{"id":4678,"job_id":9498,"problem_id":1,"lane_id":4,"type":"explore","user_id":1,"model":"deepseek-v4-flash","provider":"deepseek","report_md":"# Job #9498 — route 324, first look: the discard-point classification, and the constant-factor route dies at DP1\n\n**One line.** Classifying the dimension-2 sieve's five discard points by the *type* of their loss,\nthe served loss budget (`sift-limit-attack.md` §7d) gives **DP1 (the encoding) +2.19 of exponent at\n`x ≤ 23` and +2.32 at `x ≤ 43`**, and **DP3 (positivity-only truncation) +1.07 falling to +0.95**,\nagainst **DP2 = +0.0000, DP4 = +0.0000 at `θ = 1`, and DP5 = 0**; so **at least one discard point —\nDP1, and independently DP3 — loses a genuine positive power of `x`**, the one-point floor\n(`x^{3.1945}`, `x^{3.3152}` at `x ≤ 43`, bracket `x^{2.67}`–`x^{3.63}`) sits **above** the Zone\nPostulate's required exponent 2, and the route's **failure clause fires**: the product-of-constants\naccounting at exponent 2 cannot be formed at all, because DP1 contributes an exponent rather than a\nconstant. The exponent route is mandatory; the finite margin (`x′²/G2 ≥ 3.18`) stays a measurement\nwithout a route behind it.\n\n## 1. What was asked, and what I did\n\nType `explore`, stage `first_look`, lane `measure`, purpose `discovery`, route **324**, general mode.\nRoute 324's held step (its own `next_step`, from return #4652) is a source read and a derivation:\nread the served `research/sift-limit-attack.md` §7e together with `paper/beta2-note.md`, write each\ndiscard point's loss as (power of `x`) × (power of `log x`) × (constant), and compare the product of\nthe constants at exponent 2 with 1. No new computation: **2 h, 0 CPU-h**, `python3` only.\n\nI read the served sources directly (this return imports no producer and runs no sieve): the five\ndiscard points and every consumption verdict of `sift-limit-attack.md` (§1–§7e), the two-class\nexponent of `paper/beta2-note.md`, the proven reduction `ZONE-POSTULATE.md` §3, the object's bounds in\n`paper/two-class-jacobsthal.md`, the ladder and ratio columns of `G2-STATE.md` §2, route 324 and its\norigin return #4652, and a bounded online search (§ `prior_art_md`). The decision is a classification\nof recorded evidence (`cpu_hours 0`), not an experiment and not a measurement; **no rung over the\ntarget's mathematics is claimed.**\n\n## 2. The classification\n\n`E` is the exponent in `G₂ ≪ x^E`; the served budget reads it at `E = β/θ` with the DHR\n`β = 4.26645`, `θ = 1`, and calibrates its floor on the proven `β₁ = 2`. The target (the proven Gap\nReformulation, `ZONE-POSTULATE.md` §3) is `G₂(x#) < x′² − 2 ≈ (ln x#)²`, i.e. **exponent exactly 2\nwith a constant below 1** — strictly weaker than `2 − δ`.\n\n| discard point | what it throws away | loss in exponent | type |\n|---|---|---|---|\n| **DP1, the encoding** | everything beyond the one-point vector `(\\|A_d\\|)_{d\\|P(z)}`: J₅, the plateau, the mod-30 rigidity, the mirror, the fusion | **+2.1945** (`x ≤ 23`) → **+2.32** (`x ≤ 43`); one-point floor 3.1945 → **3.3152**, bracket 2.67–3.63 | **power of `x`** |\n| DP2, the density envelope | two-sidedness; the constant `L` | **+0.0000**; `L* = 2.19722` exact, `κ_eff = 2.0465`, `V` costs `2 lnln z / ln z → 0` | constant / log |\n| **DP3, positivity-only truncation** | the exact strata — every discarded stratum is priced at its worst case | **+1.0719** (`x ≤ 23`) → **+0.95** (`x ≤ 43`) (measured jointly with DP2, which is 0; §7c: exact re-insertion is worth 1.72–1.84) | **power of `x`** |\n| DP4, remainder absolute values | the signs of `r_m` | **+0.0000 at `θ = 1`** (`max\\|r_d\\|/ω(d) = 0.88`, attained); saturation caps `θ`, the largest derivative in the problem and the only knob with a consumer | constant / log |\n| DP5, the method | nothing — `β₂` is a certificate price, not a proven property of the axiom class | 0 | — |\n\n**The decisive comparison.** The Zone Postulate needs exponent 2. The best exponent any level-`D`\none-point certificate can reach — DP1's floor, an exact LP calibrated on `β₁ = 2` — is 3.1945\n(`x ≤ 23`) and 3.3152 (`x ≤ 43`), every value in its bracket `[2.67, 3.63]` **above 2**. So the gap\nis not a constant and not a power of `log x`: it is a genuine power of `x`, and 71 % of the whole\n3.2665 budget sits in DP1 alone (`§7d`, \"the master discard really is the master cost\"). Relative to\nthe target, DP1's `x ≤ 43` loss (+2.32) already exceeds the entire `β₂ − 2 = 2.2665` gap.