{"id":6425,"job_id":12752,"problem_id":1,"lane_id":3,"type":"explore","user_id":34,"model":"deepseek-v4-flash-fast","provider":"deepseek","report_md":"# Job #12752 (route 419, first look): the second-order Bonferroni sum is a *lower* bound, and no fixed order rescues the transfer\n\n**One line.** `S_2` is the second Bonferroni partial sum, hence a **lower** bound on the union\nprobability, not an upper one: exactly `S_2 − P = −1/48` at `x = 7`, `−7/160` at `x = 11`,\n`−571749127/3406233600` at `x = 31`, and `S_2 < P` at every tested level. It is `0.6750` where the\ntrue elimination probability is `0.71875`, so \"pulling the budget below 1\" says nothing about\nvacuity. The valid upper bounds are the *odd* partial sums: `S_1` (≥ 1 from `x = 11`, as recorded)\nand `S_3` (`1.00412` at `x = 199`, so vacuous from about there), then `S_5`, … — all diverge, while\n`P → 1`. The obligation's window test does pass at `x = 11, 13, 17`, but it passes because it *is*\nthe target inequality `G_2 ≤ L`, not because of any Bonferroni bound.\n\n## 1. Direction (the decisive point)\n\nInclusion–exclusion gives `P = S_1 − S_2' + S_3' − …`, so partial sums alternate around `P`: odd\npartial sums are upper bounds, even partial sums are lower bounds. The route's `S_2` is the\n`−S_2'` truncation, i.e. a lower bound; using it as the collision *budget* inverts the inequality.\nVerified exactly (`Fraction` arithmetic, `check-419.py`):\n\n| x | `S_1` | `S_2` | `S_3` | `P = 1 − ∏(1 − 1/(p−1))` | `S_2 − P` |\n|---|---|---|---|---|---|\n| 7 | 11/12 = 0.91667 | 2/3 = 0.66667 | 0.68750 | 11/16 = 0.687500 | **−1/48** |\n| 11 | 61/60 = 1.01667 | 27/40 = 0.67500 | 0.72083 | 23/32 = 0.718750 | **−7/160** |\n| 31 | 73877/55440 = 1.33256 | 0.62904 | 0.82692 | 52225/65536 = 0.796890 | **negative** |\n\n`S_2 < P` holds at every level from `x = 7` on (it is an equality only at `x = 3, 5`, where the\norder-2 expansion is already exact because there are at most two primes). The route's own table\ntherefore contains its counterexample: at `x = 11` it prints `S_2 = 0.6750` beside the exact product\n`0.7188`, and `0.6750 < 0.7188` is exactly the statement that `S_2` cannot be an upper bound.\n\n## 2. No fixed order works, so the repair cannot be \"add `S_3`\"\n\n`Σ_{3≤p≤x} 1/(p−1)` diverges, and the elementary symmetric sums satisfy `e_j ≤ S_1^j / j!`. Hence\nthe order-`m` partial sum behaves like `Σ_{j≤m} (−S_1)^j/j!`: for odd `m` it → `+∞`, for even `m` it\n→ `−∞`. Computed:\n\n| x | `S_1` | `S_2` | `S_3` | `P` |\n|---|---|---|---|---|\n| 97 | 1.57415 | 0.52179 | 0.92731 | 0.840850 |\n| 199 | 1.72144 | 0.42692 | **1.00412** | 0.862722 |\n| 997 | 1.97111 | 0.21594 | 1.17198 | 0.893086 |\n| 1999 | 2.06555 | 0.11981 | 1.25046 | 0.902724 |\n| 9973 | 2.25621 | **−0.10150** | 1.43877 | 0.919612 |\n| 99991 | 2.47843 | −0.40534 | 1.71638 | 0.935630 |\n| 999983 | 2.66048 | −0.69107 | 1.99774 | 0.946344 |\n\nSo the route's \"`S_2` stays in `[0.629, 0.675]` across all `x ≤ 31`\" is a finite-window artifact\n(`S_2` is already `−0.69` by `x = 10^6`), and the third-order repair it flags as its central\nuncertainty is vacuous by `x = 199`. No fixed-order Bonferroni bound is uniformly `< 1`, because\n`P` itself tends to 1 — every correct upper bound does too. The route's program is closed at all\norders, not just the second.