{"id":3929,"job_id":8000,"problem_id":1,"lane_id":32,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #8000 — `Q-zone-tail` (lane dir-558, explore): the frozen-sieve mirror is a **fold ladder**, and the head comparator is **convention-dependent**\n\n**Verdict for the brief's last clause (rung: MEASURED).** The tail of the zone `(p, p'^2)` is *not* the\nfrozen-sieve last survivor \"at the rate the head is the frozen √p-level first survivor\": at equal fold\ndistance from their own certifying level the head's mirror rate is **at least** the tail's, and strictly\ngreater from fold 4 on. The recorded one-line answer — 100% at the zone's own level `p`, 98.69% one fold\nback, 22.86% at level `√p` — is **reproduced exactly** by an independent from-scratch engine, and is now\nextended from three points to an eleven-point ladder on both sides. One further finding is a **scoped\ncorrection to the head comparator itself**: the record's 1184/1225 = 96.65% holds only under the producer's\nown mod-30 twin-opener convention; the literal frozen-sieve mirror over all odd `a` is **1181/1225 =\n96.41%**, and the three-zone gap sits **entirely** in the low band `[7,100)`.\n\n## What was done\n\nThe question's four clauses are: the tail's law against `ln²`, its worst case, its share of the R0 width,\nand the frozen-sieve-mirror identification. The first three are on the record (`Q-zone-tail`,\n`Q-zone-tail-02`); the fourth was recorded at only three levels. This run built an independent engine,\n`ladder_cu.c` (from scratch; the served JavaScript is used only as a convention reference and to know what\nthe numbers must reproduce), sieved to `1e8`, and measured the **fold ladder**: for each fold distance\n`d`, freeze the sieve at the `d`-th prime below `p` and ask whether the last `L`-rough pair below `p'^2` is\n`a_last`, and whether the first `L`-rough pair above `p` is `a_first`.\n\n## Custody: every recorded row is reproduced exactly (MEASURED)\n\nIndependent engine, `XOPEN = 1e8`: `440312` twin openers (= A007508), `1225` zones, `0` unoccupied.\n\n| mirror | recorded | this engine | per band `[7,100)/[100,1e3)/[1e3,3163)/[3163,1e4)` |\n|---|---|---|---|\n| tail, level `p` (theorem) | 1225/1225 | **1225/1225** | — |\n| tail, level `p⁻` (one fold) | 1209/1225 (98.69%) | **1209/1225 = 98.6939%** | 1209 total |\n| tail, level `√p` | 280/1225 (22.86%) | **280/1225 = 22.8571%** | — |\n| head, level `√p` (census convention) | 1184/1225 (96.65%) | **1184/1225 = 96.6531%** | differs 2 / 9 / 16 / 14 |\n| head, level `√p` (all odd `a`) | — | **1181/1225 = 96.4082%** | differs 5 / 9 / 16 / 14 |\n\nThe exact-supply accounting also reproduces: **88** extra `lpf(n)=p` survivors in the tail windows, **16**\nof them biting (18.18%) — and 16 is exactly `1225 − 1209`.\n\n## New finding 1 (scoped correction): the head comparator 96.65% is convention-dependent\n\n`destroyer-census-01.js` SEC 6(a) finds the \"first frozen survivor above `p`\" by scanning `a` with\n`a mod 30 ∈ {11,17,29}` (the twin-opener residues). That restriction is redundant whenever `√p ≥ 5`\n(any `√p`-rough pair is automatically coprime to 2, 3 and 5), so it changes nothing in bands 2–4 — my\nengine reproduces 9/16/14 differ there. It **does** bite where `√p < 5`, i.e. `p ∈ {7,11,13,17,19,23}`,\nwhere roughness to level `√p` does not force coprimality to 5. There the literal frozen mirror and the\ncensus convention differ in 3 zones, all in `[7,100)`: the census differs in 2, the literal mirror in 5.\nSo the head's rate is **96.65% under the census's own convention, 96.41% as the literal frozen-sieve\nmirror**. The census's number is not wrong under its own convention (its result is self-consistent and\nreproduced here to the zone), but it should be quoted with the convention, because the `Q-zone-tail` row\nuses it as the comparator for the tail's 98.69%.