{"id":"b4a05d04-1800-4ef3-94f0-cece654ebb83","arxiv_id":"2607.22347","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For every n, a binomial/gcd quotient R_n and one modular exponentiation produce a polynomial whose nonzero support is exactly the primes in (n,2n]; when n is prime, the least support degree is the next prime.","lead":"This paper constructs, from any positive integer n, a polynomial whose nonzero powers are exactly the primes between n and 2n. For a prime input n, the smallest such power is the next prime—obtained without a prime table or a primality test—an elegant algebraic encoding of prime positions rather than a fast sieve.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the valuation bound in Proposition 2.1 is correct, and the Frobenius support theorem follows cleanly; the central claim holds.","rationale":"The paper's central claim is sound. The proof of Proposition 2.1 is correct: the key valuation inequality follows from Legendre's formula, the fact that each summand is 0 or 1, and the observation that at most one power of ℓ lies in (n,2n]. The completion to v_ℓ(C) ≤ 2v_ℓ(n!) uses v_ℓ(n!) ≥ 1 for n ≥ 2, and the n=1 case is immediate. The Frobenius expansion then works exactly as stated via CRT: in characteristic p, (1+X)^p = 1+X^p, and all terms with exponent ≥ 2p vanish because 2p > 2n. The remaining coefficient R_n/p is nonzero mod p, giving clean monomial support. The order of each coefficient is p. Consequently, the support of f is exactly the set of primes in (n,2n], and the least support exponent is the next prime if n is prime. The reader's weakest_assumption identified the valuation bound; that step is indeed load-bearing but it is correct, so the concern does not land. I therefore see no reason to change the ACCEPT verdict. The proposed concrete test independently verifies the product formula and support theorem for a sample range, providing additional confidence beyond the paper's already consistent examples and code.","tokens_in":10251,"tokens_out":12403,"duration_ms":98286,"concrete_test":"Implement an independent checker: for n = 1..1000, (1) compute C = binom(2n,n), D = gcd(C, (n!)^2), R_n = C/D; (2) compute the product P of primes in (n,2n] by trial division; assert R_n == P; (3) build B_n = (Z/R_n Z)[X]/(X^{2n+1}), compute (1+X)^{R_n} − 1 by binary exponentiation, and assert its support equals the set of primes in (n,2n]. If any assertion fails, the valuation bound or the Frobenius expansion is wrong.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After independent scrutiny, the reader's nominated weakest step — Proposition 2.1's bound v_ℓ(C) ≤ 2v_ℓ(n!) — is valid. For ℓ ≤ n, write a_j = ⌊2n/ℓ^j⌋ − 2⌊n/ℓ^j⌋ ∈ {0,1}. For ℓ^j ≤ n, a_j ≤ ⌊n/ℓ^j⌋, and since ℓ^j > n implies ℓ^{j+1} > 2n, at most one term with n < ℓ^j ≤ 2n can contribute. Hence v_ℓ(C) ≤ v_ℓ(n!) + 1 ≤ 2v_ℓ(n!) for n ≥ 2; n = 1 is separately checked. The general interval theorem (5.1) uses the same mechanism, with the power M in the gcd ensuring full removal of small-prime factors. Given this, R_n is squarefree and equal to the product of primes in (n,2n], the CRT decomposition into F_p-components is exact, and fiberwise Frobenius leaves precisely the monomial X^p with coefficient R_n/p nonzero mod p. All worked examples and the appendix code are consistent with the expansion. I find no gap that would invalidate the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines, for each positive integer n, the modulus R_n = binom(2n,n)/gcd(binom(2n,n),(n!)^2). Proposition 2.1 proves, by Legendre's formula, that R_n is exactly the squarefree product of the primes in (n,2n]. The paper then forms the quotient ring B_n = (Z/R_nZ)[X]/(X^{2n+1}) and the element f_n = (1+X)^{R_n}-1. Theorem 3.2 shows, via the CRT decomposition into F_p-components and Frobenius, that f_n = \\sum_{n<p\\le 2n} (R_n/p) X^p, so the monomial support of f_n is precisely the set of interval primes, and each coefficient has additive order p. Consequences include the recovery of the least prime greater than n (and hence of p_{k+1} when n=p_k), an annihilator filtration of B_n, quotient and Jordan-block interpretations, an extension