{"id":1880,"job_id":2777,"problem_id":1,"lane_id":null,"type":"explore","user_id":1,"model":"claude-opus-5-5","provider":"anthropic","report_md":"# Job #2777: route 120 pursuit: F/V at 1e7..1e12. The limit is 1 - gamma, not 1/2 (result)\n\nF/V does not tend to 1/2. It tends to 1 - gamma = 0.4227843, the classical mean of {x/n} (Dirichlet; de la Vallee Poussin 1898 for {x/p} over primes). The drift that route 120 read as \"1/2 not resolved at 1e8\" is a parameter-free second-order term. Neither branch of the step holds as written: the one-parameter law fails, and the non-vanishing gap 1/2 - F/V -> gamma - 1/2 = 0.0772 is not a correlation bias. It is the generic value for any summation set of slowly varying density.\n\nDerivation (partial summation). Write V(u) = u L(u) with L slowly varying. Then F = N int_1^oo {t} [L(N/t) + (N/t)L'(N/t)] t^-2 dt. Use I(s) = int_1^oo {t} t^(-s-1) dt = 1/(s-1) - zeta(s)/s = (1-gamma) - (1-gamma-gamma_1)(s-1) + ... This gives F/V = (1-gamma) + gamma_1 dlogL/dlogN + O(2nd order). With Landau's L = (lnln N + B1)/ln N:\nP(N) = (1-gamma) - gamma_1 (1 - 1/(lnln N + B1))/ln N, where gamma_1 = -0.0728158 (Stieltjes).\nThis is a derivation sketch with the error term not written out; the numbers below test it.\n\nMeasured (exact pair enumeration, C, integer N mod pq). Gates: V and F at 1e5..1e8 equal #1375's recorded values, and V(N) = A066265(N) - pi(sqrt N) at 1e7..1e12.\nN     F/V          P(N)         F/V-P      c_eff=(1/2-F/V)lnlnN\n1e7   0.425823234  0.425816617  +6.6e-6    0.20621\n1e8   0.425474835  0.425492243  -1.7e-5    0.21713\n1e9   0.425226627  0.425230952  -4.3e-6    0.22666\n1e10  0.425020596  0.425016068  +4.5e-6    0.23518\n1e11  0.424848890  0.424836264  +1.3e-5    0.24288\n1e12  0.424699207  0.424683603  +1.6e-5    0.24992\n\nPre-registered tests (prereg2777.md, sha fc0f4bd0, written before any run past 1e8):\nH0, the step's law 1/2 - F/V = c/lnln N, with c constant within 2% over 1e7..1e10: FAILS. c_eff rises monotonically, 6.8% max deviation from its mean.\nH1, ours, |F/V - P| <= 1e-4 at 1e9 and 1e10 and F/V falling: PASSES (4e-6, 5e-6).\nOut of sample at 1e12, the step's fits predict 0.433619 (one-parameter) and 0.425363 (c/lnlnN + c'/lnN, limit 1/2). The two-parameter fit has 9.5e-5 residual on the four step scales. P(1e12) = 0.424684 has no fitted parameter. Measured: 0.424699. The four-point two-parameter fit cannot tell the limits apart, and 1e12 rejects it by 6.6e-4, about 40x P's error.\n\nConsequence for the route's claim (2): C_F = lim F ln N/(N lnln N) = (1-gamma) lim V ln N/(N lnln N) = 1 - gamma = 0.42278. It is not 0.4696 +/- 0.001, and it is not 1/2. The recorded 0.47 is (1-gamma)(1 + B1/lnln N + ...) at N <= 1e8, a factor that decays like 1/lnln N. The closed form the route asks for (item ii) is therefore 1 - gamma, and (K') and the barrier theorem are untouched.\n\n## Method\nFirst came an online prior-art search. The pre-registration (prereg2777.md) was written and hashed before any run past 1e8. fv2777.c is a segmented odd sieve over q <= N/2 with pthreads; for every pair p < q, pq <= N it adds (N mod pq)/(pq) exactly in integers, and sums in double with Kahan. This is direct enumeration in place of #1375's pair-free Phi - A formula; the gates tie the two together. Measured runtimes on 6 threads: 1e11 15 s, 1e12 178 s; smaller scales take seconds. The step listed python3 + numpy. I rebuilt the computation as a small C scanner instead of locating the proposer's script, since no >1e8 script is served (#1877). analyze2777.py (stdlib) evaluates the pre-registered tests.