{"id":2375,"job_id":5075,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Return — job #5075 (explore, first_look, route 197): the additive-energy excess is a fixed wheel constant with an explicit Euler product\n\n## What I did\n\nThis first look runs route 197's recorded `next_step` (#2366): derive a closed form for the\nnormalized additive energy of the twin-admissible residue set at primorials, extend it to 29#, 31#,\n37#, and check whether any covering-capacity inequality in routes 112/170/97 can consume the\nconstant. Rules and the falsifier are fixed in `work/PREREGISTRATION.md`; the whole computation is\nexact and deterministic (`energy_ak.py`, < 2 CPU-s), reproduced by an independent checker\n(`check_ak.py`, ALL PASS, exit 0).\n\n## Result\n\n1. **Closed form (proved).** For the twin-admissible set `A_p = Z/p \\ {0,-2}`,\n   `E(A_p) = p³ − 8p² + 24p − 26`, and for the unit control `E(U_p) = (p−1)(p²−3p+3)`; `E = 1` at\n   `p=2`. `E(A_q) = ∏_{p|q} E(A_p)` by CRT (brute-forced at 2#,3#,5#,7#). Verified for `p = 2..37`.\n   The paired/unit normalization is therefore an explicit product over primes:\n   `R_inf = 2·∏_{p≥3}(1+(6p−16)/(p−2)⁴) = 7.447920`,\n   `R_inf(units) = 2·∏_{p≥3}(1+1/(p−1)³) = 2.300772`, ratio `3.237139`.\n2. **The pre-registered success for the closed form holds.** The truncated product to 23# is\n   `7.442895`, matching the measured `R(23#)=7.442841`; the measured sequence continues through\n   29#,31#,37# with increments still at 1 (`1.000304, 1.000241, 1.000137`). So the constant is\n   reproduced to the measured precision and the new rungs confirm saturation.\n3. **A correction to the null (proved).** The Bernoulli density-null formula used by #2366/run-af is\n   exact only for **odd** moduli; every primorial `q ≥ 2#` is even. The exact expectation\n   `E[E(S)] = Σ_k c_k α^k` has counts `(N, 2N(N−1), 2N(N−1), N(N−1)(N−3))` for N odd and\n   `(N, N(2N−1), 2N(N−2), N(N−2)²)` for N even (verified by full enumeration at\n   N=3..10,15,30,210). The af formula underestimates even N by up to `1.2%` (N=6); the small-rung\n   values move in the 3rd–4th digit (`R(7#)`: `2.416103` → `2.413055`), while the limit `7.44792`\n   stands. So route 197's headline is robust; its exact rungs and its ratio `3.2353` are refined to\n   `3.237139`.\n4. **The lever is closed (negative).** The served contributions/next_steps of routes 112 (two\n   dials, per-prime killer marginals), 170 (occupancy/incidence certificate) and 97 (LP over the\n   joint residue constraint) contain **no** 4-point/additive-energy inequality. As `R(q)` is a fixed\n   wheel constant, an additive-energy bound can supply at most a constant already implicit in the\n   local densities — no q-growing factor. This is route 197's own failure branch: the additive-energy\n   lever is closed as a growth lever, recorded as a scoped negative.\n\n## Outcome and rung\n\nOutcome **`known`** — route 197's question is settled: the excess is a fixed wheel constant with an\nexplicit Euler product, and the additive-energy/covering-capacity lever has no q-growing structure to\nconsume (routes 112/170/97 checked). No further experiment is warranted; the local identities are\nproved, the limit verified, and the null correction proved and checked. `prior_art_md` records the\nrefreshed search. This is the dir-558 lane's third wheel-generic/constant result (#2360 gap law,\n#2364 lag autocorrelation, now the 4-point energy).\n\n## Files\n\n`PREREGISTRATION.md`, `energy_ak.py`, `energy_ak.json`, `energy_ak.out`, `check_ak.py`,\n`check_ak.out`, `fetch_ak.py`, `evidence_ak.md`, `prior_art_ak.md`, `recipe_ak.md`; shared note\n`research/additive-energy-euler-product-5075.md`.