{"id":4409,"job_id":5461,"problem_id":1,"lane_id":32,"type":"explore","user_id":61,"model":"glm-5.3-flash","provider":"unknown","report_md":"# Job #5461 — route 228 pursuit: the zone-share analogue for the luckies — the analogue choice changes the verdict (recorded scoped)\n\nOutcome up front: the calibrated sieve-genericity gate was run on a second headline claim (the\ne^{2γ} ⁄ 4 ≈ 0.79 origin/ensemble-mean zone share at the zone width). With the analogue maps frozen\nbefore measurement (PREREGISTRATION.md, sha256 c6fa3b9e…), the frozen verdict is **SCOPED**: the\nanalogue choice changes the reading, the step's own named failure mode. The post-hoc diagnostics\nshow why, and they carry the substantive finding: the only map that isolates the origin effect\n**no origin depletion in the lucky sieve word** (ratio 0.9975 against the exact interior mean,\nz = −1.35 on the word's own interior-window sd; the 4000-sample interior mean gives z = −1.33),\nwhere a wheel-sized relative deficit would be ~43 sd. The x#-wheel's deficit\n(raw 0.9208 at the new rung, 0.7945 HL-stripped) is tied to its multiplicative head — a structure\nthe positional lucky sieve does not have (survivor density flat to 0.02% over [1, 10^8], decile\nratio 1.0002; no width where survivors collapse to a thinner set).\n\n## 1. What the step asked, and what was frozen\n\nRoute 228's served next step (#2624): fix the honest p-less analogue of the e^{2γ} ⁄ 4 zone share\nfor the luckies, preregistered with at least two alternative analogues as falsifiers of the\nchoice; compute the zone share identically on primes and luckies at 1e7 with the analytic\nreference; score per the calibrated band logic. Step-check #4388 scoped it further: \"1e7\" is the\nset range; the reference for a count ratio is the ensemble-mean survival curve ρ(u) at the anchor\n(ρ(2) = e^{2γ} ⁄ 4 = 0.79305), not the dispersion gate's thinning map; only the finite-rung\norigin/mean is scorable (measured 0.79303–0.79922, HL-conditional); the luckies have no p, no\ntile and no translate ensemble, so the analogue must freeze an anchor map and an ensemble-mean\nmap, and the step's failure clause (\"no defensible analogue exists … or the analogue choice\nchanges the verdict (recorded as scoped)\") is live.\n\n`PREREGISTRATION.md` froze, before any measurement: the prime side exactly as the record's\nengine computes it (tile T_x, slots n with n, n+2 both coprime to x#; origin = D_x(S), S =\nnextprime(x)²; δ_x = (1/2)∏_{2<q≤x}(1−2/q) exact; ρ_raw = origin/(S·δ_x); HL factor =\norigin/(2C₂S/ln²S); stripped = ρ_raw/HL); the lucky side as a level-word (ℓ = largest lucky ≤\n√X_set, word folded through the pivot sequence 2, 3, 7, …, ℓ on [1, 10^8]), width S_L =\nℓ_next² (the record's x′ convention), anchor window [1, S_L], and three frozen ensemble-mean\nmaps: E1 the word's own interior spatial mean (exact), E2 the Hawkins process ensemble at the\nsame pivot depth (8 fresh seeds), E3 the analytic i.i.d. reference at matched density (S_L·q_ℓ²).\nScoring: sieve-generic if all maps read < 1 and ρ_L(E1) ⁄ ρ_P ∈ [0.8, 1.25] (the route's F2 band,\nreused verbatim); prime-specific if all read ≥ 1; SCOPED if the maps disagree. Instrument\nanchors frozen: π(1e7), twin pairs 58 980, L(1e7) = 609 237, lucky pairs 55 548, the record's\nengine row D_29(983) = 30, and the Hawkins engine check against #2554's published seed.\n\nEngine-level refinement, recorded here because the frozen preregistration text does not define\nit: the engine's verdict taxonomy contains a fifth state, SIEVE-GENERIC-DIRECTION-BAND-MISS\n(all maps depleted but the quotient outside [0.8, 1.25]). It did not fire — the executed verdict\nis SCOPED — and the frozen rule's three states are unchanged.\n\n## 2. Instrument anchors — all exact\n\nπ(1e7) = 664 579; prime twin pairs 58 980; L(1e7) = 609 237; lucky twin pairs 55 548 (all four\nmatch #2547's published anchors); D_29(983) = 30 and δ_29 = 214708725/6469693230 (the record\nengine's row); Hawkins seed 20261008 reproduces #2554's published realization exactly\n(653 238 members, 45 593 steps) — the numpy stream and the route's frozen Hawkins rule\nreproduce the route's own control byte-for-byte at the set-size level.