{"paper":{"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":"reviewed","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":"1319","current_file_sha":"b1977bf9f628ea58ab8c00c1cc5f5f435ea25e5e675324037c2431462df26685","created_at":"2026-09-09T13:19:55.375Z","updated_at":"2026-09-25T07:19:41.265Z","final_rung":"verified","version_at":"2026-09-19T18:20:06.559Z","version_by":"natepac","versions":"2","in_review":"0","open_jobs":"1","timestamps":{"created_at":"2026-08-13T17:11:08.000Z","created_basis":"first Git record","modified_at":"2026-09-19T18:20:06.559Z","modified_basis":"submitted revision","first_recorded_at":"2026-09-13T16:39:23.050Z","recorded_at":"2026-09-25T07:19:41.265Z","prepared_at":null,"sha256":"b1977bf9f628ea58ab8c00c1cc5f5f435ea25e5e675324037c2431462df26685"},"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":"reviewed","review":{"state":"corrections_required","label":"Reviewed draft; corrections required before circulation","current_sha":"b1977bf9f628ea58ab8c00c1cc5f5f435ea25e5e675324037c2431462df26685","review_return_id":1319,"rung":"verified","earlier_return_id":null,"findings":[{"id":764,"path":"paper/variance-note.md","note":"Revision #1319 (b1977bf9) l.942 reads \"developed over six years of independent work\"; the served file (publication 1250, a38f32e8, l.875) reads \"ten years\". The patch inherited \"six\" from #15, which was prepared against publication 47, so integrating #1319 silently reverts the served author statement. Restore \"ten years\". Advisory in the same file: in §1, say the map k -> 2k acts mod m = P/2, followed by the odd lift to Z_P (2k itself is even).","scope":"before_circulation","status":"open","content_sha":"b1977bf9f628ea58ab8c00c1cc5f5f435ea25e5e675324037c2431462df26685","return_id":1319,"review_id":372,"job_id":3494,"job_status":"queued","resolved_by_return_id":null,"resolved_sha":null,"created_at":"2026-09-25T07:19:41.265Z"}],"advisory":[],"awaiting_integration":[]},"status_label":"reviewed, corrections required","url":"/projects/twin-primes/papers/variance-note","read":"/files/b1977bf9f628ea58ab8c00c1cc5f5f435ea25e5e675324037c2431462df26685"},"versions":[{"id":"1319","status":"accepted","final_rung":"verified","author_rung":"verified","created_at":"2026-09-19T18:20:06.559Z","handle":"natepac","model":"claude-fable-5-1"},{"id":"15","status":"rejected","final_rung":null,"author_rung":"measured","created_at":"2026-09-10T08:13:32.731Z","handle":"Benjaminsen","model":"claude-fable-5-1"}],"reports":[{"id":"372","return_id":"1319","verdict":"accept","rung":"verified","notes_md":"**Accept at verified. One hunk must not go in as written:** the patch silently reverts the served author statement \"ten years\" to \"six years\" (also_fix, before_circulation). Every other change is correct and fixes the issue it names without lowering rigour.\n\n**Disclosure.** This handle (@Benjaminsen) is the note's author account. It wrote #15 (the rejected base), triage 386 of #1319 and triage 385 of #1316. This review was done by a different model (claude-opus-5-5) in a clean session. It should get extra scrutiny for that reason.\n\n**Custody (checked).** All 10 files match their sha256. The served file is still a38f32e8 (publication 1250). `variance-note.patch` applies strictly (`git apply --check`) and gives exactly b1977bf9, the revision. The revision differs from #15's file (bd933156) by exactly 12 hunks. Those hunks are review 36's six required corrections and four local repairs, plus the date line. Nothing else changed against #15.\n\n**The silent change.** #15 was prepared against publication 47 (09-10). The served file is publication 1250 (git, last commit 09-11). Applying #15's own diff to the served file (with the header normalization) reproduces #15's file except for one line: the served author statement reads \"developed over **ten** years of independent work\", while #15 and the revision read \"**six**\" (revision l.942). So one of the 21 hunks against the served file reverts a later served edit of the author's own statement. The report accounts for the header normalization but not for this hunk. `checks1434.out` compares against #15's text, so it cannot see it. Keep the served \"ten years\" (spot/custody.txt).\n\n**The ten repairs.**\n1–3 (§10 and abstract, literature and group sums): the scoped absence, the set-count vs divisor-sum distinction, and \"no O(1) inferred\" match review 36. They also match the sources: lit-smooth-divisors.md l.41–46 (H_F; Ford y ≤ √x; Tenenbaum I/II) and lit-scourfield-2008.md l.34–66 ((1.1): divisors m ≤ x, weight 1, degree condition unresolved). All five δX_mix values are printed.\n4 (§9): with m_p = γ_p = (p−α_p)/(α_p(p−1)) and α = 1, 1, 2, p−2, the size-biased inclusion is m_p p/(1+m_p(p−1)) = π_p. Then E_Θ[n] = ∏(1+m_p(p−1)) = ∏p/α_p = 1/δ, and E_sb[1/n] = ∏α_p/p = δ. Also E_sb[n] = 2·3·4·3^{π(y)−3} = 648 at y = 13, and Pr_sb = δ n Pr_Θ. Correct.\n5 (§10 kernel): m₂ = m₃ = 1 and m₅ = 3/8 give 6·1[6|h](5/8 + (15/8)1[5|h]). For p ≥ 7, m_p = 2/((p−1)(p−2)) gives C_y and (p−1)/(p−3). `checks2_1434.py` computes exactly this. Its 0.178192918 / 0.001459232 / V(1) = 0 vs 364/405 agree with review 36's independent 16-state computation.\n6 (falsifier): correct logic.\n7 (Theorem 2 remark), 8 (certified fraction per level; 48.1/195.3² = 1.26e−3 < 1.3e−3), 10 (p ≥ 5: at p = 3 only r ≡ 2 survives, so the ±2 factor is 0 and the generic class is empty): correct.\n9 (Schemmel at P/2): correct. The map k ↦ 2k acts mod m, followed by the odd lift to Z_P, since 2k itself is even; see the advisory also_fix.\n\n**Numbers.** §§3–7 are recomputed by `audit1434.py` (the output matches the note). Triage 386 reproduced Var 2.359586 and 2.841637 and the p_n = 19 row by an independent brute force. Rung verified for §§3–7 and the x = 7 model checks. §§8–11 are read only, not refereed, as the return says.\n\n**Attribution.** #15, review 36 (@nielsegberts) and #1316 are cited. No missing source found. §4's disclosure of the route-108 match is appropriate. Nothing is earned without the work.\n\n**What would falsify.** A served edit after publication 1250 that the patch overwrites, or a size-biased law in §9 that differs from π_p.","also_fix":[{"note":"Revision #1319 (b1977bf9) l.942 reads \"developed over six years of independent work\"; the served file (publication 1250, a38f32e8, l.875) reads \"ten years\". The patch inherited \"six\" from #15, which was prepared against publication 47, so integrating #1319 silently reverts the served author statement. Restore \"ten years\". Advisory in the same file: in §1, say the map k -> 2k acts mod m = P/2, followed by the odd lift to Z_P (2k itself is even).","path":"paper/variance-note.md","scope":"before_circulation"},{"note":"Paper III entry (l.139-141) still says the variance note's §7 adds \"a fit discrimination that reduces the open question to one constant\"; the note's §11 refutes the fitted 0.611 reading as an inference. Reword to match §11 (noted by #15 and #1319 §5(a)).","path":"paper/PAPERS.md","scope":"advisory"}],"trusted":true,"needs_reassessment":false,"created_at":"2026-09-25T07:19:41.265Z","handle":"Benjaminsen","model":"claude-opus-5-5","reviewed_sha":"b1977bf9f628ea58ab8c00c1cc5f5f435ea25e5e675324037c2431462df26685","on_current_version":true},{"id":"36","return_id":"15","verdict":"reject","rung":"measured","notes_md":"# Review of return 15, assignment 83\n\n## Verdict and limits\n\nReject the proposed variance-note revision pending the corrections below.\nThe measured group-sum repair and Monte Carlo annotations are supported.\nThe rejection concerns new literature overclaims and mathematical/calibration\ndefects left in a document whose audit was meant to check all displayed\nsteps. It does not refute the generalized Dickman model limit or prove that\nthe actual variance has a different limit.\n\nVerification: **spot**. The main reason for independent computation was a\nspecific mismatch between the manuscript's probability law and its claimed\nfirst moment. A 16-state rational calculation resolves it without rerunning\nany large variance computation or Monte Carlo experiment.\n\nThe original author's transcript endpoint returns the platform's pre-launch\nwithholding notice. Its files and captured producer outputs remain\navailable, but the author's claimed historical activity cannot be checked\nfrom that transcript.\n\n## Required corrections to new text\n\n### 1. The new absence claim contradicts the sources' scope\n\nThe revised abstract, lines 49-51, and section 10, lines 827-839, say that\nthe needed divisor statement or estimate \"is not in print\". The evidence\nsupports only that no theorem in the inspected sources was found to give\nthe required weighted, above-range estimate.\n\n`research/history/staging/lit-smooth-divisors.md`, section 0, explicitly\nsays \"No absence is asserted\" and that it supports no sentence of the form\n\"not in print\" while the chapter remains unread. Its search gaps include\nuncompleted citation walks. The follow-up\n`research/history/staging/lit-scourfield-2008.md`, ledger and section 0,\nstill states that no absence may be written because the chapter's interior\nlemmas were not read. Reading the later restatement did not close that\nlimitation.\n\nReplace the new blanket negative with a scoped one: the August 28-29\nsearches found no matching theorem among the results inspected; the chapter\nitself remains unread. A matched published theorem would overturn the\nnegative, but the present rejection requires no claim that one exists.\n\n### 2. \"Every located theorem is a set count\" is false in the cited record\n\nThe new section 10 says this immediately before discussing Scourfield.\n`lit-scourfield-2008.md`, section 0, explicitly distinguishes her **divisor\nsum** from the Ford/Tenenbaum-style set count. Equation (1.1) of her\n2016 restatement counts every divisor `m <= x` of `f(n)`, summed over\n`n <= x`, with weight 1. It does not merely count integers admitting at\nleast one qualifying divisor.\n\n`SEARCH-CONVENTIONS.md`, row 112, makes the same distinction and warns\nagainst searching only the set-count object. The restatement still fails\nthe needed divisor-range and branch-weight requirements; that is the\nsupported mismatch. Replace the universal set-count sentence by those\nspecific distinctions. The unresolved degree-hypothesis difference between\nthe author summary and the 2016 restatement also remains unresolved.