{"id":5864,"job_id":9814,"problem_id":1,"lane_id":2,"type":"explore","user_id":93,"model":"gemini-4-argon","provider":"google","report_md":"# Job #9814: Hybrid/M3 Model Tail Replacement for $B_{\\text{geo}}$ Across the 40-Cell Step-2 Grid (Route #82 Pursuit)\n\n## 1. Executive Summary\n\nThis work resolves the open obligation from Return `#4897` (`pursue` on Route `#82`, adversarial lane) and addresses Review `#1747`:\n\n> **Obligation (`#4897`)**: *\"Can the $B_{\\text{geo}}$ tail proxy be replaced by the actual hybrid/M3 model tail at the four tiles, settling the `p29@109`/`p29@113` dominance bookkeeping and shrinking the tail band?\"*\n\nWe re-implemented the hybrid small-prime / M3 second-cumulant model (`h4321.c` in Return `#1934`) in [`h4321_tail_9814.c`](files_to_upload/h4321_tail_9814.c) (`NMAX = 200`, half-gaps $A = b/6 \\in [1, 200]$, gaps $b \\in [6, 1200]$), validated it against all 54 pinned fold entries ($x \\in [23, 101], y \\in \\{11, 13, 17\\}$, 639 fold values) and all 8 extension entries ($x \\in \\{103, 107, 109, 113\\}, y \\in \\{13, 17\\}$, 104 fold values) in `hyb4321.json` to **`max_abs_diff = 0.000000`** ($\\le 0.001$ gate passed with $>1000\\times$ margin), and extracted the per-size histogram $h_y(A)$ beyond $b = 360$ ($A \\in [61, 200]$) at $x \\in \\{103, 107, 109, 113\\}$ for $y \\in \\{0, 11, 13, 17, 19\\}$.\n\nReplacing the geometric proxy $B_{\\text{geo}}(x, p)$ with the model kill-class tail resolves both open questions:\n1. **Why $B_{\\text{geo}}$ underestimated `p29@109` and `p29@113`**: For $p = 29$, the kill residue set $C_{29} = \\{8, 17, 27\\}$ mod $29$ starts at $j = 8$ ($b = 408$). Featureless geometric decay anchored at $b = 360$ gave $B_{\\text{geo}} / T = 0.04905$ ($x = 109$) and $0.05043$ ($x = 113$), missing the endpoint sieve-survival boost on kill-class gaps ($F_0$ multiplied by $(p-2)/(p-4) = 1.08$ at $b \\equiv 0 \\pmod p$ and $(p-3)/(p-4) = 1.04$ at $b \\equiv \\pm 2 \\pmod p$) and primorial resonance. In the actual model ($y = 17$), the conditional kill-class tail fraction is **`0.081479`** ($x = 109$, $1.6610 \\cdot B_{\\text{geo}}$) and **`0.082710`** ($x = 113$, $1.6402 \\cdot B_{\\text{geo}}$).\n2. **Wide-cell dominance (`p >= 29`, 7 cells) settled**:\n   - Against $M_{17}$, the pure null ingredient $N(B_{\\text{hyb}, 17}) \\ge M_{17}$ holds at **7/7 wide cells** under both `B_cond` and `B_norm` (up from 6/7 under $B_{\\text{geo}}$, flipping `p29@109` from $0.006127 < 0.006622$ to $0.007900 > 0.006622$ under `B_cond17` and $0.008487 > 0.006622$ under `B_norm17`).\n   - Against $\\max(M_{17}, M_{13})$, `B_norm17` achieves **7/7 wide cells** `N`-dominated (`p29@109`: $0.008487 > 0.008122$; `p29@113`: $0.010195 > 0.008675$), while `B_cond17` achieves **6/7 wide cells** (`p29@113` flips to `N` at $0.009493 > 0.008675$, while `p29@109` reaches $0.007900$, just $0.000222$ below $M_{13}/E = 0.008122$).\n3. **Tail band shrunk by $4.7\\times \\text{ to } 690\\times$**: Across the 7 wide cells, the $6.0000\\times$ proxy band $[B_{\\text{geo}}/2, 3 B_{\\text{geo}}]$ shrinks to **`1.0087x–1.0140x`** across $y \\in \\{11, 13, 17\\}$ within `B_cond`, **`1.0844x–1.1121x`** within `B_norm`, and **`1.2332x–1.2832x`** across the combined `[B_cond17, B_norm11]` envelope ($\\max \\text{hw}_{\\text{sub}} = 0.00144$ within `B_cond` and $0.01528 \\le 0.05$ across the combined envelope).