{"id":"4e57a3df-ae50-4081-956c-2f3311896a89","arxiv_id":"2606.30188","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves that both the set of integers of form p + F_{2^k} + F_q (p,q prime) and its complement have positive lower asymptotic density.","lead":"The paper proves that integers of the form prime plus Fibonacci number at power-of-two index plus Fibonacci at prime index have positive lower asymptotic density, and so does the complement set. A generalist might read it to understand how classical density results for primes plus powers of two extend to Fibonacci sequences.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Adaptation of Romanoff-type density arguments may fail due to modular restrictions on F_{2^k} + F_q arising from the restricted indices.","rationale":"The reader's weakest_assumption directly identifies the same potential zero-density obstruction tied to the choice of indices; the load-bearing risk is therefore unchanged from the provisional verdict.","tokens_in":1578,"tokens_out":398,"duration_ms":61129,"concrete_test":"Fix M = 2·3·5·7 = 210. Compute the set of all attainable values of F_{2^k} + F_q mod M for k ≤ 10 and all primes q ≤ 30; determine whether this set is a proper subset of Z/MZ. If it misses a residue class r, check whether the arithmetic progression r + Mℤ can be shown to contain infinitely many non-representable integers (or forces the Romanoff lower-density constant to vanish).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that the set A = {F_{2^k} + F_q : k ≥ 0, q prime} is sufficiently well-distributed to yield positive lower density via a Romanoff-style sieve estimate on the union of (primes + a) for a ∈ A, while simultaneously permitting a positive-density complement (via an Erdős-style arithmetic progression or density bound). Because gcd(F_m, F_n) = F_{gcd(m,n)} and the Fibonacci sequence is periodic modulo any m (Pisano period), the subsequence at indices 2^k and at prime indices q may lie in a proper subset of residues modulo small m. If this causes A to miss residue classes modulo the primorial, the sieve upper bound on the uncovered density could reach 1 (zero lower density for the representable set) or the complement construction could fail. The abstract gives no indication that these index-specific obstructions have been ruled out.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper proves that the set of positive integers representable in the form p + F_{2^k} + F_q (p, q prime, k ≥ 0) has positive lower asymptotic density, and that its complement likewise has positive lower asymptotic density, by adapting Romanoff-type sieve arguments for the representable set and an Erdős-style arithmetic-progression construction for the complement.","tokens_in":1763,"tokens_out":371,"duration_ms":28650,"significance":"If correct, the result extends the classical Romanoff–Erdős dichotomy to a sumset involving two restricted subsequences of the Fibonacci numbers. It would show that the indices 2^k and prime q are flexible enough, despite the gcd(F_m, F_n) = F_{gcd(m,n)} relation and Pisano periodicity, to produce a set A whose translates by primes cover a positive-density subset while still leaving a positive-density uncovered set.","major_comments":[{"comment":"The central claim rests on showing that A = {F_{2^k} + F_q} is sufficiently dense in residue classes modulo the primorial to permit a positive lower bound via the Romanoff sieve; the manuscript must explicitly rule out the possibility that the restricted indices force A into a proper subset of residues modulo small m (via the Pisano period or gcd properties), as this would collapse the lower density to zero. No such verification is visible in the abstract or the sketched argument.","section":"Main theorem and § on the sieve estimate"}],"minor_comments":[{"comment":"Notation for the Fibonacci sequence and the range of k should be stated once at the beginning rather than repeated.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and for identifying the need for an explicit verification in the sieve argument. The concern is well-taken and will be addressed by adding the required check to the manuscript.","responses":[{"response":"We agree that the manuscript must contain an explicit verification that the restricted indices do not confine A to a proper subset of residue classes modulo small primorials. In the revised version we will insert a new lemma (placed immediately before the application of the Romanoff sieve) that uses the Pisano period of the Fibonacci sequence modulo m together with the fact that the set of primes q is positive-density in the arithmetic progressions compatible with the period. This lemma will show that F_{2^k} + F_q occupies a positive proportion of the residue classes modulo the primorial that are coprime to the small primes appearing in the sieve, thereby ensuring the lower-density bound remains positive. The argument will be self-contained and will not rely on the abstract.","revision_made":"yes","referee_comment":"[Main theorem and § on the sieve estimate] The central claim rests on showing that A = {F_{2^k} + F_q} is sufficiently dense in residue classes modulo the primorial to permit a positive lower bound via the Romanoff sieve; the manuscript must explicitly rule out the possibility that the restricted indices force A into a proper subset of residues modulo small m (via the Pisano period or gcd properties), as this would collapse the lower density to zero. No such verification is visible in the abstract or the sketched argument."}],"tokens_in":1215,"tokens_out":342,"duration_ms":18127,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main claim is that the integers of the form p + F_{2^k} + F_q have positive lower asymptotic density, and so do the integers outside that set. This is a direct swap of the power-of-two term in Romanoff's 1934 theorem for a sum of two Fibonacci numbers, one at a power-of-two index and one at a prime index.