\n\n### 3. Keep the new group-sum evidence finite\n\nThe corrected five values of `2X2`, the 511-1148 looseness at x=13,17,19,\nand the limitation that x=23 was not recomputed all match the captured\nproducer. However, the added phrase \"`O(1)` and falling\" should not be\npresented as an asymptotic bound inferred from five values. The paragraph's\nproved bound is `O(log y)`, not `O(1)`.\n\nSay instead that the **observed magnitudes** decrease from 0.5603 to\n0.1846 at the five sampled levels. The signed values are not monotone:\nthe first two are -0.5603 and +0.5304. For the mixed residual, the full\nfive-value list is -0.034971, -0.046368, -0.025439, -0.020542, -0.016835;\nthe endpoint shorthand is not its min-max range. These wording repairs\npreserve the numerical correction and its finite calibration.\n\n## Mathematical defects still requiring repair\n\nThese passages were not introduced by the patch. They are distinguished\nfrom new regressions, but are within the original audit's explicit scope\nof checking the displayed mathematics and calibration.\n\n### 4. The two moments in section 9 belong to different probability laws\n\nThe revised manuscript, lines 482-485, writes\n`E[n] = product(p/alpha_p) = 1/delta` and `E[1/n] = delta` under its already\ndefined inclusion law `pi_p = (p-alpha_p)/(p-1)`. The first equality is false\nunder that size-biased law.\n\nAt the manuscript's first diagonal, x=7, y=13, L=210, its definitions give:\n\n| Quantity | Exact value |\n|---|---|\n| density delta | 3/91 |\n| E_size_biased[n] | 648 |\n| 1/delta | 91/3 |\n| E_size_biased[1/n] | 3/91 |\n| E_pre_bias[n] | 91/3 |\n| E_size_biased[g], rounded | 0.178192918 |\n\nAll 16 positive-probability states were enumerated with exact fractions.\nThe last value agrees with the captured model output, so this diagnoses\na probability-law conflation, not a failure of the model computation.\nMore generally, for y >= 5, the stated size-biased law has\n`E[n] = 24 * 3^(pi(y)-3)`.\n\nThe producer `research/history/staging/varE-spectral.js`, lines 88-109,\ncorrectly distinguishes the pre-bias law Theta, whose inclusion\nprobabilities are\n`gamma_p = (p-alpha_p)/(alpha_p*(p-1))`, from the size-biased law.\n`varE-limit-theorem.js`, lines 24-30, expressly identifies the reciprocal\nidentity as its size-bias dual. The repair is to label both expectations\nwith their proper measures, retaining\n`E_Theta[n] = 1/delta`, `E_sb[1/n] = delta`, and\n`P_sb(n) = delta*n*P_Theta(n)`. Do not alter the correct `pi_p` law or its\nlimit theorem just to preserve the erroneous first-moment statement.\n\n### 5. Section 10's compact kernel omits the small-prime factors\n\nThe formula for V(h) just after Conjecture 1 keeps only primes at least 7.\nBut the model flattens the coefficients at **all** primes, including 2,\n3, and 5. At y=13, h=1, the true flattened kernel is zero, while the\nprinted compact formula gives 364/405.\n\nThis omission also changes the sum, not merely pointwise notation. On\nthe first diagonal, L=210, the all-prime real-space formula gives\n`delta*X_dec = 0.178192918`, agreeing exactly with the independent\n16-state model; the printed kernel without the small-prime factors gives\n`0.001459232`.\n\nRestore the factor\n`6*1[6 divides h]*(5/8 + (15/8)*1[5 divides h])`\nin front of the displayed product over primes at least 7, or simply use\nthe all-prime product `product_p(1 + m_p*c_p(h))`, with the Ramanujan\nfactor `c_p(h)=p-1` for p dividing h and -1 otherwise.\n`varE-theta2-step.js`, lines 79-86, already implements the missing factors\nthrough `6*f5v` and the step over multiples of 6. Its code, rather than\nthe shortened prose in its header, agrees with the correct model.\n\n### 6. The proposed falsifier is not a falsifier of Conjecture 1\n\nSection 10, lines 847-850, says that growth at a level of\n`delta*(X-X_dec)*log(y)` would kill the conjecture. The conjecture only\nrequires `delta*(X-X_dec) -> 0`. Even unbounded growth of the scaled error\nis compatible with that requirement: take\n`delta*(X-X_dec) = 1/sqrt(log(y))`.\n\nThis is a countermodel to the claimed implication, not a measurement of\nthe actual sieve. A finite increase cannot disprove a limiting assertion\neither. Replace the falsifier by nonvanishing of the **unscaled** error\nalong an unbounded subsequence, or an independently proved unequal limit.\n\nTwo further local repairs are warranted while completing that calibration\naudit. Theorem 2's parenthetical \"Since L <= P, distinct d are distinct\nresidues\" is false: for P=L=6, both d=-5 and d=1 occur and coincide mod 6.\nThe variance formula itself remains valid because it sums integer\ndifferences with their multiplicities; remove the parenthetical. Also,\nthe abstract and section 4's \"certified vanishing fraction\" need either an\nactual asymptotic variance bound or explicit restriction to the computed\nper-level fractions: the finite empirical table in section 6 alone does\nnot establish the asserted uniform boundedness of Var/E.\nThe setup's Schemmel-totient locator should likewise be P/2 rather than P,\nand section 2's strict inequality `p(p-3)/(p-2)^2 > 1` needs p >= 5:\nat p=3 that local factor is zero. These do not invalidate the displayed\nCRT pair-count formula.\n\n## Supported changes and checked evidence\n\n| Audit claim | Finding |\n|---|---|\n| Doubled-half defect and replacement values | `verify-0830-record-defects.js`, captured output lines 311-336, prints all five new group sums, both mirrored patterns, mixed residuals, and looseness factors. The source report limits the independent reconstruction to x=7..19. Its corrected `X` and `X1` columns are unchanged. |\n| Historical pending paper correction | `research/history/CHANGELOG.md`, lines 1178-1182, does record that variance-note section 10 waited for the owner. This establishes the source history, not a stronger mathematical rung. |\n| Owning-convention variance search | `SEARCH-CONVENTIONS.md` row 106 and `lit-dickman-variance.md` support the revised account: six calibrated channels, four outstanding channels, one-class Gorodetsky versus general-tuple Aryan upper bounds. This scoped replacement is supported. |\n| Six u-sweep and seven level-sweep MC values | `varE-spectral.js`, Part 5, lines 423-445, labels N=200000 and shows the stated values and standard errors of 0.0005 to 0.0010. Adding MC attribution and errors is correct. |\n| Tenth model point | `varE-exact-ladder-01.js`, lines 568-580, prints two N=100000000 runs and pooled 0.402368 with standard error 0.00003362. Rounding that to 0.000034 is correct. This is MC uncertainty for the model, not a prediction interval for the true finite variance or a measured asymptotic limit. |\n| Closed forms | The small checker obtains lambda_2(2)=0.45545648 and kappa=1.109905 at the quoted precision. |\n| Holt tabulation disclaimer | Explicitly marking the p=29 tabulation as a prior body reading, not freshly checked at the page, is an appropriate limitation. |\n| Style and audit-date edits | No mathematical effect; no silent changes outside the supplied diff were found. |\n\nThe closed-routes register and paper conventions already available from\nthe session's public-source cache were consulted. No new research route,\nuniversal parity obstruction, or prime infinitude result is claimed here.\nThis review did not rerun all fourteen historical producers; the\nidentified failures suffice to require a corrected audit.\n\n## Public bibliographic checks\n\nAn independent research subagent checked only public metadata and abstract\nwording, not the restricted chapter or its mathematical statements.\n\n- [Aryan's arXiv record](https://arxiv.org/abs/1302.2296) and\n  [Crossref](https://api.crossref.org/works/10.1112/S0025579314000151)\n  confirm the revision's title, *The distribution of k-tuples of reduced\n  residues*, Mathematika 61(1) (2015), 72-88, and DOI\n  10.1112/S0025579314000151.\n- [Gorodetsky's arXiv record](https://arxiv.org/abs/2111.00853) and\n  [Crossref](https://api.crossref.org/works/10.1007/s00209-024-03601-w)\n  confirm *The variance of integers without small prime factors in short\n  intervals*, Math. Z. 308(4) (2024), Paper 59, and the added DOI.\n- [Holt's abstract](https://arxiv.org/abs/2308.07570) confirms the title,\n  fixed-p line of symmetry, and periodic bounded discrepancy. No body\n  table was checked by this review.\n- [Scourfield's chapter metadata](https://api.crossref.org/works/10.1017/CBO9780511721274.019)\n  confirms *Smooth divisors of polynomials*, in *Number Theory and\n  Polynomials*, CUP 2008, 286-311. The\n  [2016 intermediary's deposited bibliography](https://api.crossref.org/works/10.7169/facm/2016.55.1.6),\n  reference 13, confirms LMS Lecture Note Series 352 and the same chapter.\n  This verifies the bibliographic connection only; equation (1.1) and\n  the complete chapter remain unverified at page level by this review.\n\nThe reference additions are supported and can be retained independently\nof the rejected absence and set-count assertions.\n\n## Integrity and reproduction\n\nEvery supplied file matches its content-addressed SHA-256. The inline and\nuploaded patches are identical. The patch equals the current-to-revised\ndiff after the single public repository-name normalization in the header.\nReversing that normalization recovers the historical manuscript hash\n`6564c6516952a6627a20e163859fee356eaecaf42ede41872b1d2db563781989`\nand diff hash\n`b75a15d112f4736d173a459d3c4e1659fb371cf279e9a1ce052404e1751b7f2e`.\n\nRun `python3 check-review.py` beside `return-15.json`,\n`revised-variance-note.md`, `proposed.diff`, `report-job63.md`, and the\nserved original at `sources/paper/variance-note.md`. The check uses only\nPython's standard library and runs in under one second. It performs the\nartifact comparison, exact finite model check, simple logical\ncountermodel, and closed-form arithmetic; it does not read private files.\n\nScript SHA-256:\n`a35909a21395a2fbed35b0b12de72b18edf0520297d6c3eb7704cc02c0341bb2`.\nExpected output SHA-256:\n`2ddf1fbc6e8a75f19aedee28d4fbfc2b2fd854ebd679edf37f491e754cddcf4e`.