\n4. **Review `#1747` narrow-cell shortfall explained**: Decomposing $M_y / E_{\\text{lo}} = \\Delta_{\\text{cov}, y} + \\Delta_{\\text{unc}, y}$ proves that narrow-cell cases where $N < M_{17}$ (11/40 under `B_cond17`, 9/40 under `B_norm17`, down from 18/40 under $B_{\\text{geo}}$) are caused by the model's covered-region positive bias $\\Delta_{\\text{cov}, 17} = \\Lambda_{\\text{cov}, 17}^{\\text{model}} - \\Lambda_{\\text{cov}}^{\\text{exact}}$ on $a+b \\le 360$ (e.g., $+0.004520$ at `p13@103`). Against the model's uncovered tail shift $\\Delta_{\\text{unc}, 17} = \\Lambda_{1, 17}^{\\text{model}} - \\Lambda_{\\text{cov}, 17}^{\\text{model}}$, $N(B_{\\text{hyb}, 17})/E_{\\text{lo}} \\ge \\Delta_{\\text{unc}, 17}$ holds on **40/40 cells** for both `B_cond17` and `B_norm17`.\n\n---\n\n## 2. 54+8 Fold Gate & Covered-Window ($b \\in [246, 360]$) Verification\n\n[`h4321_tail_9814.c`](files_to_upload/h4321_tail_9814.c) evaluates $h_y(A) = w_0(A) F_0(A) F_1(A)^{A-1} \\exp(\\ell_2(A))$ for $A \\in [1, 200]$ and matches all 54 pinned entries (639 fold values) and 8 extension entries (104 fold values) in `hyb4321.json` with **`max_abs_diff = 0.000000`**.\n\nFor each $(x, p, y)$ with kill set $C_p = \\{A \\bmod p : 6A \\equiv 0, \\pm 2 \\pmod p\\}$, let $\\text{tail\\_h}_y = \\sum_{A=61}^{200} h_y(A)$, $\\text{kill\\_tail\\_h}_y = \\sum_{A=61, A \\bmod p \\in C_p}^{200} h_y(A)$, and $p_{\\text{kill}, y} = \\text{kill\\_tail\\_h}_y / \\text{tail\\_h}_y$:\n- **`B_cond`**: $B_{\\text{cond}, y}(x, p) = T(x, p) \\cdot p_{\\text{kill}, y}(x, p)$, where $T = E_{\\text{lo}} - \\sum_{b \\le 360} n_b(1)$.\n- **`B_norm`**: $B_{\\text{norm}, y}(x, p) = E_{\\text{lo}}(x, p) \\cdot \\text{kill\\_tail\\_h}_y(x, p) / \\text{Emcov}_y(x)$.\n\nOn the covered late window $b \\in [246, 360]$ ($A \\in [41, 60]$) where exact census gap counts $n_b(1)$ are known, `B_cond17` matches the exact kill-class sum within **`1.095%` max (`0.465%` mean)** across all 40 cells (`+0.375%` at `p29@109`, `+0.313%` at `p29@113`), whereas $B_{\\text{geo, win}}$ errs by up to **`206.924%`** (`-65.075%` at `p29@109`, `-64.458%` at `p29@113`).\n\n---\n\n## 3. Wide-Cell ($p \\ge 29$) Dominance and 40-Cell Summary\n\n| Cell | $B_{\\text{cond}, 17} / B_{\\text{geo}}$ | $B_{\\text{norm}, 17} / B_{\\text{geo}}$ | $N(B_{\\text{geo}})/E$ | $N(B_{\\text{cond}, 17})/E$ | $N(B_{\\text{norm}, 17})/E$ | $M_{17}/E$ | $M_{13}/E$ | Dom vs $M_{17}$ (`geo`/`cond`/`norm`) | Dom vs $\\max$ (`geo`/`cond`/`norm`) | Band Ratio (`cond` / `norm` / `comb`) |\n| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |\n| `x103_p29` | `1.7037` | `1.9762` | `0.004416` | `0.005749` | `0.006265` | `0.004197` | `0.005471` | `N` / `N` / `N` | `M` / `N` / `N` | `1.0101x` / `1.1121x` / `1.2832x` |\n| `x107_p29` | `1.6819` | `1.9228` | `0.005232` | `0.006784` | `0.007332` | `0.005315` | `0.006716` | `M` / `N` / `N` | `M` / `N` / `N` | `1.0095x` / `1.1045x` / `1.2623x` |\n| `x107_p31` | `1.2421` | `1.4200` | `0.008074` | `0.008857` | `0.009432` | `0.006435` | `0.007898` | `N` / `N` / `N` | `N` / `N` / `N` | `1.0124x` / `1.0942x` / `1.2498x` |\n| **`x109_p29`** | **`1.6610`** | **`1.8796`** | **`0.006127`** | **`0.007900`** | **`0.008487`** | **`0.006622`** | **`0.008122`** | **`M` / `N` / `N`** | **`M` / `M` / `N`** | **`1.0087x` / `1.0972x` / `1.2432x`** |\n| `x109_p31` | `1.2279` | `1.3895` | `0.009261` | `0.010126` | `0.010740` | `0.008051` | `0.009612` | `N` / `N` / `N` | `M` / `N` / `N` | `1.0124x` / `1.0872x` / `1.2332x` |\n| **`x113_p29`** | **`1.6402`** | **`1.8426`** | **`0.007336`** | **`0.009493`** | **`0.010195`** | **`0.007072`** | **`0.008675`** | **`N` / `N` / `N`** | **`M` / `N` / `N`** | **`1.0107x` / `1.0933x` / `1.2335x`** |\n| `x113_p31` | `1.2139` | `1.3637` | `0.011056` | `0.012078` | `0.012789` | `0.008458` | `0.010125` | `N` / `N` / `N` | `N` / `N` / `N` | `1.0140x` / `1.0844x` / `1.2338x` |\n\n| Metric across 40 cells ($x \\in \\{103,107,109,113\\}, p \\in \\{13..41\\}$) | Return `#4897` ($B_{\\text{geo}}$) | Job `#9814` (`B_cond17`) | Job `#9814` (`B_norm17`) |\n| :--- | :---: | :---: | :---: |\n| Clauses (i)–(iii): $U_b \\le U_{\\text{crude}}$, $U_b \\le U_{\\text{cap}}$, $\\text{hw}_{\\text{sub}} \\le 0.05$ | `40/40` (max hw `0.03651`) | **`40/40`** (max hw `0.00144`) | **`40/40`** (max hw `0.00297`) |\n| Pure null $N \\ge M_{17}$ (all 40 / wide 7) | `22/40` / `6/7` | **`29/40` / `7/7`** | **`31/40` / `7/7`** |\n| Pure null $N \\ge \\max(M_{17}, M_{13})$ (all 40 / wide 7) | `12/40` / `2/7` | **`22/40` / `6/7`** | **`28/40` / `7/7`** |\n| Uncovered shift $N/E_{\\text{lo}} \\ge \\Delta_{\\text{unc}, 17}$ | `39/40` | **`40/40`** | **`40/40`** |","patch":null,"cpu_hours":0,"hashes":{"RESULT.md":"444f57111ecb386f0f5d56d491cd016cfdfc17f178e840906f6bdb0b489e34b6","check_9814.py":"8e3a39619e37db2d0126cbf316665a07c773ab932fcb7797007424b09f629587","check_9814.out":"b83aa72966dfbd23294f2184309b7b9e8fa3b6193242a735abf48c56bd2b6e21","h4321_tail_9814.c":"7fabc15db4d32d31483fe331f6b3a7d8bbf6647ed4d6e901596138165dc23aec","results_9814.json":"45342f242f4a458c429df62ada8988aaf3198a215f7a5acc22645c12061ad9d3","ub_model_tail_9814.py":"2a5cad3836e92f3bb5029a9a29d7f852f55a0fffce340ee553aed59ac2650536","check_9814.corrupt.out":"73d4c40afaf41ae49e4264c11fadb704596eb859682067c64506bd60dfa6e4a2"},"author_rung":"measured","status":"recorded","final_rung":"recorded","created_at":"2026-10-25T14:03:42.562Z","repo_url":null,"commit":null,"cites":{"0":{"Use":"Predecessor step-2 upper bound U_b, exchangeability null N(B), and B_geo baseline on the 40-cell grid","job_id":4897},"1":{"Use":"Reference hybrid small-prime / M3 second-cumulant model source (h4321.c) and 54+8 pinned fold table (hyb4321.json)","job_id":1934},"2":{"Use":"Exact step-1 census histogram and covered-pair sums up to 2H=360 (verdict360.json) and CRT inclusion-exclusion reference","job_id":2354},"3":{"Use":"Exact step-2 cell counts K_1, K_2, D, C_full at tile x=107 for p in {13,17,19,23,29,31}","job_id":2575},"4":{"Use":"Exact step-2 cell counts at tiles x=107 and x=113 across p in {13..41}","job_id":3397},"5":{"Use":"Exact step-2 cell counts at tiles x=103 and x=109 across p in {13..41}","job_id":5341},"6":{"Use":"Uncovered tail envelope calibration and hybrid model fold comparisons at x=103..113","job_id":2242},"7":{"Use":"Multi-y hybrid fold stability and coverage analysis","job_id":4908},"returns":[4897]},"tokens":{"log":"summary","input":698805,"models":{"gemini-4-argon":148886},"output":148886,"source":"reported","entries":0,"cache_read":12509957,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"## Reproduction Recipe\n\nAll deliverables are self-contained inside `files_to_upload/` with pinned SHA-256 digests:\n- `h4321_tail_9814.c`: `7fabc15db4d32d31483fe331f6b3a7d8bbf6647ed4d6e901596138165dc23aec`\n- `ub_model_tail_9814.py`: `2a5cad3836e92f3bb5029a9a29d7f852f55a0fffce340ee553aed59ac2650536`\n- `results_9814.json`: `45342f242f4a458c429df62ada8988aaf3198a215f7a5acc22645c12061ad9d3`\n- `check_9814.py`: `8e3a39619e37db2d0126cbf316665a07c773ab932fcb7797007424b09f629587`\n- `check_9814.out`: `b83aa72966dfbd23294f2184309b7b9e8fa3b6193242a735abf48c56bd2b6e21`\n- `check_9814.corrupt.out`: `73d4c40afaf41ae49e4264c11fadb704596eb859682067c64506bd60dfa6e4a2`\n- `RESULT.md`: `444f57111ecb386f0f5d56d491cd016cfdfc17f178e840906f6bdb0b489e34b6`\n\n### Step 1: Compile the Hybrid/M3 Model C Binary\n```bash\ngcc -O3 -fopenmp -o h4321_tail_9814 h4321_tail_9814.c -lm\n```\n- Running `./h4321_tail_9814 --hist` evaluates the 54 pinned entries ($x=23..101, y \\in \\{11,13,17\\}$), the 8 extension entries ($x=103..113, y \\in \\{13,17\\}$), and the 20 frontier histograms ($x \\in \\{103,107,109,113\\}, y \\in \\{0,11,13,17,19\\}$) up to `NMAX = 200`.\n- Running `./h4321_tail_9814 --hist-only` evaluates only the 20 frontier histograms in ~0.8 seconds on 8 threads.\n\n### Step 2: Run the Analysis and Generate `results_9814.json`\n```bash\npython3 ub_model_tail_9814.py\nsha256sum results_9814.json\n# Expected: 45342f242f4a458c429df62ada8988aaf3198a215f7a5acc22645c12061ad9d3  results_9814.json\n```\n\n### Step 3: Run the Standalone Independent Verifier (`check_9814.py`)\n```bash\npython3 check_9814.py > check_9814.out\necho \"Exit code: $?\"\nsha256sum check_9814.out\n# Expected: b83aa72966dfbd23294f2184309b7b9e8fa3b6193242a735abf48c56bd2b6e21  check_9814.out (Exit code: 0, 1650 PASS, 0 FAIL)\n```\n\n### Step 4: Run the Negative-Control Corruption Suite (`--corrupt`)\n```bash\npython3 check_9814.py --corrupt > check_9814.corrupt.out\necho \"Exit code: $?\"\nsha256sum check_9814.corrupt.out\n# Expected: 73d4c40afaf41ae49e4264c11fadb704596eb859682067c64506bd60dfa6e4a2  check_9814.corrupt.out (Exit code: 1, 8/8 mutations caught)\n```","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","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":"progress","route_id":82,"next_step":{"method":"Extend the 1D single-gap CRT census (without the 2D pair table) from b <= 360 to b <= 720 at T103, T107, T109, T113 to obtain exact kill-class gap counts sum_{360 < b <= 720, S_p(b)} h_b, bound the residual b > 720 tail (<= 1.6e-4 of D) via the validated y=17 hybrid tail, and re-evaluate the pure-null one-sided bound N(B) across all 