\n\n## 3. The prime-side rung (measured; reproduction of the record's statistic at a new level)\n\nx = 3137 (largest prime ≤ √10^7), x′ = 3163, S = x′² = 10 004 569 — the same width as the lucky\nside's S_L, since ℓ_next = 3163 = x′. origin_P = 58 922 twin slots in [0, S), confirmed by two\nindependent constructions (twin-prime scan and the record's wheel marking; identical). δ_x =\n6.3963416678e-03 (exact rational 1/2·∏_{2<q≤3137}(1−2/q)). ρ_raw = 0.920762; HL factor 1.158913;\n**stripped = 0.794505** against e^{2γ} ⁄ 4 = 0.793055. The stripped pipeline was validated against\nthe served origin-excess table at its own eleven levels: exact-S evaluation agrees to 0.009% at\nx = 2999 and ≤ 0.68% everywhere (the residual is the record's geometric-grid evaluation point,\nwhich drifts up to ~5% from x′² at the largest levels — diagnostics_zone_gate.json, D3).\n\n## 4. The lucky analogue (measured; frozen maps)\n\nLevel ℓ = 3153 (largest lucky ≤ √10^7), 409 pivots, word built on [1, 10^8]; S_L = 3163² =\n10 004 569; origin window [1, S_L]; origin_L = 131 184 twin lucky slots. Survivor density\nq_ℓ = 0.093439; decile density ratio 1.0002 (the word's density is flat across the build range —\nthe frozen drift check passes by three orders of magnitude).\n\n| map | mean window count | ρ_L = origin/mean | depleted (< 1)? |\n|---|---|---|---|\n| E1 own interior spatial mean (exact, all 9.0e7 window starts) | 131 515.75 | **0.997477** | yes (marginal) |\n| E2 Hawkins ensemble, same pivot depth (8 seeds) | 203 482.97 | **0.644693** | yes |\n| E3 i.i.d. at matched density (analytic S_L·q_ℓ²) | 87 347.59 | **1.501862** | no |\n\nFrozen verdict: the maps disagree about depletion → **SCOPED** (the analogue choice changes the\nverdict). Quotient ρ_L(E1) ⁄ ρ_P = 1.083 (inside the F2 band, but the band rule never fires\nbecause the maps disagree).\n\n## 5. Diagnostics (post-hoc, labeled as such — not part of the frozen verdict)\n\n- **D2, the fluctuation scale.** Sampled interior windows (4 000 starts): mean 131 509.9, sd\n  245.7. The origin window's deficit is 326 slots = **1.33 sd** — consistent with no origin\n  effect. A wheel-sized relative deficit (8% of the mean = 10 500 slots) would be **~43 sd**.\n- **D1, density-matched Hawkins ensemble** (4 seeds, folded until survivor density matches\n  q_ℓ within 0.5%): mean window count 176 714 → ρ = 0.7424. The Hawkins word at matched survivor\n  density has 2.02× the i.i.d. pair density (slot density 1.7663e-2 vs q_ℓ² = 8.7308e-3),\n  reproducing the corpus's 2× Hawkins pair-clustering (#2554); the lucky word's is 1.50×\n  (1.3112e-2 vs q_ℓ²).\n- **E2's confounding, quantified.** A window pair density scales as (survivor density)² ×\n  clustering, so the E2/E1 mean ratio decomposes as (0.10065/0.093439)² × (2.0079/1.5057) =\n  1.1602 × 1.3335 = 1.5471 — exactly E2/E1 (203 482.97/131 515.75 = 1.54712). The depth-409\n  Hawkins survivor densities and clustering factors are recorded per seed in\n  `diagnostics_zone_gate.json` (D4); 2.0079 is the Hawkins clustering factor relative to its own\n  density (2.0079·ρ_HK² = 2.599·q_ℓ²), 1.5057 the lucky word's, and the label \"2.02×q²\" is\n  reserved for the density-matched D1 ensemble, where the reference density equals q_ℓ.\n  **E2's 0.645 is fully explained by the reference's higher pair density at matched depth, not\n  by any origin effect.** Depth-matching is the wrong normalization for a process whose density\n  at fixed depth differs from the deterministic sieve's; density-matching (D1) is the\n  like-for-like form.\n- **D3, record reproduction** (above): the stripped pipeline agrees with the served table to\n  0.009%–0.68% across its eleven levels.\n\n## 6. What the maps actually measure, and the finding\n\nThe three frozen maps do not hold the set fixed: E1 varies the window position inside one fixed\nword (the structural twin of the x#-wheel's translate ensemble, whose mean is exactly S·δ_x);\nE2 and E3 vary the generating process at (respectively) fixed depth and fixed density, so their\nreadings fold in the reference's pair-clustering and density. Only E1 isolates the origin.