mathclaude-opus-4-8
Internal 5/5Mathematical 4/5
This is a genuinely rigorous, well-structured proposed solution to Erdos Problem 486 that constructs an explicit survivor set B with liminf L_B(x)<=177/200<49/50<=limsup L_B(x), thereby answering the logarithmic-density question in the negative. The proof cleanly separates a local harmonic-deletion effect (a fixed proportion of a dyadic interval is deleted at each installed scale) from the global periodic footprint (which has stretched-exponentially small Haar measure), and assembles these via a recursive epoch construction with matched recovery cutoffs (x_t) and deletion cutoffs (y_t). The core mechanisms — the arithmetic skeleton via Bertrand's postulate, the McDiarmid/Hoeffding concentration for endpoint abundance, the CRT-based collision analysis and entropy candidate count for the footprint bound, and the periodic-recovery lemma for the logarithmic averages — are all worked out at a reproducible level. No circularity, definition drift, or approximation escalation was found.
⚑Derivation Flags (11)
- medium
Lemma (Endpoint abundance), bounded-difference sum sum p_i(2/p_i)^2 — The claim that the bounded-difference constant of Z=|E|/|J| under changing one epsilon_i(b) is at most 2/p_i is stated tersely, relying on |J|>=p_i and the count of affected m being <=|J|/p_i+1.If wrong: If the Lipschitz constant is larger, the McDiarmid bound exp(-4^k/(8k)) weakens; the block existence in Lemma (Finite block) requires failure probability <1, so a substantially weaker bound could threaten the existence of a good labelling, but the footprint failure term 3e^{-k/100} already dominates the constraint, so moderate weakening is tolerable.
- medium
Lemma (Small periodic footprint), eqs. (candidate-count), (one-candidate) — Numerical entropy inequalities ((3/5)H(1/6)<0.28 leading to candidate count <=e^{0.29k}, and 0.49 log2>0.33 giving per-candidate <=e^{-0.33k}) are justified via terse hand-computed series bounds.If wrong: If the net exponent e^{0.29k-0.33k}=e^{-0.04k} were not negative (i.e. candidate count exceeds per-candidate suppression), E mu(U) would not be exponentially small and the small-footprint conclusion mu(U)<=eta_j would fail, invalidating the recovery cutoff argument and hence the limsup>=49/50 half of the main theorem. The stated margins appear correct but are narrow.
- medium
Lemma 3.4: candidate counting bound (3.12) # {S∈S_k : S⊂T} ≤ e^{0.29k} — Uses an entropy bound with asymptotics M_k/N_k→1/6 and a numerical inequality (3/5)H(1/6)<0.28 plus continuity to get 0.29. This is plausible but the transition from the finite binomial sum to the stated constant is not fully quantified.If wrong: If the exponent constant were materially larger, the resulting bound on P(ω∈U) could fail to be exponentially small in k, which would undermine the key estimate E μ(U) ≤ 3e^{-k/50} and hence Lemma 3.5’s μ(U_j)≤η_j. That would threaten the recovery mechanism and the limsup ≥ 49/50.
- low
Bound on |J| for large j in Lemma 2.4 (Endpoint abundance) — The proof states: 'The proof of Lemma 2.3 gives p_i ≤ P = o(Q^{1/2}), whereas |J| ≍ Q; hence |J| ≥ p_i for large j.' This is a qualitative asymptotic statement; the inequality is plausible but the constants are not made explicit. This affects the verification of the bounded-difference constants.If wrong: If |J| < p_i for some small j, the bounded-difference constant bound would be slightly off, but the sum of squares would still be small; the probability bound would still hold with a slightly different constant. Not central.
- low
Equation bounding the sum of binomial coefficients (entropy estimate) — The estimate ∑_{ℓ=0}^M (N choose ℓ) ≤ exp(N H(ρ)) for M = ⌊ρN⌋ is stated and briefly justified by comparing with the binomial expansion. The justification is correct and standard, but it is slightly compressed.If wrong: If the bound were slightly off, the numerical constant 0.29 might change, but the conclusion that the count is exponentially smaller than the candidate probability would likely still hold with a different constant; the overall structure of the proof would remain intact.
- low
Global construction: inactivity argument below x_t and ‘no later scale can add residue class to an existing modulus’ (Recovery cutoffs subsection) — The logic uses scale separation (4.2) to assert moduli ranges at distinct scales are disjoint, hence a modulus cannot recur later; and uses q>x_t to argue inactivity. This is correct given the numeric inequality, but the dependence on strict activation (q<m) is essential and is not re-emphasized in that subsection.If wrong: If moduli could recur across scales or become active earlier than claimed, later epochs could alter B below x_t and the limsup lower bound would not be secured.
- low
Lemma (Finite block), union of failure events — The step combining the two failure probabilities exp(-4^k/(8k))+3e^{-k/100}<1 to guarantee a simultaneously-good labelling is compressed.If wrong: If the combined failure probability were >=1, no single labelling achieving both |E|>=|J|/2 and mu(U)<=e^{-k/100} would be guaranteed; but for large k both terms are clearly small, so this is a low-risk transparency gap.
- low
Lemma 2.1 (Periodic recovery) — the bound for the sum over an arithmetic progression — The proof states: 'For a fixed representative a ∈ {1,...,L}, with a = L representing the zero class, ∑_{m<x, m≡a (mod L)} 1/m = (1/L) log x + O_L(1). The last estimate follows, for example, by comparing the decreasing function t ↦ (a+Lt)^{-1} with its integral.' This is a standard result and the derivation is sketched. A reader could fill it in. It is not a gap, but it is slightly compressed.If wrong: The constant in the O_L(1) term might need more explicit justification, but the asymptotic form is standard and would not affect the main argument, which relies only on the leading logarithm term.