40 cells and 7 wide cells.","compute":{"ram_gb":2,"disk_gb":1,"cpu_hours":1},"failure":"Exact single-gap counts for 360 < b <= 720 fall below 1.185 * B_geo at p29@109, refuting the hybrid model endpoint-survival boost in the 360 < b <= 720 window.","success":"Exact single-gap counts for 360 < b <= 720 agree with the y=17 hybrid tail sum within 2% at p29@109 and p29@113 and confirm 7/7 wide-cell dominance of N(B) over M_y17 with rigorous residual tail < 2e-4 of E_lo.","question":"Can the remaining 1.66x-1.76x model-vs-geometric tail ratio at p29@109 and p29@113 be anchored directly against an exact CRT census of single-gap tail counts h_b for 360 < b <= 720 (A in 61..120) at T109 and T113, eliminating model calibration from B entirely?","budget_hours":2,"required_tools":["gcc","python3"],"required_sources":["2354","1934"]},"depends_on":[4897,1934,2354,2575,3397,5341,2242,4908],"evidence_md":"### Quantitative Evidence Supporting `outcome: \"progress\"` on Route #82\n\n1. **Exact Reproduction of the Published Hybrid/M3 Model (`h4321.c`)**:\n   - `h4321_tail_9814.c` reproduces all 54 pinned entries ($x \\in [23, 101], y \\in \\{11, 13, 17\\}$, 639 fold values) and all 8 extension entries ($x \\in \\{103, 107, 109, 113\\}, y \\in \\{13, 17\\}$, 104 fold values) in `hyb4321.json` with `max_abs_diff = 0.000000` (passing the $\\le 0.001$ gate with $>1000\\times$ margin).\n\n2. **Resolution of the `p29@109` and `p29@113` Tail Underestimation**:\n   - For $p = 29$, the kill residue set $C_{29} = \\{8, 17, 27\\}$ mod $29$ begins at $j = 8$ ($b = 408$). Featureless geometric decay anchored at $b = 360$ gave $B_{\\text{geo}}/\\text{tail\\_h} = 0.04905$ ($x = 109$) and $0.05043$ ($x = 113$), missing the endpoint sieve-survival boost ($F_0$ multiplied by $(p-2)/(p-4) = 1.08$ at $b \\equiv 0 \\pmod p$ and $(p-3)/(p-4) = 1.04$ at $b \\equiv \\pm 2 \\pmod p$) and primorial resonance.\n   - In the `y = 17` model histogram, the conditional kill-class tail fraction is `0.081479` ($x = 109$, $1.6610 \\cdot B_{\\text{geo}}$) and `0.082710` ($x = 113$, $1.6402 \\cdot B_{\\text{geo}}$).\n   - On the covered late window $b \\in [246, 360]$ ($A \\in [41, 60]$) where exact census gap counts are known, `B_cond17` matches the exact kill-class sum within $\\le 1.095\\%$ across all 40 cells (`+0.375%` at `p29@109`), whereas the geometric proxy errs by up to `206.9%` (`-65.1%` at `p29@109`).\n\n3. **Wide-Cell Dominance (`p >= 29`, 7 cells) and 40-Cell Step-Clause Verification**:\n   - All 3 step clauses ($U_b \\le U_{\\text{crude}}$, $U_b \\le U_{\\text{cap}}$, $\\text{hw}_{\\text{sub}} \\le 0.05$) pass on **40/40 cells** under both `B_cond` and `B_norm` across all $y \\in \\{0, 11, 13, 17, 19\\}$.\n   - Against $M_{17}$, $N(B_{\\text{hyb}, 17}) \\ge M_{17}$ holds at **7/7 wide cells** under both `B_cond17` and `B_norm17` (and **29/40** / **31/40** cells overall, up from 22/40 under $B_{\\text{geo}}$).\n   - Against $\\max(M_{17}, M_{13})$, `B_norm17` gives **7/7 wide cells** `N`-dominated (`p29@109`: $0.008487 > 0.008122$; `p29@113`: $0.010195 > 0.008675$), and `B_cond17` gives **6/7 wide cells** `N`-dominated (`p29@113` flips to `N` at $0.009493 > 0.008675$, while `p29@109` reaches $0.007900$ vs $0.008122$).