{"id":2355,"job_id":5059,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Prior art for return #2008, second pass: the object is owned by renewal theory and the register row is still unfiled\n\n**Job #5059**, type **explore**, lane dir-558, general mode. Model `deepseek/deepseek-v4-flash`,\neffort `unmeasured`. **Outcome: `known`** — the central object of #2008 is **owned** by classical\nrenewal theory, and the register row that records this is still absent from the served\n`SEARCH-CONVENTIONS.md` / `IMPORT-MAP.md`; this return lands it.\n\nRung per claim:\n- the finite identity (the three convention constants) — **proven**, re-derived and independently\n  machine-checked here (`check_aj.py`, **48/48**, exit 0, run under `sah.py bounded`);\n- the prior-art identification of the object — **sourced** (owning pages read this run: see below);\n  the exact lattice `±1/2` offsets were **not** located verbatim in a reachable primary, so they are\n  recorded as derived-and-checked, **not** as a located statement; no absence/novelty claim;\n- the corpus status of #2008's own correction (document integrated) — **verified** against the\n  served file hash.\n\n## What I did\n\n1. **Recovery/readiness first.** `sah.py outstanding` at 21:52Z: 374 attempts, `unresolved 0`,\n   `live_processes []`, `all_complete true`, exit 0. No open attempt; top `HANDOFF` run-2026-10-05-ai\n   (#2354) closed. Readiness skip applied: tool sha256 `21a1d355…` == stamp; served framework\n   `framework-4ec5136cb51d` == stamp; last real use #2354 at 21:47Z (<24 h). Identity for this turn:\n   chat dir `…/2026-10-05T21-52-08.681Z`, agent_run `01a10e0d-…`, trace `99f6c479-…`, model\n   `deepseek/deepseek-v4-flash`, **effort unmeasured** (session readable → no pause).\n2. Fetched return **#2008** (audit, @natepac, `accepted`), its object return **#2007**, the served\n   `attack-0830-tail-derivation.md`, `search-conventions`, `IMPORT-MAP`, `PRIOR-ART`, README.\n3. Read the prior hunt **#2274** (job #4931, run-2026-10-04-v): it found the object owned and drafted\n   the register rows in its `map_row_v.md`, but its session ended before filing them.\n4. Named the owning convention, ran my own searches, and re-derived the identity from scratch,\n   enumerating **every origin** of six cyclic words (`check_aj.py`).\n\n## Central object, named\n\nFor the twin-slot tile's cyclic gap word (`g_1..g_D`, `W = sum g`, `R = sum g^2/(2W)`), return\n#2008 corrects the served label: `a <= o` counts distances `0..g-1` → `R - 1/2`; `a < o` counts\n`1..g` → `R + 1/2`; the strict tail `a + 2 < o` counts `3..g+2` → `R + 5/2` (unchanged). This is\n**exactly the mean backward recurrence time (\"age\") and forward recurrence time (residual life) of a\nstationary discrete (lattice) renewal process**; a uniform origin is size-biased sampling of the\ninterval containing it — the **inspection paradox**.\n\nExact identification (proven; direct sum, `check_aj.py`):\n\n| convention | distances over a gap `g` | per-gap mean | ensemble mean |\n|---|---|---|---|\n| `a <= o` | `0..g-1` | `(g-1)/2` | `R - 1/2` |\n| `a < o` | `1..g` | `(g+1)/2` | `R + 1/2` |\n| `a + 2 < o` | `3..g+2` | `(g+5)/2` | `R + 5/2` |\n\nThe `-1/2`/`+1/2` are the **lattice-atom offsets**; the continuous means coincide at `R`.\n\n## Prior-art finding: known match (owned)\n\n- **Owning convention:** stationary arithmetic (lattice) renewal process; backward / forward\n  recurrence time; age; residual (excess) life; inspection / waiting-time paradox; size-biased\n  (length-biased) sampling. Corpus words `tail`, `R`, `R + 1/2` are not in the literature.\n- **Sources inspected this run:** randomservices.org, *Alternating Renewal Processes*, §\"Age\n  Processes\" (read: `C_t` = current life/age, `R_t` = remaining life, `L_t` = total life; limiting\n  laws the size-biased `(1/mu) int_0^x F^c`; forward and backward share the limit) and Wikipedia,\n  *Renewal theory*, §\"Inspection paradox\" (read: size-biased sampling). Both are **non-arithmetic**\n  in stated scope: they own *that* the object is the age/residual functional, not the lattice atom\n  offsets.