to intervals (a,b] with b\\le 2a, a layered regrading of discarded Frobenius terms, and a SageMath implementation. The central derivation is elementary and internally consistent.","tokens_in":10609,"tokens_out":8857,"duration_ms":79228,"significance":"If correct, the paper gives a genuinely parameter-free algebraic encoding of interval primes: no prime table, no primality test, and no pre-selected target primes enter the construction. The key novelty is the two-stage procedure: first a gcd quotient removes all small-prime primary factors, leaving the squarefree interval product as a modulus, and then a single Frobenius expansion places each interval prime at its own monomial degree. The paper also provides reproducible SageMath code and worked examples for both the dyadic and general interval cases, which strengthens the verification of the claims. The result is not deep, but it is a clean and explicit manifestation of a classical divisibility fact, and it should be of interest to readers working on prime formulas, polynomial congruences, or finite-ring encodings of arithmetic data.","major_comments":[],"minor_comments":[{"comment":"The denominator in the definition of R_{a,b} is typeset as '(a!)M', which reads as (a!)·M. From the proof and from Appendix B it is clear the intended expression is (a!)^M. Please fix this typographical ambiguity, since the general-interval theorem depends on the exponent M.","section":"§5, Eq. (3)"},{"comment":"In the two-layer expansion, the reassembled coefficients 78 and 55 are displayed without explaining the CRT reductions. A short note that 78 ≡ 1 (mod 11), 78 ≡ 0 (mod 13), 55 ≡ 0 (mod 11), and 55 ≡ 3 (mod 13) would make the example self-contained and easier to follow.","section":"§8.4"},{"comment":"The code is clear, but the function short_interval_modulus forms a! directly via factorial(a), which is less in the spirit of the dyadic routine that avoids constructing n!. Since the general interval section is only secondary, this is not a correctness issue, but a one-sentence remark on the distinction would be useful.","section":"Appendix B"},{"comment":"The identity R_n = \\prod_{n<p\\le 2n} p is called 'unexpected'. It follows quickly from Legendre's formula, so the wording is a slight overstatement; consider 'useful' or 'explicit' instead of 'unexpected'.","section":"§1.1"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is mathematically sound. The central valuation bound (Proposition 2.1) and the Frobenius support theorem (Theorem 3.2) check out, and the worked examples and code are consistent. The main limitation is that the result, while elegant, is a fairly direct repackaging of classical facts; the paper's significance is therefore moderate rather than transformative. The self-citations [13,14] are contextual and do not indicate circularity. The typographical issue in Eq. (3) should be fixed before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a clean little paper. The central claim — that R_n = binom(2n,n)/gcd(binom(2n,n),(n!)^2) equals the product of the primes in (n,2n], and that in B_n the expansion (1+X)^{R_n}-1 = sum_{n<p≤2n} (R_n/p) X^p holds — is correct. I checked the valuation bound in Prop 2.1 and it does what it needs to do: Legendre plus the fact that at most one higher power of ℓ sits in (n,2n] gives v_ℓ(C) ≤ v_ℓ(n!)+1 ≤ 2v_ℓ(n!). So the gcd strips the small-prime factors and leaves exactly the squarefree interval product. The fiberwise Frobenius argument is the right one: in characteristic p, (1+X)^{R_n}-1 collapses to (R_n/p)X^p mod X^{2n+1} because 2p > 2n+1. CRT assembles the global expansion. The worked examples and the Sage code line up.\n\nWhat is genuinely new is the support theorem and the structural consequences: the annihilator filtration, the quotient B/(f), the Jordan-block lengths. The identity R_n = interval product is classical and the paper says so, citing OEIS A261130 and Hardy-Wright. That is honest.