\n\n## Output (analyze2777.out)\n```\n     N              V          F/V         P(N)      F/V-P    c_eff  (F/V-(1-g))lnN\n1e7           1903878  0.425823234  0.425816617  +6.62e-06  0.20621        0.048981\n1e8          17426029  0.425474835  0.425492243  -1.74e-05  0.21713        0.049561\n1e9         160785135  0.425226627  0.425230952  -4.32e-06  0.22666        0.050612\n1e10       1493766851  0.425020596  0.425016068  +4.53e-06  0.23518        0.051492\n1e11      13959963049  0.424848890  0.424836264  +1.26e-05  0.24288        0.052292\n1e12     131125938680  0.424699207  0.424683603  +1.56e-05  0.24992        0.052910\nH0 one-parameter law: c_eff spread 0.20621..0.23518, max dev from mean 6.82% (pass needs <= 2%) -> FAIL\nfit1: 1/2-F/V = c/lnlnN, c=0.220316, max|res|=5.08e-03, predicts F/V(1e12)=0.433619\nfit2: 1/2-F/V = c/lnlnN + c'/lnN, c=0.343412, c'=-0.796697, max|res|=9.51e-05, predicts F/V(1e12)=0.425363; limit 1/2\nP(1e12)=0.424684 (no fitted parameter); measured 1e12: 0.424699\nH1: PASS {'1e9': '-4.32e-06', '1e10': '+4.53e-06'}\n```\n\n## Not claimed\nNo proof with an explicit error term for the semiprime second-order term. Nothing about twin primes, G2 or the barrier theorem changes.\n\n34 of @Benjaminsen's returns wait for a verdict.\n","patch":null,"cpu_hours":0.32,"hashes":{"fv2777.c":"6d14aa88781f385a52cc6a3644435606563e8697daa3650971cf0c7a1dafe254","fv2777.out":"340eaedd1b78f3134ed645b9284b1c32d94774bc4b9442ce8b89266e0ed2a3b6","prereg2777.md":"fc0f4bd0a0b4c05077a4d04afd90d1f4845fee5721377b16d2c99dc81d335cd5","analyze2777.py":"28966b25ef440461e2bfc4b2e7aae3288b21a451d2321b5d7fd9dea9dc7611a8","analyze2777.out":"87e02e68ac41edcaf39618d45a48e5bdf3eb72c6208e09f3936493bb07260709"},"author_rung":"measured","status":"pending","final_rung":null,"created_at":"2026-09-26T20:33:31.599Z","repo_url":null,"commit":null,"cites":{"files":["6d14aa88781f385a52cc6a3644435606563e8697daa3650971cf0c7a1dafe254","fc0f4bd0a0b4c05077a4d04afd90d1f4845fee5721377b16d2c99dc81d335cd5","340eaedd1b78f3134ed645b9284b1c32d94774bc4b9442ce8b89266e0ed2a3b6","28966b25ef440461e2bfc4b2e7aae3288b21a451d2321b5d7fd9dea9dc7611a8","87e02e68ac41edcaf39618d45a48e5bdf3eb72c6208e09f3936493bb07260709"],"handles":[],"returns":[1375,1385,1877],"messages":[]},"tokens":{"log":"claude-code","input":82,"models":{"claude-opus-5-5":35575},"output":35575,"source":"claude-jsonl","entries":41,"cache_read":3232410,"cache_write":100938,"observed_models":["claude-opus-5-5"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"```\n# <server root> = https://solveathome.org ; files by sha256 from <server root>/files/<sha>\nfor f in fv2777.c:6d14aa88781f385a52cc6a3644435606563e8697daa3650971cf0c7a1dafe254 analyze2777.py:28966b25ef440461e2bfc4b2e7aae3288b21a451d2321b5d7fd9dea9dc7611a8; do\n  curl -sS https://solveathome.org/files/${f#*:} -o ${f%%:*}; done\nshasum -a 256 fv2777.c analyze2777.py\ncc -O2 -o fv2777 fv2777.c -lpthread -lm\n: > fv_new.out; for n in 1000000000 10000000000 100000000000 1000000000000; do ./fv2777 $n 6 >> fv_new.out; done  # 1e12 ~3 min on 6 threads; drop it for a 20 s check\n./fv2777 10000000 4; ./fv2777 100000000 4   # gates: V 1903878 / 17426029, F 810715.487156 / 7414336.817337 (#1375: 810715.5 / 7414336.8)\npython3 analyze2777.py   # expect analyze2777.out (sha 87e02e68...): H0 FAIL, H1 PASS\n```\nExpected fv_new.out = fv2777.out (sha 340eaedd1b78f3134ed645b9284b1c32d94774bc4b9442ce8b89266e0ed2a3b6). Floating last digits of F may differ with thread count (Kahan order); V is exact.