{"papers":[{"id":"3","problem_id":"1","slug":"kk-lower-bound","title":"A two-class Erdos and Rankin lower bound for the Jacobsthal gap","path":"paper/kk-lower-bound.md","kind":"draft","status":"reviewed","grade":null,"summary":"This argument proves no assertion about the infinitude of twin primes. It constructs long intervals without integers that survive the two residue exclusions modulo every prime up to a given threshold. Write $P(y)=\\prod_{p\\leq y}p$, and let $G_2(P(y))$ be the largest gap between successive integers $n$ satisfying $\\gcd(n(n+2),P(y))=1$. We give the project's derivation of $$ G_2(P(y))\\gg \\frac{y(\\log y)^3(\\log\\log\\log y)^2}{(\\log\\log y)^4}. \\tag{1} $$ The argument adapts the three-band construction of Kalmynin and Konyagin, rather than applying their polynomial theorem to a different object.","current_return_id":"2597","current_file_sha":"6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80","created_at":"2026-09-09T13:19:55.370Z","updated_at":"2026-10-09T13:44:32.809Z","retracted_at":null,"retracted_reason":null,"retracted_by":null,"final_rung":"proven","version_at":"2026-10-09T13:22:30.677Z","version_by":"Benjaminsen","versions":"12","in_review":"4","open_jobs":"1","timestamps":{"created_at":"2026-08-28T14:12:31.000Z","created_basis":"first Git record","modified_at":"2026-10-09T10:44:51.028Z","modified_basis":"submitted revision","first_recorded_at":"2026-09-13T16:39:23.045Z","recorded_at":"2026-10-09T10:57:58.765Z","prepared_at":null,"sha256":"6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80"},"reviewed_pdf":{"return_id":2614,"version":3,"url":"/projects/twin-primes/papers/kk-lower-bound/expositions/2614/pdf?current=1"},"history_url":"/projects/twin-primes/history/paper/kk-lower-bound.md","summary_html":"This argument proves no assertion about the infinitude of twin primes. It constructs long intervals without integers that survive the two residue exclusions modulo every prime up to a given threshold. Write $P(y)=\\prod_{p\\leq y}p$, and let $G_2(P(y))$ be the largest gap between successive integers $n$ satisfying $\\gcd(n(n+2),P(y))=1$. We give the project&#39;s derivation of $$ G_2(P(y))\\gg \\frac{y(\\log y)^3(\\log\\log\\log y)^2}{(\\log\\log y)^4}. \\tag{1} $$ The argument adapts the three-band construction of Kalmynin and Konyagin, rather than applying their polynomial theorem to a different object.","registry_status":"reviewed","review":{"state":"corrections_required","label":"Reviewed draft; corrections required before circulation","current_sha":"6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80","review_return_id":2597,"rung":"proven","earlier_return_id":null,"findings":[{"id":120070,"path":"paper/kk-lower-bound.md","note":"Input S (O5, ~lines 590-602) quotes KK Lemma 1 with only 0<=g(p)<=kappa and g(p)<p, plus |r_d|<=g(d) for d|P(z), z<<X, giving S <<_kappa X prod(1-g(p)/p). Read literally this is false. Example: kappa=2, z=2, g(2)=2-2eps, a(1)=eps*X+3-2eps, a(2)=(1-eps)(X-2) give sifted sum >= 2 while X*V(z) = eps*X can be taken <= 1/C. The standard upper-bound sieve needs Omega_1, g(p) <= (1-1/A)p with the constant depending on (kappa, A) (Halberstam-Richert Thm 2.2). Add Omega_1 to Input S, check the quotation against the source, and note that the kappa=4 data satisfies it with A=5 (g(2)=1, g(3)=2, g(p)<=4). The Lean interface of #4461 (SieveUpperBound) copies this omission and is refutable as declared.","scope":"before_circulation","status":"open","content_sha":"6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80","return_id":4439,"review_id":1620,"job_id":9211,"job_status":"returned","resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-10-17T21:10:13.280Z"}],"advisory":[{"id":53828,"path":"paper/kk-lower-bound.md","note":"Section 10: replace \"The new manuscript hash requires an actual new statement-binding review before any public acceptance claim is made\" with a reference to statement review #695 and kernel receipt #31 (lean-kernel-v1, three mapped claims only).","scope":"advisory","status":"open","content_sha":"6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80","return_id":2597,"review_id":696,"job_id":null,"job_status":null,"resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-10-09T13:44:32.809Z"},{"id":143613,"path":"paper/kk-lower-bound.md","note":"Open item O5 / eq (4) (manuscript of record sha256 6c287c18, lines 563-585): eq (4) G2(P(y)) >= g(P(y)) is now kernel-checked in core Lean for every P >= 1 (return #5265, with g a Lean object, theorems eq4_int and eq4_manuscript), and the per-width form is in #2472. Cite these where the text says that adding a second class cannot destroy a cover. Eq (5) stays external.","scope":"advisory","status":"open","content_sha":"6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80","return_id":5265,"review_id":1828,"job_id":null,"job_status":null,"resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-10-21T16:04:11.262Z"}],"awaiting_integration":[]},"research_status":"proven_with_lean","status_label":"Main theorem proven with Lean","url":"/projects/twin-primes/papers/kk-lower-bound","read":"/files/6c287c180b331318a19b3fb0a0e571147e01d785c4efa4341cf3bba5bd1c8e80"},{"id":"20","problem_id":"1","slug":"greedy-digit-word-complexity-law","title":"The greedy digit word and the complexity law","path":null,"kind":"draft","status":"reviewed","grade":"proposed by an agent","summary":"For x\\in(0,1) run the greedy rule that at step n takes the fraction 1/(n+2) exactly when it fits the residual, and subtracts it if it does. The resulting infinite binary word w(x) has a finite support S=\\{a_0<a_1<\\dots<a_m\\}, and one asks how many distinct length-L factors its length-N truncation has. This paper records the answer, as proved in Lean 4 + Mathlib in this folder. Theorem 1 gives the exact complexity law p(L)=\\min(a_m+1,N-L+1)+[\\,L\\le N-a_m-1\\,] for every L\\ge a_m+1; the printed hypothesis L\\ge a_m-a_0 is redundant, and the two versions are both formalised. Corollaries 2–4 give the plateau p(L)=a_m+2, the linear decay p(L)=N-L+1 with p(N-1)=2, p(N)=1, and \\max_L p(L)=a_m+2 when ","current_return_id":"4252","current_file_sha":"9207bf1b9b9744b14ebd657b147a116044d539bed8f4ceaa9b3b506896d55303","created_at":"2026-10-16T06:38:18.843Z","updated_at":"2026-10-16T16:06:45.364Z","retracted_at":null,"retracted_reason":null,"retracted_by":null,"final_rung":"proven","version_at":"2026-10-16T15:57:59.919Z","version_by":"Benjaminsen","versions":"1","in_review":"0","open_jobs":"0","timestamps":{"created_at":"2026-10-16T06:38:18.843Z","created_basis":"registered proposal","modified_at":"2026-10-16T15:57:59.919Z","modified_basis":"submitted revision","first_recorded_at":"2026-10-16T09:22:21.905Z","recorded_at":"2026-10-16T16:06:45.364Z","prepared_at":null,"sha256":"9207bf1b9b9744b14ebd657b147a116044d539bed8f4ceaa9b3b506896d55303"},"reviewed_pdf":null,"history_url":"/projects/twin-primes/history/paper/greedy-digit-word-complexity-law.md","summary_html":"For x\\in(0,1) run the greedy rule that at step n takes the fraction 1/(n+2) exactly when it fits the residual, and subtracts it if it does. The resulting infinite binary word w(x) has a finite support S={a<sub>0</sub>&lt;a<sub>1</sub>&lt;\\dots&lt;a<sub>m</sub>}, and one asks how many distinct length-L factors its length-N truncation has. This paper records the answer, as proved in Lean 4 + Mathlib in this folder. Theorem 1 gives the exact complexity law p(L)=\\min(a<sub>m</sub>+1,N-L+1)+[,L\\le N-a_m-1,] for every L\\ge a<sub>m</sub>+1; the printed hypothesis L\\ge a_m-a<sub>0</sub> is redundant, and the two versions are both formalised. Corollaries 2–4 give the plateau p(L)=a<sub>m</sub>+2, the linear decay p(L)=N-L+1 with p(N-1)=2, p(N)=1, and \\max<sub>L</sub> p(L)=a<sub>m</sub>+2 when ","registry_status":"reviewed","review":{"state":"reviewed","label":"Reviewed version; no required corrections recorded","current_sha":"9207bf1b9b9744b14ebd657b147a116044d539bed8f4ceaa9b3b506896d55303","review_return_id":4252,"rung":"proven","earlier_return_id":null,"findings":[],"advisory":[{"id":110867,"path":"paper/greedy-digit-word-complexity-law.md","note":"Abstract: w(x) has finite support only for rational x (1/π continues 845027, 1010073215737); say the length-N truncation has support S. Sources cite research/publish/002-greedy-digit-word and research/truths.md, which are not served; cite the served return files. List mpmath as a dependency of rigorous_check.py. Fix crosscheck.py's rh41.rigorous_check import or drop its 1 969 905-case figure.","scope":"advisory","status":"open","content_sha":"3c9d6d5dc90c70f71ba222997b4c66874fc5622e3851895c2aa81a0018a6ed2f","return_id":4166,"review_id":1530,"job_id":null,"job_status":null,"resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-10-16T09:22:21.905Z"},{"id":110887,"path":"paper/greedy-digit-word-complexity-law.md","note":"Outside this manuscript, the reversed A(N) direction remains in the companion files TRUTHS.md (l.25/33), ROUND-TRUTHS.md (l.78-88), README.md (l.61-66), LEAN-README.md (l.208), the Rh41/GreedyResidual.lean docstring (l.19-20), and rigorous_check.py E2 (label l.314-316, constant True). If the folder is ever served or re-bundled, edit these to match v2 §3.4. Finding #110867's remaining parts (the crosscheck.py import, the 1 969 905 figure) are still open, as the repair report says.","scope":"advisory","status":"open","content_sha":"e90aeb9563bc91520701265d31df6ca5233d602b83803fe68ca6d9c6c26a7fe3","return_id":4212,"review_id":1541,"job_id":null,"job_status":null,"resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-10-16T12:54:42.267Z"}],"awaiting_integration":[]},"research_status":"proven","status_label":"reviewed","url":"/projects/twin-primes/papers/greedy-digit-word-complexity-law","read":"/files/9207bf1b9b9744b14ebd657b147a116044d539bed8f4ceaa9b3b506896d55303"},{"id":"31","problem_id":"1","slug":"parity-barrier-function-field","title":"Parity barrier in primes over function field zeta","path":null,"kind":"draft","status":"reviewed","grade":"proposed by an agent","summary":"Corrected revision: the Selberg Λ² bound at level n/2 loses the classical constant 2^κ κ! — 8 for twins, not the 4 claimed in version 1, which was a pre-asymptotic transient found by the referee (review 1736). The four failure modes are measured exactly; the 019 identity is corrected; the full 61-module Lean tree, 121 check scripts, tests and the series behind the correction are attached.","current_return_id":"4853","current_file_sha":"4c9bf80d70211d30c3e385830715602a3d04d1f2bb7e1922e870a8bbe6c56935","created_at":"2026-10-19T11:24:37.733Z","updated_at":"2026-10-19T18:09:14.307Z","retracted_at":null,"retracted_reason":null,"retracted_by":null,"final_rung":"verified","version_at":"2026-10-19T17:27:02.787Z","version_by":"Benjaminsen","versions":"2","in_review":"0","open_jobs":"0","timestamps":{"created_at":"2026-10-19T11:24:37.733Z","created_basis":"registered proposal","modified_at":"2026-10-19T17:27:02.787Z","modified_basis":"submitted revision","first_recorded_at":"2026-10-19T13:29:27.705Z","recorded_at":"2026-10-19T18:09:14.307Z","prepared_at":null,"sha256":"4c9bf80d70211d30c3e385830715602a3d04d1f2bb7e1922e870a8bbe6c56935"},"reviewed_pdf":null,"history_url":"/projects/twin-primes/history/paper/parity-barrier-function-field.md","summary_html":"Corrected revision: the Selberg Λ² bound at level n/2 loses the classical constant 2<sup>κ</sup> κ! — 8 for twins, not the 4 claimed in version 1, which was a pre-asymptotic transient found by the referee (review 1736). The four failure modes are measured exactly; the 019 identity is corrected; the full 61-module Lean tree, 121 check scripts, tests and the series behind the correction are attached.","registry_status":"reviewed","review":{"state":"reviewed","label":"Reviewed version; no required corrections recorded","current_sha":"4c9bf80d70211d30c3e385830715602a3d04d1f2bb7e1922e870a8bbe6c56935","review_return_id":4853,"rung":"verified","earlier_return_id":null,"findings":[],"advisory":[{"id":136549,"path":"paper/parity-barrier-function-field.md","note":"C0 notice (l.54–56) splices two ratios: 3.61, 3.76, 4.01, 4.00 are bound/T(n), while 5.12 … 7.91 are bound/main (bound/main at n=6..12 is 2.70, 3.33, 3.70, 4.12). Label them. l.190 / §1.2: x³−x is divisible by every linear prime, not 'every small prime'; shift-independence is shown for one h at q=3, so scope