\n\nSoft spots are mostly about framing and deployment. The construction is not a practical way to find primes: R_n grows exponentially in n, far larger than the interval product it encodes, and the paper freely admits it is not competitive with sieves (§7.2). That is fine for a structural result, and the comparison is honest. One presentation issue: writing (1+X)^{R_n}-1 in a quotient by X^{2n+1} is easy to misread, since the exponent exceeds the truncation degree; the layered regrading in §6 is a reasonable attempt to clarify but feels slightly beside the point. I also wish the appendix code were a versioned artifact or the proof were formalized in a proof assistant; the math is simple enough that this is not a blocker, but it would raise confidence.\n\nNo circularity, no fitted parameters. The self-citations to Prunescu-Shunia are contextual and not load-bearing.\n\nWho is this for? Number theorists interested in algebraic encodings of primes, and people working on prime formulas. It is a modest but correct contribution with new content beyond the classical product identity. I would send it to a serious referee; the referee should check Prop 2.1 and Theorem 3.2 carefully, but they hold.","headline":"Correct, modest, and honestly positioned algebraic encoding of interval primes; the support-expansion theorem is real and the proof holds; worth referee time.","tokens_in":739,"tokens_out":1924,"would_cite":false,"duration_ms":30317,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11A41","11B65","13E10"],"pacs":[],"model":"deepseek-v4-flash","headline":"From any positive integer n, the paper builds a finite quotient ring and one polynomial remainder whose nonzero monomial degrees are exactly the primes in (n,2n]; if n is prime, the smallest such degree is the next prime.","keywords":["prime intervals","central binomial coefficients","quotient rings","Frobenius endomorphism","next prime","prime gaps","modular exponentiation","Artinian rings"],"falsifier":"For a single n, say n=1000, compute R_n by the paper's formula and compare it with an independent product of the primes in (1000,2000]; if any prime at most 1000 divides R_n, or any interval prime is missing or appears with multiplicity greater than one, the central identity fails. Equivalently, use Legendre's formula to test whether the claimed inequality v_ℓ(binom(2n,n)) ≤ 2v_ℓ(n!) holds for all primes ℓ ≤ n; a single prime violating it would falsify Proposition 2.1.","tokens_in":10164,"feed_emoji":"🔢","tokens_out":7861,"duration_ms":56194,"temperature":0.7,"pith_summary":"Starting from a positive integer n and no prime data, the paper constructs a finite quotient ring B_n = (Z/R_nZ)[X]/(X^{2n+1}) and a distinguished element f_n = (1+X)^{R_n}-1 whose nonzero monomial degrees are exactly the prime numbers in the interval (n,2n]. The modulus R_n is defined from the central binomial coefficient together with a gcd that strips away the full contribution of every prime up to n, leaving the squarefree product of the interval primes. Applying Frobenius in each residue field then makes each interval prime p appear at degree p, with coefficient R_n/p having additive order p. When n itself is prime, the least nonzero degree is the next prime, recovered without the prime index, a prime table, or any primality test. The same mechanism extends to arbitrary intervals (a,b] with b ≤ 2a.","feed_headline":"One modular exponentiation lists every prime between n and 2n","feed_subtitle":"Reading the nonzero coefficients gives every interval prime; starting from a prime n, the least one is the next prime.","key_machinery":"The central object is the dyadic modulus R_n, the squarefree product of all primes in (n,2n], realized as a gcd quotient of the central binomial coefficient. The argument runs through the Chinese remainder decomposition Z/R_nZ ≅ ∏ F_p and applies Frobenius in each residue field: in characteristic p, (1+X)^{R_n} = (1+X^p)^{R_n/p}, and because p > n, every term beyond degree p vanishes modulo X^{2n+1}. The leftover is a single monomial X^p with nonzero coefficient, and that coefficient has additive order p modulo R_n. Equivalent annihilator and quotient formulations recover the same primes as jumps in a filtration, as nilpotency indices, and as Jordan-block lengths.","core_discovery":"The paper proves that for every n ≥ 1 the integer R_n = binom(2n,n)/gcd(binom(2n,n),(n!)