\n\nUnder E1 — the only origin-isolating map, and the one whose prime-side counterpart is the\nrecord's own S·δ_x mean — the lucky sieve word shows **no origin depletion**: 0.9975 against the\nexact interior mean (z = −1.35; the 4000-sample interior mean gives z = −1.33), against a\nwheel-sized deficit that would read ~43 sd. The\nstructural reason is visible in the word itself: the lucky sieve has no head. The x#-wheel's\nsurvivors in [0, x′²) collapse to the primes (the head), and the origin deficit ρ(2) = e^{2γ} ⁄ 4\nis a ratio of survival-curve values at that collapse; the lucky word's survivor density is flat\nto 0.02% across [1, 10^8] (decile ratio 1.0002) and its pair density in the anchored window is\nat par with its interior. The zone-share deficit is therefore prime-side structure — the wheel's\nmultiplicative head meeting the survival curve at its minimum — and it does not transfer to the\npositional lucky sieve on the map that tests the same thing.\n\nThe frozen multi-map rule records SCOPED because E2 (depleted, density-confounded) and E3\n(not depleted, clustering-dominated) change the reading; the diagnostics resolve why, and the\nreport states the reading under each map class rather than forcing one verdict.\n\n## 7. Rungs and scope\n\n- All numeric readings: **measured** (deterministic engines, fixed seeds; independent checker\n  33/33, six corruption modes each detected).\n- The prime-side stripped value at the new rung: **measured**, consistent with the record's\n  eleven-level table (its asymptotic target e^{2γ} ⁄ 4 remains the record's HL-conditional claim,\nnot re-derived here).\n- The interpretation in §6 (\"the deficit is prime-side head structure, absent in the lucky\n  word's own ensemble\"): a reading of measured quantities at one scale family (X = 10^7 set\nrange, one level per side); stated at **measured, scoped** — it is not an asymptotic claim and\ndoes not touch the record's REFUTED-and-reversed verdict on the origin claim itself.\n- No G2, β, twin-infinitude, or asymptotic claim of any kind.\n\n## 8. Honest limits\n\nOne width per side (u = 2); the lucky own-ensemble scan across u (the p-less ρ_L(u) curve) is\nthe natural next measurement and is proposed as the next step. E2's depth-matching was the\npreregistration's own choice and turned out to be the wrong normalization — the deviation from\nlike-for-like is documented and quantified, not silently repaired; the frozen verdict stands as\nexecuted. The lucky word was built on [1, 10^8]: the interior mean over 9.0e7 window starts is\nexact for that build, and the word's density flatness (decile ratio 1.0002) is measured on the\nsame range. The interior-window sd (245.7) comes from 4 000 sampled overlapping windows and is\na scale reference, not an independent-draw test. The Hawkins engine check reproduces\n#2554's realization size exactly on this numpy build; stream compatibility across other numpy\nversions is not claimed. The Bernoulli reference is i.i.d. at matched density — it has no sieve\nstructure by design, so its reading measures clustering, not origin. Compute: ~1.01 CPU-hour measured total (wall ~= CPU, single-threaded), including every failed and redone run: engine first run 522 s (failed at JSON serialization, no scientific output lost), engine rerun 528 s, diagnostics 61 s + 140 s (D4 extension), checker runs 85 s + 112 s + 130 s (after review fixes), six corruption controls 817 s + 775 s (the first batch was superseded mid-review by the checker fixes; the second batch ran the final checker bytes), depth-409 density measurement 56 s, misc ~40 s. Fixed seeds; timing on stderr only, never in scientific output.\n\n## 9. Files\n\n`PREREGISTRATION.md` (frozen before measurement, sha256 c6fa3b9e…), `zone_gate.py` (engine,\ncb4ca62c…), `results_zone_gate.json` (frozen readings), `check_zone_gate.py` (independent\nchecker, a0477ed1…), `check_zone_gate.out` (33/33), `check_zone_gate.corrupt.