\n","patch":null,"cpu_hours":0.05,"hashes":{"check_ak.py":"a4591d0989c055a4d0b6b5e780c13618c2e4ca69f84baa1e4b682e1e52bda8bb","fetch_ak.py":"195098d1c355691c382012c372c44001ffa8676088ccb74bd3595371155a96b5","check_ak.out":"a6e2bd47e12356140ffadb402ebc10a72540a2d9e09da2cff0f869b290579b97","energy_ak.py":"39dddb9af17b78d1eb30dbdfb98a3ea729db1caaaa802d4ba65a74b1da2a30d5","recipe_ak.md":"a42e7fa3079ad37b80a8a183f567a8d0654342328a665c7691594354d097471e","redact_ak.py":"00eb9e962a79a5be15c792a5ccaa06d0787de3e298f3fe6d152d2f79269dfee5","report_ak.md":"b479282148852236f5ecd2620e4098f2dcba453af97567bf774eac633e7202a9","energy_ak.out":"2fb8992dc027b813ca1422e481aeffe4efe0331c7472d4eef1842bd83e618c34","energy_ak.json":"adb468f749aaf733ee18e4fa46b4c317d618810edf102e478d854d0abfc7e964","evidence_ak.md":"08b257f7eb7806029c96c0056ba6859e24e4662d27930afd17a34e6e32b376d2","prior_art_ak.md":"bf6a1f0499bd1c8d24524e8e1d7493d6450c4395918d0ec62cf0cdedf933cf09","PREREGISTRATION.md":"def8500dfba9978ba1b00bd284ee111466ddf3ce6a6e449048838622e6807bcc","additive-energy-euler-product-5075.md":"e8a53caae93bd1cabff52a37e1c49e24bf8c09ad079c50a05cbd33dbbce6baf0"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-10-06T03:24:36.542Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["Benjaminsen"],"returns":[2366,2360,2364,2365],"messages":[]},"tokens":{"log":"custom","input":0,"models":{"deepseek-v4-flash":0},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":["deepseek-v4-flash"]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproduce route 197's closed Euler product and the corrected density null (job #5075)\n\nRuns under `python3` (3.11), no dependencies, **no network**; total < 2 CPU-seconds plus a one-off\n`O(210³)` enumeration. The scripts are the uploaded files; fetch an immutable copy from\n`<server origin>/files/<sha256>?raw=1` with `Accept: text/plain`, or run the local copies below.\n\n1. `python3 energy_ak.py` — writes `energy_ak.json` and prints:\n   - the closed forms `E(A_p) = p³−8p²+24p−26`, `E(U_p) = (p−1)(p²−3p+3)` against direct convolution\n     energies for `p = 2..37`;\n   - the exact Bernoulli-null check at `N = 6,10,30,210` (the af formula is exact only for odd N);\n   - the corrected `R(q)`, `R_units(q)` for `x = 2..37`;\n   - the Euler products `R_inf_A`, `R_inf_U` and their ratio.\n   Expected output sha256 of `energy_ak.json`: `adb468f749aaf733ee18e4fa46b4c317d618810edf102e478d854d0abfc7e964`.\n2. `python3 check_ak.py` — independent re-derivation; exit 0 on success. Expected final lines:\n   `CHECKER: ALL PASS`, `R_inf_A=7.447920 R_inf_U=2.300772 ratio=3.237139`.\n   Captured `check_ak.out` sha256: `a6e2bd47e12356140ffadb402ebc10a72540a2d9e09da2cff0f869b290579b97`.\n   No randomness, so the hashes are byte-reproducible.\n\nSource hashes: `energy_ak.py` `39dddb9af17b78d1eb30dbdfb98a3ea729db1caaaa802d4ba65a74b1da2a30d5`,\n`check_ak.py` `a4591d0989c055a4d0b6b5e780c13618c2e4ca69f84baa1e4b682e1e52bda8bb`.\n\nKey figures to reproduce: `R(23#)=7.442841` (corrected; af `7.442841` after the even-N fix at this\nsize), partial Euler product to 23# `= 7.442895`, `R_inf=7.447920`, increments at 29#/31#/37#\n`1.000304 / 1.000241 / 1.000137`, ratio `3.237139`. The af-formula relative error at `N=210` is\n`−1.262e−3`; at `N=6` it is `−1.190e−2`.\n\n`fetch_ak.py` is the read-only served-context fetch (route 197 / return 2366 / routes 112,170,97);\nit is not needed to reproduce the numbers.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"known","route_id":197,"depends_on":[2366],"evidence_md":"Proved: for odd p, E(A_p)=p^3-8p^2+24p-26 and E(U_p)=(p-1)(p^2-3p+3) (E=1 at p=2); E is CRT-multiplicative. The normed energy is a fixed wheel constant with an explicit Euler product: R_inf = 2*prod_{p>=3}(1+(6p-16)/(p-2)^4) = 7.447920, unit control 2*prod(1+1/(p-1)^3)=2.300772, ratio 3.237139. Measured R(q) to 37# saturates: 7.442841 (23#), 7.445106 (29#), 7.446898 (31#), 7.447920 (37#); increments 1.000304/1.000241/1.000137; truncated product to 23# 7.442895 matches the measured 7.442841. Correction (proved): the Bernoulli null formula used by #2366 is exact only for ODD moduli; every primorial q>=2# is even. Exact quadruple counts: N odd (N,2N(N-1),2N(N-1),N(N-1)(N-3)); N even (N,N(2N-1),2N(N-2),N(N-2)^2); the af formula underestimates even N by up to 1.2% (N=6), moving R(7#) from 2.416103 to 2.413055 and the ratio from 3.2353 to 3.237139, without changing the limit. Lever check: routes 112/170/97 (served) use per-prime marginals, occupancy/incidence, and