\n\nThe new element is the choice of those specific indices rather than arbitrary or power-of-two indices alone. The paper sets up the problem cleanly by recalling the classic positive-density result and the Erdős arithmetic-progression complement, then states the analogous statement for the Fibonacci case.\n\nThe technique appears to be an adaptation of the usual sieve or density estimates used for prime-plus-power-of-two sums. That part is straightforward in outline.\n\nThe soft spot is the distribution of A = {F_{2^k} + F_q}. Fibonacci numbers obey gcd(F_m, F_n) = F_gcd(m,n) and are periodic modulo any fixed m. Restricting the indices to powers of two and to primes can force the possible residues of those sums into a proper subset of the residues modulo small primes or primorials. If A misses too many classes, the lower-density sieve bound can fail or the construction of a positive-density complement can break. The abstract gives no indication that this modular obstruction was checked or bypassed.\n\nIf the full argument contains explicit estimates showing that A still hits enough residue classes, the result is fine. Otherwise the claim rests on an unverified adaptation.\n\nThis is for people who work on additive bases with restricted sequences such as Fibonacci numbers. A reader already familiar with Romanoff-type arguments would see the value in the index choice and could check the details.\n\nIt deserves a serious referee to verify the sieve estimates and the handling of the modular constraints.","headline":"Gao gives a Fibonacci-index variant of Romanoff's density result, but the restricted indices raise real questions about whether the sums stay dense enough modulo small primes.","tokens_in":2194,"tokens_out":455,"would_cite":false,"duration_ms":29486,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The set of integers of the form p + F_{2^k} + F_q has positive lower asymptotic density, as does its complement.","keywords":["Fibonacci numbers","primes","asymptotic density","Romanoff theorem","additive number theory","powers of two"],"falsifier":"The discovery of a long arithmetic progression consisting entirely of integers that cannot be expressed as p + F_{2^k} + F_q would falsify the positive density for the representable set.","tokens_in":2478,"feed_emoji":"","tokens_out":531,"duration_ms":40461,"temperature":0.7,"pith_summary":"The paper establishes that positive integers of the form prime plus a Fibonacci number with power-of-two index plus another prime possess positive lower asymptotic density. It further establishes that the integers not of this form also possess positive lower asymptotic density. This extends the classical result of Romanoff on primes plus powers of two by incorporating Fibonacci numbers at specific indices. A sympathetic reader cares because it indicates that the selected Fibonacci numbers do not disrupt the density properties when added to prime sums.","feed_headline":"Both representable and non-representable numbers have positive density","feed_subtitle":"Integers of the form prime plus F with power-of-two index plus prime, and their complements, both have positive lower asymptotic density.","key_machinery":"Romanoff-type density arguments adapted to the Fibonacci sequence with indices restricted to powers of two.","core_discovery":"The set of positive integers of the form p + F_{2^k} + F_q, where p and q are primes and k ≥ 0, has positive lower asymptotic density. The same holds for the set of integers not of this form.","pith_inferences":["Similar results may hold if the power-of-two restriction is relaxed to other sparse sets of indices.","The approach could apply to other binary recurrences like Lucas sequences.","This suggests the possibility of density results for sums involving multiple Fibonacci numbers."],"forward_implications":["The representable set has positive lower asymptotic density.","The non-representable set has positive lower asymptotic density.","Both sets are therefore infinite.","The Fibonacci terms at the chosen indices do not force zero density in either set."],"fun_headline_variants":["Both prime plus Fib sums and complements have positive density","Prime-Fibonacci sums and non-representables both have positive density","Prime plus two Fibonaccis and complements share positive density","Both Fibonacci prime sum sets and non-representable sets are dense"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The prime distribution and Fibonacci properties allow the density arguments to succeed without the specific indices introducing zero-density sets of exceptions.","fun_headline_variants_meta":{"raw":{"variants":["Both prime plus Fib sums and complements have positive density","Prime-Fibonacci sums and non-representables both have positive density","Prime plus two Fibonaccis and complements share positive density","Both Fibonacci prime sum sets and non-representable sets are dense"]},"model":"grok-4.3","cost_usd":0.007334,"raw_usage":{"total_tokens":3304,"prompt_tokens":526,"num_sources_used":0,"completion_tokens":70,"cost_in_usd_ticks":73337000,"prompt_tokens_details":{"text_tokens":526,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2708,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":526,"tokens_out":70,"duration_ms":32100,"temperature":1.0,"reasoning_tokens":2708,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T05:16:00.914378+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"The discovery of a long arithmetic progression consisting entirely of integers that cannot be expressed as p + F_{2^k} + F_q would falsify the positive density for the representable set.","supporting_citations":[],"review_version":1}