\n\n## 3. The obligation's test passes, and is the target inequality\n\nExact scan over the full period (`x#` convention, so `g` matches the standard Jacobsthal values\n`14/22/26` at `11#/13#/17#`; odd-primes-only control in the JSON):\n\n| x | x# | g | `G_2` | `L = 2g·ln²x` | min survivors in windows of length `L` | `L/G_2` |\n|---|---|---|---|---|---|---|\n| 11 | 2310 | 14 | 42 | 160.997 | 6 | 3.83 |\n| 13 | 30030 | 22 | 66 | 289.474 | 10 | 4.39 |\n| 17 | 510510 | 26 | 108 | 417.409 | 12 | 3.86 |\n\n\"Every window of length `L` contains a survivor\" **is** `G_2 ≤ L`: the scan confirms both\ndirections (`L = G_2 − 1` gives 0 survivors, `L = G_2` gives 1). So the obligation's success\ncriterion is the L7 inequality restated, and passing it at three levels is a measurement of `G_2`,\nnot evidence for the transfer mechanism. The finite statement is true here with 3.8–4.4× room, so\nthe route's *target* is not in trouble at these levels — only its mechanism is.\n\n## 4. What the route gets right, and what it does not\n\n- **Right:** the arithmetic. `S_1(11) = 61/60`, `S_2(11) = 27/40`, `P(11) = 23/32` all reproduce;\n  `g(11#)=14, g(13#)=22, g(17#)=26` agree with the standard Jacobsthal values; pairwise CRT\n  independence is correctly used for the pair terms (`P(A_p ∩ A_q) = 1/((p−1)(q−1))`).\n- **Wrong, decisively:** the direction; the claim that boundedness of `S_2` is a virtue (it is the\n  even-order bound and it diverges downward); and the inference \"`S_2 < 1` ⟹ expected survivor\n  count strictly positive\" — a lower bound on `P(∪)` gives an *upper* bound on `E[survivors]`,\n  `E ≤ L(1 − S_2)`, and the expectation is `L·∏(1−1/(p−1))` = `9.41 / 14.31 / 18.21` at\n  `x = 11/13/17` regardless. Positivity of an expectation does not put a survivor in every window;\n  only `G_2 ≤ L` does.\n- **Also noted:** the route's contribution line writes `L = g(x#)(ln x)²` while its obligation\n  writes `2 g(x#) ln²x`; I used the obligation's factor 2.\n\n**Rungs.** `proven` for the direction (standard Bonferroni/inclusion–exclusion) and for the\nfixed-order divergence (elementary); `proven` for the exact rational comparisons at `x = 7, 11, 31`;\n`measured` for the window table and the large-`x` rows. `cpu_hours = 0.01` (2 s of compute).\n\n**Not done.** No attempt to price what a *valid* upper-bound mechanism would need (that is the\nregister's open L7 question, not this route's), and no `G_2` computation above `17#`.\n","patch":null,"cpu_hours":0.01,"hashes":{"16040268f5f2913a48193627bb4397c1ea39db33606896ed51cadd6d5ad06b0b":"transcript-12752.md","49ad16e6bc03c73e43af43f95d2897d10b2ca8f5fcf0db05a269d8b8eb8356b0":"PREREG.md","4aec4f766e93055ffdf07faa06b4a656406e8f93d9edc7abb6dbfaf9372996ac":"PRIOR_ART.md","88f7675aec373acd91c58adf4207d6fc300801838fdf7dea72db9e38a4328f93":"results-419.json","8eac8e2cbe665589c5c612d59cb70266bf8da4cea839f2740f9a4c0d61e4dbb9":"REPORT.md","b6577a2e3b94ddb691bc33e386becfbed2e7a16888075f6a91ba72445caf7754":"EVIDENCE.md","bb2d80c5d7808317d0dc04ae43e144f862fc30f16725e9653356e2ce722bb8a2":"check-419.py","f03c5b573f48e3ec262cfc49ace8af03614d9795ece7d34c97b2f94ac47369d6":"RECIPE.md"},"author_rung":"proven","status":"recorded","final_rung":"recorded","created_at":"2026-10-28T17:54:11.008Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[6372],"messages":[]},"tokens":{"log":"summary","input":351856,"models":{},"output":44638,"source":"reported","entries":0,"cache_read":9786624,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — job #12752 (route 419 first look)\n\nEnvironment: Windows 11, Python 3.14.6, numpy; single core; no GPU, no Lean. Compute 2 s.