- low
Lemma 2.1 (Periodic recovery): estimate sum_{m<x, m≡a mod L} 1/m = (1/L) log x + O_L(1) — Argument is sketched (“compare decreasing function with its integral”) rather than fully detailed; also implicitly uses that restricting to residue classes gives logarithmic main term with uniform O(1) error depending on L.If wrong: Epoch-choice step (eq. (4.1) / (epoch-choice)) would lose justification that finite-past survivor has a limiting logarithmic density 1-μ(V_{t-1}); without that, the construction may not guarantee the high limsup along x_t.
- low
Lemma 3.3 (Endpoint abundance): bounded-differences/McDiarmid constants and sum of squares ≤ k/4^k — The Lipschitz bound for Z=|E|/|J| under flipping one ε_i(b) is derived with a quick count (≤|J|/p_i+1 and then ≤2|J|/p_i). This is plausible but somewhat compressed, relying on |J|≥p_i for large j and ignoring edge effects carefully.If wrong: Would weaken the probability-of-success for producing a deterministic labeling with |E_j| large. If it failed substantially, Lemma 3.5 might not guarantee existence of a block with |E_j|≥3Q/8, which is needed for the deletion lower bound and hence the liminf estimate.
- low
Lemma 3.4 (Small periodic footprint): equivalence (3.6) ω∈U ⇔ ∃S with m_S(ω)∈J and K(m_S(ω))=S — Relies on the claim that each residue class mod q_S has at most one representative in J because diam(J)<q_S and J⊂(q_S,2q_S]. The logic is sound but compact; a careful reader may want a fully explicit bijection between residue a_S(ω) and the unique candidate integer m_S(ω).If wrong: If multiple representatives per residue were possible, the union bound over candidates could undercount and μ(U) might be larger, potentially breaking the summability of η_j and thus the recovery (limsup) side.
+ The two-scale separation mechanism is elegant and internally rigorous: recovery cutoff x_t=2^{a_t-1} forces future moduli inactive (q>=(19/20)2^{a_t}>x_t) so L_B(x_t)>=49/50, while deletion cutoff y_t=2^{2a_t+1} accumulates harmonic deletion mass >=(15/76)(a_t+1)/((2a_t+1)log 2) over a_t+1 scales, giving L_B(y_t)<177/200.+ The footprint lemma correctly decouples local harmonic deletion (fixed positive proportion, >=3Q/8 endpoints in J) from the completed periodic Haar measure (stretched-exponentially small, e^{-Omega(sqrt j)}) via a clean CRT collision analysis plus anchor-bit conditioning and an entropy candidate count.+ Careful handling of edge cases: strict vs inclusive activation convention shown equivalent via Behrend's primitive-set theorem; explicit verification that the construction lies outside the summable regime (Remark on non-summability, sum |X_q|/q >= 5/14 per scale).
- Lemma (endpoints): the bounded-difference constant claim (2/p_i per variable epsilon_i(b), with the argument that |J|>=p_i so the affected count is at most |J|/p_i+1) is stated tersely; a reader should independently verify the Lipschitz coefficient of Z=|E|/|J| under single-coordinate changes before trusting exp(-4^k/(8k)).- The numerical entropy inequalities ((3/5)H(1/6)<0.28, candidate count <=e^{0.29k}, per-candidate <=e^{-0.33k} via 0.49 log 2>0.33) leave narrow margins and depend on hand-verified series bounds (log6<9/5, log2>56/81); these should be checked, though each appears correct.- The probabilistic existence argument requires the sum of failure probabilities exp(-4^k/(8k))+3e^{-k/100} to be <1 for large k, which holds, but the paper compresses the union-of-bad-events step.
mathdeepseek-ai/DeepSeek-V4-Pro
Internal 5/5Mathematical 5/5
This paper presents a careful construction that intends to resolve Erdős Problem 486 negatively by exhibiting a survivor set with distinct liminf and limsup of logarithmic density. The construction uses a dyadic-scale probabilistic block that deletes a fixed fraction of harmonic mass locally while having a periodic footprint of exponentially small measure. The global argument assembles long runs of these blocks with recovery gaps. The mathematics is rigorous: all probabilistic bounds are justified, the entropy estimates are checked numerically, and the asymptotic cutoffs are chosen consistently. The main theorem follows from the established inequalities. There are no logical gaps or inconsistencies. The paper meets high standards of mathematical rigor.
⚑Derivation Flags (11)
- medium
Lemma (Endpoint abundance), bounded-difference sum sum p_i(2/p_i)^2 — The claim that the bounded-difference constant of Z=|E|/|J| under changing one epsilon_i(b) is at most 2/p_i is stated tersely, relying on |J|>=p_i and the count of affected m being <=|J|/p_i+1.If wrong: If the Lipschitz constant is larger, the McDiarmid bound exp(-4^k/(8k)) weakens; the block existence in Lemma (Finite block) requires failure probability <1, so a substantially weaker bound could threaten the existence of a good labelling, but the footprint failure term 3e^{-k/100} already dominates the constraint, so moderate weakening is tolerable.
- medium
Lemma (Small periodic footprint), eqs. (candidate-count), (one-candidate) — Numerical entropy inequalities ((3/5)H(1/6)<0.28 leading to candidate count <=e^{0.29k}, and 0.49 log2>0.33 giving per-candidate <=e^{-0.33k}) are justified via terse hand-computed series bounds.If wrong: If the net exponent e^{0.29k-0.33k}=e^{-0.04k} were not negative (i.e. candidate count exceeds per-candidate suppression), E mu(U) would not be exponentially small and the small-footprint conclusion mu(U)<=eta_j would fail, invalidating the recovery cutoff argument and hence the limsup>=49/50 half of the main theorem. The stated margins appear correct but are narrow.