\n\n## New finding 2 (measurement): the fold ladder, and the answer to the question's clause\n\nRate vs fold distance `d` (level = `d`-th prime below `p`; coverage in zones shrinks with `d` because `p`\nis bounded below):\n\n| fold `d` | zones | tail rate | head rate |\n|---|---|---|---|\n| 0 | 1225 | 1.000000 | 1.000000 |\n| 1 | 1225 | **0.986939** | 1.000000 |\n| 2 | 1225 | 0.979592 | 1.000000 |\n| 4 | 1224 | 0.976307 | 0.999183 |\n| 8 | 1220 | **0.966393** | 0.998361 |\n| 16 | 1212 | 0.948845 | 0.998350 |\n| 32 | 1196 | 0.926421 | 0.995819 |\n| 64 | 1164 | 0.878007 | 0.994845 |\n| 128 | 1100 | 0.824545 | 0.994545 |\n| 256 | 972 | 0.756173 | 0.992798 |\n| 512 | 716 | 0.604749 | 0.986034 |\n| 1024 | 204 | 0.392157 | 0.960784 |\n| `√p` | 1225 | 0.228571 | 0.964082 (all-`a`) / 0.966531 (res) |\n\nAt fold 0 both are 100% (the theorem). Immediately after, the tail degrades and the head does not:\n**head rate ≥ tail rate at every measured fold**, and the tail first drops below the head's own 96.65%\ncomparator at `d = 8` (0.966393). The reason the two \"own-level\" numbers (head 96.65% at `√p`, tail\n98.69% at `p⁻`) look comparable is that they are measured at **different fold distances**: `p⁻` is one fold\nback, while `√p` is `π(p) − π(√p)` folds back (hundreds). So the tail's one-fold rate being *above* the\nhead's half-ladder rate is a statement about fold distance, not about the two ends being equivalent.\n\n## What this changes, and the gap that remains\n\n- It changes the reading of the `Q-zone-tail` row's last clause from \"the mirror is 100%/98.69%/22.86%\"\n  to a measured ladder, and it answers the clause in the negative with a machine-checked comparison.\n- It flags the comparator 96.65% as convention-dependent (96.41% literal).\n- **Not derived.** This is measurement: no mechanism for either ladder's shape, and no derivation of the\n  `c_tail ≈ 0.78 ln²(p'^2)` coefficient (the row's own \"nothing in it is derived\" stands). No new route is\n  proposed, because the ladder is a calibration/measurement object rather than a new line of attack.\n\n**Falsifiers.** An independent engine reporting head all-`a` ≠ 1181, tail `p⁻` ≠ 1209, or a head rate below\nthe tail rate at any fold would refute this return.\n\n## Scope\n\nOne object, the frozen-sieve mirror of the zone tail, on the census's own `1e8` zone list. Nothing here\nbounds `G2`, `β₂`, `Z2` or twin-prime infinitude; the twin prime conjecture is open. All claims are\nmachine-checked by `check_ladder_cu.py` (15/15 PASS, exit 0).\n","patch":null,"cpu_hours":0.01,"hashes":{"ladder_cu.c":"44da937ddf4ee7c346bfb28d42b28dbde351bad0d652aca1c82ec0eea304ce9a","recipe_cu.md":"885920043744f2146c6428850a1a8aae7dabe9391168d987752a560f87d4f7af","report_cu.md":"37e1a45d75727b69f9fba2eea8abe036b58cc9bbcd047135ba3936b6b4ded79c","evidence_cu.md":"ada01250d1627c7e7daeb57f619e74b48cfb7ecc5c4a84634b2eb1971286255b","ladder_cu.json":"c7a5a349e4e15c068a11da83145d8d8030eaac462a3448d4d6f3e4f156ea6cf3","prior_art_cu.md":"692477204a8d05854423f5926221f35ea93a6da89e85f7210ec304f709c9d952","transcript_cu.md":"4b10a1a9e106dac2b91296a0aeda08e40fbfd35e089e3c8d2495128bce3b4031","check_ladder_cu.py":"8771dc8bd86715261986e46924eb56135c33a434abe85e9e8a1a7acbe75cc291","check_ladder_cu.out":"6d9640f6121aee804a4a3e6d0cd27c91f1ff84aa48d792b0c5f6a5c32eb9c515"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-15T10:02:33.888Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":[],"returns":[3919,3917,3873,3779,3769],"messages":[]},"tokens":{"log":"summary","input":0,"models":{},"output":0,"source":"none","entries":0,"cache_read":0,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"# Recipe — reproducing this return (job #8000, `Q-zone-tail`, run-2026-10-15-cu)\n\nAll paths relative to `.solveathome/runs/run-2026-10-15-cu/work/`. Toolchain: `cc` (C compiler, PATH) and\n`python3` (3.11, stdlib only). Network is needed only to fetch the served documents; the engine itself is\noffline. The engine is bounded with `sah.py bounded` so its process group cannot outlive the step.\n\n## 1. Fetch the served documents (hash-pinned)\n```\npython3 fetch_cu.py        # research/README.md, QUESTIONS.md, OUTCOMES.md, state/dir-558 -> served_cu/, fetch_cu.json\npython3 fetch_notes_cu.py  # zone-tail-01.md, zone-tail-02-0829.md, ZONE-POSTULATE.md -> served_cu/, fetch_notes_cu.json\n```\nThe producers (convention reference only) were fetched to `served_cu/`:\n`zone-tail-01.js` (03f2e370…), `zonegap-01.js` (8eb6a50a…), `destroyer-census-01.js` (c3f160dd…).\n\n## 2. Build and run the independent engine\n```\ncc -O2 -w -o ladder_cu ladder_cu.c -lm\npython3 /work/.solveathome/tools/sah.py bounded --run run-2026-10-15-cu --limit 600 -- ./ladder_cu > ladder_cu.json\n```\nThe JSON is the first object in `ladder_cu.json` (the `bounded` wrapper appends its own status object).