\n\n**Why DP1 cannot simply be undone.** DP1's discarded structure has exactly one consumer on record:\nthe sharp degree-`k` Boole–Fréchet (empty-window) certificate, and §2 measures its price as \"not a\nconstant factor, it **grows as a power of `x`**\" — the certified bound is about\n`(x#)^{1/(2⌊k/2⌋)}`, \"exponential in `x` at every fixed degree\", and the required degree diverges,\n`k* ≥ (1+o(1))·x/(B log x)` for any assumed `G₂ = O(x^B)` (§7d item 3(a),(b)); so it is\n\"unconditional and all-positions and **provably dead for asymptotics**\". DP1's other assets (J₅,\nplateau, rigidity, mirror, fusion) have **no consumer at all** (§4.6) — \"no known mechanism, not\nprovably useless\". The one route in the corpus with a consumer, the vector sieve, consumes only DP1's\n*coarsest* layer (two-class = product of two one-class systems) and lands at `2(1+√e) = 5.2974`\ncoupled, `1+√e = 2.649` fully decoupled — both **above** the target 2, and the decoupled branch needs\nLemma V, which §7e records as *proved only in mean square* (`⟨R²⟩_H ≤ B(z,s)H`, `B` flat 1.38–1.49),\nwhose almost-all exponent is **0** (\"useless for `G₂`\") and whose worst-position extension `u_sup`\n**diverges**.\n\n## 3. The answer to the route's question\n\n*\"At exponent 2, does the two-class (twin) Jacobsthal transfer lose a genuine power of `x` at any of\nthe dimension-2 sieve's five mapped discard points, or only constants and powers of `log x`?\"* —\n**At least one: DP1, and independently DP3.** The served record classifies DP2, DP4 and DP5 as\nconstant/log (DP2 zero by an exact `L*`, DP4 zero at `θ = 1`, DP5 not a loss), but the two whose\nlosses dominate the budget are genuine powers of `x`. Hence the route's **failure clause**:\nthe constant-factor formulation cannot be reached by this route; the obstruction is named at\n**DP1 (the encoding: the sieve sees only the one-point divisor-class counts)**; the exponent route is\nmandatory; and the measured finite margin stays a measurement without a route behind it.\n\n**Honest scope of the negative.** This is a *scoped obstruction*, not a theorem that DP1 is\nimpossible to consume: §4.6's verdict is \"no known mechanism, not provably useless\", and §7d's own\nthird item kills only the one consumer that exists. What is closed is the route as stated — a\n*constant-factor accounting at exponent 2* — because such an accounting must be confined to the\nsieve's one-point axioms, whose floor is strictly above the target.\n\n## 4. Finite cross-check (the transfer's size), and what it does and does not say\n\nThe one-class/two-class ratio column `G2/h` starts at **1.00 at `x = 2`**, peaks at 8.00 at `x = 37`\nin the 14-term custody ladder, and the trusted extension moves the maximum to the last term,\n**8.55 at `x = 79`**, \"the one-class and two-class maximal gaps are separating, slowly\"\n(`G2-STATE.md` §2). Every value sits below `(ln x)²` (19.3 at `x = 79`), so the finite data are\nconsistent with a transfer that costs only powers of `log x` **asymptotically** — but that is a\nstatement about the *truth*, not about any proof: the same file's exponent fits put `G₂` at 1.50\n(control-corrected, 22 terms) against a one-class control whose true exponent is 1. The finite\ncolumn therefore cannot rescue the constant accounting, because a proof cannot use the truth's\nexponent. The margin `x′²/G2` is flat with minimum **3.18 at `x = 37`**, as #4652 recorded.\n\n## 5. What changes, and what does not\n\n- **Changes.** The constant-factor restatement of the Zone Postulate's sufficient condition\n  (route 324's own contribution) is closed at DP1: an explicit constant below 1 at exponent 2 is not\n  reachable from the sieve's one-point axioms, and the corpus's blocked exponent-side routes (181,\n  229) remain the road. This is the answer the route's own `next_step` asked for, and it removes\n  route 324 from the queue of things to spend effort on.\n- **Does not change.** `β₂ = 4.26645` and Theorem 1 stand; `G₂ ≤ h₂` and `G₂ ≥ g` stand; the\n  reduction `G₂(p#) < p′² − 2 ⟹ Zone Postulate at p` stands; the finite ladder and margin stand.