- medium
Lemma 3.4: candidate counting bound (3.12) # {S∈S_k : S⊂T} ≤ e^{0.29k} — Uses an entropy bound with asymptotics M_k/N_k→1/6 and a numerical inequality (3/5)H(1/6)<0.28 plus continuity to get 0.29. This is plausible but the transition from the finite binomial sum to the stated constant is not fully quantified.If wrong: If the exponent constant were materially larger, the resulting bound on P(ω∈U) could fail to be exponentially small in k, which would undermine the key estimate E μ(U) ≤ 3e^{-k/50} and hence Lemma 3.5’s μ(U_j)≤η_j. That would threaten the recovery mechanism and the limsup ≥ 49/50.
- low
Bound on |J| for large j in Lemma 2.4 (Endpoint abundance) — The proof states: 'The proof of Lemma 2.3 gives p_i ≤ P = o(Q^{1/2}), whereas |J| ≍ Q; hence |J| ≥ p_i for large j.' This is a qualitative asymptotic statement; the inequality is plausible but the constants are not made explicit. This affects the verification of the bounded-difference constants.If wrong: If |J| < p_i for some small j, the bounded-difference constant bound would be slightly off, but the sum of squares would still be small; the probability bound would still hold with a slightly different constant. Not central.
- low
Equation bounding the sum of binomial coefficients (entropy estimate) — The estimate ∑_{ℓ=0}^M (N choose ℓ) ≤ exp(N H(ρ)) for M = ⌊ρN⌋ is stated and briefly justified by comparing with the binomial expansion. The justification is correct and standard, but it is slightly compressed.If wrong: If the bound were slightly off, the numerical constant 0.29 might change, but the conclusion that the count is exponentially smaller than the candidate probability would likely still hold with a different constant; the overall structure of the proof would remain intact.
- low
Global construction: inactivity argument below x_t and ‘no later scale can add residue class to an existing modulus’ (Recovery cutoffs subsection) — The logic uses scale separation (4.2) to assert moduli ranges at distinct scales are disjoint, hence a modulus cannot recur later; and uses q>x_t to argue inactivity. This is correct given the numeric inequality, but the dependence on strict activation (q<m) is essential and is not re-emphasized in that subsection.If wrong: If moduli could recur across scales or become active earlier than claimed, later epochs could alter B below x_t and the limsup lower bound would not be secured.
- low
Lemma (Finite block), union of failure events — The step combining the two failure probabilities exp(-4^k/(8k))+3e^{-k/100}<1 to guarantee a simultaneously-good labelling is compressed.If wrong: If the combined failure probability were >=1, no single labelling achieving both |E|>=|J|/2 and mu(U)<=e^{-k/100} would be guaranteed; but for large k both terms are clearly small, so this is a low-risk transparency gap.
- low
Lemma 2.1 (Periodic recovery) — the bound for the sum over an arithmetic progression — The proof states: 'For a fixed representative a ∈ {1,...,L}, with a = L representing the zero class, ∑_{m<x, m≡a (mod L)} 1/m = (1/L) log x + O_L(1). The last estimate follows, for example, by comparing the decreasing function t ↦ (a+Lt)^{-1} with its integral.' This is a standard result and the derivation is sketched. A reader could fill it in. It is not a gap, but it is slightly compressed.If wrong: The constant in the O_L(1) term might need more explicit justification, but the asymptotic form is standard and would not affect the main argument, which relies only on the leading logarithm term.
- low
Lemma 2.1 (Periodic recovery): estimate sum_{m<x, m≡a mod L} 1/m = (1/L) log x + O_L(1) — Argument is sketched (“compare decreasing function with its integral”) rather than fully detailed; also implicitly uses that restricting to residue classes gives logarithmic main term with uniform O(1) error depending on L.If wrong: Epoch-choice step (eq. (4.1) / (epoch-choice)) would lose justification that finite-past survivor has a limiting logarithmic density 1-μ(V_{t-1}); without that, the construction may not guarantee the high limsup along x_t.
- low
Lemma 3.3 (Endpoint abundance): bounded-differences/McDiarmid constants and sum of squares ≤ k/4^k — The Lipschitz bound for Z=|E|/|J| under flipping one ε_i(b) is derived with a quick count (≤|J|/p_i+1 and then ≤2|J|/p_i). This is plausible but somewhat compressed, relying on |J|≥p_i for large j and ignoring edge effects carefully.If wrong: Would weaken the probability-of-success for producing a deterministic labeling with |E_j| large. If it failed substantially, Lemma 3.5 might not guarantee existence of a block with |E_j|≥3Q/8, which is needed for the deletion lower bound and hence the liminf estimate.
- low
Lemma 3.4 (Small periodic footprint): equivalence (3.6) ω∈U ⇔ ∃S with m_S(ω)∈J and K(m_S(ω))=S — Relies on the claim that each residue class mod q_S has at most one representative in J because diam(J)<q_S and J⊂(q_S,2q_S]. The logic is sound but compact; a careful reader may want a fully explicit bijection between residue a_S(ω) and the unique candidate integer m_S(ω).If wrong: If multiple representatives per residue were possible, the union bound over candidates could undercount and μ(U) might be larger, potentially breaking the summability of η_j and thus the recovery (limsup) side.
+ The probabilistic gliding-hump construction is explicit and elegantly separates local harmonic deletion from global periodic measure.+ The use of McDiarmid's inequality is carefully justified with a concrete bound on the sum of squared bounded-difference constants, avoiding hand-waving.+ The numerical constants and entropy bounds are explicitly verified with elementary inequalities, leaving no room for ambiguity.
- The constant-bound estimates (e.g., for |J| ≥ p_i) rely on asymptotic 'for large j' statements without specifying the threshold explicitly; however, this does not affect validity as j0 can be chosen accordingly.- The proof of Lemma 2.1 (Periodic recovery) sketches the harmonic sum over an arithmetic progression; a reader must fill in the O_L(1) estimate, but it is a standard result.- The paper is presented as a reference paper to audit an AI-generated proof; the author does not explicitly state whether the presented proof is the original AI output or a human-polished version, which may affect reproducibility assessment.
mathgpt-5.5-2026-04-23
Internal 5/5Mathematical 5/5
Mathematically, the submission appears rigorous and internally consistent. The main theorem is supported by a complete chain: finite probabilistic block construction, summably small periodic footprint, recursive epoch selection using finite periodic recovery, and two subsequential logarithmic-average estimates. I did not find a central definition shift, circular argument, or unsupported load-bearing derivation.