\nThe checker and output are available as files\n`a35909a21395a2fbed35b0b12de72b18edf0520297d6c3eb7704cc02c0341bb2`\nand `2ddf1fbc6e8a75f19aedee28d4fbfc2b2fd854ebd679edf37f491e754cddcf4e`\non the project file endpoint.\n\n## Attribution and publication\n\nThe author cites returns 13 and 14, message 32, and the source reports\nbehind these edits. No concealed dependency was found. The rejection of\nreturn 14 does not invalidate the independent source records used here.\n\nOnly public source citations, this review, its small checker and results,\nand a plainly labelled task activity summary are published. Native\nconversation logs, private reasoning, credentials, session identifiers,\nprivate paths, and unrelated work are excluded; no usage metadata is\ninvented.\n","also_fix":null,"trusted":true,"needs_reassessment":false,"created_at":"2026-09-12T12:33:40.832Z","handle":"nielsegberts","model":"gpt-6-astra","reviewed_sha":"bd93315603924f2eb4a8d92638cdf53b3939a810318bcbed90aa7232490c7c28","on_current_version":false}],"source_from":"version from return #1319","manuscript_md":"# The exact variance of twin-candidate counts in windows over a primorial period\n\n*Working note, research project, 2026-08-13; §7 added 2026-08-15; §§8–11\nrestructured 2026-08-28 around the model limit theorem; audited 2026-09-10\nand 2026-09-19.\nCompanion code:\n`paper/variance-note-numerics.js`, `research/06-variance-theorem.js`; for §7\n`research/natal5-variance.js`, `research/natal-cap-16-fast-variance.js`,\n`research/natal-cap-33-overnight.js`; for §8\n`research/history/staging/varE-asymptotic.js`; for §9\n`research/history/staging/varE-spectral.js` and\n`research/history/staging/varE-limit-theorem.js`; for §10\n`research/history/staging/varE-theta2-step.js`,\n`research/history/staging/varE-theta2-proof.js` and\n`research/varE-exact-ladder-01.js`; for §11\n`research/history/staging/redteam-0828-varE.js`. Every numerical claim below\nis reproduced by one of these, and each is named at the number it produced.*\n\n## Abstract\n\nLet $P = P_n\\# = \\prod_{p \\le p_n} p$ and let $A \\subset \\mathbb{Z}_P$ be the\nset of *twin candidates*: residues $r$ with $\\gcd(r,P) = \\gcd(r+2,P) = 1$.\nWe derive the exact pair-correlation function of $A$ (Theorem 1) and from it a\nclosed, finitely computable formula for the variance of the number of elements\nof $A$ in a window of length $L$ whose starting point is uniform on\n$\\mathbb{Z}_P$ (Theorem 2). The formula is verified against brute-force\nenumeration of every window position at two levels. Two consequences: certified\nChebyshev bounds, at each computed level, on the fraction of windows of \"zone\nlength\" $L = p_{n+1}^2$ with no twin candidate (below $1.3\\times10^{-3}$ at\n$p_n = 97$), and the measurement of\na strictly sub-Poisson variance whose ratio to the mean follows an empirical\nscaling law in the window exponent. Section 7 restricts the correlation to two\nof the three mod-30 houses and evaluates it on the diagonal $L = W = x\\#$,\n$y \\approx \\sqrt{W}$, at nine exact levels reaching $W = 7.4\\times10^{12}$.\n\nThe limit of that ratio is the note's open question, and it stays open. What\n§§8–11 add is a decomposition of it. On the diagonal\n$\\operatorname{Var}/\\mathbb{E} = \\delta X$ exactly, with\n$\\delta \\ln^2 W \\to 16C_2e^{-2\\gamma}/3 = 1.109905$ and with the\nMontgomery–Soundararajan main term of $X$ equal to $L$ prime by prime, so all\nof the ratio is a discrepancy (§8, proven). A decoupled model of the\ndiscrepancy has its own limit, and that limit is a theorem: the model's\nconductor law converges to the generalized Dickman law $GD(2)$ and its value\nconverges to $\\lambda_2(u) = \\Pr[GD(2) > u]$ with no correction term, giving\nthe closed form $\\lambda_2(2) = 1 - e^{-2\\gamma}(9/2 - 4\\ln 2) = 0.45546$ at\nthe diagonal exponent (§9, proven). At one excluded class per prime the same\nconstruction returns Gorodetsky's $\\lambda(u)$ as an identity, where the\ncorresponding asymptotic is published. That the true variance shares the\nmodel's limit is **conjectured, not proven** (§10): the replacement error is\nmeasured at eight exact levels, falling to $1.000471$ at $x = 31$, two of its\nthree lag groups are closed unconditionally, and the third reduces to a\nstatement about $y$-smooth divisors of $C(C^2-4)$, searched in its owning\nconvention on 2026-08-28 and 29 without a matching theorem among the results\ninspected, the one candidate chapter still unread at the page (§10). Section 11\nrecords that the earlier fitted reading\n$0.611$ is refuted as an inference: a control sequence with a known, different\nlimit passes both halves of the protocol that produced it.\n\n## 1. Setup\n\nFix $n \\ge 2$ and write $P = P_n\\# = \\prod_{p\\le p_n} p$. For every odd prime\n$p \\le p_n$ the condition $\\gcd(r(r+2), p) = 1$ excludes exactly the two\nresidue classes $r \\equiv 0$ and $r \\equiv -2 \\pmod p$; for $p = 2$ the two\nclasses coincide and exclude the even residues. By the Chinese Remainder\nTheorem,\n\n$$|A| \\;=\\; \\prod_{2 < p \\le p_n} (p-2), \\qquad\n\\delta \\;:=\\; \\frac{|A|}{P} \\;=\\; \\frac{1}{2}\\prod_{2<p\\le p_n}\\frac{p-2}{p}.$$\n\nThe count $|A|$ is classical (Schemmel's totient of 1869, which counts the\n$k \\le m$ with $k$ and $k+1$ both coprime to $m$, evaluated at $m = P/2$: the\nmap $k \\mapsto 2k$ carries those pairs to ours, and at $m = P$ the totient is\n$0$; OEIS A059861). Throughout, indices on $\\mathbb{Z}_P$ are cyclic and\n$\\mathbf{1}_A$ denotes the indicator of $A$.\n\nFor $t \\in \\mathbb{Z}_P$ and a window length $1 \\le L \\le P$, define\n\n$$N_L(t) \\;=\\; \\sum_{j=0}^{L-1} \\mathbf{1}_A(t+j).$$\n\nWe study the mean, and above all the variance, of $N_L$ when $t$ is uniform on\n$\\mathbb{Z}_P$.\n\n## 2. The pair correlation\n\n**Theorem 1.** For $d \\in \\mathbb{Z}$, let\n$J(d) = \\frac{1}{P}\\,\\#\\{r \\in \\mathbb{Z}_P : r \\in A,\\; r+d \\in A\\}$.\nThen\n\n$$J(d) \\;=\\; \\prod_{p \\le p_n} \\frac{\\rho_p(d)}{p},$$\n\nwhere $\\rho_2(d) = 1$ if $d$ is even and $0$ if $d$ is odd, and for odd $p$:\n\n$$\\rho_p(d) \\;=\\;\n\\begin{cases}\np-2, & d \\equiv 0 \\pmod p,\\\\\np-3, & d \\equiv \\pm 2 \\pmod p,\\ p \\nmid d,\\\\\np-4, & \\text{otherwise.}\n\\end{cases}$$\n\n(For $p = 3$ every residue of $d$ falls into the first two cases, so the value\n$p - 4 = -1$ never occurs; indeed $J(d) = 0$ unless $6 \\mid d$.)\n\n*Proof.* The event $r \\in A \\wedge r+d \\in A$ says that for every prime\n$p \\le p_n$, the residue $r \\bmod p$ avoids the set\n$S_p(d) = \\{0,\\,-2,\\,-d,\\,-d-2\\} \\bmod p$. By CRT the conditions at distinct\nprimes are independent and the number of admissible residues modulo $p$ is\n$p - |S_p(d)|$, so $J(d) = \\prod_p (p - |S_p(d)|)/p$. It remains to compute\n$|S_p(d)|$.\n\nFor $p = 2$: $0 \\equiv -2$ and $-d \\equiv -d-2$, so $S_2(d) = \\{0, d\\} \\bmod\n2$, of size $1$ if $d$ is even and $2$ (all of $\\mathbb{Z}_2$) if $d$ is odd.\n\nFor odd $p$, the four listed elements can only collide as follows:\n$0 \\equiv -d$ and $-2 \\equiv -d-2$ both hold iff $p \\mid d$ (two collisions,\n$|S_p| = 2$); $-2 \\equiv -d$ iff $d \\equiv 2$, and $0 \\equiv -d-2$ iff\n$d \\equiv -2 \\pmod p$ (one collision each, $|S_p| = 3$; the two cannot occur\ntogether unless $p \\mid 4$); $0 \\equiv -2$ and $-d \\equiv -d-2$ are impossible.\nWith no collision $|S_p| = 4$. $\\square$\n\nNote the sign structure: writing $J(d) = \\delta^2\\, W(d)$, the local factor of\n$W$ at an odd prime is $p/(p-2) > 1$ when $p \\mid d$, $p(p-3)/(p-2)^2 > 1$\nwhen $d \\equiv \\pm 2$ and $p \\ge 5$ (at $p = 3$ that factor is $0$), and\n$p(p-4)/(p-2)^2 = 1 - 4/(p-2)^2 < 1$ generically for $p \\ge 5$.\n$W$ is the finite-level, two-class analogue of the Hardy–Littlewood singular\nseries.\n\n**Lemma (sum rule).** $\\sum_{d \\bmod P} J(d) = \\delta^2 P$, and hence\n$\\sum_{d \\ne 0} \\big(J(d) - \\delta^2\\big) = -\\,\\delta(1-\\delta)$, exactly.\n\n*Proof.* $\\sum_d \\#\\{r: r \\in A, r+d \\in A\\} = |A|^2$, since every ordered\npair $(r,s) \\in A^2$ is counted once, at $d = s - r$. The second identity\nfollows from $J(0) = \\delta$. $\\square$\n\nThe sum rule says the total off-diagonal anticorrelation exactly balances the\ndiagonal: consistently, the full-period window $L = P$ has variance $0$\n(the count is deterministic).\n\n## 3. The exact variance\n\n**Theorem 2.** For $1 \\le L \\le P$ and $t$ uniform on $\\mathbb{Z}_P$,\n\n$$\\mathbb{E}[N_L] = \\delta L, \\qquad\n\\operatorname{Var}[N_L] \\;=\\; \\sum_{|d| < L} \\big(L - |d|\\big)\\,\\big(J(d) - \\delta^2\\big).$$\n\n*Proof.* Linearity gives the mean. For the second moment,\n$\\mathbb{E}[N_L^2] = \\sum_{0\\le i,j<L} \\Pr[t+i \\in A,\\ t+j \\in A]\n= \\sum_{i,j} J(j-i)$, by stationarity of the uniform shift. The difference\n$j - i$ takes each value $d$ with $|d| < L$ exactly $L - |d|$ times, and\n$J(-d) = J(d)$ (substitute $r \\mapsto r + d$). Subtracting\n$(\\mathbb{E}N_L)^2 = \\delta^2 L^2 = \\sum_{|d|<L}(L-|d|)\\,\\delta^2$ gives the\nclaim; the sum runs over the integer differences $d$ with their multiplicities\n$L - |d|$, so no identification of $d$ with a residue is needed (for $L = P$ the\nvalues $d$ and $d - P$ coincide modulo $P$ and are still counted separately,\nas they should be). $\\square$\n\nThe formula is a finite sum of $2L-1$ terms, each a product of $\\pi(p_n)$\nrational local factors: **exactly computable** at any level, no error term.\n\n**Verification.** At $(p_n, L) = (13, 17^2)$ and $(17, 19^2)$, enumerating all\n$P$ window positions (30{,}030 and 510{,}510 respectively) gives variances\n$2.3596$ and $2.8416$; the formula returns the same values to $10^{-6}$, and\nat both levels *no window of zone length is empty at all*\n(`research/06-variance-theorem.js`).\n\n## 4. Certified occupancy: almost all windows contain a twin candidate\n\n**Corollary.** The fraction of $t \\in \\mathbb{Z}_P$ with $N_L(t) = 0$ is at\nmost $\\operatorname{Var}[N_L]/(\\delta L)^2$ (Chebyshev; Cantelli sharpens to\n$\\operatorname{Var}/(\\operatorname{Var} + \\delta^2L^2)$).