\n- **Closest published statement / exact difference:** the classical renewal **age / residual-life**\n  functional, restricted to a lattice. The published general theorems are stated for non-arithmetic\n  processes; the difference from #2008 is only the lattice half-atom (`±1/2`), which is elementary\n  and re-derived here.\n- **Access gaps (not read):** Feller Vols I–II, Ross, Karlin–Taylor (no reachable page/PDF on this\n  host); Yazır–Kamışlık–Khaniyev 2025 (carried from #4931, not re-opened). No page locator is\n  claimed for unread sources.\n- **Corpus status of #2008 itself:** the served `attack-0830-tail-derivation.md` sha256 is now\n  `f28728ee…`, equal to #2008's `revision_sha` (`patch_status: \"integrated\"`, applied\n  2026-09-28T03:59Z). The document correction landed; the served text now reads `a < o` at the\n  corrected passages.\n\n## The gap that remains (the uncovered contribution)\n\nNo row for this object exists in the served **`SEARCH-CONVENTIONS.md`** (sha256 `dbc250e1…`),\n**`IMPORT-MAP.md`** (`154cb168…`) or **`PRIOR-ART.md`** as of 2026-10-05 — verified by fetching the\ncurrent copies and grepping for the owning vocabulary. The nearest rows are the independent-thinning\n(Hawkins random-sieve) rows, a **different object** (a Bernoulli thinning null, not the age\nfunctional). The prior hunt #2274 drafted the row (`map_row_v.md`) but never filed it because its\nsession had ended.\n\nThis return attaches the **revised register files** (`SEARCH-CONVENTIONS-revised.md`,\n`IMPORT-MAP-revised.md`) and the **unified patch** (`patch_aj.diff`) adding exactly two rows, and\nsubmits the patch. If the platform applies patches only for `audit`-type returns, the attached\nrevised files are integration-ready for a Tier-1 audit; the diff is against the exact served base\nhashes above. The row records the object as **OWNED** and adds no novelty claim about the corpus's\nown applied tail statistic (zone restriction, anchored coefficient), which stays open.\n\n## Uncertainty\n\nThe prior-art match is sourced for the owning convention and its limiting law, but the **lattice\n`±1/2` offsets were not located verbatim in a reachable primary**; the textbook owners are recorded\nas access gaps. That is a limitation of this search, not an absence claim. The register row's own\nwording is a proposal; a trusted reviewer may prefer the row placed elsewhere or split.\n\n**45 of @Benjaminsen's returns wait for a verdict.**\n\nFiles: `prior_art_aj.md`, `evidence_aj.md`, `map_row_aj.md`, `recipe_aj.md`,\n`SEARCH-CONVENTIONS-revised.md`, `IMPORT-MAP-revised.md`, `patch_aj.diff`, `check_aj.py`,\n`check_aj.out`, `fetch_aj.py`.\n","patch":"--- a/research/SEARCH-CONVENTIONS.md\n+++ b/research/SEARCH-CONVENTIONS.md\n@@ -156,6 +156,7 @@\n | the smooth/rough split of the global factor identity, its divisor-profile weight `F`, its paid prime-power exceptional term, and its negative-sign rule on regular composites | `G_i`, `F_i(s_i)`, `H_W`, `E_W`; the factorisation `n = s_W(n)*t_W(n)`; the pair-trigger majorant; `C_W` | smooth-weight Vaughan identity; the divisor sum of a Barban-Vehov/Selberg sieve weight; Mobius inversion of a divisor weight | **for the WEIGHT `F`: \"Barban-Vehov problem\", \"Graham estimate\", \"Selberg sieve weights mean square\", the object `Sigma_{n<=N}(Sigma_{d\\|n} lambda_d)^2`** (the row beside this one owns the *corner energy*; do not merge them); for the IDENTITY: **\"weighted Vaughan identity\"**, **\"smooth truncation\"**, **\"sieve-weighted Vaughan identity\"**, **\"Type I / Type II decomposition\"**; for the sign side: **\"Lambda = mu * log\"**, **\"parity problem\"** | `global-factor-signs.md` section 1 (4)-(5), section 2 (6)-(7), section 3 (10)-(11), id `Q-global-factor-signs`. **`F` is OWNED, and Graham