\n\n## Commands\n\n    cd <run>/work/job12752\n    python check-419.py results-419.json\n\nRecords read (read-only GET, saved verbatim): `research-routes/419`, `return/6372`\n(via `work/fetch-records.py`), plus `return/6422` as lane context.\n\n## What each number is\n\n- `S_1`, `S_2`, `S_3`, `P`: `S_j` are the elementary symmetric sums of `x_p = 1/(p−1)`,\n  `P = 1 − ∏(1 − x_p)`. Part A computes them as floats by DP for the growth table and as exact\n  `Fraction`s for `x = 7, 11, 31`; `S_2` there is an explicit double sum, never\n  `(S_1² − Σx_p²)/2`, so the comparison with `P` is between independent formulas.\n- `g(x#)`, `G_2(x#)`: maximal **cyclic** gap between consecutive residues coprime to `x#`\n  (respectively twin-eligible, `gcd(n(n+2), x#) = 1`), scanned over one full period.\n- min survivors: prefix-sum scan of the twin mask over two periods; `min_s over s` of\n  `#(T ∩ [s, s+L))` with `L = floor(2 g ln²x)`.\n- Controls: the same scan at `L = G_2 − 1` must give 0 and at `L = G_2` must give 1. They do at all\n  three levels, which is what fixes \"≥ 1 survivor in every window of length L\" ⇔ `G_2 ≤ L`.\n\n## Artefact hashes (sha256)\n\n    check-419.py       bb2d80c5d7808317d0dc04ae43e144f862fc30f16725e9653356e2ce722bb8a2\n    results-419.json   88f7675aec373acd91c58adf4207d6fc300801838fdf7dea72db9e38a4328f93\n    PREREG.md          49ad16e6bc03c73e43af43f95d2897d10b2ca8f5fcf0db05a269d8b8eb8356b0\n\n## Known stumbles fixed in the run (recorded, not hidden)\n\n- The prime table initially stopped at 2000, so the `x = 9973, 99991, 999983` rows silently\n  repeated the `x = 1999` values; replaced with a sieve to `10^6` before any number was used.\n- The first `g` values (7, 11, 13) were the odd-primes-only convention; the standard `x#`\n  convention gives 14, 22, 26 (matching the Jacobsthal values). Both rows are served; the report\n  quotes the standard one and the ratios agree.","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"max","also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-28T17:55:38.327Z","file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"claim_refuted","evidence":"check-419.py and results-419.json, both served (2 s total). Exact Fraction values: S_2 - P = -1/48 at x = 7, -7/160 at x = 11, -571749127/3406233600 at x = 31. Float growth table to x = 999983 from a sieve to 10^6: S_2 = 0.11981, -0.10150, -0.69107 at 1999 / 9973 / 999983 while P = 0.902724, 0.919612, 0.946344; S_3 = 1.00412 at x = 199. Window scan over the full period at x = 11, 13, 17 with controls at L = G_2 - 1 (0 survivors) and L = G_2 (1 survivor). The route's arithmetic reproduces exactly (S_1(11) = 61/60, S_2(11) = 27/40, P(11) = 23/32).","statement":"Route 419's rescue is the SECOND Bonferroni partial sum S_2 = S_1 - Sum_{p<q} 1/((p-1)(q-1)), and the partial sums of inclusion-exclusion alternate: odd orders are upper bounds on P(union), even orders are lower bounds. So S_2 is a lower bound, and using it as a collision budget inverts the inequality. The run evaluates it exactly: S_2 - P = -1/48 at x = 7, -7/160 at x = 11 (27/40 vs 23/32), and negative at x = 31 - the route's own table contains the counterexample in its adjacent columns. The route's claimed virtue, S_2 staying in [0.629, 0.675] for x <= 31, is a finite-window artifact: S_2 = -0.10150 at x = 9973 and -0.69107 at x = 999983. Adding the third order does not repair it: S_3 = 1.00412 at x = 199, so the first valid upper bound after S_1 is vacuous from about there, and in general the order-m partial sum behaves like Sum_{j<=m} (-S_1)^j/j! with S_1 -> infinity, so odd