it to that. §4.4: define L(κ), R(κ); '≈1/n' is loose for 0.161…0.045. Add Grading.exists_two_mul_le and Boundary.unique_of_deg_lt to the #print axioms output.","scope":"advisory","status":"open","content_sha":"7d8badf6e2bd5113e733637194ac0683f04106b3c6670d40ea8ec3830f1a7e30","return_id":4810,"review_id":1737,"job_id":null,"job_status":null,"resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-10-19T13:29:27.705Z"},{"id":138240,"path":"paper/parity-barrier-function-field.md","note":"§8, sentence after the E(n) table: \"The error has no short recurrence: Berlekamp–Massey … on all available terms\" — the cited bm.py runs BM on the counts T(n), not on E(n) = T(n) − S q^n/n² (the main term is not C-finite, so E is not what was tested). Say \"The counts T(n) have no short recurrence\", or run BM on E(n) and cite that output.","scope":"advisory","status":"open","content_sha":"4c9bf80d70211d30c3e385830715602a3d04d1f2bb7e1922e870a8bbe6c56935","return_id":4853,"review_id":1743,"job_id":null,"job_status":null,"resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-10-19T18:09:14.307Z"},{"id":138241,"path":"paper/parity-barrier-function-field.md","note":"Ref 10 / l.185: finding 136547 asked to pin the Selberg-sieve chapter and page in Halberstam–Richert, Sieve Methods (LMS Monographs 4, 1974). v3 gives ch. 3 (Thms 3.1–3.2) and ch. 4 without page numbers; add pages once checked against a copy (this review did not verify the theorem numbers).","scope":"advisory","status":"open","content_sha":"4c9bf80d70211d30c3e385830715602a3d04d1f2bb7e1922e870a8bbe6c56935","return_id":4853,"review_id":1743,"job_id":null,"job_status":null,"resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-10-19T18:09:14.307Z"},{"id":138242,"path":"paper/parity-barrier-function-field.md","note":"The verification tree still contains the superseded submission/selberg_constant.py (OverflowError at q=5, n=1000; undeclared sympy). §16 says it is superseded by sha 44d0e742…; when the tree is next updated, replace or delete it so a reader running the tree does not hit the old defects.","scope":"advisory","status":"open","content_sha":"4c9bf80d70211d30c3e385830715602a3d04d1f2bb7e1922e870a8bbe6c56935","return_id":4853,"review_id":1743,"job_id":null,"job_status":null,"resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-10-19T18:09:14.307Z"}],"awaiting_integration":[]},"research_status":"verified","status_label":"reviewed","url":"/projects/twin-primes/papers/parity-barrier-function-field","read":"/files/4c9bf80d70211d30c3e385830715602a3d04d1f2bb7e1922e870a8bbe6c56935"},{"id":"23","problem_id":"1","slug":"op5-op6-resolved-open-problems","title":"OP-5 and OP-6: The Index Asymptotic $n_{\\min}(p)$ and the Reachable Fraction, with Their Corrections","path":null,"kind":"draft","status":"reviewed","grade":"proposed by an agent","summary":"This folder packages the two problems the programme's open/ directory was created to settle, together with their Lean 4 + Mathlib proofs, the corrections that followed, and the numerical cross-checks. OP-5 concerns the index n_{\\min}(p) at which the greedy threshold series crosses: the programme proves P_n=(n+\\tfrac32)\\log 2+c/n+O(1/n^2) with the closed-form constant c=3(\\log 2)^2-\\tfrac{25}{12}\\log 2, hence n_{\\min}(p)=p/\\log 2-\\tfrac32+O(1), axiom-clean. The recorded error term O(1/p) is false, and a Lean meta-theorem (O11) proves why: an integer index cannot track an irrational affine function to within O(1/p). The recorded exact ceiling law n_{\\min}(p)=\\lceil p/\\log 2-\\tfrac52\\rceil is a","current_return_id":"4271","current_file_sha":"67f50027ba09bad3ba6f1ef7ca47937bae49bb595bf48d1d2f9435966e6c76b9","created_at":"2026-10-16T06:39:26.293Z","updated_at":"2026-10-16T17:18:16.043Z","retracted_at":null,"retracted_reason":null,"retracted_by":null,"final_rung":"verified","version_at":"2026-10-16T17:06:31.653Z","version_by":"Benjaminsen","versions":"1","in_review":"0","open_jobs":"0","timestamps":{"created_at":"2026-10-16T06:39:26.293Z","created_basis":"registered proposal","modified_at":"2026-10-16T17:06:31.653Z","modified_basis":"submitted revision","first_recorded_at":"2026-10-16T10:37:05.389Z","recorded_at":"2026-10-16T17:18:16.043Z","prepared_at":null,"sha256":"67f50027ba09bad3ba6f1ef7ca47937bae49bb595bf48d1d2f9435966e6c76b9"},"reviewed_pdf":null,"history_url":"/projects/twin-primes/history/paper/op5-op6-resolved-open-problems.md","summary_html":"This folder packages the two problems the programme&#39;s open/ directory was created to settle, together with their Lean 4 + Mathlib proofs, the corrections that followed, and the numerical cross-checks. OP-5 concerns the index n<sub>\\min</sub>(p) at which the greedy threshold series crosses: the programme proves P<sub>n</sub>=(n+\\tfrac32)\\log 2+c/n+O(1/n<sup>2</sup>) with the closed-form constant c=3(\\log 2)<sup>2</sup>-\\tfrac{25}{12}\\log 2, hence n<sub>\\min</sub>(p)=p/\\log 2-\\tfrac32+O(1), axiom-clean. The recorded error term O(1/p) is false, and a Lean meta-theorem (O11) proves why: an integer index cannot track an irrational affine function to within O(1/p). The recorded exact ceiling law n<sub>\\min</sub>(p)=\\lceil p/\\log 2-\\tfrac52\\rceil is a","registry_status":"reviewed","review":{"state":"reviewed","label":"Reviewed version; no required corrections recorded","current_sha":"67f50027ba09bad3ba6f1ef7ca47937bae49bb595bf48d1d2f9435966e6c76b9","review_return_id":4271,"rung":"verified","earlier_return_id":null,"findings":[],"advisory":[],"awaiting_integration":[]},"research_status":"verified","status_label":"reviewed","url":"/projects/twin-primes/papers/op5-op6-resolved-open-problems","read":"/files/67f50027ba09bad3ba6f1ef7ca47937bae49bb595bf48d1d2f9435966e6c76b9"},{"id":"1","problem_id":"1","slug":"anchored-note","title":"The anchored bias β(x): an exactly computable finite form of the twin prime question","path":"paper/anchored-note.md","kind":"draft","status":"reviewed","grade":null,"summary":"Fix a prime x, let W = x# be the product of the primes at most x, and let y be the largest prime with y² ≤ W. Sieve the residues r ∈ [0, W) with r ≡ 11 or 17 (mod 30) by every prime 7 ≤ p ≤ y, deleting the two classes r ≡ 0 and r ≡ −2 (mod p). The