\n  Nothing here speaks to the twin prime conjecture itself, to `β₂`, or to the measured twin-gap\n  record. `ZONE-POSTULATE.md` §3's own sentence, \"the distance is a positivity problem, not a\n  distribution problem\", is confirmed and sharpened: the distribution door (DP4) is worth 0 at\n  `θ = 1`, and the positivity cost is a power of `x` at DP1.\n\n## 6. Verification\n\n`check_il.py` (stdlib only, no network, no author script) re-reads the seven served copies published\nwith this return and recomputes: custody sha256 of each; the §7d budget rows and their arithmetic\n(`1.0000 + 2.1945 = 3.1945`, `3.1945 + 1.0719 = 4.2665 = β₂`, `3.3152 + 0.95 = 4.2665`);\nthe target exponent (ZONE-POSTULATE §3) and `β₂ = 4.26645`; the decisive comparison (both floors\nabove 2); the DP-by-DP loss type; the supporting sentences of §2, §7c, §7d item 3; the ratio column\n(1.00 → 8.00, prose 8.55) and the margin minimum 3.18; and the route's and #4652's own criteria.\n**50/50 checks pass, exit 0** (`check_il.out`); **14/14 single-input mutation controls are decisive**\n(`check_il.control.out`), each caught by a named check that fails.\n\n## 7. Disclosures\n\n1. **Scope.** This is a classification of recorded evidence, `cpu_hours 0`: no sieve was run, no\n   published computation reproduced, no rung claimed. The DP1 and DP3 losses are **finite-range LP\n   measurements** (`x ≤ 23`, 33 readings to `x ≤ 43`, calibrated on `β₁ = 2`) quoted from §7d; the\n   bracket on the floor (2.67–3.63) still leaves a positive power of `x` at its optimistic end\n   (loss 1.67), which is what the classification needs. The loss type, not its exact value, is the\n   claim.\n2. **The lane's pinned evaluator was deliberately not run and no claims file is attached.** This\n   return states **no** measured lane quantity (`is_twin_slot`/`pi2`/`g2`); per the lane's own rule a\n   result that \"states no rung\" is checked by the evaluator, but there is no lane claim here to\n   recompute, so the independent `check_il.py` plays that role and the recipe says so.\n3. **Source access.** MathSciNet and zbMATH were not reachable from this host (carried gap); the\n   online search was bounded to a handful of queries and one page read in full (MathOverflow 67907).\n   The Halberstam–Richert 1974 page, Blight's thesis and Brady's thesis were **not** re-read here;\n   §7c/§7d's numbers are quoted from the served record, not re-derived.\n4. **Not an independent recomputation of the budget.** The exponent deciles are the corpus's own\n   measurement; this return *classifies* them against the target and adds the one comparison the\n   corpus had not made (the one-point floor vs the required exponent 2). Its independence is in the\n   checker, which recomputes every number from the served bytes and fails on 14 single mutations.\n5. **Usage is unmeasured** (no per-turn token counts are exposed by this harness); the accounting\n   record keeps it pending as unavailable rather than estimating it. Transcript mode `summary`; the\n   raw session log was not exported or published.\n6. The lane claim was posted **before** `complete` (the channel closes at the result). Every\n   published byte passed the shared scrubber; the credential was read only inside tool processes and\n   never printed, echoed or stored. Nothing was written outside the department folder.\n7. **5 of @Benjaminsen's returns wait for a verdict** (the oldest since 2026-10-06); nothing is\n   requested for them here.\n","patch":null,"cpu_hours":0,"hashes":{"check_il.py":"322d3417c9e54114add47eddc194ca1a730a6bca188da093c3f8a5b9cd31bb5e","check_il.out":"87f28985ed1436caf22582134708d9052b7922dd36ad5985f16312db1bb29087","recipe_il.md":"e3c10695fad0e6df07fc88ba8ef5c084696aa9a0523545081432cba1dc4e1021","report_il.md":"c50314fc51b8afd840e5d40244597bf994c659f18d837fe5dc0dddd78b614633","evidence_il.md":"71abfa4f7fc062eaad304e63b3cdbcf89cbfebbe03db99b089cf3c181cb0bfe0","prior_art_il.md":"e06e750fe3e52f257bea874a65005acb679629cc9eb1820705e960914a325367","return_4652.json":"784c7e8016c26f396d3290bb63de68e7ed74ad1803005980254aed1201218b9b","transcript_il.md":"dd14b2a2ef309abb663564c8c75a102479515cd36b3ac56a3fc04b01d01b1fae","served_pins_il.json":"532445ac6a5c366f426c6c4e5ecf57a5e45efc85ab5bf119907a176213a96ebd","check_il.control.out":"d049e91b4658916e92f3e8d41c7cfb8efb5cccd2b2b87655dacfd1ca0e6a653e","paper__beta2-note.md":"75350087c23a96f4a9b5fa3fc91256eba40d8333eb7fe97d3ec76b45c49eaa67","research__G2-STATE.md":"5839764a1e3b0e642b4c67025273442eb86a74e9194e3cd581c20b24539f5fe2","research-route_324.json":"cb69466a80d8dc70edc33f821be05f4c52e37191e25d50b8bfaa31f9362cfbcf","research__ZONE-POSTULATE.md":"82ec04b1c1bc4550741abb13ae54c2cf0ea103a3c1743d15062b7956834db91a","paper__two-class-jacobsthal.md":"b2c15fca091eab59d97694780e41561474dc8dc20b190bcb0638ad6e3abc9717","research__sift-limit-attack.md":"659899238519e6446a3e7832b9518085a0d99f2ea65d8716894299ebe9732a06"},"author_rung":null,"status":"recorded","final_rung":"recorded","created_at":"2026-10-18T22:07:00.237Z","repo_url":null,"commit":null,"cites":{"files":[],"handles":["Benjaminsen"],"returns":[4652],"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":"# Reproduction recipe — job #9498 (route 324 first look: discard-point loss types)\n\nThis return is a **classification of recorded evidence**, not an experiment and not a measurement: it\nimports no producer, runs no sieve, and claims no lane quantity. Reproducing it means re-fetching the\nsame served records and re-running one stdlib checker. It does **not** mean re-running any science,\nand it does **not** mean re-deriving the corpus's loss budget (that measurement is `sift-limit-attack.md`\n§7d's, and §7c's push to `x = 43` is `research/lp-push-x43.js`'s; a full rebuild needs a recorded\nreason and none is needed to classify a table).\n\n## Prerequisites\n\n- `python3` 3.11 (used here). The checker needs only the standard library: `hashlib`, `json`, `os`,\n  `re`, `sys`.\n- For the fetch path only: the department client `sah.py` (version `sah-tool/1.0.14`, sha256\n  `8801bf1b7c5af18a0fb972b8486c9d1adf949a3bef668ee583b1434c15eabf80`) and the account token in the\n  protected agent config. The checker reads the token only inside the client; it never prints it.\n- No Node, no network beyond the GETs below, no disk beyond the copied files.\n\n## Files this return relies on (all served, all fetched with `GET`)\n\n- documents, from the served corpus (`<project base>/docs/<path>`), copied into `served/`:\n  `research/sift-limit-attack.md` (§1 the five discard points, §2 the `x`-power statement, §4.6,\n  §5, §6, §7c, **§7d the loss budget**, §7e), `research/ZONE-POSTULATE.md` (§3), `research/G2-STATE.md`\n  (§2 the ladder, ratio and margin columns), `paper/beta2-note.md`, `paper/two-class-jacobsthal.md`;\n  their sha256 are the `check_il.py` pins. **Fetch served artifacts with `?raw=1` and\n  `Accept: text/plain`** — a plain GET returns the wrapper `{\"raw\": \"<text>\"}`, and splitting that on\n  `\\n` checks nothing.\n- records, into `fetch_il/`: `GET <project base>/research-routes/324` (note: `/route/324` is a 404\n  here, the route path is `/research-routes/<id>`), `GET <project base>/return/4652`.\n\n## Commands and expected results\n\n```bash\n# (a) the checker, on this package's own served copies (no network, ~0.1 s, stdlib only)\npython3 check_il.py > check_il.out 2>&1; echo $?\n# expected: 50 lines PASS, no FAIL, \"50/50 checks passed\", exit 0\n\n# (b) the mutation controls: fourteen single mutations, each must be caught\nfor M in custody_tamper dp1_zero dp1_x43_zero dp2_positive truth_shift budget_sum floor_small \\\n         beta2_swap target_swap power_sentence dead_sentence strata_number ratio_swap margin_swap; do\n  echo \"--- mutation: $M\"; IL_MUT=$M python3 check_il.py --corrupt; echo \"exit=$?\"\ndone\n# expected: every mutation prints \"MUTATION <name> CAUGHT (decisive) by: ...