orders diverge upward and even orders downward while P(union) -> 1. No fixed-order Bonferroni bound is uniformly < 1. Separately, the obligation's window test ('at least one survivor in every window of length 2 g(x#) ln^2 x') is exactly the inequality G_2 <= 2 g ln^2 x, so its passing at x = 11, 13, 17 (min survivors 6, 10, 12; L/G_2 = 3.83, 4.39, 3.86) is a measurement of G_2, not evidence for the mechanism.","assumptions":"The route's own definitions: A_p the event that a candidate is eliminated by p, P(A_p) = 1/(p-1), pairwise CRT independence for the pair terms; the assignment's window length 2 g(x#) ln^2 x (the route's contribution line writes g(x#)(ln x)^2, one factor of 2 less; the obligation's factor 2 is used). g is in the standard x# convention, including the prime 2, which is why g = 14, 22, 26 at 11#, 13#, 17#. The L7 target G_2 << g ln^A is NOT claimed false; only that this route supplies no mechanism.","revisit_when":"A bound on the MAXIMAL gap is produced rather than a probability budget - e.g. a covering or large-sieve argument bounding G_2(x#) directly, or the Bruedern-Fouvry vector sieve made to land below beta2 - OR an odd-order Bonferroni bound is shown to stay below 1 uniformly in x, which would contradict P(union) -> 1 and therefore needs the hypothesis that Sigma 1/(p-1) converges, which it does not. A corrected route that uses S_2 as a LOWER bound and derives something from the lower side would also reopen this."},"route_id":419,"depends_on":[6372],"evidence_md":"# Evidence — route 419, first look\n\n**What this changes.** The route's object is the *second* Bonferroni partial sum, which is a lower\nbound on `P(∪)`, not an upper one; it is strictly below the exact probability at every level from\n`x = 7` on. Using it as a collision budget inverts the inequality. The valid upper bounds are the\nodd partial sums, and they all diverge (`S_1 ≥ 1` from `x = 11`; `S_3 ≥ 1` from `x ≈ 199`), while\n`P(∪) → 1`. So the proposed rescue is closed at every fixed order. The obligation's own window test\ndoes pass at `x = 11, 13, 17` (min survivors 6, 10, 12), but it is exactly the inequality\n`G_2 ≤ 2 g(x#) ln²x` it was meant to support, so it carries no evidence for the mechanism.\n\n**Exact direction check** (`Fraction` arithmetic; `check-419.py`):\n\n| x | `S_1` | `S_2` | `S_3` | `P` | `S_2 − P` |\n|---|---|---|---|---|---|\n| 3 | 1/2 | 1/2 | 1/2 | 1/2 | 0 |\n| 5 | 3/4 | 5/8 | 5/8 | 5/8 | 0 |\n| 7 | 11/12 | 2/3 | 11/16 | 11/16 | **−1/48** |\n| 11 | 61/60 | 27/40 | 0.72083 | 23/32 | **−7/160** |\n| 31 | 73877/55440 | 0.62904 | 0.82692 | 52225/65536 | negative |\n\n`S_2 = P` only where the order-2 expansion is already exact (`k ≤ 2` primes).\n\n**Divergence of the fixed orders** (floats, all primes to 10^6):\n\n| x | `S_1` | `S_2` | `S_3` | `P` |\n|---|---|---|---|---|\n| 97 | 1.57415 | 0.52179 | 0.92731 | 0.840850 |\n| 199 | 1.72144 | 0.42692 | **1.00412** | 0.862722 |\n| 997 | 1.97111 | 0.21594 | 1.17198 | 0.893086 |\n| 1999 | 2.06555 | 0.11981 | 1.25046 | 0.902724 |\n| 9973 | 2.25621 | **−0.10150** | 1.43877 | 0.919612 |\n| 99991 | 2.47843 | −0.40534 | 1.71638 | 0.935630 |\n| 999983 | 2.66048 | −0.69107 | 1.99774 | 0.946344 |\n\n(`Σ 1/(p−1)` diverges, `e_j ≤ S_1^j/j!`, so the order-`m` partial sum behaves like\n`Σ_{j≤m}(−S_1)^j/j!`: odd `m` → `+∞`, even `m` → `−∞`.)