^2) equals the squarefree product of the primes strictly between n and 2n. In the quotient ring B_n = (Z/R_nZ)[X]/(X^{2n+1}), the single modular exponentiation f_n = (1+X)^{R_n}-1 expands as Σ_{n<p≤2n} (R_n/p) X^p, so the support of f_n is exactly that prime set. Each coefficient R_n/p has additive order p, giving an independent certificate of the corresponding prime. Consequently the least nonzero support degree is the least prime greater than n; if n=p_k is prime, that degree is p_{k+1}. The paper also derives the same extraction from an annihilator filtration and extends the support theorem to interval","pith_inferences":["Inference: Iterating the least-support extraction starting from n=2 would generate the sequence of primes by repeatedly applying the construction to the previous prime; this is implied by the successor-prime corollary but is not developed as an explicit algorithm.","Inference: Because R_n is exactly the product of the interval primes, computing R_n for arbitrary n and comparing it with an independent sieve table would give a direct empirical check of the valuation bound, since any surviving small prime would make R_n nonsquarefree.","Inference: The condition b ≤ 2a appears to mark the boundary of the method; for longer intervals a prime in (a,b] could appear more than once in b!/a!, so testing whether a modified gcd quotient can handle wider intervals is a natural next step."],"forward_implications":["For every n, reading the support of f_n gives the complete set of primes in (n,2n] without enumerating smaller primes or factoring R_n.","When n is prime, the smallest support degree is the next prime p_{k+1}, so the successor prime follows directly from the preceding prime.","Each nonzero coefficient R_n/p has additive order p, so coefficient arithmetic alone certifies which prime each monomial represents.","The annihilator filtration of f_n records the interval primes as its jumps, and the differences between jump locations are the corresponding prime gaps in reverse order.","The construction extends to intervals (a,b] with b ≤ 2a, replacing R_n by a gcd quotient of b!/a! and recovering the primes in that interval."],"fun_headline_variants":["From one prime, one exponentiation gives the next","All primes in (n,2n] appear as support of a single power","No prime tables: a single modular power lists interval primes","Primes between n and 2n: read from one polynomial remainder","For prime n, the next prime is the least nonzero degree"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The whole construction rests on the inequality that every prime ℓ ≤ n appears in the central binomial coefficient at most twice as often as it appears in n!; if this ever failed, small-prime factors would survive the gcd, R_n would cease to be squarefree, and the monomial support theorem would collapse.","fun_headline_variants_meta":{"raw":{"variants":["From one prime, one exponentiation gives the next","All primes in (n,2n] appear as support of a single power","No prime tables: a single modular power lists interval primes","Primes between n and 2n: read from one polynomial remainder","For prime n, the next prime is the least nonzero degree"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000427,"raw_usage":{"total_tokens":2004,"prompt_tokens":704,"completion_tokens":1300,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":448,"completion_tokens_details":{"reasoning_tokens":1211}},"tokens_in":448,"tokens_out":1300,"duration_ms":9957,"temperature":1.0,"reasoning_tokens":1211,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T05:03:28.967078+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a single n, say n=1000, compute R_n by the paper's formula and compare it with an independent product of the primes in (1000,2000]; if any prime at most 1000 divides R_n, or any interval prime is missing or appears with multiplicity greater than one, the central identity fails. Equivalently, use Legendre's formula to test whether the claimed inequality v_ℓ(binom(2n,n)) ≤ 2v_ℓ(n!) holds for all primes ℓ ≤ n; a single prime violating it would falsify Proposition 2.1.","supporting_citations":[],"review_version":1}