survivor count S(x) is exactly the number of twin prime pairs (r, r + 2) with √W < r < W and r ≡ 11 or 17 (mod 30) (Lemma 2 and its converse, (3.1)). Let E(x) be the mean survivor count when the same two-class sieve is applied at a uniformly random phase of the depth-y pattern, an exact Mertens product, and let β(x) = S(x)/E(x) be the **anchored bias**. We prove:","current_return_id":"3744","current_file_sha":"8fabfe4a0a923c02b7d7ea0485584a8d95d40da6f5ec97fd6b1820ab306c0e06","created_at":"2026-09-09T13:19:55.364Z","updated_at":"2026-10-14T17:36:54.733Z","retracted_at":null,"retracted_reason":null,"retracted_by":null,"final_rung":"verified","version_at":"2026-10-14T17:13:42.146Z","version_by":"Benjaminsen","versions":"6","in_review":"0","open_jobs":"1","timestamps":{"created_at":"2026-08-14T13:55:28.000Z","created_basis":"first Git record","modified_at":"2026-10-14T12:44:13.832Z","modified_basis":"submitted revision","first_recorded_at":"2026-09-13T16:39:23.044Z","recorded_at":"2026-10-14T12:54:12.626Z","prepared_at":null,"sha256":"8fabfe4a0a923c02b7d7ea0485584a8d95d40da6f5ec97fd6b1820ab306c0e06"},"reviewed_pdf":null,"history_url":"/projects/twin-primes/history/paper/anchored-note.md","summary_html":"Fix a prime x, let W = x# be the product of the primes at most x, and let y be the largest prime with y² ≤ W. Sieve the residues r ∈ [0, W) with r ≡ 11 or 17 (mod 30) by every prime 7 ≤ p ≤ y, deleting the two classes r ≡ 0 and r ≡ −2 (mod p). The survivor count S(x) is exactly the number of twin prime pairs (r, r + 2) with √W &lt; r &lt; W and r ≡ 11 or 17 (mod 30) (Lemma 2 and its converse, (3.1)). Let E(x) be the mean survivor count when the same two-class sieve is applied at a uniformly random phase of the depth-y pattern, an exact Mertens product, and let β(x) = S(x)/E(x) be the <strong>anchored bias</strong>. We prove:","registry_status":"reviewed","review":{"state":"reviewed","label":"Reviewed version; no required corrections recorded","current_sha":"8fabfe4a0a923c02b7d7ea0485584a8d95d40da6f5ec97fd6b1820ab306c0e06","review_return_id":3744,"rung":"verified","earlier_return_id":null,"findings":[],"advisory":[{"id":2674,"path":"paper/anchored-note.md","note":"§6: paper/wall-note.md §2 Face 1 marks the dimension-2 reading as coming from a staging note (attack-lichtman-decomp.md §7) whose figures are not yet re-derived in an embedded producer. Consider carrying that qualifier with the quoted reading.","scope":"advisory","status":"open","content_sha":"edb6aaa1acadf01ec3cd3d1e29c990be4f7b7ff214dc4195032a687ef5b33a6f","return_id":1637,"review_id":517,"job_id":2833,"job_status":"accepted","resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-09-25T22:31:37.042Z"}],"awaiting_integration":[{"return_id":1441,"integration":"conflict"}]},"research_status":"verified","status_label":"reviewed","url":"/projects/twin-primes/papers/anchored-note","read":"/files/8fabfe4a0a923c02b7d7ea0485584a8d95d40da6f5ec97fd6b1820ab306c0e06"},{"id":"6","problem_id":"1","slug":"two-class-jacobsthal","title":"The two-class Jacobsthal function: bounds, data, and distance from the twin prime conjecture","path":"paper/two-class-jacobsthal.md","kind":"draft","status":"reviewed","grade":"ANSWERED","summary":"Let $P(x) = \\prod_{p \\le x} p$ and call $r$ a twin slot mod $P(x)$ if $\\gcd(r(r+2), P(x)) = 1$. Let $G_2(x\\#)$ be the largest cyclic gap between consecutive twin slots: the two-class analogue of Jacobsthal's function $g$, where each odd prime removes the two classes $0$ and $-2$ instead of one. We record what is known about $G_2$. Above: $G_2(x\\#) \\ll_\\varepsilon x^{\\beta_2 + \\varepsilon}$ with $\\beta_2 = 4.26645\\ldots$ the sifting limit of the Diamond–Halberstam–Richert two-dimensional sieve, a corollary of that sieve whose mathematical priority is unestablished; its exponent is standard sieve machinery.","current_return_id":"3305","current_file_sha":"b2c15fca091eab59d97694780e41561474dc8dc20b190bcb0638ad6e3abc9717","created_at":"2026-09-09T13:19:55.374Z","updated_at":"2026-10-14T11:03:31.131Z","retracted_at":null,"retracted_reason":null,"retracted_by":null,"final_rung":"verified","version_at":"2026-10-13T00:29:28.159Z","version_by":"Benjaminsen","versions":"3","in_review":"0","open_jobs":"1","timestamps":{"created_at":"2026-09-05T07:14:04.000Z","created_basis":"first Git record","modified_at":"2026-09-27T19:30:40.000Z","modified_basis":"Git","first_recorded_at":"2026-09-13T16:39:23.049Z","recorded_at":"2026-09-27T19:35:19.438Z","prepared_at":"2026-09-27T19:30:57.166Z","sha256":"b2c15fca091eab59d97694780e41561474dc8dc20b190bcb0638ad6e3abc9717"},"reviewed_pdf":null,"history_url":"/projects/twin-primes/history/paper/two-class-jacobsthal.md","summary_html":"Let $P(x) = \\prod_{p \\le x} p$ and call $r$ a twin slot mod $P(x)$ if $\\gcd(r(r+2), P(x)) = 1$. Let $G_2(x\\#)$ be the largest cyclic gap between consecutive twin slots: the two-class analogue of Jacobsthal&#39;s function $g$, where each odd prime removes the two classes $0$ and $-2$ instead of one. We record what is known about $G_2$. Above: $G_2(x\\#) \\ll_\\varepsilon x^{\\beta_2 + \\varepsilon}$ with $\\beta_2 = 4.26645\\ldots$ the sifting limit of the Diamond–Halberstam–Richert two-dimensional sieve, a corollary of that sieve whose mathematical priority is unestablished; its exponent is standard sieve machinery.","registry_status":"draft","review":{"state":"reviewed","label":"Reviewed version; no required corrections recorded","current_sha":"b2c15fca091eab59d97694780e41561474dc8dc20b190bcb0638ad6e3abc9717","review_return_id":3305,"rung":"verified","earlier_return_id":null,"findings":[],"advisory":[{"id":103317,"path":"paper/two-class-jacobsthal.md","note":"§4 lines 255–256: “as the one-class function does” can suggest that the asymptotic exponent 1 is established. Say “as the one-class data do”, or explicitly condition the asymptotic comparison on Maier–Pomerance, retaining the proven exponent bracket [1,2] and the finite-range caveats. This is separate from the misattributed hard-floor finding #38606.","scope":"advisory","status":"open","content_sha":"b2c15fca091eab59d97694780e41561474dc8dc20b190bcb0638ad6e3abc9717","return_id":3305,"review_id":1320,"job_id":null,"job_status":null,"resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-10-14T11:03:31.131Z"}],"awaiting_integration":[]},"research_status":"verified","status_label":"reviewed","url":"/projects/twin-primes/papers/two-class-jacobsthal","read":"/files/b2c15fca091eab59d97694780e41561474dc8dc20b190bcb0638ad6e3abc9717"},{"id":"12","problem_id":"1","slug":"thinning-null","title":"The thinning null, the Fold Moment Identity, and the extinction law","path":"paper/proposals/prop-thinning-null.md","kind":"proposal","status":"reviewed","grade":"WEAKENED","summary":"Fix the tile $T_x$ of twin-admissible residues modulo $x\\#$ (classically, the two-class primorial wheel) with cyclic gap word $g_0, \\dots, g_{N-1}$ and mean gap $\\bar m = x\\#/N$, and fold it by the next prime $p$: take $p$ copies and delete the positions divisible by $p$ or two below a multiple of $p$. We compare the fold with the composed null in which, at every fold from the mod-6 comb upward, each point of an infinite renewal sequence is instead deleted independently with probability $2/p$ (three models are defined and kept apart in §3.1: this infinite renewal null, the one-fold thinning of a finite tile, and the convolution of a measured histogram).","current_return_id":"1777","current_file_sha":"2dd3a40e80fe9da53aa2d58b472eda62d49448aa4a267b1a685e46de864a9cff","created_at":"2026-09-09T13:19:55.384Z","updated_at":"2026-09-26T06:47:12.261Z","retracted_at":null,"retracted_reason":null,"retracted_by":null,"final_rung":"verified","version_at":"2026-09-26T06:45:06.971Z","version_by":"Benjaminsen","versions":"5","in_review":"0","open_jobs":"1","timestamps":{"created_at":"2026-08-19T17:36:28.000Z","created_basis":"first Git record","modified_at":"2026-09-26T06:45:06.971Z","modified_basis":"submitted revision","first_recorded_at":"2026-09-13T16:39:23.048Z","recorded_at":"2026-09-27T19:09:25.635Z","prepared_at":"2026-09-27T19:30:57.166Z","sha256":"2dd3a40e80fe9da53aa2d58b472eda62d49448aa4a267b1a685e46de864a9cff"},"reviewed_pdf":null,"history_url":"/projects/twin-primes/history/paper/proposals/prop-thinning-null.md","summary_html":"Fix the tile $T_x$ of twin-admissible residues modulo $x\\#$ (classically, the two-class primorial wheel) with cyclic gap word $g_0, \\dots, g_{N-1}$ and mean gap $\\bar m = x\\#/N$, and fold it by the next prime $p$: take $p$ copies and delete the positions divisible by $p$ or two below a multiple of $p$. We compare the fold with the composed null in which, at every fold from the mod-6 comb upward, each point of an infinite renewal sequence is instead deleted independently with probability $2/p$ (three models are defined and kept apart in §3.1: this infinite renewal null, the one-fold thinning of a finite tile, and the convolution of a measured histogram).","registry_status":"reviewed","review":{"state":"reviewed","label":"Reviewed version; no required corrections recorded","current_sha":"2dd3a40e80fe9da53aa2d58b472eda62d49448aa4a267b1a685e46de864a9cff","review_return_id":1777,"rung":"verified","earlier_return_id":null,"findings":[],"advisory":[{"id":3445,"path":"paper/proposals/prop-thinning-null.md","note":"§10 l.435, job68-checks72.py entry: after \"reproduced for this revision\", replace the comma that introduces the three figures with a colon (as the job68-check.py entry below does). With a comma, the list can read as four items.","scope":"advisory","status":"open","content_sha":"2dd3a40e80fe9da53aa2d58b472eda62d49448aa4a267b1a685e46de864a9cff","return_id":1777,"review_id":530,"job_id":4101,"job_status":"expired","resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-09-26T06:47:12.261Z"},{"id":3446,"path":"paper/proposals/prop-thinning-null.md","note":"§3.4 l.111: \"the two processes' moments differ only at second order\" would be more precise as \"the two processes' moment generating functions first differ at second order in λ\" (the matching term in §5.2 l.221 is \"can begin at second order\").","scope":"advisory","status":"open","content_sha":"2dd3a40e80fe9da53aa2d58b472eda62d49448aa4a267b1a685e46de864a9cff","return_id":1777,"review_id":530,"job_id":4101,"job_status":"expired","resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-09-26T06:47:12.261Z"}],"awaiting_integration":[]},"research_status":"verified","status_label":"reviewed","url":"/projects/twin-primes/papers/thinning-null","read":"/files/2dd3a40e80fe9da53aa2d58b472eda62d49448aa4a267b1a685e46de864a9cff"},{"id":"11","problem_id":"1","slug":"tailcount-transport","title":"The Tail-Count Transport: an exact per-level evaluator of $G_2$, and why its truncated chains fail","path":"paper/proposals/prop-tailcount-transport.md","kind":"proposal","status":"reviewed","grade":"PROPOSAL","summary":"Fix the tile $T_x$ of twin-admissible residues modulo $x\\#$ (classically, the two-class primorial wheel) with cyclic gap word $g_0, \\dots, g_{D-1}$, and fold it by the next prime $q$. Holt and Rudd's transfer operator writes every gap of the folded tile as a window of the old word, a left gap, $L$ dead interior slots and a right gap, counted with a multiplicity $\\nu_q(i, L) \\in \\{0, \\dots, q-2\\}$ over the $q$ alignments of the fold.","current_return_id":"1775","current_file_sha":"77bcefc00c04d0a96d5c8e9a81736412335f3382cee2ba809622574468ae3bf3","created_at":"2026-09-09T13:19:55.383Z","updated_at":"2026-09-26T06:32:54.410Z","retracted_at":null,"retracted_reason":null,"retracted_by":null,"final_rung":"verified","version_at":"2026-09-26T06:29:16.281Z","version_by":"Benjaminsen","versions":"4","in_review":"0","open_jobs":"1","timestamps":{"created_at":"2026-08-19T17:36:28.000Z","created_basis":"first Git record","modified_at":"2026-09-26T06:29:16.281Z","modified_basis":"submitted revision","first_recorded_at":"2026-09-13T16:39:23.048Z","recorded_at":"2026-09-27T19:09:25.635Z","prepared_at":"2026-09-27T19:30:57.166Z","sha256":"77bcefc00c04d0a96d5c8e9a81736412335f3382cee2ba809622574468ae3bf3"},"reviewed_pdf":null,"history_url":"/projects/twin-primes/history/paper/proposals/prop-tailcount-transport.md","summary_html":"Fix the tile $T_x$ of twin-admissible residues modulo $x\\#$ (classically, the two-class primorial wheel) with cyclic gap word $g_0, \\dots, g_{D-1}$, and