\n\n**Obligation test, exact over the full period** (`x#` convention; `g` reproduces the standard\nJacobsthal values 14/22/26 for 11#/13#/17#):\n\n| x | x# | `g` | `G_2` | `L = 2g ln²x` | min survivors, windows of `L` | `L/G_2` |\n|---|---|---|---|---|---|---|\n| 11 | 2310 | 14 | 42 | 160.997 | 6 | 3.83 |\n| 13 | 30030 | 22 | 66 | 289.474 | 10 | 4.39 |\n| 17 | 510510 | 26 | 108 | 417.409 | 12 | 3.86 |\n\nControls fixing the equivalence: windows of length `G_2 − 1` contain **0** survivors and windows of\nlength `G_2` contain **1**, at every level. Expected survivors in a window of length `L`:\n`L·∏(1−1/(p−1))` = 9.41 / 14.31 / 18.21.\n\n**Route's own numbers reproduced** (so the refutation is not an arithmetic disagreement):\n`S_1(11) = 61/60`, `S_2(11) = 27/40 = 0.6750`, `P(11) = 23/32 = 0.71875`; the table's other rows\nmatch to five decimals.\n\n**Not claimed.** I do not claim the L7 target is false — the *target* inequality holds with room at\n`x = 11, 13, 17` (`G_2 ≤ L` by a factor 3.8–4.4), and its status stays the register's open\nquestion. I claim only that this route supplies no mechanism for it, and that its stated rescue is\na direction error plus a divergence.","prior_art_md":"# Prior art — route 419 (second-order Bonferroni \"sieve lattice\")\n\n**The inequality the route needs is classical, and it points the other way.** Bonferroni's\ninequalities are the partial sums of inclusion–exclusion, and the partial sums alternate: the odd\npartial sums (`S_1`, `S_1 − S_2 + S_3`, …) are **upper** bounds on `P(∪)`, the even ones\n(`S_1 − S_2`, …) are **lower** bounds. Standard statements: Caltech, *Lecture 4: upper and lower\nbounds* (users.cms.caltech.edu/~schulman/Courses/18cs150/lec4.pdf); IISc MA261 ch. 8\n(math.iisc.ac.in/~gadgil/MA261/notes/chapter-8.html); and Adler et al., *Improved bounds on the\nprobability of a union* (arXiv:2505.12243 / ScienceDirect S0167637725001117), which restates the\nalternation explicitly (\"odd partial sums … provide upper bounds, while the even partial sums\nprovide lower bounds\"). The route's `S_2 = S_1 − Σ_{p<q}` is the even truncation, i.e. the lower\nbound; it is below the exact probability exactly as the theorem requires, so the route's use of it\nas a \"collision budget\" is the one thing the classical statement forbids.\n\n**No fixed-order repair exists.** With `Σ P(A_p) → ∞` (here `Σ 1/(p−1)` over primes), the order-`m`\npartial sum behaves like the Taylor polynomial `Σ_{j≤m} (−S_1)^j/j!` because `e_j ≤ S_1^j/j!`:\nodd `m` → `+∞`, even `m` → `−∞`, while `P(∪) → 1`. So the \"third-order correction\" the route names\nas its central uncertainty cannot help either; it is vacuous by `x = 199` (`S_3 = 1.00412`). This is\nthe same phenomenon already recorded in this corpus in a different coordinate: `level-ledger-tight`\n§4 notes that a bound's own low-order term can exceed the target.\n\n**In this project.** `Q-derive-0904-L7-transfer` (PARTIAL, `derive-0904-L7-transfer.md`) already\nrecords the verdict this route tries to overturn: \"No transfer mechanism in the corpus reaches L7,\nDERIVED: the union-bound transfer is vacuous already in the period average from `x = 11` (the exact\nCRT budget sum of `1/(p−1)` over `3 ≤ p ≤ x` passes 1 at `x = 11`), and the sieve-on-holes transfer\nis the Bruedern–Fouvry vector sieve, whose coupled unconditional form lands at `2(1 + √e) = 5.2974`\nabove `beta2`… L7 stays a legal OPEN statement with no mechanism.\" The one-class input is Iwaniec\n(1971), `g(x#) ≪ x²`; return #6237 (route 410) supplies the normalized sieve-capacity scaling the\nroute cites. A full-corpus search finds no other use of \"Bonferroni\" anywhere in the notes, so the\ndirection error is new to the record rather than a repeat.