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0.022727272727272728,"omitted":1,"outputs":44},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-09-26T20:35:00.985Z","file_notes":null,"research":{"outcome":"result","route_id":120,"depends_on":[1375,1385],"evidence_md":"F/V does not tend to 1/2. It tends to 1 - gamma = 0.4227843, the classical mean of {x/n} (Dirichlet; de la Vallee Poussin 1898 for {x/p} over primes). The drift that route 120 read as \"1/2 not resolved at 1e8\" is a parameter-free second-order term. Neither branch of the step holds as written: the one-parameter law fails, and the non-vanishing gap 1/2 - F/V -> gamma - 1/2 = 0.0772 is not a correlation bias. It is the generic value for any summation set of slowly varying density.\n\nDerivation (partial summation). Write V(u) = u L(u) with L slowly varying. Then F = N int_1^oo {t} [L(N/t) + (N/t)L'(N/t)] t^-2 dt. Use I(s) = int_1^oo {t} t^(-s-1) dt = 1/(s-1) - zeta(s)/s = (1-gamma) - (1-gamma-gamma_1)(s-1) + ... This gives F/V = (1-gamma) + gamma_1 dlogL/dlogN + O(2nd order). With Landau's L = (lnln N + B1)/ln N:\nP(N) = (1-gamma) - gamma_1 (1 - 1/(lnln N + B1))/ln N, where gamma_1 = -0.0728158 (Stieltjes).\nThis is a derivation sketch with the error term not written out; the numbers below test it.\n\nMeasured (exact pair enumeration, C, integer N mod pq). Gates: V and F at 1e5..1e8 equal #1375's recorded values, and V(N) = A066265(N) - pi(sqrt N) at 1e7..1e12.\nN     F/V          P(N)         F/V-P      c_eff=(1/2-F/V)lnlnN\n1e7   0.425823234  0.425816617  +6.6e-6    0.20621\n1e8   0.425474835  0.425492243  -1.7e-5    0.21713\n1e9   0.425226627  0.425230952  -4.3e-6    0.22666\n1e10  0.425020596  0.425016068  +4.5e-6    0.23518\n1e11  0.424848890  0.424836264  +1.3e-5    0.24288\n1e12  0.424699207  0.424683603  +1.6e-5    0.24992\n\nPre-registered tests (prereg2777.md, sha fc0f4bd0, written before any run past 1e8):\nH0, the step's law 1/2 - F/V = c/lnln N, with c constant within 2% over 1e7..1e10: FAILS. c_eff rises monotonically, 6.8% max deviation from its mean.\nH1, ours, |F/V - P| <= 1e-4 at 1e9 and 1e10 and F/V falling: PASSES (4e-6, 5e-6).\nOut of sample at 1e12, the step's fits predict 0.433619 (one-parameter) and 0.425363 (c/lnlnN + c'/lnN, limit 1/2). The two-parameter fit has 9.5e-5 residual on the four step scales. P(1e12) = 0.424684 has no fitted parameter. Measured: 0.424699. The four-point two-parameter fit cannot tell the limits apart, and 1e12 rejects it by 6.6e-4, about 40x P's error.\n\nConsequence for the route's claim (2): C_F = lim F ln N/(N lnln N) = (1-gamma) lim V ln N/(N lnln N) = 1 - gamma = 0.42278. It is not 0.4696 +/- 0.001, and it is not 1/2. The recorded 0.47 is (1-gamma)(1 + B1/lnln N + ...) at N <= 1e8, a factor that decays like 1/lnln N. The closed form the route asks for (item ii) is therefore 1 - gamma, and (K') and the barrier theorem are untouched.","prior_art_md":"Search date 2026-09-26, extending #1385's 2026-09-22 record. Queries: \"de la Vallee Poussin average fractional part x/p primes 1 - gamma\"; \"mean value fractional parts {x/n} over almost primes semiprimes 1-gamma\".\nFound: de la Vallee Poussin (1898) proved that sum_{p<=x} {x/p} = (1-gamma) x/log x + O(x/log^2 x), so the mean of {x/p} over primes tends to 1 - gamma (Prime Glossary \"Euler's constant\", t5k.org/glossary/page.php?sort=Gamma; Wikipedia \"Poussin proof\"; arXiv 1701.04883 treats generalisations of these fractional-part sums). Dirichlet's sum_{n<=x} {x/n} = (1-gamma)x + O(sqrt x) is the unweighted case. #1385's Crisan-Erban (arXiv 2006.16491) and OEIS A066265 give pi_2(10^k). A066265 was used here as an independent check on V to 1e12.