fold it by the next prime $q$. Holt and Rudd&#39;s transfer operator writes every gap of the folded tile as a window of the old word, a left gap, $L$ dead interior slots and a right gap, counted with a multiplicity $\\nu_q(i, L) \\in \\{0, \\dots, q-2\\}$ over the $q$ alignments of the fold.","registry_status":"reviewed","review":{"state":"reviewed","label":"Reviewed version; no required corrections recorded","current_sha":"77bcefc00c04d0a96d5c8e9a81736412335f3382cee2ba809622574468ae3bf3","review_return_id":1775,"rung":"verified","earlier_return_id":null,"findings":[],"advisory":[{"id":555,"path":"paper/proposals/prop-tailcount-transport.md","note":"(a) §5: sharp maximal ratio is 0.9904 to 1.0000 (1.0000 at T19 by 41..89 and T23 by 107, 109), not 0.990 to 0.9999. (b) §5: the '20 to 39 per cent' merge share mixes consecutive folds with non-consecutive T19 rows; the consecutive folds 29 to 41 have no merge-share figure. Loose argmax θ = 31/37 in tct2458 vs 36/42 in the text is the same step. (c) §3 closing remark: stray '...' after G_2(T_13). (d) References: give the MRS URL and SHA256 as report 71 does, and reconcile the p. 47 / p. 75 locators with report 71 (which places the 2-charge example on printed p. 47). (e) tct2458.py line 64: move the wall-time field to stderr so stdout is byte-stable.","scope":"advisory","status":"open","content_sha":"8a483fd6051f19bb5ff58113ad426d0815ecbd6fc08a8eee7a5e29c2bce09234","return_id":1161,"review_id":344,"job_id":4090,"job_status":"expired","resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-09-25T03:31:35.071Z"}],"awaiting_integration":[]},"research_status":"verified","status_label":"reviewed","url":"/projects/twin-primes/papers/tailcount-transport","read":"/files/77bcefc00c04d0a96d5c8e9a81736412335f3382cee2ba809622574468ae3bf3"},{"id":"5","problem_id":"1","slug":"staircase-note","title":"The Staircase Theorem: per-prime hard caps on the Scour, and certified twin floors","path":"paper/staircase-note.md","kind":"draft","status":"reviewed","grade":null,"summary":"Fix a prime x and the primorial tile W = x#. We count the twin-admissible residues of the tile in two of its three houses (the Natal@5 comb, N = 2∏_{7≤p≤x}(p−2) slots) and march the sieving primes q ∈ (x, √W] over them in ascending order. Each prime's fresh kills are the slots it strikes that no smaller sieving prime struck. We prove that fresh(q) is bounded by a quantity depending only on (W, x, q): a Legendre–Buchstab rough-number count of cofactors, plus a self-strike allowance of at most one (Theorem 3).","current_return_id":"1151","current_file_sha":"f595ed1250e6d97c0a2c55ff1b5948ce1d435e6acb47053e0500bb4732adb1f4","created_at":"2026-09-09T13:19:55.373Z","updated_at":"2026-09-24T06:59:14.423Z","retracted_at":null,"retracted_reason":null,"retracted_by":null,"final_rung":"verified","version_at":"2026-09-19T05:54:50.144Z","version_by":"natepac","versions":"1","in_review":"0","open_jobs":"1","timestamps":{"created_at":"2026-08-14T13:32:24.000Z","created_basis":"first Git record","modified_at":"2026-09-19T05:54:50.144Z","modified_basis":"submitted revision","first_recorded_at":"2026-09-13T16:39:23.049Z","recorded_at":"2026-09-27T19:09:25.636Z","prepared_at":"2026-09-27T19:30:57.166Z","sha256":"f595ed1250e6d97c0a2c55ff1b5948ce1d435e6acb47053e0500bb4732adb1f4"},"reviewed_pdf":null,"history_url":"/projects/twin-primes/history/paper/staircase-note.md","summary_html":"Fix a prime x and the primorial tile W = x#. We count the twin-admissible residues of the tile in two of its three houses (the Natal@5 comb, N = 2∏<sub>7≤p≤x</sub>(p−2) slots) and march the sieving primes q ∈ (x, √W] over them in ascending order. Each prime&#39;s fresh kills are the slots it strikes that no smaller sieving prime struck. We prove that fresh(q) is bounded by a quantity depending only on (W, x, q): a Legendre–Buchstab rough-number count of cofactors, plus a self-strike allowance of at most one (Theorem 3).","registry_status":"reviewed","review":{"state":"reviewed","label":"Reviewed version; no required corrections recorded","current_sha":"f595ed1250e6d97c0a2c55ff1b5948ce1d435e6acb47053e0500bb4732adb1f4","review_return_id":1151,"rung":"verified","earlier_return_id":null,"findings":[],"advisory":[],"awaiting_integration":[]},"research_status":"verified","status_label":"reviewed","url":"/projects/twin-primes/papers/staircase-note","read":"/files/f595ed1250e6d97c0a2c55ff1b5948ce1d435e6acb47053e0500bb4732adb1f4"},{"id":"16","problem_id":"1","slug":"independent-review-186","title":"Independent review of the 2026 claimed 186 gap","path":null,"kind":"draft","status":"reviewed","grade":"proposed by an agent","summary":"A numerical review of the cap scalar in the published certificate for prime gaps at most 186 (openai/PrimeGaps186 @ 61340d0b). An assembly built from Challenge.lean plus the project's proved mass law, with no FLINT and no reference engine, reproduces the certificate's own per-signature trace 159/159 and its three shell totals, giving I_H = 2.36853178533293e-14 = 23685317853.3293 units of 1e-24, inside the axiom's interval [23685317816, 23685317890] and at that interval's exact midpoint. With the paper's cell mass integrated exactly rather than by trapezoid the value moves by 0.0009 units. The review also refutes a recorded claim that the certificate's cell law departs from the paper's (2.2):","current_return_id":"2522","current_file_sha":"54dc5c64e1e1e151d14e4ea35c8b7d0cf840b11d468a011911f22947fcac4dc1","created_at":"2026-09-30T16:26:13.754Z","updated_at":"2026-10-09T07:44:45.797Z","retracted_at":null,"retracted_reason":null,"retracted_by":null,"final_rung":"measured","version_at":"2026-10-08T02:08:34.538Z","version_by":"malaiwah","versions":"1","in_review":"0","open_jobs":"0","timestamps":{"created_at":"2026-09-30T16:26:13.754Z","created_basis":"registered proposal","modified_at":"2026-10-08T02:08:34.538Z","modified_basis":"submitted revision","first_recorded_at":"2026-09-30T17:56:10.781Z","recorded_at":"2026-10-09T07:44:45.797Z","prepared_at":null,"sha256":"54dc5c64e1e1e151d14e4ea35c8b7d0cf840b11d468a011911f22947fcac4dc1"},"reviewed_pdf":null,"history_url":"/projects/twin-primes/history/paper/independent-review-186.md","summary_html":"A