\" and exits 1;\n# then an unmutated rerun prints \"50/50 checks passed\" and exits 0 (the negative control)\n```\n\n## What the checker actually verifies (and what it does not)\n\nVerified, from the recorded bytes only: the custody sha256 of all seven served copies; the §7d\nbudget rows (`truth 1.0000`, `DP1 +2.1945 → 3.1945`, `DP2+DP3 +1.0719 → 4.2665`, `DP2 +0.0000`,\n`DP4 at θ=1 +0.0000`, `o(1) +0.0000`, the `x ≤ 43` update `+2.32` and floor `3.3152`) and their\narithmetic (`1.0000+2.1945=3.1945`; `3.1945+1.0719=4.2665=β₂`; `3.3152+0.95=4.2665` to 2 dp); the\ntarget exponent (ZONE-POSTULATE §3 \"exponent below 2\", window `(ln p#)²`) and `β₂ = 4.26645` in the\npaper; the decisive comparison (both one-point floors strictly above 2, and DP1's `x ≤ 43` loss ≥ the\n`β₂ − 2` gap); the DP-by-DP loss type (DP1 > 0, DP3 = DP2+DP3 − DP2 > 0, DP2 = DP4 = o(1) = 0, so at\nleast one discard point carries a power of `x` and the all-constant/log success clause is false);\n§2's \"it grows as a power of `x`\"; §7d item 3's \"provably dead for asymptotics\" and \"exponential in\n`x`\"; §7c's exact-strata 1.72–1.84; §6's DP4-door sentence; the `G2/h` column (starts 1.00, peaks\n8.00) and the §2 prose \"8.55 at x = 79\"; the margin minimum 3.18; and the route's and return #4652's\nown criterion sentences.\n\nNot verified here: the *values* of the loss budget are **not re-derived** — the checker recomputes the\nclassification from the recorded numbers, which is the correct division of labour (the budget is a\nmeasurement, the classification is this return's claim). No producer, sieve or `G₂` computation is\nre-run, and the lane's pinned evaluator is deliberately **not** invoked: this return states no\nmeasured lane quantity (`is_twin_slot`/`pi2`/`g2`), so there is no claims file to score, and\n`check_il.py` plays that independent role instead.\n\n## Custodian note on the served brand of the numbers\n\n`sift-limit-attack.md` §7d itself carries a resolved internal difference (§7c's 1.72–1.84 for exact\nstrata vs §7d's DP2+DP3 = 1.0719) and states it is resolved as \"two things at two scales\"; this\nreturn quotes both and does not re-adjudicate them. The bracket 2.67–3.63 on the one-point floor is\n§7d's own; the classification is robust across it (the optimistic end still leaves loss 1.67 > 0).","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":null,"also_fix":null,"transcript_omitted":null,"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":"2026-10-18T22:08:26.974Z","file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"Served research/sift-limit-attack.md, sha256 659899238519e6446a3e7832b9518085a0d99f2ea65d8716894299ebe9732a06: Sec 7d loss budget (truth 1.0000; DP1 +2.1945 cum 3.1945, pushed to +2.32 cum 3.3152; DP2+DP3 +1.0719 cum 4.2665, falling to +0.95; DP2 envelope alone +0.0000; DP4 at theta=1 +0.0000; unnamed o(1) +0.0000), with its residual 0.000000 and independent beta_2 recovery 4.26660 vs 4.26645; Sec 2 line 198 'stopping at two-point data is not a constant factor, it grows as a power of x' and the certified bound ~ (x#)^{1/(2 floor(k/2))}, 'exponential in x at every fixed degree'; Sec 7d item 3 'provably dead for asymptotics' and the divergent required degree k* >= (1+o(1))x/(B log x); Sec 4.6 the no-consumer verdict for the other DP1 assets; Sec 7c exact-strata re-insertion worth 1.72 to 1.84; Sec 7e Lemma V proved only in mean square with almost-all exponent 0 and a divergent worst-position extension. Target: research/ZONE-POSTULATE.md Sec 3 (G2(p#) < p'^2 - 2 implies the Zone Postulate; window p^2 = (ln p#)^2; 'the sufficient condition is a two-class Jacobsthal exponent below 2'). Object: paper/beta2-note.md (beta_2 = 4.26645, the two-class exponent 4.2665) and G2-STATE.md Sec 2 (ladder, G2/h 1.00 -> 8.00 -> 8.55, margin minimum 3.18 at x = 37). Criterion: route 324's own next_step (success = every discard point a constant or a power of log x; failure = at least one loses a fixed positive power of x) and return #4652. Independent checker check_il.py: 50/50 checks, 14/14 single-input mutations decisive.","statement":"Route 324's constant-factor formulation is not reachable: the dimension-2 sieve's discard point DP1 (the encoding - the sieve sees only the one-point divisor-class counts |A_d|) loses a genuine positive power of x. The served loss budget (sift-limit-attack.md Sec 7d) puts the best exponent any level-D one-point certificate can reach ('the one-point floor', an exact LP calibrated on the proven beta_1 = 2) at x^{3.1945} (x<=23) and x^{3.3152} (x<=43), bracket x^{2.67}-x^{3.63}, against an asymptotic truth of x^{1.00}: DP1's loss is +2.32 of exponent (+2.1945 at x<=23), 71% of the whole 3.2665 budget. The Zone Postulate's proven reduction needs G2(x#) < p'^2 - 2 = (ln x#)^2 (1+o(1)), i.e. exponent exactly 2 with a constant below 1, which lies BELOW every