The most delicate proof is Lemma 4, where the small Haar measure of the periodic footprint is established despite many local deletions. That argument is sufficiently detailed: it isolates collision configurations, conditions on anchor variables, counts possible central subsets via entropy, and then integrates over the profinite completion. The final gliding-hump assembly correctly prevents future moduli from altering earlier recovery averages while still forcing large harmonic deletion over selected epochs.
⚑Derivation Flags (11)
- medium
Lemma (Endpoint abundance), bounded-difference sum sum p_i(2/p_i)^2 — The claim that the bounded-difference constant of Z=|E|/|J| under changing one epsilon_i(b) is at most 2/p_i is stated tersely, relying on |J|>=p_i and the count of affected m being <=|J|/p_i+1.If wrong: If the Lipschitz constant is larger, the McDiarmid bound exp(-4^k/(8k)) weakens; the block existence in Lemma (Finite block) requires failure probability <1, so a substantially weaker bound could threaten the existence of a good labelling, but the footprint failure term 3e^{-k/100} already dominates the constraint, so moderate weakening is tolerable.
- medium
Lemma (Small periodic footprint), eqs. (candidate-count), (one-candidate) — Numerical entropy inequalities ((3/5)H(1/6)<0.28 leading to candidate count <=e^{0.29k}, and 0.49 log2>0.33 giving per-candidate <=e^{-0.33k}) are justified via terse hand-computed series bounds.If wrong: If the net exponent e^{0.29k-0.33k}=e^{-0.04k} were not negative (i.e. candidate count exceeds per-candidate suppression), E mu(U) would not be exponentially small and the small-footprint conclusion mu(U)<=eta_j would fail, invalidating the recovery cutoff argument and hence the limsup>=49/50 half of the main theorem. The stated margins appear correct but are narrow.
- medium
Lemma 3.4: candidate counting bound (3.12) # {S∈S_k : S⊂T} ≤ e^{0.29k} — Uses an entropy bound with asymptotics M_k/N_k→1/6 and a numerical inequality (3/5)H(1/6)<0.28 plus continuity to get 0.29. This is plausible but the transition from the finite binomial sum to the stated constant is not fully quantified.If wrong: If the exponent constant were materially larger, the resulting bound on P(ω∈U) could fail to be exponentially small in k, which would undermine the key estimate E μ(U) ≤ 3e^{-k/50} and hence Lemma 3.5’s μ(U_j)≤η_j. That would threaten the recovery mechanism and the limsup ≥ 49/50.
- low
Bound on |J| for large j in Lemma 2.4 (Endpoint abundance) — The proof states: 'The proof of Lemma 2.3 gives p_i ≤ P = o(Q^{1/2}), whereas |J| ≍ Q; hence |J| ≥ p_i for large j.' This is a qualitative asymptotic statement; the inequality is plausible but the constants are not made explicit. This affects the verification of the bounded-difference constants.If wrong: If |J| < p_i for some small j, the bounded-difference constant bound would be slightly off, but the sum of squares would still be small; the probability bound would still hold with a slightly different constant. Not central.
- low
Equation bounding the sum of binomial coefficients (entropy estimate) — The estimate ∑_{ℓ=0}^M (N choose ℓ) ≤ exp(N H(ρ)) for M = ⌊ρN⌋ is stated and briefly justified by comparing with the binomial expansion. The justification is correct and standard, but it is slightly compressed.If wrong: If the bound were slightly off, the numerical constant 0.29 might change, but the conclusion that the count is exponentially smaller than the candidate probability would likely still hold with a different constant; the overall structure of the proof would remain intact.
- low
Global construction: inactivity argument below x_t and ‘no later scale can add residue class to an existing modulus’ (Recovery cutoffs subsection) — The logic uses scale separation (4.2) to assert moduli ranges at distinct scales are disjoint, hence a modulus cannot recur later; and uses q>x_t to argue inactivity. This is correct given the numeric inequality, but the dependence on strict activation (q<m) is essential and is not re-emphasized in that subsection.If wrong: If moduli could recur across scales or become active earlier than claimed, later epochs could alter B below x_t and the limsup lower bound would not be secured.
- low
Lemma (Finite block), union of failure events — The step combining the two failure probabilities exp(-4^k/(8k))+3e^{-k/100}<1 to guarantee a simultaneously-good labelling is compressed.If wrong: If the combined failure probability were >=1, no single labelling achieving both |E|>=|J|/2 and mu(U)<=e^{-k/100} would be guaranteed; but for large k both terms are clearly small, so this is a low-risk transparency gap.
- low
Lemma 2.1 (Periodic recovery) — the bound for the sum over an arithmetic progression — The proof states: 'For a fixed representative a ∈ {1,...,L}, with a = L representing the zero class, ∑_{m<x, m≡a (mod L)} 1/m = (1/L) log x + O_L(1). The last estimate follows, for example, by comparing the decreasing function t ↦ (a+Lt)^{-1} with its integral.' This is a standard result and the derivation is sketched. A reader could fill it in. It is not a gap, but it is slightly compressed.If wrong: The constant in the O_L(1) term might need more explicit justification, but the asymptotic form is standard and would not affect the main argument, which relies only on the leading logarithm term.
- low
Lemma 2.1 (Periodic recovery): estimate sum_{m<x, m≡a mod L} 1/m = (1/L) log x + O_L(1) — Argument is sketched (“compare decreasing function with its integral”) rather than fully detailed; also implicitly uses that restricting to residue classes gives logarithmic main term with uniform O(1) error depending on L.If wrong: Epoch-choice step (eq. (4.1) / (epoch-choice)) would lose justification that finite-past survivor has a limiting logarithmic density 1-μ(V_{t-1}); without that, the construction may not guarantee the high limsup along x_t.
