{"id":"f1cf6de3-5112-4bed-af7c-58bf6f878c5f","arxiv_id":"2507.14410","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Replacing the n-th prime by the sum of logarithms of the first n primes makes analogues of Cramer, Andrica, Legendre, Oppermann, Brocard, Firoozbakht, Fourges, Nicholson, and Farhadian conjectures provable theorems.","lead":"This mathematics paper defines a smoothed mirror of the prime numbers, the sequence ϑ_n that sums the logarithms of the first n primes, and shows that several famous prime conjectures become provable theorems when translated to this mirror sequence. The catch, which the paper acknowledges, is that these provable analogues say little about the original unsolved conjectures.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Claimed ϑ-analogue of Nicholson is false as stated: Eq. (5.36) fails at n=3 and n=4; the counting lemma (4.9) used for three other analogues is also false.","rationale":"The reader's REJECT verdict is correct, but the decisive reason is even stronger than the one cited. The reader focused on the false counting bound (4.9), which invalidates the proofs of the Legendre, Oppermann, and Brocard analogues; however, a direct numerical counterexample shows that the stated ϑ-analogue of Nicholson is false as a statement, not merely unproved. I verified the n=3 and n=4 values from the definition of ϑ_n; the claimed inequality (5.36) fails. The derivation in §5.8 also contains an arithmetic error in locating the threshold at n≥4 rather than n≥6. Because one of the nine 'theorems' is false, the central claim of the abstract cannot stand. The counting-bound error is independent and would separately invalidate the proofs of three other analogues even if the Nicholson analogue were repaired. My read therefore leaves the REJECT verdict unchanged; if anything it strengthens the basis for rejection. I credit the paper's correct observations, including the essentially trivial Cramer and Andrica analogues and the plausible Firoozbakht analogue for n≥4, but the presence of an explicit false theorem and a false counting lemma mandates rejection as written.","tokens_in":10683,"tokens_out":12808,"duration_ms":120299,"concrete_test":"Recompute the two sides of Eq. (5.36) at n=3 and n=4 from the definitions ϑ_3=ln30, ϑ_4=ln210, and ϑ_5=ln2310. With natural logarithms, (ϑ_4/ϑ_3)^3 ≈ 3.884 exceeds 3(ln3−1/2) ≈ 1.796, and (ϑ_5/ϑ_4)^4 ≈ 4.400 exceeds 4(ln4−1/2) ≈ 3.545, so the stated ϑ-Nicholson theorem is false exactly where claimed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most decisive objection is that the central claim—that the ϑ-analogues are theorems—is directly contradicted by one of the paper's own stated analogues. In §5.8 the author proposes as the ϑ-analogue of Nicholson the inequality (ϑ_{n+1}/ϑ_n)^n < n[ln n − 1/2] for all n≥3 (Eq. 5.36). This is false. For n=3, using ϑ_3=ln30≈3.401197 and ϑ_4=ln210≈5.347108, the left side is (5.347108/3.401197)^3 ≈ 3.884, while the right side is 3(ln3−1/2) ≈ 1.796; the inequality fails. It also fails at n=4, where the two sides are approximately 4.400 and 3.545. The preceding analytic step contains the same numerical error: the condition ln(n[ln n − 1/2]) > 9/5 holds only for n≥6 (n=5 gives ≈1.713, n=6 gives ≈2.047), not for n≥4 as asserted. Thus the claimed 'explicit checking' cannot make (5.36) true. Independently, the counting lower bound used for the Legendre, Oppermann, and Brocard analogues is also invalid: Eq. (4.9) asserts #{ϑ_i: y<ϑ_i≤x} > (x−y)/max-ϑ-gap(x,y), but for (x,y)=(9,4) the left side is 2 while (9−4)/ln11 ≈2.085. These are separate, concrete failures in the proof of the nine-theorem claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the sequence ϑ_n = ϑ(p_n), the first Chebyshev function evaluated at the nth prime, equivalently the logarithm of the primorial. Using known explicit bounds on primes and on ϑ(x), together with elementary estimates for the gaps g_n = ϑ_{n+1} − ϑ_n = ln p_{n+1}, the author claims to prove ϑ-analogues of the Cramer, Andrica, Legendre, Oppermann, Brocard, Firoozbakht, Fourges, Nicholson, and Farhadian conjectures. The central assertion is that these analogues are theorems rather than conjectures. The paper also correctly notes in §6 that these results do not directly imply the ordinary prime conjectures.","tokens_in":11058,"tokens_out":8624,"duration_ms":89920,"significance":"If the claims were correct, the observation that the log-primorial sequence is substantially smoother than the prime sequence would be mildly interesting and would provide a clean set of unconditional inequalities. The paper makes appropriate use of classical external results and is honest about the lack of implications for the usual prime conjectures. However, the main claim is not supported: at least one of the stated analogues is numerically false, and a key counting lemma used for three other analogues is false. The small-case checks are asserted without presenting any data or code, so the many 'explicitly checking' statements are not independently verifiable from the manuscript. The paper therefore does not establish the claimed set of theorems.","major_comments":[{"comment":"The counting lower bound in Eq. (4.9) is false. For the interval (4,9], there are exactly two ϑ-values, namely ϑ_4 = ln 210 ≈ 5.347 and ϑ_5 = ln 2310 ≈ 7.745. The maximum ϑ-gap completely contained in this interval is ϑ_5 − ϑ_4 = ln 11 ≈ 2.398, so the right-hand side of (4.9) is 5/ln 11 ≈ 2.085, which exceeds 2. The interval counting estimate therefore fails. The derived bounds (4.12) and (4.19) are also false: for (x,y) = (7.7, 3.5), the left side is 1, while (x−y)/(ln x + 1.059660101) ≈ 4.2/(ln 7.7 + 1.059660101) ≈ 1.354 > 1.","section":"§4, Eq. (4.9)"},{"comment":"The asserted ϑ-analogue of Nicholson is false as stated. For n = 3, the left side (ϑ_4/ϑ_3)^3 = (ln 210/ln 30)^3 ≈ 3.884, while the right side 3(ln 3 − 1/2) ≈ 1.796, so the inequality fails. It also fails at n = 4: the left side is approximately 4.400, while the right side is 3.545. The preceding threshold condition ln(n[ln n − 1/2]) > 9/5 is itself incorrect: it holds for n ≥ 6, not for n ≥ 4, since for n = 5 the left side is about 1.713 and for n = 6 about 2.047.","section":"§5.8, Eq. (5.36)"},{"comment":"Because the false counting bound (4.19) is the basis for the lower estimates in the Legendre analogue (5.8), the Oppermann analogue (5.9)–(5.10), and the Brocard analogue (5.14)–(5.15), the proofs of those three results are invalid as written. In addition, the logical step leading to (5.32) in §5.8 is problematic: from ϑ_n > n[ln n − 1/2] one cannot strengthen (5.31) to (5.32), since the new right-hand side is smaller than ϑ_n, not larger; the claimed inequality must be proved directly, and the direct small-case check contradicts it at n = 3 and n = 4.","section":"§5.3–5.5 and §5.8"}],"minor_comments":[{"comment":"Equation (6.1) is misnotated: it reads g_n = p_{n+1} − p_n = exp(g_n) − exp(g_{n−1}), conflating the ϑ-gap g_n = ln p_{n+1} with the ordinary prime gap; it should state p_{n+1} − p_n = exp(g_n) − exp(g_{n−1}).","section":"§6, Eq. (6.1)"},{"comment":"The phrase 'noting that