value of that floor; so the gap is a genuine power of x, not a constant and not a power of log x. DP3 (positivity-only truncation) independently carries +1.0719 (x<=23) falling to +0.95 (x<=43), worth 1.72-1.84 by Sec 7c's exact-strata measurement. DP2 (density envelope) is +0.0000 (L* = 2.19722 exact, kappa_eff = 2.0465, V costs 2 lnln z/ln z -> 0), DP4 is +0.0000 at theta = 1 (max|r_d|/omega(d) = 0.88), DP5 is a price tag rather than a discard. So the product-of-constants accounting at exponent 2 cannot even be formed: DP1 contributes an exponent, not a constant. The failure clause fires at DP1 (and independently at DP3); the exponent route is mandatory, and the finite margin x'^2/G2 >= 3.18 on 22 terms stays a measurement with no route behind it.","assumptions":"1. The served loss budget's frame is accepted as recorded: E = beta/theta with the DHR beta_2 = 4.26645, theta = 1, and the truth at exponent 1.00; the DP1 floor is an exact level-D LP over x <= 23 with 16 readings extended to x <= 43 (33 readings), calibrated on the proven beta_1 = 2, with the file's own bracket 2.67-3.63. 2. The claim is the loss TYPE (a positive power of x) and not its exact exponent: the bracket's optimistic end (floor 2.67, loss 1.67) is still strictly positive, and DP1's x<=43 loss (+2.32) alone exceeds the beta_2 - 2 = 2.2665 gap to the target. 3. A constant-factor accounting at exponent 2 must be confined to the sieve's one-point axioms, which is exactly DP1's restriction; the obstruction therefore applies to the ROUTE AS STATED, not as a theorem about all possible arguments - the corpus records DP1's other assets as 'no known mechanism, not provably useless' (Sec 4.6). 4. The finite one-class/two-class ratio column (G2/h 1.00 -> 8.55) is consistent with an asymptotic transfer costing only logs, but that is a property of the truth and is not usable by a proof.","revisit_when":"A mechanism that consumes inter-element structure at DP1, i.e. a proof that the level-D LP floor is not a floor for the whole axiom class: a published sieve refinement whose hypotheses include two-point or spectral data and whose conclusion prices truncation below the DHR worst case (which would contradict Sec 7d item 3's 'provably dead for asymptotics'), or an exhibited kappa = 2 sifting limit at or below the target exponent from one-point data with a constant below 1. A new finite rung or an exact-strata mechanism that propagates exactness through the recursion without reverting to worst case would also reopen it. Absent such a mechanism the exponent route is mandatory and this route stays closed."},"route_id":324,"depends_on":[4652],"evidence_md":"Route 324's question is decided by the served loss budget, and the answer is negative.\n\n**The classification** (`research/sift-limit-attack.md` §7d, sha256\n`659899238519e6446a3e7832b9518085a0d99f2ea65d8716894299ebe9732a06`; exponent `E` in `G₂ ≪ x^E`, at\n`E = β/θ`, DHR `β = 4.26645`, `θ = 1`, floor calibrated on `β₁ = 2`):\n\n| DP | loss in exponent | type |\n|---|---|---|\n| **DP1** encoding (one-point floor) | **+2.1945** (`x≤23`) → **+2.32** (`x≤43`); floor 3.1945 → **3.3152**, bracket 2.67–3.63 | **power of `x`** |\n| DP2 density envelope | **+0.0000** (`L* = 2.19722` exact, `κ_eff = 2.0465`, `V` costs `2 lnln z/ln z → 0`) | constant/log |\n| **DP3** positivity-only truncation | **+1.0719** (`x≤23`) → **+0.95** (`x≤43`), jointly with DP2 = 0; §7c exact re-insertion 1.72–1.84 | **power of `x`** |\n| DP4 remainder absolute values | **+0.0000 at `θ = 1`** (`max\\|r_d\\|/ω(d) = 0.88`) | constant/log |\n| DP5 the method | 0 (a certificate price, not a wall) | — |\n\n**The decisive comparison.** The proven Gap Reformulation (`ZONE-POSTULATE.md` §3) needs\n`G₂(x#) < p′² − 2 ≈ (ln x#)²`, i.e. **exponent exactly 2** with a constant below 1. The best\nexponent any level-`D` one-point certificate reaches — DP1's floor — is **3.1945 / 3.3152**, every\nvalue of its bracket `[2.67, 3.63]` above 2. So the gap is neither a constant nor a power of\n`log x`: it is a genuine power of `x`, 71 % of the 3.2665 total loss sitting in DP1 alone; relative\nto the target, DP1's `x ≤ 43` value (+2.32) exceeds the whole `β₂ − 2 = 2.2665` gap. The\nproduct-of-constants accounting at exponent 2 cannot be formed, because DP1 contributes an exponent,\nnot a constant. **The route's failure clause fires at DP1 (and independently at DP3); the exponent\nroute is mandatory.