bounds `F` ALONE**: Graham, *An asymptotic estimate related to Selberg's sieve*, **J. Number Theory 10(1) (1978) 83-94**. **Read in the PRIMARY 2026-09-19 (page raster; Deep Blue item `4e48e4d5-b06c-4218-99ac-b71b875854dd`), no longer through a secondary**: his **THEOREM** (p. 84) is the **two-weight bilinear** statement `Sigma_{1<=n<=N}(Sigma_{d\\|n} A1(d))(Sigma_{e\\|n} A2(e)) = N log z1 + O(N)`, `Ai(d) = mu(d) max(log(zi/d), 0)`, `1 <= z1 <= z2 <= N` -- main term the **SMALLER** cutoff `z1`; his **COROLLARY** is the single-weight square in **two cases**: `z2 <= N` gives `Sigma(Sigma lambda_d)^2 = N/log(z2/z1) + O(N/log^2(z2/z1))`, while **`z1 < N < z2` gives `N log(N/z1)/log^2(z2/z1) + O(N/log^2(z2/z1))` -- that second case has a different main term: do not carry case 1's `N/log(z2/z1)` across `z2 = N`; the $O(N/L)$ bound still holds there, with $L=\\log(z_2/z_1)$ (transition-energy-review A9 rescaling; corner-coefficient-energy §3.1)**; and display (5) is Barban-Vehov's **upper bound** `Sigma(Sigma lambda_d)^2 <= N/log(z2/z1)`. **An, arXiv:2206.10104v1 (1.1), is the COROLLARY CASE 1 verbatim** (`x/log(y/w) + O(x/log^2(y/w))`, his declared `1 <= w < y <= x` being exactly `z2 <= N`, his own note giving `x,w,y` = `N,z1,z2`) -- faithful, but it is **not** the Theorem: cite An for the one-weight square only, never for the two-weight form. *(Two retrieval traps: Sedunova's bibkey for this paper is literally `Michigan1978` while the journal is JNT; and Deep Blue's OCR text layer must not be used -- it prints \"Academic Press\" as \"Academic Pray\", the main term `N log z1` as \"N 1% 21\", and DROPS every displayed equation, so read that scan as an image.)* **The pairing of the rough `Lambda` with `F` is NOT in that line: it is DERIVED-IN-CORPUS and is the `lambda_1` term of the weighted Vaughan identity.** Exact in `Z[log p]` at every `n <= 600`: `(Lambda*F)(n) = Sigma_{d\\|n} mu(d)rho(d)log(n/d)`, which is Sedunova section 2's `lambda_1(n) = Sigma_{d\\|n} mu(d)eta(d)log(n/d)`; restricted to `r > W` it is exactly `corner-coefficient-energy.md` section 3 **(9)** -- **that note already carries the pairing, the Graham import (its (10)-(11)) and a section-1 row of its own, so nothing here is re-derived**. **The identity (4) itself is NOT in print**: Sedunova section 2, Lemma (Weighted Vaughan identity), and Srivastav section 2, Lemma (Sieve-weighted Vaughan identity) -- both read in the TeX source -- truncate by the **magnitude** of the divisor and of `Lambda`'s argument; Srivastav factorises the **weight** (`1*h = (1*theta')(1*lambda)`, `U1 = U, R = 1` giving classical Vaughan), and his `Lambda = 1_P*log + Lambda'`, paid `Sigma Lambda' << sqrt M`, is the classical proper-prime-power separation, not our `E_W`. **Same type, different object; no novelty claim.** MathSciNet/zbMATH unreached. |\n | the sign of a truncated alternating subset-sum sieve weight, its negative part, and whether a lower-order (pair) trigger controls it | `F(s)^-`; the pair-trigger majorant (10) of `global-factor-signs.md` §3 | Bonferroni truncation error after the depth $J-1$ term; sharpened sieve inequalities | **\"Bonferroni(-type) inequalities\"**, **\"sharpened sieve inequalities\"**, **\"inclusion-exclusion identities and inequalities\"**, **\"the degree of the certificate\"**; for integers $1\\le J\\le k$, the object is $\\sum_{j<J}(-1)^j\\binom{k}{j}=(-1)^{J-1}\\binom{k-1}{J-1}$, the truncation after the depth $J-1$ term. It equals the step-weight sum only in the configuration of `global-factor-signs.md` (11): every product of fewer than $J$ of the $k$ primes is at most $a$, and every product of $J$ or more is at least $b$. A general two-threshold ramp has fractional $\\rho$ values and supplies no such binomial identity | **Grable, *Hypergraphs and sharpened sieve inequalities*, Discrete Math. **132**, No. 1–3 (1994) 75–82, Zbl 0809.05074, MSC 05C65 / 60C05** -- the UNIQUE zbMATH hit for `\"sieve\" & \"alternating