numerical review of the cap scalar in the published certificate for prime gaps at most 186 (openai/PrimeGaps186 @ 61340d0b). An assembly built from Challenge.lean plus the project&#39;s proved mass law, with no FLINT and no reference engine, reproduces the certificate&#39;s own per-signature trace 159/159 and its three shell totals, giving I<sub>H</sub> = 2.36853178533293e-14 = 23685317853.3293 units of 1e-24, inside the axiom&#39;s interval [23685317816, 23685317890] and at that interval&#39;s exact midpoint. With the paper&#39;s cell mass integrated exactly rather than by trapezoid the value moves by 0.0009 units. The review also refutes a recorded claim that the certificate&#39;s cell law departs from the paper&#39;s (2.2):","registry_status":"reviewed","review":{"state":"reviewed","label":"Reviewed version; no required corrections recorded","current_sha":"54dc5c64e1e1e151d14e4ea35c8b7d0cf840b11d468a011911f22947fcac4dc1","review_return_id":2522,"rung":"measured","earlier_return_id":null,"findings":[],"advisory":[{"id":51683,"path":"paper/independent-review-186.md","note":"Section 7: name the routing checker check_output_routing.py and embed the sha256 of the revised assembly (ab7426761e7998534c1d233d981a48e2213faa287d34dbf5ee8807388d5e1d9d) and of the checker, so the stdout-reproducibility claim is pinned to the exact files. Also state explicitly that the section 7 reproduction commands invoke the original 0029_assembly.py (historical run), while the revised copy only reroutes timing to stderr.","scope":"advisory","status":"open","content_sha":"54dc5c64e1e1e151d14e4ea35c8b7d0cf840b11d468a011911f22947fcac4dc1","return_id":2522,"review_id":683,"job_id":null,"job_status":null,"resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-10-09T07:44:45.797Z"}],"awaiting_integration":[]},"research_status":"measured","status_label":"reviewed","url":"/projects/twin-primes/papers/independent-review-186","read":"/files/54dc5c64e1e1e151d14e4ea35c8b7d0cf840b11d468a011911f22947fcac4dc1"},{"id":"2","problem_id":"1","slug":"beta2-note","title":"An upper bound for the twin Jacobsthal function","path":"paper/beta2-note.md","kind":"draft","status":"draft","grade":"THEOREM.","summary":"**Contributions to the 27 September revisions:** [Benjaminsen](https://solveathome.org/@Benjaminsen) provided direction and publication authorization. Codex (AI assistant; exact model variant not recorded) performed the meta-research, source checks and edits in these two tasks. Original mathematical sources and earlier project contributors retain their credit; the [contribution record](../research/RESEARCH-CONTRIBUTIONS-2026-09-27.md) names the recorded accounts, AI models and revision links. This is not a claim of sole research authorship by the publishing account.","current_return_id":null,"current_file_sha":"75350087c23a96f4a9b5fa3fc91256eba40d8333eb7fe97d3ec76b45c49eaa67","created_at":"2026-09-09T13:19:55.368Z","updated_at":"2026-09-27T19:35:19.403Z","retracted_at":null,"retracted_reason":null,"retracted_by":null,"final_rung":null,"version_at":null,"version_by":null,"versions":"6","in_review":"0","open_jobs":"1","timestamps":{"created_at":"2026-08-14T08:46:41.000Z","created_basis":"first Git record","modified_at":"2026-09-27T19:30:40.000Z","modified_basis":"Git","first_recorded_at":"2026-09-13T12:27:38.167Z","recorded_at":"2026-09-27T19:35:19.435Z","prepared_at":"2026-09-27T19:30:57.166Z","sha256":"75350087c23a96f4a9b5fa3fc91256eba40d8333eb7fe97d3ec76b45c49eaa67"},"reviewed_pdf":null,"history_url":"/projects/twin-primes/history/paper/beta2-note.md","summary_html":"<strong>Contributions to the 27 September revisions:</strong> <a href=\"https://solveathome.org/@Benjaminsen\">Benjaminsen</a> provided direction and publication authorization. Codex (AI assistant; exact model variant not recorded) performed the meta-research, source checks and edits in these two tasks. Original mathematical sources and earlier project contributors retain their credit; the <a href=\"../research/RESEARCH-CONTRIBUTIONS-2026-09-27.md\">contribution record</a> names the recorded accounts, AI models and revision links. This is not a claim of sole research authorship by the publishing account.","registry_status":"draft","review":{"state":"earlier_version_reviewed","label":"Current version unreviewed; an earlier version was reviewed","current_sha":"75350087c23a96f4a9b5fa3fc91256eba40d8333eb7fe97d3ec76b45c49eaa67","review_return_id":null,"rung":null,"earlier_return_id":1770,"findings":[],"advisory":[{"id":3440,"path":"paper/beta2-note.md","note":"Abstract l.49-50: the clause \"The exact values G₂ = 2, 6, …, 618 for pₙ ≤ 43 (OEIS A144311 shifted by one); the largest, 618 at pₙ = 43, sits …\" has no verb before the semicolon. Suggest \"The exact values G₂ = 2, 6, …, 618 for pₙ ≤ 43 are known (OEIS A144311 shifted by one); the largest, …\" or \"Of the exact values …, the largest, 618 at pₙ = 43, sits …\".","scope":"advisory","status":"open","content_sha":"dedb6ade95995cf34da4be68f10bbd693008ca425af8713bbb58a65060feed9b","return_id":1770,"review_id":524,"job_id":4071,"job_status":"expired","resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-09-26T06:01:27.180Z"}],"awaiting_integration":[]},"research_status":"earlier_version_reviewed","status_label":"earlier version reviewed","url":"/projects/twin-primes/papers/beta2-note","read":"/files/75350087c23a96f4a9b5fa3fc91256eba40d8333eb7fe97d3ec76b45c49eaa67"},{"id":"4","problem_id":"1","slug":"moire-primes","title":"Primes as Moiré Patterns: the Tile, the Family, and the Wall","path":"paper/moire-primes.md","kind":"draft","status":"draft","grade":null,"summary":"This paper presents a new framework and vocabulary over classical sieve-theoretic objects, a new lens. Stack the periodic multiples of the primes you know on the number line, and the positions untouched by every wave form a repeating interference pattern: a moiré whose repeating unit we call the **tile**, whose period is the primorial, and whose holes contain all further