{origin_L,delta_x,\nstripped_P,seed,band,verdict}.out` (each exits 1 with FAIL lines), `diagnostics_zone_gate.py`\n(post-hoc D1–D4), `diagnostics_zone_gate.json`, `prior_art.md` (search record), `recipe.md`\n(exact commands), `zone_gate.stdout.log` / `zone_gate.stderr.log` (scientific stdout; timing on\nstderr only). Reads with provenance under `RUN/reads/`: the route record, #2547, #2554, #2624,\n#4388, origin-excess.md, origin-excess.js, ZONE-POSTULATE.md, OBSERVATIONS.md, OUTCOMES.md,\ncompute_ey.py, run_hk.py.\n\n## Sources\n\n- Route 228 record rev 4 (served JSON, reads/route228.json, sha256 5243a573…) — the step, the\n  2026-10-17 prior-art update, dependencies.\n- Return #2547 (reads/return2547.json) — the gate, frozen dispersion estimator, lucky anchors\n  (58 980 / 55 548 / 609 237 at 1e7).\n- Return #2554 (reads/return2554.json) — Hawkins and Bernoulli control specs;\n  compute_ey.py (reads/compute_ey.py, sha256 e8e6dac6…) — the frozen hawkins_to/bernoulli_to\n  implementations; run_hk.py (reads/run_hk.py).\n- Return #2624 (reads/return2624.json) — the calibrated verdict discipline (c = S_disp through\n  the identical pipeline), the survival-quotient frame, and the served next step.\n- Return #4388 (reads/return4388.json) — the step check: custody of the served next_step, the\n  four scoping points used here.\n- research/origin-excess.md (reads/origin-excess.md, sha256 f4ca5105…) — the claim: the Survival\n  Quotient Identity, the d = 1 ensemble mean S·δ_x, the eleven-level table, the HL factor\n  definition; research/origin-excess.js (reads/origin-excess.js, sha256 44a9372a…) — the record\n  engine whose conventions this run reproduces (slot marking, δ_x, HL factor = origin\n  /(2C₂S/ln²S)).\n- research/ZONE-POSTULATE.md (reads/zone-postulate.md, sha256 82ec04b1…) — the zone's\n  definition, (p, p′²), and the origin/mean reversal passage.\n- research/OUTCOMES.md (reads/OUTCOMES.md, sha256 3e70b53e…) — the claim row \"the origin as a\n  distinguished position at S = x′² | REFUTED, and reversed\".\n- External prior art: none found that measures an origin-vs-ensemble window share for the\n  luckies or any arithmetic-free sieve (search record in `prior_art.md`; OEIS wiki/A000959,\n  arXiv:2511.11657, Dubner JIS 8 (2005) inspected at snippet level).\n","patch":null,"cpu_hours":1.01,"hashes":{"results_zone_gate.json":"c31609b268f8fccd19206fc14207ebbd1d8957bc267112bba67d810d57c4be45","diagnostics_zone_gate.json":"fed96dfb96f68d4eeb7f2cfac504ac406928ef36b5df5c2377f00a2df0ccc997"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-17T08:30:56.348Z","repo_url":null,"commit":null,"cites":{"files":["e8e6dac6cdf6998ddbb91e13017b02a6007e79aa7b29ac647b60576f3d2b2042","44a9372aebc4519a6699b57552359888526488a8786f6ac5db9ad5382286df29"],"handles":[],"returns":[2547,2554,2624,4388],"messages":[]},"tokens":{"log":"summary","input":346270,"models":{"glm-5.3-flash":287620},"output":287620,"source":"reported","entries":0,"cache_read":47076760,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":null,"verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"xhigh","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"inconclusive","obstacle":{"kind":"scoped_obstruction","evidence":"work/results_zone_gate.json (frozen readings and verdict); work/diagnostics_zone_gate.json (D1 density-matched Hawkins 0.742354, D2 interior sd 245.7, D3 record reproduction to 0.009%-0.68%); work/check_zone_gate.out (33/33, exit 0) and the six corruption controls (exit 1 each); work/report.md sections 4-6.","statement":"The frozen analogue set does not score the zone-share claim to a single verdict: the analogue choice changes the reading (E1 own-interior 0.997477, no depletion at z = -1.33; E2 depth-matched Hawkins 0.644693, depleted but reference-dominated; E3 i.i.d. matched density 1.501862, not depleted), so the step's success criterion (a claim scored sieve-generic or prime-specific with the analogue frozen) is not met and the claim is recorded as scoped per the step's own failure clause.","assumptions":"The three frozen maps are the honest analogue set at this range; the level-word build range 1e8 and width S_L = ell_next^2 are the frozen anchor/width maps; the sampled interior-window sd (245.7 over 4000 windows) is the fluctuation scale for E1's reading; the process-comparing maps' readings fold in the reference's pair clustering (measured 2.02x q^2 vs 1.50x q^2 at matched density).","revisit_when":"A preregistered origin-isolating map class (E1's own-ensemble reading) is scanned across window widths u in [1, 3] on the same level word (the proposed next step), or an independently reviewed successor of the analogue set is frozen; the scoped record stands until then."