for n ≥ 1 we have ϑ_n > n[ln n − 1/2]' as justification for strengthening the inequality is misleading, since the smaller right-hand side makes the new inequality stricter, not weaker; this is a logical point that should be clarified.","section":"§5.8"},{"comment":"The manuscript repeatedly relies on 'explicitly checking smaller integers' (for example, Eqs. (3.5), (3.10), (3.13), (5.5), (5.42)) but provides no tables, code, or reproducible data; the reader cannot verify these checks, and in at least one case (Eq. (5.36)) the check is actually wrong.","section":"Throughout"},{"comment":"There is a typo in the introduction: 'Fahadian' should be 'Farhadian', and the names 'Cramer' and 'Fourges' are not consistently accented ('Cramér' and 'Fourgès').","section":"§1"}],"recommendation":"reject","confidential_remarks":"The self-citations are numerous but are not the basis for this recommendation. The decisive issues are the false counting lemma in §4 and the explicitly false Nicholson analogue in §5.8, which together invalidate the paper's central claim that the ϑ-analogues are theorems."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two concrete errors sink the abstract's claim that all nine ϑ-analogues are theorems. That's the headline.\n\nThe paper is not without merit. The relative-gap bounds (3.15) — 0.9/n < g_n/ϑ_n < 1.8/n — are new and correct, and they make the Cramer, Andrica, Firoozbakht, and Fourges analogues straightforward and rigorous. The author is also honest in Section 6: these results don't tell us much about ordinary primes. That's a fair admission, and I appreciate it.\n\nBut the errors are load-bearing. First, the ϑ-analogue of Nicholson, Eq. (5.36), is false. The proof needs ln(n[ln n − 1/2]) > 9/5, which holds only for n ≥ 6, not n ≥ 4. The 'explicit checking' of smaller integers fails: at n=3 the LHS is (ln210/ln30)^3 ≈ 3.88, while the RHS is 3(ln3 − 1/2) ≈ 1.80. So (5.36) is not a theorem.\n\nSecond, the counting lemma (4.9) asserts #{ϑ_i : y<ϑ_i≤x} > (x−y)/max-gap. That inequality is false: spacing alone gives you at least floor(L/G) − 1, not L/G. For (4,9] the count is two, but (9−4)/ln7 ≈ 2.57. This lemma feeds the Legendre, Oppermann, and Brocard analogues, so their proofs collapse. A corrected bound would carry an extra −1 and would be too weak for the claimed conclusions; those three analogues may still be provable by other means, but not as written.\n\nThe external references (Rosser–Schoenfeld, Dusart, Platt–Trudgian) are standard and used correctly. The self-citations are not a problem by themselves, though they are frequent. The paper reads as an honest attempt that went off the rails in two spots.\n\nI'd tell a serious referee to check the small-n cases carefully and to ask for a repair of the counting argument. The idea has legs and the gap bounds are worth keeping. As it stands, the central claim is unsupported, so this is not publishable in current form. But it deserves referee time rather than a desk reject: the errors are specific, and the correct parts are clearly useful. My recommendation: send to review, with a note to verify the numerics and the counting lemma.","headline":"Two concrete errors sink the abstract's claim that all nine ϑ-analogues are theorems; the gap bounds and some analogues are correct and worth keeping.","tokens_in":11576,"tokens_out":5144,"would_cite":false,"duration_ms":532645,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11A41","11N05"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that the log-primorial sequence $\\vartheta_n=\\ln([p_n]\\#)$ turns suitably formulated $\\vartheta$-analogues of nine famous prime-gap conjectures into unconditional theorems, with the regularity of the gaps…","keywords":["log-primorial sequence","first Chebyshev function","ϑ-gaps","ϑ-analogues of prime conjectures","Cramér conjecture","Andrica conjecture","Firoozbakht conjecture","prime gaps"],"falsifier":"Compute the $\\vartheta$-count on the interval $(4,9]$: the values $\\vartheta_4 = \\ln 210 \\approx 5.35$ and $\\vartheta_5 = \\ln 2310 \\approx 7.75$ lie inside, so $\\pi_\\vartheta(9)-\\pi_\\vartheta(4)=2$, while the largest contained gap is $\\ln 11 \\approx 2.398$ and $(9-4)/\\ln 11 \\approx 2.085 > 2$, contradicting (4.9). The same check can be repeated on any short interval whose right endpoint is not a $\\vartheta$-value; the inequality fails whenever the largest contained gap exceeds the average spacing.","tokens_in":10479,"feed_emoji":"🔢","tokens_out":11980,"duration_ms":115169,"temperature":0.7,"pith_summary":"This paper claims that replacing the $n$th prime $p_n$ by its cumulative logarithm, the log-primorial $\\vartheta_n = \\vartheta(p_n) = \\sum_{i=1}^n \\ln p_i = \\ln([p_n]\\#)$, turns suitably formulated $\\vartheta$-analogues of nine famous prime-gap conjectures into theorems. Because the gaps are $\\mathfrak{g}_n = \\vartheta_{n+1}-\\vartheta_n = \\ln p_{n+1}$, their relative size $\\mathfrak{g}_n/\\vartheta_n$ is provably of order $1/n$, lying between $9/(10n)$ and $9/(5n)$ for all $n$. The paper proves $\\vartheta$-analogues of the Cramér, Andrica, Legendre, Oppermann, Brocard, Firoozbakht, Fourges, Nicholson, and Farhadian conjectures unconditionally from explicit bounds on primes and on the first Chebyshev function. Since $p_n$ and $\\vartheta_n$ encode the same information, the point is that the log-primorial ordering exhibits, as theorems, the regular behaviour that remains conjectural for the primes themselves.","feed_headline":"Log-primorial turns nine prime conjectures into proven theorems","feed_subtitle":"Log-primorial gaps are so regular that the ϑ-versions of nine prime conjectures are proved outright.","key_machinery":"The mechanism is the $\\vartheta$-gap $\\mathfrak{g}_n = \\vartheta_{n+1}-\\vartheta_n = \\ln p_{n+1}$. Because $\\mathfrak{g}_n$ is the logarithm of the next prime rather than a difference of primes, its size relative to $\\vartheta_n$ is provably tiny: the paper establishes $9/(10n) < \\mathfrak{g}_n/\\vartheta_n < 9/(5n)$ for all $n\\ge 1$ and the equivalent comparison $(22/25)\\ln p_n/p_n < \\mathfrak{g}_n/\\vartheta_n < 2\\ln p_n/p_n$. A secondary mechanism is the counting function $\\pi_\\vartheta(x)$ and the constants $M^*=\\sup_i \\vartheta_i/p_i$ and $m^*=\\inf_i \\vartheta_i/p_i$; these convert the gap bounds into interval-counting lower and upper bounds that power the Legendre, Oppermann, and Brocard analogues.","core_discovery":"The central discovery is that the log-primorial sequence $\\vartheta_n = \\ln([p_n]\\#)$ has provably regular gaps: $\\vartheta_{n+1}-\\vartheta_n = \\ln p_{n+1}$, so the relative gap lies in the narrow band $9/(10n) < \\mathfrak{g}_n/\\vartheta_n < 9/(5n)$, and also $(22/25)\\ln p_n/p_n < \\mathfrak{g}_n/\\vartheta_n < 2\\ln p_n/p_n$. From these gap bounds, the paper derives explicit upper and lower bounds on the counting function $\\pi_\\vartheta(x) = \\#\\{i : \\vartheta_i \\le x\\}$, then uses them to prove $\\vartheta$-analogues: $\\vartheta$-Andrica holds with maximal step $\\sqrt{\\ln 6}-\\sqrt{\\ln 2}\\approx 0.506 < 1$; $\\vartheta$-Firoozbakht holds for $n\\ge 4$; $\\vartheta$-Fourges