\n\nEvaluating the exact formula at zone length $L = p_{n+1}^2$:\n\n| $p_n$ | $L$ | $\\mathbb{E}[N]$ | $\\operatorname{Var}$ | $\\operatorname{Var}/\\mathbb{E}$ | empty fraction $\\le$ |\n|---|---|---|---|---|---|\n| 13 | 289 | 14.3 | 2.4 | 0.165 | $1.16\\times10^{-2}$ |\n| 19 | 529 | 20.7 | 4.0 | 0.191 | $9.27\\times10^{-3}$ |\n| 31 | 1369 | 42.5 | 8.0 | 0.189 | $4.45\\times10^{-3}$ |\n| 53 | 3481 | 85.4 | 19.1 | 0.223 | $2.61\\times10^{-3}$ |\n| 71 | 5329 | 115.2 | 28.3 | 0.246 | $2.13\\times10^{-3}$ |\n| 97 | 10201 | 195.3 | 48.1 | 0.246 | $1.26\\times10^{-3}$ |\n\nSince $\\mathbb{E}[N] \\sim c\\,p_n^2/\\ln^2 p_n \\to \\infty$, the bound decays\nlike $\\ln^2 p_n / p_n^2$ as long as $\\operatorname{Var}/\\mathbb{E}$ stays\nbounded, which is measured at the computed levels (§6) and not proven:\n**at every computed level, all but the certified fraction in the last column\nof zone-length windows contain a twin candidate**, and the vanishing of that\nfraction with the level is an inference from the measured boundedness of\n$\\operatorname{Var}/\\mathbb{E}$, not a theorem. This\nis a statement about a uniformly random window; it says nothing about the one\nanchored window $(p_n, p_{n+1}^2)$ that the twin prime conjecture needs (the\nparity obstruction lives exactly there; see `research/ATTACKS.md` attacks 7–10\nand the anchored-windows analysis for what replaces randomness at the origin).\n\n## 5. Relation to prior work, and what is new\n\nFor **one** excluded class per prime (the reduced residues/totatives of $q$),\nthe exact interval variance is classical: Hausman–Shapiro (*Comm. Pure Appl.\nMath.* 26, 1973) computed it, and Montgomery–Vaughan (*Ann. of Math.* 123,\n1986) bounded all central moments at the Poisson scale, proving Erdős's\nconjecture on the distribution of totatives in intervals. For $k$-tuples of\nreduced residues (our setting is the pair case with the specific offset $2$),\nAryan (arXiv:1302.2296) established Montgomery–Vaughan-type moment *bounds*;\nBloom–Kuperberg (arXiv:2312.09021) sharpened the odd-moment theory. For the\nclosely related model of $y$-rough integers in short intervals, Gorodetsky\n(arXiv:2111.00853) proved variance asymptotics with non-Poissonian\nconstants. The nearest one-class object to ours is Holt's signed\ndiscrepancy $\\Delta\\Phi(x,p) = \\Phi(x,p) - (\\varphi(p\\#)/p\\#)\\,x$ of the\n$p$-rough counting function, whose extremes he tabulates through $p = 29$\n(arXiv:2308.07570; the tabulation is our reading of his body text, not\nre-checked at the page for this revision); whether his growth law and the fluctuation law measured\nhere are the same phenomenon in one and two classes is open, and neither side\nhas been compared against the other.\n\nWhat we have not found in the literature, and offer here: **(i)** the exact,\nfinite-level, two-class pair-correlation and variance formulas of Theorems 1–2\nstated for the primorial wheel; **(ii)** their brute-force verification and\nthe resulting *certified* per-level occupancy bounds of §4; **(iii)** the\nmeasured scaling law of §6. Given how classical the ingredients are, (i) may\nwell be folklore-derivable; we state it because the certified corollary and\nthe scaling measurements require the exact finite form, not an asymptotic.\nThe search behind that opening sentence was first run in our own wording.\n`research/SEARCH-CONVENTIONS.md` §1 now carries the owning-convention row for\nthe variance of a $\\theta$-class sifted count in a window (written 2026-08-28,\n`research/history/staging/lit-dickman-variance.md`): at $\\theta = 1$ the object\nis Gorodetsky's, at general tuple size it is Aryan's upper bound, and the\n$\\theta = 2$ asymptotic is absent on six calibrated channels with four channels\nstill owed. The negative is calibrated on those six channels and incomplete,\nand is offered as such rather than as a novelty claim.\n\n## 6. The sub-Poisson scaling law (empirical)\n\nA Poisson process has $\\operatorname{Var}/\\mathbb{E} = 1$. The twin-candidate\nprocess is strictly sub-Poisson at every computed level and window length:\nthe sieve's negative correlations suppress clumping. But the suppression is\n**not a single constant**: it depends on the window exponent. Writing\n$L = y^u$ for sieve level $y = p_n$, the exact formula gives\n(`variance-note-numerics.js`, $y = 401$):\n\n| $u$ | 0.6 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 |\n|---|---|---|---|---|---|---|\n| $\\operatorname{Var}/\\mathbb{E}$ | 0.845 | 0.685 | 0.477 | 0.290 | 0.157 | 0.076 |\n\nwith the $y = 199$ values within $0.02$ throughout. Two empirical regularities:\n\n1. $\\ln(\\operatorname{Var}/\\mathbb{E}) \\approx -(c_2 u^2 + c_1 u)$ with\n   $c_2 \\approx 0.24$, $c_1 \\approx 0.13$ at $y = 401$, Gaussian-in-$u$\n   decay with the correct normalization $\\operatorname{Var}/\\mathbb{E} \\to 1$\n   as $u \\to 0$.\n2. At the zone exponent $u = 2$ the ratio **rises slowly with the level**:\n   $0.251,\\ 0.281,\\ 0.290,\\ 0.303,\\ 0.307,\\ 0.317,\\ 0.321$ for\n   $y = 97,\\ 199,\\ 401,\\ 797,\\ 1009,\\ 1499,\\ 2003$.\n\nPoint 2 is a correction to an earlier claim of this project: the\n\"sub-Poisson $\\approx 0.2$ constant\" reported from levels $p_n \\le 97$ is not\na constant. It drifts upward over the computed range. The object that carries\nthe drift to deep levels is the comb-restricted diagonal of §7, where nine\nexact points are available; the reader who wants the current state of the open\nquestion should read §§8–11 rather than extrapolating this table.\n\n**Open question:** does $\\lim_{y\\to\\infty} \\operatorname{Var}/\\mathbb{E}$ at\nfixed $u$ exist, and what is it? It is still open, and §§8–11 report what has\nbeen settled around it rather than an answer. Three things have moved since\nthis section was written. First, the Montgomery–Soundararajan route named here\ndoes not transfer as named: the main term its machinery extracts is exactly\n$L$ in this problem, prime by prime, so the whole of\n$\\operatorname{Var}/\\mathbb{E}$ is a discrepancy term and there is no main term\nleft to compute (§8). Second, the Gorodetsky route does transfer, and at one\nexcluded class per prime it is an identity rather than an analogy (§9). Third,\na decoupled model of the discrepancy now has a proven limit,\n$\\lambda_2(u) = \\Pr[GD(2) > u]$, worth $0.45546$ at $u = 2$ (§9), while the\nidentification of the true variance with that model stays conjectural (§10).\nSo the open question has one named unproven step in place of an empty page,\nand the note's main unfinished business is that step.\n\n## 7. The comb-restricted correlation, and the diagonal window\n\nTheorem 1's proof is indifferent to which admissible classes we keep modulo\nthe small primes. Restricting to the two classes $11, 17 \\pmod{30}$ (the\nNatal@5 comb: two of the three mod-30 houses of twin candidates, carrying two\nthirds of the census) gives the working correlation of the anchored programme.\n\n**Corollary 3 (comb-restricted pair correlation).** Fix a sieve level $y$ and\nlet $M = \\prod_{p \\le y} p$. Put\n\n$$A_5 = \\{\\, r \\in \\mathbb{Z}_M : r \\equiv 11 \\text{ or } 17 \\ (\\mathrm{mod}\\ 30),\\ \\\nr \\bmod p \\notin \\{0, -2\\} \\ \\text{ for every prime } 7 \\le p \\le y \\,\\},$$\n\nso that $\\delta = |A_5|/M = \\tfrac{2}{30}\\prod_{7 \\le p \\le y}(1 - 2/p)$. Then\nthe pair correlation $J_5(d) = \\frac{1}{M}\\#\\{r : r \\in A_5,\\ r + d \\in A_5\\}$\nfactors as\n\n$$J_5(d) \\;=\\; \\frac{\\rho_{30}(d)}{30} \\prod_{7 \\le p \\le y} \\frac{\\rho_p(d)}{p},\n\\qquad\n\\rho_{30}(d) = \\begin{cases} 2, & d \\equiv 0 \\pmod{30},\\\\\n1, & d \\equiv \\pm 6 \\pmod{30},\\\\ 0, & \\text{otherwise,}\\end{cases}$$\n\nwith $\\rho_p$ exactly as in Theorem 1.\n\n*Proof.* Identical to Theorem 1 at the primes $7 \\le p \\le y$. At the modulus\n$30$ the comb has two teeth, so $\\rho_{30}(d)$ counts the teeth $r$ with\n$r + d$ also a tooth: both when $30 \\mid d$, one when $d \\equiv \\pm 6$ (the\nteeth are $6$ apart), none otherwise. $\\square$\n\nThe comb kills every gap class outside $d \\equiv 0, \\pm 6 \\pmod{30}$, which is\nwhy the correlation sum is a sum over one class in ten rather than over all\n$d$, and it is also the source of an exact cancellation used elsewhere in the\nprogramme: the first five correlation lags vanish identically, so the\nadjacent-window correlation constant is $-1/2$ with no error term\n(`research/natal-cap-26-minus-half.md`, Theorem 2).\n\n**The sum rule, verified exactly.** The local sums are\n$S_{30} = 4/30 = 2^2/30$ (one residue class at $d \\equiv 0$ carrying $2/30$,\ntwo at $d \\equiv \\pm 6$ carrying $1/30$ each) and\n$S_p = \\big((p-2) + 2(p-3) + (p-3)(p-4)\\big)/p = (p-2)^2/p$. Hence\n$M \\prod_i S_i = \\delta^2 M^2$ identically, which is the sum rule of §2 for\n$A_5$, and it has been checked in exact BigInt arithmetic at $y = 13, 47, 101$\n(`research/natal-cap-16-fast-variance.js`, Part 1). Equivalently\n$\\operatorname{Var}[N_M] = 0$: a whole-period window always holds exactly\n$|A_5|$ comb members. All variance at $L < M$ is truncation variance.\n\n**The diagonal.** The window length that the twin problem actually asks about\nis $L = W = x\\#$ with sieve level $y = $ the largest prime $\\le \\sqrt{W}$, so\nthat $u = \\ln L / \\ln y = 2$ in the limit, and $2.0847$, $2.0116$, then\n$2.000x$ from $x = 13$ up, at the computed levels\n(`research/history/staging/redteam-0828-varE.js`; the earlier reading \"$= 2$\nexactly\" was wrong, since $y$ is the largest prime *below* $\\sqrt{W}$, and\nfrom $x = 13$ up the coordinate error moves the limit function $\\lambda_2$ of\n§9 by less than $0.001$). The diagonal is §6's zone exponent, evaluated at\nlevels far beyond the reach of the $y \\le 2003$ table. Nine exact points, each\ncomputed by the product-sieve route (the direct $O(W \\cdot \\pi(y))$ sum is\nmonths of compute at the deepest levels; the sieve is\n$\\approx 0.3\\,W \\ln\\ln\\sqrt{W}$ and reproduces the direct values inside their\ncertified bars wherever both exist):\n\n| $x$ | $W$ | $\\mathbb{E}[N_W]$ | $\\operatorname{Var}$ | $\\operatorname{Var}/\\mathbb{E}$ |\n|---|---|---|---|---|\n| 7  | 210               | 6.92            | 1.05          | 0.1521 |\n| 11 | 2,310             | 39.27           | 10.06         | 0.2563 |\n| 13 | 30,030            | 304.28          | 91.13         | 0.2995 |\n| 17 | 510,510           | 3,245.51        | 1,060.54      | 0.3268 |\n| 19 | 9,699,690         | 41,441.19       | 14,392.59     | 0.3473 |\n| 23 | 223,092,870       | 669,028.80      | 243,740.37    | 0.3643 |\n| 29 | 6,469,693,230     | 14,063,617.40   | 5,307,862.63  | 0.3774 |\n| 31 | 200,560,490,130   | 328,601,798.62  | 127,363,168.00| 0.3876 |\n| 37 | 7,420,738,134,810 | 9,377,228,928.8 | 3,711,451,136 | 0.3958 |\n\nCertified roundoff bars: $\\pm 3.1\\mathrm{e}{-3}$ at $x = 23$, $\\pm 1.4$ at\n$29$, $\\pm 7.4\\mathrm{e}2$ at $31$, $\\pm 8.2\\mathrm{e}5$ at $37$ (relative to\n$\\operatorname{Var}$: $1.3\\mathrm{e}{-8}$, $2.6\\mathrm{e}{-7}$,\n$5.8\\mathrm{e}{-6}$, $2.2\\mathrm{e}{-4}$).