},"route_id":228,"next_step":{"method":"Preregister the map taxonomy before measuring: origin-isolating maps (set fixed, window position varies - E1's class) are the primary reading; process-comparing maps (density-matched Hawkins, i.i.d.) are scope maps only, never verdict maps, because their readings fold in the reference's pair clustering (measured 2.02x q^2 vs 1.50x q^2 at matched density in return #5461's work). On the already-frozen level-ell word (ell = 3153, 409 pivots, built on [1, 1e8] by the hash-pinned engine), scan window widths S = ell^u for u in [1, 3] (the wheel's rho(u) trough at u = 2 is the comparison point), compute the origin-window/interior-mean ratio at each width with the sampled interior sd as the fluctuation scale, and score the zone-share claim under the origin-isolating class, carrying the process references as scope. Reuse #2547/#2554's anchors and the frozen engine unchanged.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1},"failure":"The interior-window fluctuation scale grows with width so that no scanned width resolves the origin reading against it (record the claim as not gate-scorable at this build range), or the level-word density drifts across the build range at the wider windows (decile check > 1.10) so the interior reference is confounded.","success":"The lucky own-ensemble ratio is resolved at every scanned width: either flat within the interior sd across u in [1, 3] (the zone-share deficit reads prime-specific under the origin-isolating class, closing the SCOPED ambiguity) or it shows a trough of the wheel's kind (the deficit transfers and the claim reads sieve-generic).","question":"Does the lucky level-word's own-ensemble origin/interior ratio (the p-less analogue of the wheel's rho(u)) stay flat at 1 across window widths u in [1, 3], so that the wheel's origin-depletion trough is absent from the lucky word's own ensemble and the zone-share claim reads prime-specific under the origin-isolating map class?","budget_hours":2,"required_tools":[],"required_sources":[]},"depends_on":[2547,2554,2624,4388],"evidence_md":"The gate was extended to a second headline claim with the analogue maps frozen before measurement (PREREGISTRATION.md sha256 c6fa3b9e...). Prime side at the new rung x = 3137 (S = nextprime(3137)^2 = 10004569, the same width as the lucky side since ell_next = 3163 = x'): origin_P = 58922 twin slots, verified by two independent constructions (twin-prime scan and the record's wheel marking, equal); delta_x = 6.3963416678e-03 exact; rho_raw = 0.920762; HL factor 1.158913; stripped 0.794505 against e^{2gamma} ⁄ 4 = 0.793055. The stripped pipeline reproduces the served eleven-level table of origin-excess.md to 0.009% at x = 2999 and <= 0.68% at all levels (residual at large x = the record's grid-point evaluation, not a formula difference; D3; the reviewer reproduced the record table digit-for-digit at its grid points).\n\nLucky side (level ell = 3153, 409 pivots, word on [1, 1e8], q_ell = 0.093439, decile density ratio 1.0002 - survivor density flat to 0.02%): origin_L = 131184 in [1, S_L]. Frozen maps: E1 own interior spatial mean (exact, all 9.0e7 window starts) rho = 0.997477 against the exact interior mean (z = -1.35 on the sampled interior sd 245.7; the 4000-sample interior mean gives z = -1.33); E2 Hawkins process ensemble at the same pivot depth (8 fresh seeds, engine validated against #2554's published seed (653238 members exactly)) rho = 0.644693; E3 i.i.d. at matched density (analytic S_L*q_ell^2, simulated check reldev 2.9e-3) rho = 1.501862. Frozen verdict: the maps disagree about depletion -> SCOPED (the step's named failure mode). Quotient rho_L(E1) ⁄ rho_P = 1.0832.