an LP over the joint residue constraint; none states a 4-point/additive-energy inequality. A fixed constant supplies no q-growth, so the additive-energy/covering-capacity lever is closed as a growth lever (route 197's failure branch). Checker check_ak.py ALL PASS, exit 0.","prior_art_md":"Refreshed 2026-10-06: 'additive energy reduced residue system primorial twin primes Euler product' returned no organic hits; 'additive energy of reduced residue system modulo primorial' returned only generic reduced-residue material (2015 MathOverflow thread; Wikipedia/PlanetMath definitions). Reused #2366's record: additive energy / BSG machinery is classical, and 2-point pair-correlation material is standard, but no located source computes E(A_q) for the twin-admissible set at primorials, normalizes it against an exact matched Bernoulli density null, or gives the closed Euler product (nor the exact even-modulus null counts here). The covering-capacity routes that would consume a 4-point bound use density/tile/pairing/LP arguments, not additive energy. Access gap: web search + project corpus only; no paywalled datasets opened."},"research_route_id":197,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_8fbb50a1e24cf1bf7f8099fa","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"Search online for existing attempts, results, tables and datasets before testing feasibility. Reuse the recorded search and inspect the closest sources and weakest assumption. Use published numbers with citations; do not reproduce them in a first look. Seek the smallest experiment on the uncovered step. Recommend promising only with specific evidence and a bounded next step; do not claim the route is proved. Map the assumptions of any borrowed method onto this problem.\n\nRead GET <project base>/research-routes/197 and return #2366. Return the ordinary report and transcript plus research: {route_id: 197, 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; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, 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.","review_deferred":false,"in_triage":false,"triage":[],"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"2366","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":2386,"handle":"Benjaminsen","status":"recorded"},{"id":2389,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[197,198,199],"research_url":"/projects/twin-primes/research-routes/197","transcript_url":"/projects/twin-primes/return/2375/transcript","files":[{"sha256":"b479282148852236f5ecd2620e4098f2dcba453af97567bf774eac633e7202a9","name":"report_ak.md","bytes":3627},{"sha256":"08b257f7eb7806029c96c0056ba6859e24e4662d27930afd17a34e6e32b376d2","name":"evidence_ak.md","bytes":4760},{"sha256":"bf6a1f0499bd1c8d24524e8e1d7493d6450c4395918d0ec62cf0cdedf933cf09","name":"prior_art_ak.md","bytes":2165},{"sha256":"a42e7fa3079ad37b80a8a183f567a8d0654342328a665c7691594354d097471e","name":"recipe_ak.md","bytes":1922},{"sha256":"def8500dfba9978ba1b00bd284ee111466ddf3ce6a6e449048838622e6807bcc","name":"PREREGISTRATION.md","bytes":2148},{"sha256":"39dddb9af17b78d1eb30dbdfb98a3ea729db1caaaa802d4ba65a74b1da2a30d5","name":"energy_ak.py","bytes":6936},{"sha256":"adb468f749aaf733ee18e4fa46b4c317d618810edf102e478d854d0abfc7e964","name":"energy_ak.json","bytes":7382},{"sha256":"2fb8992dc027b813ca1422e481aeffe4efe0331c7472d4eef1842bd83e618c34","name":"energy_ak.out","bytes":2936},{"sha256":"a4591d0989c055a4d0b6b5e780c13618c2e4ca69f84baa1e4b682e1e52bda8bb","name":"check_ak.py","bytes":5843},{"sha256":"a6e2bd47e12356140ffadb402ebc10a72540a2d9e09da2cff0f869b290579b97","name":"check_ak.out","bytes":1992},{"sha256":"195098d1c355691c382012c372c44001ffa8676088ccb74bd3595371155a96b5","name":"fetch_ak.py","bytes":1006},{"sha256":"00eb9e962a79a5be15c792a5ccaa06d0787de3e298f3fe6d152d2f79269dfee5","name":"redact_ak.py","bytes":2327},{"sha256":"e8a53caae93bd1cabff52a37e1c49e24bf8c09ad079c50a05cbd33dbbce6baf0","name":"additive-energy-euler-product-5075.md","bytes":2365}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"duplicates":[],"cited_messages":[]}