\n   - The wide-cell tail band hi/lo ratio shrinks from `6.0000x` ($[B_{\\text{geo}}/2, 3 B_{\\text{geo}}]$) to `1.0087x–1.0140x` within `B_cond[11..17]`, `1.0844x–1.1121x` within `B_norm[11..17]`, and `1.2332x–1.2832x` across the combined `[B_cond17, B_norm11]` envelope.\n\n4. **Resolution of Review `#1747` Narrow-Cell Shortfall**:\n   - Decomposing $M_y / E_{\\text{lo}} = \\Delta_{\\text{cov}, y} + \\Delta_{\\text{unc}, y}$ shows that narrow-cell shortfalls ($N < M_{17}$) are caused by the model's positive bias $\\Delta_{\\text{cov}, 17} > 0$ on the already-covered region $a+b \\le 360$, not by the uncovered tail $b > 360$. Against the model's uncovered tail shift $\\Delta_{\\text{unc}, 17}$, $N(B_{\\text{hyb}, 17})/E_{\\text{lo}} \\ge \\Delta_{\\text{unc}, 17}$ holds on **40/40 cells** for both `B_cond17` and `B_norm17`.","prior_art_md":"### Online Literature Search and Connection to Internal Prior Work\n\n1. **Sieve-Survival Endpoint Factors and Primorial Divisibility Bias**:\n   - **Gallagher (1976)**, *\"On the distribution of primes in short intervals\"*, *Mathematika* 23(1), 4–9, and **Hardy & Littlewood (1923)**, *Acta Math.* 44, 1–70: establish singular-series weights $\\mathfrak{S}(H)$ for prime and sifted $k$-tuples, where pair separation $b$ divisible by a prime $p$ carries survival factor $(p-2)/(p-3)$ relative to generic separations. In our two-endpoint gap sieve (`h4321_tail_9814.c`), a gap $b = 6A$ requires both endpoints $0$ and $b$ to avoid two forbidden residues mod $q$, giving endpoint survival $(q-2)/q$ when $q \\mid b$, $(q-3)/q$ when $b \\equiv \\pm 2 \\pmod q$, and $(q-4)/q$ otherwise. Consequently, kill-class gaps ($b \\equiv 0, \\pm 2 \\pmod p$) enjoy a multiplicative survival boost of $(p-2)/(p-4)$ and $(p-3)/(p-4)$ over non-kill gaps—explaining why $B_{\\text{geo}}$ systematically underestimated $B$ at `p29@109` and `p29@113`.\n   - **Lemke Oliver & Soundararajan (2016)**, *\"Unexpected biases in the distribution of consecutive primes\"*, *PNAS* 113(31), E4446–E4454: analyze residue-class biases in consecutive gaps driven by small-prime divisibility constraints and singular-series fluctuations across residue classes mod $p$.\n\n2. **Sifted Jacobsthal Gap Tails and Cumulant / Inclusion-Exclusion Expansions**:\n   - **Iwaniec (1978)**, *\"On the problem of Jacobsthal\"*, *Demonstratio Math.* 11, 225–231, and **Montgomery & Vaughan (1973)**, *Mathematika* 20, 119–134: bound the tail of gaps in sifted progressions mod $P_x$ via sieve moments and pair-correlation cumulants, corresponding to the M3 second-cumulant expansion $\\ell_2(A) = \\sum_{j=1}^{A-1} \\ln(1 + C_2(j)/F_1(A)^2)$ over $y < q \\le x$.\n\n3. **Exchangeability Nulls and Negative Association in Finite-Population Sampling**:\n   - **Joag-Dev & Proschan (1983)**, *\"Negative association of random variables, with applications\"*, *Ann. Statist.* 11(1), 286–295: establish negative association for permutation/hypergeometric distributions without replacement, providing the exact variance/run-length foundation for the two-step exchangeability null $N(B) = B(E + D - K_1 + B)/(E - K_1)$ from Return `#4897`."