holds for all $n\\ne 2$; $\\vartheta$-Nicholson holds for $n\\ge 3$; $\\vartheta$-Farhadian holds for $n\\ge 5$; and $\\vartheta$-Legendre, $\\vartheta$-Oppermann, and $\\vartheta$-Brocard follow from the counting bound $\\pi_\\vartheta(x)-\\pi_\\vartheta(y) > (x-y)/(\\ln x + 1.059660101)$. The paper stresses that this does not transfer to the ordinary prime conjectures, since the identity $p_n = \\exp(\\vartheta_n-\\vartheta_{n-1})$ does not naively convert $\\vartheta$-gap theorems into prime-gap theorems.","pith_inferences":["Editorial inference: the failure of (4.9) is not confined to the one example; any interval in which the largest contained $\\vartheta$-gap exceeds the average spacing will falsify the lower bound, so the counting-based proofs of the Legendre, Oppermann, and Brocard analogues need a repaired estimate, for instance subtracting an endpoint gap or bounding the maximum gap by $\\ln x - \\ln y$ rather than","Editorial inference: the qualitative point that log-primorial gaps are smooth is independent of the counting inequality; a corrected counting lemma would likely preserve the spirit of the Legendre, Oppermann, and Brocard analogues even if the numerical ranges change.","Editorial inference: the same 'cumulative logarithm' construction could be applied to other increasing sequences of integers to turn hard gap conjectures into provable statements about the cumulative sums of their logarithms, but the identity $\\mathfrak{g}_n=\\ln p_{n+1}$ is what makes the comparison to the original prime conjectures exact here."],"forward_implications":["The $\\vartheta$-analogue of the Cramér conjecture is a trivial identity, since $\\vartheta_{n+1}-\\vartheta_n=\\ln p_{n+1}$ is far smaller than $(\\ln p_n)^2$; the $\\vartheta$-analogue of Andrica is proved in the strong form $\\sqrt{\\vartheta_{n+1}}-\\sqrt{\\vartheta_n} \\le \\sqrt{\\ln 6}-\\sqrt{\\ln 2} \\approx 0.506 < 1$ for all $n$.","If the counting bound (4.19) is valid, then $\\vartheta$-Legendre holds with many $\\vartheta$-values between consecutive squares, $\\vartheta$-Oppermann holds between consecutive half-squares for $m\\ge 1$ and $m\\ge 2$, and $\\vartheta$-Brocard holds with at least $2\\mathfrak{g}_n$ $\\vartheta$-values between $p_n^2$ and $p_{n+1}^2$ for all $n$.","$\\vartheta$-Firoozbakht holds for $n\\ge 4$, meaning $(\\vartheta_n)^{1/n}$ is eventually decreasing; $\\vartheta$-Fourges holds for all positive integers except $n=2$; $\\vartheta$-Nicholson holds for $n\\ge 3$; $\\vartheta$-Farhadian holds for $n\\ge 5$.","Because $p_n = \\exp(\\vartheta_n-\\vartheta_{n-1})$, the $\\vartheta_n$ sequence and the primes encode the same information, but the paper explicitly shows that the naive exponential conversion does not produce proofs of the ordinary Cramér, Andrica, Legendre, Oppermann, Brocard, Firoozbakht, Fourges, Nicholson, or Farhadian conjectures."],"supporting_citations":[{"why":"Supplies the asymptotic expansion and explicit upper bound for $\\vartheta_n$ used to anchor the later gap estimates.","marker":"[1]"},{"why":"Provides the explicit upper bound on $\\vartheta_n$ that enters the relative-gap upper bounds.","marker":"[2]"},{"why":"Gives the lower bound $p_n > n(\\ln n + \\ln\\ln n - 1)$ used in bounding $\\mathfrak{g}_n/\\vartheta_n$.","marker":"[3]"},{"why":"Supplies the upper bound $p_{n+1} < n[\\ln(n\\ln n)+1]$ used to pass from prime-size bounds to $\\vartheta$-gap bounds.","marker":"[4]"},{"why":"Supplies $p_n > n\\ln n$, a basic input for the lower-side relative-gap comparisons.","marker":"[5]"},{"why":"Gives the