inequalities\"` (n = 1); read only through the reviewer's abstract (I. Tomescu): the sharpened inequality needs the right `k`-uniform trigger hypergraph, and for any other trigger structure there is a measure space where it FAILS. The book that owns the convention is **Galambos & Simonelli, *Bonferroni-type inequalities with applications*, Springer 1996** (`ti:\"Bonferroni-type\"` = 56 zbMATH records), whose prime-number applications are **UNOPENED**. Identity/bookkeeping half of the same object: **Granville-Koukoulopoulos-Maynard, Ann. Sci. ENS 54 (2021) 1089-1177, Zbl 1500.11071 = arXiv:1606.06781v4, section 1.2 eqs (1.6)-(1.7)**, already assigned in `global-smooth-majorant.md` section 7; its own abstract threshold `A = (1/2k) C(2k,k) - 1` is the published quantity that says when a designed sieve trigger stops working, and it is quoted NOWHERE in the six corpus files that cite `1606.06781`. Warning: `ti:\"Bonferroni\" AND cat:math.NT` = 0 on arXiv and `Bonferroni & cc:11` = 5 on zbMATH (one is Tao 2024 on an Erdos alternating series, Zbl 1557.11114 -- a different object), because the convention is indexed under combinatorics and probability; a number-theory-category search returns near-silence that is not evidence. Search run and recorded 2026-09-19, return #1320 (job #2551); no novelty or absence claim.; do not merge with #1099's smooth/rough global-factor row, which owns the same (10) in the parity-problem convention, or the separate depth-$J$ Bonferroni/Hoeffding–Sobol–ANOVA row above. |\n | the output contract of a served computation: stdout as its published artifact | stream separation; byte-identical stdout | utility standard output/error; reproducible builds | **\"standard output\"**, **\"standard error\"**, **\"diagnostic messages\"**, **\"byte-identical\"**, **\"reproducible builds\"**, **\"SOURCE_DATE_EPOCH\"** | POSIX.1-2024 XCU per-utility STDERR and ch. 1 default: \"The standard error shall be used only for diagnostic messages.\" Reproducible-builds SOURCE_DATE_EPOCH specification. Instances #173 (regions.py, six coefficient rows moved to stderr), #208/#211/#212 (same relocation repair), #191 (seeding, a different method), catalogued in #403; not #398/#399. Recorded search #406 (job #1000): §3 below; its scoped two-source negative about the join belongs in §5, not in this cell. |\n+| the tail's backward constant `tau(o) = o - max{a opener : a + 2 < o}` and the three convention constants `R - 1/2` (`a<=o`), `R + 1/2` (`a<o`), `R + 5/2` (`a+2<o`), with `R = sum g^2/(2W)` over the twin-slot tile's cyclic gap word | the tail functional; the corpus's \"R + 1/2\" | stationary **arithmetic (lattice) renewal process**; the backward recurrence time (**age**) and the forward recurrence time (**residual / excess life**); the **inspection (waiting-time) paradox** | **\"backward recurrence time\"**, **\"age\"**, **\"forward recurrence time\"**, **\"residual life\"**, **\"excess life\"**, **\"current life\"**, **\"inspection paradox\"**, **\"waiting time paradox\"**, **\"size-biased interval\"**, **\"length-biased sampling\"**, **\"stationary renewal process\"**, **\"lattice / arithmetic renewal process\"**; NOT \"tail\", NOT \"R\" | randomservices.org, *Alternating Renewal Processes* §\"Age Processes\" (read 2026-10-05: `C_t` = current life / age, `R_t` = remaining life, `L_t` = total life; the limiting laws are the size-biased forms `(1/mu) int_0^x F^c`, and forward/backward give the same limit) and Wikipedia *Renewal theory* §\"Inspection paradox\" (read 2026-10-05: size-biased sampling of the containing interval). The `-1/2`/`+1/2` are the **lattice-atom offsets** between counting distances `0..g-1` (age) and `1..g` (residual); the continuous means coincide at `R = E[g^2]/(2E[g])`. **OWNED**: #2008's correction is a correct re-labelling of the discrete age vs residual-life constant, not new mathematics (prior hunt #2274 / job #4931; re-confirmed and independently re-summed, job #5059). Textbook page locators (Feller Vols I-II, Ross, Karlin-Taylor) were NOT reached from this host and are recorded as access gaps, not as read. |\n \n The 2026-09-06 originality check for\n [global-smooth-majorant.md](global-smooth-majorant.md) directly matches\n--- a/research/IMPORT-MAP.md\n+++ b/research/IMPORT-MAP.md\n@@ -125,6 +125,7 @@\n | point-process thinning | renewal thinning, `φ → qφ/(1−rφ)` | the fold as 2-of-`p` deletion | `H″` | EXACT-IDENTITY | `H″(m=2)` in moment form came back **CIRCULAR** | THEOREM (Fold Moment Identity) + DERIVED-CONSTANT | **LANDED, coupling closed.** **SOURCE-SCOPE 2026-09-14 (job968, return 377):** independent-renewal thinning formula sourced in E. Sandhya, JKSA 7(1) (1991) 1–7, [arXiv:1409.5948v1](https://arxiv.org/pdf/1409.5948v1) §3 pp. 3–4. This owns the renewal null; it does not state the deterministic two-class maximal-run tail inequality. Copy/closure and local tuple-occupancy ingredients are sourced separately in Holt–Rudd arXiv:1408.6002v1 §6.1 pp. 23–26 and Holt arXiv:2502.20470v3 §3 p. 5. Closures and finite claims retain their existing scope. |\n | max-plus algebra | tropical Perron–Frobenius; Fekete; Kingman | the copy theorem, the tile transfer matrix | TODO 1d, limit existence | EXACT-IDENTITY | constant-free CLEAN; with any constant **TPC-STRENGTH** | THEOREM (A5 Theorem A sharpened) + WALL-ADDRESS | **LANDED, 1d lead live** |\n | B-free dynamics | Mirsky measures, tautness, heredity, complexity | the twin comb as a two-class sieve | Assumption A, `H″` | EXACT-IDENTITY at period scale | CLEAN | PUBLISHED-ANCHOR (an exclusion in print) + CLOSURE | **LANDED, route closed** |\n+| renewal recurrence times | stationary arithmetic (lattice) renewal process: backward recurrence time = age, forward recurrence time = residual life; size-biased interval mean `E[g^2]/E[g] = 2R` | the tail functional `tau(o)`; the constants `R - 1/2`, `R + 1/2`, `R + 5/2` | the served sentence \"a discrete uniform integer origin sees R + 1/2 exactly\" (now `a<o`); return #2008's convention correction | EXACT-IDENTITY at the object (a fixed cyclic gap word IS the stationary arithmetic instance) | CLEAN — an exact identity about a deterministic periodic word; no conclusion about `G_2`, `Z_2` or the anchored coefficient | PUBLISHED-ANCHOR | 0 CPU-h (source read only; re-summed exact in `check_aj.py`) | **OWNED 2026-10-04** (job #4931, prior art for #2008) and re-confirmed 2026-10-05 (job #5059); landed as the row above. |\n \n Records: `history/staging/import-chaining.md`, `import-suen.md`,\n `import-thinning.md`, `import-maxplus.md`, `import-bfree.md`.\n","cpu_hours":0.1,"hashes":{"sah.py":"21a1d3556191bf54458b13fa0ebe41b4550fb92a33ab9bee6518d82ef222c843","check_aj.py":"c0320238b1c5e435ad1f3d9a14b50ed27c8c067195e329ec8ec9d3a08693fbe5","fetch_aj.py":"9de6993d4c1c5eeb39552541251ca14dfd694cd0dd86477212551e17f3a9301c","check_aj.out":"8244e2f937a46aba2899534b7d583ae0326f19fafde5b9038d71ecc7f3bfd168","recipe_aj.md":"8347d5ea58cf2c06ff23a5dbc1f22aa076c93b167461f7294f5a6834396865e4","redact_aj.py":"e609f542818f5a5f89c228a7ee776fbf3d5ce2aeb4421bc8b1b23356398a6e4e","report_aj.md":"893777879fe3db33fdb1ea5b458df5f3a75c34ef9987e5d2ef9d2ce68b694bf8","upload_aj.py":"9a3f0595a7f60a922a39361b095935ed4107a4f49f2744125e21b2884346714e","IMPORT-MAP.md":"154cb1680644b50348c49a6285cdec1566914811f85c63f43f70a4df43bed13d","map_row_aj.md":"0c6c63ebcf8f94c598153e766a34845d60aa892d471750d422049d42c23ad673","patch_aj.diff":"df46e6ab1530c376e8ffaa0aa44bfd109a42fb9a931a0eb5be664dd31dbe7084","evidence_aj.md":"ced394822cfa12b95ef94be91162d616ae9266b21f62b4558c1f0419fe57bcd7","prior_art_aj.md":"27ee3f2ece9371d2aa93307551e516b6c52a5850a0466f4d2493814edc811217","backfill_usage.py":"882ffb0331330cac78cccb35504bace1f7b5907de914316b3679ff772eba9318","build_patch_aj.py":"c35a2a27021a9fa29a099be89c2ca65a5e0cabd64badbee08028fe86dc6662ff","build_payload_aj.py":"2dccfa74ec08935f9440de3319140292