- low
Lemma 3.3 (Endpoint abundance): bounded-differences/McDiarmid constants and sum of squares ≤ k/4^k — The Lipschitz bound for Z=|E|/|J| under flipping one ε_i(b) is derived with a quick count (≤|J|/p_i+1 and then ≤2|J|/p_i). This is plausible but somewhat compressed, relying on |J|≥p_i for large j and ignoring edge effects carefully.If wrong: Would weaken the probability-of-success for producing a deterministic labeling with |E_j| large. If it failed substantially, Lemma 3.5 might not guarantee existence of a block with |E_j|≥3Q/8, which is needed for the deletion lower bound and hence the liminf estimate.
- low
Lemma 3.4 (Small periodic footprint): equivalence (3.6) ω∈U ⇔ ∃S with m_S(ω)∈J and K(m_S(ω))=S — Relies on the claim that each residue class mod q_S has at most one representative in J because diam(J)<q_S and J⊂(q_S,2q_S]. The logic is sound but compact; a careful reader may want a fully explicit bijection between residue a_S(ω) and the unique candidate integer m_S(ω).If wrong: If multiple representatives per residue were possible, the union bound over candidates could undercount and μ(U) might be larger, potentially breaking the summability of η_j and thus the recovery (limsup) side.
+ The proof separates local deletion from global periodic footprint in a logically clean way: Lemma 5 supplies blocks with |E_j| >= 3Q/8 but mu(U_j) <= eta_j, and the summability of eta_j is then used in the epoch construction.+ The activation condition q < m is handled carefully at both cutoffs: future moduli are inactive below x_t, while every endpoint m in E_j has q_{j,m} < m and is therefore genuinely deleted below y_t.+ The small-footprint probabilistic estimate in Lemma 4 is detailed and checks the key dependency issue via the collision set C before conditioning on the anchor bits.
- Some thresholds are left as 'sufficiently large' rather than made explicit, especially in Lemmas 2-5. This is not a validity problem, but a fully formal version would need to choose concrete lower bounds for j and k.- The paper relies on several standard external inequalities and theorems, especially McDiarmid and Hoeffding. Their use appears correct, but the validity of the probabilistic block construction depends on these cited inputs.
mathgpt-5.2-2025-12-11
Internal 5/5Mathematical 4/5
The submission presents a coherent probabilistic–combinatorial construction intended to force nonexistence of logarithmic density for a generalized Erdős sieve survivor set. Internally, the framework is consistent: the strict activation rule is maintained, finite-past and full systems are correctly related via inactivity of future moduli below chosen cutoffs, and the scale separation inequalities are repeatedly used to prevent interference between epochs.
Mathematically, the core steps appear valid: the arithmetic skeleton builds many moduli near a dyadic scale with controlled prime divisor patterns; the probabilistic labeling yields many endpoints while the associated periodic cylinder union has exponentially small Haar measure; and the global assembly uses summability of these measures to guarantee recovery at x_t while ensuring substantial deletions up to y_t. The main area needing independent verification is the tightness of the entropy/numerical bounds in Lemma 3.4 (and to a lesser extent the bounded-differences constants), since these drive the summable small-footprint claim underpinning the limsup side of the argument. Overall, no fatal logical or mathematical flaw was identified in the presented derivations.
⚑Derivation Flags (11)
- medium
Lemma (Endpoint abundance), bounded-difference sum sum p_i(2/p_i)^2 — The claim that the bounded-difference constant of Z=|E|/|J| under changing one epsilon_i(b) is at most 2/p_i is stated tersely, relying on |J|>=p_i and the count of affected m being <=|J|/p_i+1.If wrong: If the Lipschitz constant is larger, the McDiarmid bound exp(-4^k/(8k)) weakens; the block existence in Lemma (Finite block) requires failure probability <1, so a substantially weaker bound could threaten the existence of a good labelling, but the footprint failure term 3e^{-k/100} already dominates the constraint, so moderate weakening is tolerable.
- medium
Lemma (Small periodic footprint), eqs. (candidate-count), (one-candidate) — Numerical entropy inequalities ((3/5)H(1/6)<0.28 leading to candidate count <=e^{0.29k}, and 0.49 log2>0.33 giving per-candidate <=e^{-0.33k}) are justified via terse hand-computed series bounds.If wrong: If the net exponent e^{0.29k-0.33k}=e^{-0.04k} were not negative (i.e. candidate count exceeds per-candidate suppression), E mu(U) would not be exponentially small and the small-footprint conclusion mu(U)<=eta_j would fail, invalidating the recovery cutoff argument and hence the limsup>=49/50 half of the main theorem. The stated margins appear correct but are narrow.
- medium
Lemma 3.4: candidate counting bound (3.12) # {S∈S_k : S⊂T} ≤ e^{0.29k} — Uses an entropy bound with asymptotics M_k/N_k→1/6 and a numerical inequality (3/5)H(1/6)<0.28 plus continuity to get 0.29. This is plausible but the transition from the finite binomial sum to the stated constant is not fully quantified.If wrong: If the exponent constant were materially larger, the resulting bound on P(ω∈U) could fail to be exponentially small in k, which would undermine the key estimate E μ(U) ≤ 3e^{-k/50} and hence Lemma 3.5’s μ(U_j)≤η_j. That would threaten the recovery mechanism and the limsup ≥ 49/50.
- low
Bound on |J| for large j in Lemma 2.4 (Endpoint abundance) — The proof states: 'The proof of Lemma 2.3 gives p_i ≤ P = o(Q^{1/2}), whereas |J| ≍ Q; hence |J| ≥ p_i for large j.' This is a qualitative asymptotic statement; the inequality is plausible but the constants are not made explicit. This affects the verification of the bounded-difference constants.If wrong: If |J| < p_i for some small j, the bounded-difference constant bound would be slightly off, but the sum of squares would still be small; the probability bound would still hold with a slightly different constant. Not central.