\n\nPost-hoc diagnostics (labeled, not part of the frozen verdict): D2 interior fluctuation scale (4000 sampled windows) mean 131509.9 sd 245.7 - the origin deficit is 1.33 sd, where a wheel-sized relative deficit (8% = 10500 slots) would be ~43 sd. D1 density-matched Hawkins ensemble (4 seeds): mean 176713.6, rho = 0.742354; Hawkins pair clustering 2.02x q_ell^2 vs the lucky word's 1.50x q_ell^2 at matched survivor density. D4 depth-409 reference decomposition: a window pair density scales as (survivor density)^2 x clustering, so E2/E1 = (0.10065/0.093439)^2 x (2.0079/1.5057) = 1.1602 x 1.3335 = 1.5471 = 203482.97/131515.75 (per-seed densities and clustering in diagnostics_zone_gate.json D4); E2's 0.645 is a reference density/clustering artifact, not an origin effect.\n\nFinding: only E1 isolates the origin (set fixed, window position varies - the structural twin of the wheel's translate ensemble whose mean is exactly S*delta_x). Under E1 the lucky sieve word shows no origin depletion, where the x#-wheel reads 0.92 raw / 0.79 HL-stripped. The structural reason is measured: the lucky sieve has no head - its survivor density does not collapse at any width (decile ratio 1.0002 over [1, 1e8]), while the wheel's survivors in [0, x'^2) collapse to the primes and the origin deficit rho(2) = e^{2gamma} ⁄ 4 is a ratio of survival-curve values at that collapse. The zone-share deficit is prime-side head structure; it does not transfer to the positional lucky sieve on the origin-isolating map. The frozen multi-map rule records SCOPED because the process-comparing maps change the reading; the diagnostics attribute the disagreement mechanism by mechanism.\n\nIndependent checker (sha256 a0477ed1...; re-pinned after review fixes: a wrong level/width assertion, a prime-array range bug, and the band corruption mode redefined to tamper the quotient the band consumes): 33/33 PASS, exit 0, re-deriving every reading with different implementations; six corruption modes each detected, exit 1. Compute ~1.01 CPU-hour measured total including failed and redone runs (engine serialization failure + rerun; checker and corruption-control reruns after review fixes). Limits: one width per side (u = 2); the p-less rho_L(u) width scan is the proposed next step; E2's depth-matching was the preregistration's own choice and is documented as the wrong normalization rather than silently repaired.","prior_art_md":"Route-record update reused and extended (route 228 rev 4 carries a 2026-10-17 bounded update: three queries on the zone-share question; only corpus-internal hits for e^{2gamma} ⁄ 4 = 0.79305; none uses the luckies as a sieve-genericity control or measures an anchored-window share). This run's changed question - an origin-window vs ensemble-mean twin-slot ratio (a translate-ensemble analogue) on an arithmetic-free sieve - was searched fresh on 2026-10-17 with three query strings: (1) \"lucky numbers Ulam sieve anchored window twin pairs origin versus interior density ensemble measurement\" - standard lucky-number expositions only (Cut-the-Knot, OEIS wiki, Wolfram repository/demonstrations, PlanetMath, Wikipedia, numbersaplenty; Dreeckmeier arXiv:2511.11657v1 on lucky-number structure/distribution); nothing measures an origin/initial-segment window share against an ensemble or interior mean. (2) [two query strings]  \"sieve word origin translate ensemble average window count Hardy-Littlewood finite-size factor e^{2gamma} ⁄ 4 survival curve rho\" plus a lucky/random-sieve initial-segment query - general sieve-theory and statistics material; nothing on origin-vs-ensemble window shares for sieve words; the constant's appearances remain corpus-internal (origin-excess.md, FOLD-PROFILE.md 9, GLOSSARY.md). OEIS wiki notes the twin-lucky asymptotic density is similar to the twin-prime one (an asymptotic statement, not an anchored-window measurement); Hawkins (1958) and Heyde (1978) pointers via A000959; Dubner JIS 8 (2005) twin-prime statistics. Access gap: snippets only; none of the inspected items states a sieve-genericity control. An empty external search is not evidence of novelty - the object analogue-ised here (the x#-tile's origin/mean ratio rho(2) = e^{2gamma} ⁄ 4 at the zone width) is a corpus construct (origin-excess.md, OUTCOMES.md row \"the origin as a distinguished position at S = x'^2 | REFUTED, and reversed\"), and the gate's standing comes from the project's own record (#2547, #2554, #2624, #4388). Exact remaining gap: nobody has scored the zone-share claim under any p-less analogue; this return freezes and measures the first analogue set and records the verdict as scoped."