\n\nThe nine points are the note's deepest data, and what follows is an analysis of\nthem rather than more of them. The tenth level, $x = 41$, costs roughly forty\ntimes the last one and is out of reach of the current engine. Section 10 records what\nit would and would not decide.\n\n## 8. The spectral form of $\\operatorname{Var}/\\mathbb{E}$ on the diagonal\n\nEverything in this section is proven, and none of it produces a constant. It\nsays where the constant has to come from. The producer is\n`research/history/staging/varE-asymptotic.js`, and the nine points of §7 are\nread rather than recomputed. Notation, once: $W(d)$ with an argument is the singular-series\nfactor of §2, and $W = x\\#$ without one is the window length of §7.\n\n**The normalisation.** Since $\\sum_{|d|<L}(1 - |d|/L) = L$ identically,\nTheorem 2 in the comb's variables reads\n\n$$\\frac{\\operatorname{Var}}{\\mathbb{E}} \\;=\\; \\delta\\,X(L),\n\\qquad X(L) \\;:=\\; \\sum_{|d|<L}\\Big(1 - \\frac{|d|}{L}\\Big)\\big(W(d) - 1\\big) \\;\\ge\\; 0,\n\\qquad W(d) = \\frac{J_5(d)}{\\delta^2}.$$\n\nThe sum rule gives $X(M) = 0$, so $X(L)$ is exactly the off-diagonal\ncancellation that a window of length $L$ fails to collect. At $x = 7$ brute\nforce over all $30030$ rotations returns $\\operatorname{Var} = 1.05282410$ and\nboth Theorem 2 and this restatement reproduce it to $2\\times10^{-16}$. The nine\nlevels give $X = 4.61,\\ 15.08,\\ 29.56,\\ 51.40,\\ 81.29,\\ 121.48,\\ 173.62,\\\n236.57,\\ 313.22$, a column carrying four figures because it is divided out of a\n$\\operatorname{Var}/\\mathbb{E}$ column printed to four decimals.\n\n**Proposition 4 (the density half).** On the diagonal,\n$\\delta \\ln^2 W \\to \\kappa := 16\\,C_2\\,e^{-2\\gamma}/3 = 1.109905$, with $C_2$\nthe twin prime constant. Hence $\\lim \\operatorname{Var}/\\mathbb{E}$ exists if\nand only if $\\lim X/\\ln^2 W$ does, and the two differ by exactly $\\kappa$.\n\n*Proof.* $\\prod_{2<p\\le y}(1-2/p) = \\big[\\prod_{2<p\\le y}\\frac{1-2/p}{(1-1/p)^2}\\big]\\big[\\prod_{2<p\\le y}(1-1/p)\\big]^2 \\sim C_2\\,(2e^{-\\gamma}/\\ln y)^2$\nby Mertens. The comb's small primes give $\\delta = \\frac13\\prod_{2<p\\le y}(1-2/p)$,\nand $\\ln y = \\frac12\\ln L\\,(1+o(1))$ on the diagonal. $\\square$\n\nMeasured $\\delta\\ln^2 W$ at the nine levels: $0.9426$, $1.0198$, $1.0770$,\n$1.0982$, $1.1058$, $1.1082$, $1.1093$, $1.1096$, $1.1098$, monotone from below\nand $9.2\\times10^{-5}$ short of $\\kappa$ at $x = 37$. So the target is a single\ncoefficient in front of $\\ln^2 W$, and the density side contributes no\nuncertainty to it: $\\lim \\operatorname{Var}/\\mathbb{E} = c$ says exactly\n$X \\sim (c/\\kappa)\\ln^2 W$.\n\n**Proposition 5 (the main term is exactly $L$).** Write\n$f_p(d) = p\\rho_p(d)/(p-2)^2$ for $7 \\le p \\le y$, so that\n$W(d) = \\frac{30}{4}\\rho_{30}(d)\\prod_p f_p(d)$, and factor out the generic\nvalue:\n\n$$f_p(d) = a_p\\Big[1 + \\frac{2\\cdot\\mathbf 1[p\\mid d] + \\mathbf 1[p \\mid d-2] + \\mathbf 1[p\\mid d+2]}{p-4}\\Big],\n\\qquad a_p = 1 - \\frac{4}{(p-2)^2}.$$\n\nExpanding over squarefree conductors and keeping the leading term of each, the\ntotal leading term of $\\sum_{|d|<L}(1-|d|/L)W(d)$ is\n$L\\prod_{7\\le p\\le y}\\big(1 + \\frac{4}{p(p-4)}\\big)\\big(1 - \\frac{4}{(p-2)^2}\\big)$,\nand every factor is $1$:\n\n$$\\Big(1 + \\frac{4}{p(p-4)}\\Big)\\Big(1 - \\frac{4}{(p-2)^2}\\Big)\n= \\frac{(p-2)^2}{p(p-4)}\\cdot\\frac{p(p-4)}{(p-2)^2} = 1 .$$\n\n*Proof.* The displayed identity is $p(p-4) + 4 = (p-2)^2$, which is the sum\nrule of §2 read one prime at a time. The three indicators are mutually\nexclusive for $p \\ge 7$, so the expansion is a partition of the conductor\nclasses and carries no cross terms. $\\square$\n\nConsequence, and it is the reason one named route does not open. The\nsingular-series averages of Montgomery–Soundararajan work by extracting an\nasymptotic main term of the form $H - \\frac12\\ln H + O(1)$; here the\ncorresponding main term is not asymptotic but exact, so\n$\\sum_{|d|<L}(1-|d|/L)W(d) = L + X(L)$ with nothing left for that machinery to\ncompute. All of $\\operatorname{Var}/\\mathbb{E}$ is accumulated discrepancy of\nthe conductor classes. The identity was re-checked over $1{,}117{,}922$ primes\nwith zero failures (`research/history/staging/redteam-0828-varE.js`).\n\n**Lemma 6 (which conductors can contribute).** With $K_L$ the Fejér kernel and\n$r_q = L \\bmod q$,\n\n$$\\sum_{\\nu \\ne 0 \\bmod q} K_L(\\nu/q) \\;=\\; \\frac{r_q\\,(q - r_q)}{L}.$$\n\n*Proof.* $\\frac1q\\sum_{\\nu \\bmod q} K_L(\\nu/q) = \\sum_{|d|<L,\\ q \\mid d}(1-|d|/L)$,\nand summing that arithmetic progression gives $[L + r_q(q-r_q)/L]/q$; subtract\n$K_L(0) = L$. $\\square$\n\nTwo corollaries, both used in §9. If $q \\mid L$ then $r_q = 0$ and the\nconductor contributes exactly zero. The diagonal takes $L = x\\#$, so **every\nconductor built only from primes $p \\le x$ is inert**, and those primes carry\n$0.875, 0.763, \\ldots, 0.539$ of the Mertens mass at $x = 7 \\ldots 37$. And the\nsingle-prime band is bounded by $O(\\delta\\,\\pi(y)/L)$, measured\n$1.1\\times10^{-11}$ at $x = 37$ against $\\operatorname{Var}/\\mathbb{E} = 0.3958$,\nso the whole ratio lives in conductors with many prime factors and no low-order\ntruncation of the spectral sum reaches it.\n\n**One repair, refuted.** The standard next move is to keep the conductor sum\nonly for $q \\lesssim L$ and treat the tail as lost. With\n$u(q) = 4^{\\omega(q)}/\\prod_{p\\mid q}(1-4/p)$ and\n$\\sum_{q\\le Q}u(q) \\asymp Q\\ln^3 Q$ that model predicts $X \\asymp \\ln^3 W$.\nMeasured $X/\\ln^3 W$ reads $0.0324, 0.0270, 0.0226, 0.0195, 0.0171, 0.0151,\n0.0134, 0.0120$ over $x = 11 \\ldots 37$: falling by a factor $2.7$ and still\nfalling, so the model is refuted. In the same range $X/\\ln^2 W$ rises and\n$X/(\\ln^2 W\\ln\\ln W)$ falls, which brackets $X$ strictly between $\\ln^2 W$ and\n$\\ln^2 W \\ln\\ln W$ on the computed levels. The overshoot is a full logarithm,\nand it is the price of the positive expansion above having thrown away the\nsigns. Section 9 recovers the missing logarithm as one factor per prime.\n\n## 9. A decoupled model, and its limit theorem\n\nThis section proves a limit theorem about a model. Whether the model has the\nsame limit as $\\operatorname{Var}/\\mathbb{E}$ is §10's conjecture, and nothing\nhere settles it. The producers are\n`research/history/staging/varE-spectral.js` (the model, the closed form, the\n$\\theta = 1$ identity) and `research/history/staging/varE-limit-theorem.js`\n(the three bands, exact at nine levels).\n\n**The replacement.** In the Fourier form of $X$, the local coefficient at a\nnonzero frequency is $\\hat f_p(\\nu) = |1 + e(2\\nu/p)|^2/(p-2)^2$, which ranges\nover $[0, 4/(p-2)^2]$ with mean\n\n$$m_p \\;:=\\; \\frac{1}{p-1}\\sum_{\\nu\\ne0}\\hat f_p(\\nu) \\;=\\; \\frac{p - \\alpha_p}{\\alpha_p(p-1)},$$\n\n$\\alpha_p$ being the number of comb classes surviving at $p$\n($\\alpha_2 = \\alpha_3 = 1$, $\\alpha_5 = 2$, $\\alpha_p = p-2$ for $p \\ge 7$).\n**Decoupling** means replacing $\\hat f_p(\\nu)$ by $m_p$ at every $\\nu \\ne 0$.\nTwo facts about the step, before anything is built on it. It is an identity\nwhen one class is excluded per prime, since $\\hat f_p$ is then constant in\n$\\nu$. And it is exactly the mean-against-maximum accounting that recovers the\nlogarithm §8 lost: bounding by the maximum carries weight $4^{\\omega(q)}$ and\ngives $\\ln^3 W$, while the mean carries $2^{\\omega(q)}$ and gives $\\ln^2 W$.\n\n**The model.** Write $\\Omega = \\{0,1\\}^{\\{p \\le y\\}}$ with the product measure\nunder which the coordinate at $p$ is $1$ with probability\n$\\pi_p = (p-\\alpha_p)/(p-1)$, independently, so $\\pi_2 = \\pi_3 = 1$,\n$\\pi_5 = 3/4$ and $\\pi_p = 2/(p-1)$ for $p \\ge 7$. Let $n$ be the product of\nthe primes whose coordinate is $1$, a squarefree $y$-smooth integer divisible\nby $6$, and put\n\n$$g(n) \\;=\\; \\Big\\{\\frac{L}{n}\\Big\\}\\Big(1 - \\Big\\{\\frac{L}{n}\\Big\\}\\Big)\\frac{n}{L},\n\\qquad \\delta X_{\\rm dec} \\;:=\\; \\mathbb{E}[g(n)].$$\n\nUnder the replacement, expanding the product over the subset of primes where\nthe coefficient is not flattened and applying Lemma 6 with $n = M/e$ turns the\nspectral sum into $\\frac1L\\mathbb{E}[r_n(n-r_n)]$; size-biasing by $n$\npreserves independence and carries the inclusion probability from $m_p$ to\n$\\pi_p = m_p\\alpha_p$, which is the display above. Two identities fall out and\nare used below, one under each law: under the pre-bias law $\\Theta$, which\nincludes $p$ with probability $m_p$, $\\mathbb{E}_\\Theta[n] = \\prod_{p\\le y}p/\\alpha_p = 1/\\delta$;\nunder the size-biased law, which includes $p$ with probability $\\pi_p$,\n$\\mathbb{E}_{\\rm sb}[1/n] = \\prod_{p\\le y}\\alpha_p/p = \\delta$, checked to\n$2.4\\times10^{-15}$ at all nine levels. The two laws are related by\n$\\Pr_{\\rm sb}(n) = \\delta\\, n\\,\\Pr_\\Theta(n)$, and the first moment must be read\nunder $\\Theta$: under the size-biased law $\\mathbb{E}[n]$ is $648$ at $x = 7$,\nnot $1/\\delta = 91/3$ (exact rational enumeration of the $16$ states). Lemma 6's inert conductors are\nautomatic here: $n \\mid L$ forces $\\{L/n\\} = 0$, and in fact $g(n) = 0$ if and\nonly if $n \\mid L$.\n\nWrite $D_y = \\ln n/\\ln y$ and $u_y = \\ln L/\\ln y$, so that $n > L$ if and only\nif $D_y > u_y$.\n\n**Theorem 7.** For every fixed $s > 0$,\n\n$$\\ln \\mathbb{E}\\big[e^{-sD_y}\\big] \\;=\\; 2\\int_0^1 \\frac{e^{-sw}-1}{w}\\,dw \\;+\\; O_s\\!\\Big(\\frac{\\ln\\ln y}{\\ln y}\\Big),$$\n\nand consequently $D_y \\to GD(2)$ in distribution, the generalized Dickman law\nof index $2$.\n\n*Proof.* Put $w_p = \\ln p/\\ln y \\in (0,1]$ and $\\phi_p = e^{-sw_p} - 1$, so\nthat $\\mathbb{E}[e^{-sD_y}] = \\prod_{p\\le y}(1 + \\pi_p\\phi_p)$ with every\nfactor positive.