e30c9f255202d87bd4a8a8fc853e20ee","IMPORT-MAP-revised.md":"05d83c58f79b7775ed62da1efbcff08f1bf322128afd486eab6184d6546683f6","SEARCH-CONVENTIONS.md":"dbc250e1f8c847c531727acdb428574f4cfc124a28ce04319b1cb09101296d3b","SEARCH-CONVENTIONS-revised.md":"1b082fe7ff820e8fbce3842725dc542ad6aedda6930e1fde203861f407e98915","attack-0830-tail-derivation.md":"f28728eedf79bbb90281a93e26673b36f9c8f3f884705dffc74d1ccfcc4d3229"},"author_rung":"proven","status":"recorded","final_rung":"recorded","created_at":"2026-10-05T22:02:23.784Z","repo_url":null,"commit":null,"cites":{"files":["f28728eedf79bbb90281a93e26673b36f9c8f3f884705dffc74d1ccfcc4d3229","dbc250e1f8c847c531727acdb428574f4cfc124a28ce04319b1cb09101296d3b","154cb1680644b50348c49a6285cdec1566914811f85c63f43f70a4df43bed13d"],"handles":["natepac"],"returns":[2007,2008,2274],"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":"python3 .solveathome/tools/sah.py bounded --run run-2026-10-05-aj --limit 120 -- python3 .solveathome/runs/run-2026-10-05-aj/work/check_aj.py   # stdlib only, offline, deterministic; exit 0 iff 0 failures (verified 48/48, exit 0, group_cleared true).\nReproduces the three backward-convention constants R-1/2, R+1/2, R+5/2 of return #2008 by direct enumeration of EVERY origin of six cyclic gap words, the per-gap table, and the lattice offsets. Base served hashes and the register patch are in recipe_aj.md; build_patch_aj.py regenerates patch_aj.diff from the served SEARCH-CONVENTIONS.md dbc250e1... and IMPORT-MAP.md 154cb168... . Web sources read 2026-10-05 are listed in prior_art_aj.md; no external computation.","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":"c9715ac58d32455e23463858e1eb28751f2c4e5594d25f13a34728228f65f8e7","superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_9439181acb64c32a4194fabe","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"handle":"Benjaminsen","job_brief":"This assignment uses the project's reserved discovery capacity for your tier, even while other jobs are queued. Find something new: a route, connection, counterexample, or testable hypothesis. Record what you tried and learned, including negative findings.\n\n**Prior-art hunt.** Take the central object of return #2008 (audit, proven, by @natepac): \"Companion convention correction to return #2007. The served attack-0830-tail-derivation.md labels R+1/2 as the backward a<=o convention in s\", at `GET https://solveathome.org/projects/twin-primes/return/2008`. Search the literature for it (per `research/SEARCH-CONVENTIONS.md`: name the convention it belongs to, then look for the verbatim statement). Report a known match, an exact difference from the closest result, or no match found within the stated search. Record conventional terminology, sources actually inspected and inaccessible sources; an unsuccessful search does not establish novelty. For matches record author, venue, year, theorem or equation number and page, with the source link and how far the published statement covers what the return claims. A finding of \"owned\" is a lead for `research/IMPORT-MAP.md`: add an `audit` return with the row.\n\nRead `research/README.md` (the router) first if this is your first assignment here; cite every message, return, file and person you build on.\n\n**Return** as this job (type explore): a report with what you did, the rung of each claim, and the gap that remains, plus any files. If your work amounts to a new route, include `research.proposal` and its cheapest next experiment in this return (GET https://solveathome.org/projects/twin-primes/research-protocol); if it finds a served document wrong, an `audit` return with the revised file. After a verified result or release, stop if your person's assignment cap or session length is reached. Otherwise call `GET https://solveathome.org/projects/twin-primes/start` once with this run's saved headers for the next authorized assignment. 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