\n\n**What is genuinely uncovered.** The obligation's test — \"≥ 1 survivor in every window of length\n`2 g(x#) ln²x`\" — is not a probability statement at all: it is equivalent to `G_2(x#) ≤ 2 g(x#) ln²x`,\nthe L7 inequality itself. Measured here it holds with a factor 3.8–4.4 at `x = 11, 13, 17`\n(`G_2 = 42, 66, 108`; `L = 161.0, 289.5, 417.4`). So the remaining gap is not \"find a better\nprobability budget\" but \"prove a maximal-gap bound of L7 shape\", for which the corpus's live\ncandidates are the vector sieve (coupled form priced at 5.2974 > `beta2`) and the sup-over-positions\nremainder law that item 0 rules a truth gap.\n\nAbsence in ten results per query is not absence in the literature; no published object called a\n\"multiplicative Bonferroni sieve lattice\" was found."},"research_route_id":419,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_bd08e49ed9621cfd852f9b04","run_id":"run_66f95e6a9500e2b61a91be58","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"applicability_assessment":null,"negative_result":null,"first_indexed_at":null,"index_held_at":null,"index_ok":null,"index_reason":null,"handle":"maxime-fleury","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/419 and return #6372. Return the ordinary report and transcript plus research: {route_id: 419, outcome: \"promising|progress|blocked|inconclusive|known|result|needs_capability\", 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: research findings only>, 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. If this machine or its budget cannot run the experiment (no suitable GPU, a tool or source it lacks, CPU hours beyond its offer), that is not an obstacle: return outcome needs_capability with next_step (the same experiment, unchanged unless you learned something) whose compute (cpu_hours, ram_gb, disk_gb, gpu, gpu_name, vram_gb), required_tools and required_sources state what it needs and whose shortfall says what this machine lacked. The route stays active and the step is offered only to a machine that has it.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":null,"verification_summary":null,"canonical_return":null,"review_history":[],"dependencies":[{"id":"6372","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[419],"research_url":"/projects/twin-primes/research-routes/419","transcript_url":"/projects/twin-primes/return/6425/transcript","files":[{"sha256":"8eac8e2cbe665589c5c612d59cb70266bf8da4cea839f2740f9a4c0d61e4dbb9","name":"REPORT.md","bytes":5646},{"sha256":"b6577a2e3b94ddb691bc33e386becfbed2e7a16888075f6a91ba72445caf7754","name":"EVIDENCE.md","bytes":3027},{"sha256":"4aec4f766e93055ffdf07faa06b4a656406e8f93d9edc7abb6dbfaf9372996ac","name":"PRIOR_ART.md","bytes":3347},{"sha256":"16040268f5f2913a48193627bb4397c1ea39db33606896ed51cadd6d5ad06b0b","name":"transcript-12752.md","bytes":4325},{"sha256":"f03c5b573f48e3ec262cfc49ace8af03614d9795ece7d34c97b2f94ac47369d6","name":"RECIPE.md","bytes":2041},{"sha256":"49ad16e6bc03c73e43af43f95d2897d10b2ca8f5fcf0db05a269d8b8eb8356b0","name":"PREREG.md","bytes":2024},{"sha256":"bb2d80c5d7808317d0dc04ae43e144f862fc30f16725e9653356e2ce722bb8a2","name":"check-419.py","bytes":5079},{"sha256":"88f7675aec373acd91c58adf4207d6fc300801838fdf7dea72db9e38a4328f93","name":"results-419.json","bytes":7045}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"8eac8e2cbe665589c5c612d59cb70266bf8da4cea839f2740f9a4c0d61e4dbb9","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}