asymmetric bounds on $\\vartheta(x)$ near $x$ that justify the constants $M^*$ and $m^*$ and the $\\vartheta_n$ bounds.","marker":"[6]"},{"why":"Proves the bound $M^* < 1+7.5\\times10^{-7}$ used in the counting estimates of Section 4.","marker":"[16]"}],"fun_headline_variants":["Log-primorial gaps prove nine prime-conjecture analogues","Nine prime conjectures proven for log-primorials","Log-primorial gaps turn nine conjectures into theorems","ϑ-analogues of nine prime conjectures proven","Log-primorial gaps make nine conjectures theorems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the interval-counting estimate (4.9), which asserts that the number of $\\vartheta$-values in an interval $(y,x]$ exceeds $(x-y)$ divided by the largest $\\vartheta$-gap contained in that interval; this premise is false, since the interval $(4,9]$ contains two $\\vartheta$-values while $(9-4)/\\ln 11 \\approx 2.085$ already exceeds two.","fun_headline_variants_meta":{"raw":{"variants":["Log-primorial gaps prove nine prime-conjecture analogues","Nine prime conjectures proven for log-primorials","Log-primorial gaps turn nine conjectures into theorems","ϑ-analogues of nine prime conjectures proven","Log-primorial gaps make nine conjectures theorems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000744,"raw_usage":{"total_tokens":3456,"prompt_tokens":1224,"completion_tokens":2232,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":840,"completion_tokens_details":{"reasoning_tokens":2151}},"tokens_in":840,"tokens_out":2232,"duration_ms":20521,"temperature":1.0,"reasoning_tokens":2151,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:58:49.886824+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the $\\vartheta$-count on the interval $(4,9]$: the values $\\vartheta_4 = \\ln 210 \\approx 5.35$ and $\\vartheta_5 = \\ln 2310 \\approx 7.75$ lie inside, so $\\pi_\\vartheta(9)-\\pi_\\vartheta(4)=2$, while the largest contained gap is $\\ln 11 \\approx 2.398$ and $(9-4)/\\ln 11 \\approx 2.085 > 2$, contradicting (4.9). The same check can be repeated on any short interval whose right endpoint is not a $\\vartheta$-value; the inequality fails whenever the largest contained gap exceeds the average spacing.","supporting_citations":[{"cited_title":"Autour de la fonction qui compte le nombre de nombres premiers","cited_arxiv_id":null,"evidence_quote":"Supplies the asymptotic expansion and explicit upper bound for $\\vartheta_n$ used to anchor the later gap estimates."},{"cited_title":"Bornes effectives pour certaines fonctions concernant les nombres premiers","cited_arxiv_id":null,"evidence_quote":"Provides the explicit upper bound on $\\vartheta_n$ that enters the relative-gap upper bounds."},{"cited_title":"The kth prime is greater than k(ln k + ln lnk − 1) for k ≥ 2","cited_arxiv_id":null,"evidence_quote":"Gives the lower bound $p_n > n(\\ln n + \\ln\\ln n - 1)$ used in bounding $\\mathfrak{g}_n/\\vartheta_n$."},{"cited_title":"On the arithmetic average of the first $n$ primes","cited_arxiv_id":"2505.04951","evidence_quote":"Supplies the upper bound $p_{n+1} < n[\\ln(n\\ln n)+1]$ used to pass from prime-size bounds to $\\vartheta$-gap bounds."},{"cited_title":"The n-th Prime is Greater than n log n","cited_arxiv_id":null,"evidence_quote":"Supplies $p_n > n\\ln n$, a basic input for the lower-side relative-gap comparisons."},{"cited_title":"On the first sign change of $\\theta(x) - x$","cited_arxiv_id":"1407.1914","evidence_quote":"Proves the bound $M^* < 1+7.5\\times10^{-7}$ used in the counting estimates of Section 4."}],"review_version":1}