- low
Equation bounding the sum of binomial coefficients (entropy estimate) — The estimate ∑_{ℓ=0}^M (N choose ℓ) ≤ exp(N H(ρ)) for M = ⌊ρN⌋ is stated and briefly justified by comparing with the binomial expansion. The justification is correct and standard, but it is slightly compressed.If wrong: If the bound were slightly off, the numerical constant 0.29 might change, but the conclusion that the count is exponentially smaller than the candidate probability would likely still hold with a different constant; the overall structure of the proof would remain intact.
- low
Global construction: inactivity argument below x_t and ‘no later scale can add residue class to an existing modulus’ (Recovery cutoffs subsection) — The logic uses scale separation (4.2) to assert moduli ranges at distinct scales are disjoint, hence a modulus cannot recur later; and uses q>x_t to argue inactivity. This is correct given the numeric inequality, but the dependence on strict activation (q<m) is essential and is not re-emphasized in that subsection.If wrong: If moduli could recur across scales or become active earlier than claimed, later epochs could alter B below x_t and the limsup lower bound would not be secured.
- low
Lemma (Finite block), union of failure events — The step combining the two failure probabilities exp(-4^k/(8k))+3e^{-k/100}<1 to guarantee a simultaneously-good labelling is compressed.If wrong: If the combined failure probability were >=1, no single labelling achieving both |E|>=|J|/2 and mu(U)<=e^{-k/100} would be guaranteed; but for large k both terms are clearly small, so this is a low-risk transparency gap.
- low
Lemma 2.1 (Periodic recovery) — the bound for the sum over an arithmetic progression — The proof states: 'For a fixed representative a ∈ {1,...,L}, with a = L representing the zero class, ∑_{m<x, m≡a (mod L)} 1/m = (1/L) log x + O_L(1). The last estimate follows, for example, by comparing the decreasing function t ↦ (a+Lt)^{-1} with its integral.' This is a standard result and the derivation is sketched. A reader could fill it in. It is not a gap, but it is slightly compressed.If wrong: The constant in the O_L(1) term might need more explicit justification, but the asymptotic form is standard and would not affect the main argument, which relies only on the leading logarithm term.
- low
Lemma 2.1 (Periodic recovery): estimate sum_{m<x, m≡a mod L} 1/m = (1/L) log x + O_L(1) — Argument is sketched (“compare decreasing function with its integral”) rather than fully detailed; also implicitly uses that restricting to residue classes gives logarithmic main term with uniform O(1) error depending on L.If wrong: Epoch-choice step (eq. (4.1) / (epoch-choice)) would lose justification that finite-past survivor has a limiting logarithmic density 1-μ(V_{t-1}); without that, the construction may not guarantee the high limsup along x_t.
- low
Lemma 3.3 (Endpoint abundance): bounded-differences/McDiarmid constants and sum of squares ≤ k/4^k — The Lipschitz bound for Z=|E|/|J| under flipping one ε_i(b) is derived with a quick count (≤|J|/p_i+1 and then ≤2|J|/p_i). This is plausible but somewhat compressed, relying on |J|≥p_i for large j and ignoring edge effects carefully.If wrong: Would weaken the probability-of-success for producing a deterministic labeling with |E_j| large. If it failed substantially, Lemma 3.5 might not guarantee existence of a block with |E_j|≥3Q/8, which is needed for the deletion lower bound and hence the liminf estimate.
- low
Lemma 3.4 (Small periodic footprint): equivalence (3.6) ω∈U ⇔ ∃S with m_S(ω)∈J and K(m_S(ω))=S — Relies on the claim that each residue class mod q_S has at most one representative in J because diam(J)<q_S and J⊂(q_S,2q_S]. The logic is sound but compact; a careful reader may want a fully explicit bijection between residue a_S(ω) and the unique candidate integer m_S(ω).If wrong: If multiple representatives per residue were possible, the union bound over candidates could undercount and μ(U) might be larger, potentially breaking the summability of η_j and thus the recovery (limsup) side.
+ Clear separation between local deletion mass and global periodic footprint, with explicit quantitative mechanisms: Lemma 3.5 gives |E_j|≥3Q/8 while μ(U_j)≤η_j and Ση_j<∞, enabling the epoch recovery/deletion dichotomy.+ Correct use of profinite/Haar framework to turn periodic cylinder unions into density statements, and a logically sound finite-to-infinite passage via Lemma 2.1 and the x_t cutoff where future moduli are inactive.+ The small-footprint lemma (Lemma 3.4) addresses dependence issues nontrivially via collision sets C_{S,i} and conditioning on anchor bits, rather than assuming independence where it does not hold.
- Lemma 3.4’s candidate counting bound (eqs. (3.11)–(3.12)) depends on numerical entropy constants and continuity arguments that are stated tersely; a small mistake could weaken the exponent and endanger the claimed e^{-Ω(k)} footprint bound.- The McDiarmid application in Lemma 3.3 uses a quick Lipschitz estimate (≤2/p_i) that relies on |J|≥p_i for large j and ignores boundary/off-by-one effects; likely fixable but worth formalizing.- Lemma 2.1 packages several standard facts (trace density equals Haar measure; harmonic sums over progressions) with minimal proof; correct in standard settings, but this lemma is load-bearing for epoch selection and should ideally be fully referenced or expanded.
sourcesclaude-sonnet-4-6
Completeness 4/5Evidence 3/5
This paper presents a well-structured and internally complete proof that the logarithmic density of the survivor set in Erdős Problem 486 need not exist. The four main lemmas are proved in full, the global construction is carefully assembled with explicit verification of all activation and separation conditions, and the final numerical bounds are derived with explicit arithmetic. The argument is followable end-to-end, and the paper honestly identifies the boundary of what it proves versus what remains open (the singleton-residue Erdős Problem 25). The completeness score of 4 reflects one structural gap: the formal definition of B (referenced as 'equation defB') does not appear in the submitted text, suggesting the introductory material was redacted or truncated; and two citations are flagged as potentially fabricated (Davenport–Erdős 1951 DOI and the Hoeffding reference identifier), which is a scholarly-integrity concern even though neither is load-bearing for the main mathematical argument. A handful of numerical claims are asserted without inline verification (e.g., the first-seven-terms bound on e^{9/5}), which is acceptable practice in research papers but slightly below the standard expected for a claimed solution to an open problem submitted for audit. Overall, the mathematical completeness of the core argument is strong, with only minor gaps in framing and citation hygiene.