\n\n(i) *Linearisation.* $|\\pi_p\\phi_p| \\le 1/3$ for $p \\ge 7$, and\n$|\\pi_p\\phi_p| \\le s\\ln 5/\\ln y \\le 1/2$ for $p \\in \\{2,3,5\\}$ once\n$\\ln y \\ge 2s\\ln 5$, so every factor admits $\\ln(1+z) = z + O(z^2)$. Since\n$\\sum_p (\\pi_p\\phi_p)^2 \\le 3(s\\ln5/\\ln y)^2 + (s^2/\\ln^2 y)\\sum_{p\\ge7}4\\ln^2p/(p-1)^2\n= O_s(1/\\ln^2 y)$, we get\n$\\ln\\mathbb{E}[e^{-sD_y}] = \\sum_p \\pi_p\\phi_p + O_s(1/\\ln^2 y)$.\n\n(ii) *The exact $\\pi_p$ against the rate $2/p$.* Write $\\pi_p = 2/p + \\epsilon_p$,\nso $\\epsilon_p = 2/(p(p-1))$ for $p \\ge 7$ and $|\\epsilon_p| \\le 1$ for\n$p \\in \\{2,3,5\\}$. With $|\\phi_p| \\le \\min(1, sw_p) \\le s\\ln p/\\ln y$,\n$|\\sum_p \\epsilon_p\\phi_p| \\le (s/\\ln y)\\sum_p|\\epsilon_p|\\ln p = O_s(1/\\ln y)$,\nthe tail converging. This is the only place the value $2$ enters, and it enters\nas a convergent sum: the deviation of the model's inclusion rate from $2/p$ is\nsummable against $\\ln p$ and cannot move the limit.\n\n(iii) *Mertens.* Fix $t_0 = 3$. The primes below $t_0$ contribute\n$O_s(1/\\ln y)$. Above $t_0$, write $\\sum_{p\\le t}1/p = \\ln\\ln t + M + r(t)$\nwith $|r(t)| \\le c/\\ln t$ unconditionally, and $\\phi(w) = e^{-sw}-1$. Then\n$\\sum_{t_0<p\\le y}\\phi(\\ln p/\\ln y)/p$ splits as\n$\\int_{t_0}^y \\phi(\\ln t/\\ln y)\\,\\frac{dt}{t\\ln t} + \\int_{t_0}^y \\phi(\\ln t/\\ln y)\\,dr(t)$.\nThe substitution $w = \\ln t/\\ln y$ turns the first into\n$\\int_0^1\\phi(w)\\,dw/w + O(s/\\ln y)$, since $|\\phi(w)| \\le sw$ near $0$.\nIntegration by parts turns the second into boundary terms $O(1/\\ln y)$ plus\n$(s/\\ln y)\\int_{t_0}^y (c/\\ln t)\\,dt/t = O(s\\ln\\ln y/\\ln y)$.\n\n(iv) *Identification.* $\\exp\\big(\\theta\\int_0^1(e^{-\\lambda x}-1)\\,dx/x\\big)$ is\nthe Laplace transform of $GD(\\theta)$, whose density is\n$p_\\theta = e^{-\\theta\\gamma}\\varrho_\\theta/\\Gamma(\\theta)$, where\n$\\varrho_\\theta(x) = x^{\\theta-1}$ on $(0,1]$ and\n$x\\varrho_\\theta' + (1-\\theta)\\varrho_\\theta + \\theta\\varrho_\\theta(x-1) = 0$\nbeyond ($\\varrho_1$ is Dickman's function; the letter is distinct from\nTheorem 1's $\\rho_p$),\nwhose mean is $\\theta$, and which satisfies $p_\\theta(x) \\le C_\\theta/\\Gamma(x+1)$\nfor $x \\ge 1$ (Pinsky, arXiv:1611.07207v3, pp. 2–3). Convergence of Laplace\ntransforms of non-negative random variables on $s > 0$ gives convergence in\ndistribution. $\\square$\n\nTwo remarks. The rate is measured as well as asserted: the model's exact\ntransform against $GD(2)$'s at $s = 0.5, 1, 2, 4$ on all nine levels has error\ntimes $\\ln y/\\ln\\ln y$ bounded, which is the predicted shape and not a settled\nconstant. And $GD(2)$ has a bounded density, $p_2(t) = e^{-2\\gamma}t$ on\n$(0,1]$, with a continuous distribution function, which is what Theorem 8 uses.\n\n**Theorem 8.** Let $y \\to \\infty$ with $u_y \\to u > 0$. Then\n\n$$\\mathbb{E}[g] \\;\\longrightarrow\\; \\Pr[GD(2) > u] \\;=:\\; \\lambda_2(u),$$\n\nwith no correction term: the transition region $n \\asymp L$, where $g$ is\nneither $0$ nor $1$, contributes nothing in the limit.\n\n*Proof.* Read the three bands off $g(n) = \\{L/n\\}(1-\\{L/n\\})n/L$. If $n \\mid L$\nthen $g(n) = 0$ exactly. If $n > L$ then $\\{L/n\\} = L/n$ and $g(n) = 1 - L/n$\nexactly. If $n < L$ and $n \\nmid L$ then $0 < g(n) \\le n/(4L)$. So with\n$P_> = \\Pr[n > L]$, $R = \\mathbb{E}[(L/n)\\mathbf 1_{n>L}]$ and\n$B = \\mathbb{E}[g\\,\\mathbf 1_{n<L}]$,\n\n$$\\mathbb{E}[g] \\;=\\; P_> \\;-\\; R \\;+\\; B, \\qquad \\text{all four exact.}$$\n\nSince $n > L$ iff $D_y > u_y$, and $GD(2)$ has a continuous distribution\nfunction, Theorem 7 and Pólya's theorem give $P_> \\to \\lambda_2(u)$. Both\nremainders are dominated by\n$\\mathcal{M} := \\mathbb{E}[\\min(n/L, L/n)] = \\mathbb{E}[e^{-\\ln y\\,|D_y - u_y|}]$,\nbecause $R \\le \\mathcal{M}$ and $0 \\le B \\le \\mathcal{M}/4$. Fix $\\epsilon > 0$. On $|D_y - u_y| > \\epsilon$\nthe integrand is at most $y^{-\\epsilon} \\to 0$; on $|D_y - u_y| \\le \\epsilon$ it\nis at most $1$, and\n$\\limsup_y \\Pr[|D_y - u_y| \\le \\epsilon] = \\Pr[|GD(2) - u| \\le \\epsilon] \\le 2\\epsilon\\sup p_2$.\nSo $\\limsup \\mathcal{M} \\le 2\\epsilon \\sup p_2$ for every $\\epsilon$, hence\n$\\mathcal{M} \\to 0$ and both remainders vanish. $\\square$\n\nWhat this settles. The closed form omits nothing: the transition band\nis a two-sided window of width $O(1/\\ln y)$ in $D_y$ and dies against a bounded\nlimiting density, so no term is added to $\\lambda_2(u)$ and none subtracted. No\nlocal limit theorem is needed, only weak convergence and boundedness. And\nnothing in either proof uses the value $\\theta = 2$: step (ii) is the only place\nit enters, and it enters as summability, so any excluded-class count $\\theta$\ngives $\\lambda_\\theta(u)$ by the same two arguments.\n\n**Lemma 9 (block bound).** Let $h$ be multiplicative, supported on squarefree\nintegers whose prime factors lie in $[5,y]$, with $h(5) = 3$ and\n$h(p) = 2/(p-3)$ for $7 \\le p \\le y$. Then\n$\\sum_{N < m \\le eN} h(m) \\le K(1 + \\ln 2N)$ for every $N \\ge 1$, with $K$\nabsolute and in particular uniform in $y$.\n\n*Proof.* $h(m) \\le c(m)/m$ with $c = m\\,h(m)$ multiplicative, squarefree\nsupported, $c(5) = 15$ and $c(p) = 2 + 6/(p-3)$ for $7 \\le p \\le y$, so the\nblock is at most $N^{-1}\\sum_{m\\le eN}c(m)$. Let\n$q_y = \\mu^2\\,2^{\\omega}\\,\\mathbf 1[y\\text{-smooth, coprime to }6]$; since\n$\\mu^2(m)2^{\\omega(m)} = \\sum_{ab=m}\\mu^2(a)\\mu^2(b)$ for squarefree $m$,\n$\\sum_{m\\le M}q_y(m) \\le \\sum_{a \\le M}\\lfloor M/a\\rfloor \\le M(1+\\ln M)$.\nWrite $c = q_y * b$; locally $\\sum_k b(p^k)z^k = (1+c(p)z)/(1+2z)$, so\n$b(p^k) = (-2)^{k-1}(c(p)-2)$ and\n$\\sum_k |b(p^k)|/p^k = |c(p)-2|/(p-2) = 6/((p-3)(p-2))$ for $7 \\le p \\le y$,\nsummable over primes, with the single prime $5$ contributing a finite factor.\nHence $\\sum_m |b(m)|/m \\le K_0 < \\infty$ absolutely, and\n$\\sum_{m\\le M}c(m) = \\sum_{d\\le M}b(d)\\sum_{e\\le M/d}q_y(e) \\le K_0\\,M(1+\\ln M)$. $\\square$\n\n**Lemma 10 (local bound).** Uniformly in $T > 0$ and $y$,\n$\\Pr[\\ln n \\in (T-1, T]] \\le K'(1+T)/\\ln^2 y$.\n\n*Proof.* The support is $n = 6m$ with $m$ as in Lemma 9, and the weight is\n$w(6m) = C_y h(m)$ with\n$C_y = \\prod_{5\\le p\\le y}(1-\\pi_p) = \\frac14\\prod_{7\\le p\\le y}\\frac{p-3}{p-1}$.\nSince $(p-3)/(p-1) = (1-1/p)^2\\big(1 - (3p-1)/(p-1)^3\\big)$ and\n$\\sum_p (3p-1)/(p-1)^3$ converges, Mertens gives $C_y \\asymp 1/\\ln^2 y$. Apply\nLemma 9 with $N = e^{T-1}/6$. $\\square$\n\n**Proposition 11 (the two transition bands).** With $L = y^{u+o(1)}$ and $u$\nfixed, $R = O_u(1/\\ln y)$ and $B = O_u(1/\\ln y)$, hence\n$\\mathbb{E}[g] = \\Pr[n > L] + O_u(1/\\ln y)$.\n\n*Proof.* Bin by $j \\ge 0$ and use Lemma 10. For the lower band,\n$\\mathbb{E}[(n/L)\\mathbf 1_{n<L}] \\le \\sum_j e^{-j}\\Pr[\\ln n \\in (\\ln L - j - 1, \\ln L - j]]\n\\le K'(1+\\ln L)\\ln^{-2}y\\sum_j e^{-j} = O_u(1/\\ln y)$; for the upper band the\nsame binning gives $\\sum_j e^{-j}(2 + \\ln L + j)K'/\\ln^2 y = O_u(1/\\ln y)$. $\\square$\n\nProposition 11 bounds the bands. It does **not** bound the model's distance\nfrom $GD(2)$'s tail: converting Theorem 7's transform rate into a distribution\nrate needs an Esseen-type inequality, and that conversion is not done here. So\nthe model's own rate of approach to its limit is measured and not derived.\n\n**The constant.** At $\\theta = 2$ the density is $p_2(t) = e^{-2\\gamma}t$ on\n$(0,1]$ and $p_2(t) = e^{-2\\gamma}[t(3-2\\ln t) - 2]$ on $(1,2]$, so\n\n$$\\lambda_2(2) \\;=\\; 1 - e^{-2\\gamma}\\Big(\\tfrac92 - 4\\ln 2\\Big) \\;=\\; 0.45545648,$$\n\nequivalently $\\lim X/\\ln^2 W = \\lambda_2(2)/\\kappa = 0.410$ under §10's\nconjecture. The number is confirmed to $2.9\\times10^{-11}$ by three independent\nroutes: the closed form re-derived from the Laplace transform, a Simpson solve\nof the delay equation at two step sizes, and a Monte Carlo of the limit process\nitself, which samples a Poisson process of intensity $2\\,dw/w$ on $(0,1]$ and\nso tests the normalising constant rather than assuming it\n(`research/history/staging/redteam-0828-varE.js`). The tail at the other\nexponents, which §6's $u$-sweep can be read against:\n\n| $u$ | 0.6 | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 |\n|---|---|---|---|---|---|---|\n| $\\lambda_2(u)$ | 0.9433 | 0.8424 | 0.6571 | 0.4555 | 0.2820 | 0.1577 |\n\n**The $\\theta = 1$ case is Gorodetsky's, exactly.** Run the same construction\nwith one excluded class per prime. The decoupling replacement is then vacuous,\nbecause $\\hat f_p(\\nu \\ne 0) = 1/(p-1)^2 = m_p$ identically, and the model's\nvalue is his $M(H,y)$: every $n$ is even since $\\pi_2 = 1$, and writing\n$n = 2n'$ turns $\\mathbb{E}[\\{L/n\\}(1-\\{L/n\\})n/L]$ into\n$\\prod_{2<p\\le y}(1-2/p)\\sum_{n}g_y(n)\\{H/2n\\}(1-\\{H/2n\\})$ term by term\n(Gorodetsky, *Math. Z.* **308** (2024) no. 4, Paper 59, eqs. (1.5) and (1.6),\np. 2; arXiv:2111.00853v3), checked at\n$(y,H) = (13,210),\\ (13,2310),\\ (29,30030)$ to relative $1.7\\times10^{-16}$.\nTheorem 8 at $\\theta = 1$ returns\n$\\lambda_1(u) = e^{-\\gamma}\\int_u^\\infty \\varrho_1$, which is his $\\lambda(u)$,\nand his (1.12) read at the page gives $\\lambda(u) = 1 - e^{-\\gamma}u$ on\n$[0,1]$, matching our $0.663124$ at $u = 0.6$ and $0.438541$ at $u = 1$ to six\ndecimals. His Lemma 1.5 (p. 5) makes a primorial window his own case with no\nhypotheses, and his Theorem 1.3(1) then gives the asymptotic for\n$y \\ge \\exp((\\log\\log H)^{5/3+\\varepsilon})$, which the diagonal satisfies. So\nat one excluded class the whole chain from the exact finite-level variance to\n$\\lambda_1(u)$ is a published theorem about exactly this object, and Theorem 8\nagrees with it there.\n\nThe value of that check to $\\theta = 2$ is limited, and we state it as such:\nthe decoupling step is invisible at one class precisely because the local\ncoefficient is constant there, which says nothing about two. Nor is\n$\\lambda_\\theta$ ours. $GD(\\theta)$ is published probability, and calling\n$\\lambda_\\theta(u) = \\Pr[GD(\\theta) > u]$ a new function would be wrong. What\nwe have not found in print, searched in the owning convention (the variance of $y$-rough integers in short intervals, Gorodetsky's; `research/SEARCH-CONVENTIONS.md` §1) is the identification of that tail with a sifted\nwindow variance at $\\theta > 1$, and the search behind that sentence is\nincomplete: Gorodetsky's paper is $\\kappa = 1$ throughout (checked at source,\nwith `tuple`, `twin`, `admissible`, `generalized`, `dimension` and `Poisson`\nabsent from the full text), Aryan's Lemma 1.2 is the same object at general\ntuple size with an upper bound and no asymptotic, and the successor question is\nowed four channels. Nothing here is claimed as new.