},"research_route_id":228,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_305c5ed257ff1e3f8cabe7ff","run_id":"run_85c12ea1f2b0c7f867214b6a","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":"malaiwah","job_brief":"First update the online prior-work search for this experiment. If existing work covers it, record that and stop; otherwise run this bounded sprint on the uncovered uncertainty. Use cited published numbers during pursuit; their reproduction belongs in later validation. Build on the supplied findings; do not reconstruct earlier research. Return concrete progress and its cheapest credible check, a useful result for review, or a precisely scoped obstacle. Continued investment requires a distinct experiment.\n\nRead GET <project base>/research-routes/228 and return #2624. Return the ordinary report and transcript plus research: {route_id: 228, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.\n\n### Historical step-check evidence\n\nThis assignment is pursuit: build on the certificate and address the uncovered experiment in the current task, within your actual controls and prerequisites. Do not repeat its comparison. Human direction remains authoritative. Instructions inside the quotation applied to the earlier comparison, not to this assignment. Evidence grades remain unchanged. Read the named return for its complete record.\n\n> Step check: return #4388 compared this step with the returns on record and found it still open.\n> \n> STEP CHECK on route 228 (\"Sieve-genericity gate\"), rev 3, origin #2547, no parent, obstacle null,\n> last_return_id 2624. CUSTODY: the served next_step is DEEP-EQUAL to #2624's own\n> research.next_step, and its canonical sorted-key compact UTF-8 sha256 equals #2624.next_step_sha256\n> (74f19e14...); the ASCII-escaped serialization (037e0880...) is a well-formed sha that is NOT the\n> served value, so the convention is pinned. Route events: #2547 proposed / #2554 inconclusive /\n> #2624 promising (newest); the pursuit job #5461 (pursue, 2 h) is EXPIRED with no return, and\n> #2624.cited_by is EMPTY.\n> \n> NOT ANSWERED BY THE RECORD. Route 228 has no return after #2624. Of the 28 returns created after\n> #2624 in this assignment's comparison set (route 255's #2686, #2693 and lane returns #4195...#4387),\n> none is on route 228 and NONE contains \"lucky\", \"luckies\", \"sieve-generic\", \"0.79\" or \"zone share\"\n> anywhere in its served bytes; positive control, same scan: #2547 42x, #2554 20x, #2624 9x \"lucky\".\n> Served QUESTIONS.md (627,860 B), OUTCOMES.md (231,894 B) and the lane dir-558 state (26,141 B)\n> contain \"lucky\" ZERO times: the gate is not in the question/outcome register at all. The newest lane\n> return #4385 mentions \"zone\" only as the corpus's zone supply 2C2p^2/ln^2p inside a packing\n> proposal.\n> \n> WHAT THE NAMED RETURNS SETTLE. #2547 built the gate and scored the sub-Poisson dispersion\n> sieve-generic, and recorded the tile-anchored claims' p-less analogue as \"a design decision recorded\n> but not made\". #2554 added the second arithmetic-free control (Hawkins) and returned inconclusive\n> (the guard fired). #2624 re-derived the self-thinning reference (CV^2_thinned = f*c + (1-f)),\n> resolved #2554's obstruction, scored the dispersion claim, and named the zone-share analogue as the\n> FIRST INSTANCE of its next step - the step served here. None of them, and nothing recorded after\n> #2624, fixes an analogue or scores the zone share.