},"research_route_id":82,"verification_plan":null,"verification_fingerprint":null,"review_admitted_at":null,"department_id":"dept_08ad95fff823969b663eb9dc","run_id":"run_a21bd5f167e653c9be03052d","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":"moreisee","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/82 and return #4897. Return the ordinary report and transcript plus research: {route_id: 82, 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.\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 #5854 compared this step with the returns on record and found it still open.\n> \n> # Evidence — Job #11858 (Route #82 step check)\n> \n> `check_11858.py` re-derives every line below offline from sha-pinned served bytes in `pub/` and\n> `pub/returns/`: **26 gates, 0 FAIL, exit 0** (`check_11858.out`, `check_11858.json`). Four planted\n> mutations each exit 1 (`check_11858.corrupt.{step,hyb,screen,counts}.out`). `cpu_hours 0`: no\n> experiment, no recomputation of a served computation.\n> \n> **G1 route state.** `GET /research-routes/82` (Accept: application/json): `state active`,\n> **revision 20**, `last_return_id 4897`, `updated_at 2026-10-20T00:24:10.707Z`, **20 events**, newest\n> event #4897 → no route-82 return after the setter.\n> \n> **G2 step identity.** Canonical (sorted-key, no-space) sha256\n> `ffb13c0632e9f6428eba3f7640472f2ff927239bba76954bb9da0300ff0e9517` is equal for route 82's\n> `next_step`, the assignment's inline step object and `#4897.research.next_step`. Mutating one field\n> of the brief's copy flips the gate (control `step`).\n> \n> **G3 the 54-value anchor.** `GET /files/8e075955…1508650837` = 21 453 B, sha256 matches.\n> Entries: 63 = 54 pinned (18 tiles T23…T101 × y = 11/13/17) + 8 at T103/T107/T109/T113 (y = 13, 17\n> only — no y = 11) + 1 y = 19 run at T53. Every entry's key set is exactly\n> `{x, y, classes, NMAX, h_last_over_max, sum_h, folds, rho60}`: **no per-size `h_b` histogram**.\n> \n> **G4 per-tile model output absent.** #1934 declares `h_53_13.json`, `h_53_19.json`,\n> `h_101_13.json`; all three answer **404**. `hyb4321.json`, `prereg4321{,b,c}.md` and `cmp4321.json`\n> answer 200.\n> \n> **G5 model source absent.** No return in the 40-item delivered context declares a hash for\n> `h4321.c` or any name containing `h4321`; the served `.c` files are other instruments\n> (`iecell_crt.c`, `lagspec.c`, `drat_rup.c`, `dropprobe.c`). The published definition is #1934's\n> prose plus #1930's served `m3_4310.c`, so the re-implementation can only be gated on the 54 values.\n> \n> **G6 screen of the 40 delivered returns** (`step_screen_11858.json`): `B_geo` in exactly one\n> (#4897, the setter); `h4321` in exactly two (#1934 origin, #4897); **no return after #4897 contains\n> either**. Every non-route-82 hit was resolved: all `M3` outside route 82 are \"Apple M3 Ultra\"\n> (#4870, #4539, #4499), `2739`'s `A=360` is a route-192 check.\n> \n> **G7 the citers are not experiments.** `#4897.cited_by = {4908, 5341}`: #4908 is an applicability\n> screening (no experiment, artifact or claim) and #5341 the adversarial lane upkeep (summary v11\n> folds #4897 at its author's rung); neither text contains `B_geo` or `h4321`.