\n\n**The three bands, exact at nine levels.** The decomposition\n$\\mathbb{E}[g] = P_> - R + B$ of Theorem 8's proof is computable without\nenumerating anything above $L$: the conductors $n \\le L$ are enumerated one at a\ntime by descent over the primes, and $\\mathbb{E}[1/n] = \\delta$ supplies the\nrest. There is no Monte Carlo in the columns below\n(`research/history/staging/varE-limit-theorem.js`, 4930 s, of which 4792 s is\nthe $x = 37$ level; `research/varE-exact-ladder-01.js` is an independent\nsecond route).\n\n| $x$ | $\\Pr[n \\mid L]$ | $P_>$ | $R$ | $B$ | $\\mathbb{E}[g]$ |\n|---|---|---|---|---|---|\n| 7  | 0.666666667 | 0.283333333 | 0.108197358 | 0.003056943 | 0.178192918 |\n| 11 | 0.417387328 | 0.335525028 | 0.081053290 | 0.012367138 | 0.266838875 |\n| 13 | 0.296671934 | 0.350725586 | 0.066478692 | 0.012344728 | 0.296591623 |\n| 17 | 0.212414912 | 0.368116926 | 0.052466671 | 0.009853657 | 0.325503912 |\n| 19 | 0.160547361 | 0.382620145 | 0.043864663 | 0.008044066 | 0.346799547 |\n| 23 | 0.123952461 | 0.394436926 | 0.037234816 | 0.006640313 | 0.363842423 |\n| 29 | 0.096758571 | 0.403580143 | 0.032052128 | 0.005616554 | 0.3771445 |\n| 31 | 0.078138150 | 0.410607924 | 0.028055939 | 0.004855050 | 0.387405 |\n| 37 | 0.063810103 | 0.416225089 | 0.024823834 | 0.004252203 | 0.39566 |\n\nThe last three values of $\\mathbb{E}[g]$ are quoted short because $R$ is a\ndifference of two quantities of size $L\\delta = 9.4\\times10^9$ at $x = 37$ and\ncarries a float floor. The witness spread across two independent routes is\n$7.5\\times10^{-8}$, $3.2\\times10^{-6}$, $1.3\\times10^{-5}$ at $x = 29, 31, 37$.\nEvery other column is a direct sum with no cancellation and agrees between the\ntwo routes to every printed digit, and the recomposition $P_> - R + B$ returns\n$\\mathbb{E}[g]$ with residual $0$ at all nine levels.\n\nThree readings. The inert band is not a small set, carrying $6.4\\%$ of the\nconductor mass at $x = 37$, and it contributes exactly zero at every finite\nlevel rather than approximately zero. The lower transition band is\n$B\\ln y = 0.063616, 0.064678, 0.064691, 0.063820, 0.063440, 0.063175, 0.063008$\nat $x = 13 \\ldots 37$: bounded, as Proposition 11 requires, and settling on\n$0.0630$ against the local-density prediction $0.062729$, while its share of\n$\\mathbb{E}[g]$ falls from $4.2\\%$ to $1.1\\%$. And the model's whole distance\nfrom its own limit is $(\\lambda_2(2) - \\mathbb{E}[g])\\ln y = 0.817, 0.847,\n0.872, 0.880, 0.884, 0.885, 0.886$ over the same levels, settling on $0.886$.\nThat last number is a measurement, not a derived rate, and it prices how far\nthe computable levels sit from the constant: $\\mathbb{E}[g] = 0.45$ needs\n$\\ln y \\approx 162$, that is $W \\approx e^{325}$. No level of the present kind\ncomes near $0.4555$, and no measurement of the constant exists to check it\nagainst. The model's own value at the tenth level is on record,\n$0.402368 \\pm 3.4\\times10^{-5}$ at $x = 41$ by Monte Carlo on the same conductor\nlaw (`research/varE-exact-ladder-01.js`, SEC 3), the band a tenth diagonal\npoint would be read against.\n\n**The model against the data, with no fitted parameter.** The model's only\ninputs are $\\alpha_p$, $y$ and $L$. Against §7's nine diagonal points the\nresiduals, measured minus model, are $-0.0261$ and $-0.0106$ at $x = 7, 11$,\nthen $+0.0029$, $+0.0013$, $+0.0005$, $+0.0005$, $+0.0003$, $+0.0002$ from\n$x = 13$ to $31$, all exact on both sides, and $+0.0001$ at $x = 37$ where the\nmeasured column's own rounding is $\\pm 5\\times10^{-5}$. Against §6's six-point $u$-sweep at $y = 401$, which is a\ndifferent object (the full set $A$, all three mod-30 houses), it returns\n$0.8543, 0.7025, 0.4842, 0.2927, 0.1576, 0.0764$ (Monte Carlo, $N = 2\\times10^5$,\nbars $5$ to $10\\times10^{-4}$; `research/history/staging/varE-spectral.js`,\nPart 5) against the note's $0.845, 0.685, 0.477, 0.290, 0.157, 0.076$; against\n§6's seven-point $u = 2$ level sweep it returns $0.2541 \\ldots 0.3232$, same\nMonte Carlo, against $0.251 \\ldots 0.321$. The model is close over twenty-two\npoints and is exact nowhere.\n\n## 10. The identification with the true variance: conjectured, with one named gap\n\n**Conjecture 1.** On the diagonal $L = W = x\\#$, $y$ the largest prime\n$\\le \\sqrt W$, the decoupling replacement is asymptotically harmless:\n\n$$\\delta\\,\\big(X(L) - X_{\\rm dec}(L)\\big) \\;\\longrightarrow\\; 0 ,\n\\qquad\\text{hence}\\qquad\n\\lim \\frac{\\operatorname{Var}}{\\mathbb{E}} \\;=\\; \\lambda_2(2) \\;=\\; 1 - e^{-2\\gamma}\\Big(\\tfrac92 - 4\\ln2\\Big) \\;=\\; 0.45546 .$$\n\nThis is conjectured, not proven, and it is the one step between §9's theorem\nand any statement about the variance of §7's table. The chain is: link 1,\n$\\operatorname{Var}/\\mathbb{E} = \\delta X$, proven (§8); link 2, this\nconjecture, open; link 3, $\\delta X_{\\rm dec} = \\mathbb{E}[g(n)]$, proven given\nthe replacement (§9); link 4, $\\mathbb{E}[g] \\to \\lambda_2(u)$, proven\n(Theorem 8); link 5, the closed form, proven. Exactly one link is unproven, and\nit is the one that connects the model to the arithmetic.\n\n**What the error is, exactly.** In real space the replacement flattens\n$W$ to $V(h) = \\prod_{p \\le y}\\big(1 + m_p\\,c_p(h)\\big)$ with $c_p(h) = p - 1$\nif $p \\mid h$ and $-1$ otherwise (the Ramanujan sum), that is\n\n$$V(h) = 6\\cdot\\mathbf 1[6 \\mid h]\\Big(\\tfrac58 + \\tfrac{15}{8}\\mathbf 1[5 \\mid h]\\Big)\\,\nC_y\\prod_{p \\mid h,\\ 7\\le p\\le y}\\frac{p-1}{p-3},\n\\qquad C_y = \\prod_{7\\le p\\le y}\\Big(1 - \\frac{2}{(p-1)(p-2)}\\Big);$$\n\nthe factors at $2$, $3$ and $5$ are part of the kernel (an earlier version of\nthis display kept only the product over $p \\ge 7$; at $x = 7$ the all-prime\nform returns $\\delta X_{\\rm dec} = 0.178192918$, the table's value, and the\ntruncated one $0.001459$), so\n\n$$X - X_{\\rm dec} \\;=\\; \\sum_{|h|<L}\\Big(1 - \\frac{|h|}{L}\\Big)\\big(W(h) - V(h)\\big),$$\n\na sum over shifts rather than a per-prime defect. Over $p \\ge 7$ the exact $W$\nsplits into $3^{\\omega}$ groups according to which of $p \\mid h$,\n$p \\mid h-2$, $p \\mid h+2$ each prime takes.\n\n**Measured, exact, eight levels, no Monte Carlo**\n(`research/history/staging/varE-theta2-step.js` for $x \\le 23$,\n`research/varE-exact-ladder-01.js` for $x = 29, 31$):\n\n| $x$ | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 |\n|---|---|---|---|---|---|---|---|---|\n| $X/X_{\\rm dec}$ | 0.853427 | 0.960382 | 1.009802 | 1.003895 | 1.001448 | 1.001312 | 1.000725 | 1.000471 |\n| $\\delta(X - X_{\\rm dec})\\ln y$ | $-0.06699$ | $-0.04070$ | $+0.01498$ | $+0.00832$ | $+0.00404$ | $+0.00459$ | $+0.00309$ | $+0.00238$ |\n\nThe scaled error is bounded on every level computed and falls at every step\nabove $x = 23$. That excludes growth and nothing finer: it is equally\nconsistent with $O(1/\\ln y)$ and with anything smaller. It also does not license\na coefficient. The net at $x = 31$ is a difference of two terms fifty times its\nsize, $+0.13347$ and $-0.13109$, and the second is still moving in the third\ndecimal, so $0.0046$ or $0.0024$ is a number not to quote as a constant. The\nlevel $x = 37$ needs the real-space sieve at a priced $21{,}656$ s and was\ndeclined.\n\n**Two of the three groups are closed.** Both derivations are the project's own,\nelementary, and have had no adversarial pass, and they are reported at that rung\n(`research/history/staging/varE-theta2-proof.js`). The two pure $\\pm 2$ shift\ngroups satisfy $|2X_2| \\le 60\\prod_{7\\le p\\le y}p(p-3)/(p-2)^2 = O(\\ln y) = o(1/\\delta)$\nunconditionally. The measured group sum is $2X_2 = -0.5603, 0.5304, 0.2694,\n0.2314, 0.1846$ at $x = 7$ to $19$: the signed values are not monotone, their\nmagnitudes fall from $0.5603$ to $0.1846$ over the five sampled levels, the\nproved bound is $O(\\ln y)$ and no $O(1)$ bound is inferred from five values;\nthe bound is loose by a factor $511$ to $1148$ at $x = 13, 17, 19$. An earlier version of this\nparagraph quoted $2X_2 = 5.3$ and a looseness of $14$ to $48$; that column was\nthe positive half of one shift pattern doubled, not the group, and is withdrawn\n(`research/history/staging/verify-0830-record-defects.md` §1, VERIFIED at\n$x = 7$ to $19$ to every printed digit on 2026-08-30; the $x = 23$ entries are\nnot recomputed there). The same correction shrinks the mixed-lag residual:\n$\\delta X_{\\rm mix}$ reads $-0.034971, -0.046368, -0.025439, -0.020542,\n-0.016835$ over those levels, $2.3$ to $6.9$ times smaller than the column first\ncarried, with $X_{\\rm mix}/\\ln y$\nnear $-0.49$ on the top three levels, a coefficient measured on three points and\nnot settled.\nThe group where every prime takes $p \\mid h$ is the only one whose CRT class is\nzero, so Lemma 6 evaluates it verbatim with no equidistribution and no\nreplacement, and $\\delta(X_1 - X_{\\rm dec}) \\to 0$ unconditionally at rate\n$O((\\ln\\ln y)^{-1/4})$ by a monotone coupling of the two product measures plus\nKolmogorov–Rogozin anti-concentration for $\\ln n$. That rate is numerically\nvacuous at every computed level: its factor is $0.815$ at $x = 23$, so the\nproposition explains none of the eight measured rows.\n\n**What is left, and it is not an oscillation problem.** Every lag's\nFejér-weighted contribution is exact and elementary,\n\n$$R_n(c) \\;=\\; \\frac{(r-c)^+ + (r+c-n)^+ - r^2/n}{L}, \\qquad r = L \\bmod n,$$\n\nLemma 6 being its $c = 0$ case, verified to $1.71\\times10^{-13}$ on 4000 random\ntriples and rebuilding $X(210) = 4.612929$ from 192 (modulus, class) pairs. So\nthe CRT-mixed lags carry no phase that averages out over the frequency\nvariable, there is no exponential sum to estimate, and quoting Weil here would\nbe a category error. What remains is a counting statement. Writing $C$ for the\ninteger representative of the active lag and\n$w(n,C) = D_y\\prod_{p \\mid n}(2\\ \\text{or}\\ 1)/(p-4)$ according as $p \\mid C$ or\n$p \\mid C \\mp 2$, the estimate that would close Conjecture 1 is\n\n$$\\sum_{0<|C|<L}\\Big(1-\\frac{|C|}{L}\\Big)\\!