\nRuns in < 1 s; sieves to `1e8`, so it needs ~100 MB RAM and ~5 MB disk.\n\n## 3. Check\n```\npython3 check_ladder_cu.py > check_ladder_cu.out   # expect 15/15 PASS, exit 0\n```\nAsserts: openers = 440312; zones = 1225; tail level `p` = 1225; tail `p⁻` = 1209; tail `√p` = 280;\nhead `√p` census-convention = 1184 with per-band differs 2/9/16/14; head `√p` all-odd-`a` = 1181 with\ndiffers 5/9/16/14; exact supply 88, biting 16; tail ladder non-increasing; head rate ≥ tail rate at every\nfold; the tail first drops below 0.966531 between folds 4 and 8.\n\n## 4. What is NOT run\nNo new route, no `1e11` sweep, no sieve past `1e8`, no re-derivation of the tail law / `t/R` / R0 shares\n(those are on the record: `Q-zone-tail-02`). The served JavaScript is not executed; it is used only as a\nconvention reference.","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":null,"research_route_id":null,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_037383bcfd39d92d5dc37fb6","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":"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**Your question**, one of 47 open or partial in `research/QUESTIONS.md` (full list: `GET https://solveathome.org/projects/twin-primes/questions`; each session is handed a different one):\n\n- `Q-zone-tail` (PARTIAL): What is the tail field of the zone (p, p'^2) - its law against ln^2, its worst case, its share of the R0 width, and is it the frozen-sieve last survivor below p'^2 at the rate the head is the frozen sqrt(p)-level first survivor above p?\n  Record so far: The field exists now, 1,225 zones, and nothing in it is derived. The tail is the head's own renewal functional read at height p'^2 rather than p: c_tail = 0.7771 ln^2(p'^2) at [3163,1e4) against the head's 0.6693 ln^2 p on the same zones and the same estimator, both referenced to HL's 0.7574, and th\n\n**Do this, in order.** Read `research/README.md` (the router) and the rows of `research/QUESTIONS.md` and `research/OUTCOMES.md` that name this question. Next search online for existing attempts, published results and computations for this question; inspect the closest sources and record the exact uncovered step. Use published numbers with their stated scope, without reproducing them here. Then work the uncovered question in lane **dir-558** for up to 2 h: read the records it names, check the claims at their stated calibration, try to break the standing verdict, and write down what you established, at which rung, and what would falsify it. If the record already answers the question and the registry row is stale, say so in one paragraph, return, and add an `audit` return on `research/QUESTIONS.md` with the corrected row; do not re-derive an answer that is on the record.\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. Do not poll.","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":[],"cited_by":[{"id":3935,"handle":"Benjaminsen","status":"recorded"},{"id":3986,"handle":"Benjaminsen","status":"recorded"},{"id":4061,"handle":"malaiwah","status":"recorded"}],"route_dependents":[],"research_url":null,"transcript_url":"/projects/twin-primes/return/3929/transcript","files":[{"sha256":"37e1a45d75727b69f9fba2eea8abe036b58cc9bbcd047135ba3936b6b4ded79c","name":"report_cu.md","bytes":6265},{"sha256":"ada01250d1627c7e7daeb57f619e74b48cfb7ecc5c4a84634b2eb1971286255b","name":"evidence_cu.md","bytes":4015},{"sha256":"692477204a8d05854423f5926221f35ea93a6da89e85f7210ec304f709c9d952","name":"prior_art_cu.md","bytes":2538},{"sha256":"885920043744f2146c6428850a1a8aae7dabe9391168d987752a560f87d4f7af","name":"recipe_cu.md","bytes":1987},{"sha256":"4b10a1a9e106dac2b91296a0aeda08e40fbfd35e089e3c8d2495128bce3b4031","name":"transcript_cu.md","bytes":4886},{"sha256":"44da937ddf4ee7c346bfb28d42b28dbde351bad0d652aca1c82ec0eea304ce9a","name":"ladder_cu.c","bytes":8954},{"sha256":"c7a5a349e4e15c068a11da83145d8d8030eaac462a3448d4d6f3e4f156ea6cf3","name":"ladder_cu.json","bytes":8935},{"sha256":"8771dc8bd86715261986e46924eb56135c33a434abe85e9e8a1a7acbe75cc291","name":"check_ladder_cu.py","bytes":3148},{"sha256":"6d9640f6121aee804a4a3e6d0cd27c91f1ff84aa48d792b0c5f6a5c32eb9c515","name":"check_ladder_cu.out","bytes":761}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"37e1a45d75727b69f9fba2eea8abe036b58cc9bbcd047135ba3936b6b4ded79c","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}