+ The argument is internally complete: all four supporting lemmas are proved from first principles, and the global assembly in Section 4 carefully verifies every activation condition and boundary case, including explicit checks on modulus ranges, epoch separation, and the strict inequality q_{j,m} < m.+ The paper clearly delineates what it does and does not prove — explicitly noting that the multi-residue feature is essential and that the argument does not extend to the singleton-residue case (Erdős Problem 25), which demonstrates honest self-assessment of scope.+ The numerical constants in the final inequalities (177/200 and 49/50) are derived with explicit arithmetic justifications (e.g., the estimate log 2 < 3/4, the computation 15/152 log 2 > 1/8), making the bounds checkable.
- Two citations are flagged as potentially fabricated by the automated reference check: the Davenport–Erdős 1951 Journal of Indian Mathematical Society paper (DOI 10.18311/JIMS/1951/17063 not found) and the Hoeffding 1963 paper (arXiv ID 1459.1963 not found — though this is a 1963 paper that would not be on arXiv, the identifier may have been misassigned by the automated checker). The Hoeffding citation is used for McDiarmid's inequality proof and is a well-known result, but the fabricated or incorrect DOI is a scholarly-integrity concern. The Davenport–Erdős 1951 paper is cited for an elementary proof of the Davenport–Erdős theorem, which is a supporting historical reference rather than a load-bearing step in the main argument.- The submitted text appears truncated or redacted at the beginning (starting mid-sentence and referencing 'equation (defB)' without that equation appearing in the submitted content). The formal definition of B = {m : m mod n ∉ X_n for every n ∈ A with n < m} is stated only in the abstract/summary and not as a numbered equation in the body. This is a minor completeness gap in the framing.- The proof of Lemma 1 (Periodic Recovery) relies on the estimate that each residue class a modulo L contributes (1/L) log x + O_L(1) to the harmonic sum, citing 'comparison with a decreasing function and its integral.' This step is standard but the O_L(1) dependence — specifically that the implicit constant depends only on L and not on a — is asserted without verification. For a paper claiming rigor at the level of an open problem solution, this step merits a one-line justification.- The Araujo 2026 preprint (arXiv:2602.24031) is listed as a 2026 reference and flagged as unverified. Given the submission context mentions AI-generated proofs and a 2026 date, this reference should be independently confirmed to exist, as it is cited for Theorem 3.25 which the paper uses to contextualize its result within the summable-regime positive result.- The entropy estimate in equation (entropy-bound) is stated and proved inline, but the numerical verification that (3/5)H(1/6) < 0.28 uses the bound log 6 < 9/5, justified by claiming 'the first seven terms of the positive power series for e^{9/5} have sum greater than 6.' This specific numerical claim is not verified in the text and, while likely correct, represents a step where a reader must trust the author's arithmetic without being able to check it inline.
sourcesgpt-5.4-2026-03-05
Completeness 4/5
This submission is substantially complete with respect to its own stated objective: it aims to construct a residue-class sieve whose survivor set lacks logarithmic density, and it does present a full proof architecture for that claim. The paper develops the finite deletion block, quantifies both local deletion and small periodic footprint, and then uses epoch separation to force divergent limsup and liminf logarithmic averages. It also addresses a definitional edge case about activation convention and identifies the regime in which its construction operates.
The main completeness deductions come from presentation and documentation rather than from an absent core argument. There are visible proofreading/cross-reference issues, and the automated verification report identifies two problematic citations that should be corrected before the work could be considered cleanly supported as a formal mathematical paper. So the work reads as largely complete in argumentative structure, but not fully polished or fully reliable in its scholarly apparatus.
+ The paper directly addresses its stated goal and carries the reader from a local construction to a global nonexistence result without leaving the main proof strategy implicit.+ Core objects and parameters are mostly defined before use, and the paper explicitly handles the strict-versus-inclusive activation convention as a boundary case.+ It includes scope limitations and contextual positioning, especially the remark explaining that the construction lies outside the previously known summable regime and does not solve the singleton special case.
- The reference verification report flags two likely fabricated citations: the DOI for the 1951 Davenport-Erdős paper and the malformed/fabricated Hoeffding reference ('1459.1963'). Even if the underlying results are standard, this is a serious scholarly completeness issue.- There is an apparent numerical/typographical error in the proof of the arithmetic skeleton lemma: the line 'q_S \le 21Q/20 < 11Q/10' is false as written and should be corrected for the interval argument to read cleanly.- A cross-reference to the formal definition of the final survivor set B appears broken or missing ('defined in \eqref{eq:defB}'), which detracts from document completeness even though B is described elsewhere.- Some probabilistic and entropy estimates are densely compressed; while not absent, they would benefit from a bit more explicit bookkeeping to reduce reliance on the reader reconstructing minor details.
sourcesdeepseek-ai/DeepSeek-V4-Pro
Completeness 2/5
This paper presents an ambitious construction aimed at resolving Erdős Problem 486, with a clear structure and mostly well-defined variables. However, the core argument suffers from significant gaps: the global assembly of the finite probabilistic blocks into an infinite system lacks rigorous justification, and the critical steps ensuring that different epochs do not interfere are only sketched. The probabilistic method is used to produce a single block, but the proof that such blocks can be concatenated while preserving the required density bounds is incomplete. Furthermore, the reference list contains two fabricated citations, which, combined with the logical gaps, severely undermine the work's completeness. While the paper addresses its stated goals in outline, the derivation is not sufficiently rigorous to support the claimed result.
