_N=∑_{j≥1} g_{N+j} 2^{-j} satisfy T_{N+1}=2T_N−g_{N+1}. That recurrence characterises rationality and yields finite tests with explicit remainder bounds. An exact finite certificate shows that a rational value of the prime series would have denominator at least 2^{39998}>10^{12040}. The bound does not establish irrationality.

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the complete problem-specific reasoning surface for Erdős #251, including all registered result families and their boundaries.

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the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #251, which remains open.

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long251:sec:tail, long251:sec:obstructions, long251:sec:open, erdos 251 complete family map (our selection, not the paper's own advice)

A weighted summability criterion proves irrationality of reciprocal Mersenne subseries: if h is the part of a supported on a finite nonempty prime set P, every infinite A with ∑ h(a)/(a(b^{h(a)}−1)) finite has irrational ∑ (b^a−1)^{-1}. The base-two condition gives the conclusion at every integer base for every infinite subset, and permits an example with divergent reciprocal sum, while excluding full support and all odd exponents. The proof averages over a finite part of the support and over dyadic observation lengths. Erdős's weaker reciprocal-summable theorem is proved in full, and a common window extends both arguments to unions of weighted and positive-cover supports. Finite-denominator and achievement-set calculations formulate tests for the targets 1/2 and 1/21; those tests do not decide either target or the universal question. Prime support at base 2 and squarefree support at power-of-two bases are attributed as existing work.

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a claim-bounded problem-specific reasoning surface for Erdős #257: selected checked premises, finite computations, closed routes, surviving obligations, and their exact evidence bands.

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the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #257, whose universal and half-value questions remain open.

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257 problem, 257 wall, 257 survivors, 257 howto (our selection, not the paper's own advice)

The reciprocal three-prime running-LCM kernel has nonsingular minors of every order: row and column rescaling leaves a two-valued matrix, and density supplies the required threshold pattern. The record computes the rank of every finite restriction and the least uniform approximation error over all matrices of finite separated rank. For the repeated {2,3,5} sum it derives an integer-coefficient tail recurrence and a residue criterion equivalent to irrationality. The unproved step is to find such windows for every positive multiplier coprime to 30 and after every prescribed start. It also gives the full two-prime comparison, following Fan's earlier Hecke–Mahler reduction, and identifies the boundary terms that survive a proposed finite-difference cancellation.

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the complete problem-specific reasoning surface for Erdős #269, including all registered result families and their boundaries.

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the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #269, which remains open.

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long269:sec:problem, long269:sec:rank, long269:sec:open, erdos 269 complete family map (our selection, not the paper's own advice)

For a monic trinomial with roots in the open unit disc, any two distinct roots are joined inside {|f|<1} by a path of length less than 2. For any squarefree monic polynomial of degree n≥2, an area-growth argument gives such a path when the least critical-value modulus μ is at most 13/25; scaling gives length less than (5/2)μ^{1/n} inside {|f|<(25/13)μ}. The record develops this argument, an inverse-ray averaging estimate, and a construction using an isolated simple critical value, together with results for collinear roots and several coefficient families. A weighted Poisson identity gives a sharp mean bound on critical values; examples distinguish that bound from path-length control. The final sections examine inverse-sheet topology and refute an earlier spanning-tree estimate. These results do not prove the unrestricted length-2 assertion.

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the complete problem-specific reasoning surface for Erdős #1041, including all registered result families and their boundaries.

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the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #1041, which remains open.

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low critical closure, solved polynomial families, gap, erdos 1041 complete family map (our selection, not the paper's own advice)

Specialising Zudilin's 2004 linear forms shows F(a/b) irrational for coprime a>b≥1 when θ=log b / log a is less than θ^*=0.4056830213840605…, with irrationality-exponent bound (1−θ)/(θ^*−θ) for every positive integral power. In particular the bound is less than 301 for every positive integral power of 31/4. The constants are Zudilin's; the proof cancels common cyclotomic factors before rational specialisation. For the 2016 Hankel construction with x=z=1, the first nonzero term of the normalised determinant is computed at every rank. A separate positive-measure argument proves positivity and a size estimate for each fixed real 0<q<1. The final sections explain the limits of the stated clearing and congruence arguments at 3/2: no small nonzero sequence of divided remainders is constructed there, and the full rational-base conjecture remains open.

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the complete problem-specific reasoning surface for Erdős #1049, including all registered result families and their boundaries.

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the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #1049, which remains open.

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long1049:sec:region, long1049:sec:hankel order, long1049:sec:open, erdos 1049 complete family map (our selection, not the paper's own advice)