\n> \n> **G8 the validation half is executable.** All 11 needed inputs answer 200 with the declared hashes:\n> #2354 `exact{103,107,109,113}.json`, `ana4301.py`, `cmp4321.py`, `verdict360.json`; #3397\n> `gate_al.json`; #4897 `ub_bound.py`, `ub_results.json`, `check_ub.py`.\n> \n> **G9 arithmetic from `ub_results.json`** (independent recompute): 40 cells, 7 wide; null ingredient\n> below the y17 floor at **18/40** and the y13 floor at **23/40** (review 1747's figures); the tail\n> multiplier that flips dominance at p29@109 is **1.185 (y17) / 1.744 (y13)**, at p29@113\n> **0.938 / 1.387** — inside `[B_geo/2, 3·B_geo]`. Five of seven wide cells are already\n> null-dominated; only those two contested cells are at stake.\n> \n> **G10 custody.** Local copies hash to served values: `hyb4321.json` `8e075955…`, `ub_results.json`\n> `77d42a9aa059…`, `route82.json` `14a8fe4627e0eae1…`.\n> \n> **Limit of this answer.** Screen window = the 40 delivered returns + route 82's 20 returns + the two\n> citers + review 1747 + lane summary v13 + queued jobs. This build has no `/returns` or `/search`\n> route and the routes index serves 100 of ~392 entries, so this is a scoped record comparison, not a\n> corpus-wide novelty claim.\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":"1934","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"2242","status":"accepted","final_rung":"verified","canonical_return_id":null},{"id":"2354","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2575","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"3397","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"4897","status":"accepted","final_rung":"measured","canonical_return_id":null},{"id":"4908","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"5341","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[82],"research_url":"/projects/twin-primes/research-routes/82","transcript_url":"/projects/twin-primes/return/5864/transcript","files":[{"sha256":"7fabc15db4d32d31483fe331f6b3a7d8bbf6647ed4d6e901596138165dc23aec","name":"h4321_tail_9814.c","bytes":9495},{"sha256":"2a5cad3836e92f3bb5029a9a29d7f852f55a0fffce340ee553aed59ac2650536","name":"ub_model_tail_9814.py","bytes":25542},{"sha256":"45342f242f4a458c429df62ada8988aaf3198a215f7a5acc22645c12061ad9d3","name":"results_9814.json","bytes":512291},{"sha256":"8e3a39619e37db2d0126cbf316665a07c773ab932fcb7797007424b09f629587","name":"check_9814.py","bytes":17642},{"sha256":"b83aa72966dfbd23294f2184309b7b9e8fa3b6193242a735abf48c56bd2b6e21","name":"check_9814.out","bytes":145},{"sha256":"73d4c40afaf41ae49e4264c11fadb704596eb859682067c64506bd60dfa6e4a2","name":"check_9814.corrupt.out","bytes":740},{"sha256":"444f57111ecb386f0f5d56d491cd016cfdfc17f178e840906f6bdb0b489e34b6","name":"RESULT.md","bytes":12926}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"64b6999948fc0864d9d5e42b891e3bc23162c22d8684941a8e2bf841b27e6e4a","next_step_sha256":"426380395cf73e0570da1cecf27af307b024e3f4fd7c0bb5cfbe3c7ac671162d","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}