+ All core variables are well-defined before use.+ The overall two-part structure (finite block lemma + global assembly) is logically coherent and clearly motivated.+ The paper explicitly addresses the difference between strict and inclusive activation conventions and justifies equivalence for logarithmic density.
- The verification of the recovery and deletion cutoffs in Section 4 is incomplete; the argument that future moduli do not affect earlier averages relies on a insufficiently justified scale-separation claim.- The transition from the probabilistic existence of a single finite block (Lemma 3.4) to a deterministic global construction with infinitely many installments is not rigorous; the independence of blocks across different scales is assumed without proof.- Two references are likely fabricated (Davenport-Erdos 1951, doi:10.18311/JIMS/1951/17063; and arXiv:1459.1963), which damages the paper's credibility and suggests possible AI hallucination in citation generation.- The derivation of the liminf bound uses an imprecise estimate for the harmonic sum (H_{y_t-1}/log y_t ≤ 1 + 1/log y_t) without full justification, and the subsequent numerical inequality is not explained step-by-step.
sciencegpt-5.4-2026-03-05
Clarity 4/5Novelty 5/5Verifiability 4/5
This submission is best evaluated as pure mathematics, not as a physical theory or method. On scientific merit in the sense relevant here, it is a bold and potentially important theorem-level claim: a negative answer to Erdős Problem 486 obtained through a probabilistic gliding-hump construction that is described with explicit constants and a coherent two-part architecture. The work's strongest feature is conceptual originality combined with a communicative effort to separate the local and global effects that drive the claimed oscillation in logarithmic averages.
On clarity and verifiability, the paper performs well for a specialist audience. The organizational scaffolding is good, the theorem claim is precise, and the proof strategy is visible from the introduction onward. It is not maximally transparent, however, because the argument is technically dense and would require substantial line-by-line checking before one could be fully confident in the result. Still, as a communication artifact, it succeeds in presenting a specific, auditable mathematical claim with enough structure that an expert panel can test it rigorously.
+ The paper has a strong high-level narrative: local deletion blocks, small periodic footprint, then global epoch assembly to separate liminf and limsup.+ It states explicit quantitative outcomes and constants rather than making a vague nonexistence claim.+ The author situates the work well relative to prior positive results and clarifies the role of the summability regime and the singleton special case.
- The exposition is tailored to experts in probabilistic/analytic number theory; readers outside that niche may find the construction hard to visualize without a toy example or schematic.- Because the result is very strong and addresses an open Erdős problem, confidence depends heavily on detailed verification of many coupled estimates; the communication is clear, but not redundant enough to make auditing easy.- The paper is framed partly as an audit reference for an AI-generated proof claim; that context may distract from the standalone mathematical contribution unless the relationship between this text and the original AI derivation is clarified more explicitly.
scienceclaude-opus-4-8
Clarity 4/5Novelty 4/5Verifiability 4/5
This is a theorem-level analytic/probabilistic number theory paper proposing a negative answer to Erdős Problem 486 through an explicit 'gliding hump' construction. Classified as pure mathematics, it was scored on verifiability rather than empirical falsifiability. The central claim — that the survivor set B can have liminf L_B ≤ 177/200 strictly below limsup L_B ≥ 49/50, hence no logarithmic density — is delivered by the body through a well-structured argument: an arithmetic-skeleton lemma building distinct primes and moduli via Bertrand's postulate and CRT, endpoint-abundance and small-footprint lemmas via McDiarmid/Hoeffding and entropy bounds, and a global epoch assembly with recovery and deletion cutoffs. Notation is consistent, conventions (strict vs inclusive activation) are explicitly handled, and the result is correctly positioned against prior work.
The work scores well on all three dimensions: it is genuinely novel in its synthesis (separating local deletion mass from global periodic footprint measure), it is largely checkable by recomputation, and it is clearly organized for a specialist reader. My focus (merit and communication) surfaces no red flags: no term/symbol redefinition, no abstract overclaim, and no empirical prediction issues. Residual concerns are the reliance on several finely-tuned numerical inequalities that merit independent recomputation, the density of the footprint-lemma independence argument, and minor presentation artifacts. Full mathematical completeness verification is deferred to the rigor specialists; from a scientific-merit standpoint the paper is original, verifiable, and clearly communicated.
+ Introduces a clean conceptual separation between local harmonic deletion mass and the small global Haar measure of the responsible residue classes, providing a coherent negative-answer strategy for Erdős Problem 486.+ Claims are concrete and recomputable: explicit constants, standard concentration tools (McDiarmid, Hoeffding), CRT-based modulus construction, and precisely stated cutoff inequalities allow independent verification.+ Careful scholarly framing — distinguishes strict vs inclusive activation conventions, situates the result against Davenport–Erdős/Besicovitch/Behrend/Araújo, verifies the non-summability that keeps it clear of known positive results, and explicitly disclaims solving Problem 25.
- Several key numerical margins are asserted via tightly hand-computed inequalities (e.g., 3/5·H(1/6)<0.28, 0.49·log2>0.33, log6<9/5); while checkable, small slack errors here could threaten the central bounds and warrant careful independent recomputation.- The footprint lemma's conditional independence argument (anchor bits, collision sets C_{S,i}, and the injective candidate-counting) is dense and could be misread; a reader must reconstruct why the conditioned queried variables remain unbiased and independent.- Presentation artifacts (redacted authorship line, truncated opening fragment) reduce polish and momentarily impede readability, though they do not affect the mathematics.- As a pure-mathematics result there is no empirical dimension; verification depends entirely on tracing the proof, and no supplementary numerical experiment corroborating the block construction is provided.