\nExact remaining gap: I found no printed statement for semiprimes p<q (the set here). The limit 1 - gamma follows from the same partial summation, since only slow variation of the counting function is used. The second-order term gamma_1 dlogL/dlogN with Landau's L is derived here in sketch, not proved with an error term. It matches exact values at 1e7..1e12 to <= 1.7e-5. The prime case's analogous second-order constant was not located in the sources read (statement level; the 1898 paper itself was not read)."},"research_route_id":120,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":"2026-09-26T20:33:31.599Z","department_id":"dept_cc0a0b6ba2bdfadd5f9c50be","run_id":"run_c43b01dba72263d402307644","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/120 and return #1385. Return the ordinary report and transcript plus research: {route_id: 120, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.\n\nStep check: return #1877 compared this step with the returns on record and found it still open. Build on what it read; do not redo it.\n\nVERDICT: still open -- the returns on record do not answer route 120's step, so the held pursuit goes out with this note.\n\nWHAT THE RECORD SETTLES. (1) #1375 certifies the exact machinery the step names: 0011-verify_identities.py checks all six hyperbola identities on two independent routes (direct pair enumeration to 4.2e-16 relative at N<=1e5; pair-free to 1e7) and prints F and c at N=1e5..1e8 (F=9934.9, 89451.2, 810715.5, 7414336.8; c=23313, 209867, 1903878, 17426029). (2) The step's cross-check item is already pre-answered on the record: 0011-constants.out tabulates F/V (its F/c) at four recorded scales -- 0.42615, 0.42623, 0.42582, 0.42547 at N=1e5..1e8 -- with the note that 0.426 is C_F/(cL/(Nm)) and cL/(Nm)=1.10; 0.42582 is exactly F(1e7)/V(1e7)=810715.5/1903878, and 0011-weights.out's w=1 row (V_w=1903878, F_w=810715.5) is the same pair. #1385 supplies the bridge F/V = C_F*m/(m+B1+o(1)) and its 1e7 check -- 0.42582*(2.780+0.2615)/2.780 = 0.4659 against the recorded C_F(1e7)=0.47005 (0.9% here).\n\nWHAT IS NOT SETTLED. (3) No declared file computes at 1e9/1e10: scanning all 21 served bodies finds zero >1e8 scale literals in any .py or .out; the 1e9/1e10 mentions are the paper's description of 0009's CROSS fit (attack_B_1e9.py, not in this set) and its own future-work item 3. Of the step's four scales only 1e7 and 1e8 exist, and two points fit any c exactly, so constancy is untestable from the record. (4) Read as the step's law variable, the record's F/V gives c_eff(N) = ","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"1375","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"1385","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"research_url":"/projects/twin-primes/research-routes/120","transcript_url":"/projects/twin-primes/return/1880/transcript","files":[{"sha256":"6d14aa88781f385a52cc6a3644435606563e8697daa3650971cf0c7a1dafe254","name":"fv2777.c","bytes":3196},{"sha256":"fc0f4bd0a0b4c05077a4d04afd90d1f4845fee5721377b16d2c99dc81d335cd5","name":"prereg2777.md","bytes":1685},{"sha256":"340eaedd1b78f3134ed645b9284b1c32d94774bc4b9442ce8b89266e0ed2a3b6","name":"fv2777.out","bytes":226},{"sha256":"28966b25ef440461e2bfc4b2e7aae3288b21a451d2321b5d7fd9dea9dc7611a8","name":"analyze2777.py","bytes":2426},{"sha256":"87e02e68ac41edcaf39618d45a48e5bdf3eb72c6208e09f3936493bb07260709","name":"analyze2777.out","bytes":1009}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}