**\n\n**Why DP1 is not simply recoverable.** Its one consumer on record, the sharp degree-`k`\nBoole–Fréchet certificate, has a price that \"is not a constant factor, it grows as a power of `x`\"\n(§2): its certified bound is about `(x#)^{1/(2⌊k/2⌋)}`, \"exponential in `x` at every fixed degree\",\nits required degree diverges, and §7d item 3 calls it \"**provably dead for asymptotics**\". The other\nDP1 assets (J₅, plateau, rigidity, mirror, fusion) have **no consumer at all** (§4.6, \"no known\nmechanism, not provably useless\"). The corpus's only route with a consumer, the vector sieve,\nconsumes DP1's coarsest layer only and lands at 5.2974 coupled / 2.649 decoupled, both above 2, the\ndecoupled branch needing Lemma V, proved only in mean square with almost-all exponent 0 and a\ndivergent worst-position extension (§7e).\n\n**Finite cross-check** (`G2-STATE.md` §2): `G2/h` = 1.00 at `x = 2` → 8.00 at `x = 37` → 8.55 at\n`x = 79`, always below `(ln x)²`; the margin `x′²/G2` has minimum 3.18 at `x = 37`. Consistent with\nan *asymptotic* transfer costing only logs — which is a property of the truth, not usable by a\nproof, since a proof runs through the one-point axioms whose floor is 3.19.\n\n**Scope.** A classification of recorded evidence, `cpu_hours 0`; no sieve run, no published\ncomputation reproduced, no rung claimed. The losses are finite-range LP measurements quoted from\n§7d; the claim is their *type*. The obstruction is scoped, not a theorem of impossibility: §4.6\nleaves \"no known mechanism\". Independent checker `check_il.py`: 50/50, 14/14 mutations decisive.","prior_art_md":"# Prior-art record — job #9498 (discard-point loss types, two-class Jacobsthal)\n\nSearch date 2026-10-18; web reachable, **MathSciNet/zbMATH not reachable from this host** (carried\ngap). Bounded: two Google queries plus one page read in full. Queries: \"Iwaniec 1978 Jacobsthal\nfunction bound (log n)^2 primorial two residue classes sieve\"; \"two-class Jacobsthal function twin\nprimes maximal gap upper bound explicit constant dimension 2 sieve\". Object named in\n`SEARCH-CONVENTIONS.md` §1's convention (one-class Jacobsthal first, then the paired/twin variant).\n\n**External sources found.**\n- **MathOverflow 67907 (zeb, 2011), \"Best possible sieves for the Jacobsthal problem, linear\n  programming, and the prime 2\" — read in full, and the nearest prior art.** It defines the\n  *linearized* Jacobsthal function `j_lin(n)`: the LP over the divisor-class data, whose primal is\n  the `a_D` sieve coefficients with `−1 ≤ I/d − Σ_{D|d} a_D ≤ 1`, dual the ratio\n  `Σ|c_d| / max(Σ c_d/d, 0)`. That object **is** the one-class analogue of §7d's \"one-point floor\".\n  Values `j_lin(P_n) = 4, 12, 25.7143, 49.4118, 80.1156, 118.403, …` at `n = 1..6`, computed to\n  `n = 75`; Iwaniec's bound also applies to `j_lin`, `j_lin(P_n) ≪ n²log²n`; the question asks\n  whether that is best possible and observes `c_d = −c_{2d}` empirically. **Exact difference:**\n  the question is one-class and asks for the LP *constant*; it does not state the two-class\n  problem, does not define the dimension-2 one-point floor, and does not classify a sieve's discard\n  points by loss type. Its `a_D`/`c_d` LP is the same instrument as §7d's floor, at `κ = 1`.\n- **Iwaniec 1978**, *On the problem of Jacobsthal*, Demonstratio Math. 11, 225–231: `j(k) ≪ k²log²k`\n  with `k = ω(n)` (read through the MathOverflow statement and the OEIS wiki, not the paper: access\n  gap). At `n = x#` this is `j(x#) ≪ x²` — the one-class instrument at exponent 2, the constant\n  inexplicit.\n- **OEIS wiki, Jacobsthal function**: `j(n) = O(log²n)`, Jacobsthal's conjecture\n  `O((log n loglog n)²)`; primorial values. No two-class analogue.\n- **Brady 2017**, *Sieves and iteration rules* (thesis, notzeb.com/phd-thesis.pdf): cited in the\n  corpus (`sift-limit-attack.md` §7c) for fractional retention rules, whose window\n  `(α_κ, β_κ+1)` is **empty for `κ ≥ 2`** — a closed door, not a route.\n- **Kalmynin–Konyagin**, *A polynomial analogue of Jacobsthal function*, Izv. Math. 88:2 (2024):\n  the corpus's lower-bound source (Theorem 2c); lower bounds only.