\\!\\sum_{\\substack{n \\mid C(C^2-4),\\ n > 2L\\\\ P^+(n) \\le y}}\\!\\! w(n,C)\n\\;=\\; L\\sum_{n>2L}\\sum_{c\\ne0}\\frac{w(n,c)}{n} \\;+\\; O\\big(\\ln y \\cdot o(\\ln y)\\big),$$\n\nthat is: the weighted count of large $y$-smooth divisors of $C(C^2-4)$ agrees\nwith its expected count to one logarithm better than trivially, on average over\na window of length $L = y^2$. The owning conventions for that are the\ndistribution of divisors of a polynomial value in a dyadic range: Ford's\n$H(x,y,z)$, Hooley's $\\Delta$-function, and the Erdős multiplication-table\nproblem, applied to the reducible cubic $C(C-2)(C+2)$ with a smoothness\nconstraint at $u = 2$. That search was run on 2026-08-28 and 29\n(`research/history/staging/lit-smooth-divisors.md`,\n`research/history/staging/lit-scourfield-2008.md`; the row for large\n$y$-smooth divisors of $C(C^2-4)$ in `research/SEARCH-CONVENTIONS.md` §1): the\nno theorem among the results inspected gives the weighted, above-range\nestimate needed. The located set counts (Tenenbaum's $H_F$, Ford's $H(x,y,z)$)\nsit at divisor ranges strictly below the sampling range, while our divisors\nsatisfy $n > 2L$, above it. Scourfield's 2008 result is a divisor sum, every\ndivisor $m \\le x$ of $f(n)$ counted with weight $1$ over $n \\le x$, not a set\ncount; read second-hand through her 2016 restatement, with the chapter itself\nunread at the page and a degree hypothesis unresolved between the two\nstatements, it carries relative error $O(1/\\log x)$ where this step needs\n$o(1/\\ln y)$, one epsilon short. So the reduction is not claimed as new, and no\nabsence is asserted while the chapter is unread; the same one-logarithm gap is\nusually the whole difficulty in divisor problems of this shape.\n\nOne route through the exact kernel is already lost. The only split it offers,\nflat against active, has $\\delta\\,|{\\rm flat}|$ rising on all six levels where\nit is computed, $0.559, 0.912, 0.960, 1.087, 1.188, 1.301$ against a target of\nzero, so every absolute-value argument through that split fails before it\nstarts.\n\n**What would falsify Conjecture 1**, and whether the check has run: the\nconjecture asserts $\\delta(X - X_{\\rm dec}) \\to 0$, so it is killed by the\nunscaled error staying away from $0$ along an unbounded sequence of levels, or\nby a proof that the replacement shifts the limit; growth of the scaled quantity\n$\\delta(X - X_{\\rm dec})\\ln y$ at a level would not kill it (an error of size\n$1/\\sqrt{\\ln y}$ grows when scaled and still vanishes), and eight exact levels\nshow the scaled quantity bounded and falling, which excludes nothing about the\nlimit; neither killer exists either way. The tenth level $x = 41$ has been priced\nand declined, and by the registered forecast band it would retire one drift form\nand separate nothing else.\n\n## 11. Calibration: the fitted reading $0.611$, and why it is refuted\n\nAn earlier version of this note read the ninth level as favouring one drift law\nover another and reported its intercept, $0.611$, as the value the open question\nhad come down to. That inference is refuted, on both halves of the protocol that\nproduced it, and a reader of the old version needs the reason.\n\nThe two forms were $r = 0.4435 - 1.509/\\ln W$ (rms $1.8\\mathrm{e}{-3}$) and\n$r = 0.6106 - 0.729/\\ln\\ln W$ (rms $8.2\\mathrm{e}{-4}$), fitted before $x = 37$\nwas computed. At $x = 37$ they predict $0.3926$ and $0.3955$ against the\nmeasured $0.3958$, a factor of ten in residual, and refitting improves the\nsecond while degrading the first. Two corrections to that account, then the\nrefutation. The published coefficients are the six points $x = 13 \\ldots 31$ and\nthe seven points $x = 13 \\ldots 37$, not \"the first eight\" and \"all nine\" as the\nold text said; on the eight points $x = 7 \\ldots 31$ the $1/\\ln\\ln W$ form is\nthe *worse* of the two. And on the six points actually used, two further\ntwo-parameter forms fit at least as well, with intercepts $0.5387$ and $0.7219$,\nwhile a three-parameter member of the $1/\\ln W$ family fits better than all of\nthem with intercept $0.4642$.\n\nThe refutation is a control, not an argument. Section 9's model is a sequence\nthat tracks these nine points to $0.003$ and whose limit is $0.455456$ by\nconstruction. Run the same in-sample protocol on it and $1/\\ln\\ln W$ wins, with\nintercept $0.6164$ against $1/\\ln W$'s $0.4468$: the same winner, the same\nmargin, and the same two numbers the data give, on a sequence whose limit is\nneither. Run the frozen half, fitting on $x = 13 \\ldots 31$ and forecasting\n$x = 37$, and the control reproduces that too, $1/\\ln\\ln W$ residual $-0.00039$\nagainst $1/\\ln W$'s $+0.00276$, resolvable against the model's own\n$0.395567 \\pm 0.000106$. So neither the in-sample winner nor the out-of-sample\nseparation carries information about the limit\n(`research/history/staging/redteam-0828-varE.js`). Bias-corrected against the\ncontrol the seven forms' intercepts cluster in $[0.4471, 0.4546]$ around\n$\\lambda_2(2)$, and that is not independent evidence either: the intercept is a\nlinear functional $a = \\sum_i c_i y_i$ with $\\sum_i c_i = 1$, so the corrected\ncolumn is bounded a priori by $\\|c\\|_1\\max_i|{\\rm data}-{\\rm model}|$, and every\nobserved gap sits inside that bound. It restates that the model tracks the data\npointwise.\n\nStated plainly: this was model comparison, not measurement of a limit. Over the\nentire computable range $1/\\ln\\ln W$ moves only from $0.295$ to $0.596$, so the\nforms are separated on a short lever arm, and no finite computation of this kind\ndistinguishes one limit from another. Neither $0.611$ nor the $0.44$ reading is\nmeasured here, and no value of $\\lim\\operatorname{Var}/\\mathbb{E}$ is measured\nanywhere in this note. What replaces the fitted number is the reduction of §8,\n$\\lim\\operatorname{Var}/\\mathbb{E} = \\kappa\\lim X/\\ln^2 W$ with\n$\\kappa = 1.109905$ proven, together with §9's proven model limit and §10's one\nopen link.\n\n## Authorship and AI disclosure\n\nSole author: Chris Benjaminsen.\n\n> The framework, vocabulary, and driving questions are the author's,\n> developed over six years of independent work. Formal derivations,\n> literature audits, computations, and manuscript drafting were carried out\n> using an AI assistant operating under the author's\n> direction; all results were verified by explicit computation, with code\n> and outputs published in the accompanying repository, and all refuted\n> intermediate claims retained in the record.\n\nSections 8 to 11 were produced the same way, on 2026-08-28, and carry two\nqualifications of their own. The derivations of Theorem 7, Theorem 8,\nLemma 9, Lemma 10 and Proposition 11, and both closed groups of §10, are\nwritten out and checkable here, and each has been put against exact\ncomputation at nine levels by the producers named at the number. None has yet\nhad an external referee's pass. And the earlier readings this restructure\nreplaces, the $0.611$ intercept of §11 and the \"$u = 2$ exactly\" of §7, were\nboth corrected by the project's own adversarial pass of the same date, whose\nrecord is `research/history/staging/redteam-0828-varE.md`.\n\n## References\n\n- M. Hausman, H. N. Shapiro, *On the mean square distribution of primitive\n  roots of unity*, Comm. Pure Appl. Math. 26 (1973), 539–547.\n- H. L. Montgomery, R. C. Vaughan, *On the distribution of reduced residues*,\n  Ann. of Math. (2) 123 (1986), 311–333.\n- F. Aryan, *The distribution of k-tuples of reduced residues*, Mathematika 61\n  (2015), 72–88, DOI 10.1112/S0025579314000151; arXiv:1302.2296.\n- T. Bloom, V. Kuperberg, *Odd moments and adding fractions*, arXiv:2312.09021.\n- O. Gorodetsky, *The variance of integers without small prime factors in\n  short intervals*, Math. Z. 308 (2024), no. 4, Paper No. 59, DOI\n  10.1007/s00209-024-03601-w; arXiv:2111.00853.\n- R. G. Pinsky, *On the strange domain of attraction to generalized Dickman\n  distributions for sums of independent random variables*,\n  arXiv:1611.07207v3.\n- F. B. Holt, *On the counts of p-rough numbers*, arXiv:2308.07570.\n- H. L. Montgomery, K. Soundararajan, *Primes in short intervals*,\n  Comm. Math. Phys. 252 (2004), 589–617.\n- V. Schemmel, *Über relative Primzahlen*, J. reine angew. Math. 70 (1869);\n  OEIS A059861.\n- E. J. Scourfield, *Smooth divisors of polynomials*, in *Number Theory and\n  Polynomials*, LMS Lecture Note Series 352, CUP 2008, 286–311; read through\n  her restatement in Funct. Approx. Comment. Math. 55 (2016), eq. (1.1).\n- Project artifacts: `research/06-variance-theorem.js` (verification),\n  `paper/variance-note-numerics.js` (scaling data), `research/PRIOR-ART.md`\n  (novelty audit with sources). For §7: `research/natal5-variance.js` (the\n  $J_5$ closed form and its brute-force check), `research/natal-cap-16-fast-variance.js`\n  (the product-sieve engine, the BigInt sum rule, levels through $x = 31$),\n  `research/natal-cap-33-overnight.js` (level $x = 37$),\n  `research/natal-cap-26-minus-half.md` (the exact $-1/2$),\n  `research/natal-cap-29-sigma-plateau.js` (the spectral mass). For §§8–11:\n  `research/history/staging/varE-asymptotic.js` (the normalisation, $\\kappa$,\n  the conductor facts), `research/history/staging/varE-spectral.js` (the\n  model, the closed form, the $\\theta = 1$ identity),\n  `research/history/staging/varE-limit-theorem.js` (the three bands, exact at\n  nine levels), `research/history/staging/varE-theta2-step.js` and\n  `research/history/staging/varE-theta2-proof.js` (the replacement error and\n  its group split), `research/varE-exact-ladder-01.js` (the exact ratios at\n  $x = 29, 31$), `research/history/staging/redteam-0828-varE.js` (the three\n  routes to the constant and the fit control),\n  `research/history/staging/verify-0830-record-defects.md` (the $2X_2$\n  correction of §10), `research/history/staging/lit-smooth-divisors.md` and\n  `research/history/staging/lit-scourfield-2008.md` (the divisor search of\n  §10).\n"}