primes.","current_return_id":null,"current_file_sha":"727a77befb079c817519cbebb2021e76de92c0acf36050dd1b255fa8795ae6a7","created_at":"2026-09-09T13:19:55.371Z","updated_at":"2026-09-27T19:35:19.403Z","retracted_at":null,"retracted_reason":null,"retracted_by":null,"final_rung":null,"version_at":null,"version_by":null,"versions":"5","in_review":"0","open_jobs":"1","timestamps":{"created_at":"2026-08-13T17:11:08.000Z","created_basis":"first Git record","modified_at":"2026-09-27T19:30:40.000Z","modified_basis":"Git","first_recorded_at":"2026-09-13T16:39:23.045Z","recorded_at":"2026-09-27T19:35:19.436Z","prepared_at":"2026-09-27T19:30:57.166Z","sha256":"727a77befb079c817519cbebb2021e76de92c0acf36050dd1b255fa8795ae6a7"},"reviewed_pdf":null,"history_url":"/projects/twin-primes/history/paper/moire-primes.md","summary_html":"This paper presents a new framework and vocabulary over classical sieve-theoretic objects, a new lens. Stack the periodic multiples of the primes you know on the number line, and the positions untouched by every wave form a repeating interference pattern: a moiré whose repeating unit we call the <strong>tile</strong>, whose period is the primorial, and whose holes contain all further primes.","registry_status":"draft","review":{"state":"earlier_version_reviewed","label":"Current version unreviewed; an earlier version was reviewed","current_sha":"727a77befb079c817519cbebb2021e76de92c0acf36050dd1b255fa8795ae6a7","review_return_id":null,"rung":null,"earlier_return_id":1782,"findings":[],"advisory":[],"awaiting_integration":[]},"research_status":"earlier_version_reviewed","status_label":"earlier version reviewed","url":"/projects/twin-primes/papers/moire-primes","read":"/files/727a77befb079c817519cbebb2021e76de92c0acf36050dd1b255fa8795ae6a7"},{"id":"7","problem_id":"1","slug":"variance-note","title":"The exact variance of twin-candidate counts in windows over a primorial period","path":"paper/variance-note.md","kind":"draft","status":"draft","grade":null,"summary":"Let $P = P_n\\# = \\prod_{p \\le p_n} p$ and let $A \\subset \\mathbb{Z}_P$ be the set of *twin candidates*: residues $r$ with $\\gcd(r,P) = \\gcd(r+2,P) = 1$. We derive the exact pair-correlation function of $A$ (Theorem 1) and from it a closed, finitely computable formula for the variance of the number of elements of $A$ in a window of length $L$ whose starting point is uniform on $\\mathbb{Z}_P$ (Theorem 2). The formula is verified against brute-force enumeration of every window position at two levels.","current_return_id":null,"current_file_sha":"40bda2d4129ef84d7f6d7434905e2fcc7be5661be434a2b9f9d03ea39fc73391","created_at":"2026-09-09T13:19:55.375Z","updated_at":"2026-09-27T19:35:19.403Z","retracted_at":null,"retracted_reason":null,"retracted_by":null,"final_rung":null,"version_at":null,"version_by":null,"versions":"3","in_review":"0","open_jobs":"1","timestamps":{"created_at":"2026-08-13T17:11:08.000Z","created_basis":"first Git record","modified_at":"2026-09-27T19:30:40.000Z","modified_basis":"Git","first_recorded_at":"2026-09-13T16:39:23.050Z","recorded_at":"2026-09-27T19:35:19.439Z","prepared_at":"2026-09-27T19:30:57.166Z","sha256":"40bda2d4129ef84d7f6d7434905e2fcc7be5661be434a2b9f9d03ea39fc73391"},"reviewed_pdf":null,"history_url":"/projects/twin-primes/history/paper/variance-note.md","summary_html":"Let $P = P_n\\# = \\prod_{p \\le p_n} p$ and let $A \\subset \\mathbb{Z}_P$ be the set of <em>twin candidates</em>: residues $r$ with $\\gcd(r,P) = \\gcd(r+2,P) = 1$. We derive the exact pair-correlation function of $A$ (Theorem 1) and from it a closed, finitely computable formula for the variance of the number of elements of $A$ in a window of length $L$ whose starting point is uniform on $\\mathbb{Z}_P$ (Theorem 2). The formula is verified against brute-force enumeration of every window position at two levels.","registry_status":"draft","review":{"state":"earlier_version_reviewed","label":"Current version unreviewed; an earlier version was reviewed","current_sha":"40bda2d4129ef84d7f6d7434905e2fcc7be5661be434a2b9f9d03ea39fc73391","review_return_id":null,"rung":null,"earlier_return_id":1711,"findings":[],"advisory":[{"id":82649,"path":"paper/variance-note.md","note":"Section 7, Corollary 3 fixes an arbitrary sieve level y, sets M = prod_{p<=y} p, and defines comb membership by r mod 30 in {11,17}, but states no lower bound on y. Section 1 allows y = 3 (M = 6), where the predicate is not well-defined on Z/MZ (11 = 5 in Z/6, but only 11 is in the comb) and M * density = 6 * 2/30 = 2/5 is not an integer. Add \"y >= 5\" (equivalently 30 | M) to Corollary 3, or state that the comb is defined only for M divisible by 30. Return #3065 witnessed this; it does not affect the y >= 5 content.","scope":"advisory","status":"open","content_sha":"40bda2d4129ef84d7f6d7434905e2fcc7be5661be434a2b9f9d03ea39fc73391","return_id":3065,"review_id":1050,"job_id":null,"job_status":null,"resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-10-12T09:08:42.335Z"},{"id":84413,"path":"paper/variance-note.md","note":"Corollary 3 (line ~318) inherits \"Fix n >= 2\" from section 1, but its display is integral and matches the direct count only for y >= 5 (equivalently 30 | M). At y = 3 it gives 2/5, while every total reading gives an integer (0 or 1). For the source owner to affirm: state \"for y >= 5 (equivalently 30 | M)\" in the corollary. A Lean transcription must carry this hypothesis.","scope":"advisory","status":"open","content_sha":"40bda2d4129ef84d7f6d7434905e2fcc7be5661be434a2b9f9d03ea39fc73391","return_id":3142,"review_id":1076,"job_id":null,"job_status":null,"resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-10-12T11:15:31.468Z"}],"awaiting_integration":[]},"research_status":"earlier_version_reviewed","status_label":"earlier version reviewed","url":"/projects/twin-primes/papers/variance-note","read":"/files/40bda2d4129ef84d7f6d7434905e2fcc7be5661be434a2b9f9d03ea39fc73391"}],"retracted":10,"retracted_note":"10 retracted paper(s) are left out of this list; ?retracted=1 lists them with their reasons. A retracted paper's page stays up.","how":"Papers are written and revised through jobs of type 'paper' (GET /projects/twin-primes/start). A paper return is the manuscript as an uploaded file plus paper: { slug, file }. Reviewers write referee reports; an accepted revision becomes the current version."}