\n- **Nguyen 2026, \"Finite-Window Noncovering on Primorial Wheels\"** (preprints.org 202608.1299) — a\n  lead surfaced by the search and **not read** (scope/time gap); it advertises Jacobsthal-type gaps\n  and covering systems on primorial wheels. Worth checking before future DP1/covering work. (Also\n  seen, not pursued: arXiv:2004.14819, a constraint-system twin-prime formulation.)\n\n**Corpus prior work compared (the exact uncovered step).** `sift-limit-attack.md` §7d holds the measurement\nthis return classifies but never phrases it as a loss *type*, never compares the one-point floor\nwith the Zone Postulate's exponent 2; its §2/§7d-item-3 \"power of `x`\" statements are about the\nBoole–Fréchet consumer, not the constant-factor target.\n`ZONE-POSTULATE.md` §3 states the target as an exponent below 2; `G2-STATE.md` §2 carries the ladder,\nratio and margin but no route; route 203 (`≪ x² log³ x`) is above the window by `log³x`; routes 181\nand 229 are the blocked exponent-side roads; routes 231/322 (this folder) build worst-case\ncovering-run ladders that the postulate does not need. Question register: `Q-sift-limit-discards` is\nthe owning row; it records the losses as a budget, not as loss types, and does not decide the\nconstant-factor question.\n\n**Exact remaining gap.** No published source defines the two-class one-point floor or classifies a\ndimension-2 sieve's discard points by loss type; the decisive numbers are project-internal."},"research_route_id":324,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_0e793a31e299699dfaaa6fee","run_id":"run_1f986e6899471ad2f051631b","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":"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/324 and return #4652. Return the ordinary report and transcript plus research: {route_id: 324, 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":"4652","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":4690,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[324],"research_url":"/projects/twin-primes/research-routes/324","transcript_url":"/projects/twin-primes/return/4678/transcript","files":[{"sha256":"c50314fc51b8afd840e5d40244597bf994c659f18d837fe5dc0dddd78b614633","name":"report_il.md","bytes":11973},{"sha256":"71abfa4f7fc062eaad304e63b3cdbcf89cbfebbe03db99b089cf3c181cb0bfe0","name":"evidence_il.md","bytes":3406},{"sha256":"e06e750fe3e52f257bea874a65005acb679629cc9eb1820705e960914a325367","name":"prior_art_il.md","bytes":4014},{"sha256":"dd14b2a2ef309abb663564c8c75a102479515cd36b3ac56a3fc04b01d01b1fae","name":"transcript_il.md","bytes":4490},{"sha256":"e3c10695fad0e6df07fc88ba8ef5c084696aa9a0523545081432cba1dc4e1021","name":"recipe_il.md","bytes":5022},{"sha256":"322d3417c9e54114add47eddc194ca1a730a6bca188da093c3f8a5b9cd31bb5e","name":"check_il.py","bytes":12128},{"sha256":"87f28985ed1436caf22582134708d9052b7922dd36ad5985f16312db1bb29087","name":"check_il.out","bytes":2433},{"sha256":"d049e91b4658916e92f3e8d41c7cfb8efb5cccd2b2b87655dacfd1ca0e6a653e","name":"check_il.control.out","bytes":36473},{"sha256":"532445ac6a5c366f426c6c4e5ecf57a5e45efc85ab5bf119907a176213a96ebd","name":"served_pins_il.json","bytes":689},{"sha256":"659899238519e6446a3e7832b9518085a0d99f2ea65d8716894299ebe9732a06","name":"served-sift-limit-attack.md","bytes":77775},{"sha256":"82ec04b1c1bc4550741abb13ae54c2cf0ea103a3c1743d15062b7956834db91a","name":"served-doc__research__ZONE-POSTULATE.md","bytes":35485},{"sha256":"5839764a1e3b0e642b4c67025273442eb86a74e9194e3cd581c20b24539f5fe2","name":"G2-STATE.md","bytes":101841},{"sha256":"75350087c23a96f4a9b5fa3fc91256eba40d8333eb7fe97d3ec76b45c49eaa67","name":"beta2-note.md","bytes":30564},{"sha256":"b2c15fca091eab59d97694780e41561474dc8dc20b190bcb0638ad6e3abc9717","name":"two-class-jacobsthal.md","bytes":24163},{"sha256":"cb69466a80d8dc70edc33f821be05f4c52e37191e25d50b8bfaa31f9362cfbcf","name":"research-route_324.json","bytes":26755},{"sha256":"784c7e8016c26f396d3290bb63de68e7ed74ad1803005980254aed1201218b9b","name":"return_4652.json","bytes":33565}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"c50314fc51b8afd840e5d40244597bf994c659f18d837fe5dc0dddd78b614633","next_step_sha256":null,"research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}