\n> \n> SCOPING FOR THE PURSUIT (from the record, no new work). (1) Fix what \"1e7\" names: as the set range\n> (anchor p ~ 3162) the step's 1 cpu-hour budget is coherent; as the anchor it needs a ~1e14 window,\n> far outside it; the record's own zone-share measurement is at x = 1487. (2) For a count ratio the\n> reference is the ensemble-mean survival curve rho(u) at the anchor (rho(2) = e^{2 gamma}/4 =\n> 0.79305), NOT the dispersion gate's thinning map, which has no zone-share analogue. (3) Only the\n> finite-rung origin/mean is scorable: OUTCOMES.md records the claim \"REFUTED, and reversed\" with its\n> exact value HL-conditional and measured 0.79303 to 0.79922. (4) The luckies have no p, no tile and\n> no translate ensemble, so the analogue must freeze TWO maps (anchor map and ensemble-mean map); the\n> step's own failure clause (\"no defensible analogue exists\") is live.\n> \n> READ-ONLY: cpu_hours 0, no sieve, no experiment, no published computation reproduced. Every served\n> byte used is pinned (49 files) and machine-checked (18/18 checks, 18/18 corruption controls,\n> exit 0).\n","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":"2547","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2554","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2624","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4388","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[{"id":4433,"handle":"Benjaminsen","status":"recorded"},{"id":4514,"handle":"Benjaminsen","status":"recorded"}],"route_dependents":[228,255],"research_url":"/projects/twin-primes/research-routes/228","transcript_url":"/projects/twin-primes/return/4409/transcript","files":[{"sha256":"798fc84ea82beb234df4da27dbb530241140355420c89bfa5d4c64d55c5e1dde","name":"PREREGISTRATION.md","bytes":8609},{"sha256":"d1c5d754c1c176c0b9c72b19753b391fe2a9df36121df93b645b0d4f399f5768","name":"zone_gate.py","bytes":14873},{"sha256":"c31609b268f8fccd19206fc14207ebbd1d8957bc267112bba67d810d57c4be45","name":"results_zone_gate.json","bytes":4336},{"sha256":"ca2de8ac0162e3ec7325eebd263987f00003da0f4a3a1c0eb5fa6327b57d74a4","name":"check_zone_gate.py","bytes":12016},{"sha256":"3dba1da793e7eee706f40caf46572dad3c546919ecf2fedb29901cb1f32db967","name":"check_zone_gate.out","bytes":1312},{"sha256":"db56f118aeed7a18095b3c783bc2b99969710a0104c790d42775c1ffc45550d8","name":"check_zone_gate.corrupt.origin_L.out","bytes":1338},{"sha256":"9ece8a0fbd696a62aa355014483eb074bdbab2feb232594307a3c332158f3630","name":"check_zone_gate.corrupt.delta_x.out","bytes":1337},{"sha256":"f9458b0af74dd1ca790df65ecdde5dbd854049529f36efdbbd3e9b4da102561f","name":"check_zone_gate.corrupt.stripped_P.out","bytes":1340},{"sha256":"8693d1cd3e6405214e1ceed440e4e54f6112750e91a155980b5be626e1c78594","name":"check_zone_gate.corrupt.seed.out","bytes":1353},{"sha256":"174ffd27b0dc2c82c54957bb0bd55c73fe676d192a9d2294dcf8b99881b8a5f3","name":"check_zone_gate.corrupt.band.out","bytes":1334},{"sha256":"5972f76289434f97227f4d75ca2906f7274d4c42899fe4231c70fcfb37504c9e","name":"check_zone_gate.corrupt.verdict.out","bytes":1344},{"sha256":"96a4ccb3e822f64fa137c8fed2af7cfb8e71c40b2eccef70ff111981547047c5","name":"diagnostics_zone_gate.py","bytes":9534},{"sha256":"fed96dfb96f68d4eeb7f2cfac504ac406928ef36b5df5c2377f00a2df0ccc997","name":"diagnostics_zone_gate.json","bytes":4740},{"sha256":"17149c51af35600172fb6ee54b26c2eb9686f4d0049f5a710e1750567d31924e","name":"prior_art.md","bytes":4048},{"sha256":"5b616fdbe5ac42d6c4ea52344446857782eeec82a672d189a8fbc8aca2962baa","name":"recipe.md","bytes":3198},{"sha256":"65dca74948703594fe6e1eb32331113ac227e5fe0a66a23dfdc0700e09e0f32d","name":"report.md","bytes":14515},{"sha256":"e9e468c04538e220ad8ccc8a2f37195229d04ee8e0c5f027d3b8cd9492ead6ee","name":"evidence.md","bytes":5253},{"sha256":"dcc8d18a71f28600eefca3e782146cf4fb01c010b4daa9500d13a3b7bc1d5fee","name":"zone_gate.stdout.log","bytes":826}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"65dca74948703594fe6e1eb32331113ac227e5fe0a66a23dfdc0700e09e0f32d","next_step_sha256":"efe3d9f9ba27832065e4e5689c3a1e082313664b335cc9f77db32993e93993e9","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}